Hydraulics - Formulae for Partially Filled Circular Pipes (Akgiray, 2004)

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ENVIRONMENTAL ENGINEERING SCIENCEVolume 21, Number 3, 2004 Mary Ann Liebert, Inc.

Simple Formulae for Velocity, Depth of Flow, and SlopeCalculations in Partially Filled Circular Pipes

mer Akgiray*

Department of Environmental EngineeringFaculty of Engineering

Marmara UniversityIstanbul, Turkey

ABSTRACT

The application of the Manning equation to partially filled circular pipes is considered. Three differentapproaches based on the Manning equation are analyzed and compared: (1) using a constant value for theroughness coefficient n and defining the hydraulic radius as the flow area divided by the wetted perime-ter. (2) Taking the variation of n with the depth of flow into account and employing the same definitionof the hydraulic radius. (3) Defining the hydraulic radius as the flow area divided by the sum of the wet-ted perimeter and one-half of the width of the airwater surface and assuming n is constant. It is shownthat the latter two approaches lead to similar predictions when 0.1 # h/D # 1.0. With any one of theseapproaches, tedious iterative calculations become necessary when diameter (D), slope (S), and flow rate(Q) are given, and one needs to find the depth of flow (h/D) and the velocity (V ). Simple explicit for-mulas are derived for each of the three approaches. These equations are accurate enough to be used in de-sign and sufficiently simple to be used with a hand calculator.

Key words: hydraulic radius; Manning equation; roughness coefficient; sewer design

371

*Corresponding author: Faculty of Engineering, Department of Environmental Engineering, Marmara University, Goztepe81040, Istanbul, Turkey. Phone/Fax: (90) 216-3481369; E-mail: [email protected]

INTRODUCTION

SEWERS ARE COMMONLY DESIGNED to flow full only un-der maximum conditions. In sewer design, therefore,it is necessary to be able to predict the velocity and dis-charge when a sewer is partly filled. Mannings equationhas been the most commonly used formula in sewer de-sign because of its simplicity and the generally satisfac-tory results it has given. In metric units, the Manningequation can be written as follows:

V 5 Rh2/3S1/2 (1)

or

Q 5 Rh2/3S1/2 (2)

where V 5 the velocity (m/s), S 5 the slope of the en-ergy grade line, Rh 5 the hydraulic radius defined as theflow area divided by the wetted perimeter (m), A 5 thecross-sectional area of flow (m2), Q 5 the discharge(m3/s), and n 5 Mannings coefficient of roughness. The

A}n

1}n

following additional notation will be used in what fol-lows: h 5 the depth of the water (m), D 5 the pipe di-ameter (m), u 5 the water surface angle in radians (seeFig. 1). The values of the hydraulic radius and the flowarea are fixed once h/D or u is specified [Equations(3)(5)].

If D, S, and h/D are known, one can calculate the othervariables A, Rh, Q, and V in a straightforward manner. If,on the other hand, D, S, and Q are given, iterative cal-culations are required to find h/D and V. Saat (1990,1992) and Giroud et al. (2000) devised solutions to thisproblem, eliminating the need for iterative calculationsin the range 0 # h/D , 0.938. There is no doubt that,with modern software, the mentioned iterative calcula-tions can be completed quickly and accurately. Be thatas it may, an explicit solution is always preferred overan implicit equation that must be solved iteratively. Theusefulness of the explicit solutions of the Manning equa-tion and possible applications were discussed by Wheeler(1992).

The Manning equation is usually applied by assumingthat the roughness coefficient n is constant (Metcalf &Eddy, 1981; Benefield et al., 1984). The equations pre-sented by Saat (1990, 1992) and Giroud et al. (2000),for example, are based on this assumption. On the otherhand, it is reported that n changes with the ratio h/D(Camp, 1946). Therefore, it is desirable to have equationssimilar to those given by Saat and Giroud et al. in casethe designer wants to account for the dependence of n onthe depth of flow (h/D).

For completely filled pipe flow, the use of the Darcy-Weisbach equation in conjunction with the Colebrookformula (or its equivalent, the Moody diagram) is wellestablished. By equating the head loss calculated by theManning equation (with Rh 5 D/4 and n 5 nf) to thatgiven by the Colebrook formula, it is possible to showthat nf depends on the Reynolds number, the relativeroughness e/D, and the pipe diameter D (see Appendix

for details). As discussed by Massey (1989), the depen-dence on Reynolds number can be ignored only at highvalues of the Reynolds number. It is assumed here thata proper value for nf (corresponding to given e/D and D)is known; the effect of the ratio h/D on the value of theratio n/nf is considered in detail in what follows.

Saat (1990) and Wheeler (1992) mentioned the pos-sibility of using an alternative definition of the hydraulicradius that was recommended by Escritt (1984). In thisapproach, the hydraulic radius is defined as the flow areadivided by the sum of the wetted perimeter and one-halfof the width of the airwater surface. Saat (1992) pro-posed an explicit solution applicable with this approachas well, allowing the direct estimation of h/D when D, S,and Q are given. This solution method can be used when0 # h/D # 0.80.

Another situation that necessitates iterative calcula-tions is when D, V, and Q are known, and one needs tocalculate S and h/D. Esen (1993) presented an explicitsolution applicable in the range 0.02 , h/D , 0.40.

The following is an outline of this paper:In the first part, equations applicable to partially filled

pipes are reviewed. This helps introduce the notation andthe terminology used here, and lays down the basis of thematerial presented later in the paper.

Second, the problem of finding h/D when D, S, and Qare given and n is taken as a constant is examined. Here,an explicit solution is presented to replace the two equa-tions given by Saat (1990, 1992) over the range 0 ,h/D , 0.938. The equation is slightly more accurate andsimpler in form. Furthermore, another equation is devel-oped for the range 0.938 # h/D # 1.0, facilitating the ex-plicit calculation (without iterations) of h/D over the en-tire range 0 , h/D # 1.0, provided that n is independentof the depth of flow.

In the third part, the effect of the dependence of n onh/D is examined. Using the data presented by Camp(1946), an expression for n/nf as a function of the surfaceangle u is derived. Explicit equations are presented to fa-cilitate the direct calculation of h/D and V when D, S,and Q are known.

Next, the use of Escritts definition of the hydraulic ra-dius in the Manning equation is considered. In this ap-proach, n is assumed to be constant. Again, explicit equa-tions are developed allowing the direct calculation of h/Dand V when D, S, and Q are known.

