Art s Fluid Flow

Art Montemayor

Nov 01, 2005 Rev: 0

Fluid FlowThis workbook will serve to present information for the estimating of the pressure drop as well as the friction loss of fluids flowing through pipe, fittings, and valves. In order to understand and be able to employ the many theories and mathematical equations that are the mainstay tools of Fluid Mechanics, it is important to also understand their origin and their limitations as well as their strengths. The pressure loss suffered by a length of straight pipe is primarily due to the effect of friction generated by the flowing fluid against the internal pipe wall. In 1883, Osborne Reynolds published his famous paper entitled "An experimental investigation of the circumstances which determine whether motion of water shall be direct or sinuous and of the law of resistance in parallel channels" (Paper 44). This paper, published in the Philosophical Transactions of the Royal Society, proved to be a classic in the literature of the science of fluid motion and had a profound effect on the development of fluid mechanics in the widest sense. It contained the enunciation of the dimensionless group, the Reynolds Number, which is given:

Re = D v / where, Re D v = The Reynolds Number, a dimensionless number = the internal diameter of the pipe, usually in feet = the mean fluid velocity in the pipe, usually in ft/sec = the density of the fluid, usually in lb/ft3 = the absolute viscosity of the fluid, usually in lbs mass per foot-second

Note: viscosity is usually measured and given in centipoise (cP) and these are converted as 1.0 cP = 0 lbm/ft-sec = 2.42 lbm/ft-hr

This was the turning point in fluid flow. It set in motion all that followed to date. The Reynolds number basically defined two fluid flow regions: 1. the region of Viscous (or Laminar) Flow; and, 2. the region of Turbulent Flow. For engineering purposes, flow in pipes is usually considered to be laminar if the Reynolds number is less than 2,000. It is considered turbulent if the Reynolds number is greater than 4,000. Between these two basic regions lies the so-called "critical zone" where the flow - varying between laminar and turbulent, or in the process of change - is considered unpredicatable. The friction loss of a fluid flowing through a pipe is but a special case of a general law of the resistance between a solid and fluid in relative motion. Consider a solid body, of any desired shape, immersed in a stream of flowing fluid. Let the length of this body, measured perpendicular to the general direction of flow of the fluid, be D, and let the area of contact between the solid and the fluid be A. If the velocity of the fluid past the body be small in comparision to the velocity of sound, it has been found experimentally that the resisting force depends only on the roughness, size, and shape of the solid and on the velocity, density , and viscosity of the fluid. Through the use of Dimensional Analysis as applied to these quantities, it can be shown that:

F v2 Dv = f A gC where, F = Total resisting force; v = velocity of fluid past the body; Page 1 of 148

FileName: 83236863.xls WorkSheet: Fluid Flow Background

Art Montemayor = fluid density gC = gravitational constant, lbm-ft/lbf-sec2 f = a function whose precise value must be determined, case by case Note the appearance of the Reynolds Number in this theoretical equation.

Nov 01, 2005 Rev: 0

In the particular case of a fluid flowing through a pipe of length L and with a circular cross-sectional flow area, the total force resisting the flow must equal the product of the area of contact between the fluid and the pipe wall and the F/A of the above equation. The pressure drop will equal this product divided by the pipe's cross-sectional area, since pressure is measured in force per unit area. This is expressed as:

The pressure loss in pipes due to friction arises from two mechanisms: 1) surface friction existing at the point of fluid contact with the parent metal of the pipe; Page 2 of 148 FileName: 83236863.xls WorkSheet: Fluid Flow Background

Art Montemayor 2) turbulent dissipation due to sudden changes in mean fluid velocity.

Nov 01, 2005 Rev: 0

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FileName: 83236863.xls WorkSheet: Fluid Flow Background

Art Montemayor

Valve Resistance to Flow

November 24, 1997 Rev: 0

DURCO VALVESDurco SLEEVELINE non-lubricated plug valves are used as block valves and as 3-way directional valves. One style of the straightway model, V-PORT, has a triangular-shaped port that offers better control valve characteristics. The sleeve used is made of PTFE for positive shut-off, self-lubrication and extended service life. Another model is the T-LINE, which has the plug and all other wetted internal parts sleeved with PTFE as well as the plug proper itself. The Chemical service butterfly control valve is 100% PTFE-sleeved on all internal wetted parts.

