Chapter Water Resources Engineering

21M. Kent Loftin

PresidentSynint, Inc.

Hobe Sound, Florida

WATER RESOURCES

ENGINEERING*

Water resources engineering is con-cerned with the protection, devel-opment, and efficient managementof water resources for beneficial

purposes. It involves planning, design, and con-struction of projects for supply of water fordomestic, commercial, public, and industrialpurposes, flood protection, hydroelectric power,control of rivers and water runoff, and conserva-tion of water resources, including prevention ofpollution.

Water resources engineering primarily dealswith water sources, collection, flow control, trans-mission, storage, and distribution. For efficientmanagement of these aspects, water resourcesengineers require a knowledge of fluid mecha-nics; hydraulics of pipes, culverts, and openchannels; hydrology; water demand, qualityrequirements, and treatment; production of waterfrom wells, lakes, rivers, and seas; transmissionand distribution of water supplies; design ofreservoirs and dams; and production of hydroelec-tric power. These subjects are addressed in thefollowing articles.

21.1 Dimensions and Units

A list of symbols and their dimensions used in thissection is given in Table 21.1. Table 21.2 lists

conversion factors for commonly used quantities,including the basic equivalents between theEnglish and metric systems. For additional con-versions to the metric system (SI) of units, see theappendix.

Fluid Mechanics

Fluid mechanics describes the behavior of waterunder various static and dynamic conditions. Thistheory, in general, has been developed for an idealliquid, a frictionless, inelastic liquid whose par-ticles follow smooth flow paths. Since water onlyapproaches an ideal liquid, empirical coefficientsand formulas are used to describe more accuratelythe behavior of water. These empiricisms areintended to compensate for all neglected andunknown factors.

The relatively high degree of dependence onempiricism, however, does not minimize the im-portance of an understanding of the basic theory.Since major hydraulic problems are seldom iden-tical to the experiments from which the empiricalcoefficients were derived, the application of funda-mentals is frequently the only means available foranalysis and design.

21.2 Properties of Fluids

Specific weight or unit weight w is defined asweight per unit volume. The specific weight ofwater varies from 62.42 lb/ft3 at 32 8F to 62.22 lb/ft3

* Revised and updated from Sec. 21, Water Engineering, bySamuel B. Nelson, in the third edition.

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)Copyright © 2004 The McGraw-Hill Companies. All rights reserved.

Any use is subject to the Terms of Use as given at the website.

Source: Standard Handbook for Civil Engineers

Table 21.1 Symbols, Dimensions, and Units Used in Water Engineering

Symbol Terminology Dimen-sions

Units

A Area L2 ft2

C Chezy roughness coefficient L1/2/T ft1/2/sC1 Hazen-Williams roughness coefficient L0.37/T ft0.37/sd Depth L ftdc Critical depth L ftD Diameter L ftE Modulus of elasticity F/L2 psiF Force F lbg Acceleration due to gravity L/T2 ft/s2

H Total head, head on weir L fth Head or height L fthf Head loss due to friction L ftL Length L ftM Mass FT2/L lb . s2/ftn Manning’s roughness coefficient T/L1/3 s/ft1/3

P Perimeter, weir height L ftP Force due to pressure F lbp Pressure F/L2 psfQ Flow rate L3/T ft3/sq Unit flow rate L3/T . L ft3/(s . ft)r Radius L ftR Hydraulic radius L ftT Time T st Time, thickness T, L s, ftV Velocity L/T ft/sW Weight F lbw Specific weight F/L3 lb/ft3

y Depth in open channel, distance from solid boundary L ftZ Height above datum L ft1 Size of roughness L ftm Viscosity FT/L2 lb . s/ftn Kinematic viscosity L2/T ft2/sr Density FT2/L4 lb . s2/ft4

s Surface tension F/L lb/ftt Shear stress F/L2 psi

Symbols for dimensionless quantities

Symbol Quantity

C Weir coefficient, coefficient of dischargeCc Coefficient of contractionCn Coefficient of velocityF Froude numberf Darcy-Weisbach friction factorK Head-loss coefficientR Reynolds numberS Friction slope—slope of energy grade lineSc Critical slopeh Efficiency

Sp. gr. Specific gravity

21.2 n Section Twenty-One



WATER RESOURCES ENGINEERING*

at 80 8F but is commonly taken as 62.4 lb/ft3 for themajority of engineering calculations. The specificweight of sea water is about 64.0 lb/ft3.

Density r is defined asmass per unit volume andis significant in all flow problemswhere accelerationis important. It is obtained by dividing the specificweight w by the acceleration due to gravity g. Thevariation of g with latitude and altitude is smallenough to warrant the assumption that its value isconstant at 32.2 ft/s2 in hydraulics computations.

The specific gravity of a substance is the ratio ofits density at some temperature to that of purewater at 68.2 8F (20 8C).

Modulus of elasticity E of a fluid is defined asthe change in pressure intensity divided by thecorresponding change in volume per unit volume.Its value for water is about 300,000 psi, varyingslightly with temperature. The modulus of elasticityof water is large enough to permit the assumptionthat it is incompressible for all hydraulics problemsexcept those involving water hammer (Art. 21.13).

Surface tension and capillarity are a result ofthe molecular forces of liquid molecules. Surfacetension s is due to the cohesive forces betweenliquid molecules. It shows up as the apparent skinthat forms when a free liquid surface is in contactwith another fluid. It is expressed as the force in theliquid surface normal to a line of unit length drawnin the surface. Surface tension decreases with in-creasing temperature and is also dependent on thefluid with which the liquid surface is in contact.The surface tension of water at 70 8F in contact withair is 0.00498 lb/ft.

Capillarity is due to both the cohesive forcesbetween liquid molecules and adhesive forces ofliquid molecules. It shows up as the difference inliquid surface elevations between the inside andoutside of a small tube that has one end submer-ged in the liquid. Since the adhesive forces ofwater molecules are greater than the cohesive forcesbetween water molecules, water wets a surface andrises in a small tube, as shown in Fig. 21.1. Capillarity

Table 21.2 Conversion Table for Commonly Used Quantities

Area Discharge1 acr-

e ¼ 43,56-0 ft2

1 m-i2 ¼ 640 a-cres

1 ft3/s ¼ 449 gal/min ¼ 646,000 gal/day1 ft3/s ¼ 1.98 acre-ft/day ¼ 724 acre-ft/year1 ft3/s ¼ 50 miner’s inches in Idaho, Kansas,

Nebraska, New Mexico, NorthDakota, and South Dakota

1 ft3/s ¼ 40 miner’s inches in Arizona,California, Montana, and Oregon

1 MGD* ¼ 3.07 acre-ft/day ¼ 1120 acre-ft/year1 MGD* ¼ 1.55 ft3/s ¼ 694 gal/min1 million acre-ft/year ¼ 1380 ft3/s

Volume1 ft3 ¼ 7.4805 gal

1 acre-ft ¼ 325,850 gal1 MG ¼ 3.0689 acre-ft Power1 hp ¼ 550 ft . lb/s1 hp ¼ 0.746 kW1 hp ¼ 6535 kWh/yearWeight of waterPressure1 ft3 weighs 62.4 lb1 psi ¼ 2.31 ft of water1 gal

weighs 8.34 lb ¼ 51.7 mm of mercury1 in of mercury ¼ 1.13 ft of water1 ft of water ¼ 0.433psi1 atm† ¼ 29.9 in of mercury ¼ 14.7 psiMetric equivalents Length:1 ft ¼ 0.3048 m Area: 1 acre ¼ 4046.9m2 Volume: 1 gal ¼ 3.7854L 1 m3 ¼ 264.17gal Weight: 1 lb ¼ 0.4536 kg

* Prefix M indicates million; for example, MG ¼ million gallons.† atm indicates atmospheres.

Water Resources Engineering n 21.3




is commonly expressed as the height of this rise.In equation form,

h ¼ 2s cos u

(w1 � w2)r(21:1)

where h ¼ capillary rise, ft

s ¼ surface tension, lb/ft

w1 and w2 ¼ specific weights of fluids below andabove meniscus, respectively, lb/ft

u ¼ angle of contact

r ¼ radius of capillary tube, ft

Capillarity, like surface tension, decreases withincreasing temperature. Its temperature variation,however, is small and insignificant in mostproblems.

Surface tension and capillarity, although negli-gible in many water engineering problems, aresignificant in others, such as capillary rise and flowof liquids in narrow spaces, formation of spray fromwater jets, interpretation of the results obtained onsmall models, and freezing damage to concrete.

Atmospheric pressure is the pressure due to theweight of the air above the earth’s surface. Its valueat sea level is 2116 psf or 14.7 psi. The variation inatmospheric pressure with elevation from sea levelto 10,000 ft is shown in Fig. 21.2.Gage pressure, psi,

is pressure above or below atmospheric. Absolutepressure, psia, is the total pressure includingatmospheric pressure. Thus, at sea level, a gagepressure of 10 psi is equivalent to 24.7 psia. Gagepressure is positive when pressure is greater thanatmospheric and is negative when pressure is lessthan atmospheric.

Vapor pressure is the partial pressure caused bythe formation of vapor at the free surface of aliquid. When the liquid is in a closed container, thepartial pressure due to the molecules leavingthe surface increases until the rates at which themolecules leave and reenter the liquid are equal.The vapor pressure at this equilibrium condition iscalled the saturation pressure. Vapor pressureincreases with increasing temperature, as shown inFig. 21.3.

Cavitation occurs in flowing liquids at press-ures below the vapor pressure of the liquid.Cavitation is a major problem in the design ofpumps and turbines since it causes mechanicalvibrations, pitting, and loss of efficiency throughgradual destruction of the impeller. The cavitationphenomenon may be described as follows:

Because of low pressures, portions of the liquidvaporize, with subsequent formation of vaporcavities. As these cavities are carried a shortdistance downstream, abrupt pressure increasesforce them to collapse, or implode. The implosionand ensuing inrush of liquid produce regions ofvery high pressure, which extend into the pores of

Fig. 21.2 Atmospheric pressure decreases withelevation above mean sea level. The curve is basedon the ICAO standard atmosphere.

Fig. 21.1 Capillary action raises water in asmall-diameter tube. Meniscus, or liquid surface,is concave upward.





the metal. (Pressures as high as 350,000 psi havebeen measured in the collapse of vapor cavitiesby the Fluid Mechanics Laboratory at StanfordUniversity.) Since these vapor cavities form andcollapse at very high frequencies, weakening ofthe metal results as fatigue develops, and pittingappears.

Cavitation may be prevented by designingpumps and turbines so that the pressure in theliquid at all points is always above its vaporpressure.

Viscosity, m of a fluid, also called the coefficientof viscosity, absolute viscosity, or dynamic vis-cosity, is a measure of its resistance to flow. It isexpressed as the ratio of the tangential shearingstresses between flow layers to the rate of change ofvelocity with depth:

m ¼ t

dV=dy(21:2)

where t ¼ shearing stress, lb/ft2

V ¼ velocity, ft/s

y ¼ depth, ft

Viscosity decreases as temperature increases butmay be assumed independent of changes in pres-sure for themajority of engineeringproblems.Waterat 70 8F has a viscosity of 0.00002050 lb . s/ft2.

Kinematic viscosity n is defined as viscosity mdivided by density r . It is so named because itsunits, ft2/s, are a combination of the kinematicunits of length and time. Water at 70 8F has akinematic viscosity of 0.00001059 ft2/s.

In hydraulics, viscosity is most frequently en-countered in the calculation of Reynolds number(Art. 21.8) to determine whether laminar, transi-tional, or completely turbulent flow exists.

21.3 Fluid Pressures

Pressure or intensity of pressure p is the force perunit area acting on any real or imaginary surfacewithin a fluid. Fluid pressure acts normal to thesurface at all points. At any depth, the pressure actsequally in all directions. This results from theinability of a fluid to transmit shear when at rest.Liquid and gas pressures differ in that the variationof pressure with depth is linear for a liquid andnonlinear for a gas.

Hydrostatic pressure is the pressure due todepth. It may be derived by considering a sub-merged rectangular prism of water of height Dh, ft,and cross-sectional areaA, ft2, as shown in Fig. 21.4.The boundaries of this prism are imaginary. Sincethe prism is at rest, the summation of all forces inboth the vertical and horizontal directions must bezero. Let w equal the specific weight of the liquid,lb/ft3. Then, the forces acting in the verticaldirection are the weight of the prism wADh, theforce due to pressure p1, psf, on the top surface, andthe force due to pressure p2, psf, on the bottomsurface. Summing these vertical forces and settingthe total equal to zero yields

p2A� wADh� p1A ¼ 0 (21:3a)

Fig. 21.3 Vapor pressure of water increases rapidly with temperature.





Division of Eq. (21.3a) by A yields

p2 ¼ wDhþ p1 (21:3b)

For the special case where the top of the prismcoincides with the water surface, p1 is atmosphericpressure. Since most hydraulics problems involvegage pressure, p1 is zero (gage pressure is zero atatmospheric pressure). Taking Dh to be h, the depthbelow thewater surface, ft, then p2 is p, the pressure,psf, at depth h. Equation (21.3b) then becomes

p ¼ wh h ¼ p

w(21:4)

Equation (21.4) gives the depth of water h ofspecific weight w required to produce a gage pres-sure p. By adding atmospheric pressure pa to Eq.(21.4), absolute pressure pab is obtained as shown inFig. 21.4. Thus,

pab ¼ whþ pa (21:5)

21.3.1 Pressures on SubmergedPlane Surfaces

This is important in the design of weirs, dams,tanks, and other water control structures. For

horizontal surfaces, the pressure-force determi-nation is a simple matter since the pressure isconstant. For determination of the pressure forceon inclined or vertical surfaces, however, thesummation concepts of integral calculus mustbe used.

Figure 21.5 represents any submerged planesurface of negligible thickness inclined at anangle u with the horizontal. The resultant pressureforce P, lb, acting on the surface is equal to

Ðp dA.

Since p ¼ wh and h ¼ y sin u, where w is the specificweight of water, lb/ft3,

P ¼ w

ðy sin u dA (21:6)

Equation (21.6) can be simplified by settingÐy dA ¼ �yyA, where A is the area of the submerged

surface, ft2; and �yy sin u ¼ �hh, the depth of thecentroid, ft. Therefore,

P ¼ w�hhA ¼ pcgA (21:7)

where pcg is the pressure at the centroid, psf.The point on the submerged surface at which

the resultant pressure force acts is called the centerof pressure (c.p.). It is below the center of gravity

Fig. 21.4 Hydrostatic pressure varies linearly with depth.





because the pressure intensity increases withdepth. The location of the center of pressure,represented by the length yp , is calculated bysumming the moments of the incremental forcesabout an axis in the water surface through pointW (Fig. 21.5). Thus, Pyp ¼

Ðy dP. Since dP ¼

wy sin u dA and P ¼ wÐy sin u dA,

yp ¼Ðy2 dAÐy dA

(21:8)

The quantityÐy2 dA is the moment of inertia of

the area about the axis through W. It also equals

AK2 þ A�yy2, where K is the radius of gyration, ft, ofthe surface about its centroidal axis. The denomi-nator of Eq. (21.8) equals �yyA. Hence

yp ¼ �yyþ K2

�yy(21:9)

and K2=�yy is the distance between the centroid andcenter of pressure.

Values of K2 for some common shapes are givenin Fig. 21.6 (see also Fig. 6.29). For areas for whichradius of gyration has not been determined, yp maybe calculated directly from Eq. (21.8).

Fig. 21.5 Total pressure on submerged plane surface depends on pressure at the center of gravity (c.g.)but acts at a point (c.p.) that is below the c.g.

Fig. 21.6 Radius of gyration and location of centroid (c.g.) of common shapes.





The horizontal location of the center of pressuremay be determined as follows: It lies on the verticalaxis of symmetry for surfaces symmetrical aboutthe vertical. It lies on the locus of the midpoints ofhorizontal lines located on the submerged surface,if that locus is a straight line. Otherwise, thehorizontal location may be found by takingmoments about an axis perpendicular to the onethrough W in Fig. 21.5 and lying in the plane of thesubmerged surface.

Example 21.1: Determine the magnitude andpoint of action of the resultant pressure force on a5-ft-square sluice gate inclined at an angle u of 53.28to the horizontal (Fig. 21.7).

From Eq. (21.7), the total force P ¼ w�hhA, with

�hh ¼ [2:5þ 1 =

2 (5)] sin 53:2W ¼ 5:0� 0:8 ¼ 4:0 ft

A ¼ 5� 5 ¼ 25 ft2

Thus, P ¼ 62:4� 4� 25 ¼ 6240 lb. From Eq. (21.9),its point of action is a distance yp ¼ �yyþ K2=�yyfrom point G, and �yy ¼ 2:5þ 1 =

2 (5:0) ¼ 5:0 ft. Also,K2 ¼ b2=12 ¼ 52=12 ¼ 2:08. Therefore, yp ¼ 5:0þ2:08=5 ¼ 5:0þ 0:42 ¼ 5:42 ft.

21.3.2 Pressure on SubmergedCurved Surfaces

The resultant pressure force on submerged curvedsurfaces cannot be calculated from the equationsdeveloped for the pressure force on submergedplane surfaces because of the variation in directionof the pressure force. The resultant pressure forcecan be calculated, however, by determining itshorizontal and vertical components and combiningthem vectorially.

A typical configuration of pressure on asubmerged curved surface is shown in Fig. 21.8.Consider ABC a 1-ft-thick prism and analyze it as afree body by the principles of statics. Note:

1. The horizontal component PH of the resultantpressure force has a magnitude equal to thepressure force on the vertical projection AC ofthe curved surface and acts at the centroid ofpressure diagram ACDE.

2. The vertical component PV of the resultantpressure force has a magnitude equal to the sumof the pressure force on the horizontal projectionAB of the curved surface and the weight of thewater vertically above ABC. The horizontal

Fig. 21.7 Sluice gate (crosshatched) is subjected to hydrostatic pressure. (See Example 21.1.)





location of the vertical component is calculatedby taking moments of the two vertical forcesabout point C.

When water is below the curved surface, such asfor a taintor gate (Fig. 21.9), the vertical componentPV of the resultant pressure force has a magnitudeequal to theweight of the imaginaryvolumeofwatervertically above the surface. PVacts upward throughthe center of gravity of this imaginary volume.

Example 21.2: Calculate the magnitude anddirection of the resultant pressure on a 1-ft-widestrip of the semicircular taintor gate in Fig. 21.9.

The magnitude of the horizontal componentPH of the resultant pressure force equals thepressure force on the vertical projection of thetaintor gate. From Eq. (21.7), PH ¼ w�hhA ¼ 62:4�2:5� 5 ¼ 780 lb.

The magnitude of the vertical component ofthe resultant pressure force equals the weight of theimaginary volume of water in the prismABC abovethe curved surface. The volume of this prism ispR2/4 ¼ 3.14 � 25/4 ¼ 19.6 ft3, so the weight of thewater is 19.6w ¼ 19.6 � 62.4 ¼ 1220 lb ¼ PV.

The magnitude of the resultant pressure forceequals

P ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2H þ P2

V

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi7802 þ 12202

p¼ 1450 lb

The tangent of the angle the resultant pressureforce makes with the horizontal ¼ PV/PH ¼ 1220/780 ¼ 1.564. The corresponding angle is 57.48.

The positions of the horizontal and verticalcomponents of the resultant pressure force are notrequired to find the point of action of the resultant.Its angle with the horizontal is known, and for aconstant-radius surface, the resultant must actperpendicular to the surface.

21.4 Submerged and FloatingBodies

The principles of buoyancy govern the behavior ofsubmerged and floating bodies and are important indetermining the stability and draft of cargo vessels.

The buoyant force acting on a submerged body equalsthe weight of the volume of liquid displaced.

Fig. 21.8 Hydrostatic pressure on a submergedcurved surface. (a) Pressure variation over thesurface. (b) Free-body diagram.

Fig. 21.9 Taintor gate has submerged curvedsurface under pressure. Vertical component ofpressure acts upward. (See Example 21.2.)





Fig. 21.10 Stability of a ship depends on the location of itsmetacenter relative to its center of gravity (c.g.).

A floating body displaces a volume of liquid equal toits weight.

The buoyant force acts vertically through the centerof buoyancy c.b., which is located at the center of gravityof the volume of liquid displaced.

For a body to be in equilibrium, whether float-ing or submerged, the center of buoyancy and cen-ter of gravity must be on the same vertical line AB(Fig. 21.10a). The stability of a ship, its tendency notto overturn when it is in a nonequilibrium position,is indicated by the metacenter. It is the point atwhich a vertical line through the center of buoy-ancy intersects the rotated position of the linethrough the centers of gravity and buoyancy for theequilibrium condition A0B0 (Fig. 21.10b). The ship isstable only if the metacenter is above the center ofgravity since the resulting moment for this condi-tion tends to right the ship.

The distance between the ship’s metacenterand center of gravity is called the metacentric heightand is designated by ym in Fig. 21.10b. Given in feetby Eq. (21.10) ym is a measure of degree of stabilityor instability of a ship since the magnitudes ofmoments produced in a roll are directly propor-tional to this distance.

ym ¼ I

V+ ys (21:10)

where I ¼ moment of inertia of ship’s cross sectionat waterline about longitudinal axisthrough 0, ft4

V ¼ volume of displaced liquid, ft3

ys ¼ distance, ft, between centers of buoyancyand gravity when ship is in equilibrium

The negative sign should be used when the centerof gravity is above the center of buoyancy.

21.5 Manometers

A manometer is a device for measuring pressure.It consists of a tube containing a column of one ortwo liquids that balances the unknown pressure.The basis for the calculation of this unknownpressure is provided by the height of the liquidcolumn. All manometer problems may be solvedwith Eq. (21.4), p ¼ wh. Manometers indicate h, thepressure head, or the difference in head.

The primary application of manometers ismeasurement of relatively low pressures, for whichaneroid and Bourdon gages are not sufficientlyaccurate because of their inherent mechanical limi-tations. However, manometers may also be used inprecise measurement of high pressures by arran-ging several U-tube manometers in series (Fig.21.12c). Manometers are used for both static andflow applications, although the latter is most com-mon.

Three basic types are used (shown in Fig. 21.11):piezometer, U-tube manometer, and differentialmanometer. Following is a brief discussion of thebasic types.

The piezometer (Fig. 21.11a) consists of a tubewith one end tapped flush with the wall of the





container in which the pressure is to be measuredand the other end open to the atmosphere. The onlyliquid it contains is the one whose pressure is beingmeasured (the metered liquid). The piezometer isa sensitive gage, but it is limited to the measure-ment of relatively small pressures, usually heads of5 ft of water or less. Larger pressures would createan impractically high column of liquid.

Example 21.3: The gage pressure pc in the pipeof Fig. 21.11a is 2.17 psi. The liquid is water withw ¼ 62.4 lb/ft3. What is hm?

hm ¼ pcw

¼ 2:17� 144

62:4¼ 5:0 ft

For pressures greater than 5 ft of water, theU-tube manometer (Fig. 21.11b) is used. It is simi-lar to the piezometer except that it contains anindicating liquid with a specific gravity usuallymuch larger than that of the metered liquid. Theonly other criteria are that the indicating liquidshould have a good meniscus and be immisciblewith the metered liquid.

The U-tube manometer is used when pressuresare either too high or too low for the piezometer.High pressures can be measured by arrangingU-tube manometers in series (Fig. 21.12c). Very lowpressures, including negative gage pressures, canbe measured if the bottom of the U tube extendsbelow the center line of the container of the

Fig. 21.11 Basic types of manometers. (a) Piezometers; (b) U-tube manometer; (c) differentialmanometer.





metered liquid. The most common use of theU-tube manometer is measurement of the press-ures of flowing water. In this application, the usualindicating liquid is mercury.

A movable scale, as opposed to a fixed scale,facilitates reading the U-tube manometer. The scaleis positioned between the two vertical legs andmoved to adjust for the variation in distance hmfrom the center line of the pressure vessel to theindicating liquid. This zero adjustment enables adirect reading of the heights hi and hm of the liquid

columns. The scale may be calibrated in anyconvenient units, such as ft of water or psi.

The differential manometer (Fig. 21.11c) isidentical to the U-tube manometer but measuresthe difference in pressure between two points.(It does not indicate the pressure at either point.)The differential manometer may have either thestandardU-tube configuration or an invertedU-tubeconfiguration, depending on the comparative speci-fic gravities of the indicating and metered liquids.The inverted U-tube configuration (Fig. 21.12b) is

Fig. 21.12 Manometer shapes: (a) Sump in manometer to damp flow disturbances. (b) Inverted U formeasuring pressures on liquids with low specific gravity. (c) Series arrangement for measuring highpressures.





used when the indicating liquid has a lowerspecific gravity than the metered liquid.

Example 21.4: A differential manometer (Fig.21.11c) is measuring the difference in pressurebetween two water pipes. The indicating liquid ismercury (specific gravity ¼ 13.6), hi is 2.25 ft, hm1 is9 in, and z is 1.0 ft. What is the pressure differentialbetween the two pipes?

hm2 ¼ hi þ hm1 � z ¼ 2:25þ 0:75� 1:0 ¼ 2:0 ft

The pressure at B, psf, is

pB ¼ pc2 þ w2hm2 ¼ pc2 þ 62:4� 2:0 ¼ pc2 þ 125

The pressure at A, psf, is

pA ¼ pc1 þ w1hm1 þ wihi

¼ pc1 þ 62:4� 0:75þ 13:6� 62:4� 2:25

¼ pc1 þ 1957

Since the pressure at A must equal that at B,

pc2 þ 125 ¼ pc1 þ 1957

Hence, the pressure differential between the pipesis

pc2 � pc1 ¼ 1832 psf ¼ 12:7 psi

When small pressure differences in water aremeasured, if the specific gravity of the indicatingliquid is between 1.0 and 2.0 and the points at whichthe pressure is being measured are at the same level,the actual pressure difference, when expressedin feet of water, is magnified by the differentialmanometer. For example, if the actual difference is0.50 ft of water and the indicating liquid has aspecific gravity of 1.40, the magnification will be 2.5;that is, the height of the liquid column hi will be1.25 ft ofwater. The closer the specific gravities of themetered and indicating liquids, the greater themagnification and sensitivity. This is true only up toa magnification of about 5. Above 5, the increasedsensitivity may be deceptive because the meniscusbetween the two liquids becomes poorly definedand sluggish in movement.

Many factors affect the accuracy of manometers.Most of them, however, may be neglected in themajority of hydraulics applications since they aresignificant only in precise reading of manometers,such as might be required in laboratories. Onefactor, however, is significant: the existence ofsurges in the manometer caused by the pulsations

and disturbances in the flow of water resultingfrom turbulence. These surges make reading ofthe manometer difficult. They may be reduced oreliminated by installing a large-diameter section,or sump, in the manometer, as shown in Fig. 21.12a.This sump will damp the pulsations and keep thedistance from the center line of the conduit to theindicating liquid essentially at a constant value.

21.6 Fundamentals ofFluid Flow

For fluid energy, the law of conservation of energyis represented by the Bernoulli equation:

Z1 þ p1w

þ V21

2g¼ Z2 þ p2

wþ V2

2

2g(21:11)

where Z1 ¼ elevation, ft, at any point 1 of flowingfluid above an arbitrary datum

Z2 ¼ elevation, ft, at downstream point influid above same datum

p1 ¼ pressure at 1, psf

p2 ¼ pressure at 2, psf

w ¼ specific weight of fluid, lb/ft3

V1 ¼ velocity of fluid at 1, ft/s

V2 ¼ velocity of fluid at 2, ft/s

g ¼ acceleration due to gravity, 32.2 ft/s2

The left side of the equation sums the total energyper unit weight of fluid at 1, and the right side, thetotal energy per unit weight at 2. Equation (21.11)applies only to an ideal fluid. Its practical userequires a term to account for the decrease in totalhead, ft. through friction. This term hf, when addedto the downstream side of Eq. (21.11), yields the formof the Bernoulli equation most frequently used:

Z1 þ p1w

þ V21

2g¼ Z2 þ p2

wþ V2

2

2gþ hf (21:12)

The energy contained in an elemental volume offluid thus is a function of its elevation, velocity, andpressure (Fig. 21.13). The energy due to elevation isthe potential energy and equalsWZa, whereW is theweight, lb, of the fluid in the elemental volume andZa is its elevation, ft, above some arbitrary datum.The energy due to velocity is the kinetic energy. Itequals WV2

a=2g, where Va is the velocity, ft/s. Thepressure energy equals Wpa/w, where pa is the





pressure lb/ft2, and w is the specific weight of thefluid, lb/ft3. The total energy, in the elementalvolume of fluid is

E ¼ WZa þWpaw

þWV2a

2g(21:13)

Dividing both sides of the equation by W yields theenergy per unit weight of flowing fluid, or the totalhead ft:

H ¼ Za þ paw

þ V2a

2g(21:14)

pa/w is called pressure head; V2a=2g, velocity head.

As indicated in Fig. 21.13, Zþ p=w is constantfor any point in a cross section and normal to theflow through a pipe or channel. Kinetic energy atthe section, however, varies with velocity. Usually,Z þ p/w at the midpoint and the average velocityat a section are assumed when the Bernoulliequation is applied to flow across the section orwhen total head is to be determined. Averagevelocity, ft/s ¼ Q/A, where Q is the quantity offlow, ft3/s, across the area of the section A, ft2.

Example 21.5: Determine the energy lossbetween points 1 and 2 in the 24-in-diameter pipein Fig. 21.14. The pipe carries water flowing at31.4 ft3/s.

Average velocity in the pipe ¼ Q/A ¼ 31.4/3.14 ¼ 10 ft/s. Select point 1 far enough from thereservoir outlet that V1 can be assumed to be 0.Since the datum plane passes through point 2,Z2 ¼ 0. Also, since the pipe has free discharge,p2 ¼ 0. Thus substitution in Eq. (21.12) yields

30þ 20þ 0 ¼ 0þ 0þ 102

64:4þ hf

where hf is the friction loss, ft. Hence, hf ¼ 50�1:55 ¼ 48:45 ft.

Note that in this example hf includes minorlosses due to the pipe entrance, gate valve, and anybends.

Fig. 21.13 Energy in a liquid depends on ele-vation, velocity, and pressure.

Fig. 21.14 Flow from an elevated reservoir—application of the Bernoulli equation. (See Example 21.5.)





The Bernoulli equation and the variation ofpressure may be represented graphically, res-pectively, by energy and hydraulic grade lines(Fig. 21.15). The energy grade line, sometimes calledthe total head line, shows the decrease in totalenergy per unit weight H in the direction of flow.The slope of the energy grade line is called theenergy gradient or friction slope. The hydraulicgrade line lies a distance V2/2g below the energygrade line and shows the variation of velocity orpressure in the direction of flow. The slope of thehydraulic grade line is termed the hydraulicgradient. In open-channel flow, the hydraulicgrade line coincides with the water surface, whilein pressure flow, it represents the height to whichwater would rise in a piezometer (see also Example21.7, Art. 21.9).

Momentum is a fundamental concept that mustbe considered in the design of essentially allwaterworks facilities involving flow. A changein momentum, which may result from a change in

either velocity, direction, or magnitude of flow, isequal to the impulse, the force F acting on the fluidtimes the period of time dt over which it acts.Dividing the total change in momentum by thetime interval over which the change occurs givesthe momentum equation, or impulse-momentumequation:

Fx ¼ pQDVx (21:15)

where Fx ¼ summation of all forces in X directionper unit time causing change in mom-entum in X direction, lb

r ¼ density of flowing fluid, lb . s2/ft4

(specific weight divided by g)

Q ¼ flow rate, ft3/s

DVx ¼ change in velocity in X direction, ft/s

Similar equations may be written for the Y and Zdirections. The impulse-momentum equation often

Fig. 21.15 Energy grade line and hydraulic grade line indicate variations in energy and pressure head,respectively, in a liquid as it flows along a pipe or channel.





is used in conjunction with the Bernoulli equation[Eq. (21.11) or (21.12)] but may be used separately.

Example 21.6: Calculate the resultant force onthe reducer elbow in Fig. 21.16. The pipe centerline lies in a horizontal plane. The pipe reducesfrom 48 in in diameter to 16 in. The pressure atthe upstream side of the reducer bend (point 1)is 100 psi, and the water flow is 100 ft3/s. (Neglectfriction loss at the bend.)

Velocity at points 1 and 2 is found by dividingQ ¼ 100 ft3/s by the respective areas: V1 ¼ 100 �4/42p ¼ 7.96 ft/s andV2 ¼ 100 � 4/1.332p ¼71.5 ft/s.

With p1 known, the Bernoulli equation for theflow in the elbow is:

0þ 100� 144

62:4þ 7:962

2� 32:2¼ 0þ p2

62:4þ 71:52

2� 32:2

Solution of the equation yields the pressure at 2:

p2 ¼ 9500 psf

The total pressure force at 1 is P1 ¼ p1A1 ¼181,000 lb, and at 2, P2 ¼ ppA2 ¼ 13,200 lb.

Let R be the force, lb, exerted by the pipe on thefluid (equal and opposite in direction to the forceagainst the pipe, which is to be determined). Then,the force F changing the momentum of the fluidequals the vector sum P1 2 P2 þ R. To find F, applyEq. (21.15) first in the X direction, then in the Ydirection, and determine the resultant of the forces:

In the X direction, since DVx ¼ 2(7.96 sin53.28 2 71.5) ¼ 65.1 and the density r ¼ 62.4/32.2 ¼ 1.94,

Fx ¼ 181,000 cos 53:28� 13,200þ Rx

¼ 1:94� 100� 65:1

Rx ¼ �82,600 lb

In the Y direction, since DVy ¼ �(�7:96; cos 53:28�0) ¼ 4:78,

Fy ¼ �181,000 sin 53:28þ Ry

¼ 1:94� 100� 4:78

Ry ¼ 145,700 lb

The resultant R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2x þ R2

y

q¼ 167,500 lb. It acts

at an angle u with the horizontal such that tan u ¼145,700=82,600; so u ¼ 60:5

W

. The force against thepipe acts in the opposite direction.

21.7 Water ResourcesModeling

A model is a tool that can be used to determine thelikely response of a system to a given set of stimuliwithout having to actually impose those stimuli onthe system. In water resources engineering, modelsare used to determine the likely response of asystem, such as a river, aquifer, or drainage basin, to

Fig. 21.16 Flow induces forces in a pipe at bends and at changes in size of section—application ofmomentum equation. (See Example 21.6.)





a given set of stimuli, such as storm rainfall,droughts, alternative management schemes, orproposed works. Models are cost-effective andconvenient for such investigations. See also Art. 1.7.

Models can typically be categorized as one ofthree major types:

Physical Models n The system (prototype) ismodeled with physical components that representcomponents of the system. Usually, scale factorsare applied to set the model at only a fraction of thesize and cost of the prototype. Physical modelsare expensive to build, operate, and maintain butare especially useful in analyzing complex phe-nomena that are not easy or presently possible toexpress mathematically.

Analog Models n The system (prototype) ismodeled with electronic circuits that representcomponents of the system. Some conveyance andresistance phenomena such as those found intransmission networks and groundwater analysesare easily modeled with analog techniques inas-much as electric current flow and water flowbehave similarly in certain instances. Analogmodels are an abstraction of the prototype. Popularbefore the advent of digital computers, analogmodels are now infrequently used in view of theefficiency and portability of mathematical models.

Mathematical Models n The system (proto-type) is modeled with sets of mathematicalexpressions that represent components of thesystem. Mathematical models are normally pro-grammed in an appropriate computer language,and through execution of the computer program,simulations of prototype behavior are possible.Mathematical models are limited only by themodel creator’s ability to describe the prototypemathematically, the capability of the computingresources, or availability of data to support themodeling effort. They can be as simple or ascomplex as a given analysis requires and areamong the most cost-effective means to performcertain analyses.

A fourth mode of modeling, hybrid modeling,employs both physical and mathematical models.It exploits the advantages of these types of modelswhile avoiding their limitations. For instance,complex three-dimensional flow patterns, ero-sional scour, and sediment deposition occurring

in the immediate vicinity of a bridge pier or watercontrol structure can be best modeled with aphysical model while the overall water surface,momentum, and velocity profile over the encom-passing river reach can be best modeled by anappropriate mathematical model.

With hybrid models, one model often providesinput to or verification of the other model. In thepreceding example, the mathematical modelwould provide depth and velocity profile input tothe physical model, and the physical model may beable to provide a more accurate estimate of localhead loss at the pier or structure. In this way, thetwo models can be executed interactively until allcommon boundary conditions synchronize. Theresulting hybrid model will consist of a mathemat-ical model that properly accounts for overallhydraulic effects and local head loss at the pier orstructure and a physical model that properlyaccounts for localized forces affecting the stabilityor performance of the pier or structure.

As computers become faster, numerical model-ing in three dimensions, utilizing the full set ofequations for hydrodynamic flow ismakingpossiblereliable solutions for very complex problems. Even-tually, these sophisticated models may eliminate theneed for physical modeling in most cases.

21.7.1 Similitude for PhysicalModels

A physical model is a system whose operation canbe used to predict the characteristics of a similarsystem, or prototype, usually more complex orbuilt to a much larger scale. A knowledge of thelaws governing the phenomena under investi-gation is necessary if the model study is to yieldaccurate quantitative results.

Forces acting on the model should be pro-portional to forces on the prototype. The four forcesusually considered in hydraulic models are inertia,gravity, viscosity, and surface tension. Because ofthe laws governing these forces and because themodel and prototype are normally not the samesize, it is usually not possible to have all four forcesin the model in the same proportions as they are inthe prototype. It is, however, a simple procedure tohave two predominant forces in the same pro-portion. In most models, the fact that two of thefour forces are not in the same proportion as theyare in the prototype does not introduce serious





error. The inertial force, which is always a pre-dominant force, and one other force are made pro-portional.

Ratios of the forces of gravity, viscosity, andsurface tension to the force of inertia are designated,respectively, Froude number, Reynolds number,and Weber number. Equating the Froude numberof the model and the Froude number of the proto-type ensures that the gravitational and inertialforces are in the same proportion. Similarly, equa-ting the Reynolds numbers of the model and proto-type ensures that the viscous and inertial forceswill be in the same proportion. And equating theWeber numbers ensures proportionality of surfacetension and inertial forces.

The Froude number is

F ¼ VffiffiffiffiffiffiLg

p (21:16)

where F ¼ Froude number (dimensionless)

V ¼ velocity of fluid, ft/s

L ¼ linear dimension (characteristic, such asdepth or diameter), ft


For hydraulic structures, such as spillways andweirs, where there is a rapidly changing water-surface profile, the two predominant forces areinertia and gravity. Therefore, the Froude numbersof the model and prototype are equated:

Fm ¼ Fp

VmffiffiffiffiffiffiffiffiLmg

p ¼ VpffiffiffiffiffiffiffiLpg

p (21:17a)

where subscriptm applies to the model and p to theprototype. Squaring both sides of Eq. (21.17a) andgrouping like terms yields

V2m

V2p

¼ LmLp

(21:17b)

Let Vr ¼ Vm/Vp and Lr ¼ Lm/Lp. Then

V2r ¼ Lr Vr ¼ L1=2r (21:18)

The subscript r indicates ratio of quantity in modelto that in prototype.

If the ratios of all the physical dimensions of amodel to all the corresponding physical dimen-sions of the prototype are equal to the length ratio,the model is termed a true model. In a true modelwhere the Froude number is the governing designcriterion, the length ratio is the only variable.

Once the length ratio has been set, all the physicaldimensions of the model are fixed. The dischargeratio is determined as follows:

Qr ¼ VrAr (21:19a)

Since Vr ¼ L1=2r and Ar ¼ area ratio ¼ L2r ,

Qr ¼ VrAr ¼ L5=2r (21:19b)

By this method all the necessary characteristics of aspillway or weir model can be determined.

The Reynolds number is

R ¼ VL

n(21:20)

R is dimensionless, and n is the kinematic viscosityof fluid, ft2/s. The Reynolds numbers of model andprototype are equated when the viscous andinertial forces are predominant. Viscous forces areusually predominant when flow occurs in a closedsystem, such as pipe flow where there is no freesurface. The following relations are obtained byequating Reynolds numbers of the model andprototype:

VmLmnm

¼ VpLp

np(21:21a)

Vr ¼ nrLr

(21:21b)

The variable factors that fix the design of a truemodel when the Reynolds number governs are thelength ratio and the viscosity ratio.

The Weber number is

W ¼ V2Lr

s(21:22)

where r ¼ density of fluid, lb . s2/ft4 (specificweight divided by g)

s ¼ surface tension of fluid, psf

The Weber numbers of model and prototype areequated in certain types of wave studies, theformation of drops and air bubbles, entrainment ofair in flowing water, and other phenomena wheresurface tension and inertial forces are predominant.The velocity ratio is determined as follows:

V2mLmrmsm

¼ V2pLprp

sp(21:23a)

V2r ¼ sr

rrLr(21:23b)





The fluid properties and the length ratio fix thedesign of a model governed by the Weber number.

In some cases, such as a morning-glory spillway,inertial, viscous, and gravity forces all have animportant effect on the flow. In these cases it isusually not possible to have both the Reynolds andFroude numbers of the model and prototype equal.The solution to this type of problem is mostlyempirical andmay consist of an attempt to evaluatethe effects of viscosity and gravity separately.

For the flow of water in open channels andrivers where the friction slope is relatively flat,model designs are often based on the Manningequation. The relations between the model andprototype are determined as follows:

Vm

Vp¼ (1:486=nm)R

2=3m S1=2m

(1:486=np)R2=3p S1=2p

(21:24)

where n ¼ Manning roughness coefficient (T/L1/3,T representing time)

R ¼ hydraulic radius (L)

S ¼ loss of head due to friction per unitlength of conduit (dimensionless)

¼ slope of energy gradient

For true models, Sr ¼ 1, Rr ¼ Lr. Hence,

Vr ¼ L2=3r

nr(21:25)

In models of rivers and channels, it is necessaryfor the flow to be turbulent. The U.S. WaterwaysExperiment Station has determined that flow willbe turbulent if

VR

n� 4000 (21:26)

where V ¼ mean velocity, ft/s

R ¼ hydraulic radius, ft

n ¼ kinematic viscosity, ft2/s

If the model is to be a true model, it may have to beuneconomically large for the flow to be turbulent.Another problem also encountered in true modelsis surface tension. In a true model of a wide riverwhere the depth may be only a fraction of an inch,the surface tension will distort the flow to such anextent that the model may be useless. To overcomethe effect of surface tension and to get turbulentflow, the depth scale is often made much larger

than the length scale. This type of model is called adistorted model.

The relations between a distorted model of achannel and a prototype are determined in thesame manner as was Eq. (21.24). The only dif-ference is that the slope ratio Sr equals the depthratio dr and the hydraulic-radius ratio is a functionof the width ratio and depth ratio.