In the fourth part, three different approaches employ-ing the Manning equation are compared: (1) assumingconstant n and using the conventional definition of thehydraulic radius; (2) taking the variation of n with thedepth of flow into account and using the conventionaldefinition of the hydraulic radius; (3) assuming constantn and using Escritts hydraulic radius. It is shown that

372 AKGIRAY

Figure 1. Cross-section of a partially filled pipe.

the predictions based on the latter two approaches areclose to each other, but are significantly different fromthose of the first approach.

Finally, as an alternative to the calculation method pre-sented by Esen (1993), simple approximate equationsgiving u, h/D, and S as functions of A/D2 are presented.These equations are not restricted to h/D , 0.40, and fa-cilitate the solution of the problem without iterations. An-other equation is derived to estimate S directly from thegiven values of Q, V, and D. This equation accounts forthe variation of n with h/D, a factor not considered byEsen.

THEORY

For a partially filled circular pipe with diameter D, thefollowing equations apply (see Fig. 1):

u 5 2 cos21 11 2 2 (3)A 5 (u 2 sin u) (4)

where h is the depth of the water in the pipe, u is the wa-ter surface angle in radians, and A is the area of flow. Itmay be noted that some authors use the symbol u to de-note half of this angle. Referral to Fig. 1 should helpavoid any possible confusion.

The hydraulic radius Rh is usually defined as the flowarea divided by the wetted perimeter. It can thus be writ-ten as

Rh 5 1 2 (5)Escritt (1984) noted that defining the hydraulic radius

as the flow area divided by the sum of the wetted pe-rimeter and one-half of the width of the airwater sur-face resulted in improved accuracy in calculations. Thehydraulic radius based on this definition will be denotedby Re here. It can be calculated by using the followingexpression:

Re 5 1 2 (6)Escritt (1984) noted that turbulent flow in open chan-

nels is different from that in fully charged pipes or cul-verts for as soon as there is a free surface, waves form,and these dissipate energy. Escritt explained his defini-tion of hydraulic radius by stating that the dissipation ofenergy by waves is directly related to the surface overwhich this dissipation can take place.

It is common practice to apply the Manning equationby defining a coefficient K as follows:

u 2 sin u}}u 1 sin(u/2)

D}4

u 2 sin u}}

u

D}4

D2}8

2h}D

Q 5 D8/3S1/2 (7)

Substituting Equations (4) and (5) into Equation (2) andcomparing the resulting expression with Equation (7)yields the following relation between K and u:

Kh 5 5 2213/3 (u 2 sin u)5/3u22/3 (8)

where 2213/3 < 0.0496. The subscript h is used to em-phasize that this expression is based on the usual defini-tion of the hydraulic radius Rh [Equation (5)]. Equation(8) holds regardless of whether n is a constant or a func-tion of the depth of flow. To study the effect of the de-pendence of n on the flow depth, however, it will be con-venient to define a new coefficient Khf based on the valueof n at full flow conditions:

Kh f 5 5 5 (9)

where nf is the value of n when h/D 5 1 and u 5 2p. Thefunction f(u) 5 n/nf gives the dependence of n on thedepth of flow.

Consider next the approach recommended by Escritt(1984) for defining the hydraulic radius. If Re is used in-stead of Rh in the Manning equation, that is, when Equa-tion (6) is substituted into Equation (2), the following isobtained:

Ke 5 5 2213/3 (u 2 sin u)5/3

1u 1 sin( )222/3

5 Kh1 22/3

(10)

This equation holds regardless of whether n is a constantor a function of h/D. An additional coefficient Kef anal-ogous to Kh f could be defined. This will not be needed,however, because n is assumed to be constant (i.e., n 5nf) in Escritts approach.

In this and the next two sections, attention is focusedon the use of the Manning equation with the usual defi-nition of the hydraulic radius. The use of Escritts hy-draulic radius will be considered later in the paper.

Two possibilities will be considered here: (1) n 5 nf isassumed to be a constant, or (2) dependence of n on h/Dis taken into account. With either approach, for given val-ues of nf, D, S, and h/D, one can calculate the other vari-ables directly in the following order.

1. u [using Equation (3)]2. A [using Equation (4)], Rh [using Equation (5)], and

Kh [using Equation (8)].3. n [using a relation of the form n/nf 5 f(u)]4. Q [using Equations (2) or (7)] and velocity V [using

Equation (1) or V 5 Q/A].

u}}u 1 sin(u/2)

u}2

Qn}D8/3S1/2

Kh}f(u)

n fKh}

nQnf}

D8/3S1/2

Qn}D8/3S1/2

K}n

SIMPLE FORMULAE IN CIRCULAR PIPES 373

ENVIRON ENG SCI, VOL. 21, NO. 3, 2004

It should be noted that these computations can be com-pleted easily without the use of any nomographs or time-consuming iterative calculations.

Another situation of interest is when nf, D, S, and Qare known, and one needs to estimate h/D and V. Sinceh/D and u are not known a priori, one cannot directlycalculate A, Rh, K, or n (when n is assumed to depend onh/D). Iterative calculations are carried out in this case asfollows.

1. Substitute Equation (8) and the relation n/nf 5 f(u)into Equation (7) to obtain:

2213/3 (u 2 sin u)5/3u22/3 5 n f f(u)QD28/3S21/2 (11)

2. Solve Equation (11) iteratively (using, e.g., Newtonsmethod) to obtain u.

3. Calculate h/D [using Equation (3)], A [using Equation(4)], and V (using V 5 Q/A).

To eliminate the need for the mentioned iterative cal-culations, Saat (1990) proposed the following approx-imate relation, giving u explicitly as a function of Kh:

u < 1212pKh (12)

Saat (1990) noted that this equation is applicable foru # 265 (Kh # 1/p). In response to Wheeler (1992),Saat (1992) presented a second equation applicable inthe range 265 # u # 302.41 (1/p # Kh # 0.33528):

u < (1 2 0.2765 2 0.824Kh) (13)

As pointed out by Wheeler (1992), Equations (12) and(13) are very useful in that they obviate the need for it-erative calculations. The calculation method becomesvery simple, making easy manual computation feasible,provided that n is constant:

1. Calculate Kh using Equation (7) [n 5 nf, f(u) 5 1, andtherefore K 5 Kh 5 Khf in this case.]

2. Calculate u using either Equation (12) or Equation(13).

3. Calculate h/D [using Equation (3)], A [using Equation(4)], and V (using V 5 Q/A).

Giroud et al. (2000) developed the following explicitequation for the direct calculation of velocity without cal-culating u or h/D first:

V < 0.759111 2 2Kh4/131 2 (14)It should be noted that Equations (12), (13), and (14)

make the direct solution of the problem possible only

D2/3S1/2}

nf

Kh}2

302p}180

3p}2

when n is constant [i.e., when f(u) 5 1] and then only foru , 302.41.