Valve size, in

Standard SLEEVELINE model Max. K Cv

V-Port SLEEVELINE model Max. Cv K

Standard T-LINE model Max. Cv 5.6 12.0 33.9 95.4 199.0 343.0 813.0 1,105.0 1,424.0 K 1.7817 1.9644 0.7779 0.4973 0.3612 0.6155 0.3463 0.9489 1.8058

V-Port T-LINE model Max. Cv K

BIG MAX Butterfly model Max. Cv K

CHEMICAL Service Butterfly model Max. Cv K

1/2 3/4 1.0 1 1/2 2.0 3.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 24.0

48.8 83.5 153.4 322.0 555.0 955.0 1,410.0 2,130.0 3,430.0 7,000.0

0.3754 0.6491 0.6079 0.6984 0.7430 1.2704 1.8419 1.9705 2.9192 1.1957

27.2 31.3 53.8 121.0 190.0 400.0

1.2084 4.6197 4.9419 4.9460 6.3397 7.2414

23.3 39.6 43.8 66.3

1.6467 2.8861 7.4561 16.4739

210.0 415.0 1,115.0 2,020.0 3,230.0 4,825.0 6,800.0 8,800.0 11,500.0 14,000.0 20,500.0

1.6420 1.3289 0.9320 0.8974 0.8569 0.7963 0.7427 0.7566 0.7096 0.7298 0.7058

172.0 300.0 670.0 1,190.0 1,830.0 2,650.0 3,690.0 4,842.0 6,129.0 7,566.0 10,895.0

2.4477 2.5429 2.5810 2.5859 2.6695 2.6398 2.5223 2.4990 2.4983 2.4988 2.4988

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Electronic FileName: 83236863.xls WorkSheet: Durco Valves

Art Montemayor

Jamesbury Valves


JAMESBURY Valves ---- Cv values This valve flow resistance data is taken from the Jamesbury Valve Catalog. Art Montemayor

MODEL CLINCHER, Type 2000; screwed Style A DOUBLE-SEAL; screwed Series 4000, REDUCED PORT Series 4000, FULL PORT; Le in ft Type 5150 & 530S; flanged Floating Ball; flanged Trunnion model; flanged Typr 6150 & 6300; Full Port Floating Ball; Full Port Trunnion Model; Full Port 3-Way Flanged Ball Valve FIRE-TITE Butterfly Series F815W & F815L Series F830W & F830L

1/4 11 8.3

3/8 16 8.3

1/2 16 8.3 13 0.13 9

3/4 37 14 33 0.19 19

1 49 35 44 0.24 45

1 1/4 49 55 46 0.27

1 1/2 100 94 95 0.31 125

VALVE SIZE, in inches 2 3 4 6 115 115 111 350

8

10

12

14

16

18

20

165

350

690 765 1890 1890 3900 3900 6700 5100 8100 11000 16000

50

100

270 500 1330 2560 5600 10000 10000 16400 23800 27000 37000 47000 60000 440 880 1500 2300

50

155

270

165 165

400 400

950 950

1800 1800

2900 2900

4300 4300

5800 5200

8000 10500 14000 6900 9300 11300

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Electronic File: 83236863.xls WorkSheet: Jamesbury Valves

Art Montemayor ORBIT Valves --- Cv values Source: Orbit Valve Catalog

Valve Flow Resistance


For each Orbit valve there is a flow coefficient Cv. This is defined as the flow in gallons per minute of water at 60 oF with a pressure drop of 1 psi across the valve. By using the following simplified formulas, the pressure drop for a given set of flow conditions may be calculated: FOR LIQUIDS:

p = G(Q/Cv)2

FOR GASES:

p = 541 x 10-9 (Q/Cv)2(GT/P)

Where, p = PSI drop across valve G = Specific gravity of liquid (water=1) Q = Flow in gpm Cv = Valve coefficient

p = PSI drop across valve G = Specific gravity (Air @ 14.7 psi & 60 oF = 1.0) T = Absolute temperature, oR P = Line pressure, psia Q = Flow in SCFH (@ 14.7 psi & 60 oF) Cv = Valve coefficient

MODEL ANSI, 150# flange Regular Port; flanged Full Port; flanged ANSI, 300# flange Regular Port; flanged Full Port; flanged ANSI, 1500# flange Regular Port; flanged Full Port; flanged

1

1.5

2

3

4

6

SIZE, in inches 8

10

12

14

16

18

20

49

123

153 439

200 1,018

572 2,080

762 4,865

2,090 6,905

4,537 11,106

7,513 17,148

13,119 30,556

10,129 41,551

21,217

15,167

49

123

160 407

243 920

610 2,080

760 4,865

1,560 6,778

4,513 11,106

7,513 17,148

13,119 30,556

10,129 41,551

21,217

15,167

31

267

205 676

505 1,359

921 2,951

2,081 7,301

3,986 11,603

5,912 17,030

11,185

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Electronic File: 83236863.xls WorkSheet: Orbit Valves