One type of model, called amovable-bedmodel,is used to study erosion and transportation of silt inriverbeds. Because the laws governing the trans-portation of material are not fully understood,movable-bed models are built largely on the basisof experience and give only qualitative results.

21.7.2 Types and Applications ofMathematical Models

Used in many applications of water resourcesengineering, mathematical models are, in particu-lar, applied in hydrologic and hydraulic investi-gations of man-made and natural systems for bothsurface-water and groundwater purposes. Thesystem (prototype) is modeled with sets ofmathematical expressions that represent compo-nents of the system. These expressions, in turn,are linked together to represent the system as awhole.

Mathematical models are used for both analysisand design. They are normally programmed in anappropriate computer language, and throughexecution of the computer program, simulationsof prototype behavior are possible. They may besingle-purpose (for a specific site) or generalpurpose (applicable to a variety of sites).

Single-purpose models typically represent thespecific temporal and spatial descriptions ofthe prototype directly in the computer code.For instance, the logical representation of proto-types, such as flow networks, catchment areas, andinfiltration parameters, may be part of the sourcecode and is said to be hardwired into the computerprogram. For such models, the software (thecomputer program code) and the application inputcodes (hydrologic and hydraulic parameters) arebound into one entity. This, however, usually hasmore disadvantages than advantages, especiallywhen modifications of the model are requiredor when the model has to be applied by engineerswho were not involved in the original programcoding. The preferred approach in modeling is





instead to develop general-purpose models bywriting software that is essentially independent ofapplication input code.

General-purpose models are used for specificanalytical tasks. These may be as simple as deter-mination of excess rainfall, given rainfall andrainfall-loss parameters, or as complex as long-period simulation of flow andpollutant transport incombined groundwater and surface-water systems.

Advances are continually being made in com-puter resources and use of models is becomingmore widespread. As a result, the desirability ofmore uniformity of software packages and ofobject-oriented software has become apparent.In object-oriented software, every program com-ponent is generalized as much as feasible and theentire program is essentially a collection ofmodular software components. This approach,when fully implemented, will provide completecompatibility among all types of water resourcessoftware. Also, this approach will provide nearlycomplete compatibility of all databases, of all data-bases and software, and among water resourcesmodelers in the government, academia, andprivate sectors. The result will be a reduction induplication of the efforts of software developersand modelers and an increase in the efficiency ofwater-resources engineering investigations.

Typical applications of mathematical modelsinclude the following: stochastic processes; evap-oration and irrigation; hydrodynamics; hydrologicforecasting; watershed hydrology; design of hy-draulic structures; reservoir regulation; flood ordrought impacts; flow routing; channel and riverhydraulics; sediment or pollutant transport; quan-tity and quality of water supply; ecosystem impactsand restoration; impacts of dam breaks; wave ortidal analyses; landfill leachate analyses; andgroundwater yield, seepage, or pollution.

Several different models varying in complexityor sophistication, or both, and in application typemay be required in many types of investigations.As a general rule, if comparisons of different plansare required, the fewer the number of modelsemployed in a given study, the greater the chancethat meaningful results will be produced. Theavailability and quality of data for calibration andverification, the model output required for designor evaluation, and the general acceptance by theengineering community should be considered inselection of a model or group of models for anyinvestigation.

Mathematical modeling is one of the fastestchanging fields in engineering. Applicationsshould be upgraded accordingly if their continueduse is expected.

(D. R. Maidment, “Handbook of Hydrology,”D. H. Hoggan, “Computer-Assisted FloodplainHydrology and Hydraulics,” N. S. Grigg, “WaterResources Planning,” V. J. Zipparo and H. Hasen,“Davis’ Handbook of Applied Hydraulics,”McGraw-Hill,NewYork (books.mcgraw-hill.com).)

Pipe Flow

The term pipe flow as used in this section refers toflow in a circular closed conduit entirely filled withfluid. For closed conduits other than circular,reasonably good results are obtained in theturbulent range with standard pipe-flow formulasif the diameter is replaced by four times thehydraulic radius. But when there is severedeviation from a circular cross section, as inannular passages, this method gives flows signifi-cantly underestimated. (J. F. Walker, G. A. Whan,and R. R. Rothfus, “Fluid Friction in NoncircularDucts,” Journal of the American Institute of ChemicalEngineers, vol. 3, 1957.)

21.8 Laminar Flow

In laminar flow, fluid particles move in parallellayers in one direction. The parabolic velocitydistribution in laminar flow, shown in Fig. 21.17,creates a shearing stress t ¼ m dV/dy, where dV/dyis the rate of change of velocity with depth and m isthe coefficient of viscosity (see Viscosity, Art. 21.2).As this shearing stress increases, the viscous forcesbecome unable to damp out disturbances, andturbulent flow results. The region of change is

Fig. 21.17 Velocity distribution for lamellarflow in a circular pipe is parabolic. Maximum velo-city is twice the average velocity.





dependent on the fluid’s velocity, density, andviscosity and the size of the conduit.

A dimensionless parameter called the Reynoldsnumber has been found to be a reliable criterionfor the determination of laminar or turbulent flow.It is the ratio of inertial forces to viscous forces,and is given by

R ¼ VDr

m¼ VD

n(21:27)

where V ¼ fluid velocity, ft/s

D ¼ pipe diameter, ft

r ¼ density of fluid, lb . s2/ft4 (specificweight divided by g, 32.2 ft/s2)

m ¼ viscosity of fluid lb . s/ft2

n ¼ m/r ¼ kinematic viscosity, ft2/s

For a Reynolds number less than 2000, flow islaminar in circular pipes. When the Reynolds num-ber is greater than 2000, laminar flow is unstable;a disturbance will probably be magnified, causingthe flow to become turbulent.

In laminar flow, the following equation for headloss due to friction can be developed by consider-ing the forces acting on a cylinder of fluid in a pipe:

hf ¼ 32mLV

D2rg¼ 32mLV

D2w(21:28)

where hf ¼ head loss due to friction, ft

L ¼ length of pipe section considered, ft


w ¼ specific weight of fluid, lb/ft3

Substitution of the Reynolds number yields

hf ¼ 64

R

L

D

V2

2g(21:29)

For laminar flow, Eq. (21.29) is identical to theDarcy-Weisbach formula Eq. (21.30) since in lam-inar flow the friction f ¼ 64/R.

(E. F. Brater, “Handbook of Hydraulics,” 6th ed.,McGraw-Hill Book Company, New York (books.mcgraw-hill.com).)

21.9 Turbulent Flow

In turbulent flow, the inertial forces are so greatthat viscous forces cannot dampen out disturb-ances caused primarily by the surface roughness.

These disturbances create eddies, which haveboth a rotational and translational velocity. Thetranslation of these eddies is a mixing actionthat affects an interchange of momentum acrossthe cross section of the conduit. As a result, thevelocity distribution is more uniform, as shown inFig. 21.18, than for laminar flow (Fig. 21.17).

For a Reynolds number greater than 2000 but tothe left of the dashed line in Fig. 21.19, there is atransition from laminar to turbulent flow. In thisregion, there is a laminar film at the boundariesthat covers some of the smaller roughnessprojections. This explains why the friction loss inthis region has both laminar and turbulent char-acteristics. As the Reynolds number increases, thislaminar boundary layer decreases in thicknessuntil, at completely turbulent flow, it no longercovers any of the roughness projections. To theright of the dashed line in Fig. 21.19, the flow iscompletely turbulent, and viscous forces do notaffect the friction loss.

Because of the random nature of turbulent flow,it is not practical to treat it analytically. Therefore,formulas for head loss and flow in the turbulentregions have been developed through experimen-tal and statistical means. Experimentation in tur-bulent flow has shown that:

The head loss varies directly as the length of the pipe.

The head loss varies almost as the square of thevelocity.

The head loss varies almost inversely as thediameter.

The head loss depends on the surface roughness ofthe pipe wall.

The head loss depends on the fluid’s density andviscosity.

The head loss is independent of the pressure.

Fig. 21.18 Velocity distribution for turbulentflow in a circular pipe is more nearly uniform thanthat for lamellar flow.





21.9.1 Darcy-Weisbach Formula

One of the most widely used equations for pipeflow, the Darcy-Weisbach formula satisfies theabove condition and is valid for laminar or tur-bulent flow in all fluids.

hf ¼ fL

D

V2

2g(21:30)

where hf ¼ head loss due to friction, ft

f ¼ friction factor (see Fig. 21.19)

L ¼ length of pipe, ft

D ¼ diameter of pipe, ft

V ¼ velocity of fluid, ft/s


It employs the Moody diagram (Fig. 21.19) for ev-aluating the friction factor f. (L. F. Moody, “FrictionFactors for Pipe Flow,” Transactions of the AmericanSociety of Mechanical Engineers, November 1944.)

Because Eq. (21.30) is dimensionally homogen-eous, it can be used with any consistent set of unitswithout changing the value of the friction factor.

Roughness values 1 (ft) for use with the Moodydiagram to determine the Darcy-Weisbach frictionfactor f are listed in Table 21.3.

The following formulas were derived for headloss in waterworks design and give good results forwater-transmission and -distribution calculations.They contain a factor that depends on the surfaceroughness of the pipe material. The accuracy ofthese formulas is greatly affected by the selection ofthe roughness factor, which requires experience inits choice.

Fig. 21.19 Chart relates friction forces for flow in pipe to Reynolds numbers and condition of pipes.

Table 21.3 Typical Values of Roughness for Usein the Moody Diagram (Fig. 21.19) to Determine f

1, ft

Steel pipe:Severe tuberculation and incrustation 0.03–0.008General tuberculation 0.008–0.003Heavy brush-coat asphalts, enamels,

and tars 0.003–0.001

Light rust 0.001–0.0005New smooth pipe, centrifugally

applied enamels 0.0002–0.00003

(Table continued )





21.9.2 Chezy Formula

This equation holds for head loss in conduits andgives reasonably good results for high Reynoldsnumbers:

V ¼ CffiffiffiffiffiffiRS

p(21:31)

where V ¼ velocity, ft/s

C ¼ coefficient, dependent on surfaceroughness of conduit

S ¼ slope of energy grade line or head lossdue to friction, ft/ft of conduit


Hydraulic radius of a conduit is the cross-sec-tional area of the fluid in it divided by the perim-eter of the wetted section.

21.9.3 Manning’s Formula

Through experimentation, Manning concludedthat the C in the Chezy equation [Eq. (21.31)]should vary as R1/6

C ¼ 1:486R1=6

n(21:32)

where n ¼ coefficient, dependent on surface rough-ness. (Although based on surface roughness, n inpractice is sometimes treated as a lumped parameterfor all head losses.) Substitution into Eq. (21.31) gives

V ¼ 1:486

nR2=3S1=2 (21:33a)

Upon substitution of D/4, where D is the pipediameter, for the hydraulic radius of the pipe, the

following equations are obtained for pipes flowingfull:

V ¼ 0:590

nD2=3S1=2 (21:33b)

Q ¼ 0:463

nD8=3S1=2 (21:33c)

hf ¼ 4:66n2LQ2

D16=3(21:33d)

D ¼ 2:159Qn

S1=2

� �3=8

(21:33e)

where Q ¼ flow, ft3/s.Tables 21.4 and 21.11 (p. 21.47) give values of

n for the foot-pound-second system. See alsoTable 22.3 for velocity and flow at various slopes.

21.9.4 Hazen-Williams Formula

This is oneof themostwidelyused formulas forpipe-flow computations of water utilities, although it wasdeveloped for both open channels and pipe flow:

V ¼ 1:318C1R0:63S0:54 (21:34a)

For pipes flowing full:

V ¼ 0:55C1D0:63S0:54 (21:34b)

Q ¼ 0:432C1D2:63S0:54 (21:34c)

hf ¼ 4:727

D4:87L

Q

C1

� �1:85

(21:34d)

D ¼ 1:376

S0:205Q

C1

� �0:38

(21:34e)

Table 21.4 Values of n for Pipes, to Be Used with the Manning Formula

Material of PipeVariation Use in designing

From To From To

Clean cast iron 0.011 0.015 0.013 0.015

Dirty or tubercu-lated cast iron

0.015 0.035

Riveted steel orspiral steel

0.013 0.017 0.015 0.017

Welded steel 0.010 0.013 0.012 0.013Galvanized iron 0.012 0.017 0.015 0.017

(Table continued )





where V ¼ velocity, ft/s

C1 ¼ coefficient, dependent on surfaceroughness


S ¼ head loss due to friction, ft/ft of pipe


L ¼ length of pipe, ft

Q ¼ discharge, ft3/s

hf ¼ friction loss, ft

The C1 terms in Table 21.5 are in the foot-pound-second system.

Determination of flow in branching pipes illus-trates the use of friction-loss equations and the hy-draulic-grade-line concept.

Example 21.7: Figure 21.20 shows a typicalthree-reservoir problem. The elevations of thehydraulic grade lines for the three pipes are equalat point D. The Hazen-Williams equation for fric-tion loss [Eq. (21.34d)] can be written for each pipe

meeting at D. With the continuity equation forquantity of flow, there are as many equations asthere are unknowns:

Za ¼Zd þ pD

wþ 4:727LA

D4:87A

QA

CA

� �1:85

(21:35a)

Zb ¼Zd þ PD

wþ 4:727LB

D4:87B

QB

CB

� �1:85

(21:35b)

Zc ¼Zd þ PD

wþ 4:727LC

D4:87C

QC

CC

� �1:85

(21:35c)

QA þQB ¼ QC (21:36)

where pD ¼ pressure at D

w ¼ unit weight of liquid

With the elevations Z of the three reservoirs and thepipe intersection known, the easiest way to solvethese equations is by trying different values ofpD/w in Eqs. (21.35) and substituting the valuesobtained for Q into Eq. (21.36) for a check. If thevalue of Zd þ pD/w becomes greater than Zb, thesign of the friction-loss term is negative instead ofpositive. This would indicate water is flowing fromreservoir A into reservoirs B and C. Flow in pipenetwork is easily determined with available com-puter programs, many of which are specialized tosolve specific pipe design problems efficiently.

21.10 Minor Losses in Pipes

Energy losses occur in pipe contractions, bends,enlargements, and valves and other pipe fittings.These losses can usually be neglected if the lengthof the pipeline is greater than 1500 times the pipe’sdiameter. However, in short pipelines, becausethese losses may exceed the friction losses, minorlosses must be considered.

Table 21.5 Values of C1 in Hazen and WilliamsFormula

Type of pipe C1

Cast iron:New All sizes, 1305 years old All sizes up to 24 in, 120

24 in and over, 11510 years old 12 in, 110

4 in, 10530 in and over, 85

40 years old 16 in, 804 in, 65

Welded steel Values the same as for cast-ironpipe, 5 years older

Riveted steel Values the same as for cast-ironpipe, 10 years older

Wood stave Average value, regardless ofage, 120

Concrete orconcrete-

lined

Large sizes, good workmanship,steel forms, 140

Large sizes, good workmanship,wood forms, 120

(Table continued )

Fig. 21.20 Flow between reservoirs. (See Ex-ample 21.7)





21.10.1 Sudden Enlargements

The following equation for the head loss, ft, acrossa sudden enlargement of pipe diameter has beendetermined analytically and agrees well withexperimental results:

hL ¼ (V1 � V2)2

2g(21:37)

where V1 ¼ velocity before enlargement, ft/s

V2 ¼ velocity after enlargement, ft/s

g ¼ 32.2 ft/s2

It was derived by applying the Bernoulli equationand the momentum equation across an enlarge-ment.

Another equation for the head loss caused bysudden enlargements was determined experimen-tally by Archer. This equation gives slightly betteragreementwithexperimental results thanEq. (21.37):

hL ¼ 1:1(V1 � V2)1:92

2g(21:38)

A special application of Eq. (21.37) or (21.38) isthe discharge from a pipe into a reservoir. The waterin the reservoir has no velocity, so a full velocity headis lost.

21.10.2 Gradual Enlargements

The equation for the head loss due to a gradualconical enlargement of a pipe takes the followingform:

hL ¼ K(V1 � V2)2

2g(21:39)

where K ¼ loss coefficient (see Fig. 21.21).

Since the experimental data available on grad-ual enlargements are limited and inconclusive, thevalues of K in Fig. 21.21 are approximate. (A. H.Gibson, “Hydraulics and Its Applications,” Con-stable & Co., Ltd., London.)

21.10.3 Sudden Contraction

The following equation for the head loss across asudden contraction of a pipe was determined bythe same type of analytical studies as Eq. (21.37):

hL ¼ 1

Cc� 1

� �2V2

2g(21:40)

where Cc ¼ coefficient of contraction (see Table 21.6)

V ¼ velocity in smaller-diameter pipe, ft/s

This equation gives best results when the headloss is greater than 1 ft. Table 21.6 gives Cc values

Fig. 21.21 Head-loss coefficients for a pipe with diverging sides depend on the angle of divergence ofthe sides.





for sudden contractions, determined by JuliusWeisbach (“Die Experiments-Hydraulik”).

Another formula for determining the loss ofhead caused by a sudden contraction, determinedexperimentally by Brightmore, is

hL ¼ 0:7(V1 � V2)2

2g(21:41)

This equation gives best results if the head loss isless than 1 ft.

A special case of sudden contraction is the en-trance loss for pipes. Some typical values of the losscoefficient K in hL ¼ KV2/2g, whereV is the velocityin the pipe, are presented in Table 21.7.

21.10.4 Bends and Standard FittingLosses

The head loss that occurs in pipe fittings, such asvalves and elbows, and at bends is given by

hL ¼ KV2

2g(21:42)

Table 21.8 gives some typical K values for theselosses.

The values in Table 21.8 are only approximate.K values vary not only for different sizes of fittingbut with different manufacturers. For these reas-ons, manufacturers’ data are the best source for losscoefficients.

Experimental data available on bend lossescover a rather narrow range of laboratory experi-ments utilizing small-diameter pipes and do notgive conclusive results. The data indicate the lossesvary with surface roughness, Reynolds number,ratio of radius of bend r to pipe diameter D, and

angle of bend. The data are in agreement that thehead loss, not including friction loss, decreasessharply as the r/D ratio increases from zero toaround 4 or 5. When r/D increases above 4 or 5,there is disagreement. Some experiments indicatethat the head loss, not including friction loss in thebend, increases significantly with an increasing r/D. Experiments on smooth pipes, indicate that thisincrease is very slight and that above an r/D of 4,the bend loss essentially remains constant. (H. Ito,“Pressure Losses in Smooth Pipe Bends,” Trans-actions of the American Society of Civil Engineers,series D, vol. 82, no. 1, 1960.)

Because experiments have produced suchwidely varying data, bend-loss coefficients giveonly an approximation of losses to be expected.Figure 21.22 gives values of K for 908 bends for usewith Eq. (21.42). (K. H. Beij, “Pressure Losses forFluid Flow in 908 Pipe Bends,” Journal of Research,National Bureau of Standards, vol. 21, July 1938.)

To obtain losses in bends other than 908, thefollowing formula may be used to adjust the Kvalues given in Fig. 21.22:

K0 ¼ K

ffiffiffiffiffiD

90

r(21:43)

where D ¼ deflection angle, deg

The K0 value may be used in place of K inEq. (21.42).

Table 21.6 Cc for Contractions in Pipe Area from A1 to A2

A2/A1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Cc 0.62 0.63 0.64 0.66 0.68 0.71 0.76 0.81 0.89 1.0

Table 21.7 Coefficients for Entrance Losses

Pipe projecting intoreservoir

K ¼ 0.80

(Table continued)

Table 21.8 Coefficients for Fitting Losses andLosses at Bends

Fitting K

Globe valve, fully open 10.0

Angle valve, fully open 5.0Swing check valve,fully open

2.5

Gate valve, fully open 0.2Closed-return bend 2.2

(Table continued )





Minor losses are often given as the equivalentlength of pipe that has the same energy loss for thesame discharge. (V. J. Zipparo and H. Hasen, “Davis’Handbook ofAppliedHydraulics,” 4th ed.,McGraw-Hill, Inc., New York (books.mcgraw-hill.com).)

21.11 Orifices

An orifice is an opening with a closed perimeterthrough which water flows. Orifices may have any

shape, although they are usually round, square, orrectangular.

21.11.1 Orifice Discharge intoFree Air

Discharge through a sharp-edged orifice may becalculated from

Q ¼ Caffiffiffiffiffiffiffiffi2gh

p(21:44)

where Q ¼ discharge, ft3/s

C ¼ coefficient of discharge

a ¼ area of orifice, ft2

g ¼ acceleration due to gravity, ft/s2

h ¼ head on horizontal center line oforifice, ft

Coefficients of discharge C are given in Table21.9 for low velocity of approach. If this velocity issignificant, its effect should be taken into account.Equation (21.44) is applicable for any head forwhich the coefficient of discharge is known. Forlow heads, measuring the head from the center lineof the orifice is not theoretically correct; how-ever,this error is corrected by the C values.

Fig. 21.22 Recommended values of head-losscoefficients K for 908 bends in closed conduits.

Table 21.9 Smith’s Coefficients of Discharge for Circular and Square Orifices with Full Contraction*

Dia. of circular orifices, ftHead,

Side of square orifices, ft

0.02 0.04 0.1 1.0 ft 0.02 0.04 0.1 1.0

0.637 0.618 0.4 0.643 0.6210.655 0.630 0.613 0.6 0.660 0.636 0.6170.648 0.626 0 610 0.590 0.8 0.652 0.631 0.615 0.5970.644 0.623 0.608 0.591 1 0.648 0.628 0.613 0.5990.637 0.618 0.605 0.593 1.5 0.641 0.622 0.610 0.601

0.632 0.614 0.604 0.595 2 0.637 0.619 0.608 0.6020.629 0.612 0.603 0.596 2.5 0.634 0.617 0.607 0.6020.627 0.611 0.603 0.597 3 0.632 0.616 0.607 0.6030.623 0.609 0.602 0.596 4 0.628 0.614 0.606 0.6020.618 0.607 0.600 0.596 6 0.623 0.612 0.605 0.602

0.614 0.605 0.600 0.596 8 0.619 0.610 0.605 0.6020.611 0.603 0.598 0.595 10 0.616 0.608 0.604 0.6010.601 0.599 0.596 0.594 20 0.606 0.604 0.602 0.6000.596 0.595 0.594 0.593 50 0.602 0.601 0.600 0.5990.593 0.592 0.592 0.592 100 0.599 0.598 0.598 0.598

* Hamilton Smith, Jr., “Hydraulics,” 1886.





The coefficient of discharge C is the product ofthe coefficient of velocity Cn and the coefficient ofcontraction Cc. The coefficient of velocity is theratio obtained by dividing the actual velocity at thevena contracta (contraction of the jet discharged)by the theoretical velocity. The theoretical velocitymay be calculated by writing Bernoulli’s equationfor points 1 and 2 in Fig. 21.23.

V21

2gþ p1

wþ Z1 ¼ V2

2

2gþ p2

wþ Z2 (21:45)

With the reference plane through point 2, Z1 ¼ h,V1 ¼ 0, p1/w ¼ p2/w ¼ 0, and Z2 ¼ 0, and Eq.(21.45) becomes

V2 ¼ffiffiffiffiffiffiffiffi2gh

p(21:46)

The actual velocity, determined experimentally, isless than the theoretical velocity because of theenergy loss from point 1 to point 2. Typical valuesof Cn range from 0.94 to 0.99.

The coefficient of contraction Cc is the ratio ofthe smallest area of the jet, the vena contracta, to thearea of the orifice. Contraction of a fluid jet willoccur if the orifice is square-edged and so locatedthat some of the fluid approaches the orifice at anangle to the direction of flow through the orifice.This fluid has a momentum component perpen-dicular to the axis of the jet which causes the jet to

contract. Typical values of the coefficient of con-traction range from 0.61 to 0.67.

If the water entering the orifice does not havethis momentum, the contraction is completelysuppressed. Figure 21.24a is an example of a partlysuppressed contraction; no contraction occurs atthe bottom of the jet. In Fig. 21.24b, the edges ofthe orifice have been rounded to reduce or elimi-nate the contraction. With a partly suppressedorifice, the increased area of jet caused by sup-pressing the contraction on one side is partly offsetbecause more water at a higher velocity enters onthe other sides. The result is a slightly greatercoefficient of contraction.

21.11.2 Submerged Orifices

Flow through a submerged orifice may be compu-ted by applying Bernoulli’s equation to points 1and 2 in Fig. 21.25.

V2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g h1 � h2 þ V2

1

2g� hL

� �s(21:47)

where hL ¼ losses in head, ft, between 1 and 2.Assuming V1 � 0, setting h1 2 h2 ¼ Dh, and

using a coefficient of discharge C to account forlosses, Eq. (21.48) is obtained.

Q ¼ Caffiffiffiffiffiffiffiffiffiffiffi2gDh

p(21:48)

Fig. 21.23 Fluid jet takes a parabolic path.





Values of C for submerged orifices do not differgreatly from those for nonsubmerged orifices. (Fortable of values of coefficients of discharge forsubmerged orifices, see E. F. Brater, “Handbook ofHydraulics,” 6th ed., McGraw-Hill Book Company,New York (books.mcgraw-hill.com).)

21.11.3 Discharge underFalling Head

The flow from a reservoir or vessel when the in-flow is less than the outflow represents a conditionof falling head. The time required for a certain

quantity of water to flow from a reservoir can becalculated by equating the volume of water thatflows through the orifice or pipe in time dt to thevolume decrease in the reservoir (Fig. 21.26):

Caffiffiffiffiffiffiffiffi2gy

pdt ¼ Ady (21:49)

Solving for dt yields

dt ¼ Ady


p (21:50)

where a ¼ area of orifice, ft2

A ¼ area of reservoir, ft2

Fig. 21.26 Discharge from a reservoir withdropping water level.

Fig. 21.24 Types of orifices: (a) Sharp-edged with partly suppressed contraction. (b) Round-edgedwith no contraction.

Fig. 21.25 Discharge through a submergedorifice.





y ¼ head on orifice at time t, ft



Expressing the area as a function of y [A ¼ F(y)]and summing from time zero, when y ¼ h1, to timet, when y ¼ h2, Eq. (21.50) becomes

t ¼ðh1h2

F(y) dy


p (21:51)

If the area of the reservoir is constant as y varies,Eq. (21.51) upon integration becomes

t ¼ 2A

Caffiffiffiffiffi2g

p (ffiffiffiffiffih1

p�

ffiffiffiffiffih2

p) (21:52)

where h1 ¼ head at the start, ft

h2 ¼ head at the end, ft

t ¼ time interval for head to fall from h1 toh2, s

21.11.4 Fluid Jets

Where the effect of air resistance is small, a fluiddischarged through an orifice into the air willfollow the path of a projectile. The initial velocity ofthe jet is

V0 ¼ Cn

ffiffiffiffiffiffiffiffi2gh

p(21:53)

where h ¼ head on center line of orifice, ft

Cn ¼ coefficient of velocity

The direction of the initial velocity depends onthe orientation of the surface in which the orifice islocated. For simplicity, the following equationswere determined assuming the orifice is located ina vertical surface (Fig. 21.23). The velocity of the jetin the X direction (horizontal) remains constant.

Vx ¼ V0 ¼ Cn


p(21:54)

The velocity in the Y direction is initially zero andthereafter a function of time and the acceleration ofgravity:

Vy ¼ gt (21:55)

The X coordinate at time t is

X ¼ Vxt ¼ tCn


p(21:56)

The Y coordinate is

Y ¼ Vavgt ¼ gt2

2(21:57)

whereVavg ¼ average velocity over period of time t.The equation for the path of the jet [Eq. (21.58)],obtained by solving Eq. (21.57) for t and substitut-ing in Eq. (21.56), is that for a parabola:

X2 ¼ C2n4hY (21:58)

Equation (21.58) can be used to determine Cn

experimentally. Rearranging Eq. (21.58) gives

Cn ¼ffiffiffiffiffiffiffiffiX2

4hY

r(21:59)

The X and Y coordinates can be measured in alaboratory and Cn calculated from Eq. (21.59).

21.11.5 Orifice Discharge intoShort Tubes

When water flows from a reservoir into a pipe ortube with a sharp leading edge, the same type ofcontraction occurs as for a sharp-edged orifice. Inthe tube or pipe, however, the water contracts andthen expands to fill the tube. If the tube isdischarging at atmospheric pressure, a partialvacuum is created at the contraction, as can beseen by applying the Bernoulli equation acrosspoints 1 and 2 in Fig. 21.27. This reduced pressurecauses the flow through a short tube to be greaterthan that through a sharp-edged orifice of the samedimensions. If the head on the tube is greater than50 ft and the tube is short, the water will shootthrough the tube without filling it. When thishappens, the tube acts as a sharp-edged orifice.

For a short tube flowing full, the coefficient ofcontraction Cc ¼ 1.00 and the coefficient of velocityCn ¼ 0.82. Therefore, the coefficient of dischargeC ¼ 0.82. Solving for head loss as a proportion offinal velocity head, a K value for Eq. (21.42) of 0.5 isobtained as follows: The theoretical velocity headwith no loss is V2

T=2g. Actual velocity head isV2

a=2g ¼ (0:82VT)2=2g ¼ 0:67V2

T=2g. The head losshL ¼ 1:00V2

T=2g�0:67V2T=2g¼ 0:33V2

T=2g. From hL ¼KV2

a=2g, where V2a=2g is the actual velocity head,

K¼ 2ghLV2

a

¼ (0:33V2T�2g)

(2g�0:67V2T)¼ 0:5

For a reentrant tube projecting into a reservoir(Fig. 21.28), the coefficients of velocity and dischargeequal 0.75, and the loss coefficient K equals 0.80.





21.11.6 Orifice Discharge intoDiverging Conical Tubes

This type of tube can greatly increase the flowthrough an orifice by reducing the pressure at theorifice below atmospheric. Equation (21.60) for thepressure at the entrance to the tube is obtained bywriting the Bernoulli equation for points 1 and 3and points 1 and 2 in Fig. 21.29.

p2 ¼ wh 1� a3a2

� �2" #

(21:60)

where p2 ¼ gage pressure at tube entrance, psf

w ¼ unit weight of water, lb/ft3

h ¼ head on center line of orifice, ft

a2 ¼ area of smallest part of jet (vena con-tracta, if one exists), ft2

a3 ¼ area of discharge end of tube, ft2

Discharge is also calculated by writing the Ber-noulli equation for points 1 and 3 in Fig. 21.29.

For this analysis to be valid, the tube must flowfull, and the pressure in the throat of the tube mustnot fall to the vapor pressure of water. Experimentsby Venturi show the most efficient angle u to bearound 58.

21.12 Siphons

A siphon is a closed conduit that rises above thehydraulic grade line and in which the pressure atsome point is below atmospheric (Fig. 21.30). Themost common use of a siphon is the siphon spillway.

Fig. 21.28 Flow from a reservoir through areentrant tube resembles that through a flush tube(Fig. 21.27) but the head loss is larger.

Fig. 21.29 Diverging conical tube increasesflow from a reservoir through an orifice by redu-cing the pressure below atmospheric.

Fig. 21.27 Flow from a reservoir through a tube with a sharp-edged inlet.

Fig. 21.30 Siphon between reservoirs risesabove hydraulic grade line yet permits flow ofwater between them.





Flow through a siphon can be calculated bywriting the Bernoulli equation for the entrance andexit. But the pressure in the siphon must bechecked to be sure it does not fall to the vaporpressure of water. This is accomplished by writingthe Bernoulli equation across a point of knownpressure and a point where the elevation head orthe velocity head is a maximum in the conduit.If the pressure were to fall to the vapor pres-sure, vaporization would decrease or totally stopthe flow.

The pipe shown in Fig. 21.31 is also commonlycalled a siphon or inverted siphon. This is amisnomer since the pressure at all points in thepipe is above atmospheric. The American Societyof Civil Engineers recommends that the invertedsiphon be called a sag pipe to avoid the falseimpression that it acts as a siphon.

21.13 Water Hammer

Water hammer is a change in pressure, either aboveor below the normal pressure, caused by avariation of the flow rate in a pipe. Every time theflow rate is changed, either increased or decreased,it causes water hammer. However, the stresses arenot critical in small-diameter pipes with flows atlow velocities.

The water flowing in a pipe has momentumequal to the mass of the water times its velocity.When a valve is closed, this momentum drops tozero. The change causes a pressure rise, whichbegins at the valve and is transmitted up the pipe.The pressure at the valve will rise until it is highenough to overcome the momentum of the waterand bring the water to a stop. This pressure

buildup travels the full length of the pipe to thereservoir (Fig. 21.32).

At the instant the pressure wave reaches thereservoir, the water in the pipe is motionless, but ata pressure much higher than normal. The differ-ential pressure between the pipe and the reservoirthen causes the water in the pipe to rush back intothe reservoir. As the water flows into the reservoir,the pressure in the pipe falls.

At the instant the pressure at the valve reachesnormal, the water has attained considerablemomentum up the pipe. As the water flows awayfrom the closed valve, the pressure at the valvedrops until differential pressure again brings thewater to a stop. This pressure drop begins atthe valve and continues up the pipe until it reachesthe reservoir.

Fig. 21.31 Sag pipe permits flow between two reservoirs despite a dip and a rise.

Fig. 21.32 Variation with time of pressure atthree points in a penstock, for water hammer frominstantaneous closure of a valve.





The pressure in the pipe is now below normal,so water from the reservoir rushes into the pipe.This cycle repeats over and over until frictiondamps these oscillations. Because of the highvelocity of the pressure waves, each cycle may takeonly a fraction of a second.

The equation for the velocity of a wave in apipe is

U ¼ffiffiffiE

r

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

1þ ED=Ept

s(21:61)

where U ¼ velocity of pressure wave along pipe,ft/s

E ¼ modulus of elasticity of water, 43.2 �106 psf

r ¼ density of water, 1.94 lb . s/ft4 (specificweight divided by acceleration due togravity)


Ep ¼ modulus of elasticity of pipe mater-ial, psf

t ¼ thickness of pipe wall, ft

21.13.1 Instantaneous Closure

The magnitude of the pressure change that resultswhen flow is varied depends on the rate of changeof flow and the length of the pipeline. Any gradualmovement of a valve that is made in less time thanit takes for a pressure wave to travel from the valveto the reservoir and be reflected back to the valveproduces the same pressure change as an instan-taneous movement. For instantaneous closure:

T ,2L

U(21:62)

where L ¼ length of pipe from reservoir to valve, ft

T ¼ time required to change settingof valve, s

A plot of pressure vs. time for various pointsalong a pipe is shown in Fig. 21.32 for the instan-taneous closure of a valve. Equation (21.63a) for thepressure rise or fall caused by adjusting a valve wasderived by equating the momentum of the water inthe pipe to the force impulse required to bring thewater to a stop.

Dp ¼ �rUDV (21:63a)

In terms of pressure head, Eq. (21.63a) becomes

Dh ¼ �UDV

g(21:63b)

where Dp ¼ pressure change from normal dueto instantaneous change of valve set-ting, psf

Dh ¼ head change from normal due toinstantaneous change of valve set-ting, ft

DV ¼ change in the velocity of water causedby adjusting valve, ft/s

If the closing or opening of a valve is instan-taneous, the pressure change can be calculated inone step from Eq. (21.63).

21.13.2 Gradual Closure

The following method of determining the pressurechange due to gradual closure of a valve gives aquick, approximate solution. The pressure rise orhead change is assumed to be in direct proportionto the closure time:

Dhg ¼ tiDh

T¼ 2LDV

Tg(21:64)

where Dhg ¼ head change due to gradual closure, ft

ti ¼ time for wave to travel from the valveto the reservoir and be reflected backto valve, s

T ¼ actual closure time of valve, s

Dh ¼ head rise due to instantaneous clos-ure, ft

L ¼ length of pipeline, ft

DV ¼ change in velocity of water due toinstantaneous closure, ft/s


Arithmetic integration is a more exact methodfor finding the pressure change due to gradualmovement of a valve. The calculations can be read-ily programmed for a computer and are availablein software packages. Integration is a direct meansof studying every physical element of the processof water hammer. The valve is assumed to close in aseries of small movements, each causing anindividual pressure wave. The magnitude of thesepressure waves is given by Eq. (21.63). The





individual pressure waves are totaled to give thepressure at any desired point for a certain time.

The first step in this method is to choose the timeinterval for each incremental movement of thevalve. (It is convenient to make the time intervalsome submultiple of L/U, such as L/aU, where aequals any integer, so that the pressure wavesreflected at the reservoir will be superimposedupon the newwaves being formed at the valve. Thewave formed at the valve will be opposite in signto the water reflected from the reservoir, so therewill be a tendency for the waves to cancel out.)Assuming a valve is fully open and requires Tseconds for closing, the number of incrementalclosing movements required is T/Dt, where Dt, theincrement of time, equals L/aU.

Once the time interval has been determined, anestimate of the velocity change DV during each timeinterval must be made, to apply Eq. (21.63). A roughestimate for the velocity following the incrementalchange is Vn ¼ Vo(An/Ao), where Vn is the velocityfollowing a certain incremental movement, Vo theoriginal velocity, An the area of the valve openingafter the corresponding incremental movement, andAo the original area of the valve opening.

The change in head can now be calculated withEq. (21.63). With the head known, the estimated velo-city Vn can be checked by the following equation:

Vn ¼ VoAn

Ao

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHo þ SDh

Ho

s(21:65)

where Ho ¼ head at valve before any movement ofvalve, ft

Ho þ SDh ¼ total pressure at valve after particularmovement; this includes pressurechange caused by valve movementplus effect of waves reflected fromreservoir, ft

An ¼ area of valve opening after n incre-mental closings; this area can bedetermined from closure character-istics of valve or by assuming itscharacteristics, ft2

If the velocity obtained from Eq. (21.65) differsgreatly from the estimated velocity, then that ob-tained from Eq. (21.65) should be used to recalcu-late Dh.

(V. J. Zipparo and H. Hasen, “Davis’ Handbookof Applied Hydraulics,” 4th ed., McGraw-Hill, Inc.,New York (books.mcgraw-hill.com).)

Example 21.8: The following problem illustratesthe use of the preceding methods and comparesthe results: Steel penstock, length ¼ 3000 ft, diam-eter ¼ 10 ft, area ¼ 78.5 ft2, initial velocity ¼10 ft/s, penstock thickness ¼ 1 in, head at turbinewith valve open ¼ 1000 ft, and modulus of elastic-ity of steel ¼ 43.2 � 108 psf.

(For penstocks as shown in Fig. 21.32, thicknessand diameter normally vary with head. Thus, thevelocity of the pressure waves is different in eachsection of the penstock. Separate calculations forthe velocity of the pressure wave should be madefor each thickness and diameter of penstock toobtain the time required for a wave to travel to thereservoir and back to the valve.)

Velocity of pressure wave, from Eq. (21.61), is

U ¼ffiffiffiE

r

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

1þ ED=Ept

s

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi43:2� 106

1:94

r

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

1þ 43:2� 106 � 10� 12=(1� 43:2� 108)

s

¼ 3180 ft=s

The time required for the wave to travel tothe reservoir and be reflected back to the valve ¼2L/U ¼ 6000/3180 ¼ 1.90 s.

If closure time T of the valve is less than 1.90 s,the closure is instantaneous, and the pressure rise,from Eq. (21.63), is

Dh ¼ UDV

g¼ 3180� 10

32:2¼ 990 ft

Assuming T ¼ 4.75 s, approximate equation(21.64) gives the following result:

Dhg ¼ tiDV

T¼ 1:90� 990

4:75¼ 396 ft

21.13.3 Surge Tanks

It is uneconomical to design long pipelines forpressures created by water hammer or to operatea valve slowly enough to reduce these pressures.Usually, to prevent water hammer, a surge tank isinstalled close to valves at the end of long conduits.A surge tank is a tank containing water and con-nected to the conduit. The water column, in effect,floats on the line.





When a valve is suddenly closed, the water inthe line rushes into the surge tank. The water levelin the tank rises until the increased pressure inthe surge tank overcomes the momentum of thewater. When a valve is suddenly opened, thesurge tank supplies water to the line whenthe pressure drops. The section of pipe betweenthe surge tank and the valve (Fig. 21.33) must stillbe designed for water hammer; but the closuretime to reduce the pressures for this section willbe only a fraction of the time required without thesurge tank.

Although a surge tank is one of the mostcommonly used devices to prevent water hammer,it is by no means the only one. Various types ofrelief valves and air chambers are widely used onsmall-diameter lines, where the pressure of waterhammer may be relieved by the release of arelatively small quantity of water.

Pipe Stresses

21.14 Pipe StressesPerpendicular to theLongitudinal Axis

The stresses acting perpendicular to the longitudi-nal axis of a pipe are caused by either internal orexternal pressures on the pipe walls.

Internal pressure creates a stress commonlycalled hoop tension. It may be calculated by takinga free-body diagram of a 1-in-long strip of pipe cutby a vertical plane through the longitudinal axis(Fig. 21.34). The forces in the vertical directioncancel out. The sum of the forces in the horizontaldirection is

pD ¼ 2F (21:66)

where p ¼ internal pressure, psi

D ¼ outside diameter of pipe, in

F ¼ force acting on each cut of edge ofpipe, lb

Hence, the stress, psi, on the pipe material is

f ¼ F

A¼ pD

2t(21:67)

where A ¼ area of cut edge of pipe, ft2

t ¼ thickness of pipe wall, in

From the derivation of Eq. (21.67), it wouldappear that the diameter used for calculationsshould be the inside diameter. However, Eq. (21.67)is not theoretically exact and gives stresses slightlylower than those actually developed. For thisreason the outside diameter often is used (see alsoArt. 6.10).

Equation (21.67) is exact for all practicalpurposes when D/t is equal to or greater than 50.If D/t is less than 10, this equation will usually bequite conservative and therefore will yield anuneconomical design. For steel pipes, Eq. (21.67)gives directly the thickness required to resist in-ternal pressure.

For concrete pipes, this analysis is approximate,however, since concrete cannot resist large tensilestresses. The force F must be carried by steel rein-forcing. The internal diameter is used in Eq. (21.67)for concrete pipe.

When a pipe has external pressure acting onit, the analysis is much more complex becausethe pipe material no longer acts in direct tension.The external pressure creates bending and com-pressive stresses that cause buckling.

(S. P. Timoshenko and J. M. Gere, “Theory ofelastic Stability,” 2nd ed., McGraw-Hill BookCompany, New York (books.mcgraw-hill.com).)

Fig. 21.34 Internal pipe pressure produceshoop tension.

Fig. 21.33 Surge tank is placed near a valve ona penstock to prevent water hammer.





21.15 Pipe Stresses Parallelto the Longitudinal Axis

If a pipe is supported on piers, it acts like a beam.The stresses created can be calculated from thebending moment and shear equations for acontinuous circular hollow beam. This stress isusually not critical in high-head pipes. However,thin-walled pipes usually require stiffening to pre-vent buckling and excessive deflection from theconcentrated loads.