Consider next the case when nf, D, V, and Q are known,and one needs to calculate S and h/D. The calculationsin this case would proceed as follows:

1. Calculate A using A 5 Q/V.2. Solve Equation (4) iteratively to obtain u.3. Calculate h/D [using Equation (3)] and Rh [using

Equation (5)].4. Calculate n [using a relation of the form n/nf 5 f(u)].5. Calculate S [using Equations (1) or (2)].

To eliminate the iterative calculations in this case, Esen(1993) proposed the following equations applicable in therange 30 , u , 160 (0.02 , h/D , 0.40):

5 0.61481 20.4256

(15a)

and

5 0.91061 20.6876

(15b)

EQUATIONS FOR CONSTANT n

Consider the problem studied by Saat (1990): theManning equation and the classical definition of the hy-draulic radius are to be used to estimate V and h/D fromknown values of Q, D, S, and nf. When the dependenceof Mannings n on the depth of flow is ignored, Equa-tion (11) can be written as follows:

2213/3 (u 2 sin u)5/3u22/3 5 nf QD28/3S21/2 5 Kh (16)

Since Q, D, S, and nf are known, Kh is first calculatedvia the equality on the right. The value of u is next cal-culated. The development of the new explicit solution foru is explained below.

Kh has the maximum value 0.3353 at u 5 302.41(5.278 radians). The functional relationship in Equation(16) has the following additional properties (Fig. 2): whenu is considered as a function of Kh, Equation (16) hastwo roots in the range 0.3117 # Kh , 0.3352. It can beshown that dKh/du 5 0 and du/dKh 5 ` at u 5 0 and u 5302.41.

The first equation proposed by Saat (1990) [Equa-tion (12) here] satisfies the condition du/dKh 5 ` at u 50, whereas his second equation [Equation (13) here] sat-isfies du/dKh 5 ` at u < 302.41. It appears desirable toreplace these two equations by a simple equation that sat-isfies both conditions at once. It is easy to see that thefunction cos21(1 2 2K/Kmax) has the desired properties.By multiplying this expression with aKb, and carrying

Q}VD2

h}D

Q}VD2

Vnf}D2/3S1/2

374 AKGIRAY

out a least-squares analysis to determine the values of aand b that give the best agreement with the exact solu-tion at the discrete points 10, 20, . . . , 290, 300, and302.41, the following expression is obtained (with r2 50.9997):

u 5 1.28Kh20.26 cos21(1 2 5.965Kh) (17)

This single equation is applicable over the entire range0 , u # 302.41 (0 , h/d # 0.938), and is recom-mended to replace Equations (12) and (13) (see Fig. 2).

Saat did not present an equation to represent the up-per portion of the curve, that is, beyond 302.41 (5.278radians). The following approximation matches the ex-act solution very closely (with a maximum error of0.04%), and can be used to find the second root in therange for 302.41 # u # 360 and 0.311686 # Kh #0.335282:

u 5 5.2781 (22.1512 50.087Kh) 0.335282 2 Kh (18)Figure 2 compares Equation (17) with the curve obtained

using Equation (16). Since the curve derived from Equa-tion (18) is not visually distinguishable from the solutionof Equation (16) in the range 302.41 # u # 360, it is notseparately shown in this figure. The error (uextimated 2uexact) is plotted in Fig. 3. It is seen that Equation (17),which is proposed here to replace Equations (12) and(13), although simpler in form, has slightly better accu-racy than those equations. On the other hand, as far asthe calculation of u is concerned, Equations (12) and (13)are already sufficiently accurate in the range they are ap-plicable (i.e., u # 302.41). The advantage of Equation(17) is that it is simpler (a single equation instead of twoover the entire range u # 302.41) and easier to use.

Once u is calculated, the expression V 5 8(Q/D2)/(u 2sinu) can be used to calculate the velocity. Small errors

in u, however, can lead to relatively large errors in thequantity (u 2 sinu), especially when u # 10 (h/D ,0.02). The following are recommended: (1) if the valuesof u and h/D are not needed, the equation developed byGiroud et al. (2000) [Equation (14) herein] should beused to estimate V directly. (2) If both u (or h/D) and Vare needed, Equation (17) should be used to calculate uand h/D, whereas Equation (14) should be employed tocalculate V. (3) To study conditions close to full flow(u . 302.41), Equation (18) should be used to calculateu and all the dependent quantities including V 58(Q/D2)/(u 2 sinu).

EQUATIONS FOR VARIABLE n

Camp (1946) presented data in graphical form show-ing the ratio n/nf as a function of h/D. This graph hasbeen reproduced in many widely used texts (e.g., Chow,1959; Metcalf & Eddy, 1981; Benefield et al., 1984). Itwill be employed here to derive expressions similar toEquations (14), (17), and (18). These expressions can beused to take the dependence of n on depth of flow intoaccount.

A regression equation was first derived from Campscurve by carrying out a least-squares analysis. The val-ues of n/nf at several discrete values of h/D were readfrom Camps curve, and a fifth degree polynomial wasfound to pass through these data points with negligi-ble error (r2 5 0.9996). The constraint n/nf 5 1 at h/D 51 was imposed in the derivation of this equation:

5 20.8627X5 1 0.4281X4 1 0.7626X3

2 1.02X2 1 0.8057X 1 1 (19a)

n}nf



Figure 2. Proposed approximate explicit solution and the exact solution.

where

X 5 1 2 h/D (20)

and

h/D 5 (1 2 cos(u/2))/2 (21)

When plotted against h/D (solid line in Fig. 4), Equation(19) yields a curve that is indistinguishable from thatgiven by Camp (1946). It may be noticed that Equation(19) is rather bulky, and it has not been developed forend usage. It is an intermediate result that is utilized inthis paper as explained in what follows.

It should be mentioned that some authors (e.g., Bene-field et al., 1984; Grant, 1992) extend Camps curve to

h/D 5 0 (i.e., 1 2 h/D 5 1) by assuming n 5 nf there, al-though Camp (1946) did not indicate the shape of thecurve for h/D , 0.025 [Equation (19a) predicts n/nf 51.11 at h/D 5 0.) As an alternative to Equation (19a), thefollowing expression has been developed, and will be pre-sented here:

5 1 1 0.37X0.58[sin(pX)]0.36 (19b)

Equation (19b) has the advantage of being simpler thanEquation (19a), while at the same time satisfying the con-dition n/nf 5 1 at h/D 5 0. Equation (19a) will be em-ployed in the rest of this paper as an intermediate equa-tion in the development of other equations [i.e., Equations

n}nf

376 AKGIRAY

Figure 3. Difference between estimated and exact values of u.

Figure 4. Data points were read from the curve given by Camp (1946). The curve drawn with the solid line is the polynomialfitted to these points. The dashed lined is obtained from Equation (36).

(23), (24), (25), and (46)], as it agrees with the data pointsin Camp (1946) better. Equation (19b), however, may beeasier to use when the value of n/nf is needed.

It should be noted that when Equations (20) and (21)are substituted into Equation (19), one obtains the func-tional relation n/nf 5 f(u). Note also that, if the value ofn (say, n1) at a u value (say, u1) other than 2p is known,the value of nf can be calculated (nf 5 n1/f(u1)).

An inspection of Equation (9) shows that, since nf, D,S, and Q are known in the situation considered here, onecan directly calculate Khf. The following equation thenneeds to be solved for u:

Khf 5 5 5 (22)

where f(u) is the expression given by Equations (19)(21).Again, this is a nonlinear algebraic equation and could besolved using an iterative procedure.

To find an explicit expression for u, it is first noted thatKhf has the maximum value 0.324642 at u 5 321.46136(5.61056 radians). Note also that Equation (22) has tworoots in the range 0.311686 # Khf , 0.324642.

To eliminate iterative calculations, the following ex-plicit approximate solution has been developed:

u 5 1.3645Khf20.2537cos21(1 2 6.1606Khf) (23)

This equation is applicable in the range 0 , u #321.46 (which corresponds to 0 , h/D # 0.972) and0 , Khf # 0.324642. The mean error at the discretepoints 10, 20, . . . , 310, 320, and 321.46 is lessthan 1.5%.

For completeness, an explicit equation applicable inthe range 321.46 # u # 360 and 0.311686 # Khf #0.324642 will also be given here:

u 5 5.6106 1 (18.9 2 41.6Khf)0.324642 2 Khf (24)Equation (24) is very accurate, and can be used to cal-

culate all the quantities that depend on u, including thevelocity V 5 8(Q/D2)/(u 2 sinu), provided that u .321.46. The following equation has been developed herefor the calculation of velocity when u # 321.46o:

V 5 (10.18 2 9.61Khf0.0036)Khf0.3361 2 (25)This explicit equation yields values that are within62.0% of the exact values over the entire range 1 #u # 321.46 (0.00002 # h/D # 0.972).

EQUATIONS BASED ON ESCRITTSHYDRAULIC RADIUS

An inspection of the leftmost equality in Equation (10)shows that, if nf, D, S, and Q are known, one can calcu-

D2/3S1/2}

nf

2213/3(u 2 sinu)5/3u22/3}}}

f(u)Kh}f(u)

Qnf}D8/3S1/2

late Ke. (It should be remembered that n 5 nf is assumedto be constant in this approach.) The equation to be solvedin this case is:

Ke5 52213/3 (u 2 sin u)5/3

1u 1 sin( )222/3

(10)

Ke has the maximum value 0.3193 at u 5 321.08(5.604 radians), and Equation (10) has two roots in therange 0.3117 # Ke , 0.3193. Saat (1992) presented anapproximate explicit solution applicable in the range 0 #u # 255.0, which corresponds to 0 # h/D # 0.80:

u 5 12123.636Ke (26)The following explicit solution developed here is ap-

plicable in the range 0 , u # 321.08 (which corre-sponds to 0 , h/D # 0.9714) and 0 , Ke # 0.3193:

u 5 1.3291Ke20.2587 cos21(1 2 6.263Ke) (27)

The mean error at the discrete the points 10, 20, . . . ,310, 320, and 321.08 is ,1.1%. Equation (27) isslightly more accurate than Equation (26) in the range0 , u # 255.0. The main motivation for introducingEquation (27) is that it is applicable over the entire range0 , u # 321.0839.

An explicit formula applicable in the range 321.08 #u # 360 and 0.311686 # Ke # 0.319314 is also de-rived:

u 5 5.604 1 (58.92 2 164Ke)0.31932 2 Ke (28)Equation (28) is very accurate, and can be used to

calculate all the quantities that depend on u, including the velocity V 5 8(Q/D2)/(u 2 sinu), provided that u .321.08. The following equation has been developed herefor the calculation of velocity when u # 321.08:

V 5 (0.625 2 0.815K e2.71)K e0.3071 2 (29)This explicit equation yields values that are within 61.0% of the exact values in the entire range 1 # u #321.08 (0.00002 # h/D # 0.9714).

COMPARISON OF DIFFERENTAPPROACHES

Saat (1990) and Wheeler (1992) refer to the work ofEscritt (1984), who recommended an alternative defini-tion of the hydraulic radius [Equation (6) herein].Wheeler (1992) states that Escritts definition of the hy-draulic radius agrees with the observations in Fig. 24 ofthe Manual of Practice (ASCE, 1976) within an averageof about 3%. The purpose of this section is to examine

D2/3S1/2}

nf

3p}2

u}2

Qnf}D8/3S1/2



Escritts approach and compare it with the use of the clas-sical definition of the hydraulic radius in the Manningequation.

While Saat (1990, 1992) and Wheeler (1992) wereconsidering the use of Equation (6) in conjunction withthe Manning formula, Escritt (1984) actually recom-mended the use of the following equation instead of theManning equation:

Q 5 76.253ARe0.62S1/2 (30)

where Re is the hydraulic radius defined as the flow areadivided by the sum of the wetted perimeter and one-halfof the width of the airwater surface:

Re 5 1 2 (6)Based on Equations (6) and (30), Escritt (1984) pre-

sented the values listed in Table 1 (Table 3 in Escritt,1984). As a footnote in the table, Escritt states that thesevalues are based on numerous tests on partially-filledsmall pipes and large sewers, and now accepted as moreaccurate than previous figures. Escritt goes on to statethe following: In the past, tables and diagrams were pre-pared showing partially-filled circular pipes and culverts,and on the basis of these theoretical calculations, it wasthought that at a depth of flow of 0.94 of the diameter,the discharge would be about 1.0757 times the flowing-full discharge. For the reasons just described this has beenproved a fallacy, as is shown in Table 3. These state-ments seem to require a closer scrutiny.