21.16 Temperature Expansionof Pipe

If a pipe is subject to a wide range of temperatures,the pipe should be the stress due to temperaturevariation designed for or expansion joints shouldbe provided. The stress, psi, due to a temperaturechange is

f ¼ cEDT (21:68)

where E ¼ modulus of elasticity of pipe material,psi

DT ¼ temperature change from installationtemperature

c ¼ coefficient of thermal expansion of pipematerial

The movement that should be allowed for, ifexpansion joints are to be used, is

DL ¼ LcDT (21:69)

where DL ¼ movement in length L of pipe

L ¼ length between expansion joints

21.17 Forces Due toPipe Bends

It is common practice to use thrust blocks in pipebends to take the forces on the pipe caused by themomentum change and the unbalanced internalpressure of the water.

In all bends, there will be a slight loss of headdue to turbulence and friction. This loss will causea pressure change across the bend, but it is usuallysmall enough to be neglected. When there is achange in the cross-sectional area of the pipe, therewill be an additional pressure change that canbe calculated with the Bernoulli equation (seeExample 6, Art. 21.6). In this case, the pressuredifferential may be large and must be considered.

The force diagram in Fig. 21.35 is a convenientmethod for finding the resultant force on a bend.The forces can be resolved into X and Y compo-nents to find the magnitude and direction of theresultant force on the pipe. In Fig. 21.35:

Fig. 21.35 Forces produced by flow at a pipe bend and change in diameter.





V1 ¼ velocity before change in size of pipe, ft/s

V2 ¼ velocity after change in size of pipe, ft/s

p1 ¼ pressure before bend or size change in pipe,psf

p2 ¼ pressure after bend or size change in pipe,psf

A1 ¼ area before size change in pipe, ft2

A2 ¼ area after size change in pipe, ft2

F2m ¼ force due to momentum of water in section2 ¼ V2Qw/g

F1m ¼ force due to momentum of water in section1 ¼ V1Qw/g

P2 ¼ pressure of water in section 2 times area ofsection 2 ¼ p2A2

P1 ¼ pressure of water in section 1 times area ofsection 1 ¼ p1A1

w ¼ unit weight of liquid, lb/ft3

Q ¼ discharge, ft3/s

If the pressure loss in the bend is neglected andthere is no change in magnitude of velocity aroundthe bend, Eqs. (21.70) and (21.71) give a quicksolution.

R ¼ 2A wV2

gþ p

� �cos

u

2(21:70)

a ¼ u

2(21:71)

where R ¼ resultant force on bend, lb

a ¼ angle R makes with F1m

p ¼ pressure, psf

w ¼ unit weight of water, 62.4 lb/ft3

V ¼ velocity of flow, ft/s


A ¼ area of pipe, ft2

u ¼ angle between pipes (08 � u � 1808)

Although thrust blocks are normally used totake the force on bends, in many cases the pipematerial takes this force. The stress caused by thisforce is directly additive to other stresses alongthe longitudinal axis of the pipe. In small pipes, theforce caused by bends can easily be carried bythe pipe material; however, the joints must also beable to take these forces.

Culverts

A culvert is a closed conduit for the passage ofsurface drainage under a highway, a railroad,canal, or other embankment. The slope of a culvertand its inlet and outlet conditions are usuallydetermined by the topography of the site. Becauseof the many combinations obtained by varying theentrance conditions, exit conditions, and slope, nosingle formula can be given that will apply to allculvert problems.

The basic method for determining dischargethrough a culvert requires application of theBernoulli equation between a point just outsidethe entrance and a point somewhere downstream.An understanding of uniform and nonuniformflow is necessary to understand culvert flow fully.However, an exact theoretical analysis, involvingdetailed calculation of drawdown and backwatercurves, is usually unwarranted because of therelatively low accuracy attainable in determiningrunoff. Neglecting drawdown and backwatercurves does not seriously affect the accuracy butgreatly simplifies the calculations.

21.18 Culverts on CriticalSlopes or Steeper

In a culvert with a critical slope, the normal depth(Art. 21.22) is equal to the critical depth (Art. 21.23).

Entrance Submerged or Unsubmergedbut Free Exit n If a culvert is on critical slopeor steeper, that is, the normal depth is equal toor less than the critical depth, the discharge will beentirely dependent on the entrance conditions(Fig. 21.36). Increasing the slope of the culvert pastcritical slope (the slope just sufficient to maintainflow at critical depth) will decrease the depth offlow downstream from the entrance. But theincreased slope will not increase the amount ofwater entering the culvert because the entrancedepth will remain at critical.

The discharge is given by the equation for flowthrough an orifice if the entrance is submerged, orby the equation for flow over a weir if the entranceis not submerged. Coefficients of discharge forweirs and orifices give good results, but they do notcover the entire range of entry conditions encoun-tered in culvert problems. For this reason, com-puter software, charts, and nomographs have been





developed and are used almost exclusivelyin design. (“Handbook of Concrete CulvertPipe Hydraulics,” EB058W, Portland CementAssociation.)

Entrance Unsubmerged but Exit Sub-merged n In this case, the submergence of the exitwill cause a hydraulic jump to occur in the culvert(Fig. 21.37). The jump will not affect the culvertdischarge, and the control will still be at the inlet.

Entrance and Exit Submerged n Whenboth the exit and entrance are submerged(Fig. 21.38), the culvert flows full, and the dischargeis independent of the slope. This is normal pipeflow and is easily solved by using the Manning

or Darcy-Weisbach formula for friction loss[Eq. (21.33d) or (21.30)]. From the Bernoulli equa-tion for the entrance and exit, and the Manningequation for friction loss, the following equation isobtained:

H ¼ (1þ Ke)V2

2gþ V2n2L

2:21R4=3(21:72)

Solution for the velocity of flow yields

V ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

H

(1þ Ke=2g)þ (n2L=2:21R4=3)

s(21:73)

Fig. 21.37 Flow through a culvert withentrance unsubmerged but exit submerged. Whenslope is less than critical, open-channel flow takesplace, and dn . dc. When slope exceeds critical,flow depends on inlet condition, and dn , dc.

Fig. 21.38 With entrance and exit of a culvertsubmerged, normal pipe flow occurs. Discharge isindependent of slope. The fluid flows under pres-sure. Discharge may be determined from Bernoulliand Manning equations.

Fig. 21.36 Flow through a culvert with free discharge. Normal depth dn is less than critical depth dc;slope is greater than the critical slope. Discharge depends on the type of inlet and the head H.





where H ¼ elevation difference between head-water and tailwater, ft

V ¼ velocity in culvert, ft/s


Ke ¼ entrance-loss coefficient (Art. 21.20)

n ¼ Manning’s roughness coefficient

L ¼ length of culvert, ft

R ¼ hydraulic radius of culvert, ft

Equation (21.72) can be solved directly since thevelocity is the only unknown.

21.19 Culverts on SubcriticalSlopes

Critical slope is the slope just sufficient to maintainflow at critical depth. When the slope is less thancritical, the flow is considered subcritical (Art. 21.23).

Entrance Submerged or Unsubmergedbut Free Exit n For these conditions, dependingon the head, the flow can be either pressure oropen-channel.

The discharge, for the open-channel condition(Fig. 21.39), is obtained by writing the Bernoulliequation for a point just outside the entrance anda point a short distance downstream from theentrance. Thus,

H ¼ KeV2

2gþ V2

2gþ dn (21:74)

The velocity can be determined from the Man-ning equation:

V2 ¼ 2:2SR4=3

n2(21:75)

Substituting this into Eq. (21.74) yields

H ¼ (1þ Ke)2:2

2gn2SR4=3 þ dn (21:76)

where H ¼ head on entrance measured from bot-tom of culvert, ft

Ke ¼ entrance-loss coefficient (Art. 21.20)

S ¼ slope of energy grade line, which forculverts is assumed to equal slopeof bottom of culvert

R ¼ hydraulic radius of culvert, ft

dn ¼ normal depth of flow, ft

To solve Eq. (21.76), it is necessary to trydifferent values of dn and corresponding values ofR until a value is found that satisfies the equation. Ifthe head on a culvert is high, a value of dn less thanthe culvert diameter will not satisfy Eq. (21.76).This means the flow is under pressure (Fig. 21.40),and discharge is given by Eq. (21.72).

When the depth of the water is slightly belowthe top of the culvert, there is a range of unstableflow fluctuating between pressure and open chan-nel. If this condition exists, it is good practice tocheck the discharge for both pressure flow andopen-channel flow. The condition that gives thelesser discharge should be assumed to exist.

Short Culvert with Free Exit n When aculvert on a slope less than critical has a free exit,there will be a drawdown of the water surface atthe exit and for some distance upstream. Themagnitude of the drawdown depends on thefriction slope of the culvert and the differencebetween the critical and normal depths. If thefriction slope approaches critical, the differencebetween normal depth and critical depth is small(Fig. 21.39), and the drawdown will not extend forany significant distance upstream. When thefriction slope is flat, there will be a large differencebetween normal and critical depth. The effect of thedrawdown will extend a greater distance upstreamand may reach the entrance of a short culvert(Fig. 21.41). This drawdown of the water level inthe entrance of the culvert will increase the

Fig. 21.39 Open-channel flow occurs in a culvertwith free discharge and normal depth dn greaterthan the critical depth dc when the entrance isunsubmerged or slightly submerged. Dischargedepends on head H, loss at entrance, and slope ofculvert.





discharge, causing it to be about the same as for aculvert on a slope steeper than critical (Art. 21.18).Most culverts, however, are on too steep a slope forthe backwater to have any effect for an appreciabledistance upstream.

Entrance Unsubmerged but Exit Sub-merged n If the level of submergence of the exit iswell below the bottom of the entrance (Fig. 21.37),the backwater from the submergence will not

extend to the entrance. The discharge for this casewill be given by Eq. (21.76).

If the level of submergence of the exit is close tothe level of the entrance, it may be assumed that thebackwater will cause the culvert to flow full and apipe flow condition will result. The discharge forthis case is given by Eqs. (21.72) and (21.73).

When the level of submergence falls betweenthese two cases and the project does not warrant atrial approach with backwater curves, it is goodpractice to assume the condition that gives thelesser discharge.

21.20 Entrance Losses forCulverts

Flow in a culvert may be significantly affected byloss in head because of conditions at the entrance(Arts. 21.18 and 21.19). Table 21.10 lists coefficientsof entrance loss Ke for some typical entrance con-ditions.

These values are for culverts flowing full. Whenthe entrance is not submerged, the coefficients areusually somewhat lower. But because of the many

Fig. 21.41 Drawdown of water surface at a freeexit of a short culvert with slope less than criticalaffects depth at entrance and controls discharge.

Fig. 21.40 Culvert with free discharge and normal depth dn greater than critical depth dc flows fullwhen the entrance is deeply submerged. Discharge is given by equations for pipe flow.





unknowns entering into determination of culvertflow, the values tabulated can be used for sub-merged or unsubmerged cases without much lossof accuracy.

Example 21.9: Given: Maximum head above thetop of the culvert ¼ 5 ft, slope ¼ 0.01, length ¼300 ft, discharge Q ¼ 40 ft3/s, n ¼ 0.013, and freeexit. Find: size of culvert.

Procedure: First assume a trial culvert; theninvestigate the assumed section to find itsdischarge. Assume a 2 � 2 ft concrete box section.Calculate Q assuming entrance control, withEq. (21.44) for discharge through an orifice. Thecoefficient of discharge C for a 2-ft-square orifice isabout 0.6. Head h on center line of entrance ¼5þ 1 =

2 � 2 ¼ 6 ft. Entrance area a ¼ 2 � 2 ¼ 4 ft2.

Q ¼ Caffiffiffiffiffiffiffiffi2gh

p¼ 0:6� 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi64:4� 6

p¼ 47:2 ft3=s

For entrance control, the flow must be supercriticaland dn must be less than 2 ft. First find dn.

To calculate the hydraulic radius, assume thedepth is slightly less than 2 ft since this will givethe maximum possible value of the hydraulicradius for this culvert.

R ¼ area of flow

wetted perimeter¼ 2� 2

6¼ 0:67 ft

Application of Eq. (21.33a) gives

V ¼ 1:486

nR2=3S1=2 ¼ 1:486

0:013� 0:672=3 � 0:011=2

¼ 8:76 ft=s

dn ¼ Q

V �width¼ 47:2

8:76� 2¼ 2:69 ft

Since dn is greater than the culvert depth, the flow isunder pressure, and the entrance will not control.

Since the culvert is under pressure, Eq. (21.72)applies. But

H ¼ 5þ 0:01� 300 ¼ 8 ft

(see Fig. 21.40). The hydraulic radius for pipe flowis R ¼ 22=8 ¼ 1 =

2 . Substitution in Eq. (21.72) yields

8 ¼ 1:5V2

2gþ 0:0575V2 ¼ 0:0808V2

V ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8=0:0808

p¼ 9:95 ft=s

Q ¼ Va ¼ 9:95� 4 ¼ 39:8 ft3=s

Since the discharge of tile assumed culvert sec-tion under the allowable head equals the maximumexpected runoff, the assumed culvert would besatisfactory.

Open-Channel Flow

Free surface flow, or open-channel flow, includesall cases of flow in which the liquid surface is opento the atmosphere. Thus, flow in a pipe is open-channel flow if the pipe is only partly full.

21.21 Basic Elements of OpenChannels

A uniform channel is one of constant cross section.It has uniform flow if the grade, or slope, of thewater surface is the same as that of the channel.Hence, depth of flow is constant throughout.Steady flow in a channel occurs if the depth atany location remains constant with time.

The discharge Q at any section is defined as thevolume of water passing that section per unit oftime. It is expressed in cubic feet per second, ft3/s,and is given by

Q ¼ VA (21:77)

where V ¼ average velocity, ft/s

A ¼ cross-sectional area of flow, ft2

When the discharge is constant, the flow is said tobe continuous and therefore

Q ¼ V1A1 ¼ V2A2 ¼ . . . (21:78)

where the subscripts designate different channelsections. Equation (21.78) is known as the con-tinuity equation for continuous steady flow.

In a uniform channel, varied flow occurs if thelongitudinal water-surface profile is not parallel with

Table 21.10 Entrance Loss Coefficients forCulverts

Inlet condition Ke

Sharp-edged projecting inlet 0.9

Flush inlet, square edge 0.5Concrete pipe, groove or bell, projecting 0.15Concrete pipe, groove or bell, flush 0.10Well-rounded entrance 0.08





the channel bottom. Varied flow exists within thelimits of backwater curves, within a hydraulic jump,and within a channel of changing slope or discharge.

Depth of flow d is taken as the vertical distance,ft, from the bottom of a channel to the watersurface. The wetted perimeter is the length, ft, of aline bounding the cross-sectional area of flow,minus the free surface width. The hydraulic radiusR equals the area of flow divided by its wettedperimeter. The average velocity of flowV is definedas the discharge divided by the area of flow,

V ¼ Q

A(21:79)

The velocity head HV, ft, is generally given by

HV ¼ V2

2g(21:80)

where V ¼ average velocity from Eq. (21.79), ft/s


Velocity heads of individual filaments of flow varyconsiderably above and below the velocity headbased on the average velocity. Since these velocitiesare squared in head and energy computations, theaverage of the velocity heads will be greater thanthe average-velocity head. The true velocity headmay be expressed as

HVa ¼ aV2

2g(21:81)

where a is an empirical coefficient that representsthe degree of turbulence. Experimental data indi-

cate that a may vary from about 1.03 to 1.36 forprismatic channels. It is, however, normally takenas 1.00 for practical hydraulic work and is evalu-ated only for precise investigations of energy loss.

The total energy per pound of water relative tothe bottom of the channel at a vertical section iscalled the specific energy head He. It is composedof the depth of flow at any point, plus the velocityhead at the point. It is expressed in feet as

He ¼ dþ V2

2g(21:82)

A longitudinal profile of the elevation of thespecific energy head is called the energy gradeline, or the total-head line. A longitudinal profileof the water surface is called the hydraulic gradeline. The vertical distance between these profiles atany point equals the velocity head at that point.

Figure 21.42 shows a section of uniform openchannel for which the slopes of the water surfaceSw and the energy grade line S equal the slope ofthe channel bottom So.

Loss of head due to friction hf in channel lengthL equals the drop in elevation of the channel DZ inthe same distance.

21.22 Normal Depth of Flow

The depth of equilibrium flow that exists in thechannel of Fig. 21.42 is called the normal depth dn.This depth is unique for specific discharge and

Fig. 21.42 Characteristics of uniform open-channel flow.





channel conditions. It may be computed by a trial-and-error process when the channel shape, slope,roughness, and discharge are known. A form of theManning equation has been suggested for this cal-culation. (V. T. Chow. “Open-Channel Hydraulics,”McGraw-Hill Book Company, New York.)

AR2=3 ¼ Qn

1:486S1=2(21:83)

where A ¼ area of flow, ft2


Q ¼ amount of flow or discharge, ft3/s

n ¼ Manning’s roughness coefficient

S ¼ slope of energy grade line or loss ofhead, ft, due to friction per lin ft ofchannel

AR2/3 is referred to as a section factor. Depth dn foruniform channels may be computed with compu-ter software or for manual computations simplifiedby use of tables that relate dn to the bottomwidth of

a rectangular or trapezoidal channel, or to thediameter of a circular channel. (See, for example,E. F. Brater, “Handbook of Hydraulics,” 6th ed.,McGraw-Hill Book Company, New York.)

In a prismatic channel of gradually increasingslope, normal depth decreases downstream, asshown in Fig. 21.43, and specific energy first de-creases and then increases as shown in Fig. 21.44.

The specific energy is high initially where thechannel is relatively flat because of the largenormal depth (Fig. 21.43). As the depth decreasesdownstream, the specific energy also decreases. It

Fig. 21.43 Prismatic channel with graduallyincreasing bottom slope. Normal depth increasesdownstream as slope increases.

Fig. 21.44 Specific energy headHe changes with depth for constant discharge in a rectangular channelof changing slope. He is a minimum for flow with critical depth.





reaches a minimum at the point where the flowsatisfies the equation

A3

T¼ Q2

g(21:84)

in which T is the top width of the channel, ft. For arectangular channel, Eq. (21.84) reduces to

d

2¼ V2

2g(21:85)

where V ¼ Q/A ¼ mean velocity of flow, ft3/s

d ¼ depth of flow, ft

This indicates that the specific energy is a mini-mum where the normal depth equals twice thevelocity head. As the depth continues to decreasein the downstream direction, the specific energyincreases again because of the higher velocity head(Fig. 21.44).

21.23 Critical Depth ofOpen-Channel Flow

Thedepth of flowthat satisfies Eq. (21.84) is called thecritical depth dc. For a given value of specific energy,the critical depth gives the greatest discharge, orconversely, for a given discharge, the specific energyis a minimum for the critical depth (Fig. 21.44).

In the section of mild slope upstream from thecritical-depth point in Fig. 21.43, the depth isgreater than critical. The flow there is calledsubcritical flow, indicating that the velocity is lessthan that at critical depth. In the section of steeperslope below the critical-depth point, the depth isbelow critical. The velocity there exceeds that atcritical depth, and flow is supercritical.

Critical depth may be computed for a uniformchannel once the discharge is known. Determi-nation of this depth is independent of the channelslope and roughness since critical depth simplyrepresents a depth for which the specific energyhead is a minimum. Critical depth may becalculated by trial and error with Eq. (21.84), or itmay be found directly from tables (E. F. Brater,“Handbook of Hydraulics,” 6th ed., McGraw-HillBook Company, New York). For rectangularchannels, Eq. (21.84) may be reduced to

dc ¼ffiffiffiffiffiffiffiQ2

b2g3

s(21:86)

where dc ¼ critical depth, ft

Q ¼ quantity of flow or discharge, ft3/s

b ¼ width of channel, ft

Critical slope is the slope of the channel bed thatwill maintain flow at critical depth. Such slopesshould be avoided in channel design because flownear critical depth tends to be unstable and exhibitsturbulence and water-surface undulations.

Critical depth, once calculated, should be plottedfor the full length of a uniform channel, regardless ofslope, to determinewhether the normal depth at anysection is subcritical or supercritical. [As indicatedby Eq. (21.85), if the velocity head is less than half thedepth in a rectangular channel, flow is subcritical,but if velocity head exceeds half the depth, flow issupercritical.] If channel configuration is such thatthe normal depth must go from below to abovecritical, a hydraulic jump will occur, along with ahigh loss of energy. Critical depth will change if thechannel cross section changes, so the possibility of ahydraulic jump in the vicinity of a transition shouldbe investigated.

For every depth greater than critical depth, thereis a corresponding depth less than critical that hasan identical value of specific energy (Fig. 21.44).These depths of equal energy are called alternatedepths. The fact that the energy is the same foralternate depths does not mean that the flow mayswitch from one alternate depth to the other andback again; flow will always seek to attain thenormal depth in a uniform channel and willmaintain that depth unless an obstruction is met.

It can be seen from Fig. 21.44 that any obstructionto flow that causes a reduction in total head causessubcritical flow to experience a drop in depth andsupercritical flow to undergo an increase in depth.

If supercritical flow exists momentarily on a flatslope because of a sudden grade change in thechannel (Fig. 21.52b, p. 21.56), depth increasessuddenly from the depth below critical to a depthabove critical in a hydraulic jump. The depthfollowing the jump will not be the alternate depth,however. There has been a loss of energy in makingthe jump. The new depth is said to be sequent to theinitial depth, indicating an irreversible occurrence.There is no similar phenomenon that allows asudden change in depth from subcritical flow tosupercritical flow with a corresponding gain inenergy. Such a change occurs gradually, withoutturbulence, as indicated in Fig. 21.45.





21.24 Manning’s Equation forOpen Channels

One of the more popular of the numerous equa-tions developed for determination of flow in anopen channel is Manning’s variation of the Chezyformula,

V ¼ CffiffiffiffiffiffiRS

p(21:87)

where R ¼ hydraulic radius, ft

V ¼ mean velocity of flow, ft/s

S ¼ slope of energy grade line or loss ofhead due to friction, ft/lin ft of channel

C ¼ Chezy roughness coefficient

Manning proposed

C ¼ 1:4861=6

n(21:88)

where n is the coefficient of roughness in the earlierGanguillet-Kutter formula (see also Art. 21.25).When Manning’s C is used in the Chezy formula,the Manning equation for flow velocity in an openchannel results:

V ¼ 1:486

nR2=3S1=2 (21:89)

Since the discharge Q ¼ VA, Eq. (21.89) may bewritten

Q ¼ 1:486

nAR2=3S1=2 (21:90)


Q ¼ quantity of flow, ft3/s

Roughness Coefficient for Open Channels.Values of the roughness coefficient n for Manning’sequation have been determined for a wide range ofnatural and artificial channel construction mate-rials. Excerpts from a table of these coefficientstaken from V. T. Chow, “Open-Channel Hydraulics,”McGraw-Hill Book Company, New York (www.mcgraw-hill.com), are in Table 21.11. Dr. Chowcompiled data for his table from work byR. E. Horton and from technical bulletins publishedby the U.S. Department of Agriculture.

Channel roughness does not remain constantwith time or even depth of flow. An unlined channelexcavated in earth may have one n value when firstput in service and another when overgrown withweeds and brush. If an unlined channel is to have areasonably constant n value over its useful lifetime,there must be a continuing maintenance program.

Shallow flow in an unlined channel will result inan increase in the effective n value if the channelbottom is covered with large boulders or ridges ofsilt since these projections would then have a largerinfluence on the flow than for deep flow. A deeper-than-normal flowwill also result in an increase in theeffective n value if there is a dense growth of brushalong the banks within the path of flow. Whenchannel banks are overtopped during a flood, theeffective n value increases as the flow spills intoheavy growth bordering the channel. (Althoughbased on surface roughness, n in practice is some-times treated as a lumped parameter for all headlosses.) The roughness of a lined channel experienceschange with age because of both deterioration of thesurface and accumulation of foreign matter; there-fore, the average n values given in Table 21.11 are

Fig. 21.45 Change in flow stage from subcritical to supercritical occurs gradually.





recommended only for well-maintained channels.(See also Art. 21.9 and Table 21.4.)

21.25 Water-Surface Profilesfor GraduallyVaried Flow

Examples of various surface curves possible withgradually varied flow are shown in Fig. 21.46.

These surface profiles represent backwater curvesthat form under the conditions illustrated in ex-amples (a) through (r).

These curves are divided into five groups,according to the slope of the channel in which theyappear (Art. 21.23). Each group is labeled witha letter descriptive of the slope: M for mild(subcritical), S for steep (supercritical), C forcritical, H for horizontal, and A for adverse. Thetwo dashed lines in the left-hand figure for each

Table 21.11 Values of the Roughness Coefficient n for Use in the Manning Equation

Min Avg Max

A. Open-channel flow in closed conduits1. Corrugated-metal storm drain 0.021 0.024 0.0302. Cement-mortar surface 0.011 0.013 0.0153. Concrete (unfinished)

a. Steel form 0.012 0.013 0.014b. Smooth wood form 0.012 0.014 0.016c. Rough wood form 0.015 0.017 0.020

B. Lined channels1. Metal

a. Smooth steel (unpainted) 0.011 0.012 0.014b. Corrugated 0.021 0.025 0.030

2. Wooda. Planed, untreated 0.010 0.012 0.014

3. Concretea. Float finish 0.013 0.015 0.016b. Gunite, good section 0.016 0.019 0.023c. Gunite, wavy section 0.018 0.022 0.025

4. Masonrya. Cemented rubble 0.017 0.025 0.030b. Dry rubble 0.023 0.032 0.035

5. Asphalta. Smooth 0.013 0.013b. Rough 0.016 0.016

C. Unlined channels1. Excavated earth, straight and uniform

a. Clean, after weathering 0.018 0.022 0.025b. With short grass, few weeds 0.022 0.027 0.033c. Dense weeds, high as flow depth 0.050 0.080 0.120d. Dense brush, high stage 0.080 0.100 0.140

2. Dredged eartha. No vegetation 0.025 0.028 0.033b. Light brush on banks 0.035 0.050 0.060

3. Rock cutsa. Smooth and uniform 0.025 0.035 0.040b. Jagged and irregular 0.035 0.040 0.050





class are the normal-depth line N.D.L. and thecritical-depth line C.D.L. The N.D.L. and C.D.L.are identical for a channel of critical slope, and theN.D.L. is replaced by a horizontal line, at an

arbitrary elevation, for the channels of horizontalor adverse slope.

There are three types of surface-profile curvespossible in channels of mild or steep slope, and two

Fig. 21.46 Typical flow profiles for channels with various slopes. N.D.L. indicates normal-depth line;C.D.L., critical-depth line.





types for channels of critical, horizontal, andadverse slope.

TheM1 curve is the familiar surface profile fromwhich all backwater curves derive their name andis the most important from a practical point ofview. It forms above the normal-depth line andoccurs when water is backed up a stream by highwater in the downstream channel, as shown inFig. 21.46a and b.

The M2 curve forms between the normal- andcritical-depth lines. It occurs under conditionsshown in Fig. 21.46c and d, corresponding to anincrease in channel width or slope.

The M3 curve forms between the channelbottom and critical-depth line. It terminates in ahydraulic jump, except where a drop-off in thechannel occurs before a jump can form. Examplesof the M3 curve are in Fig. 21.46e and f (a partlyopened sluice gate and a decrease in channelslope, respectively).

The S1 curve begins at a hydraulic jump andextends downstream, becoming tangent to ahorizontal line (Fig. 21.46g and h) under channelconditions corresponding to those for Fig. 21.46aand b.

The S2 curve, commonly called a drawdowncurve, extends downstream from the criticaldepth and becomes tangent to the normal-depthline under conditions corresponding to those forFig. 21.46i and j.

The S3 curve is of the transitional type. It formsbetween two normal depths of less than criticaldepth under conditions corresponding to those forFig. 21.46k and l.

Examples in Fig. 21.46m through r show condi-tions for the formation of C, H, and A profiles.

The curves in Fig. 21.46 approach the normal-depth line asymptotically and terminate abruptlyin a vertical line as they approach the criticaldepth. The curves that approach the bottomintersect it at a definite angle but are imaginarynear the bottom since velocity would have to beinfinite to satisfy Eq. (21.77) if the depth were zero.The curves are shown dotted near the critical-depth line as a reminder that this portion of thecurve does not possess the same degree of accu-racy as the rest of the curve because of neglect ofvertical components of velocity in the calculations.These curves either start or end at what is called apoint of control.

A point of control is a physical location in aprismatic channel at which the depth of steady

flow may readily be determined. This depth isusually different from the normal depth for thechannel because of a grade change, gate, weir, dam,free overfall, or other feature at that location thatcauses a backwater curve to form. Calculations forthe length and shape of the surface profile of abackwater curve start at this known depth andlocation and proceed either up or downstream,depending on the type of flow. For subcritical flowconditions, the curve proceeds upstream from thepoint of control in a true backwater curve. Thesurface curve that occurs under supercritical flowconditions proceeds downstream from the point ofcontrol and might better be called a downwatercurve.

The point of control is always at the down-stream end of a backwater curve in subcritical flowand at the upstream end for supercritical flow. Thisis explained as follows: A backwater curve may bethought of as being the result of some disruption ofuniform flow that causes a wave of disturbance inthe channel. The wave travels at a speed, known asits celerity, which always equals the criticalvelocity for the channel. If a disturbance waveattempts to move upstream against supercriticalflow (flow moving at a speed greater than critical),it will be swept downstream by the flow and haveno effect on conditions upstream. A disturbancewave is held steady by critical flow and movesupstream in subcritical flow.

When a hydraulic jump occurs on a mild slopeand is followed by a free overfall (Fig. 21.51),backwater curves form both before and after thejump. The point of control for the curve in thesupercritical region above the jump will be locatedat the vena contracta that forms just below thesluice gate. The point of control for the backwatercurve in the subcritical region below the jump is atthe free overfall where critical depth occurs.Computations for these backwater curves arecarried toward the jump from their respectivepoints of control and are extended across the jumpto help determine its exact location. But a back-water curve cannot be calculated through a hy-draulic jump from either direction. The surfaceprofiles involved terminate abruptly in a verticalline as they approach the critical depth, and ahydraulic jump always occurs across critical depth.See Art. 21.27.5.

(R. H. French, “Open-Channel Hydraulics,”McGraw-Hill, Inc., New York (books.mcgraw-hill.com).)





21.26 Backwater-CurveComputations

The solution of a backwater curve involves compu-tation of a gradually varied flow profile. Solutionsavailable include the graphical-integration, direc-tion-integration, and stepmethods. Explanations ofboth the graphical- and direct-integration methodsare in V. T. Chow, “Open-Channel Hydraulics,”McGraw-Hill Book Company, New York.

Two variations of the step method include thedirect or uniform method and the standardmethod. They are simple and widely used and areavailable in many software packages.

For step-method computations, the channel isdivided into short lengths, or reaches, with rela-tively small variation. In a series of steps startingfrom a point of control, each reach is solved insuccession. Step methods have been developedfor channels with uniform or varying cross sec-tions.

Direct step method of backwater computationinvolves solving for an unknown length of channelbetween two known depths. The procedure isapplicable only to uniform prismatic channels withgradually varying area of flow.

For the section of channel in Fig. 21.47, Bern-oulli’s equation for the reach between sections 1and 2 is

SoLþ d1 þ V21

2g¼ d2 þ V2

2

2gþ �SSL (21:91)

where V1 and V2 ¼ mean velocities of flow atsections 1 and 2, ft/s

d1 and d2 ¼ depths of flow at sections 1 and2, ft

g ¼ acceleration due to gravity,32.2 ft/s2

S̄ ¼ average head loss due to fric-tion, ft/ft of channel

So ¼ slope of channel bottom

L ¼ length of channel betweensections 1 and 2, ft

Note that SoL ¼ Dz, the change in elevation, ft, ofthe channel bottom between sections 1 and 2, and�SSL ¼ hf , the head loss, ft, due to friction in the samereach. (For uniform, prismatic channels, hi, theeddy loss, is negligible and can be ignored.) �SSequals the slope calculated for the average depth in

Fig. 21.47 Channel with constant discharge and gradually varying cross section.





the reach but may be approximated by the averageof the values of friction slope S for the depths atsections 1 and 2.

Solving Eq. (21.91) for L gives

L ¼ (d2 þ V22=2g)� (d1 þ V2

1=2g)

So � �SS

¼ He2 �He1

So � �SS(21:92)

where He1 and He2 are the specific energy headsfor sections 1 and 2, respectively, as given byEq. (21.82). The friction slope S at any point may becomputed by the Manning equation, rearranged asfollows:

S ¼ n2V2

2:21R4=3(21:93)

where R ¼ hydraulic radius, ft

n ¼ roughness coefficient (Art. 21.24)

Note that the slope S used in the Manningequation is the slope of the energy grade line, notthe channel bottom. Note also that the roughnesscoefficient n is squared in Eq. (21.93), and its valuemust therefore be chosen with special care to avoidan exaggerated error in the computed frictionslope. The smaller the value of n, the longer thebackwater curve profile, and vice versa. Therefore,the smallest n possible for the prevailing conditionsshould be selected for computation of a backwatercurve if knowledge of the longest possible flowprofile is required.

The first step in the direct step method involveschoosing a series of depths for the end points ofeach reach. These depths will range from the depthat the point of control to the ending depth for thebackwater curve. This ending depth is often thenormal depth for the channel (Art. 21.22) but maybe some intermediate depth, such as for a curvepreceding a hydraulic jump. Depths should bechosen so that the velocity change across a reachdoes not exceed 20% of the velocity at the be-ginning of the reach. Also the change in depthbetween sections should never exceed 1 ft.

The specific energy head He should be com-puted for the chosen depth at each of the varioussections and the change in specific energy bet-ween sections determined. Next, the friction slopeS should be computed at each section fromEq. (21.93). The average of two sections gives thefriction slope �SS between sections. Finally, the

difference between �SS and slope of channel bottomSo should be computed and the length of reachdetermined from Eq. (21.92).

Standard step method allows computation ofbackwater curves in both nonprismatic naturalchannels and nonuniform artificial channels as wellas in uniform channels. This method involvessolving for the depth of flow at various locationsalong a channel with Bernoulli’s energy equationand a known length of reach.

A surface profile is determined in the followingmanner: The channel is examined for changes incross section, grade, or roughness, and thelocations of these changes are given station num-bers. Stations are also established between theselocations such that the velocity change betweenany two consecutive stations is not greater than20% of the velocity at the former station. Data con-cerning the hydraulic elements of the channel arecollected at each station. Computation of thesurface curve is then made in steps, starting fromthe point of control and progressing from station tostation—in an upstream direction for subcriticalflow and downstream for supercritical flow. Thelength of reach in each step is given by thestationing, and the depth of flow is determined bytrial and error.

Nonprismatic channels do not have well-defined points of control to aid in determining thestarting depth for a backwater curve. Therefore,the water-surface elevation at the beginning mustbe determined as follows:

The step computations are started at a point in thechannel some distance upstream or downstreamfrom the desired starting point, depending onwhether flow is supercritical or subcritical, respec-tively. Then, computations progress toward theinitial section. Since this stepmethod is a convergingprocess, this procedure produces the true depth forthe initial section within a relatively few steps.

The energy balance used in the standard stepmethod is shown graphically in Fig. 21.47, in whichthe position of the water surface at section 1 is Z1

and at section 2, Z2, referred to a horizontal datum.Writing Bernoulli’s equation [Eq. (21.11)] for sec-tions 1 and 2 Yields

Z1 þ V21

2g¼ Z2 þ V2

2

2gþ hf þ hi (21:94)

where V1 and V2 are the mean velocities, ft/s,at sections 1 and 2; the friction loss, ft, in the





reach ( �SSL) is denoted by hf ; and the term hi is addedto account for eddy loss, ft.

Eddy loss, sometimes called impact loss, is ahead loss caused by flow running contrary to themain current because of irregularities in the chan-nel. No rational method is available for deter-mination of eddy loss, and it is therefore oftenaccounted for, in natural channels, by a slight in-crease in Manning’s n. Eddy loss depends mainlyon a change in velocity head. For lined channels, ithas been expressed as a coefficient k to be appliedas follows:

hi ¼ kV2

1

2g� V2

2

2g

� �¼ k D

V2

2g

� �(21:95)

The coefficient k is 0.2 for diverging reaches, from 0to 0.1 for converging reaches, and about 0.5 forabrupt expansions and contractions.

The total head at any section of the channel is

H ¼ Zþ V2

2g(21:96)

where Z equals the elevation of the channel bottomabove the given datum plus the depth of flow d atthat section. Friction slope S is computed fromEq. (21.93). Then, �SS, the average friction slope forthe reach, is calculated as the mean of the slope forthe section and the preceding section. Friction losshf is the product of �SS and the length of the reachL. Eddy loss hi is found from Eq. (21.95). Next, totalheadH, ft, is obtained from Eq. (21.94), which, aftersubstitution of H from Eq. (21.96), becomes

H1 ¼ H2 þ hf þ hi (21:97)

where H1 and H2 equal the total head of sections 1and 2, respectively. The value of total headcomputed from Eq. (21.97) must agree with thevalue of total head calculated previously for thesection or the assumed water-surface elevation Z1

is incorrect. Agreement is assumed if the twovalues of total head are within 0.1 ft in elevation. Ifthe two values of total head do not agree, a newwater-surface elevation must be assumed for Z1

and the computations repeated until agreement isobtained. The value that finally leads to agreementgives the correct water-surface elevation.

Backwater curves for natural river or streamchannels (irregularly shaped channels) are calcu-lated in a manner similar to that described forregularly shaped channels. However, some accountmust be taken of the varying channel roughness

and the differences in velocity and capacity in themain channel and the overbank or floodplainportions of the stream channel. The most expedi-tious way of determining the backwater curves is toplot the channel cross section to a scale convenientfor measurement of lengths and areas; subdividethe cross section into main channels and floodplainareas; and determine the discharge, velocity, andfriction slope for each subarea at selected water-surface elevations. Utilizing the above data,determine the total discharge (the sum of thesubarea discharges), the mean velocity (the totaldischarge divided by the total area), and a (theenergy coefficient or coriolis coefficient to be ap-plied to the velocity head). Many of the availablecomputer software packages that compute back-water profiles are applicable to irregular channelsand flooded overbank areas.

The backwater curve is usually started byassuming normal depth at a point some distancedownstream from the start of the reach underanalysis. Several intermediate cross sectionsshould be taken between the point where normaldepth is assumed and the start of the reach forwhich a detailed water-surface profile is required.This allows the intermediate sections to “dampenout” any minor errors in the assumed startingwater-surface elevation.

The accuracy or validity of the water-surfaceprofile is contingent on an accurate evaluation ofthe channel roughness and judicious selection ofcross-section location. A greater number of crosssections generally enhances the validity of thewater-surface profile; however, because of theextensive calculations involved with each crosssection, their number should be limited to as few asaccuracy permits.

The effect of bridges, approach roadways, bridgepiers, and culverts can be determined usingprocedures outlined in R. H. French, “Open-ChannelHydraulics,”McGraw-Hill Book Company,New York, and J. N. Bradley, “Hydraulics ofBridge Waterways,” Hydraulics Design Series no. 1,2nd ed., U.S. Department of Transportation, FederalHighway Administration, Bureau of Public Roads,1970.

21.27 Hydraulic Jump

This is an abrupt increase in depth of rapidlyflowing water (Fig. 21.48). Flow at the jump





changes from a supercritical to a subcriticalstage with an accompanying loss of kinetic energy(Art. 21.23).

A hydraulic jump is the only means bywhich thedepth of flow can change from less than critical togreater than critical in a uniform channel. A jumpwill occur either where supercritical flow exists in achannel of subcritical slope, as shown in Figs. 21.51and 21.52b, or where a steep channel enters areservoir. The first condition ismet in amild channeldownstream from a sluice gate or ogee overflowspillway, or at an abrupt change in channel slopefrom steep to mild. The second condition occurswhere flow in a steep channel is blocked by anoverflow weir, a gate, or other obstruction.

A hydraulic jump can be either stationary ormoving, depending on whether the flow is steadyor unsteady, respectively.

21.27.1 Depth and Head Loss in aHydraulic Jump

Depth at the jump is not discontinuous. The changein depth occurs over a finite distance, known as thelength of jump. The upstream surface of the jump,known as the roller, is a turbulent mass of water,which is continually tumbling erratically againstthe rapidly flowing sheet below.

The depth before a jump is the initial depth,and the depth after a jump is the sequent depth.The specific energy for the sequent depth is lessthan that for the initial depth because of the energydissipation within the jump. (Initial and sequentdepths should not be confused with the depths ofequal energy, or alternate depths.)

According to Newton’s second law of motion,the rate of loss of momentum at the jump mustequal the unbalanced pressure force acting on themovingwater and tending to retard itsmotion. Thisunbalanced force equals the difference between the

hydrostatic forces corresponding to the depthsbefore and after the jump. For rectangular channels,this resultant pressure force is

F ¼ d22w

2� d21w

2(21:98)

where d1 ¼ depth before jump, ft

d2 ¼ depth after jump, ft

w ¼ unit weight of water, lb/ft3

The rate of change of momentum at the jump perfoot width of channel equals

F ¼ MV1 �MV2

t¼ qw

g(V1 � V2) (21:99)

where M ¼ mass of water, lb . s2/ft

V1 ¼ velocity at depth d1, ft/s

V2 ¼ velocity at depth d2, ft/s

q ¼ discharge per foot width of rectangularchannel, ft3/s

t ¼ unit of time, s


Equating the values of F in Eqs. (21.98) and (21.99),and substituting V1d1 for q and V1d1/d2 for V2, thereduced equation for rectangular channels becomes

V21 ¼ gd2

2d1(d2 þ d1) (21:100)

Equation (21.100) may then be solved for thesequent depth:

d2 ¼ �d12

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2V2

1d1g

þ d214

s(21:101)

If V2d2/d1 is substituted for V1, in Eq. (21.100),

d1 ¼ �d22

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2V2

2d2g

þ d224

s(21:102)

Equation (21.102) may be used in determining theposition of the jump where V2 and d2 are known.Relationships may be derived similarly for chan-nels of any cross section.

The head loss in a jump equals the difference inspecific-energy head before and after the jump.This difference (Fig. 21.49) is given by

DHe ¼ He1 �He2 ¼ (d2 � d1)3

4d1d2(21:103)

Fig. 21.48 Hydraulic jump.





where He1 ¼ specific-energy head of stream beforejump, ft

He2 ¼ specific-energy head of stream afterjump, ft

The specific energy for free-surface flow is given byEq. (21.82).

The depths before and after a hydraulicjump may be related to the critical depth by theequation

d1d2d1 þ d2

2¼ q2

g¼ d3c (21:104)

where q ¼ discharge, ft3/s per ft of channel width

dc ¼ critical depth for the channel, ft

It may be seen from this equation that if d1 ¼ dc, d2must also equal dc.