First, although Q/Qf increases monotonically with h/Din the table presented by Escritt, his formula predicts thepresence of a maximum in Q/Qf. It is an easy exercise toshow that the following expressions follow from Escrittsformula [Equation (30)].

5 (31a)(u 2 sin u)1.62}}(u 1 sin(u/2))0.62

1}2p

Q}Qf


D}4

5 1 20.62

(31b)

The Q/Qf and V/Vf values in the table are consistent withthese equations. Differentiating Equation (31a) with re-spect to u, setting d(Q/Qf)/du 5 0 and solving for u givesthe following values: u 5 322.04 (h/D 5 0.9728) and(Q/Qf)max 5 1.022. Therefore, Escritts formula also pre-dicts a maximum in Q/Qf, although this maximum isweaker and occurs at a greater flow depth than that pre-dicted by the Manning formula with the usual definitionof the hydraulic radius. Escritts table contains Q/Qf val-ues at the discrete values h/D 5 0.9 and h/D 5 1.0, andtherefore, the maximum at h/D 5 0.9728 is not visible inthe table. Figure 5 displays the curve (labeled Curve I)obtained from Equation (31a).

Also plotted in Fig. 5 are three additional curves. Theseare explained next.

Curve II is obtained using the Manning equation and theusual definition of the hydraulic radius [Equations (2) and(5)]. It is derived by assuming that n is independent of h/D:

5 (32)

When evaluated at the discrete points h/D 5 0.01, 0.02,. . . , 0.99, 1.00, the mean deviation of Equation (32) fromEquation (31a) is 15.8%; the maximum deviation is20.9%.

Curve III is also obtained using the Manning equationand the usual definition of the hydraulic radius [Equa-tions (2) and (5)]. It is derived by assuming that n de-pends on the depth of flow as given by Equations (19),(20), and (21):

5 1 221

(33)

When evaluated at the points h/D 5 0.01, 0.02, . . . , 0.99,1.00, the average deviation of Equation (33) from Equation(31a) is 3.3%; the maximum deviation is 5.9% (see Fig. 6).

Curve IV is obtained using the Manning equation andEscritts definition of the hydraulic radius [Equations (2)and (6)]. It is derived by assuming that n is independentof h/D:

5 (34)

When evaluated at the points h/D 5 0.01, 0.02, . . . , 0.99,1.00, the average deviation of Equation (34) from Equa-tion (31a) is 2.7%. As can be seen in Fig. 6, however, thepercent difference between the two equations increaseswhen h/D , 0.1. At the same discrete points, the averagedifference between Equations (33) and (34) is 2.9%.

The following conclusions can be drawn from the aboveinformation and Figs. 5 and 6. (1) when h/D . 0.1, the Q/Qf

(u 2 sin u)5/3}}(u 1 sin(u/2))2/3

1}2p

Q}Qf

n}nf

(u 2 sin u)5/3}}

u2/31

}2p

Q}Qf

(u 2 sin u)5/3}}

u2/31

}2p

Q}Qf


V}Vf

378 AKGIRAY

Table 1. Reproduced from Table 3 given in Escritt (1984).

h/D A/Af V/Vf Q/Qf

1 1 1 10.9 0.9480 1.0394 0.98530.8 0.8576 1.0189 0.87380.7 0.7477 0.9765 0.73010.6 0.6265 0.9173 0.57470.5 0.5000 0.8425 0.42130.4 0.3735 0.7517 0.28080.3 0.2523 0.6426 0.16210.2 0.1424 0.5095 0.07250.1 0.0520 0.3373 0.0176

values predicted by three of the approaches (curves I, III,and IV) are approximately the same. Escritt (1984) statedthat the Q/Qf values reported in Table 1 are in good agree-ment with experimental data. It is thus seen that similarQ/Qf values are obtained using the Manning equation pro-vided that either Escritts definition of the hydraulic radiusis employed, or the variation of n on h/D is taken into ac-count in accordance with the data presented by Camp(1946). (2) All four curves contain a maximum at a valueof h/D less than 1. Accounting for the dependence of n ondepth of flow or using Escritts definition of hydraulic ra-dius, however, suppresses the value of (Q/Qf)max.

It should be noted that the Manning equation [Equa-tion (2)] and Escritts formula [Equation (30)] do not givethe same Qf value when applied to completely filledpipes, although the two hydraulic radii are the same whenh/D 5 1 (Re 5 Rh 5 D/4). This means that the predictionof approximately the same Q/Qf values by the two for-mulas (as in curves I and III or curves I and IV in Fig.5) does not mean that the Q values calculated will be inagreement. Note also that Escritts formula [Equation(30)] does not include a roughness coefficient similarto Mannings n. No further consideration will be givento Equation (30) here. The use of Escritts hydraulic



Figure 5. Curve I: Escritts formula [Equation (30)] with Escritts hydraulic radius [Equation (6)]. Curve II: Manning equation,constant n 5 nf, usual definition of hydraulic radius [Equation (5)]. Curve III: Manning equation, variable n, usual definition ofhydraulic radius. Curve IV: Manning equation, constant n 5 nf, Escritts hydraulic radius.

Figure 6. Percent differences of (Q/Qf)III and (Q/Qf)IV from (Q/Qf)I.

radius [Equation (6)] in the Manning equation, how-ever, will be discussed in some detail in what follows.

In practice, the Manning equation is most commonlyapplied by using a constant value for n (Chow, 1959;Metcalf & Eddy, 1981; Benefield et al., 1984). Equation(5) is normally used to define the hydraulic radius. Thisapproach yields Curve II in Fig. 5.