21.27.2 Jump in HorizontalRectangular Channels

The form of a hydraulic jump in a horizontalrectangular channel may be of several distincttypes, depending on the Froude number of theincoming flow F ¼ V/(gL)1/2 [Eq. (21.16)], where Lis a characteristic length, ft; V is the mean velocity,ft/s; and g ¼ acceleration due to gravity, ft/s2. Foropen-channel flow, the characteristic length for theFroude number is made equal to the hydraulicdepth dh.

Hydraulic depth is defined as

dh ¼ A

T(21:105)


T ¼ width of free surface, ft

For rectangular channels, hydraulic depth equalsdepth of flow.

Various forms of hydraulic jump, and theirrelation to the Froude number of the approachingflow F1, were classified by the U.S. Bureau ofReclamation and are presented in Fig. 21.49.

For F1 ¼ 1, the flow is critical and there is nojump.

For F1 ¼ 1 to 1.7, there are undulations on thesurface. The jump is called an undular jump.

For F1 ¼ 1.7 to 2.5, a series of small rollersdevelop on the surface of the jump, but thedownstream water surface remains smooth.The velocity throughout is fairly uniform and theenergy loss is low. This jump may be called a weakjump.

For F1 ¼ 2.5 to 4.5, an oscillating jet is enteringthe jump. The jet moves from the channel bottomto the surface and back again with no set period.Each oscillation produces a large wave of irregularperiod, which, very commonly in canals, can travelfor miles, doing extensive damage to earth banksand riprap surfaces. This jump may be called anoscillating jump.

For F1 ¼ 4.5 to 9.0, the downstream extremity ofthe surface roller and the point at which the high-velocity jet tends to leave the flow occur atpractically the same vertical section. The actionand position of this jump are least sensitive tovariation in tailwater depth. The jump is well-balanced, and the performance is at its best. Theenergy dissipation ranges from 45 to 70%. Thisjump may be called a steady jump.

Fig. 21.49 Type of hydraulic jump depends onFroude number.





For F1 ¼ 9.0 and larger, the high-velocity jetgrabs intermittent slugs of water rolling down thefront face of the jump, generating waves down-stream and causing a rough surface. The jumpaction is rough but effective, and energy dissipa-tion may reach 85%. This jump may be called astrong jump.

Note that the ranges of the Froude numbergiven for the various types of jump are not clear-cutbut overlap to a certain extent, depending on localconditions.

21.27.3 Hydraulic Jump as anEnergy Dissipator

A hydraulic jump is a useful means for dissipatingexcess energy in supercritical flow (Art. 21.23). Ajump may be used to prevent erosion below anoverflow spillway, chute, or sluice gate by quicklyreducing the velocity of the flow over a pavedapron. A special section of channel built to containa hydraulic jump is known as a stilling basin.

If a hydraulic jump is to function ideally as anenergy dissipator, below a spillway, for example,the elevation of the water surface after the jumpmust coincide with the normal tailwater elevationfor every discharge. If the tailwater is too low, thehigh-velocity flow will continue downstream forsome distance before the jump can occur. If thetailwater is too high, the jumpwill be drowned out,and there will be a much smaller dissipation oftotal head. In either case, dangerous erosion islikely to occur for a considerable distance down-stream.

The ideal condition is to have the sequent-depthcurve, which gives discharge vs. depth after thejump, coincide exactly with the tailwater-ratingcurve. The tailwater-rating curve gives normaldepths in the discharge channel for the range offlows to be expected. Changes in the spillwaydesign that can bemade to alter the tailwater-ratingcurve involve changing the crest length, changingthe apron elevation, and sloping the apron.

Accessories, such as chute blocks and baffleblocks are usually installed in a stilling basin tocontrol the jump. The main purpose of theseaccessories is to shorten the range within which thejump will take place, not only to force the jump tooccur within the basin but to reduce the size andtherefore the cost of the basin. Controls within astilling basin have additional advantages in that

they improve the dissipation function of the basinand stabilize the jump action.

21.27.4 Length of Hydraulic Jump

The length of a hydraulic jump L may be definedas the horizontal distance from the upstream edgeof the roller to a point on the raised surfaceimmediately downstream from cessation of theviolent turbulence. This length (Fig. 21.48) defiesaccurate mathematical expression, partly becauseof the nonuniform velocity distribution within thejump. But it has been determined experimentally.The experimental results may be summarizedconveniently by plotting the Froude number of theupstream flow F1 against a dimensionless ratio ofjump length to downstream depth L/d2. Theresulting curve (Fig. 21.50) has a flat portion inthe range of steady jumps. The curve thus mini-mizes the effect of any errors made in calculation ofthe Froude number in the range where thisinformation is most frequently needed. The curve,prepared by V. T. Chow from data gathered by theU.S. Bureau of Reclamation, was developed forjumps in rectangular channels, but it will giveapproximate results for jumps formed in trapezoi-dal channels.

For other than rectangular channels the depth d1used in the equation for Froude number is thehydraulic depth given by Eq. (21.105).

21.27.5 Location of aHydraulic Jump

It is important to know where a hydraulic jumpwill form since the turbulent energy released in ajump can extensively scour an unlined channel ordestroy paving in a thinly lined channel. Specialreinforced sections of channel must be built towithstand the pounding and vibration of a jumpand to provide extra freeboard for the added depthat the jump. These features are expensive to build;therefore, a great savings can be realized if their useis restricted to a limited area through a knowledgeof the jump location.

The precision with which the location is pre-dicted depends on the accuracy with which thefriction losses and length of jump are estimated andon whether the discharge is as assumed. Themethod of prediction used for rectangular channelsis illustrated for a sluice gate in Fig. 21.51.





The water-surface profiles of the flow approach-ing and leaving the jump, curves AB and ED inFig. 21.51, are type M3 and M2 backwater curves,respectively (Fig. 21.46e and c).

Backwater curve ED has as its point of controlthe critical depth dc, which occurs near the channeldrop-off. Critical depth does not exist exactly at theedge, as theory would indicate, but instead occursa short distance upstream. The distance is small(from three to four times dc) and can be ignored formost problems. The actual depth at the brink is71.5% of critical depth, but it is normally assumedto be 0.7dc for simplicity.

The point of control for backwater curve AB istaken as the depth at the vena contracta, which

forms just downstream from the sluice gate. Thedistance from the gate to the vena contracta Le isnearly equal to the size of gate opening h. Theamount of contraction varies with both the head onthe gate and the gate opening. Depth at thecontraction ranges from 50 to over 90% of h. Thedepth of flow at the vena contracta may be taken as0.75h in the absence of better information.

Jump location is determined as follows: Thebackwater curves AB and ED are computed in theirrespective directions until they overlap, using thestep methods of Art. 21.26. With values of d2obtained from Eq. (21.101), CB, the curve of depthssequent to curve AB, is plotted through the areawhere it crosses curve ED. A horizontal intercept

Fig. 21.51 Graphical method for locating hydraulic jump beyond a sluice gate.

Fig. 21.50 Length of hydraulic jump in a horizontal channel depends on sequent depth d2 and theFroude number of the approaching flow.





FG, equal in length to L, the computed length ofjump, is then fitted between the curves CB and ED.The jump may be expected to form between thepoints H and G since all requirements for the for-mation of a jump are satisfied at this location.

If the downstream depth is increased because ofan obstruction, the jump moves upstream and mayeventually be drowned out in front of the sluice gate.Conversely, if the downstream depth is lowered,the jump moves to a new location downstream.

When the slope of a channel has an abruptchange from steeper than critical (Art. 21.23) tomild, a jump forms that may be located eitherabove or below the grade change. The position ofthe jump depends on whether the downstreamdepth d2 is greater than, less than, or equal to thedepth d01 sequent to the upstream depth d1. Twopossible positions are shown in Fig. 21.52.

It is assumed, for simplicity, that flow isuniform, except in the reach between the jumpand the grade break. If the downstream depth d2 isgreater than the upstream sequent depth d01,computed from Eq. (21.101) with d1 given, thejump occurs in the steep region, as shown in Fig.21.52a. The surface curve EO is of the S1 type (Fig.21.46) and is asymptotic to a horizontal line at O.Line CB0 is a plot of the depth d01 sequent to thedepth of approach line AB. The jump location isfound by producing a horizontal intercept FG,equal to the computed length of the jump, betweenlines CB0 and EO. A jump will form between H andG since all requirements are satisfied for thislocation. As depth d2 is lowered, the jump movesdownstream to a new position, as shown inFig. 21.52b. If d2 is less than d01, computed fromEq. (21.102), the jump will form in the mild channeland can be located as described for Fig. 21.51.

(R. H. French, “Open-Channel Hydraulics,”McGraw-Hill, Inc., New York (books.mcgraw-hill.com).)

21.28 Flow at Entrance to aSteep Channel

The discharge Q, ft3/s, in a channel leaving areservoir is a function of the total head H, ft, on thechannel entrance, the entrance loss, ft and theslope of the channel. If the channel has a slopesteeper than the critical slope (Art. 21.23), the flowpasses through critical depth at the entrance, anddischarge is at a maximum. If the channel entranceis rectangular in cross section, the critical depthdc ¼ 2 =

3He [according to Eqs. (21.82) and (21.85)],where He is the specific energy head, ft, in thereservoir and datum is the elevation of the lip of thechannel (Fig. 21.53a).

From Q ¼ AV, with the area of flow A ¼ bdc ¼2 =

3bHe and the velocity

V ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g(He � dc)

p¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi64:4He

3

r

the discharge for rectangular channels, ignoringentrance loss, is

Q ¼ 3:087bH3=2e (21:106)

where b is the channel width, ft.If the entrance loss must be considered, or if the

channel entrance is other than rectangular, the inlet

Fig. 21.52 Hydraulic jump may occur at achange in bottom slope, or (a) above it, or (b)below it.





depth must be solved for by trial and error sincethe discharge is unknown. The procedure for find-ing the correct discharge is as follows:

A trial discharge is chosen. Then, the criticaldepth for the given shape of channel entrance isdetermined (see those in E. F. Brater, “Handbook ofHydraulics,” 6th ed., McGraw-Hill Book Company,New York.) Adding dc to its associated velocity headgives the specific energy in the channel entrance, towhich the resulting entrance loss is added. This sumthen is compared with the specific energy of thereservoir water, which equals the depth of waterabove datum plus the velocity head of flow towardthe channel. (This velocity head is normally so smallthat it may be taken as zero in most calculations.) Ifthe specific energy computed for the depth of waterin the reservoir equals the sumof specific energy andentrance loss determined for the channel entrance,

then the assumed discharge is correct; if not, a newdischarge is assumed, and the computations con-tinued until a balance is reached.

A first trial discharge may be found fromQ ¼ A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g(He � d)

p, where (He � d) gives actual

head producing flow (Fig. 21.53). A reasonablevalue for the depth d would be 2 =

3He for steepchannels and an even greater percentage of He formild channels.

The entrance loss equals the product of anempirical constant k and the change in velocityhead DHn at the entrance. If the velocity in thereservoir is assumed to be zero, then the entranceloss is k(V1

2/2g), where V1 is the velocity computedfor the channel entrance. Safe design values forthe coefficient vary from about 0.1 for a well-rounded entrance to slightly over 0.3 for one withsquared ends.

Fig. 21.53 Flow at entrance to (a) steep channel; (b) mild-slope channel.





21.29 Flow at Entrance to aChannel of Mild Slope

When water flows from a reservoir into a channelwith slope less than the critical slope (Art. 21.23),the depth of flow at the channel entrance equals thenormal depth for the channel (Art. 21.22). Theentrance depth and discharge are dependent oneach other. The discharge that results from a givenhead is that for which flow enters the channelwithout forming either a backwater or drawdowncurve within the entrance. This requirementnecessitates the formation of normal depth d sinceonly at this equilibrium depth is there no tendencyto change the discharge or to form backwatercurves. (In Fig. 21.53b, d is normal depth.)

A solution for discharge at entrance to a channelof mild slope is found as follows: A trial discharge,ft3/s, is estimated from Q ¼ A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g(He � d)

p, where

He 2 d is the actual head, ft, producing flow. He isthe specific energy head, ft, of the reservoir waterrelative to datum at lip of channel; A is the cross-sectional area of flow, ft2; and g is acceleration dueto gravity, 32.2 ft/s2. The normal depth of thechannel is determined for this discharge fromEq. (21.83). The velocity head is computed for thisdepth-discharge combination, and an entrance-losscalculation is made (see Art. 21.33). The sum of thespecific energy of flow in the channel entrance andthe entrance loss must equal the specific energy ofthe water in the reservoir for an energy balance toexist between those points (Fig. 21.53b). If the trialdischarge gives this balance of energy, then thedischarge is correct; if not, a new discharge ischosen, and the calculations continued until asatisfactory balance is obtained.

21.30 Channel Section ofGreatest Efficiency

If a channel of any shape is to reach its greatesthydraulic efficiency, it must have the shortest pos-sible wetted perimeter for a given cross-sectionalarea. The resulting shape gives the greatest hy-draulic radius and therefore the greatest capacityfor that area. This can be seen from the Manningequation for discharge [Eq. (21.83)], in which Q is adirect function of hydraulic radius to the two-thirds power.

The most efficient of all possible open-channelcross sections is the semicircle. There are practical

objections to the use of this shape because of thedifficulty of construction, but it finds some use inmetal flumes where sections can be preformed. Themost efficient of all trapezoidal sections is the halfhexagon, which is used extensively for large water-supply channels. The rectangular section with thegreatest efficiency has a depth of flow equal to one-half the width. This shape is often used for boxculverts and small drainage ditches.

21.31 Subcritical Flow aroundBends in Channels

Because of the inability of liquids to resist shearingstress, the free surface of steady uniform flow isalways normal to the resultant of the forces actingon the water. Water in a reservoir has a horizontalsurface since the only force acting on it is the forceof gravity.

Water reacts in accordance with Newton’s firstlaw of motion: It flows in a straight line unlessdeflected from its path by an outside force. Whenwater is forced to flow in a curved path, its surfaceassumes a position normal to the resultant of theforces of gravity and radial acceleration. The forcedue to radial acceleration equals the force requiredto turn the water from a straight-line path, or mV2/rc form, a unit mass of water, where V is its averagevelocity, ft/s, and rc the radius of curvature, ft, ofthe center line of the channel.

The water surface makes an angle f with thehorizontal such that

tanf ¼ V2

rcg(21:107)

The theoretical difference y, ft, in water-surfacelevel between the inside and outside banks of acurve (Fig. 21.54) is found by multiplying tan f bythe top width of the channel T, ft. Thus,

y ¼ V2T

rcg(21:108)

where the radius of curvature rc of the center of thechannel is assumed to represent the averagecurvature of flow. This equation gives values of ysmaller than those actually encountered because ofthe use of average values of velocity and radius,rather than empirically derived values more rep-resentative of actual conditions. The error will notbe great, however, if the depth of flow is well above





critical (Art. 21.23). In this range, the true value of ywould be only a few inches.

The difference in surface elevation found fromEq. (21.108), although it involves some drop insurface elevation on the inside of the curve, doesnot allow a savings of freeboard height on theinside bank. The water surface there is wavy andthus needs a freeboard height at least equal to thatof a straight channel.

The top layer of flow in a channel has a highervelocity than flow near the bottom because of theretarding effect of friction along the floor ofthe channel. A greater force is required to deflectthe high-velocity flow. Therefore, when a streamenters a curve, the higher-velocity flow movesto the outside of the bend. If the bend conti-nues long enough, all the high-velocity water willmove against the outer bank and may causeextensive scour unless special bank protection isprovided.

Since the higher-velocity flow is pressed dir-ectly against the bank, an increase in friction lossresults. This increased loss may be accounted forin calculations by assuming an increased value ofthe roughness coefficient n within the curve.Scobey suggests that the value of n be increasedby 0.001 for each 208 of curvature in 100 ft offlume. His values have not been evaluatedcompletely, however, and should be used withdiscretion. (F. C. Scobey, “The Flow of Water inFlumes,” U.S. Department of Agriculture, TechnicalBulletin 393.)

21.32 Supercritical Flowaround Bends inChannels

When water, traveling at a velocity greater thancritical (Art. 21.23), flows around a bend in a channel,a series of standing waves are produced. Twowavesform at the start of the curve. One is a positive wave,of greater-than-average surface elevation, whichstarts at the outside wall and extends across thechannel on the lineAME (Fig. 21.55). The second is anegative wave, with a surface elevation of less-than-average height, which starts at the inside wall andextends across the channel on the line BMD. Thesewaves cross at M, are reflected from oppositechannel walls at D and E, recross as shown, andcontinue crossing and recrossing.

The two waves at the entrance form at anangle with the approach channel known as thewave angle bo. This angle may be determined fromthe equation

sinbo ¼1

F1(21:109)

where F1 represents the Froude number of flow inthe approach channel [Eq. (21.16)].

The distance from the beginning of the curve tothe first wave peak on the outside bank is deter-mined by the central angle uo. This angle may befound from

tan uo ¼ T

(rc þ T=2) tanbo

(21:110)

where T is the normal top width of channel and rcis the radius of curvature of the center of channel.The depths along the banks at an angle u , uo aregiven by

d ¼ V2

gsin2 bo +

u

2

� �(21:111)

Fig. 21.54 Water-surface profile at a bend in achannel with subcritical flow.

Fig. 21.55 Plan view of supercritical flowaround a bend in an open channel.





where the positive sign gives depths along theoutside wall and the negative sign, depths alongthe inside wall. The depth of maximum heightfor the first positive wave is obtained by substitut-ing the value of uo found from Eq. (21.110) for u inEq. (21.111).

Standing waves in existing rectangular chan-nels may be prevented by installing diagonal sillsat the beginning and end of the curve. The sillsintroduce a counterdisturbance of the right magni-tude, phase, and shape to neutralize the undesir-able oscillations that normally form at the changeof curvature. The details of sill design have beendetermined experimentally.

Good flow conditions may be ensured in newprojects with supercritical flow in rectangularchannels by providing transition curves or bybanking the channel bottom. Circular transitioncurves aid in wave control by setting up counter-disturbances in the flow similar to those providedby diagonal sills. A transition curve should have aradius of curvature twice the radius of the centralcurve. It should curve in the same directionand have a central angle given, with sufficientaccuracy, by

tan ut ¼ T

2rc tanbo

(21:112)

Transition curves should be used at both the be-ginning and end of a curve to prevent disturbancesdownstream.

Banking the channel bottom is the most effectivemethod of wave control. It permits equilibriumconditions to be set up without introduction of acounterdisturbance. The cross slope required forequilibrium is the same as the surface slope foundfor subcritical flow around a bend (Fig. 21.54). Theangle f the bottom makes with the horizontal isfound from the equation

tanf ¼ V2

rcg(21:113)

21.33 Transitions in OpenChannels

A transition is a structure placed between two openchannels of different shape or cross-sectionalarea to produce a smooth, low-head-loss transferof flow. The major problems associated with designof a transition lie in locating the invert and

determining the various cross-sectional areas sothat the flow is in accord with the assumptionsmade in locating the invert. Many variables, suchas flow-rate changes, wall roughness, and channelshape and slope, must be taken into account indesign of a smooth-flow transition.

When proceeding downstream through atransition, the flow may remain subcritical orsupercritical (Art. 21.23), change from subcriticalto supercritical, or change from supercritical tosubcritical. The latter flow possibility may producea hydraulic jump.

Special care must be exercised in the design ifthe depth in either of the two channels connected isnear the critical depth. In this range, a small changein energy head within the transition may cause thedepth of flow to change to its alternate depth. Aflow that switches to its subcritical alternate depthmay overflow the channel. A flow that changes toits supercritical alternate depth may cause exces-sive channel scour. The relationship of flow depthto energy head can be shown on a plot such asFig. 21.44, p. 21.43.

To place a transition properly between two openchannels, it is necessary to determine the designflow and calculate normal and critical depths foreach channel section. Maximum flow is usuallyselected as the design flow. Normal depth for eachsection is used for the design depth. After thedesign has been completed for maximum flow,hydraulic calculations should be made to check thesuitability of the structure for lower flows.

The transition length that produces a smooth-flowing, low-head-loss structure is obtained for anangle of about 12.58 between the channel axis andthe lines of intersection of the water surface withthe channel sides, as shown in Fig. 21.56. Thelength of the transition Lt is then given by

Lt ¼1 =

2 (T2 � T1)

tan 12:5W

(21:114)

Fig. 21.56 Plan view of a transition betweentwo open channels with different widths.





where T2 and T1 are the top widths of sections 2and 1, respectively.

In design of an inlet-type transition structure,the water-surface level of the downstream channelmust be set below the water-surface level of theupstream channel by at least the sum of theincrease in velocity head, plus any transition andfriction losses. The transition loss, ft, is given byK(DV2/2g), where K, the loss factor, equals about0.1 for an inlet-type structure; DV is the velocitychange, ft/s; and g ¼ 32.2 ft/s2. The total drop inwater surface yd across the inlet-type transition isthen 1.1[D(V2/2g)], if friction is ignored.

For outlet-type structures, the average velocitydecreases, and part of the loss in velocity head isrecovered as added depth. The rise of the watersurface for an outlet structure equals the decreasein velocity head minus the outlet and frictionlosses. The outlet loss factor is normally 0.2 forwell-designed transitions. If friction is ignored, thetotal rise in water surface yr across the outletstructure is 0.8[D(V2/2g)].

Many well-designed transitions have a reverseparabolic water-surface curve tangent to the watersurfaces in each channel (Fig. 21.57). After such awater-surface profile is chosen, depth and cross-sectional areas are selected at points along thetransition to produce this smooth curve. Straight,angular walls usually will not produce a smoothparabolic water surface; therefore, a transition witha curved bottom or sides has to be designed.

The total transition length Lt is split into an evennumber of sections of equal length x. For Fig. 21.57,six equal lengths of 10 ft each are used, for anassumed drop in water surface yd of 1 ft. It isassumed that the water surface will follow parabolaAC for the length Lt/2 to produce a water-surfacedrop of yd/2 and that the other half of the surface

drop takes place along the parabola CB. The water-surface profile can be determined from the generalequation for a parabola, y ¼ ax2, where y is the ver-tical drop in the distance x, measured from A or B.

The surface drops at sections 1 and 2 are found asfollows: At the midpoint of the transition, y3 ¼ax2 ¼ yd/2 ¼ 0.5 ¼ a(30)2, fromwhich a ¼ 0.000556.Then y1 ¼ ax21 ¼ 0:000556(10)2 ¼ 0:056 ft and y2 ¼ax22 ¼ 0:000556(20)2 ¼ 0:222 ft.

21.34 Weirs

A weir is a barrier in an open channel over whichwater flows. The edge or surface over which thewater flows is called the crest. The overflowingsheet of water is the nappe.

If the nappe discharges into the air, the weir hasfree discharge. If the discharge is partly under water,the weir is submerged or drowned.

21.34.1 Types of Weirs

Aweir with a sharp upstream corner or edge suchthat the water springs clear of the crest is a sharp-crested weir (Fig. 21.58). All other weirs are classed

Fig. 21.58 Sharp-crested weir.

Fig. 21.57 Profile of reverse parabolic water-surface curve for well-designed transitions.





asweirs not sharp-crested. Sharp-crested weirs areclassified according to the shape of the weiropening, such as rectangular weirs, triangular orV-notch weirs, trapezoidal weirs, and parabolicweirs. Weirs not sharp-crested are classified ac-cording to the shape of their cross section, such asbroad-crested weirs, triangular weirs, and, asshown in Fig. 21.59, trapezoidal weirs.

The channel leading up to a weir is the channelof approach. The mean velocity in this channel isthe velocity of approach. The depth of waterproducing the discharge is the head.

Sharp-crested weirs are useful only as a meansof measuring flowing water. In contrast, weirs notsharp-crested are commonly incorporated into hy-draulic structures as control or regulation devices,with measurement of flow as their secondaryfunction.

21.34.2 Rectangular Sharp-CrestedWeirs

Discharge over a rectangular sharp-crested weir isgiven by

Q ¼ CLH3=2 (21:115)


C ¼ discharge coefficient

L ¼ effective length of crest, ft

H ¼ measured head ¼ depth of flow aboveelevation of crest, ft

The head should be measured at least 2.5H up-stream from the weir, to be beyond the drop in thewater surface (surface contraction) near the weir.

Numerous equations have been developed forfinding the discharge coefficient C. One such equa-tion, which applies only when the nappe is fully

ventilated, was developed by Rehbock and simpli-fied by Chow:

C ¼ 3:27þ 0:40H

P(21:116)

where P is the height of the weir above the channelbottom (Fig. 21.58) (V. T. Chow, “Open-ChannelHydraulics,” McGraw-Hill Book Company, NewYork).

The height of weir P must be at least 2.5H for acomplete crest contraction to form. If P is less than2.5H, the crest contraction is reduced and said to bepartly suppressed. Equation (21.116) corrects forthe effects of friction, contraction of the nappe,unequal velocities in the channel of approach, andpartial suppression of the crest contraction andincludes a correction for the velocity of approachand the associated velocity head.

To be fully ventilated, a nappe must have itslower surface subjected to full atmospheric press-ure. A partial vacuum below the nappe can resultthrough removal of air by the overflowing jet ifthere is restricted ventilation at the sides of theweir.This lack of ventilation causes increased dischargeand a fluctuation and shape change of the nappe.The resulting unsteady condition is very objection-able when the weir is used as a measuring device.

At very low heads, the nappe has a tendency toadhere to the downstream face of a rectangular weireven when means for ventilation are provided. Aweir operating under such conditions could not beexpected to have the same relationship betweenheadand discharge as would a fully ventilated nappe.

A V-notch weir (Fig. 21.60) should be used formeasurement of flow at very low heads if accuracyof measurement is required.

End contractions occur when the weir openingdoes not extend the full width of the approachchannel. Water flowing near the walls must movetoward the center of the channel to pass over theweir, thus causing a contraction of the flow. Thenappe continues to contract as it passes over thecrest. Hence, below the crest, the nappe has aminimum width less than the crest length.

The effective length L, ft, of a contracted-widthweir is given by

L ¼ L0 � 0:1NH (21:117)

where L0 ¼ measured length of crest, ft

N ¼ number of end contractions

H ¼ measured head, ftFig. 21.59 Weir not sharp-crested.





If flow contraction occurs at both ends of a weir,there are two end contractions and N ¼ 2. If theweir crest extends to one channel wall but notthe other, there is one end contraction and N ¼ 1.The effective crest length of a full-width weir istaken as its measured length. Such a weir is said tohave its contractions suppressed.

21.34.3 Triangular or V-NotchSharp-Crested Weirs

The triangular or V-notch weir (Fig. 21.60) has adistinct advantage over a rectangular sharp-crestedweir (Art. 21.34.2) when low discharges are to bemeasured. Flow over a V-notch weir starts at apoint, and both discharge and width of flowincrease as a function of depth. This has the effectof spreading out the low-discharge end of thedepth-discharge curve and therefore allows moreaccurate determination of discharge in this region.

Discharge is given by

Q ¼ C1H5=2 tan

u

2(21:118)

where u ¼ notch angle

H ¼ measured head, ft

C1 ¼ discharge coefficient

The headH is measured from the notch elevation tothe water-surface elevation at a distance 2.5Hupstream from the weir. Values of the dischargecoefficient were derived experimentally by Lenz,who developed a procedure for including the effectof viscosity and surface tension as well as the effectof contraction and velocity of approach (A. T. Lenz,“Viscosity and Surface Tension Effects on V-NotchWeir Coefficients,” Transactions of the American

Society of Civil Engineers, vol. 69, 1943). His valueswere summarized by Brater, who presented thedata in the form of curves (Fig. 21.61) (E. F. Brater“Handbook of Hydraulics,” 6th ed., McGraw-HillBook Company, New York).

AV-notch weir tends to concentrate or focus theoverflowing nappe, causing it to spring clear of thedownstream face for even the smallest flows. Thischaracteristic prevents a change in the head-dis-charge relationship at low flows and adds materi-ally to the reliability of the weir.

21.34.4 Trapezoidal Sharp-CrestedWeirs

The discharge from a trapezoidal weir (Fig. 21.62)is assumed the same as that from a rectangular weirand a triangular weir in combination.

Q ¼ C2LH3=2 þ C3ZH

5=2 (21:119)

Fig. 21.60 V-notch weir.

Fig. 21.61 Chart gives discharge coefficientsfor sharp-crested V-notch weirs. The coefficientsdepend on head and notch angle.






L ¼ length of notch at bottom, ft

H ¼ head, measured from notch bottom, ft

Z ¼ b/H [substituted for tan(u/2)in Eq. (21.118)]

b ¼ half the difference between lengths ofnotch at top and bottom, ft

No data are available for determination of coeffi-cients C2 and C3. They must be determined experi-mentally for each installation.

21.34.5 Submerged Sharp-CrestedWeirs

The discharge over a submerged sharp-crestedweir (Fig. 21.63) is affected not only by the head onthe upstream side H1 but by the head downstreamH2. Discharge also is influenced to some extent bythe height P of the weir crest above the floor of thechannel.

The discharge Qs, ft3/s, for a submerged weir is

related to the free or unsubmergeddischargeQ, ft3/s,for that weir by a function of H2/H1. Villemonteexpressed this relationship by the equation

Qs

Q¼ 1� H2

H1

� �n� �0:385(21:120)

where n is the exponent of H in the equation for freedischarge for the shape of weir used. (The value of n

is 3⁄2for a rectangular sharp-crested weir and 5⁄2for atriangular weir.) To use the Villemonte equation, firstcompute the rate of flow Q for the weir when notsubmerged, and then, using this rate and therequired depths, solve for the submerged dischargeQs. (J. R. Villemonte, “Submerged-Weir DischargeStudies,” Engineering News-Record, Dec. 25, 1947,p. 866.)

Equation (21.120) may be used to compute thedischarge for a submerged sharp-crested weir ofany shape simply by changing the value of n. Themaximum deviation from the Villemonte equationfor all test results was found to be 5%. Where greataccuracy is essential, it is recommended that theweir be tested in a laboratory under conditionscomparable with those at its point of intended use.

21.34.6 Weirs Not Sharp-Crested

These are sturdy, heavily constructed devices,normally an integral part of hydraulic projects(Fig. 21.59). Typically, a weir not sharp-crestedserves as the crest section for an overflow dam orthe entrance section for a spillway or channel. Sucha weir can be used for discharge measurement, butits purpose is normally one of control and regul-ation of water levels or discharge, or both.

The discharge over a weir not sharp-crested isgiven by

Q ¼ CLH3=2t (21:121)



L ¼ effective length of crest, ft

Ht ¼ total head on crest including velocityhead of approach, ft

The head of water producing discharge over aweir is the total of measured head H and velocityhead of approach Hn. The velocity head of ap-proach is accounted for by the discharge coefficientfor sharp-crested weirs but must be consideredseparately for weirs not sharp-crested. Thus, for

Fig. 21.62 Trapezoidal sharp-crested weir.

Fig. 21.63 Submerged sharp-crested weir.

such weirs, Eq. (21.115) is rewritten in the form

Q ¼ CL H þ V2

2g

� �3=2

(21:122)

where H ¼ measured head, ft

V ¼ velocity of approach, ft/s





V2/2g ¼ Hn, velocity head of approach, ft,neglecting degree of turbulence givenby Eq. (21.81)


Since velocity and discharge are dependent oneach other in this equation and both are unknown,discharge must be found by a series of approxi-mations, which may be done as follows: First,compute a trial discharge from the measured head,neglecting the velocity head. Then, using this dis-charge, compute the velocity of approach, velocityhead, and finally total head. From this total head,compute the first corrected discharge. This cor-rected discharge will be sufficiently accurate if thevelocity of approach is small. But the processshould be repeated, starting with the correcteddischarge, where approach velocities are high.

The discharge coefficient C must be determinedempirically for weirs not sharp-crested. If a weir ofuntested shape is to be constructed, it must becalibrated in place or a model study made todetermine its head-discharge relationship. Theproblem of establishing a fixed relation betweenhead and discharge is complicated by the fact thatthe nappe may assume a variety of shapes inpassing over the weir. For each change of nappeshape, there is a corresponding change in therelation between head and discharge. The effect ismost critical for low heads. A nappe undergoesseveral changes in succession as the head varies,and the successive shapes that appear with anincreasing stage may differ from those pertainingto similar stages with decreasing head. Therefore,care must be exercised when using these weirs forflowmeasurement to ensure that the conditions aresimilar to those at the time of calibration.

Large weirs not sharp-crested often have pierson their crest to support control gates or a roadway.These piers reduce the effective length of crest bymore than the sum of their individual widthsbecause of the formation of flow contractions ateach pier. The effective crest length for a weir notsharp-crested is given by

L ¼ L0 � 2(NKp þ Ka)Ht (21:123)

where L ¼ effective crest length, ft

L0 ¼ net crest lengths, ft ¼ measured lengthminus width of all piers

N ¼ number of piers

Kp ¼ pier-contraction coefficient

Ka ¼ abutment-contraction coefficient

Ht ¼ total head on crest including velocityhead of approach, ft

(U.S. Department of the Interior, “Design of SmallDams,” Government Printing Office, Washington,DC 20402.)

The pier-contraction coefficient Kp is affectedby the shape and location of the pier nose,thickness of pier, head in relation to design heads,and approach velocity. For conditions of designhead H, the average pier-contraction coefficientsare as shown in Table 21.12.

The abutment-contraction coefficient Ka is af-fected by the shape of the abutment, the angle be-tween the upstream approach wall and the axis offlow, the head in relation to the design head, andthe approach velocity. For conditions of designhead Hd, average coefficients may be assumed asshown in Table 21.13.

21.34.7 Submergence of Weirs NotSharp-Crested

Spillways and other weirs not sharp-crested aresubmerged when their tailwater level is highenough to affect their discharge. Because of thesurface disturbance produced in the vicinity of thecrest, such a spillway or weir is unsatisfactory foraccurate flow measurement.

Approximate values of discharge may be foundby applying the following rules proposed by E. F.Brater: (1) If the depth of submergence is notgreater than 0.2 of the head, ignore the submer-gence and treat the weir as though it had freedischarge. (2) For narrow weirs having a sharpupstream leading edge, use a submerged-weirformula for sharp-crested weirs. (3) Broad-crestedweirs are not affected by submergence up toapproximately 0.66 of the head. (4) For weirswith narrow rounded crests, increase discharge

Table 21.12 Pier-Contraction Coefficients

Condition Kp

Square-nosed piers with corners rounded on aradius equal to about 0.1 of the pier thickness

0.02

Rounded-nosed piers 0.01

Pointed-nosed piers 0





obtained by a formula for submerged sharp-crestedweirs by 10% or more. Of the above rules, 1, 2, and3 probably apply quite accurately, while 4 is simplya rough approximation.

21.34.8 The Ogee-Crested Weir

The ogee-crested weir was developed in an attemptto produce a weir that would not have the unde-sirable nappe variation normally associated withweirs not sharp-crested. A shape was needed thatwould force the nappe to assume a single path forany discharge, thus making the weir consistent forflowmeasurement.Theogee-crestedweir (Fig. 21.64)has such a shape. Its crest profile conforms closely tothe profile of the lower surface of a ventilated nappeflowing over a rectangular sharp-crested weir.

The shape of this nappe, and therefore of anogee crest, depends on the head producing thedischarge. Consequently, an ogee crest is designedfor a single total head, called the design head Hd.When an ogee weir is discharging at the design

head, the flow glides over the crest with nointerference from the boundary surface and attainsnear-maximum discharge efficiency.

For flow at heads lower than the design head,the nappe is supported by the crest and pressuredevelops on the crest that is above atmospheric butless than hydrostatic. This crest pressure reducesthe discharge below that for ideal flow. (Ideal flowis flow over a fully ventilated sharp-crested weirunder the same head H.)

When the weir is discharging at heads greaterthan the design head, the pressure on the crest isless than atmospheric, and the discharge increasesover that for ideal flow. The pressure may becomeso low that separation in flowwill occur. Accordingto Chow, however, the design head may be safelyexceeded by at least 50% before harmful cavitationdevelops (V. T. Chow, “Open-Channel Hydrau-lics,” McGraw-Hill Book Company, New York(books.mcgraw-hill.com)).

The measured headH on an ogee-crested weir istaken as the distance from the highest point of thecrest to the level of the water surface at a distance2.5H upstream. This depth coincides with thedepth measured between the upstream water leveland the bottom of the nappe, at the point ofmaximum contraction, for a sharp-crested weir.This relationship is shown in Fig. 21.65.

Discharge coefficients for ogee-crested weirs aretherefore determined from sharp-crested-weircoefficients after an adjustment for this differencein head. These coefficients are a function of theapproach velocity, which varies with the ratio ofheight of weir P to actual total head Ht, wheredischarge is given by Eq. (21.122). Figure 21.66 for

Fig. 21.64 Ogee-crested weir with verticalupstream face.

Fig. 21.65 Location of origin of coordinates forsharp-crested and ogee-crested weirs.

Table21.13 Abutment-ContractionCoefficients

Condition Ka

Square abutment with headwall at 908 todirection of flow

0.20

Rounded abutments with headwall at 908 todirection of flow when 0.5Hd � r* � 0.15Hd

0.10

Rounded abutments where r* . 0.5Hd andheadwall is placed not more than 458 todirection of flow

0

* r ¼ radius of abutment rounding.





an ogee weir with a vertical upstream face givescoefficient Cd for discharge at design head Hd. (U.S.Department of the Interior, “Design of SmallDams,” Government Printing Office, Washington,DC 20402. This manual and V. T. Chow, “Open-Channel Hydraulics,” McGraw-Hill Book Com-pany, New York (books.mcgraw-hill.com), present

methods for determining the shape of an ogee crestprofile.) When the weir is discharging at other thanthe design head, the flow differs from ideal, and thedischarge coefficient changes from the dischargecoefficient given in Fig. 21.66.

Figure 21.67 gives values of the dischargecoefficient C as a function of the ratio Ht/Hd,

Fig. 21.66 Chart gives discharge coefficients at design head Hd for vertical-faced ogee-crested weirs.(From “Design of Small Dams,” U.S. Bureau of Reclamation.)

Fig. 21.67 Chart gives discharge coefficients for vertical-faced ogee-crested weirs at heads Ht otherthan design head Hd. (From “Design of Small Dams,” U.S. Bureau of Reclamation.)





where Ht is the actual head being considered andHd is the design head.

If an ogee weir has a sloping upstream face,there is a tendency for an increase in discharge overthat for a weir with a vertical face. Figure 21.68shows the ratio of the coefficient for an ogee weirwith a sloping face to the coefficient for a weir witha vertical upstream face. The coefficient of dis-charge for an ogee weir with a sloping upstreamface, if flow is at other than the design head, isdetermined from Fig. 21.66 and is then correctedfor head and slope with Figs. 21.67 and 21.68.

21.34.9 Broad-Crested Weir

This is a weir with a horizontal or nearly horizontalcrest. The crest must be sufficiently long in thedirection of flow that the nappe is supported andhydrostatic pressure developed on the crest for atleast a short distance. A broad-crested weir isnearly rectangular in cross section. Unless other-wise noted, it will be assumed to have verticalfaces, a plane horizontal crest, and sharp right-angled edges.

Figure 21.69 shows a broad-crested weir that,because of its sharp upstream edge, has contractionof the nappe. This causes a zone of reducedpressure at the leading edge. When the headH on abroad-crested weir reaches one to two times itsbreadth b, the nappe springs free, and the weir actsas a sharp-crested weir.

Discharge over a broad-crested weir is given byEq. (21.115) since the velocity of approach was

ignored in experiments performed to determinethe coefficient of discharge. These coefficients pro-bably apply more accurately, therefore, where thevelocity of approach is not high. Values of thedischarge coefficient, compiled by King, appear inTable 21.14. (E. F. Brater, “Handbook of Hydraulics,6th ed., McGraw-Hill Book Company, New York(books.mcgraw-hill.com).)

21.34.10 Weirs of Irregular Section

This group includes those weirs whose crosssection deviates from typical broad-crested orogee-crested weirs. Weirs of irregular section, fairlycommon in waterworks projects, are used asspillways and control structures. Experimentaldata are available on the more common shapes.(See, for example, E. F. Brater, “Handbook ofHydraulics,” 6th ed., McGraw-Hill Book Company,New York (books.mcgraw-hill.com).)

Fig. 21.69 Broad-crested weir.

Fig. 21.68 Chart gives design coefficients at design head Hd for ogee-crested weirs with slopingupstream face. (From “Design of Small Dams,” U.S. Bureau of Reclamation.)





21.35 Sediment Transfer andDeposition in OpenChannels

Sediment from open channels has many undesir-able effects: Reservoirs have a reduced useful lifebecause of loss of storage through the accumu-lation of silt. Sediment causes a hazard in navigablechannels and harbors and an increase in frequencyof flooding due to aggravation of rivers and floodchannels. Silting of arable land by flooding riversdestroys fertility when the silt originates from bankor gully erosion rather than from surface, or soil,erosion. The cost of operating irrigation systemsis increased by the need for frequent dredging.Water-supply facilities have increased costs be-cause of the necessity of providing desilting worksand because of the wear on mechanical equipment,such as gates, valves, and turbines.

21.35.1 Sediment Deposition inReservoirs

Deposition of silt results when the transportingforces of a river are dissipated as the river enters a

body of still water, such as a reservoir. Heavier siltsizes, those forming the bed load, are deposited in adelta as the river enters calm water. The smaller siltsizes, those carried in suspension, travel fartherinto the reservoir before deposition.

This incoming water, with its load of suspendedsilt, has a specific gravity greater than that of theclear water in the reservoir andmay form a densitycurrent, rather than mixing immediately with theclear water. A density current, once formed,quickly moves to the bottom and flows in a densecloud down the slopes of the reservoir until it isblocked by a dam. The dense flow then spreads outin this deeper area, where the stilling effect of thebasin eventually causes deposition of the sediment.Deposits of fine sediment form about one-third ofthe volume of silt deposits in a reservoir. Much ofall of this fine sediment is transported to itsfinal location by density currents. The visibledelta formed by the coarse sediments frequentlydistracts attention from the unseen bottom depo-sits of fine sediment, which are often of equalconsequence.

Most reservoirs trap from 70 to almost 100% ofthe incoming sediment, depending on whetherthe reservoir is used for flood control or storage.