Consider the other two alternative approaches that em-ploy the Manning equation. In one approach, the depen-dence of n on h/D is taken into account, and the usual def-inition of the hydraulic radius is employed [Curve III andEquation (33)]. In the other approach, n 5 nf is assumedto be constant and Escritts definition of hydraulic radiusis employed [Curve IV and Equation (34)]. The possibil-ity of using this latter approach was mentioned by Saat(1990) and recommended by Wheeler (1992). Since the Qfvalues predicted by these two approaches are the same(both using the Manning equation), the closeness of Q/Qfvalues for flow depths h/D . 0.1 implies the closeness ofpredicted Q values in the same range.

To make the comparison concrete, assume that nf, D,S, and h/D are known, and the value of Q will be calcu-lated. Let QIII and QIV denote the Q values calculated bythe two approaches, respectively. Using Equations (2),(5), and (6), the following is obtained:

5 5 1 22/3

1 25 1 2

22/3f(u) (35)

Note that f(u) 5 n/nf is given by Equations (19)(21).Figure 7 displays QIV/QIII values obtained using Equa-tion (35). It is seen that QIII and QIV differ by at most64% in the range 0.11 # h/D # 1.0. Further insight canbe gained by forming the ratios QII/QIII and QII/QIV,

u 1 sin(u/2)}}

u

n}nf

Re}Rh

(A/nf)Re2/3 S1/2}}(A/n)Rh

2/3S1/2QIV}QIII

where QII represents the value calculated assuming con-stant n and the usual definition of Rh:

5 5 1 22/3

5 1 22/3

(36)

5 5 1 2 5 f(u) (37)The rightmost expression of Equation (36) was plot-

ted in Fig. 4 (dashed line) together with the right-handside of Equation (37) [f(u), solid line]. Again, it is seenthat the difference between these two functions is notvery large, and their ratio QIV/QIII [Equation (35)] is notfar removed from 1.0 when 0.1 # h/D # 1.0.

From Fig. 4, it is apparent that each of QIII and QIV dif-fers from QII by as much as 15 to 30% in the range 0.2 ,h/D , 1.0. (One caveat: it is assumed that n 5 nf is the valueused in calculating QII. If the constant value used for n isthe value corresponding to, say, 50% filled pipe, then asmaller value for QII will be predicted.) It is also seen that,when compared to using a constant n 5 nf value in con-junction with the usual definition of the hydraulic radius(giving QII), either taking the variation of n with h/D intoaccount (giving QIII), or using Escritts definition of hy-draulic radius (giving QIV), leads to comparable decreasesin the predicted values of Q in the range 0.1 # h/D # 1.0.

No attempt has been made here to compile past experi-mental data to determine the most accurate approach. It isapparent, however, that accounting for the variation of theroughness coefficient with flow depth as suggested byCamp (1946) leads to good general agreement with the ex-perimental data referred to by Escritt (1984) and Wheeler(1992). Simple explicit equations have been presented here

n}nf

(A/nf)Rh2/3S1/2}}(A/n)Rh

2/3S1/2QII}QIII

u 1 sin(u/2)}}

u

Rh}Re

(A/nf)Rh2/3 S1/2

}}(A/nf)Re2/3 S1/2

QII}QIV

380 AKGIRAY

Figure 7. QIII: obtained with variable n and the usual definition of Rh. QIV: obtained with constant n and Escritts definitionof the hydraulic radius Re.

to avoid iterative calculations when using any one of thethree approaches. It will be up to the judgment of the de-signer to choose the most appropriate approach. Whateverthe approach employed is, the equations presented hereinwill facilitate calculations.

CALCULATION OF SLOPE FOR GIVENVELOCITY AND FLOW RATE

Assume nf, D, V, and Q are known, and one needs tocalculate S and h/D. The calculations involved in this casewere described previously. As an alternative to themethod proposed by Esen (1993), Li (1994) presentedthe following two equations applicable over 0 , u ,360:For 0 , 8A/D2 , p:

For p , 8A/D2 , 2p:

These equations are very accurate (Esen, 1994). Tokeep things in perspective, however, it should be re-membered that the purpose here is to solve the equa-tion

5 u 2 sin u (40)

for u when the left-hand side (8A/D2 5 8(Q/V)/D2) isknown. Remembering that the Manning equation is itselfan approximate relation, and considering the possiblelack of precision in the other variables in a system (e.g.,n values), Equations (38) and (39), which are arguablyquite cumbersome for manual computations, may oftenbe more accurate than necessary. The following simplerformula is therefore presented here for use in quick man-ual calculations (see Fig. 8):

u 5 2.51Y20.34[cos21(1 2 0.3183Y)]1.33 (41)

where Y 5 8A/D2. The maximum error (in u) due to theuse of this equation will be less than 3% in the range0.05 , h/D , 1.0. The mean error at the discrete points10, 20, . . . , 350, 360 is 1.7%. Once u is computed,Equation (3) can next be used to calculate h/D. To cal-culate h/D directly, the following approximate relationmay be more convenient:

8A}D2



u 5 2p 2 (39)(2p 2 8A/D2) 1 sin 136(2p 2 8A/D2)2 2 36(2p 2 8A/D2) cos 136((2p 2 8A/D2)2}}}}}}}}}

1 2 cos 1 36(2p 2 8A/D2)2

5 0.0047Y3 2 0.0453Y2 1 0.2554Y (42)

The maximum error uhpredicted 2 hexactu/D in Equation(42) is approximately 0.02. This means that the error inthe predicted value of h will be less than ,2% of the pipediameter. The error involved in the two-step calculation[u by Equation (41), and then h/D by Equation (3)] isabout half of this value.

Next, consider the calculation of S. Since D, V, and Qare known in this case, u and h/D are uniquely fixed byEquations (3) and (40). The value of S, on the other hand,depends on the approach adopted in applying the Manningequation. Employing the usual definition of the hydraulicradius [Equation (5)], the following can be written:

S 5 1(D/2Q)4/3V10/3n2f 2 f 2(u)u4/3 (43)

h}D

Equation (19) may be used to calculate f(u). If n 5 nf isassumed, then f(u) ; 1 in Equation (43). If Equation (6)is used to define the hydraulic radius, and assuming thatn 5 nf, one gets:

S 5 1(D/2Q)4/3V10/3n2f 2 (u 1 sin(u/2))4/3 (44)Figure 9 compares the functions u4/3 (curve II), f 2(u)u4/3

(curve III), and [u 1 sin(u/2)]4/3 (curve IV). It is again seenthat accounting for the variation of n with the depth of flowor using Escritts definition of the hydraulic radius lead tosimilar predictions: The slopes calculated are within 66%of each other when 0.1 # h/D # 1.0. On the other hand,accounting for the variation of n as suggested by Camp(1946) leads to considerably higher slope values (curve III)compared to assuming n 5 nf (curve II).