Table 21.14 Values of C in Q ¼ CLH3/2 for Broad-Crested Weirs

Mea-suredHead

Breadth of crest of weir, ft

H, ft 0.50 0.75 1.00 1.50 2.00 2.50 3.00 4.00 5.00 10.00 15.00

0.2 2.80 2.75 2.69 2.62 2.54 2.48 2.44 2.38 2.34 2.49 2.680.4 2.92 2.80 2.72 2.64 2.61 2.60 2.58 2.54 2.50 2.56 2.700.6 3.08 2.89 2.75 2.64 2.61 2.60 2.68 2.69 2.70 2.70 2.700.8 3.30 3.04 2.85 2.68 2.60 2.60 2.67 2.68 2.68 2.69 2.641.0 3.32 3.14 2.98 2.75 2.66 2.64 2.65 2.67 2.68 2.68 2.63

1.2 3.32 3.20 3.08 2.86 2.70 2.65 2.64 2.67 2.66 2.69 2.641.4 3.32 3.26 3.20 2.92 2.77 2.68 2.64 2.65 2.65 2.67 2.641.6 3.32 3.29 3.28 3.07 2.89 2.75 2.68 2.66 2.65 2.64 2.631.8 3.32 3.32 3.31 3.07 2.88 2.74 2.68 2.66 2.65 2.64 2.632.0 3.32 3.31 3.30 3.03 2.85 2.76 2.72 2.68 2.65 2.64 2.63

2.5 3.32 3.32 3.31 3.28 3.07 2.89 2.81 2.72 2.67 2.64 2.633.0 3.32 3.32 3.32 3.32 3.20 3.05 2.92 2.73 2.66 2.64 2.633.5 3.32 3.32 3.32 3.32 3.32 3.19 2.97 2.76 2.68 2.64 2.634.0 3.32 3.32 3.32 3.32 3.32 3.32 3.07 2.79 2.70 2.64 2.63

(Table continued )





Flood-control reservoirs are normally emptiedshortly after a storm, so the suspended materialsare carried out with the water before settling canoccur. This procedure reduces new deposits byalmost 30% after each storm. Storage reservoirsused for water supply or power generation pur-poses, on the other hand, normally retain anyinflow long enough for settlement of all suspendedmatter to occur. Their discharges are regulated toallow generation of power or to produce a uniformflow downstreamwith no thought to the venting ofsilt-laden storm flows.

The greater part of the annual suspended siltload in a streammay be carried in a relatively shorttime. The stream runs comparatively clear duringthe remainder of the year.

Venting of much of the annual suspendedsilt load is feasible through the use of densitycurrents. These currents are stable, once formed,and often extend to the reservoir outlet. If densitycurrents are observed and their time of arrival atthe outlet determined, appropriate gates can beopened and much of the fine sediment entering astorage reservoir can be vented before it has timeto form permanent deposits. This venting oper-ation can extend the life of a reservoir by manyyears.

Numerous phenomena can destroy a reservoir,such as loss of storage capacity by landslide andloss of the dam by earthquake, landslide, over-topping, or failure of materials. The most commonmanner of destruction, however, is through loss ofstorage by deposition of silt. Redemption ofreservoir capacity lost through silting is almostalways economically unfeasible because of thewide distribution of deposits in a reservoir and thelarge quantity present. The most practicable meansof avoiding a loss in reservoir capacity are toprevent formation of permanent deposits by takingadvantage of density currents and to control rate ofsediment production from eroding areas. Whenneither can be done, sufficient storage space mustbe provided in the design of the reservoir tocompensate for depletion by silting during areasonable economic lifetime.

Sediment production and its transportation toreservoirs or navigable waters cannot be preventedat costs proportionate to the resulting benefits.However, nature may be economically improved attimes through a program of erosion control,reducing sediment production to less than thatnormally found under virgin conditions.

The deposits of silt that form in a storagereservoir are categorized into two distinct types:Delta deposits, formed from the bed load, arecoarse-grained, with an in-place weight of about80 lb/ft3. Deposits produced from the suspendedload are fine-grained, with an average weight ofabout 30 lb/ft3. They constitute about one-sixth ofthe total weight of sediment delivered but accountfor about one-third of the volume of all deposits ina storage reservoir because of their low density.If sediment deposits are periodically above water,because of fluctuations in the reservoir water level,their density increases and the volume ratios givenabove for continued submergence no longer apply.

21.35.2 Prediction of Sediment-Delivery Rate

Two methods of approach are available forpredicting the rate of sediment accumulation in areservoir; both involve predicting the rate ofsediment delivery.

One approach depends on historical records ofthe silting rate for existing reservoirs and is purelyempirical. By this method, the silting records of areservoir may be used to predict either the siltingrate for that reservoir or the probable pattern ofsilt accumulation for a proposed reservoir in asimilar area. This method allows transposition ofdata from one watershed to another because themeasured annual sediment accumulation of areservoir is expressed as a rate of sediment deli-very per unit area of its watershed. Of course, therate is not uniform during the year, or from year toyear, because of variations in rainfall, but it shouldaverage the computed annual amount over the lifeof the project. The annual silt accumulation in areservoir is determined by surveying exposeddeltas and taking depth soundings. The resultingvolume is adjusted to account for any silt lossthrough sluice gates or over the spillway and isthen expressed as silt delivery per square mile ofdrainage area. This silt-delivery figure is furtheradjusted for rainfall and runoff conditions, to give afigure that could reasonably be expected during ayear of average rainfall. If this adjusted figure is tobe transposed to a neighboring drainage basin,adjustments should be made to account for bothsoil cover and rainfall differences between thebasins. (For a discussion of the factors upon whichthis adjustment is based, see Art. 21.39.)





Silt-delivery measurements or estimates do notgive total silt production for an area because part ofthe silt produced in a basin is deposited onfloodplains and in channels before it reaches areservoir. The difference between the amount of siltproduced and that delivered increases as the sizeof the drainage area increases because of theincreased chance that the silt will be depositedbefore it reaches the reservoir. Therefore, if a silt-delivery measurement or estimate is to be tran-sposed to a basin of different size, an adjustmentshould be made to account for this discrepancy aswell. The information for this adjustment can comeonly from field reconnaissance of the two areas todetermine differences that might account for avariation in deposition of silt along the watercourses.

The second general method of calculating sedi-ment-delivery rate involves determining the rate ofsediment transport as a function of stream dis-charge and density of suspended silt. The totalsediment inflow for the year is then computedfrom these relationships and the recorded stream-discharge data.

The total quantity of sediment carried by a riveris assumed transported either as suspended load oras bed load (Art. 21.35.1). The division is based onparticle size but depends on velocity of flow aswell. There is no sharp line of demarcation betweenthe two classes. According to Witzig, about 80% ofthe volume of all sediment is produced bystreambank erosion; the remaining 20% is pro-duced by land-surface erosion. Constant erosion ofthe streambanks keeps the streambed well sup-plied with the coarse silt that travels as bed load.The fine silt that travels in suspension is producedin small amounts by streambank erosion. But forthe most part, this silt comes from land-surfaceerosion, which generally occurs only during astorm.

The total quantity of sediment in suspension isnot necessarily related directly to discharge at alltimes. The quantity is affected by seasonal varia-tions in the supply and source of fine sediment andby distribution of rainfall and runoff from thewatershed. Therefore, measurement of the sedi-ment load for a given discharge does not necessa-rily indicate the amount that may be carried by anequal discharge at another time.

The bed load consists of the silt particles toolarge to be held in suspension. This size rangeincludes particles of coarse sand, gravel, and

boulders. The bed-load particles are moved byrolling along the bed of the stream. Some of thefiner bed-load particles are moved in a series ofsteps or jumps representing a transition betweentransportation as bed load and suspended load.

The quantity of bed load is considered aconstant function of the discharge because thesediment supply for the bed-load forces is alwaysavailable in all but lined channels. An acceptedformula for the quantity of sediment transported asbed load is the Schoklitsch formula:

Gb ¼ 86:7

D1=2g

S3=2(Qi � bqo) (21:124)

where Gb ¼ total bed load, lb/s

Dg ¼ effective grain diameter, in

S ¼ slope of energy gradient

Qi ¼ total instantaneous discharge, ft3/s

b ¼ width of river, ft

qo ¼ critical discharge, ft3/s per ft of riverwidth

¼ (0.00532/S4/3)Dg

An approximate solution for bed load by theSchoklitsch formula can be made by determiningor assuming mean values of slope, discharge, and asingle grain size representative of the bed-loadsediment. A mean grain size of 0.04 in in diameter(about 1 mm) is reasonable for a river with a slopeof about 1.0 ft/mi.

The size of grains moving on the bed of a riverdepends on velocity of flow, which varies with bothslope and discharge. Therefore, the mean grain sizechanges as the flow increases during a storm or asthe river changes slope along its course. It is obviousthat considerable error could result from the use ofEq. (21.124) if it is necessary to guess at a mean graindiameter in the absence of carefully collected fielddata. Frequently, however, if insufficient data or lackofmoneypreventmore thorough investigations, thisshortcut can give results of sufficient accuracy.

Numerous formulas have been developed torepresent the condition of flow involved in tran-sportation of suspended sediment. These formulasexpress the degree of turbulent energy involved insuspension of the sediment and the mode oftransfer of this energy to the silt and other fluidparticles. The formulas require a number of em-pirical constants but are based on a sound physical





and rational foundation. They require informationas to the sediment composition by grain size, theactual quantity of silt in suspension at a givendepth, and the stream velocity. (See H. A. Einstein,“The Bed-Load Function for Sediment Transpor-tation in Open-Channel Flows,” U.S. Departmentof Agriculture.)

An approximate determination of suspendedloadmay bemadewithout using these complicatedformulas. The weight of suspended sedimenttransported by a river in an average year normallyequals about 20% of the weight transported as bedload. The total weight of material annually movedby a river is therefore equal to 120% of the weight ofmaterial transported as bed load during the year ascomputed from Eq. (21.124).

(W. H. Graf, “Hydraulics of Sediment Trans-port,” McGraw-Hill Book Company, New York(books.mcgraw-hill.com).)

21.36 Erosion Control

The various methods used in erosion control arecollectively called upstream engineering. They con-sist of soil conservation measures such as refor-estation, check-dam construction, planting ofburned-over areas, contour plowing, and regu-lation of crop and grazing practices. Also includedare measures for proper treatment of high em-bankments and cuts and stabilization of stream-banks by planting or by revetment construction.

One phase of reforestation that may be appliednear a reservoir is planting of vegetation screens.Such screens, planted on the flats adjacent to thenormal stream channel at the head of a reservoir,reduce the velocity of silt-laden storm inflows thatinundate these areas. This stilling action causesextensive deposition to occur before the silt reachesthe main cavity of the reservoir. Use of vegetationscreens, debris barriers, or desilting basins above areservoir should be planned with future develop-ment in mind. For instance, if the dam is raised at alater date, the accumulated silt in this area woulddetract from the added storage that might other-wise have been obtained.

Hydrology

Hydrology is the study of the waters of the earth,their occurrence, circulation, and distribution, their

chemical and physical properties, and their reac-tion with their environment, including theirrelation to living things. A major concern is thecirculation, on or near the land surface, of waterand its constituents throughout the hydrologiccycle. In this cycle, water evaporation from oceans,rivers, lakes, and other sources is carried over theearth and precipitated as rain or snow. Theprecipitation forms runoff on the land, infiltratesinto the soil, recharges groundwater, dischargesinto streams, and then flows into the oceans andlakes, from which evaporation restarts the cycle.Thus hydrology deals with precipitation, evapor-ation, infiltration, groundwater flow, runoff, andstream flow.

21.37 Precipitation

The primary concern with precipitation in waterresources engineering is forecasting it. The meansfor doing so are based on either current or pastdata, or a combination of the two.

Current data, in the form of synoptic weathercharts, are published daily by the U.S. WeatherBureau. These charts summarize the variousmeteorological factors, such as wind, temperature,and pressure, through whose interaction precipi-tation is produced.

Past data are primarily in the form of rainfallrecords for a standard period, such as an hour, day,or year. They are the major source of data fordetermination of the recurrence interval for stormsof a definite magnitude and the magnitude ofstorms in a definite recurrence interval.

Rainfall records are obtained from rain gages,which are of two types. The first type is a recordingor automatic gage. It continually records, by ink penand revolving drum, or digital microchip technol-ogy, the variation in rainfall intensity as well as thetotal rainfall volume. The second type is a nonrecor-ding gage; it measures only the total rain volumethat fell during the period between observations.The standard observation time for nonrecordinggages for the U.S. Weather Bureau is 24 h.

Corrections must be made to rain-gage recordsto account for the mean precipitation over theentire drainage basin, for hourly rainfall rates whenonly daily volumes are given, and for errors arisingout of the location of the gage. Most methods usedin runoff determinations are based on the assump-tion that rainfall is uniform over the entire drainage





basin. This necessitates development of a correc-tion factor to balance out the rainfall variationcaused by various topographical features in thewatershed. Rain gages tend to give rainfall vol-umes that are too small. This error is caused by themovement of wind around the gage, and itincreases as wind velocity increases. This “wind-age” error is much more pronounced when the raingage is near the top or bottom of a cliff or near otherbig obstructions. Care must be exercised in place-ment of rain gages to ensure accuracy.

The probable maximum precipitation is thegreatest rainfall intensity or volume that could everbe expected to occur in a specific drainage basin.This rainfall magnitude is frequently used as thedesign storm for major hydraulic structures toserve the basin when the rainfall records are shortand extrapolation to the desired design-stormfrequency could be grossly inaccurate. The magni-tude of probable maximum precipitation is basedon simultaneous occurrence of the maximumvalues of the meteorological factors that combineto form precipitation. The two most importantfactors are wind and air-mass moisture content. Anidea of the magnitude of the probable maximumprecipitation can also be obtained by transposingthe greatest rainfall that has occurred in a met-eorologically homogeneous region. For methodsfor determining the probable maximum precipi-tation, see D. R. Maidment, “Handbook of Hydrol-ogy,” McGraw-Hill, Inc., New York.

Not all rain reaches the ground. A portion mayevaporate as it falls, while another portion may becaught on leaves, branches, and other vegetationsurfaces. This phenomenon, called interception, isa loss from a runoff standpoint since the rainevaporates and never reaches the ground. Inter-ception may be significant for small-intensitystorms occurring with little or no wind over anarea with heavy vegetation growth.

21.38 Evaporation andTranspiration

These are processes by which moisture is returnedto the atmosphere. In evaporation, water changesfrom liquid to gaseous form. In transpiration,plants give off water vapor during synthesis ofplant tissue.

Evapotranspiration, commonly termed con-sumptive use, refers to the total evaporation from

all sources such as free water, ground, and plant-leaf surfaces. On an annual basis, the consumptiveuse may vary from 15 in/year for barren land to35 in/year for heavily forested areas and 40 in/yearin tropical and subtropical regions. Evapotrans-piration is important because, on a long-term basis,precipitation minus evapotranspiration equalsrunoff.

Evaporation may occur from free-water, plant,or ground surfaces. Of the three, free-water sur-face evaporation is usually the most important. Itmust be considered in the design of a reservoir,especially if the reservoir is shallow, has a relativelylarge surface area, and is located in a semiarid orarid region. Evaporation is a direct function of thewind and temperature and an inverse function ofatmospheric pressure and amount of soluble solidsin the water.

The rate of evaporation is dependent on thevapor-pressure gradient between the water surfaceand the air above it. This relation is known asDalton’s law. The Meyer equation [Eq. (21.125)],developed from Dalton’s law, is one of manyevaporation formulas and is popular for makingevaporation-rate calculations.

E ¼ C(ew � ea)c (21:125)

c ¼ 1þ 0:1w (21:126)

where E ¼ evaporation rate, in 30-day month

C ¼ empirical coefficient, equal to 15 forsmall, shallow pools and 11 for large,deep reservoirs

ew ¼ saturation vapor pressure, in of mer-cury, corresponding to monthly meanair temperature observed at nearbystations for small bodies of shallowwater or corresponding to water tem-perature instead of air temperature forlarge bodies of deep water

ea ¼ actual vapor pressure, in of mercury, inair based on monthly mean air tem-perature and relative humidity atnearby stations for small bodies ofshallow water or based on informationobtained about 30 ft above the watersurface for large bodies of deep water

w ¼ monthly mean wind velocity, mi/h atabout 30 ft above ground

c ¼ wind factor





As an example of the evaporation that may occurfrom a large reservoir, the mean annual evapor-ation from Lake Mead is 6 ft.

Evaporation from free-water surfaces is usuallymeasured with an evaporation pan. This pan isa standard size and is located on the groundnear the body of water whose evaporation is to bedetermined. The depth of water in this pan ischecked periodically and corrections made forfactors other than evaporation that may have raisedor lowered the water surface. A pan coefficient isthen applied to the measured pan evaporation toget the reservoir evaporation.

The standard evaporation pan of the NationalWeather Service, called a Class A Level Pan, is inwidespread use. It is 4 ft in diameter and 10 indeep. It is positioned 6 in above the ground. Its pancoefficient is commonly taken as 0.70, although itmay vary between 0.60 and 0.80, depending on thegeographical region. Annual evaporation from thepan ranges from 25 in in Maine and Washington to120 in along the Texas-Mexico and California-Arizona borders.

Evaporation rates from reservoirs may be redu-ced by spreading thin molecular films on the watersurface. Hexadeconal, or cetyl alcohol, is one suchfilm that has been effective on small reservoirswherethere is little wind. On large reservoirs, wind tendsto push the film to the shore. Since hexadeconal isremoved by wind, birds, insects, aquatic life, andbiologic attrition, it must be applied periodically formaximum effectiveness. Hexadeconal appears tohave no adverse effects on either humans orwildlife.

Evaporation from ground surfaces is usually ofminor importance, except in arid, tropical, andsubtropical regions having high water tables andwhere it pertains to the determination of initial soil-moisture conditions in a runoff analysis.

(D. R. Maidment, “Handbook of Hydrology,”McGraw-Hill, Inc., New York (books.mcgraw-hill.com).)

21.39 Runoff

This is the residual precipitation remaining afterinterception and evapotranspiration losses havebeen deducted. It appears in surface channels,natural or manmade, whose flow is perennial orintermittent. Classified by the path taken to achannel, runoff may be surface, subsurface, orgroundwater flow.

Surface flow moves across the land as overlandflow until it reaches a channel, where it continues aschannel or stream flow. After joining stream flow, itcombines with the other runoff components in thechannel to form total runoff.

Subsurface flow, also known as interflow, subsur-face runoff, subsurface storm flow, and storm seepage,infiltrates only the upper soil layers without joiningthemain groundwater body.Moving laterally, itmaycontinue underground until it reaches a channel orreturns to the surface and continues as overlandflow. The time for subsurface flow to reach a channeldepends on the geology of the area. Commonly, it isassumed that subsurface flow reaches a channelduring or shortly after a storm. Subsurface flowmaybe the major portion of total runoff for moderate orlight rains in arid regions since surface flow underthose conditions is reduced by unusually highevaporation and infiltration.

Groundwater flow, or groundwater runoff, isthat flow supplied by deep percolation. It is theflow of the main groundwater body and requireslong periods, perhaps several years, to reach achannel. Groundwater flow is responsible for thedry-weather flow of streams and remains practi-cally constant during a storm. Groundwater flow isprimarily the concern of water-supply engineers.Surface and subsurface flow are of interest to flood-control engineers.

In practice, direct runoff and base flow are theonly two divisions of runoff used. The basis for thisclassification is travel time rather than path. Directrunoff leaves the basin during or shortly after astorm, whereas base flow from the storm may notleave the basin for months or even years.

Runoff is supplied by precipitation. The portionof precipitation that contributes entirely to directrunoff is called effective precipitation, or effective rainif the precipitation is rain. That portion of theprecipitation which contributes entirely to surfacerunoff is called excess precipitation, or excess rain.Thus, effective rain includes subsurface flow,whereas excess rain is only surface flow.

The two major characteristics that affect runoffare climatic and drainage-basin factors. The num-ber of factors is an indication of the complexity ofaccurately determining runoff:

1. Climatic characteristics

a. Precipitation—form (rain, hail, snow, frost,dew), intensity, duration, time distribution,





seasonal distribution, areal distribution,recurrence interval, antecedent precipitation,soil moisture, direction of storm movement

b. Temperature—variation, snow storage, fro-zen ground during storms, extremes duringprecipitation

c. Wind—velocity, direction, duration

d. Humidity

e. Atmospheric pressure

f. Solar radiation

2. Drainage-basin characteristics

a. Topographic—size, shape, slope, elevation,drainage net, general location, land use andcover, lakes and other bodies of water, arti-ficial drainage, orientation, channels (size,shape of cross section, slope, roughness,length)

b. Geologic—soil type, permeability, ground-water formations, stratification

21.40 Sources of HydrologicData

The importance of exhausting all possible sourcesof hydrologic data, both published and unpub-lished, as the first step in design of a hydraulicproject cannot be overemphasized. The majority ofhydrologic data is collected and published bygovernment agencies, those of the Federal govern-ment being the largest and most important. Theprincipal source of precipitation data is the U.S.Weather Bureau. Its extensive system of gagessupplies complete precipitation data as well as allother types of hydrologic data. These data arecompiled and presented in monthly and yearlysummaries in the Bureau’s “Climatological Data.”In addition to the monthly and yearly summaries,special-interest items, such as rainfall intensity forvarious durations and recurrence intervals, arepublished in Weather Bureau technical papers.

Other sources are Water Bulletins of the Inter-national Boundary Commission, the U.S. Agricul-tural Research Service, and various state and localagencies.

The principal source of runoff data is the WaterSupply Papers of the U.S. Geological Survey. Thesepapers contain records of daily flow, mean flow,yearly flow volume, extremes of flow, and statis-tical data pertaining to the entire record. Alsoincluded in the Papers are lists of reports covering

unusually large floods and records of dischargecollected by agencies other than the U.S. GeologicalSurvey. The Water Supply Papers are publishedyearly in 14 parts; each part is for an area whoseboundaries coincide with natural-drainage fea-tures, as shown in Fig. 21.70.

Other agencies that collect and publish stream-flow and flood records are the Corps of Engineers,TVA, International Boundary Commission, andWeather Bureau. The Corps of Engineers publishesdata on floods in which loss of life and extensiveproperty damage occurred. Less obvious sources ofstream-flow data are water-right decrees by districtcourts, county records of water-right filings andState Engineer permits, and annual reports ofvarious interstate-compact commissions.

21.41 Methods for RunoffDeterminations

The method selected to determine runoff dependson its applicability to the area of concern, thequantity and type of data available, the detailrequired in the final answer, and the accuracydesired. Applicability depends on the character-istics of the particular area and the assumptionsfrom which the method was developed. Quantityand type of data available refer to the length, detail,and completeness of the hydrologic records, whichmay be either precipitation or stream flow. Anexample of the variation of detail in the final resultmay be found in the determination of flood runoff.Several methods yield only peak discharge; othersgive the complete hydrograph. Accuracy is limited

Fig. 21.70 Drainage subdivisions of the UnitedStates for stream-flow records published in “WaterSupply Papers,” U.S. Geological Survey.





by the cost of performing analyses and assump-tions made in the development of a method.

The methods that follow are a convenient meansfor solving typical runoff problems encountered inwater resources engineering. One method pertainsto minor hydraulic structures, the second to majorhydraulic structures. A minor structure is one oflow cost and of relativelyminor importance andpre-sents small downstream damage potential. Typicalexamples are small highway and railroad culvertsand low-capacity storm drains. Major hydraulicstructures are characterized by their high cost, greatimportance, and large downstream damage poten-tial. Typical examples of major hydraulic structuresare large reservoirs, deep culverts under vitalhighways and railways, and high-capacity stormdrains and flood-control channels.

21.41.1 Method for DeterminingRunoff for Minor HydraulicStructures

The most common means for determining runofffor minor hydraulic structures is the rationalformula

Q ¼ CIA (21:127)

where Q ¼ peak discharge, ft3/s

C ¼ runoff coefficient ¼ percentage of rainthat appears as direct runoff

I ¼ rainfall intensity, in/h

A ¼ drainage area, acres

The assumptions inherent in the rational for-mula are:

1. The maximum rate of runoff for a particularrainfall intensity occurs if the duration of rain-fall is equal to or greater than the time of concen-tration. The time of concentration is commonlydefined as the time required for water to flowfrom themost distant point of a drainage basin tothe point of flow measurement.

2. The maximum rate of runoff from a specificrainfall intensity whose duration is equal to orgreater than the time of concentration is directlyproportional to the rainfall intensity.

3. The frequency of occurrence of the peakdischarge is the same as that of the rainfallintensity from which it was calculated.

4. The peak discharge per unit area decreasesas the drainage area increases, and the inten-sity of rainfall decreases as its durationincreases.

5. The coefficient of runoff remains constant for allstorms on a given watershed.

Since these assumptions apply reasonably wellfor urbanized areas with simple drainage facilitiesof fixed dimensions and hydraulic characteristics,the rational formula has gained widespread usein the design of drainage systems for these areas.Its simplicity and ease of application have resultedin its being used for more complex urban systemsand rural areas where the assumptions are not soapplicable.

The rational formula is criticized for expressingrunoff as a fraction of rainfall rather than as rainfallminus losses and for combining all the complexfactors that affect runoff into a single coefficient.Although these and similar criticisms are valid, useof a more complicated formula is not justifiedbecause the time and money spent to obtain thenecessary data would not be warranted for minorhydraulic structures.

Numerous refinements have been developedfor the runoff coefficient. As an example, the LosAngeles County Flood Control District givesrunoff coefficients as a function of the soil andarea type and of the rainfall intensity for the timeof concentration. Other similar refinements arepossible if the resources are available. Carefulselection of the runoff coefficient C will givevalues of peak runoff consistent with projectsignificance. The values of C in Table 21.15 forurban areas are commonly recommended designvalues (V. T. Chow, “Hydrologic Determination ofWaterway Areas for the Design of DrainageStructures in Small Drainage Basins,” University ofIllinois Engineering Experimental Station Bulletin426, 1962).

After selection of the design-storm frequencyof occurrence, for example, a 50- or 100-year-frequency storm, the rainfall intensity I may bedetermined from any of a number of formulas orfrom a statistical analysis of rainfall data if enoughare available. Chow lists 24 rainfall-intensityformulas of the form

I ¼ KFn1

(tþ b)n(21:128)





where I ¼ rainfall intensity, in/h

K, b, n, and n1 ¼ respectively, coefficient, factor, andexponents depending on condi-tions that affect rainfall intensity

F ¼ frequency of occurrence of rain-fall, years

t ¼ duration of storm, min

¼ time of concentration

Perhaps the most useful of these formulas is theSteel formula:

I ¼ K

tþ b(21:129)

where K and b are dependent on the storm fre-quency and region of the United States (Fig. 21.71and Table 21.16).

Equation (21.129) gives the average maximumprecipitation rates for durations up to 2 h.

The time of concentration Tc at any point in adrainage system is the sum of the overland flowtime; the flow time in streets, gutters, or ditches;and the flow time in conduits. Overland flow timemay be determined from any number of formulasdeveloped for the purpose. (See D. R. Maidment,“Handbook of Hydrology,” McGraw-Hill, Inc.,New York.) The flow time in gutters, streets, dit-ches, and conduits can be determined from a cal-culation of the average velocity using the Manningequation [Eq. (21.89)]. The time of concentration isusually expressed in minutes.

After determining the time of concentration,calculate the corresponding rainfall intensity fromeither Eq. (21.128) or Eq. (21.129), or any equivalentmethod. Then select the runoff coefficient fromTable 21.15 and determine the peak discharge fromEq. (21.127).

Since the rational formula assumes a constantuniform rainfall for the time of concentration overthe entire area, the area A must be selected so thatthis assumption applies with reasonable accuracy.Adhering to this assumption may necessitate sub-dividing the drainage area.

21.41.2 Method for DeterminingRunoff for Major HydraulicStructures

The unit-hydrographmethod, pioneered in 1932 byLeRoy K. Sherman, is a convenient, widely accep-ted procedure for determining runoff for major

Table 21.15 Common Runoff Coefficients

Type ofDrainage Area

RunoffCoefficient C

Business:Downtown areas 0.70–0.95Neighborhood areas 0.50–0.70

Residential:Single-family areas 0.30–0.50Multiunits, detached 0.40–0.60Multiunits, attached 0.60–0.75Suburban 0.25–0.40Apartment dwelling

areas0.50–0.70

Industrial:Light areas 0.50–0.80Heavy areas 0.60–0.90Parks, cemeteries 0.10–0.25Playgrounds 0.20–0.35Railroad-yard areas 0.20–0.40Unimproved areas 0.10–0.30

Streets:Asphaltic 0.70–0.95Concrete 0.80–0.95Brick 0.70–0.85Drives and walks 0.75–0.85Roofs 0.75–0.95

Lawns:Sandy soil, flat, 2% 0.05–0.10Sandy soil, avg, 2–7% 0.10–0.15Sandy soil, steep, 7% 0.15–0.20

(Table continued)

Fig. 21.71 Regions of the United States for usewith the Steel formula.





hydraulic structures. (Leroy K. Sherman, “Stream-flow from Rainfall by Unit-Graph Method,”Engineering News-Record, vol. 108, pp. 501–505,January-June 1932.) It permits calculation of thecomplete runoff hydrograph from any rainfall afterthe unit hydrograph has been established for theparticular area of concern.

The unit hydrograph is defined as a runoffhydrograph resulting from a unit storm. A unitstorm has practically constant rainfall intensity forits duration, termed a unit period, and a runoffvolume of 1 in (water with a depth of 1 in over aunit area, usually 1 acre). Thus, a unit storm mayhave a 2-in/h effective intensity lasting 1 =

2 h or a0.2-in/h effective intensity lasting 5 h. The signifi-cant part of the definition is not the volume but theconstancy of intensity. Adjustments can be madewithin unit-hydrograph theory for situationswhere the runoff volume is different from 1 in, butcorrections for highly variable rainfall rates cannotbe made.

The unit hydrograph is similar in concept todetermining a set of factors for a specific drainagebasin. The set consists of one factor for each var-iable that affects runoff. The unit hydrograph ismuch quicker, easier, and more accurate than anysuch set of factors. The method is summarized bythe formula

Effective rain� unit hydrograph ¼ runoff

(21:130)

The unit hydrograph thus is the link betweenrainfall and runoff. It may be thought of as anintegral of the many complex factors that affectrunoff. The unit hydrograph can be derived fromrainfall and stream-flow data for a particular stormor from stream-flow data alone.

Assumptions made in the development of theunit-hydrograph theory are:

1. Rainfall intensity is constant for its duration or aspecified period of time. This requires that astorm of short duration, termed a unit storm, beused for the derivation of the unit hydrograph.

2. The effective rainfall is uniformly distributedover the drainage basin. This specifies that thedrainage area be small enough for the rainfall tobe essentially constant over the entire area. If thewatershed is very large, subdivision may berequired; the unit-hydrograph theory is thenapplied to each subarea.

3. The base of the hydrograph of direct runoff isconstant for any effective rainfall of unit dura-tion. This needs no clarification except that thebase of a hydrograph, that is, the time of stormrunoff, is largely arbitrary since it depends onthe method of base-flow separation.

4. The ordinates of the direct runoff hydrographsof a common base time are directly proportionalto the total amount of direct runoff representedby each hydrograph. Illustrated in Fig. 21.72,

Table 21.16 Coefficients for Steel Formula

Fre-quency,years

Coeffi-cients

Region

1 2 3 4 5 6 7

2 K 206 140 106 70 70 68 32b 30 21 17 13 16 14 11

4 K 247 190 131 97 81 75 48b 29 25 19 16 13 12 12

10 K 300 230 170 111 111 122 60b 36 29 23 16 17 23 13

25 K 327 260 230 170 130 155 67b 33 32 30 27 17 26 10

50 K 315 350 250 187 187 160 65b 28 38 27 24 25 21 8

100 K 367 375 290 220 240 210 77b 33 36 31 28 29 26 10





this is basically the principle of superposition orproportionality. It enables calculation of therunoff for a storm of any intensity or durationfrom a unit storm, which is of fixed intensityand duration. A given storm may be resolvedinto a number of unit storms. Then, the runoffmay be calculated by superimposing thatnumber of unit hydrographs.

5. The hydrograph of direct runoff for a givenperiod of rainfall reflects all the combinedphysical characteristics of the basin (commonlyreferred to as the principle of time invariance). Thisassumption implies that the characteristics ofthe drainage basin have not changed since theunit hydrograph was derived. Because thisapplies with varying degrees of accuracy towatersheds, the characteristics of the drainagebasin must be fixed or specified. Daily andweekly variations in initial soil moisture are

probably the greatest source of error in thismethod since they are largely unknown. Man-made alterations and stream-flow conditionscan be accounted for much more easily.

For ease of manipulation, the unit hydrograph isfrequently expressed in histogram form as a dis-tribution graph (Fig. 21.73), which illustrates thepercentages of total runoff that occur duringsuccessive unit periods. The ordinate for each unitperiod is the mean value of runoff for that period.

Since the unit hydrograph is derived for a unitstorm of specific duration, it may be used only forstorms divided into unit periods of that length.Usually, because of storm variations, the unitperiod must be different from that for which theunit hydrograph was derived. This requires therecalculation of the unit hydrograph for the newunit period. This is accomplished by offsettingtwo S hydrographs by a time equal to the duration

Fig. 21.72 Unit hydrograph (a) prepared for a unit storm is used to develop a composite hydrograph(b) for any storm.





of the desired unit period (Fig. 21.74). An Shydrograph is a representation of the cumulativepercentages of runoff that occur during a stormwhich has a continuous constant rainfall. It iscalculated by cumulatively plotting the distri-bution percentages that make up the distributiongraph. The distribution percentages for the newunit hydrograph are determined by taking thedifference between mean ordinates for the twooffset S hydrographs and dividing by the new unitperiod.

Transposition of a unit hydrograph from onebasin to another similar basin may be made bycorrelating their respective shape and slope factors.This method was developed by Franklin F. Snyder(Transactions of the American Geophysical Union, vol.19, pt. I, pp. 447–454). Also, since S hydrographsare a characteristic of a drainage basin, those from

various basins may be compared to obtain an ideaof the variations that might exist when transposingdata from one basin to another.

In the application of the unit-hydrographmethod, a loss rate must be established to deter-mine effective rain. This loss, during heavy storms,is usually considered to be entirely infiltration. Theinfiltration capacity of a soil may be determinedexperimentally by lysimeter or infiltrometer (R.K. Linsley et al., “Hydrology for Engineers,” 3rded., McGraw-Hill, Inc., New York (books.mcgraw--hill.com).)

21.42 Groundwater

Groundwater is subsurface water in porous stratawithin a zone of saturation. It supplies about 20%of the United States water demand. Wheregroundwater is to be used as a water-supplysource, the extent of the groundwater basin andthe rate at which continuing extractions may bemade should be determined.

Aquifers are groundwater formations capableof furnishing an economical water supply. Thoseformations from which extractions cannot be madeeconomically are called aquicludes.

Permeability indicates the ease with whichwater moves through a soil and determineswhether a groundwater formation is an aquifer oraquiclude.

The rate of movement of groundwater is givenby Darcy’s law:

Q ¼ KIA (21:131)

where Q ¼ flow rate, gal/day

K ¼ hydraulic conductivity, ft/day orm/day

I ¼ hydraulic gradient, ft/ft or m/m

A ¼ cross-sectional area, perpendicular todirection of flow, ft2 or m2

Hydraulic conductivity is a measure of the abilityof a soil to transmit water. It is a nonlinear functionof volumetric soil water content and varies withsoil texture. Many methods are available fordetermining hydraulic conductivity. (See D. R.Maidment, “Handbook of Hydrology,” McGraw-Hill, Inc., New York (books.mcgraw-hill.com).)

Transmissibility is another index for the rate ofgroundwater movement and equals the product of

Fig. 21.74 Distribution percentages are deter-mined from an offset S hydrograph.

Fig. 21.73 Distribution graph represents a unithydrograph as a histogram.





hydraulic conductivity and the thickness of theaquifer. Transmissibility indicates for the aquifer asa whole what hydraulic conductivity indicates forthe soil.

An aquifer whose water surface is subjected toatmospheric pressure and may rise and fall withchanges in volume is a free or unconfined aquifer. Anaquifer that contains water under hydrostaticpressure, because of impermeable layers aboveand below it, is a confined or artesian aquifer. If a wellis drilled into an artesian aquifer, the water in thiswell will rise to a height corresponding to thehydrostatic pressure within the aquifer. Frequently,this hydrostatic pressure is sufficient to cause thewater to jet beyond the ground surface into theatmosphere. An artesian aquifer is analogous to alarge-capacity conduit with full flow in thatextractions from it cause a decrease in pressure,rather than a change in volume. This is in contrastto a free aquifer, where extractions cause a decreasein the elevation of the groundwater table.

Groundwater Management n With increas-ing use being made of groundwater resources,effective groundwater management is an absolutenecessity. Adequate management should includenot only quantity but quality. Quantity manage-ment consists of effective control over extractionsand replenishment. Quality management consistsof effective control over groundwater pollutionresulting from waste disposal, recycling, poor-quality replenishment waters, or other causes.

Several steps or investigations are necessary fordeveloping an effective management program.First is a comprehensive geologic investigation ofthe groundwater basin to determine the character-istics of the aquifers. Second is a qualitative andquantitative hydrologic study of both surface waterand groundwaters to determine historical sur-pluses and deficiencies, safe yield, and overdraft.(Safe yield is the magnitude of the annualextractions from an aquifer that can continueindefinitely without bringing some undesirableresult. Deteriorating water quality, need forexcessive pumping lifts, or infringement on thewater rights of others are examples of undesirableresults that could define safe yield. Regardless ofhow it is defined, safe yield applies only to aspecific set of conditions based largely on judgmentas to what is desirable. Extractions in excess of thesafe yield are termed overdrafts.) In conjunctionwith the hydrologic study, present and future

water demands should be determined. A detailedwater-quality study should be made not only of thegroundwater within the basin but also of all surfacewaters, wastewaters, and other waters that replen-ish the groundwater basin. Undesirable water-quality and -quantity conditions should beidentified.

Following the preceding preliminary work,alternative management plans should be formu-lated. These management plans should considervariations in the quantity of extractions; ground-water levels; quality, quantity, and location ofartificial replenishment; source, quantity, andquality of water supply; and methods of waste-water disposal. All alternative plans must recog-nize all legal and jurisdictional constraints.

The final step is the operational-economicevaluation of the alternatives and the selection ofa recommended groundwater management plan.Operations and economic studies are normallyconducted by superimposing present and futureconditions in each alternative plan on historicalhydrologic conditions that occurred during a baseperiod. (A base period is a period of time, usually anumber of years, specifically chosen for detailedhydrologic analysis because conditions of watersupply and climate during the period are equival-ent to amean of long-term conditions and adequatedata for such hydrologic analysis are available.)

Economic evaluation of alternative plans shouldconsider cost of water-supply facilities, cost ofreplenishment water, cost of wastewater-disposalfacilities, cost of pumping groundwater at thevarious operational levels considered, and indirectwater-quality use costs, among others. (Indirectwater-quality use costs are those indirect costsincurred by water distributors and consumers asa result of using water of different qualities. Thesecosts include increased soap costs, water softeningcosts, and costs associated with the more rapiddeterioration of plumbing and waterworks equip-ment—all of which increase as the hardness andsalinity of the water increase.)

Operational studies should determine the mostefficient manner of joint operation of surface andgroundwater systems (conjunctive use).

Use of computers and the development of amathematical model for the groundwater basin arealmost essential because of the number ofrepetitive calculations involved.

Upon completion of the operational andeconomic studies, the most favorable management





scheme should be selected as the recommendedplan. This selection should be based not only oneconomic and operational considerations but onsocial, institutional, legal, and environmental fac-tors. The plan should be capable of being readilyimplemented, flexible enough to accommodatedifferent growth rates, financially feasible, andgenerally acceptable to the water and wastewateragencies operating in the basin.

An operating agency should be designated orformed to implement the recommended plan. Theagency should have adequate powers to control orcooperate in the control of surface-water suppliesgroundwater recharge sites, surface-water deliveryfacilities, amount and location of groundwaterextractions, and wastewater treatment and dis-posal facilities. The operating agency shoulddevelop a comprehensive monitoring network anda data collection and evaluation program todetermine the effectiveness of the managementplan and to implement any changes in the plandeemed necessary. This monitoring network mayconsist of selected wells where groundwater levelsand chemical characteristics are measured andcertain surface-water sampling locations whereboth quantitative and qualitative factors aremeasured. The program should also includequantitative evaluation of extractions, water used,wastewater disposed, and natural and artificialreplenishment. Integration of the above data withthe computer model of the groundwater basin is anefficient method of evaluating the groundwatermanagement scheme.

(“Ground Water Management,” Manual andReport on Engineering Practice, no. 40, AmericanSociety of Civil Engineers, 1987; J. Bear, “Hydrau-lics of Ground Water,” N. S. Grigg, “WaterResources Planning,” A. I. Kashef, “GroundwaterEngineering,” R. K. Linsley et al., “Hydrology forEngineers,” 3rd ed., McGraw-Hill Book Company,New York (books.mcgraw-hill.com).)

Water Supply

A waterworks system is created or expanded tosupply a sufficient volume of water at adequatepressure from the supply source to consumers fordomestic, irrigation, industrial, fire-fighting, andsanitary purposes. A primary concern of theengineer is estimation of the quantity of potablewater to be consumed by the community since the

engineer must design adequately sized com-ponents of the water-supply system. Water-supplyfacilities consist of collection, storage, transmission,pumping, distribution, and treatment works.

To assure continuous service to the consumerfor fire-fighting and sanitary purposes in the eventof an earthquake, fire, flood, or other unforeseenemergency, careful consideration must be given tothe selection of standby equipment and alternativesupplies of water. Maximum protection must begiven to power sources and pumps that must beavailable to operate continuously during emer-gency conditions. A dependable supply withsufficient pressure for fighting fires considerablyincreases capital expenditures for system construc-tion. The smaller the system, the larger thepercentage of the total cost chargeable to depend-able fire flow.

21.43 Water Consumption

The size of a proposed water-supply project isusually based on an average annual per capitaconsumption rate. Therefore, forecasts of popu-lation for the design period are of the greatestimportance and must be made with care to ensurethat components for the project are of adequatesize. Estimation of future population, however, is avery difficult task.

Several mathematical methods are available foruse in predicting populations of cities. Somemethods commonly used are arithmetical increase,percentage increase, decreasing percentage in-crease, graphical comparison with other cities, andthe ratio method of comparing a community with astate or country of which the community is a part.Great care and judgment must be exercised inpopulation prediction since many factors, such asindustrial development, land speculation, geo-graphical boundaries, and age of the city, maydrastically alter mathematical estimates.

The total water supply of a city is usuallydistributed among the following four major classesof consumers: domestic, industrial, commercial,and public.

Domestic use consists of water furnished tohouses, apartments, motels, and hotels for drink-ing, bathing, washing, sanitary, culinary, and lawn-sprinkling purposes. Domestic use accounts forbetween 30 and 60% (50 to 60 gal per capita perday) of total water consumption in an average city.





Commercial water is used in stores and officebuildings for sanitary, janitorial, and air condition-ing purposes. Commercial use of water amounts toabout 10 to 30% of total consumption.

Industrial uses of water are diverse but consistmainly of heat exchange, cooling, and cleaning. Nodirect relationship exists between the amount ofindustrial water used and the population of thecommunity, but 20 to 50% of the total quantity ofwater used per capita per day is normally chargedto industrial usage. Usually the larger-sized citieshave a high degree of industrialization and show acorrespondingly greater percentage of total con-sumption as industrial water.

Public use of water for parks, public buildings,and streets contributes to the total amount of waterconsumed per capita. Fire demands are usuallyincluded in this class of water use. The totalquantity of water used for fire fighting may not belarge, but because of the high rate at which it isrequired, it may control the design of the facilities.About 5 to 10% of all water used is for public uses.

Waste and miscellaneous usage of waterinclude that lost because of leakage in mains,meter malfunctions, reservoir evaporation, andunauthorized uses. About 10 to 15% of total con-sumption may be charged to waste and miscella-neous uses.

Water Demand Rate n Many factors, suchas the climate, size of the city, standard of living,degree of industrialization, type of service (meter-ed or unmetered), lawn sprinkling, air condition-ing, cost, pressure, and quality of the water,influence the demand rate for water.

Presence of industries usually increases the totalper capita use of water but decreases the demandfluctuation. A good estimate of the potential indus-trial water demand can bemade by relating demandto the percent of land zoned for industrial use.

Small cities frequently have a low per capitademand for water, especially if portions of the cityare unsewered. Fluctuations in demand are greaterin small cities, mainly because of the lack of largeindustries. High standards of living increase waterdemand and fluctuations in rate of use.

Warm and dry climates have a higher rate ofwater consumption because of sprinkling and airconditioning. Cold weather sometimes increasesconsumption because water is allowed to run toprevent pipes from freezing.

Demand for water is related to water-servicemeters, cost, quality, and pressure. Metering waterreduces the quantity of water consumed by 10 to25% because of the usual increase in total cost ofconsumers if they continue to use water at theunmetered rate. High water pressures increasedemand because of greater losses at leaking mains,valves, and faucets. Normally, if the cost of waterincreases, the demand for it decreases. Demand forwater usually increases with an improvement inquality.

Demand rates vary with time of day, month, andyear. Table 21.17 is a comparison between water-demand rates for the city of Los Angeles and anational average calculated from data in a U.S.Public Health Service Report. The national demand-rate data, as presented in Table 21.17, are theaverage of a range of values, including some veryhigh and very low rates due to variations inclimatic conditions, degree of industrialization,and time of day. Examples of divergent averagedaily demand rates for various United States citiesare: 230 gal per capita per day for Chicago, 210 galper capita per day for Denver, 150 gal per capita perday for Baltimore, and 135 gal per capita per dayfor Kansas City, Mo.

The “California Water Atlas,” 1979, State ofCalifornia Office of Planning and Research, pre-sents average monthly demand rates for water use

Table 21.17 Water-Demand Rates

National avg Los Angeles, Calif.

Gal per capitaper day

% of avgannual rate

Gal per capitaper day

% of avgannual rate

Avg day 160 100 175 100Max day 265 165 280 160Max h 400 250 420 240





in four coastal and four inland cities for the period1966 to 1970. In the atlas, the effect of warm, dryclimatic conditions is indicated for each location bythe ratio of average monthly use to the annualaverage gallon per capita per day. The maximummonthly water-demand rates ranged from 119 to141% of the annual rate for the coastal cities andfrom 144 to 187% of the annual rate for the dry,inland, valley cities. Moreover, the annual demandrates for the inland areas averaged 78% higher thanthose for the coastal cities. The difference is dueprimarily to the great amount of lawn sprinkling inLos Angeles. Past water-demand records of boththe city being considered and other cities of similarsize, industrialization, climate, and so on shouldbe considered and incorporated in demand-rateprojections for water systems.

The total quantity of water used for fightingfires is normally quite small, but the demand rateis high. The fire demand as established by theAmerican Insurance Association is

G ¼ 1020ffiffiffiP

p(1� 0:01

ffiffiffiP

p) (21:132)

where G ¼ fire-demand rate, gal/min

P ¼ population, thousands

The required fire flows computed from thisformula are listed in Table 21.18. When calculatingthe total flow to be used in design, fire flow shouldbe added to the average consumption for themaximum day.

21.44 Water-Supply Sources

The major sources of a water supply are surfacewater and groundwater. In the past, surface sourceshave included only the commonly occurring naturalfresh waters, such as lakes, rivers, and streams, butwith rapid population expansion and increased percapitawater use associatedwith a higher standard ofliving, consideration must be given to desalinationand waste-water reclamation as well.

In selection of a source of supply, the variousfactors to be considered are adequacy and reliabil-ity, quality, cost, legality, and politics. The criteriaare not listed in any special order since they are, toa large extent, interdependent. Cost, however, isprobably the most important because almost anysource could be used if consumers are willing topay a high enough price. In some local areas, asincreasing demands exceed the capacity of existingsources, the increasing cost of each new supplyfocuses attention on reclamation of local suppliesof wastewater and desalination.

Adequacy of supply requires that the source belarge enough to meet the entire water demand.Total dependence on a single source, however, isfrequently undesirable, and in some cases, diversi-fication is essential for reliability. The source mustalso be capable of meeting demands during poweroutages and natural or created disasters. The mostdesirable supplies from a reliability standpoint, inorder, are (1) an inexhaustible supply, whetherfrom surface or groundwater, which flows by

Table 21.18 Required Fire Flow, Hydrant Spacing, and Fire Reserve Storage*

PopulationFire Flow

Duration,h

Reservestorage,

Avg area served perhydrant

in high-value districts, ft*

gal/min MGD† MG† Directstreams

Enginestreams

1,000 1,000 1.4 4 0.3 100,000 120,0002,000 1,500 2.2 6 0.6 90,0004,000 2,000 2.9 8 1.0 85,000 110,00010,000 3,000 4.3 10 1.8 70,000 100,00017,000 4,000 5.8 10 2.4 55,000 90,00028,000 5,000 7.2 10 3.0 40,000 85,00040,000 6,000 8.6 10 3.6 40,000 80,00080,000 8,000 11.5 10 4.8 40,000 60,000125,000 10,000 14.4 10 6.0 40,000 48,000200,000 12,000 17.3 10 7.2 40,000 40,000





gravity through the distribution system; (2) agravity source supplemented by storage reservoirs;(3) an inexhaustible source that requires pumping;and (4) sources that require both storage andpumping. As demand increases and supplies be-come overtaxed, conservation practices in every-day use become a valuable management tool.

Quality of the source determines both accept-ability and cost; it varies considerably betweenregions. Preliminary estimates of quality can bemade by examining the source, geology, andculture of the area.

Legality of supply is determined by doctrinesand principles of water rights, such as appropria-tion, riparian, and ownership rights. Appropria-tion right gives the first right priority over laterrights: “first in time means first in right.” Riparianright permits owners of land adjacent to a stream orlake to take water from that stream or lake for useon their land. Ownership right gives a landownerpossession of everything below and above the land.Legality is especially important for groundwatersupplies or where there is transfer of water fromone watershed to another.

A political problem with water supply existsbecause political boundaries seldom conform tonatural-drainage boundaries. This problem isespecially acute in extensive water-importationplans, but it even exists in varying forms forwastewater reclamation and desalination projects.

Desalination processes are of two fundamentaltypes: those that extract salt from the water, such aselectrodialysis and ion exchange, and those thatextract water from the salt, such as distillation,freezing, and reverse osmosis. The energy cost ofthe former processes is dependent on the saltconcentration. Hence, they are used mainly forbrackish water. The energy costs for the water-extraction processes are essentially independent ofsalinity. These processes are used for seawaterconversion. Very large dual-purpose nuclearpower and desalination plants, which take advan-tage of the economies realized by enormousfacilities, have been proposed, but such plants arefeasible only for those large urban areas located oncoasts. Transmission and pumping costs makeinland use uneconomical. Although desalinationmay have advantages as a local source, it is not atpresent a panacea that will irrigate the deserts.

Acceptance of wastewater reclamation as awater source for direct domestic use is hindered bypublic opinion and uncertainty regarding viruses.

Much effort has been expended to solve theseproblems. But until such time as they are solved,wastewater reclamation will have only limited usefor water supply. In the meantime, reclaimed wateris being used for irrigation in agricultural andlandscaping applications.

(D.W. Prasifka, “Current Trends inWater-SupplyPlanning,” Van Nostrand Reinhold, New York.)

21.45 Quality Standardsfor Water

The Safe DrinkingWater Act of 1974 mandated thatnationwide standards be established to help ensurethat the public receives safe water throughout theUnited States. National Interim Primary DrinkingWater Standards were adopted in 1975, basedlargely on the 1962 U.S. Public Health ServiceStandards (Publication no. 956), which were usedfor control of water quality for interstate carriers.These earlier standards had been widely adoptedvoluntarily by both public and private utilitiesand received the immediate endorsement of theAmerican Water Works Association as a mini-mum standard for all public water supplies in theUnited States. Similar standardswere developed bythe World Health Organization as standardsfor drinking-water quality at international ports(“International Standards for Drinking Water,”World Health Organization, Geneva, Switzerland).Heightened concern over our changing environ-ment and its health effect on water supplies was amajor cause of the change from voluntary tomandatory water-quality standards. The law wasamended in 1986 and 1996. The law applies to everypublic water system in the United States exceptprivate wells serving fewer than 25 individuals.

The Safe Drinking Water Act defines contami-nant as any physical, chemical, biological, orradiological substance or matter in water. Maxi-mum contaminant level (MCL) indicates themaximum permissible level of a contaminant inwater that is delivered to any user of a public watersystem. The act clearly delineates between health-related quality contaminants and aesthetic-relatedcontaminants by classifying the former as primaryand the latter as secondary contaminants.

Primary Standards n Table 21.19 listsmaximum contaminant levels required by theNational Primary Drinking Water Standards.





Table 21.19 National Primary Drinking Water Standards (Safe Drinking Water Act, 1974 and asamended in 1986 and 1996 and specified in EPA 816-F-02-013, July 2002. See USEPA’s website:www.epa.gov/safewater/)

ContaminantMCLG

1

(mg/L)2MCL or TT

1

(mg/L)2Potential health effects from

exposure above the MCL

Common sources ofcontaminant indrinking water

MICROORGANISMS

Cryptosporidium zero TT3 Gastrointestinal illness (e.g.,diarrhea, vomiting, cramps)

Human and fecalanimal waste

Giardia lamblia zero TT3 Gastrointestinal illness (e.g.,diarrhea, vomiting, cramps)

Human and animalfecal waste

Heterotrophic platecount (HPC)

n/a TT3 HPC has no health effects; it isan analytic method used tomeasure the variety ofbacteria that are common inwater. The lower theconcentration of bacteria indrinking water, the bettermaintained the water systemis

HPC measures arange of bacteriathat are naturallypresent in theenvironment

Legionella zero TT3 Legionnaire’s Disease, a type ofpneumonia

Found naturally inwater; multipliesin heating systems

Total coliforms(including fecalcoliform and E. Coli)

zero 5.0%4 Not a health threat in itself; it isused to indicate whetherother potentially harmfulbacteria may be present5

Coliforms are natur-ally present in theenvironment; aswell as feces; fecalcoliforms and E.

coli only comefrom human andanimal fecal waste

Turbidity n/a TT3 Turbidity is a measure of thecloudiness of water. It is usedto indicate water quality andfiltration effectiveness (e.g.,whether disease-causingorganisms are present).Higher turbidity levels areoften associated with higherlevels of disease-causingmicroorganisms such asviruses, parasites and somebacteria. These organismscan cause symptoms such asnausea, cramps, diarrhea,and associated headaches.

Soil runoff

Viruses (enteric) zero TT3 Gastrointestinal illness (e.g.,diarrhea, vomiting, cramps)

Human and animalfecal waste

DISINFECTION BYPRODUCTS

Bromate zero 0.010 Increased risk of cancer Byproduct ofdrinking waterdisinfection

(Table continued )





Table 21.19 (Continued)

ContaminantMCLG

1

(mg/L)2MCL or TT

1




Chlorite 0.8 1.0 Anemia; infants & youngchildren: nervous systemeffects

Byproduct ofdrinking waterdisinfection

Haloacetic acids(HAA5)

n/a6 0.060 Increased risk of cancer Byproduct ofdrinking waterdisinfection

Total Trihalomethanes(TTHMs)

none7

n/a60.100.080

Liver, kidney or central nervoussystem problems; increasedrisk of cancer

Byproduct ofdrinking waterdisinfection

DISINFECTANTS MRDL1

(Mg/L)2MRDL1

(Mg/L)2

Chloramines (as Cl2) MRDLG ¼ 4 MRDL ¼ 4.01 Eye/nose irritation; stomachdiscomfort, anemia

Water additive usedto controlmicrobes

Chlorine (as Cl2) MRDLG ¼ 4 MRDL ¼ 4.01 Eye/nose irritation; stomachdiscomfort


Chlorine dioxide (asClO2)

MRDLG ¼ 0.81 MRDL ¼ 0.81 Anemia; infants & youngchildren: nervous systemeffects


INORGANIC CHEMICALS

Antimony 0.006 0.006 Increase in blood cholesterol;decrease in blood sugar

Discharge frompetroleumrefineries; fireretardants;ceramics;electronics; solder

Arsenic 07 0.010 as of1/23/06

Skin damage or problems withcirculatory systems, and mayhave increased risk of gettingcancer

Erosion of naturaldeposits; runofffrom orchards,runoff from glass& electronicsproduction wastes

Asbestos (fibers .10 micrometers)

7 million fibersper liter(MFL)

7 MFL Increased risk of developingbenign intestinal polyps

Decay of asbestoscement in watermains; erosion ofnatural deposits

Barium 2 2 Increase in blood pressure Discharge of drillingwastes; dischargefrom metalrefineries; erosionof natural deposits

(Table continued )






ContaminantMCLG

1

(mg/L)2MCL or TT

1




Beryllium 0.004 0.004 Intestinal lesions Discharge frommetal refineriesand coal-burning factories;discharge fromelectrical,aerospace, anddefense industries

Cadmium 0.005 0.005 Kidney damage Corrosion ofgalvanized pipes;erosion of naturaldeposits;discharge frommetal refineries;runoff from wastebatteries andpaints

Chromium (total) 0.1 0.1 Allergic dermatitis Discharge from steeland pulp mills;erosion of naturaldeposits

Copper 1.3 TT8; ActionLevel ¼ 1.3

Short term exposure:Gastrointestinal distressLong term exposure: Liver ofkidney damage People withWilson’s Disease shouldconsult their personal doctorif the amount of copper intheir water exceeds the actionlevel

Corrosion ofhouseholdplumbingsystems; erosionof natural deposits

Cyanide (as freecyanide)

0.2 0.2 Nerve damage or thyroidproblems

Discharge fromsteel/metal factories;discharge fromplastic andfertilizer factories

Fluoride 4.0 4.0 Bone disease (pain andtenderness of the bones);Children may get mottledteeth

Water additive whichpromotes strongteeth; erosion ofnatural deposits;discharge fromfertilizer andaluminumfactories

(Table continued )






ContaminantMCLG

1

(mg/L)2MCL or TT

1




Lead zero TT8; ActionLevel ¼0.015

Infants and children: Delays inphysical or metaldevelopment; children couldshow slight deficits inattention span and learningabilities

Adults: Kidney problems;high blood pressure

Corrosion ofhouseholdplumbingsystems; erosionof natural deposits

Mercury (inorganic) 0.002 0.002 Kidney damage Erosion of naturaldeposits;discharge fromrefineries andfactories; runofffrom landfills andcroplands

Nitrate (measured asNitrogen)

10 10 Infants below the age of sixmonths who drink watercontaining nitrate in excess ofthe MCL could becomeseriously ill and, if untreated,may die. Symptoms includeshortness of breath and blue-baby syndrome.

Runoff from fertilizeruse; leaching fromseptic tanks,sewage; erosion ofnatural deposits

Nitrate (measured asNitrogen)

1 1 Infants below the age of sixmonths who drink watercontaining nitrate in excess ofthe MCL could becomeseriously ill and, if untreated,may die. Symptoms includeshortness of breath and blue-baby syndrome.

Runoff from fertilizeruse; leaching fromseptic tanks,sewage; erosion ofnatural deposits

Selenium 0.05 0.05 Hair or fingernail loss;numbness in fingers or toes;circulatory problems

Discharge frompetroleumrefineries; erosionof naturaldeposits;discharge frommines

Thallium 0.0005 0.002 Hair loss; changes in blood;kidney, intestine, or liverproblems

Leaching from ore-processing sites;discharge fromelectronics, glass,and drug factories

ORGANIC CHEMICALS

Acrylamide zero TT9 Nervous system or bloodproblems; increased risk ofcancer

Added to waterduring sewage/wastewatertreatment

Alachlor zero 0.002 Eye, liver, kidney or spleenproblems; anemia; increasedrisk of cancer

Runoff fromherbicide used onrow crops

(Table continued )






ContaminantMCLG

1

(mg/L)2MCL or TT

1




Atrazine 0.003 0.003 Cardiovascular system orreproductive problems


Benzene zero 0.005 Anemia; decrease in bloodplatelets; increased risk ofcancer

Discharge fromfactories; leachingfrom gas storagetanks and landfills

Benzo(a)pyrene(PAHs)

zero 0.0002 Reproductive difficulties;increased risk of cancer

Leaching fromlinings of waterstorage tanks anddistribution lines

Carbofuran 0.04 0.04 Problems with blood, nervoussystem, or reproductivesystem

Leaching of soilfumigant used onrice and alfalfa

Carbon tetrachloride zero 0.005 Liver problems; increased riskof cancer

Discharge fromchemical plantsand otherindustrialactivities

Chlordane zero 0.002 Liver or nervous systemproblems; increased risk ofcancer

Residue of bannedtermiticide

Chlorobenzene 0.1 0.1 Liver or kidney problems Discharge fromchemical andagriculturalchemical factories

2,4-D 0.07 0.07 Kidney, liver, or adrenal glandproblems


Dalapon 0.2 0.2 Minor kidney changes Runoff fromherbicide used onrights of way

1,2-Dibromo-3-chloropropane(DBCP)

zero 0.0002 Reproductive difficulties;increased risk of cancer

Runoff/leachingfrom soil fumigantused on soybeans,cotton, pine-apples, andorchards

o-Dichlorobenzene 0.6 0.6 Liver, kidney, or circulatorysystem problems

Discharge fromindustrialchemical factories

p-Dichlorobenzene 0.075 0.075 Anemia; liver, kidney or spleendamage; changes in blood


1,2-Dichloroethane zero 0.005 Increased risk of cancer Discharge fromindustrialchemical factories

1,1-Dichloroethylene 0.007 0.007 Liver problems Discharge fromindustrialchemical factories

(Table continued )






ContaminantMCLG

1

(mg/L)2MCL or TT

1




cis-1,2-Dichloro-ethylene

0.07 0.07 Liver problems Discharge fromindustrialchemical factories

trans-1,2-Dichloro-ethylene

0.1 0.1 Liver problems Discharge fromindustrialchemical factories

Dichloromethane zero 0.005 Liver problems; increasedrisk of cancer

Discharge from drugand chemicalfactories

1,2-Dichloropropane zero 0.005 Increased risk of cancer Discharge fromindustrialchemical factories

Di(2-ethylhexyl)adipate

0.4 0.4 Weight loss, liver problems,or possible reproductivedifficulties

Discharge fromchemical factories

Di(2-ethylhexyl)phthalate

zero 0.006 Reproductive difficulties;liver problems; increased riskof cancer

Discharge fromrubber andchemical factories

Dinoseb 0.007 0.007 Reproductive difficulties Runoff fromherbicide used onsoybeans andvegetables

Dioxin (2,3,7,8-TCDD) zero 0.00000003 Reproductive difficulties;increased risk of cancer

Emissions fromwasteincineration andother combustion;discharge fromchemical factories

Diquat 0.02 0.02 Cataracts Runoff fromherbicide use

Endothall 0.1 0.1 Stomach and intestinalproblems

Runoff fromherbicide use

Endrin 0.002 0.002 Liver problems Residue of bannedinsecticide

Epichlorohydrin zero TT9 Increased cancer risk, andover a long periodof time, stomachproblems

Discharge fromindustrial chemicalfactories; animpurity of somewater treatmentchemicals

Ethylbenzene 0.7 0.7 Liver or kidneys problems Discharge frompetroleumrefineries

Ethylene dibromide zero 0.00005 Problems with liver, stomach,reproductive system, orkidneys; increased risk ofcancer

Discharge frompetroleumrefineries

Glyphosate 0.7 0.7 Kidney problems; reproductivedifficulties

Runoff fromherbicide use

Heptachlor zero 0.0004 Liver damage; increased risk ofcancer

Residue of bannedtermiticide

Heptachlor epoxide zero 0.0002 Liver damage; increased risk ofcancer

Breakdown ofheptachlor

(Table continued )






ContaminantMCLG

1

(mg/L)2MCL or TT

1




Hexachlorobenzene zero 0.001 Liver or kidney problems;reproductive difficulties;increased risk of cancer

Discharge frommetal refineriesand agriculturalchemical factories

Hexachlorocyclopent

adiene

0.05 0.05 Kidney or stomach problems Discharge fromchemical factories

Lindane 0.0002 0.0002 Liver or kidney problems Runoff/leachingfrom insecticideused on cattle,lumber, gardens

Methoxychlor 0.04 0.04 Reproductive difficulties Runoff/leachingfrom insecticideused on fruits,vegetables, alfalfa,livestock

Oxamyl (Vydate) 0.2 0.2 Slight nervous system effects Runoff/leaching frominsecticide used onapples, potatoes,and tomatoes

Polychlorinatedbiphenyls (PCBs)

zero 0.0005 Skin changes; thymus glandproblems; immunedeficiencies; reproductiveor nervous systemdifficulties; increased riskof cancer

Runoff from landfills;discharge of wastechemicals

Pentachlorophenol zero 0.001 Liver of kidney problems,increased cancer risk

Discharge fromwood preservingfactories

Picloram 0.5 0.5 Liver problems Herbicide runoffSimazine 0.004 0.004 Problems with blood Herbicide runoffStyrene 0.1 0.1 Liver, kidney, or circulatory

system problemsDischarge from

rubber and plasticfactories; leachingfrom landfills

Tetrachloroethylene zero 0.005 Liver problems; increasedrisk of cancer

Discharge fromfactories and drycleaners

Toluene 1 1 Nervous system, kidney,or liver problems

Discharge frompetroleumfactories

Toxaphene zero 0.003 Kidney, liver, or thyroidproblems; increased riskof cancer

Runoff/leachingfrom insecticideused on cottonand cattle

2,4,5-TP (Silvex) 0.05 0.05 Liver problems Residue of bannedherbicide

1,2,4-Trichlorobenzene 0.07 0.07 Changes in adrenal glands Discharge fromtextile finishingfactories

(Table continued )






ContaminantMCLG

1

(mg/L)2MCL or TT

1




1,1,1-Trichloroethane 0.20 0.2 Liver, nervous system, orcirculatory problems

Discharge from metaldegreasing sitesand other factories

1,1,2-Trichloroethane 0.003 0.005 Liver, nervous system, orcirculatory problems


Trichloroethylene zero 0.005 Liver problems; increasedrisk or cancer

Discharge frommetal degreasingsites and otherfactories

Vinyl chloride zero 0.002 Increased risk of cancer Leaching from PVCpipes; dischargefrom plasticfactories

Xylenes (total) 10 10 Nervous system damage Discharge frompetroleumfactories;discharge fromchemical factories

RADIONUCLIDES

Alpha particles none7 15 picocuriesper Liter(pCi/L)

Increased risk of cancer Erosion of naturaldeposits of certainminerals thatare radioactiveand may emit aform of radiationknown as alpharadiation

Beta particles andphoton emitters

none7 4 milliremsper year(mrem/yr)

Increased risk of cancer Decay of natural andman-madedeposits of certainminerals that areradioactive andmay emit forms ofradiation knownas photons andbeta radiation

Radium 226 andRadium 228(combined)

none7 5 pCi/L Increased risk of cancer Erosion of naturaldeposits

Uranium zero 30 mg/L as of12/08/03

Increased risk of cancer, kidneytoxicity

Erosion of naturaldeposits

NOTES

1—Definitions†Maximum Contaminant Level Goal (MCLG)—The level of a contaminant in drinking water below which there is no known orexpected risk to health. MCLGs allow for a margin of safety and are non-enforceable public health goals.

† Maximum Contaminant Level (MCL)—The highest level of a contaminant that is allowed in drinking water. MCLs are set asclose to MCLGs as feasible using the best available treatment technology and taking cost into consideration. MCLs areenforceable standards.

† Maximum Residual Disinfectant Level Goal (MRDLG)—The level of a drinking water disinfectant below which there is noknown or expected risk to health. MRDLGs do not reflect the benefits of the use of disinfectants to control microbialcontaminants.





† Maximum Residual Disinfectant Level (MRDL)—The highest level of a disinfectant allowed in drinking water. There isconvincing evidence that addition of a disinfectant is necessary for control of microbial contaminants.

† Treatment Technique (TT)—A required process intended to reduce the level of a contaminant in drinking water.2—Units are in milligrams per liter (mg/L) unless otherwise noted. Milligrams per liter are equivalent to parts per million (ppm).3—EPA’s surface water treatment rules require systems using surface water or ground water under the direct influence of surface

water to (1) disinfect their water, and (2) filter their water or meet criteria for avoiding filtrations so that the following contaminants arecontrolled at the following levels:

† Cryptosporidium (as of 1/1/02 for systems serving .10,000 and 1/14/05 for systems serving ,10,000) 99% removal.† Giardia lamblia: 99.9% removal/inactivation† Viruses: 99.9% removal/inactivation† Legionella: No limit, but EPA believes that if Giardia and viruses are removed/inactivated, Legionella will also be controlled.† Turbidity: At no time can turbidity (cloudiness of water) go above 5 nephelolometric turbidity units (NTU); systems that filtermust ensure that the turbidity go no higher than 1 NTU (0.5 NTU for conventional or direct filtration) in at least 95% of the dailysamples in any month. As of January 1, 2002, turbidity may never exceed 1 NTU, and must not exceed 0.3 NTU in 95% of dailysamples in any month.

† HPC: No more than 500 bacterial colonies per milliliter† Long Term 1 Enhanced Surface Water Treatment (Effective Date: January 14, 2005); Surface water systems or (GWUDI) systemsserving fewer than 10,000 people must comply with the applicable Long Term 1 Enhanced Surface Water Treatment Ruleprovisions (e.g. turbidity standards, individual filter monitoring, Cryptosporidium removal requirements, updated watershedcontrol requirements for unfiltered systems).

† Filter BackwashRecycling; The Filter BackwashRecyclingRule requires systems that recycle to return specific recycle flows throughall processes of the system’s existing conventional or direct filtration system or at an alternate location approved by the state.

4—No more than 5.0% samples total coliform-positive in a month. (For water systems that collect fewer than 40 routine samplesper month, no more than one sample can be total coliform-positive per month.) Every sample that has total coliform must be analyzedfor either fecal coliforms or E. coli if two consecutive TC-positive samples, and one is also positive for E. coli fecal coilforms, system hasan acute MCL violation.

5—Fecal coliform and E. coli are bacteria whose presence indicates that the water may be contaminated with human or animalwastes. Disease-causingmicrobes (pathogens) in these wastes can cause diarrhea, cramps, nausea, headaches, or other symptoms. Thesepathogens may pose a special health risk for infants, young children, and people with severely compromised immune systems.

6—Although there is no collective MCLG for this contaminant group, there are individual MCLGs for some of the individualcontaminants:

† Haloacetic acids: dichloroacetic acid (zero); trichloroacetic acid (0.3 mg/L)† Trihalomethanes: bromodichloromethane (zero); bromoform (zero); dibromochloromethane (0.06 mg/L)

7—MCLGs were not established before the 1986 Amendments to the Safe Drinking Water Act. The standard for this contaminantwas set prior to 1986. Therefore, there is no MCLG for this contaminant.

8—Lead and copper are regulated by a Treatment Technique that requires systems to control the corrosiveness of their water. If morethan 10% of tap water samples exceed the action level, water systems must take additional steps. For copper, the action level is 1.3 mg/L,and for lead is 0.015 mg/L.

9—Each water system must certify, in writing, to the state that when it uses acrylamide and/or epichlorohydrin to treat water, thecombination (or product) of dose and monomer level does not exceed the levels specified, as follows: Acrylamide ¼ 0.05% dosed at1 mg/L (or equivalent); Epichlorohydrin ¼ 0.01% dosed at 20 mg/L (or equivalent).





Enforcement responsibility rests with the U.S.Environmental Protection Agency or with thosestates electing to take primary responsibility forensuring compliance with the regulations. The EPAupdates standards periodically.

Secondary Standards n The aesthetic con-taminants are covered by the secondary drinkingwater regulations. The limits are called secondarymaximum contaminant levels (SMCL) and arelisted in Table 21.20. These levels represent reas-onable goals for drinking-water quality but are notFederally enforceable. The states may use theseSMCL as guidelines and establish higher or lowerlevels that may be appropriate, dependent on localconditions, such as unavailability of alternatesource waters or other compelling factors, if publichealth and welfare are not adversely affected.

Source Protection n The U.S. Public HealthService Drinking Water Standards recognized theneed for protecting the source of water supplies, asindicated by the following extract:

The water supply should be obtained from the mostdesirable source feasible, and effort should be made toprevent or control pollution of the source. If the source isnot adequately protected against pollution by naturalmeans, the supply shall be adequately protected bytreatment.

Sanitary surveys shall be made of the water-supplysystem, from the source of supply to the connection of the

customer’s service piping, to locate and correct anyhealth hazards that might exist. The frequency of thesesurveys shall depend upon the historical need.

Adequate capacity shall be provided to meet peakdemands without development of low pressures and thepossibility of backflow of polluted water from customerpiping.

Case histories and monitoring programs havebeen reported indicating that active source pro-tection can enhance water quality with minimalextra expense. Preventing contamination of drink-ing water supplies is part of EPA’s mission.(See R. B. Pojasek, “Drinking-Water Quality Enhan-cement through Source Protection,” Ann ArborScience Publishers, Inc., Ann Arbor, Mich.)

Water Treatment

Water is treated to remove disease-producingbacteria, unpleasant tastes and odors, particulateand colored matter, and hardness and to lower thelevels of any contaminants when necessary to meetwater-quality standards. Some of the more com-mon methods of treatment are plain sedimentationand storage, coagulation-sedimentation, slow andrapid sand filtration, disinfection, and softening(see also Art. 21.51).

Long-term storage of water reduces the amountof disease-producing bacteria and particulatematter. But economic conditions usually compelwater purveyors to use more efficient methods oftreatment, such as those mentioned above.

21.46 SedimentationProcesses

Sedimentation or clarification is a process ofremoving particulate matter from water throughgravity settlement in a basin by reducing the flow-through velocity. Factors that affect the settling rateof particulate matter suspended in water are size,shape, and specific gravity of the suspended par-ticles; temperature and viscosity of the water; andsize and shape of the settling basin.

The settling velocity ns of spherically shapedparticles in a viscous liquid can be found by use ofStokes’ law if the Reynolds number R ¼ nrd/m,calculated with n ¼ ns, is equal to or less than 1.

ns ¼ g(r1 � r)d2

18m(21:133)

Table 21.20 Secondary Drinking-WaterStandards

Contaminant Secondary Standard

Aluminum 0.05 to 0.2 mg/LChloride 250 mg/LColor 15 (color units)Copper 1.0 mg/LCorrosivity noncorrosiveFluoride 2.0 mg/LFoaming Agents 0.5 mg/LIron 0.3 mg/LManganese 0.05 mg/LOdor 3 threshold odor num-

berpH 6.5–8.5Silver 0.10 mg/L

(Table continued)





where ns ¼ settling velocity of particle, m/s

g ¼ acceleration due to gravity, m/s2

m ¼ absolute viscosity of the fluid, Pa . s

r1 ¼ density of particle, g/mm3

r ¼ density of fluid, g/mm3

d ¼ particle diameter, mm

If R � 2000, Newton’s law applies:

ns ¼ 4g(r1 � r)d

3rCD(21:134)

where CD is the drag coefficient. Figure 21.75shows a plot of CD values vs. Reynolds numbers, tobe used in Eq. (21.134).

In the region where 1.0 , R , 2000, there is atransition from Stokes’ law to Newton’s. Thesettling velocity in this region is somewherebetween the values given by Newton’s law andthose given by Stokes’ law; however, no exactexpression has been developed to give the velocity.

Figure 21.76 shows the relationship of settlingvelocity to diameter of spherical particles withspecific gravity S between 1.001 and 5.0.

21.46.1 Plain Sedimentation

The ideal settling basin (Fig. 21.77) is a sedimen-tation tank in which flow is horizontal, velocity is

constant, and concentration of particles of each sizeis the same at all points of the vertical cross sectionat the inlet end. The basin has a volumetric capacityC, depth ho, and width B. The surface loading rateor overflow velocity no, equal to the settlingvelocity of the smallest particle to be completelyremoved, can be determined by dividing theflow rate Q by the settling surface area A. For thisideal basin, the overflow velocity therefore is no ¼Q/A ¼ Q/BLo, whereQ ¼ BhoV and Lo is the lengthof settling zone, V the flow-through velocity.(Usually, no is expressed in gallons per day persquare foot of surface area.) The detention timet ¼ ho/no ¼ Lo/V also equals the volumetric capac-ity C divided by the rate of flow Q.

Particles with a settling velocity ns � no, andthose that enter the settling zone between f and j (atleft in Fig. 21.77) with a settling velocity ns largerthan (n1 ¼ h1V/Lo) but less than no, are removed inthis basin. The particles with a settling velocityns � n1 that enter the settling zone between f and eare not removed in this basin.

The efficiency of a sedimentation basin is theratio of the flow-through period to the detentiontime. The flow-through period is the time requiredfor a dye, salt, or other indicator to pass through thebasin. Settling-basin efficiencies are reduced bymany factors such as cross currents, short circuit-ing, and eddy currents. A well-designed tankshould have an efficiency of 30 to 50%.

Fig. 21.75 Newton drag coefficients for spheres in fluids. (Observed curves, after Camp, Transactions ofthe American Society of Civil Engineers, vol. 103, p. 897, 1946.)





Some design criteria for sedimentation tanksare:

Period of detention—2 to 8 h

Length-to-width ratio of flow-through channel—3:1 to 5 :1

Depth of basin—10 to 25 ft (15 ft average)

Width of flow-through channel—not over 40 ft(30 ft most common)

Diameter of circular tank—35 to 200 ft (mostcommon, 100 ft)

Flow-through velocity—not to exceed 1.5 ft/min(most common velocity, 1.0 ft/min)

Surface loading or overflow velocity, gal per dayper ft2 of surface area—between 500 and 2000 formost settling basins

Sedimentation tanks may be built in any ofa variety of shapes, for example, rectangular

(Fig. 21.78a) or circular (Fig. 21.78b). Multistorytanks, such as the two-story basin with a single trayin Fig. 21.78c, occupy less site area than the single-story basin. The tubular settler (Fig. 21.78d) withparallel flow upward provides very high surfaceareas.

(American Water Works Association and Ameri-can Society of Civil Engineers, “Water TreatmentPlant Design,”McGraw-Hill, Inc., New York (books.mcgraw-hill.com); G. M. Fair, J. C. Geyer, andD. A. Okun, “Water and Wastewater Engineering,”John Wiley & Sons, Inc., New York (www.wiley.com).)

21.46.2 Coagulation-Sedimentation

To increase the settling rate and remove finelydivided particles in suspension, coagulants areadded to the water. Without coagulants, finelydivided particles do not settle out because of theirhigh ratio of surface area to mass and the presence

Fig. 21.76 Chart gives settling velocities of spherical particles with specific gravities S, at 10 8C.





of negative charges on them. The velocity at whichdrag and gravitational forces are equal is very low,and the negative charges on the particles produceelectrostatic forces of repulsion that tend to keepthe particles separated and prevent agglomeration.When coagulating chemicals are mixed with water,however, they introduce highly charged positivenuclei that attract and neutralize the negativelycharged suspended matter.

Iron and aluminum compounds are commonlyused as coagulants because of their high positiveionic charge. The alkalinity of the water beingtreated must be high enough for an insoluble hy-droxide or hydrate of these metals to form. Theseinsoluble flocs of iron and aluminum, which com-bine with themselves and other suspended parti-cles, precipitate out when a floc of sufficient size isformed.

The more common coagulants are aluminumsulfate, commonly known as alum [Al2(SO4)3 .

18H2O]; ferrous sulfate (FeSO4. 7H2O); ferric chlo-

ride (FeCl3); and chlorinated copperas (a mixture offerric chloride and ferrous sulfate). The type andamount of coagulant necessary to clarify a specifiedwater depend on the qualities of water to be treated,

such as pH, temperature, turbidity, color, and hard-ness. Jar tests are usually made in a laboratory todetermine the optimum amount of coagulant.

Some organic polymers are alternatives to themetallic coagulants. Polymers are long-chain, high-molecular-weight, organic polyelectrolytes. Theyare available in three types: cationic, or positivelycharged; anionic, or negatively charged; and non-ionic, or neutral in charge. Cationic polymers aregenerally the most suitable for use as primarycoagulants. Anionic polymers, however, are oftenused as flocculant aids in conjunction with an ironor aluminum salt to cause the formation of largerfloc particles. Thereby, lesser amounts of metallicsalt are needed to effect good coagulation.

Because of differences in the characteristics ofthe suspended matter found in natural waters, notall waters can be treated with equal success withthe same polymer or the same dosages. Jar testsshould be run with several dosages of the variouspolymers available to aid in selecting the materialbest suited for each water supply, considering bothcost and performance.

There are several reasons for considering the useof polymers: increased settling rate and improved

Fig. 21.77 Longitudinal section through an ideal settling basin.





Fig. 21.78 Types of sedimentation tanks: (a) Rectangular settling basin. (b) Circular clarifier. (c) Two-story sedimentation basin. (d) Tubular settler.





filtrability of the floc, production of a smallervolume of sludge, and easier dewatering. Also,polymers have a minor effect on pH; consequently,the need for final pH adjustment in the finishedwater may be reduced.

Process Steps n The complete clarificationprocess is usually divided into three stages: (1)rapid chemical mixing; (2) flocculation or slowstirring, to get the small floc to agglomerate; and (3)coagulation-sedimentation in low-flow-velocitysettling basins. Rapid chemical mixing may beaccomplished with many devices, such as mech-anical stirrers, centrifugal pumps, and air jets. Thetime necessary for mixing ranges from a fewseconds to 20 min. Flocculation or slow stirringincreases floc size and speeds up settling. Thespeed of the agitators must be great enough,however, to cause contact between the small flocbut not so great that the larger floc is broken up.Flocculator detention time should be in the 20- to60-min range. The coagulation-sedimentation pro-cess takes place in a clarifier basin nearly identicalto a plain sedimentation basin. The detentionperiod for a clarifier should be between 2 and 8 h.

(G. L. Culp and R. L. Culp, “New Concepts inWater Purification,” Van Nostrand ReinholdCompany, New York; American Water WorksAssociation, “Water Quality and Treatment,” 4thed., T. J. McGhee, “Water Supply and Sewerage,”R. A. Corbitt, “Standard Handbook of Environ-mental Engineering,” McGraw-Hill, Inc., NewYork(books.mcgraw-hill.com).)

21.47 Filtration Processes

Passing water through a layer of sand removesmuch of the finely divided particulate matter andsome of the larger bacteria. The filtering processhas many components, such as physical straining,chemical and biological reactions, settling, andneutralization of electrostatic charges.

Direct Filtration n It is possible by use ofdirect filtration to eliminate the sedimentation step,in some instances, for treatment of raw waters thatare low in turbidity, color, coliform organisms,plankton, and suspended solids, such as paperfiber. Direct filtration is a water-treatment processin which raw water is not settled prior to thefiltration step. It usually includes addition of a

coagulant to destabilize the colloidal particles anda polymer as a flocculant aid. The process requiresrapid mixing, agitation in a well-designed floccu-lator for 10 to 30 min, addition of a polymer as afilter aid, and dual- or mixed-media filtration.

Pilot plant tests are essential for selecting thebest combination of coagulant and flocculant aid toobtain a strong floc and to provide criteria fordesign of the filtration units.

The principal advantages of direct filtration areits lower capital and operation costs. Elimination ofsettling basins can result in capital cost savings of20 to 30%, and operational cost may be cut 10 to30% by reduced chemical doses. Direct filtrationmerits investigation before construction of newfacilities if the turbidity of the source wateraverages less than 25 TU.

Slow Sand Filters n These consist of anunderdrained, watertight container containing a 2-to 4-ft layer of sand supported by a 6- to 12-in layerof gravel. The effective size of the sand should be inthe 0.25- to 0.35-mm range. (The effective size isthe size of a sieve, in millimeters, that will pass10%, by weight, of the sand. The uniformitycoefficient is the ratio of the size of a sieve that willpass 60% of the sample to the effective size.) Theuniformity coefficient of the sand should be lessthan 3. The sand is normally submerged under 4 or5 ft of water. The water passes through the filter at arate of 3 to 6 MG per acre per day, depending on theturbidity. The slow filter is not as versatile or asefficient as rapid sand filters.

Rapid Sand Filtration n This is normally pre-ceded by chemical treatment, such as flocculation-coagulation and disinfection, so the water can bepassed through the sand at a higher rate. Usually,the effluent from a rapid filter needs furtherdisinfection or chlorination because the bacteria arenot completely removed in this process. A diagramof a typical gravity-type rapid sand filter is shown inFig. 21.79.

The normal order of flow through the varyingcomponents of the filter is from the clarifiers(settling tanks) to the top of the sand layer, throughthe sand and gravel layers, through the underdrainlaterals to the main drain, and then through thecontroller to the clear well for storage. Wash(cleaning) water flow takes place in a reversedirection after the filter effluent line has beenclosed. The wash-water flow is through the main





drain to the laterals, from the laterals upwardthrough the gravel and sand to the wash-watertroughs. The troughs carry the water to the gullet,which is drained to waste.

Some common design factors for rapid sandfilters are:

Effective grain size—0.35 to 0.55 mm

Uniformity coefficient—1.20 to 1.75

Thickness of sand layer—24 to 30 in depending ongrain size

Thickness of gravel layer—15 to 24 in

Gravel size—from 1⁄8 to 11⁄2 in

Filtration rate—2 to 4 gal/min . ft2 (125 to 250 MGper acre per day)

Total depth of each basin—8 to 10 ft

Maximum head loss allowed before washingsand—8 to 10 ft

Sand expansion during washing—25 to 50%

Wash-water rate—15 to 20 gal/min . ft2

Distance from top edge of wash-water trough totop of unexpanded sand—24 to 30 in

Length of filter runs between washings—12 to 72 h

Spacing between wash-water troughs—4 to 6 ft

Ratio of length to width of each basin—1.25 to 1.35

Rapid sand filters are operated until theparticulate matter and unsettled floc cover theopenings between the sand grains, creating a highhead loss across the filter. This high head loss slowsdown the flow rate and may force some of theparticulate matter through the sand and gravellayers. Filters are usually backwashed when theparticulate-matter concentration increases in thefilter effluent or when the head loss reaches 8 to10 ft. Backwashing a filter consists of forcingfiltered water through the filter from the drainsupward to the wash-water troughs. The light-weight sediment is washed from the sand grainsby the moving water and sometimes by otheragitating devices, such as rakes, water sprays, andair jets. Filters must be washed thoroughly ordifficulties with mud balls, bed cracking, or sandincrustation will be encountered.