The function f 2(u)u4/3 (curve III) is well approximatedby the following equation in the range 0.05 , h/D , 0.98and 0.1 , (8A/D2) , 6.25:

f 2(u)u4/3 < 3.9Y 0.512 (45)

This formula can be combined with Equation (43) to es-timate S directly without having to calculate u or h/Dfirst:

S 5 4.49n2f Q20.82D0.309 V 2.82 (46)

u 5 (38)(8A/D2) 1 sin1 348A/D22 2 348A/D2 cos1 348A/D22}}}}}}

1 2 cos1 348A/D22

This equation applies if n varies with h/D as reported byCamp (1946), and yields accurate results provided that0.05 , h/D , 0.98.

APPLICATIONS

The following examples illustrate the applications ofsome of the new equations proposed in this paper.

Example 1

Determine the depth of flow and velocity in a sewerwith a diameter of 300 mm laid on a slope of 0.005 m/m

with a constant n value of 0.015 and a discharge of 0.01m3/s. Employ the usual definition of the hydraulic radius.From Equation (16),

Kh 5 nf QD28/3S21/2 5 5 0.0526

From Equation (17),

u 5 1.28(0.0526)20.26 cos21(1 2 5.965(0.0526)) 52.242 radians

From Equation (3),

5 (1 2 cos(2.242/2)) 5 0.2831}2

h}D

( 0.015)(0.01)}}0.38/30.0051/2

382 AKGIRAY

Figure 8. Angle u as a function of 8A/D2 5 8Q/(VD2).

Figure 9. S(D/2Q)24/3V210/3n22f as a function of 8A/D2 5 8Q/(VD2).

From Equation (4),

A 5 (2.242 2 sin 2.242) 5 0.0164 m2

Answers:

h 5 (0.283)(0.3 m) 5 0.084 m 5 84.8 mm (exact value is 83.4 mm)

V 5 Q/A 5 0.01/0.0164 5 0.609 m/s (exact value is 0.624 m/s).

Alternatively, Equation (14) can be used to calculate Vdirectly:

V 5 0.7591

11 2 20.05264/131 2 5 0.631 m/s

Example 2

Calculate the minimum slope of a sewer to carry a flowof 0.01 m3/s. The minimum velocity is specified as 0.5m/s and as a preliminary value the diameter of the sewerwill be assumed to be 0.45 m. Assume nf 5 0.013. [Thisexample was used by Esen (1993) and Li (1994) to il-lustrate their calculation methods.]

First calculate 8A/D2 5 8Q/(VD2) 5 8(0.01)/0.5(0.45)2

5 0.7901.From Equation (42),

5 0.0047(0.7901)3 2 0.0453(0.7901)2

1 0.2554(0.7901) 5 0.176

h 5 (0.45)(0.176) 5 0.079 m

This should be compared with the exact value0.0825m. The error is 0.35 cm, that is, ,0.8% of the pipediameter. Equation (15b) predicts 0.0834 m, which ismore accurate. It should be remembered, however, thatEquation (15) is applicable for h/D , 0.4 only, whereasEquation (42) can be used for essentially all values ofh/D. Alternatively, using Equation (41), better accuracycan be obtained albeit in two steps:

u 5 2.51(0.7901)20.34[cos21

(1 2 0.3183(0.7901))]1.33 5 1.7730 radians 5 101.58

Substituting this value into Equation (3),

5 (1 2 cos(1.7730/2)) 5 0.184, and h

5 (0.45m)(0.184) 5 0.0828 m

The exact values of u and h/D are 1.7703 radians(5101.43) and 0.183, respectively. To estimate theslope, the calculated value of u may first be used in Equa-tions (5) and (19) to calculate Rh and n, respectively, and

1}2

h}D

h}D

0.32/30.0051/2}}

0.0150.0526}

2

(0.3m)2}

8

then Equation (1) can be employed. It is simpler to useEquation (46):

S 5 4.49(0.013)2(0.5)2.82(0.45)0.309

(0.01)20.82 5 0.00367

The exact value is 0.00368, that is, the error is 0.28%. Ifthe dependence of n on h/D is ignored, and the value n 5nf 5 0.013 is used in the calculations, then Equation (43)gives [with f(u) ; 1]:

S 5 ((0.45/2 3 0.01)4/30.510/30.0132)1.7734/3 5 0.00229

The exact value is 0.00228. Li (1994) obtained0.00227, whereas Esen (1993) found 0.00232. These au-thors did not consider the variation of n with the depthof flow. This example illustrates the fact that the effectof this variation may be quite significant.

SUMMARY AND CONCLUSIONS

The application of the Manning equation to partiallyfilled circular pipes is considered. Three different ap-proaches based on the Manning equation are analyzed andcompared. The hydraulic radius that appears in the Man-ning formula is normally defined as the area of flow di-vided by the wetted perimeter. The use of this definition inconjunction with the assumption that Mannings n is con-stant is the most commonly used approach in practice. Analternative definition for the hydraulic radius, that is, flowarea divided by the sum of the wetted perimeter and one-half of the width of the airwater surface, was proposed byEscritt (1984) and recommended by Wheeler (1992). Man-nings n is assumed to be constant in this approach as well.The third alternative approach considered here is the use ofthe data presented by Camp (1946) to account for the de-pendence of n on h/D while preserving the usual definitionof the hydraulic radius. It is shown that the latter two ap-proaches give approximately the same results (predicted Qvalues are within 64%) in the range 0.1 # h/D # 1.0. Bothapproaches yield Q values about 2030% less than that ob-tained by assuming constant n 5 nf in conjunction with theusual definition of the hydraulic radius.

A significant part of this work concerns the developmentof simple and accurate explicit equations that can be usedto calculate the depth of flow (h/D) and velocity (V) whenD, S, and Q are given. These equations obviate the needfor iterative calculations. Wheeler (1992) noted that hy-draulic designers should have a choice of methods of com-putation. This not only gives them the option of exercisingjudgment about unusual conditions, but enables them to de-velop an envelop of curves bracketing upper an lower con-ditions. In line with this point of view, explicit equationsfor each of the mentioned three methods have been devel-



oped and presented. Another set of explicit relations facil-itating the estimation of u, h/D, and S from known valuesof D, Q, and V are presented. A simple explicit equationthat takes the dependence of Mannings n on h/D into ac-count is developed to estimate S from given values of D,Q, and V. Examples are provided to illustrate the applica-tions of the proposed equations. All of the formulas pro-posed in this paper are accurate enough to be used in com-puter calculations (so that simpler and more robustprograms can be written), and simple enough to be usedwith a hand calculator.