Immediately after washing, filters pass water ata high rate, which produces an undertreatedeffluent. Either manual or automatic rate controlmust be used to prevent such an occurrence. Many

Fig. 21.79 Gravity-type rapid sand filter.





treatment plants control the rate of filtration byusing venturi tube devices, which throttle the filtereffluent line when there is high-velocity flow. Asclogging begins to occur in the filter, the velocity offlow in the effluent line decreases, and the ratecontroller then opens to increase the velocity.

A negative head is produced on the filter whenthe head loss across the filter is greater than thedepth of water on the sand. Negative heads canproduce a condition known as air binding, whichis caused by removal of dissolved gases fromthe water and formation between sand grains ofbubbles that decrease filter capacity.

The underdrains of a filter are commonly madeof perforated pipe or porous plates. The under-drains should be arranged so that each area filtersand distributes its proportionate share of water.The ratio of total area of perforations to the totalfilter-bed area is normally in the 0.002:1 to 0.005:1range. The diameter of the perforations variesbetween 1⁄4and

3⁄4in.Wash-water troughs should be evenly spaced,

and water should not have to travel more than 3 fthorizontally to get to a wash-water gutter. Thedepth of water flow in a horizontal gutter may becalculated from

Q ¼ 1:72by3=2 (21:135)

where Q ¼ total flow received by trough, gal/min

b ¼ width of trough, in

y ¼ water depth at upstream end oftrough, in

The total gutter depth can be found by adding 2 or3 in of freeboard to the calculated depth y.

Other Processes n Anthracite coal may beused in place of sand in gravity-type filters. Theeffective grain size is greater than that of sand, thuspermitting higher filtration rates and longer filterruns.

Dual-media, mixed-media, or deep coarse-mediafilters, however, may be more advantageous. Theyoperate at the higher filtration rates of 4 to 8 gal/min . ft2.

A pressure filter is composed of a gravity-filtermedium enclosed in a watertight vessel. Thefiltering medium may be sand, diatomaceousearth, or anthracite coal. Pressure filters are pri-marily supplemental and are used for specialized

industrial uses and for clarifying swimming poolwater.

Filter galleries are made up of horizontal,perforated, or open-joint pipes, placed in shallowsand or gravel aquifers. Galleries typically are fedby diversion or pumping from streams intospreading basins with gravel or sand bottoms.Some, however, may be located in aquifers withhigh groundwater table. The filtered water may bepumped from the gallery or allowed to flow outone end by gravity.

(G. L. Culp and R. L. Culp, “New Concepts inWater Purification,” Van Nostrand Reinhold Com-pany, New York; American Water Works Associ-ation, “Water Quality and Treatment,” 4th ed., andAmerican Society of Civil Engineers, “WaterTreatment Plant Design,” and T. J. McGhee, “WaterSupply and Sewerage,” 6th ed., McGraw-Hill BookCompany, New York (books.mcgraw-hill.com);G. M. Fair, J. C. Geyer, and D. A. Okun, “Waterand Wastewater Engineering,” John Wiley & Sons,Inc., New York (www.wiley.com).)

21.48 Water Softening

Presence of the bicarbonates, carbonates, sulfates,and chlorides of calcium and magnesium in watercauses hardness. Three major classifications ofhardness are: (1) carbonate (temporary) hardnesscaused by bicarbonates, (2) noncarbonate (perma-nent) hardness, and (3) total hardness. Municipaltreatment plants generally use either the lime-soda(precipitation) process or the base-exchange (zeo-lite) process to reduce the hardness of the waterto below 100 mg/L (about 100 ppm) of CaCO3

equivalence.In the lime-soda process, lime (CaO), hydrated

lime [Ca(OH)2], and soda ash (Na2CO3) are addedto the water in sufficient quantities to reduce thehardness to an acceptable level. The amounts oflime and soda ash required for softening to aresidual hardness can be determined by use ofchemical-equivalent weights, taking into accountthat commercial grades of lime and hydrated limeare about 90 and 68% CaO, respectively. Residualhardness of 50 to 100 mg/L as CaCO3 remains inthe treated water because of the very slightsolubility of both CaCO3 and Mg(OH)2. Hardnessof water is normally expressed in grains pergallon (gpg) or mg/L of CaCO3, where 1 gpg ¼17.1 mg/L.





Chemical equations for the common lime-sodasoftening processes are

CO2 þ CaO ! CaCO3# (21:136)

Ca(HCO3)2 þ CaO ! 2CaCO3# þ H2O (21:137)

MgSO4 þ CaOþH2O

! Mg(OH)2 # CaSO4 (soluble) (21:138)

CaSO4 (soluble)þNa2CO3 ! CaCO3# þ NaSO4

(21:139)

Since the carbonate and magnesium hydroxideparticles settle out in sedimentation basins, facili-ties must be provided for particle removal and dis-posal. Deposition of CaCO3 and Mg(OH)2 on sandgrains, in clear wells, and in distribution pipes canbe prevented by recarbonation with CO2 beforesand-filter treatment.

Hardness in water can be reduced to zero bypassing the water through a base-exchange orzeolite material. These materials remove cations,such as calcium and magnesium, from water andreplace them with soluble sodium and hydrogencations. Calcium can be removed from water asshown by the following reaction:

Ca2þ þNa2RN CaRþ 2Naþ (21:140)

where Ca2þ is the calcium hardness ion removed,Naþ is the sodium ion replacing the Ca2þ in water,and R is the zeolite material. The reaction can bereversed (from right to left) by increasing the Naþ

concentration to a high value, as generally is donein regeneration of the softening unit.

Sodium chloride (table salt) is commonly usedto regenerate the unit. Regeneration requiresbetween 0.3 and 0.5 lb of salt per 1000 grains ofhardness removed.

Some hardness-removal capacities per cubicfoot of base-exchange material are: natural zeo-lite—2500 to 3000 grains, synthetic zeolite—5000 to30,000 grains (1 lb ¼ 7000 grains).

(American Water Works Association, “WaterQuality and Treatment,” 4th ed., and AmericanSociety of Civil Engineers, “Water Treatment PlantDesign,” McGraw-Hill Book Company, New York(books.mcgraw-hill.com).)

21.49 Disinfection withChlorine

Chlorine in either the liquid, gas, or hypochloriteform is frequently used for destroying bacteria inwater supplies. Other disinfectants are iodine,bromine, ozone, chlorine dioxide, ultraviolet light,and lime.

The reaction of chlorine with water is

Cl2 þH2ONHþ þ Cl� þHOCl (21:141)

The hypochlorous acid (HOCl) reacts with theorganic matter in bacteria to form a chlorinatedcomplex that destroys living cells. The amount ofchlorine (chlorine dose) added to the water dependson the amount of impurities to be removed and thedesired residual of chlorine in the water. Chlorineresiduals of 0.1 or 0.2 mg/L are normally main-tained in water-treatment-plant effluent streams asa factor of safety for the water as it travels to theconsumer.

The concern over trihalomethane formation fol-lowing chlorination of waters containing appreci-able amounts of natural organic materials (Art.21.62) has led to use of alternate disinfectants. Theprime candidates are ozone and chlorine dioxide.The benefits of ozone should be investigated fornew or modified treatment plants, particularly ifthere are color or taste and odor problems in theraw water.

(AmericanWaterWorks Association andAmeri-can Society of Civil Engineers, “Water TreatmentPlant Design,” and T. J. McGhee, “Water Supplyand Sewerage,” McGraw-Hill, Inc., New York(books.mcgraw-hill.com).)

21.50 Carbonate Stability

Water may either corrode or place a protectivecarbonate film on the interior surfaces of pipes.Which it does depends on the nature and amountof chemicals dissolved in the water.

An approximation of the stability of a watersupply can be obtained by adding an excess ofcalcium carbonate powder to one-half of a watersample. Stir or shake each half sample at 5-minintervals for about 1 h. Filter both solutions; then,either take the pH or determine the methyl orangealkalinity of each sample. If the untreatedwater hasa higher alkalinity or pH than the CaCO3-treatedwater, the water is saturated with carbonate and





may deposit protective films in pipes. If theuntreated water has a lower pH or alkalinity valuethan the treated water, the water is unsaturatedwith carbonate and may be corrosive. If the pH oralkalinity is the same in both samples, the water isin equilibrium in regard to carbonates.

The greater the difference in either alkalinity orpH between the two samples, the greater theamount of either unsaturation or saturation withrespect to carbonates. If the untreated water has amuch higher pH or alkalinity than the treatedwater, the water is highly saturated with carbon-ates. It can cause a problem with heavy carbonatedeposits in pipes and appurtenances of thepurveyor and consumer.

(G. M. Fair, J. C. Geyer, and D. A. Okun, “Waterand Wastewater Engineering,” John Wiley & Sons,Inc., New York (www.wiley.com).)

21.51 MiscellaneousTreatments

Many different methods of treatment are used toremove such undesirable elements as color, taste,odor, excessive fluorides, detergents, iron, manga-nese, and substances exceeding the water-qualitymaximum contaminant levels (Art. 21.45).

Activated carbon is commonly used for tasteand odor removal. The carbon can be applied as apowder to the water and later removed by a sandfilter, or the water can be passed through a bed ofcarbon to remove natural and synthetic organicchemicals.

Treatment techniques for removal of inorganiccontaminants include conventional coagulation,lime softening, cation exchange, anion exchange,activated carbon, reverse osmosis, and electro-dialysis. Concerns over the potential for leadpoisoning from lead in drinking water passingthrough lead pipes installed long ago but still inuse or from leaded solder used for pipe joints haveencouraged abandonment of such practices. Wherethe presence of lead is detected in a water supply,despite its low solubility, its concentration can benearly completely removed with lime softening oralum and ferric sulfate coagulation.

(American Water Works Association andAmerican Society of Civil Engineers, “WaterTreatment Plant Design,” McGraw-Hill, Inc., NewYork (books.mcgraw-hill.com).)

Water Collection, Storage,and Distribution

21.52 Reservoirs

The basic purpose of impounding reservoirs is tohold runoff during periods of high runoff andrelease it during periods of low runoff. The specificfunctions of reservoirs are hydroelectric, floodcontrol, irrigation, water supply, and recreation(see also Art. 21.52.1). Many large reservoirs aremultipurpose, as a consequence of which thespecific functions may dictate conflicting designand operating criteria. Also, equitable cost allo-cation is more difficult.

Sizing of a reservoir for a project where thedemand for water is much greater than the meanstreamflow is an economic balance between benefitsand costs. A preliminary study of available reservoirsites should be made to obtain the relative costs forvarious size reservoirs. The dependable flow thatcan be obtained from various-size reservoirs can bedetermined from the mass diagram for stream flow.An economic comparison should then be made ofthe benefits of various flows and the costs of variousreservoirs. The reservoir size that will give themaximum benefit should be selected.

When the demand rate is known, as is the casefor many water-supply projects, the required sizeof the reservoir can be determined directly from amass diagram of stream flow.

Themass diagram (Fig. 21.80) is a graphical plotof total stream-flow volume against time. The slopeof the curve is the rate of flow.

Selection of the critical period of years for amasscurve depends on the function of the reservoir.For a water-supply or hydroelectric development,minimum flows will be critical, whereas for flood-control reservoirs, maximum flows will govern.

Reservoir capacity for a certain demand can beobtained by drawing a line with a slope equal to thedemand tangent to the mass curve at the beginningof a selected dry period, as shown by lines AB andAC in Fig. 21.80. The ordinates d and e represent thestorage required to maintain demands AB and AC.

Once a reservoir site has been selected, area-volume curves (Fig. 21.81) are drawn to give thecharacteristics of the site. The plot of volume vs.water elevation is determined by planimetering thearea of selected contours within the reservoir siteand multiplying by the contour interval. Aerialmapping has made it possible to obtain accurate





contour maps at only a fraction of the costs of oldermethods.

Another important consideration in the designof reservoirs is deposition of sediment (see Arts.21.35 and 21.52.2).

In selection of a site for a water-supply reservoir,give special attention to water quality. If possible,the watershed should be relatively uninhabited toreduce the amount of treatment required. (Waterfrom practically all sources should be disinfected in

Fig. 21.80 Mass diagram of stream flow.

Fig. 21.81 Area-capacity curves for a reservoir.





the distribution system to ensure against pollutionand contamination.) Shallow reservoirs usuallygive more problems with color, odor, and turbiditythan deep reservoirs, particularly in warm climatesor during warm seasons of the year. Runoff heav-ily laden with silt and debris should be divertedfrom the reservoir or treated before it is mixed withthe water supply. Alum is mixed into reservoirs toreduce turbidity, and copper sulfate is used to killvegetation.

In deep reservoirs, during the summer monthsthe upper part of the reservoir will be warmed,while below a certain level the temperature may bemany degrees cooler. The zone where the abrupttemperature change takes place, which may beonly a few feet thick, is called the thermocline. Thewaters above and below the thermocline circulate,but there is no circulation across this zone. Thewater in the lower level becomes low in dissolvedoxygen and develops bad tastes and odors. Whenthe temperature drops in the fall, the water at theupper level becomes heavier than the water at thelower level and the two levels become intermixed,causing bad tastes and odors in the entire reservoir.To oxidize organic matter and prevent poor water

quality in lower levels of reservoirs during summermonths, chlorine or compressed air should bereleased at various points on the bottom of thereservoirs.

21.52.1 Distribution Reservoirs

The two main functions of distribution reservoirsare to equalize supply and demand over periods ofvarying consumption and to supply water duringequipment failure or for fire demand. Majorsources of supply for some cities, such asNew York, San Francisco, and Los Angeles, arelarge distances from the city. Because of the largecost of aqueducts, it is usually economical to sizethem for the mean annual flow and provide ter-minal storage for daily and seasonal fluctuations ofdemand. Terminal storage is also necessarybecause of the possibility of a failure along anaqueduct.

It is usually economical to have equalizingreservoirs at various points in the distributionsystem so that main supply lines, pumping plants,and treatment plants can be sized for maximumdaily instead of maximum hourly demand. During

Fig. 21.82 Chart indicates percentage of incoming sediment trapped in reservoirs.





hours of maximum demand, water flows fromthese reservoirs to the consumers. When thedemand drops off, the flow refills the reservoir. Amass diagram (Fig. 21.80) can be used to determinethe required capacity of the reservoir.

Equalizing reservoirs are usually built at theopposite end of the system from the source ofsupply, so that during peak flows the maximumdistance from the supply to the consumer is cut inhalf. It is necessary for an equalizing reservoir tohave an elevation high enough to provide adequatepressure throughout the system served. For thecorrect hydraulic grade, it is necessary to build thereservoir above the area it serves. If the topographywill not allow a surface reservoir, a standpipe or anelevated tank must be constructed. Standard ele-vated tanks are available in capacities up to 2 MG.

21.52.2 Reservoir Trap Efficiency

The methods of Art. 21.35.2 for determining thequantities of sediment delivered to a reservoirrequire knowledge of the trap efficiency of thereservoir before the percentage of the incomingsilt that will remain to reduce storage can bedetermined. Studies of trap efficiency were madeby G. M. Brune, who developed a curve to expressthe relationship between trap efficiency and whathe called the capacity-inflow ratio for a reservoir

(Fig. 21.82) (G. M. Brune, “Trap Efficiency ofReservoirs,” Transactions of the American GeophysicalUnion, vol. 34, no. 3, June 1953).

The higher the capacity-inflow ratio, acre-feet ofstorage per acre-foot of annual inflow, the greaterthe percentage of sediment trapped in a reservoir.For any given storage reservoir, the trap efficiencydecreases with time since the capacity-inflow ratiodecreases as sediment builds up. The rate of siltingof a storage reservoir decreases when the capacityis reduced to an amount such that some spillage ofsilt-laden water occurs with each major storm. Thisrate decrease occurs because an increasing percen-tage of the annual suspended silt load is ventedbefore sedimentation can occur.

21.53 Wells

A gravity well is a vertical hole penetrating anaquifer that has a free-water surface at atmosphericpressure (Fig. 21.83). A pressure or artesian wellpasses through an impervious stratum into aconfined aquifer containing water at a pressuregreater than atmospheric (Fig. 21.84). A flowingartesian well is an artesian well extending into aconfined aquifer that is under sufficient pressureto cause water to flow above the casing head. Agallery or horizontal well is a horizontal or nearly

Fig. 21.83 Gravity well in a free aquifer.





horizontal tunnel, ditch, or pipe placed normal togroundwater flow in an aquifer.

21.53.1 Drawdown

When water is pumped from a well, the water levelaround the well draws down and forms a cone ofdepression (Fig. 21.83). The line of intersectionbetween the cone of depression and the originalwater surface is called the circle of influence.

Interference between two or more wells iscaused by the overlapping of circles of influence.Drawdown for each interfering well is increasedand the rate of water flow is decreased for eachwell in proportion to the degree of interference.Interference between two or more closely spacedwells may increase to the extent that the system ofwells produces one large cone of depression.

Since nearly all soils are heterogeneous, pump-ing tests should be made in the field to determinethe value of the hydraulic conductivity K. Apermeability analysis of a soil sample that is notrepresentative of the soil throughout the aquiferwould produce an unreliable value for K.

21.53.2 Flow From Wells

The steady flow rate Q can be found for a gravitywell by using the Dupuit formula:

Q ¼ 1:36K(H2 � h2)

log (D=d)(21:142)

where Q ¼ flow, gal/day

K ¼ hydraulic conductivity, ft/day under1 :1 hydraulic gradient

H ¼ total depth ofwater frombottomofwellto free-water surface before pumping, ft

h ¼ H minus drawdown, ft

D ¼ diameter of circle of influence, ft

d ¼ diameter of well, ft

The steady flow, gal/day, from an artesian well isgiven by

Q ¼ 2:73Kt(H � h)

log (D=d)(21:143)

Fig. 21.84 Artesian well in a pressure aquifer.





where t is the thickness of confined aquifer, ft(Fig. 21.84).

A long time elapses between the beginning ofpumping and establishment of a steady-flow con-dition (a circle of influence with constant diameter).Hence, correct values for drawdown and the circleof influence can be obtained only after long periodsof continuous pumping.

A nonequilibrium formula developed by Theisand a modified nonequilibrium formula producedby Jacob are used in analyzing well flow conditionswhere equilibrium has not been established. Bothmethods utilize a storage coefficient S and thecoefficient of transmissibility T to eliminate com-plications due to the time lag before reachingsteady flow. (C. V. Theis, “The Significance of theCone of Depression in Groundwater Bodies,”Economic Geology, vol. 33, p. 889, December 1938;C. E. Jacob, “Drawdown Test to DetermineEffective Radius of Artesian Well,” Proceedings ofthe American Society of Civil Engineers, vol. 72, no. 5,p. 629, 1940.) Computer software packages areavailable for analysis of groundwater flow withfinite-element models.

21.53.3 Excavation of Wells

Wells may be classed by the method by which theyare constructed and their depth. Shallowwells (lessthan 100 ft deep) are usually dug, bored, or driven.Deep wells (depth greater than 100 ft) are usuallydrilled by either the standard cable-tool, waterjet,hollow-core, or hydraulic rotary methods.

21.53.4 Well Equipment

Essential well equipment consists of casing, screen,eductor or riser pipe, pump (Art. 21.57), and motor.The casing keeps the wall material and pollutedwater from entering the well and prevents theleakage of good water from the well.

The screen is placed below the casing to containthe walls of the aquifer, to allow water to pass fromthe aquifer into the well, and to stop movement ofthe larger sand particles into the well. The pump,motor, and eductor pipe are utilized to move thewater from the aquifer to the collecting lines at theground surface.

(G. M. Fair, J. C. Geyer, and D. A. Okun, “Waterand Wastewater Engineering,” John Wiley & Sons,Inc., New York (www.wiley.com); T. J. McGhee,

“Water Supply and Sewerage,” 6th ed., McGraw-Hill, Inc., New York (books.mcgraw-hill.com).)

21.54 Water DistributionPiping

A water-distribution system should reliably pro-vide potable water in sufficient quantity and atadequate pressure for domestic and fire-protectionpurposes. To provide adequate domestic service,the pressure in the main at house serviceconnections usually should not be below 45 psi.But if oversized plumbing is provided, 35 psi isadequate. In steep hillside areas, the system isusually divided into several different pressurezones, interconnected with pumps and pressureregulators. Since each additional zone causesincreased expenses and decreased reliability, it isdesirable to keep their number to a minimum. TheAmerican Water Works Association has rec-ommended 60 to 75 psi as a desirable range forpressures; however, in areas of steep topographywhere local elevation differences may be over1000 ft, such a narrow range is not practical.

House plumbing is designed to withstand amaximum pressure of between 100 and 125 psi.When the pressure in distribution lines is above125 psi, it is necessary to install pressure regulatorsat each house to prevent damage to appliances,such as water heaters and dishwashers.

21.54.1 Water for Fire Fighting

Pressure requirements for fire fighting depend onthe technique and equipment used. Four methodsof supplying fire protection are:

1. Use of mobile pumpers which take water from ahydrant. This method is used in most largecommunities that have full-time, well-trainedfire departments. The required pressure in theimmediate area of the fire is 20 psi.

2. Maintenance of adequate pressure at all times inthe distribution system to allow direct connec-tion of fire hoses to hydrants. This technique iscommonly used in small communities that donot have a full-time fire department and mobilepumpers. The pressure in the distributionsystem in the vicinity of a fire should bebetween 50 and 75 psi.





3. Use of stationary fire pumps located at variouspoints in the distribution system, to boost thepressure during a fire and allow direct connec-tion of hoses to hydrants. This method is not soreliable or so widely used as the first two.

4. Use of a separate high-pressure distributionsystem for fire protection only. There are onlyrare instances in high-value districts of large citieswhere this method is used because the cost of adual distribution system is usually prohibitive.

21.54.2 Hydraulic Analysis ofDistribution Piping

Distribution systems are usually laid out on agridiron system with cross connections at variousintervals. Dead-end pipes should be avoidedbecause they cause water-quality problems.

Economic velocities are usually around 3 to4 ft/s, although during fires they can be muchhigher. Two- and four-inch-diameter pipe can beused for short lengths in residential areas; however,the American Insurance Association (AIA) requires6-in pipe for fire service in residential areas. Also,maximum length between cross connections islimited to 600 ft. In high value districts, the AIArequires an 8-in pipe, with cross connection at allintersecting streets. The AIA standards also requirethat gate valves be located so that no single case ofpipe breakage, outside main arteries, requiresshutting off from service an artery or more than500 ft of pipe in high valued districts, or more than800 ft in any area. All small distribution linesbranching from main arteries should be equippedwith valves. (“Standard Schedule for GradingCities and Towns of the United States with Ref-erence to Their Fire Defenses and Physical Condi-tions,” American Insurance Association.)

Adequate service requires a knowledge of thehydraulic grade at many points in a distributionsystem for various flows. Several methods, basedon the following rules, have been developed foranalysis of complex networks:

1. The head loss in a conduit varies as a power ofthe flow rate.

2. The algebraic sum of all flows into and out ofany pipe junction equals zero.

3. The algebraic sum of all head losses betweenany two points is the same by any route, and the

algebraic sum of all head losses around a loopequals zero.

A convenient device for simplifying complexnetworks of various size pipes is the equivalent pipe.For a series of different size pipes or severalparallel pipes, one pipe of any desired diameterand one specific length or any desired length andone specific diameter can be substituted; this willgive the same head loss as the original for all flowrates if there are no take-outs or inputs betweenthe two end points. In complex networks, theequivalent pipe is used mainly to simplify calcu-lation.

Example 21.10: Determine the equivalent 8-in-diameter pipe that will have the same loss of headas the sections of pipe from A to D in Fig. 21.85a.

First, transform pipes CD, AB, and BD intoequivalent lengths of 8-in pipe; then, transform theresulting sections into a single 8-in pipe with thesame head loss. The head loss may be calculatedfrom Eqs. (21.34d).

Assume any convenient flow through CD, say500 gal/min. Equation (21.34d) indicates that loss of

Fig. 21.85 Distribution loop (a)may be replacedby equivalent loop (b).





head in 1000 ft of 6-in pipe is 32 ft and in 1000 ft of8-in pipe, 7.8 ft. Then, the equivalent length of 8-inpipe for CD is 500 � 32/7.8 ¼ 2050 ft. Similarly,the equivalent pipe for AB should be 165 ft long,and for BD, 420 ft long. The network of 8-in pipe isshown in Fig. 21.85b. It consists of pipe 1,3000 þ 2050 ¼ 5050 ft long, connected in parallelto pipe 2, 165 þ 420 ¼ 585 ft long.

To reduce the parallel pipes to an equivalent 8-inpipe, assume a flow of 1000 gal/min through pipe2. For this flow, the head loss in an 8-in pipeper 1000 ft is 29 ft. Hence the head loss in pipe 2would be 29 � 585/1000 ¼ 17 ft. Since the pipes areconnected in parallel, the head loss in pipe 1 alsomust be 17 ft, or 3.37 ft/1000 ft. The flow that willproduce this head loss in an 8-in pipe is 310 gal/min[Eq. (21.34c)]. The equivalent pipe, therefore, mustcarry 1000 þ 310 ¼ 1310 gal/min with a head lossof 17 ft. For a flow of 1310 gal/min, an 8-in pipewould have a head loss of 48 ft in 1000 ft, accordingto Eq. 21.34d. For a loss of 17 ft, an 8-in pipe wouldhave to be 1000 � 17/48 ¼ 350 ft long. So the pipesbetween A and D in Fig. 21.85a are equivalent toa single 8-in pipe 350 ft long.

Pipe Network Equations n For hydraulicanalysis of a water distribution system, it is con-venient to represent the network by a mathematicalmodel. Generally, it is useful to include in themodel only the major elements needed for amathematical description of the basic network. (Formodels that are to be used for such conditions aslow pressures in a small service region, however, itmay be necessary to include all the distributionmains in the system.) The three analysis rules onp. 21.110 can then be used to develop a system ofsimultaneous equations that can be solved for flowand pressure in the network.

Typically, either the Darcy-Weisbach or theHazen-Williams formula is used to relate thecharacteristics of each pipe in the system. Conse-quently, the equations for each pipe are nonlinear.As a result, a direct solution generally is notavailable. In practice, the equations are solved byan iteration process, in which the values of somevariables are assumed to make the equations linearand then the equations are solved for the othervariables. The initial assumptions are corrected andused to develop new linear equations, which aresolved to obtain more accurate values of thevariables.

One example of this technique is the HardyCross method, a controlled trial-and-error method,which was widely used before the advent ofcomputers. Flows are first assumed; then consecu-tive adjustments are computed to correct theseassumed values. In most cases, sufficient accuracycan be obtained with three adjustments; however,there are rare cases where the computed adjust-ments do not approach zero. In these cases, anapproximate method must be used.

Assumed flows in a loop are adjusted in accor-dance with the following equation:

DQ ¼ SKQn

SnKQn�1(21:144)

where KQn ¼ hf ¼ loss of head due to friction.When the Hazen-Williams equation, used inExample 21.10, is put in the form hf ¼ KQn thenK ¼ 4:727L=D4:87C1:85

1 and n ¼ 1.85. The expres-sion SnKQn21 equals S(nKQn/Q). In the Hazen-Williams formula n ¼ 1.85 for all pipes and cantherefore be taken outside the summation sign.Hence, the adjustment equation becomes

DQ ¼ Shf

nS(hf =Q)(21:145)

It is important that a consistent set of signs beused. The sign convention chosen for the followingexample makes clockwise flows and the lossesfrom these flows positive; counterclockwise flowsand their losses are negative.

Approaches generally used for formulating theequations for analysis of a water distributionnetwork include the following:

Flow method, in which pipe flows are theunknowns.

Node method, in which pressure heads at thepipe end points are the unknowns.

Loop method, in which the energy in eachindependent loop is expressed in terms of the flowsin each pipe in the loop. In turn, the actual flow ineach pipe is expressed in terms of an assumed flowand a flow correction factor for each loop.

Computer software packages are available foranalysis of networks by such methods. They canperform not only steady-state analyses of pressu-res and flows in pipe networks but also time-dependent analyses of pressure and flow underchanging system demands and of flow patternsand basic water quality.





(V. J. Zipparo and H. Hasen, “Davis’ Handbookof Applied Hydraulics,” McGraw-Hill, Inc.,New York (books.mcgraw-hill.com); AWWA, “Dis-tribution Network Analysis for Water Utilities,”Manual of Water Supply Practices M32, AmericanWater Works Association, Denver, Colo. (www.awwa.org); T. M. Walski, “Analysis of WaterDistribution Systems,” Van Nostrand Reinhold,New York.)

21.54.3 Cover over Buried Pipes

The cover required over distribution pipesdepends on the climate, size of main, and traffic.In northern areas, frost penetration, which may beas deep as 7 ft. is usually the governing factor. Infrost-free areas, a minimum of 24 in is required bythe AIA. If large mains are placed under heavytraffic, the stress produced by wheel loads shouldbe investigated.

21.54.4 Maintenance ofWater Pipes

Maintenance of distribution systems involveskeeping records, cleaning and lining pipe, findingand repairing leaks, inspecting hydrants andvalves, and many other functions necessary toeliminate problems in operation. Valves should beinspected annually and fire hydrants semiannually.Records of all inspections and repairs shouldbe kept.

Unlined distribution pipes, after years of usage,lose much of their capacity because of corrosionand incrustations. Cleaning and lining with cementmortar restores the original capacity. Dead-endpipes should be flushed periodically to reduce theaccumulation of rust and organic matter.

21.54.5 Economic Sizing ofDistribution Piping

When designing any major project, the designershould choose the most economical of numerousalternatives. Most of these alternatives can beseparated and studied individually. An example oftwo alternatives for a distribution system is oneserving peak hourly demands totally by pumpsand one doing it by pumps and equalizingreservoirs. The total costs of each plan should becompared by an annual or present-worth costanalysis.

A method of determining minimum cost thatcan readily be adapted tomany conditions is settingthe first derivative of the total cost, taken withrespect to the variable in question, equal to zero. Inthe sizing of pipes in a distribution system suppliedby pumps, the total costs of the pipes, pumpingplant, and energymay be expressed as an equation.To find the most economical diameter of pipe, thefirst derivative of the total cost, taken with respectto the pipe diameter, should be set equal to zero.The following equation for the most economicalpipe diameter was derived in that manner:

D ¼ 0:215fbQ3

aS

aiHa

� �1=7

(21:146)

where D ¼ pipe diameter, ft

f ¼ Darcy-Weisbach friction factor

b ¼ value of power, dollars/hp per year

Qa ¼ average discharge, ft3/s

S ¼ allowable unit stress in pipe, psi

a ¼ in-place cost of pipe, dollars/pound

i ¼ yearly fixed charges for pipeline(expressed as a fraction of total capitalcost)

Ha ¼ average head on pipe, ft

21.54.6 Pipe Materials

Cast iron, steel, concrete, and plastics, such aspolyvinyl chloride, polyethylene, polybutylene,and glass-fiber-reinforced thermosetting resins, arethe most common materials used in distributionpipes. Wood pipelines are still in existence, butwood is rarely used in new installations. Copper,lead, zinc, brass, bronze, and plastic are materialsused in small pipes, valves, pumps, and otherappurtenances. Common pipe-joint materials are:cement, sand, rubber, plastic, and sulfur com-pounds.

Cast iron is the most common material for citywater mains. Standard sizes range from 2 to 24 in indiameter. Cast iron is resistant to corrosion andusually has good hydraulic characteristics. If it iscement-lined, the Hazen-Williams C value may beas high as 145. In unlined pipes, however, irontubercles may form and seriously affect flowcapacity. Tuberculation can be prevented by liningwith cement or tar materials. The relatively highcost of cast-iron pipe is only a slight disadvantage,





largely offset by the long average life of trouble-freeservice. Bell-and-spigot and flange are the mostcommon joints in cast-iron pipe.

Steel is commonly used for large pipelines andtrunk mains but rarely for distribution mains. Steelpipes with either longitudinal or spiral jointsare formed at steel mills from flat sheets. Thetranverse joints between pipe sections are usuallymade by welding, riveting, bell-and-spigot withrubber gasket, sealed flanges, or Dresser-type coup-lings. Since steel is stronger than iron, thinner andlighter pipes can be used for the same pressures.Some disadvantages of thin steel pipe are inability tocarry high external loads, possibility of collapse dueto negative gage pressures, and high maintenancecosts due to higher corrosion rates and thinner pipewalls. Steel pipes are usually corrosion-protected onboth the outside and inside with coal tar or cementmortar. Under favorable conditions, the life of steelpipe is between 50 and 75 years.

Concrete pipe may be precast in sections andassembled on the job or cast in place. A machinethat produces a monolithic, jointless concrete pipewithout formwork has been developed for gravity-flow and low-pressure applications. Most of theprecast-concrete pipe is reinforced or prestressedwith steel. Concrete pipe may be made watertightby insertion of a thin steel cylinder in the pipewalls.High-strengthwire is frequently wound around thethin steel cylinder for reinforcement. Concrete isplaced inside and outside the steel cylinder toprevent corrosion and strengthen the pipe. Someadvantages of concrete pipe are low maintenancecost, resistance to corrosion under normal con-ditions, low transportation costs for materials ifwater and aggregate are available locally, andability to withstand external loads. Some disad-vantages to be considered are leaching of free limefrom the concrete, the tendency to leak underpressure due to the cracking and permeability ofconcrete, and corrosion in strong acids or alkalies.

21.55 Corrosion in WaterDistribution Systems

Many millions of dollars are expended every yearto replace pipes, valves, hydrants, tanks, andmeters destroyed by corrosion. Some causes ofcorrosion are the contact of two dissimilar metalswith water or soil, stray electric currents, impu-rities and strains in metals, contact between acids

and metals, bacteria in water, or soil-producingcompounds that react with metals.

Electrochemical corrosion of a metal occurswhen an electrolyte and two electrodes, an anodeand a cathode, are present. (Water may serve as anelectrolyte.) At the anode, the metal in contact withthe electrolyte changes into a positively chargedparticle, which goes into solution or forms an oxidefilm. (The ease with which a metal changes to ametallic ion when it is in contact with waterdepends on its oxidation potential or solutionpressure. Metals can be arranged in an electro-motive series of decreasing oxidation potentials.Metals high in the electromotive series corrodemore readily than metals located in a lowerposition.) For an iron pipe exposed to water, forexample, the anode reaction is Fe (metal) !Fe2þ þ 2e, where e is an electron. At the cathode,the metal having the excess electrons gives them upto a charged particle, such as hydrogen in solution:2Hþ þ 2e ! H2 (gas). If the hydrogen gas pro-duced at the cathode is removed from thecathode by reaction with oxygen to produce watermolecules or by water movement (depolarization),the corrosion process continues (Fig. 21.86). Indi-cations of corrosion in an inaccessible iron or steelpipeline are discharges of rusty-colored water (dueto the loosening of rust and scale) and metallic-tasting water.

A marked decrease in capacity and pressure in apipe section usually indicates tuberculation insidethe line. Tuberculation is caused by the depositionand growth of insoluble iron compounds inside apipe. Iron-consuming bacteria in water can pro-duce ferrous oxide directly if the iron concentrationis about 2 ppm. A continuous supply of solubleiron in the presence of iron-consuming bacteria ordissolved oxygen and basic substances in the waterincreases the size of the tubercles. Tubercles maybecome so large and decrease the capacity in thepipe to such an extent that it has to be cleaned orreplaced.

Several factors influence the type and quantityof metallic corrosion:

Presence of protective films. Some metals formoxide films that act as protective layers for themetal. Aluminum, zinc, and chromium are exam-ples of this type of metal.

Strains, cracks, and undissolved impurities in ametal act as sites for corrosion.

Agitation or movement of water increases thecorrosion rate of ametal because the oxygen supply





rate to the cathode and the removal rate of metalions from the anode are increased. The presence ofionic compounds in the water speeds up corrosionbecause the ions act as conductors of electricity,and the more ions, the faster electrons can movethrough the water.

Alternate wetting and drying tends to breakup the rust or oxide film, thus facilitating penetra-tion of the film by oxygen and water and lead toincreased corrosion.

High hydrogen-ion concentrations increasecorrosion rates because of the greater accessibilityof the hydrogen ions to the cathode.

Corrosion may be prevented or retarded byproper selection of materials, use of protectivecoatings, and treatment of the water. Whenselecting materials, the engineer should take intoaccount the characteristics of the water and soilconditions encountered. Protective coatings formetals may be metallic or nonmetallic and appliedon both the inside and outside surfaces of the pipe.Common nonmetallic coatings are cement and

asphalt. Zinc is an example of metallic coatingmaterials used. Steel pipe dipped in zinc (galva-nized) or copper tubing is commonly used forsmall service lines.

Also, to prevent corrosion, water may be treatedwith bases, such as soda ash, caustic soda, andlime, to reduce hydrogen-ion concentration and toinduce precipitation of thin films of carbonates,hydroxides, oxides, and so on on the walls of thepipes. These thin films reduce the ability of water tocorrode otherwise unprotected metal surfaces.Corrosion, however, normally precedes depositionof scale because iron must be in solution to reactwith the basic substances and dissolved oxygen inthe water to form scale.

Electrochemical corrosion of external surfaces ofpipelines and water tanks can be retarded byapplication of a direct current to the metal to beprotected and to another metal that acts as asacrificial anode (Fig. 21.87). The potential appliedto or produced by the two metal surfaces must belarge enough to make the protected metal act as acathode. The sacrificial anode corrodes and mustbe replaced periodically. Zinc, magnesium, graph-ite, and aluminum alloys are commonly used foranode materials.

(American Water Works Association, “WaterQuality and Treatment,” 4th ed., McGraw-Hill,Inc., New York.)

21.56 Centrifugal Pumps

The purpose of any pump is to transformmechanical or electrical energy into pressure

Fig. 21.86 Electrochemical corrosion of iron in low-pH water.

Fig. 21.87 Cathodic protection of a metal.





energy. The centrifugal pump, the most commonwaterworks pump, accomplishes that in two steps.The first transforms the mechanical or electricalenergy into kinetic energy with a spinning ele-ment, or impeller. The kinetic energy is thenconverted to pressure energy by diffuser vanes ora gradually diverging discharge tube, called avolute (Fig. 21.88).

Water enters at the center, or eye, of the impeller and is forced outward toward the casing bycentrifugal force. The discharge head of a centrifu-gal pump is a function of the impeller diameter andspeed of rotation.

Design factors requiring consideration in theselection of a centrifugal pump are net positivesuction head required, efficiency, horsepower, andthe head-discharge relationship.

Net positive suction head (NPSH) is the energyin the liquid at the center line of the pump. To havepractical meaning, it must be referred to as eitherthe required or available NPSH. Required NPSH isa characteristic of the pump and is given by themanufacturer. Available NPSH is a characteristic of

the system and is determined by the engineer. It isthe pressure in the liquid over and above its vaporpressure at the suction flange of the pump and isgiven, in feet, by

Available NPSH ¼ 144pa � pn

w� hf þ z (21:147)

Fig. 21.88 Volute-type centrifugal pump.

Fig. 21.89 Curves used in selection of a centrifugal pump.





where pa ¼ pressure, psia, on free-water surface orat center line of closed conduit

pn ¼ vapor pressure, psia, of water at itspumping temperature

hf ¼ friction loss in suction line, ft of water

z ¼ elevation difference, ft, between pumpcenter line and water surface

w ¼ unit weight of liquid, lb/ft3

If the suction water surface is below the pumpcenter line, z is negative. To prevent cavitation, it isnecessary to have the available NPSH alwaysgreater than the required NPSH. For that reason, itis customary to analyze a required NPSH vs.discharge curve with the brake horsepower, head,and efficiency curves when selecting a pump.

The operating point of a centrifugal pump isdetermined by the intersection of the pump’s head-capacity curve and the system head curve, as shownin Fig. 21.89. (Also included in Fig. 21.89 are theother curves used in pump selection.) A systemhead curve is a plot of the head losses in the systemvs. pump discharge. This curve shows the headdifferential that must be supplied by the pump. In atypical water-system analysis, there may be threeor four pertinent system head curves correspond-ing to various consumption rates. The intersectionof these curves with the head vs. Q curve define arange of operation rather than a single point.

Selection of a centrifugal pump is largely amatter of matching one of the many pumpsavailable to the system characteristics. An import-ant consideration is that the point of maximumefficiency should be at or near the operating point.Centrifugal pumps are available in almost anycapacity desired, with lifts of up to 700 ft per stage.Efficiencies may be as high as 93% for large pumps.

See also Art. 21.57 and check valves in Art.21.58.

(I. J. Karassik et al., “PumpHandbook,” 2nd ed.,McGraw-Hill Book Company, New York (books.mcgraw-hill.com).)

21.57 Well Pumps

These are classified as centrifugal, propeller, jet,helical, rotary, reciprocating, and air lift. Althoughcentrifugal pumps (Art. 21.56) are the mostcommon for both shallow-well and deep-well

pumps, circumstances may dictate one of the othertypes.

Centrifugal pumps are used in wells over 6 inin diameter. They have capacities up to 4000 or5000 gal/min and heads up to 1200 ft, dependingon the number of stages. Efficiencies may be ashigh as 90% for the larger capacities; however,below 200 gal/min, the maximum efficiency is 75 to80%.

Propeller pumps are an axial-flow type. Theyare used in high-capacity low-head applications.

Jet pumps (Fig. 21.90) operate by dischargingwater through a nozzle and diverging conical tube,which are located at the well bottom. The diverg-ing conical tube creates lift by converting the high-

Fig. 21.90 Section through a jet pump (simp-lified).





velocity head to pressure head. The suctionconnection is made between the nozzle and en-trance to the diverging tube. Jet pumps have lowefficiencies. They are used in small-capacity low-lift applications, especially where the watercontains sand or other impurities.

Helical pumps are a positive-displacement typewith a metal helical rotor rotating inside a rubberhelical stator. The screw action of the rotor forceswater through the pump and up the dischargepipe. Helical pumps are small-capacity high-liftpumps. They may be used in wells over 4 in ininside diameter.

Rotary pumps are also of the displacement type.They have a fixed chamber in which gears, vanes,cams, or pistons rotate with very close tolerances.These pumps have relatively constant partial-loadefficiencies. Full-load efficiencies range from 50 to85%. Because of the close tolerances, they can beused only for sediment-free water.

Reciprocating pumps, either hand- or motor-driven, utilize piston action to move water. Theirpresent-day use is primarily for small-capacitylow-lift private applications.

Air-lift pumps generate lift by using air bubblesto decrease the specific weight of the column ofwater in the discharge pipe below that of thesurrounding water in the well and create a pressuredifferential that forces the water out of the well.Air-lift pumps are the simplest and most foolproofof well pumps since they have no submergedmoving parts. They can be used in any well buthave the disadvantage of efficiencies below 50%.

Specific speed Ns is a widely used criterion forpump selection. It is the impeller speed corre-sponding to a discharge of 1 gal/min at 1 ft of headfor the most efficient design.

Ns ¼ nQ1=2h�3=4 (21:148)

where n ¼ impeller speed, r/min

Q ¼ discharge, gal/min

H ¼ head, ft

The favorable design range of Ns for radial-flow(centrifugal) pumps is from 1500 to 4100. For Ns

between 4100 and 7500, mixed-flow pumps havingboth radial and axial characteristics should beused, and for Ns above 7500, axial-flow (propeller)pumps should be used.

Shallow-well pumps have their motors andimpellers at ground level, so that the entire lift is

suction. Since excessive suction lifts cause cavita-tion, the lift is limited by atmospheric pressure andthe velocity head at the impeller, which is a functionof specific speed. At sea level, the maximumpractical lift for a shallow-well pump is about 25 ft.

Deep-well pumps have their impellers closeenough to the water surface to eliminate cavitation.The motor may be at ground level with a long shaftconnecting it to the impellers, or it may be at thebottom of the well, below and directly adjacent tothe impellers. The former type is called a deep-wellturbine pump and the latter a submersible pump.Deep-well turbine pumps can be used only forstraight wells. The pump shaft is supported atintervals of about 10 ft by rubber or metal bearings,which are water- or oil-lubricated, respectively. Ifsand is carried out with the water, an enclosed-shaft or submersible pumpmust be used to preventbearing damage. Submersible pumps may be usedin crooked wells. Other advantages include ease ofincreasing the well depth or lift and silent op-eration. One disadvantage is that the motors aredifficult to reach for repairs.

(I. J. Karassik et al., “PumpHandbook,” 2nd ed.,McGraw-Hill Book Company, New York (books.mcgraw-hill.com).)

21.58 Valves

Water facilities use many different types of valves.These are generally classified according to thefunction they perform. The two major water-valveclassifications are isolating and controlling.

Isolating valves are used for separating orshutting off sections of pipe, pumps, and controldevices from the rest of the system for inspectionand repair purposes. The major types of isolatingvalves are gate, plug, sluice gate, and butterfly.

A control valve is normally used for continu-ously controlling pressures and flow rates. Check,needle, globe, air-relief, pressure-regulating, pres-sure-relief, and altitude valves are usually con-sidered as control valves.

Gate valves are the isolating valves used mostoften in distribution systems, primarily because oftheir low cost, availability, and low head loss whenfully open. They have limited value as control orthrottling devices because of seat wear and thedownstream deflection and chatter of the gate disk.Also, the open area and rate of flow through thevalve are not proportional to the percentage





opening of the valve when partly open. Corrosion,solids deposition, tubercle formation, large press-ure differences, and thermal expansion producedifficulties in opening normally closed gate valvesor in closing normally open gate valves. Periodicinspection and operation of valves that areinfrequently operated will prevent many oper-ational difficulties. Some of the larger gate valveshave gear-reduction drives to permit manualoperation. Very large valves are operated byhydraulic and electric power.

A plug valve may be used for both control andisolation purposes. It consists of a cylindricallyshaped plug (with a rectangular slot or circularorifice) placed in a close-fitting cylindrical seatperpendicular to the direction of flow. Cone andspherical valves are special types of plug valves.Plug, cone, and spherical valves can all be fullyclosed or opened by a 908 rotation of the plug. Thevalves may or may not be lubricated (large ironvalves usually are). Hydraulic or electric power iscommonly utilized for operating the larger valves.Small plug valves are commonly used for isolationpurposes on domestic and commercial serviceconnections and are known as service, curb, orcorporation cocks. Usually, because the meter is notdirectly adjacent to the distribution pipe, threevalves must be used, one at the service connection,one just upstream of the meter, and one betweenthe meter and the customer’s service line. Plug andcone valves are also used for throttling and remote-control shutoff. Low head loss, in-service lubrica-tion features, and easy, fast operation, even in thepresence of unequal pressures across the valve,are the major advantages of plug-type valves. Butthese valves cost more than gate, globe, and but-terfly valves.

Butterfly valves can be used for throttling andisolation purposes. The butterfly-valve mechanismconsists of a relatively thin circular disk pivoted ona horizontal shaft. Hand or motor power, appliedthrough a gear-reduction device, rotates the disk.Simplicity of construction and quick, easy oper-ation are reasons why these valves are replacingsluice gates and gate valves in many locations. LosAngeles has replaced many sluice gates in reser-voir towers with butterfly valves having seats ofcorrosion-resistant metal, rubber, or Neoprene. Adisadvantage of butterfly valves is the higher costrelative to sluice gates or gate valves.

Sluice gates are mainly used on the sides ofreservoir control towers and in open-channel

structures where pressure on one side of thegate helps to seat it and prevent water leakage.Difficulties with leakage and corrosion of gateframes and stems are the main disadvantages ofsluice gates. Low cost and ease of operation inopen-channel flow conditions are the majoradvantages.

A needle valve is made of a streamlined plug orneedle that fits into a small orifice with a carefullymachined seat. Needle valves are used for accuratecontrol of water flow because a large movement ofthe needle is necessary before any measurablechange of flow rate takes place. Needle valves arenot normally used for isolating purposes becauseof the high head losses produced by water flowthrough the small orifices. Large-sized needlevalves are used for flow regulation under highheads, such as for free discharge from reservoirs.Interior-differential, tube, and hollow-jet are threeof the most common types of large needle valves.

Globe valves are commonly used in smallersizes for domestic purposes. The valve mechanismconsists of a screw-operated disk that is forceddown on a circular seat. Because of high headlosses, globe valves are rarely used for isolationpurposes, but they are commonly used for pressureregulation in water-distribution systems. Manyautomatic control valves, such as pressure regula-tors and altitude, check, and relief valves, haveglobe-valve bodies with various types of controlmechanisms.

Pressure-regulating valves are used to reducepressures automatically. An air-relief and inletvalve serves the dual purpose of allowing air toeither escape or enter a pipeline. Air that accu-mulates at high points in a pipe impedes waterflow and should be allowed to escape through anair-relief valve placed at this location. Furthermore,draining water from low elevations in a pipelinemay cause negative pressures at higher elevationsand collapse a pipe. Air should be allowed to enterthrough air-relief and inlet valves at the high pointsto prevent this.

Pressure-relief valves are used to release excesspressure in an enclosure. Often, these excess pres-sures are caused by sudden closure of a valve.

Altitude valves are used to control the waterlevel of elevated reservoirs. A pressure-activatedcontrol closes the altitude valve when the tank isfull and opens the valve to allow water to flowfrom the tank when pressure below the valvedecreases.





Check valves are used in pipelines to allow forone-directional flow only. Check valves placed incentrifugal-pump suction lines are called footvalves. These valves hold water in the suction lineand pump case so that the pump will not needmanual priming when started. The most commoncheck valve is the swing type.

21.59 Fire Hydrants

A fire hydrant normally consists of a cast-iron bar-rel and a gate or compression-type shutoff valve,which connects the barrel to the main. Two or morehose outlets are normally located in the barrelabove the ground surface. Usually, an additionalgate valve is required between the hydrant and themain to allow for shutoff and repair of the hydrant.

The number of 21⁄2 -in-diameter hose outlets ona hydrant determines its class. For example, ahydrant with two hose outlets is called a two-wayhydrant.

Fire-hydrant construction standards have beenestablished by the American Water Works Associ-ation and the American Insurance Association.These standards relate the diameter of the barrel tothe size of the main-valve opening. A barreldiameter of at least 4 in is required for a two-wayhydrant, 5 in for a three-way hydrant, and 6 in for afour-way hydrant. A minimum of two hose outletsis required on a fire hydrant.Where pumper serviceis necessary for adequate water pressure, a largepumper outletmust be furnished. Thismay take theplace of one of the smaller 21⁄2-in hose outlets. Theminimum allowable diameter for the pipe connec-tion between the main and the hydrant is 6 in.

Fire hydrants usually are either dry or wet bar-rel, depending on the location of the main valve inthe hydrant. The main valve in the dry-barrel typeshould be located below the frost line. When thevalve is in a closed position, a drain should be opento prevent freezing of water in the barrel. The wet-barrel, or California type, hydrants have the mainvalve located near the hose outlets. Many firehydrants have a safety joint above the groundsurface to permit removal of the upper part of thebarrel with a minimum loss of water.

Hose connections 31⁄16 in in diameter with71⁄2 threads per inch have been selected by theAmerican Insurance Association as standard toallow for interchange of fire-fighting equipmentbetween cities.

Friction losses should not exceed 21⁄2psi in ahydrant and 5 psi between the main and outletwhen flow is 600 gal/min.

21.60 Metering Devices

Metering devices are classified as either velocity ordisplacement types. Velocity types measure thevelocity of flow either directly by current-measur-ing devices or indirectly by venturi-principledevices and are usually calibrated to indicate theflow rate directly. The velocity-type metering de-vices are applied to measurement of flows instreams, rivers, and large pipes, such as trunk linesof distribution systems. Displacement-type meter-ing devices indicate flow rate directly, by recordingand integrating the rate at which their measuringchambers are filled and emptied. Weighing metersare also displacement-type metering devices, butthey are used primarily in laboratories. Displace-ment types are used for the smaller flows indistribution systems, such as meters for individualcustomer connections.

Criteria for selection of a type of water meterinclude accuracy and range of measurement,amount of head loss through the meter, durability,simplicity and ease of repairs, and cost.

Velocity-Type Metering Devices n Venturimeters, or modifications thereof, are the mostcommon velocity-type devices. These meters pro-duce a regular and predictable fall in the hydraulicgrade line that is related to flow rate. Three devicesthat operate on this principle are the venturi,nozzle, and orifice plate meters shown in Fig. 21.91.

Straightening vanes are installed upstream fromthese and other velocity-type meters if the pipe is ofinsufficient length to eliminate helical flow com-ponents caused by bends or other fittings.

The standard venturi meter (Fig. 21.91a) wasdeveloped to provide a device with minimum headloss. Since most of the loss is associated with thediffuser section, its angle is the major factor indetermining the head loss.

Flow through a venturi meter is given by

Q ¼ cKd22ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih1 � h2

p(21:149)

K ¼ 4

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g

1� (d2=d1)2

s(21:150)





where Q ¼ flow rate, ft3/s

c ¼ empirical discharge coefficient depen-dent on throat velocity and diameter

d1 ¼ diameter of main section, ft

d2 ¼ diameter of throat, ft

h1 ¼ pressure in main section, ft of water

h2 ¼ pressure in throat section, ft of water

(For values of c and K for various throat diametersand velocities, see E. F. Brater, “Handbook ofHydraulics,” 6th ed., McGraw-Hill Book Company,New York (books.mcgraw-hill.com).)

As in venturi meters, flows through nozzle andorifice-plate meters are calculated from the press-ure difference across the meters. Nozzle andorifice-plate meters are used where conservationof head is not the prime concern or where headdissipation is desired.

Current meters consist of either a propeller or aseries of cups or vanes mounted on a shaft free torotate under the action of the flowing water. Thepropeller type has its axis of rotation horizontaland will not give accurate measurement unless thecurrent velocity is parallel to the axis of rotation.The cup-type meter, called a Price meter, has avertical axis of rotation and measures currents

Fig. 21.91 Venturi-type metering devices: (a) Standard venturi meter. (b) Nozzle meter. (c) Orifice-plate meter.





whose velocity is in any direction in a horizontalplane. However, vertical velocity components,which do not affect propeller meters, cause thePrice meter to indicate greater-than-actual velo-cities. A clicking noise, made by the making andbreaking of an electrical contact and picked up by aset of earphones, indicates the speed of rotation ofthe meter. The clicking noise occurs either onceeach revolution or once each five revolutions.Current meters are used almost exclusively forstream flow, although the propeller type is occa-sionally substituted for a venturi meter in pipeflow.

Displacement-TypeMeters n These may bepiston, rotary, or nutating-disk types. The nutatingdisk is used, almost to the exclusion of the twoother types, for metering domestic-service connec-tions. Its widespread use stems from its simplicityof construction and long-term accuracy. Thenutating-disk meter derives its name from thedisk’s nodding motion, which is similar to that of atop before it stops. The disk is kept in motion bysuccessive volumes of water which enter aboveand below it. A hard rubber that softens at hightemperature is usually used for the disk, so abackflow-prevention device is required between anutating-disk meter and a water heater. Error ofnutating-disk meters is about 1.5% within thenormal test-flow limits.

Compound meters contain separate measuringdevices for both low and high flows. They areusually a nutating-disk meter and a propeller-typecurrent meter, respectively. An automatic pressure-sensing device directs the flow through the appro-priate meter.

21.61 Water Rates

The interests of the public and individual custo-mers of water-supply systems can best be servedby self-sustained, utility-type enterprises. Ratescharged to finance these systems should be basedon sound engineering and economic principles anddesigned to avoid discrimination between classesof customers. Gross revenue should cover operat-ing and maintenance expenses, fixed charges oncapital investment, and development of the system.Billings for water should be based on metered use

and such fixed charges as are required. Ratestructures are typically based on demand, load fac-tors, fire use, peak rates of use, seasonal use, andsimilar items. The system of accounting shouldconform to the legally established system of ac-counting prescribed for the utility, if any, or to someother recognized system.

Rates most commonly used today are flat rate,step rate, and block rate.

Flat rate is a monthly or quarterly charge thatdoes not vary with the amount of water used. Thistype of charge tends to encourage waste. Althoughit has been commonly employed in small commu-nities where water is not metered, flat rate is fallinginto disuse.

With step rate, customers are charged at one rateper 1000 gal for all water used. The rate a customerpays decreases as the total quantity used increases.The major objection to this method is that a cus-tomer who uses a quantity slightly less than thepoint of rate change will pay more than the cus-tomer who uses a little more.

The block rate schedule consists of one price per1000 gal for the first volume or block of water usedper billing period and lesser rates for additionalblocks. This type of pricing tends to discouragewaste but does not restrict usage unnecessarily.Both the step and block rates can have a monthlyservice charge.

When fixing a system of rates, the suppliershould consider the following factors: (1) cost ofcollection facilities, treatment chemicals, pumpingenergy, and, where applicable, buying water from awholesale supplier; (2) cost of distribution andtreatment facilities; and (3) cost, including meter-ing and billing, of serving an individual customer.Cost component 1, called the commodity com-ponent, is directly dependent on total usage andtherefore should be distributed equally to all watersold. Cost component 2, called the demand compo-nent, depends on the peak usage of a customer. If acustomer’s usage is zero during peak hour, it willnot appreciably affect the cost or design of distri-bution facilities. Since peak-hour demands usuallygovern the design of a distribution system, this is agood criterion for allocating distribution costs. It isgenerally recognized that residential areas, wherethe majority of small users are, have very highratios of peak demand to total usage and shouldtherefore pay a major share of the demand com-ponent. Both the step and block rates attempt toallocate this cost to the small user by charging a





higher rate for the first water sold to a customerand charging decreasing rates with increasedusage. For most distribution systems, a large shareof the demand component also should be allocatedto fire service. The portion attributed to fire serviceis usually paid by taxes. Cost component 3, calledthe customer component, is usually distributed tothe customer by a monthly service charge thatdepends only on the size of service. This charge isusually small.

Hydroelectric Power and Dams

Hydroelectric plants, which generate electricpower from water dropping a sufficient verticaldistance to drive large hydraulic turbines, supplyan appreciable portion of the electric powerconsumed in the U.S. Hydroelectric generation isan attractive power source because it is a renewableresource and a nonconsumptive use of water. Atypical hydroelectric plant consists of a dam todivert or store water from a river or stream; canals,tunnels, penstocks, and a forebay to convey waterto turbines; draft tube, tunnel, or tailrace to returnwater downstream to the river or stream; turbinesand governors; generators and exciters; equipmentsuch as protective devices and regulators; a build-ing to house the machinery and equipment; andtransformers, switching equipment, and powertransmission lines to deliver the power producedto a load center for distribution to consumers.

21.62 Hydroelectric-PowerGeneration

Hydroelectric power is electrical power obtainedfrom conversion of potential and kinetic energy ofwater. The potential energy of a volume of water isthe product of its weight and the vertical distance itcan fall:

PE ¼ WZ (21:151)

where PE ¼ potential energy

W ¼ total weight of the water

Z ¼ vertical distance water can fall

Because the kinetic energy of the supply source isvery small or zero inmost hydropower (hydroelectricpower) developments, the kinetic-energy termdoesnot appear in power formulas.

Power is the rate at which energy is produced orutilized.

1 horsepower (hp) ¼ 550 ft � lb=s1 kilowatt (kW) ¼ 738 ft � lb=s

1 hp ¼ 0:746 kW

1kW ¼ 1:341 hp

Power obtained from water flowmay be computedfrom

hp ¼ hQwh

550¼ hQh

8:8(21:152a)

kW ¼ hQwh

738¼ hQh

11:8(21:152b)

where kW ¼ kilowatts

hp ¼ horsepower

Q ¼ flow rate, ft3/s

w ¼ unit weight of water ¼ 62.4 lb/ft3

h ¼ effective head ¼ total elevation differ-ence minus line losses due to frictionand turbulence, ft

h ¼ efficiency of turbine and generator

Hydroplants can be classified on a basis ofreservoir capacity and use as run-of-river hydrowithout storage, base-load plants, run-of-riverplants with storage, and peak-load plants.

Run-of-River Hydro without Storage n

This type of plant has no storage facilities. Powergeneration is totally dependent on the flow of theriver. A development of this type is usually builtfor some other purpose, such as navigation, powerproduction being only incidental.

The economics of a run-of-river hydroplantdepend on the minimum flow of the river. If theminimum flow is very low, it will be necessary toinvest money in steam-generation facilities to pro-vide supplemental power during low-flow periods.Therefore, the value of the plant will be only thefuel saved that would otherwise be required forsteam generation.

Base-Load Hydro Plants n This type is alsoa run-of-river hydroplant without storage, but it islocated on a river that provides a minimum flowcapable of serving the power demand without





supplementary steam-generating facilities. Thereliable plant capacity is set below the expectedminimum flow in the river. This type of run-of-river hydroplant utilizes only a small proportionof the flow of a river. It must pass not only highseasonal flows but also the water it cannot utilizeduring hours of low power demand.

Cost of a base-load plant can be compared withthe cost of the steam capacity that would benecessary to serve power demands if hydrogenera-tion were not developed.

Run-of-River Plants with Storage n Asmall amount of storage can greatly increase thereliable capacity of a hydroplant. The water notrequired for generation during hours of low powerdemand can be stored and used for generationduring periods of peak demand.

Storage can be provided for a daily, weekly, orseasonal cycle. On a daily cycle, the requiredreservoir capacity is less than the river’s daily flowvolume. On a weekly cycle, the flow during theperiods of low power demand on weekends is alsostored to give additional capacity for peak periods

during the week. On a seasonal cycle, the highflood flows are stored to be used during periods oflow flow. The seasonal operation requires manytimes the storage necessary for weekly or dailyoperation and therefore may be uneconomicalunless the reservoir is multipurpose. Then, part ofits cost can be underwritten by flood-control orirrigation projects.

Peak-Load Plants n The power demand onan electrical system fluctuates from a daily high toa nightly low. Depending on the size of the utilityand type of customers served, peak demands maybe several times the magnitude of the low demandsencountered at night. These fluctuations in de-mand necessitate generation facilities whose fullcapacity is used only a few hours a day, duringperiods of peak power demand (Fig. 21.92).

Capacity factor is the percentage of the time thefull capacity of a plant is used or the ratio of theaverage power the plant produces to the plant’scapacity. It can be computed on a daily, weekly, oryearly basis.

Fig. 21.92 Daily load curves for generating plants. (Department of Water and Power, Los Angeles, Calif.)





Hydroplants that are used mainly to supplypower for the periods of peak demand are gen-erally called peak-load plants. The main classes ofpeak-load plants are pumped-storage plants andrun-of-river plants with storage.

If sufficient generating capacity and reservoirstorage are planned for a run-of-river hydroplant,only a relatively small supply of water is needed toproduce a high generation capacity for a few hoursduration. This enables a large utility to use steamgeneration at a high capacity factor where it is mostefficient and to supply peak demands from hydro-plants.

Pumped Storage n This is a means of storinglarge quantities of energy, generated duringperiods when excess generating capacity is avail-able, to be used at some future time. Water ispumped from a low reservoir to a higher one byenergy from steam or base-load hydro whenpower demand is low. When needed, the watergenerates power by flowing through a turbineback into the low reservoir. Because of friction lossin the penstock and losses due to the imperfectefficiencies of pumps and turbines, only two-thirds of the energy required to pump the water isrecovered.

The balance of energy between pumping andgenerating can be on a daily or weekly basis. Butbecause the weekly cycle requires several timesmore reservoir storage than the daily cycle, it usu-ally is not as economical.

When pumped storage is operated at a highcapacity factor to transfer large quantities ofelectric energy from off-peak to peak, the energyloss may make it uneconomical. This undesirableenergy-loss feature of pumped storage is overcomewhen it is used as reserve capacity.

Electrical systems require what is called spin-ning reserve, which is capacity above that necessaryto serve the expected maximum load, readyinstantly to generate power in case of failure ofgenerating equipment or an unanticipated highpower demand. Many utilities keep a spinningreserve capacity equal to the size of their largestsingle generating unit, or 15% of their maximumdemand (Fig. 21.92).

(V. J. Zipparo and H. Hasen, “Davis’ HandbookofAppliedHydraulics,” 4th ed.,McGraw-Hill BookCompany, New York (books.mcgraw-hill.com).)

21.63 Dams

Dams are usually classified on the basis of the typeof construction material or the method used toresist water pressure. The main classifications aregravity, arch, buttress, earth, and rock-fill.

Gravity dams are concrete or masonry damsthat resist the forces acting on them entirely by theirweight. Figure 21.93 shows the forces that act on atypical gravity dam. The largest force is usually thehydrostatic force of the water F1. Its distributionis triangular, varying from zero at the top to fullhydrostatic at the bottom. Force F2 represents siltpressure, which results fromdeposition of silt at thebase of the dam. This silt pressure can be calculatedby Rankine’s theory for earth pressure using thesubmerged weight of the silt.

Force F3 represents ice pressure against the face ofthe dam. In cold climates, ice, which forms on thereservoir surface, expands when the temperaturerises andexerts a forceon the topofadam. In thepast,ice pressures as high as 50,000 psf have been used forthe design of dams in the north; however, today it isrealized these values aremuch too high. Amethod ofcalculating these forces, presented by Edwin Rose,gives values ranging from 2000 to 10,000 psf, depen-ding on the rate of temperature rise and restrainingconditions at the edges of the reservoir. (E. Rose,“Thrust Exerted by Expanding Ice,” Proceedings of theAmerican Society of Civil Engineers, May 1946.)

Practically all regions in the United States aresubject to earthquakes of varying intensity. Earth-quakes cause vertical and horizontal accelerationsof the earth, which create forces on any objectresting on it. The magnitude of these forces equalsthe mass of the object times the acceleration fromthe earthquake. These accelerations occur in everydirection, so the effect of the forces must beanalyzed for all directions. Most dams in seismi-cally active regions in the United States have beendesigned for an acceleration equal to 0.1 g, whereg is the acceleration due to gravity. The effectof accelerations on the dam is represented in Fig.21.93 by forces F4 and F5. Force F6 represents theinertial force of the water on the face of the dam. Aclose approximation of the force, given by Eq.(21.153), was developed by von Karman. (“Press-ure on Dams During Earthquakes,” discussion byvon Karman, Transactions of the American Society ofCivil Engineers, vol. 98, p. 434, 1933.)

F6 ¼ 0:555awh2 (21:153)





where w ¼ unit weight of water, lb/ft3

a ¼ acceleration due to earthquake, ft/s2

h ¼ depth of water behind dam, ft

The force F6 acts at a point 0.425h above the base.Force F7 is due to the weight of water on an

inclined face. Gravity dams usually have aninclined upstream face to facilitate construction.

Force F8 represents an uplift force that acts onthe undersurface of any section taken through thedam or under the base of the dam. This uplift iscaused by the seepage of water through pores orimperfections in the foundations or throughimperfectly bonded construction joints in themasonry. In the past, engineers assumed that,because of bearing contact, this pressure acted onlyon some percentage of the total area. Recent belief,however, is that uplift acts on 100% of the area ofthe base.

A process used to reduce uplift pressures callsfor grouting along the heel and use of drainsbehind the grout. When the base is not drained, theuplift pressure is assumed to vary linearly from

between full and one-half hydrostatic pressure atthe heel to the full tailwater pressure at the toe.

Force F9 represents the weight of the dam. It actsat the centroid of the cross-sectional area of thedam.

Summation of the vertical forces and ofmoments about any point yields the foundationpressure. The foundation pressure at the heel of thedam should be compressive. Hence, the resultantof all forces acting on the dam should fall withinthe middle third of the base of the dam.

The basic modes of failure possible for a gravitydam are by sliding along a horizontal plane,overturning by rotating about the toe, or failure ofthe foundation material. The first two modesdepend mainly on the cross-sectional shape of thedam, whereas the third depends on both the cross-sectional shape and the foundation material.

Gravity dams can be built on earth foundations,but their height in these cases has been limited toaround 65 ft. The main reason gravity dams areused is that they can pass large flood flows overtheir crest without damage. Their first cost andmaintenance cost are usually greater than those of

Fig. 21.93 Forces acting on a concrete gravity dam.





earth or rock-fill dams of comparable height andcrest length.

Arch dams are concrete dams that carry theforce of the water through arch action. Stresses inan arch dam may be determined with computersby the finite-element method or by an approximatemethod in which the water force is dividedbetween elements: a series of horizontal archesthat span between the abutments and a series ofvertical cantilevers fixed at the foundation. Thedistribution of load between the arches and can-tilevers is determined by the trial-load method.First, a division of the load is assumed and thedeflections in the arches and cantilevers arecomputed. The deflection of an arch at any pointmust equal the deflection of the cantilever at thesame point. If the deflections are not equal, a newdivision of the load is assumed and the deflectionsrecalculated. This process is continued until equaldeflections are obtained.

The external forces an arch dam must resist arebasically the same as those on a gravity dam; how-ever, their relative importance is much different.On arch dams, uplift is not so important, but iceloads and temperature stress are much more criti-cal. Arch dams require much less concrete thangravity dams and usually have a much lower firstcost. They are not suited to most sites, however,since they must be located in a relatively narrowcanyon supported by good rock abutments.

Buttress dams consist of a watertight membranesupported by a series of buttresses at right angles tothe axis of the dam. Although there are many typesof buttress dams, those widely used are the flat-slaband the multiple-arch. These differ in that thewater-supporting membrane for the flat-slab typeis a continuous concrete slab spanning the but-tresses. In the multiple-arch, the membrane is aseries of concrete arches. The multiple-arch re-quires less reinforcing steel and can span longerdistances between buttresses, but its formwork ismore expensive.

The upstream face of a buttress dam is usuallyinclined at about 458. The weight of the water onthe face is necessary to increase the dam’s resis-tance to sliding and overturning.

The forces acting on a buttress dam are exactlythe same as those that act on a gravity dam.However, the vertical load of the water is muchgreater on a buttress dam, and uplift forces aresmaller. The modes of failure are also the same, butthe structural design is much more critical.

Although buttress dams usually require lessthan half the volume of concrete required bygravity dams, they are not necessarily less expen-sive because of the large amount of formwork andreinforcing steel required. With the rapidly increas-ing cost of labor over the past several decades, thebuttress dam has lost much of its earlier popu-larity.

Earth dams are designed to utilize materialsavailable at the dam site. They can be constructedof almost any material with very primitiveconstruction equipment. Successful earth damshave been built of gravel, sand, silt, rock flour, andclay. If a large quantity of pervious material, suchas sand and gravel, is available and clayeymaterials must be imported, the dam would havea small impervious clay core, the materialavailable locally making up the bulk of the dam.Concrete has been used for an impervious core,but it does not provide the flexibility of claymaterials. If pervious material is not available, thedam can be constructed of clayey materials withunderdrains of imported sand or gravel under thedownstream toe to collect seepage and relieve porepressures.

Slopes of an earth dam are rarely greater than 2horizontal to 1 vertical and are usually about 3 to 1.The governing criterion is usually the stability ofthe slopes against slide-out failure. Stability underthe action of seismic forces is especially critical. Forsoils in which pore pressure changes develop as aresult of shear strain induced by an earthquake,determination of appropriate values for yield accel-eration is very difficult. For some types of soil,no well-defined yield acceleration exists; displa-cements occur over a wide range of accelera-tions.

Another factor that sometimes determines thesteepness of the slopes is the amount of seepagethat can be tolerated. If the dam is on a perviousfoundation, it may be necessary to increase the basewidth to reduce seepage. The seepage may also bereduced by placing an impervious blanket on theupstream side of the dam to increase the seepagepath or by using a cutoff wall in the foundation,such as sheetpiling or a clay-filled trench.

Earth dams can be built to almost any heightand on foundations not strong enough for concretedams. Improvements in earth-moving equipmenthave resulted in a decreased cost for earth dams,and rising labor costs have increased the cost forconcrete dams.





Rock-fill dams usually consist of a dumpedrock fill, a rubble cushion of laid-up stone on theupstream face, bonding into the dumped rock, andan upstream impervious facing, bearing on therubble cushion, with a cutoff wall extending intothe foundation. The dumped rock fill may consistof rocks varying in size from small fragments toboulders weighing as much as 25 tons. The fill isusually compacted by dropping the rock, some-times from as high as 175 ft. onto the fill. Sluicing ofthe fill with high-pressure hoses is also used towash fines from between contact points of the rockand reduce settlement. The rubble cushion consistsof rocks individually placed to reduce the voidsand provide support for the impervious facing. Thefacing is usually concrete, or wood over concrete,although steel has been used occasionally. Thecutoff wall is usually concrete.

Rock-fill dams are generally designed empiri-cally. Low rock-fill dams may have an upstreamface as steep as 1 =

2 horizontal on 1 vertical. Thedownstream face is usually 1.3 on 1, the naturalangle of repose of rock. For dams over 200 ft high,both the upstream and downstream faces are usu-ally on a slope of 1.3 on 1.

The major problem encountered in rock-filldams is large settlements that occur after construc-tion when the reservoir is first filled. Verticalsettlements and horizontal displacements in excessof 5% of the height of the dam have occurred;therefore, the impervious facing must be veryflexible or damage will occur during settlement.One solution to this problem has been to put atemporary facing on the dam and to replace it witha permanent facing after settlement has takenplace. Temporary facings are usually of wood.

Rock-fill dams are used extensively in remotelocations where cement is expensive and thematerials for an earth dam are not available. Theircost compares favorably with that of concretedams. Leakage should be expected, but rock-filldams are very stable and have been overtoppedwithout suffering major damage.

(V. J. Zipparo and H. Hasen, “Davis’ Handbookof Applied Hydraulics,” 4th ed., McGraw-HillBook Company, New York (books.mcgraw-hill.com); “Design of Small Dams” and “EmbankmentDams,” U. S. Bureau of Relamation; “Earth andRockfill Dams: General Design and ConstructionConsiderations,” EM 1110-2-2300, U. S. ArmyCorps of Engineers (www.usace.army.mil/inet/usace-docs/eng-manual/em.htm).)

21.64 Hydraulic Turbines

In the past, hydraulic power-generating machinesmeant a large number of different types ofequipment. Today, however, the turbine is the onlytype of importance in hydraulic power generation.Its function is transformation of the kinetic andpotential energy of water into useful work.

Turbines are classified as impulse turbines andreaction turbines.

Impulse turbines utilize the energy of water byfirst transforming it into kinetic energy, by freedischarge of the water through a nozzle. The noz-zle is directed at buckets positioned along theperimeter of a water wheel. The force of the waterstriking these buckets causes the wheel to rotate,providing power.

The only type of water wheel used today inimpulse turbineswas developed in 1880 by Pelton—the Pelton wheel (Fig. 21.94). The wheel is coveredby a housing to prevent splashing and to guide thedischarge after the water strikes the wheel.

In most impulse turbines, the water wheelrotates on a horizontal shaft and is acted on by thedischarge from one or two nozzles. But verticalshafts may be used with as many as six nozzles, toobtain a high efficiency for very low loads. In suchinstallations, efficiencies of 92% for full load andslightly below 90% for loads as low as one-quarterof full load have been obtained.

Impulse turbines are commonly used for headsgreater than 1000 ft. (An impulse turbine at theReisseckPower Plant inAustria operates under a neteffective head of 5800 ft.) There is no lower limit ofhead for impulse turbines. They have been used forheads as low as 50 ft; however, the reaction turbine isusually better suited to low heads at large flows.

Fig. 21.94 Impulse (Pelton) type of hydraulicturbine.





Reaction Turbines n Types of reaction tur-bines include the Francis (Fig. 21.95a), the pro-peller-type (Fig. 21.95b) and the axial flow (Fig.21.95c). In these, the flow from the headwater to thetailwater is in a closed conduit system.

The Francis turbine usually consists of fouressential parts: scroll case, wicket gates, runner,and draft tube.

The scroll case transfers the water from thepenstock (supply pipe) to the wicket gates and

Fig. 21.95 Reaction types of hydraulic turbines: (a) Francis; (b) Kaplan; (c) axial flow.





runner. It distributes the water so that all pointson the perimeter of the runner receive the samequantity of water.

The wicket gates, located just outside theperimeter of the runner, control the amount ofwater that enters the turbine. When the powerdemand on the turbine changes, a governoractuates a mechanism that opens or closes thegates.

The runner is the part of the turbine thattransforms the pressure and kinetic energy of thewater into useful work. As the water flows throughthe turbine, it changes direction. This creates aforce on the runner, causing it to rotate and turn thegenerator.

The draft tube is a conical tube with diver-ging sides. It decelerates the flow dischargedfrom the runner, so that the remaining kineticenergy may be regained by conversion into suctionhead.

Francis turbines have a maximum efficiency ofabout 94% when operated at or close to full load.However, if the load drops below 50%, theirefficiency decreases rapidly. Francis turbines arecommonly used for heads between 100 and 1000 ft.At heads above 1000 ft. problems are encounteredin controlling cavitation and in building a scrollcase to take the high pressures. At heads below100 ft. the propeller-type turbine is usually moreefficient.

Propeller Turbines n There are two types ofpropeller turbines: the movable-blade type, such asthe Kaplan turbine, and the fixed-blade type. Theonly difference between the two is that the pitch ofthe propeller blades is adjustable in a Kaplanturbine.

The propeller turbine (Fig. 21.95) has the samebasic parts as the Francis turbine: scroll case, wicketgates, runner, and draft tube. The basic differencebetween the Francis turbine and the propellerturbine is in the shape of the runner. The runner ofa propeller-type turbine operates in the samemanner as a fan or a ship’s propeller: The watermoving past the blades creates a force that causesthe runner to rotate.

Propeller-type turbines are used for headsranging from a few feet to about 100 ft. The Kaplanturbine has an efficiency of about 94% for full loadand drops only to 92% for 40% load. The fixed-blade-type turbine also has an efficiency of about

94% for full load; however, its efficiency drops offrapidly below full load.

Axial-Flow Turbines n These provideenhanced performance for operation under low-head and large capacity.

(V. J. Zipparo and H. Hasen, “Davis’ Handbookof Applied Hydraulics,” 4th ed., McGraw-HillBook Company, New York (books.mcgraw-hill.com).)

21.65 Methods for Control ofFlows from Reservoirs

Any reservoir with an appreciable drainage areamust have a spillway to discharge flood flowswithout damage to the dam and to keep thereservoir water surface below some predeterminedlevel.

21.65.1 Spillways

An overflow spillway allows water to pass overthe crest of a section of the dam. This type ofspillway is widely used for concrete dams because,if designed correctly, the dam will not be damagedby the water. To use an overflow spillway for earthor rock-fill dams, it is necessary to make thespillway a concrete gravity section. This may not bepossible for high earth dams because the foun-dation may not be able to support a high concretegravity section.

The discharge over an overflow spillway isgiven by the equation for discharge over a weir(Art. 21.34). Since the discharge varies as the headto the 3⁄2 power, overflow spillways keep the waterlevel within close limits even when there is a largevariation in flows.

It is desirable for an overflow spillway to havethe form of the underside of the nappe of a sharp-crested weir. This type of spillway, called an ogeespillway, should be designed—as should all spill-ways—so that separation of the water from the faceof the spillway will not occur. Thus, the danger ofcavitation will be eliminated.

In a chute spillway, water flows over a crest intoa steeply sloping, lined, open channel. The flow ismade supercritical to keep the size and length ofthe chute to a minimum. Gradual vertical curvesshould be used in the chute to avoid separation ofthe flow from the channel bottom.





Chute spillways are commonly used for earthand rock-fill dams where the topography allows achute to carry the water away from the toe toeliminate the danger of undermining. The dis-charge over the crest is given by the equations fordischarge over a weir or the entrance to an openchannel.

In a side-channel spillway, the flow passes overa crest into a channel parallel to this crest. The crestis usually a concrete gravity section, although itcan be concrete laid on the natural embankment.Side-channel spillways are often used in narrowcanyons where it is not possible to obtain sufficientcrest length for overflow or chute spillways. Theflow in the channel parallel to the crest is deter-mined by applying the momentum principle in thedirection of flow and assuming the energy of thewater flowing over the crest is completely dis-sipated (U.S. Bureau of Reclamation, “Design ofSmall Dams,” Government Printing Office, Wash-ington, DC 20402).

In a shaft spillway, sometimes called a morn-ing-glory spillway, the water flows over a circularweir into a vertical shaft. The shaft terminates in ahorizontal conduit that carries the water past thedam. The weir can be sharp-crested, flared, or ogeein cross section. (This type of spillway should notbe constructed over or through earth dams.) If thetopography is not suitable for a chute or side-channel spillway, a shaft spillway may be the bestalternative.

There are two conditions of discharge for a shaftspillway, both depending on the head on the weir.When the head is relatively low, the discharge isgoverned by the flow over the weir, which isdirectly proportional to the 3 =

2 power of the head onthe weir. As the head increases, at some point thedischarge will no longer be controlled by theamount of water that can flow over the weir but bythe amount of water that can flow through theconduit. The discharge for this condition is directlyproportional to the 1⁄2power of the elevation dif-ference between the reservoir water level and thelevel of discharge of the spillway conduit. Once thissecond condition is reached, a large increase inhead will cause only a small increase in flow. Sinceanalytical analysis of discharge does not give goodresults on this type of spillway, model tests areusually employed.

A siphon spillway (Fig. 21.96) is a closedconduit for discharging water over or through adam. The entrance to a siphon spillway is usually

submerged below the normal water level so that itwill not clog with debris or ice. The discharge endof the siphon is usually sealed by deflecting theflow across the barrel or by submerging it so thatair cannot enter.

The air vent shown in Fig. 21.96 determines thereservoir level at which the siphon flow begins.When the reservoir water level rises above the vent,the siphon’s intake is sealed. Water flowing overthe crest of the siphon removes the air in the siphonand full flow begins. Because the flow depends onthe siphoning action, siphon spillways hold thewater level of a reservoir within close limits. Butthey are not good for handling large variations inflows because their discharge is directly pro-portional to the square root of the head. They arerelatively expensive because of the cost of formingthe barrel.

21.65.2 Intake Structures

The various functions an intake structure mayserve include permitting withdrawal of water fromvarious levels of a reservoir, controlling flow,excluding debris and ice from a conduit, andproviding support for the conduit. The type ofintake structure required depends on the functionsand characteristics of the reservoir. The simplesttype of intake is a block of concrete supporting theend of a conduit equipped with a bar screen toexclude foreign matter. In contrast, the intake

Fig. 21.96 Siphon spillway.





towers at Hoover Dam, which serve 30-ft-diameterpenstocks, are 395-ft-high concrete towers, withtwo 32-ft-diameter cylinder gates under a maxi-mum head of over 300 ft.

Intake towers are commonly used where there isa large fluctuation in the water level of a reservoiror where it is necessary to control the quality ofwater used for a domestic supply. They are usuallymade of concrete and have ports at various levels topermit selection of water from different elevations.The ports are usually provided with gates or valvesand some type of trash rack.

The main hydraulic consideration in the designof an intake is to keep losses to a minimum. To dothis, the velocities through the trash racks shouldbe kept less than 0.5 ft/s, and the standard rules forreducing hydraulic losses should be observed.

21.65.3 Crest Gates

These include a number of different types ofpermanent and temporary devices that operate onthe crest of spillways to increase the storage of areservoir temporarily while control of spillwayflows is retained. During periods of low flow whenthe full spillway capacity is not required, theadditional head and storage gained with crest gatesmay be very valuable.

Flashboards and stop logs are the most com-mon types of crest gates used for small installa-

tions under low head. Flashboards are usuallywood planks that span between vertical pipes thatcantilever above the spillway crest. When thereservoir water surface reaches some predeter-mined level, the pipes fail, allowing the fullcapacity of the spillway to be utilized. Stop logsare wood planks that span between slotted verticalpiers which cantilever above the spillway crest.

On large stop-log installations, the hydrostaticforce creates large frictional forces between thesliding element and the vertical guide, makingremoval difficult. These frictional forces make itnecessary to use a type of gate that depends onrolling rather than sliding friction and operatesfreely under hydrostatic pressure.

Taintor gates and sliding gates mounted onlow-friction roller bearings are the most widelyused types of crest gates on major installations. In ataintor gate (Fig. 21.97), the friction is concentratedin the trunnion and does not affect the operation.Since flow passes under taintor and slide gates,there is a tendency for ice and trash to pile upagainst them, causing damage and hamperingoperation.

Fig. 21.97 Taintor gate.

Fig. 21.98 Bear-trap gate.

Fig. 21.99 Drum gate.





Bear-trap and drum gates allow the flow to passover the top. The bear-trap gate consists of twoleaves hinged, as shown in Fig. 21.98. To raise abear-trap gate, water is admitted to the space underthe leaves to force the leaves up. The drum gate(Fig. 21.99) consists of a segment of a cylinder thatis lowered into a recess in the crest when not in use.

Because of the large recess required in the dam,drum gates are not suited to small dams.

(V. J. Zipparo and H. Hasen, “Davis’ Handbookof Applied Hydraulics,” 4th ed., and H. E. Babbitt,J. J. Doland, and J. L. Cleasby, “Water SupplyEngineering,” McGraw-Hill Book Company, NewYork (books.mcgraw-hill.com).)





Documents

Chapter Water Resources Engineering