APPENDIX

For completely filled pipe flow, it is widely acceptedthat the use of the Darcy-Weisbach equation in conjunc-tion with the Colebrook formula (or its equivalent, theMoody diagram) is the most accurate calculation method.The Darcy-Weisbach equation is written as follows:

hf 5 f (47)

The Colebrook formula relates the friction factor f to therelative roughness e/D and the Reynolds number NR:

5 22 log3 1 4 (48)Since the Colebrook formula is implicit in f, a number ofequivalent explicit equations have been proposed in theliterature. One such equation is the Haaland equation(Finnemore and Franzini, 2002):

5 21.8 log31 21.11

1 4 (49)6.9}NRe

}3.7D

1}f

2.51}NRf

e}3.7D

1}f

V 2}2g

L}D

When applied to completely filled pipe flow, the Man-ning equation should lead to predictions consistent withthe above equations. Setting u 5 2p, h 5 D, n 5 nf, andRh 5 D/4 in the Manning equation and combining it withEquations (47) and (49) gives, after some algebraic ma-nipulation, the following result:

nf 5 (50)

This equation shows that, for a given pipe material (givenabsolute roughness height e), nf depends on the pipe di-ameter and the Reynolds number. For large values of theReynolds number, Equation (50) simplifies to the fol-lowing:

nf 5 (51)

If the Colebrook formula and the Darcy-Weisbach equa-tion are assumed to be applicable to partially filled pipesas well, the following equation is obtained for large val-ues of Reynolds number (Massey, 1989; Finnemore andFranzini, 2002):

n 5 (52)

Using Equation (5) for the hydraulic radius and combin-ing Equations (51) and (52), the following is obtained:

5 (53)

According to this equation n/nf is approximately unitywhen h/D . 0.1 (see Fig. 10), that is, Equation (53)

log(3.7D/e)}}}log(3.7D(u 2 sin u)/(eu))

(u 2 sin u)1/6}}

u1/6n}nf

Rh1/6

}}}

42g log3}14.e8Rh} 4

0.20D1/6}}

2g log3}3.7eD

}4

0.22D1/6}}}}22g log31}3.7

eD

}21.11

1 }6N.

R

9}4

384 AKGIRAY

Figure 10. The ratio n/nf according to Equation (53).

does not explain or fit the data presented by Camp(1946). It may be noted that there is a discontinuity inEquation (53) at [(u 2 sinu)/u 5 e/(3.7D)]. The devia-tion of n/nf from unity increases as the point of dis-continuity is approached. When Equation (53) is in-serted into Equation (33), and values between 0.04 and0.0001 (typical range of e/D values in the Moody dia-gram) are employed for e/D, the resulting Q/Qf curves(not shown) are either visually indistinguishable fromCurve II of Fig. 5 or are very close to it (Curve II isobtained using constant n 5 nf and the usual definitionof Rh). The deviation of n/nf from unity for small val-ues of h/D (see Fig. 10) does not lead to a visible changein the Q/Qf curve because Q/Qf values are very closeto zero for h/D , 0.1.

NOMENCLATURE

A area of flowD pipe diametere roughness heightf functional relation between n/nf and uf (In the Appendix) friction factor defined by the

Darcy-Weisbach equationh Water depthn Mannings roughness coefficientnf the value of n when h 5 DNR Reynolds numberQ volumetric flow rateQf volumetric flow rate when the pipe is fullRh hydraulic radius based on Equation (5)Re hydraulic radius based on Equation (6)S slope of the energy grade lineV velocity of flowVf velocity when the pipe is fullY intermediate quantity defined as 8A/D2

u surface angle in radians (Fig.1)

REFERENCES

ASCE. (1976). Manual of Practice no.37: Design and Con-struction of Sanitary and Storm Sewers. New York. (Quotedin Wheeler (1992).)

BENEFIELD, L.D., JUDKINS, J.F., and PARR, A.D. (1984).Treatment Plant Hydraulics for Environmental Engineers.Englewood Cliffs, NJ: Prentice-Hall.

CAMP, T.R. (1946). Design of sewers to facilitate flow. SewageWorks J., 18, 316.

CHOW, V.T. (1959). Open Channel Hydraulics. New York:McGraw-Hill.

ESCRITT, L.B. (1984). Flow in sewers. In: Sewerage andSewage TreatmentInternational Practice. New York: JohnWiley & Sons.

ESEN, I.I. (1993). Design of sewers based on minimum ve-locity. ASCE J. Environ. Eng. 119(3), 591594.

ESEN, I.I. (1994). Closure by author. ASCE J. Environ. Eng.120(5), 1350.

FINNEMORE, E.J., and FRANZINI, J.B. (2002). Fluid Me-chanics, 10th ed. New York: McGraw-Hill, pp. 427428.

GIROUD, J.P., PALMER, B., and DOVE, J.E. (2000). Calcu-lation of flow velocity in pipes as a function of flow rate.Geosynthet. Int. 7(46), 583600.

GRANT, D.M. (1992). ISCO Open Channel Flow Measure-ment Handbook, 3rd ed. Lincoln, NE: ISCO.

LI, K.S. (1994). Discussion of design of sewers based on min-imum velocity by I.I. Esen. ASCE J. Environ. Eng. 120(5),13481350.

MASSEY, B.S. (1989). Mechanics of fluids, 6th ed. London:Chapman & Hall.

METCALF & EDDY. (1981). Wastewater Engineering: Collec-tion and Pumping of Wastewater. New York: McGraw-Hill.

SAATI, A.M. (1990). Velocity and depth of flow calculationsin partially filled pipes. ASCE J. Environ. Eng. 116(6),12021212.

SAATI, A.M. (1992). Closure by author. ASCE J. Environ.Eng. 118(3), 454.

WHEELER, W. (1992). Discussion of velocity and depth offlow calculations in partially filled pipes by A.M. Saat.ASCE J. Environ. Eng. 118(3), 451454.



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Hydraulics - Formulae for Partially Filled Circular Pipes (Akgiray, 2004)