Art Montemayor



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Electronic File: 83236863.xls WorkSheet: Orbit Valves

Art Montemayor VOGT Valves --- Cv values (Reference: VOGT Catalog F-12, 1980) TYPICAL CV FACTORS FOR VOGT VALVES & STRAINERS (FOR ALL VALVES IN FULL-OPEN POSITION)

Valve Fittings Resistance to Flow


Compact Swing Check

Full Port Swing Check

Needle Point Globe

Needle Point Angle

Meter Globe

Meter Angle

12443 Flow Control

15443 Flow Control

Nominal Valve Size inches 1/4 3/8 1/2 3/4 1 1 1/4 1 1/2 2 2 1/2 3 4

2.6 3.9 9.3 9.5 7.5 80.5 83.6 96.7 165.0 247.0 410.0

2.6 3.9 9.3 28.7 46.5 80.5 109.6 181.0 258.0 398.0 685.0

1.3 2.0 2.8 3.2 6.8 15.2 19.6 27.0

1.5 2.9 3.6 6.7 11.9 20.3 26.1 43.4 57.6 89.0 153.3

0.6 1.4 1.8 3.5 5.8 7.6 13.2 21.5

0.8 0.8 0.8 1.0 1.5 3.5 5.5 5.5

1.3 1.9 2.7 3.2 6.7 14.8 19.2 26.6

1.5 2.9 3.6 6.7 11.9 20.3 26.1 43.4 57.6 89.0 153.3

9.0 8.8 16.0 40.0 65.0

7.6 13.4 21.7 37.5 51.1 84.2

5.9 10.3 16.7 28.9 39.2 64.7

1.8 3.3 5.2 9.1 14.7 25.5 34.7 57.1 81.5

1.0 2.1 2.7 5.2 8.7

1.2 1.2 1.2 1.5 2.3 5.2 8.2 8.2

1.5 2.4 4.5 9.7 14.6

1.5 2.4 4.5 12.2 19.6

2.5 4.0 6.5 14.0 23.0 32.0 52.0

6.0 14.0 14.0 40.0 40.0

PIPE FITTINGS 45o Elbow Standard Elbow Long Sweep Elbow Medium Sweep Elbow Close Return Elbow Tee-Straight Flow Tee-Side Outlet Flow

K 0.42 0.90 0.60 0.75 2.20 0.60 1.80

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Electronic File: 83236863.xls WorkSheet: Vogt Valves

Y" Pattern

Compact Gate

Compact Globe

Compact Check

Strainers

Full Port Gate

Full Port Globe

Full Port check

Full Port Angle

H. & V. Check

Art Montemayor TYPICAL PIPE-TANK RESISTANCES Entrance Loss: From Tank to Pipe Type of Resistance Illustration K



Flush Connection

0.50

Projecting Pipe

0.78

Slightly Rounded

0.23

Well Rounded

0.04

Exit Loss: From Pipe to Tank

Any of the above, reversed1.00

d2 d1d1/d2 K 0.1 0.46 0.2 0.45 SUDDEN ("ABRUPT") CONTRACTION 0.3 0.4 0.5 0.6 0.7 0.42 0.40 0.36 0.28 0.19 0.8 0.10 0.9 0.04

d1/d2 K

0.1 0.98

0.2 0.92

SUDDEN ("ABRUPT") EXPANSION 0.3 0.4 0.5 0.6 0.7 0.83 0.71 0.56 0.41 0.28

0.8 0.13

0.9 0.04

d1 d2Page 9 of 148 Electronic File: 83236863.xls WorkSheet: Vogt Valves

Art Montemayor



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Electronic File: 83236863.xls WorkSheet: Vogt Valves

Art Montemayor KTM Ball Valves --- Cv values

Ball Valve Flow Resistance

December 1, 1997 Rev: 0

TYPICAL CV FACTORS FOR KTM BALL VALVES (FOR ALL VALVES IN FULL-OPEN POSITION) Valve Size inches 1/2 3/4 1 1 1/2 2 2 1/2 3 4 6 8 10 12 14 16 18 20 Reduced Percent Reduction Full Bore Bore Ball in Flow by using Ball Cv Cv reduced bore 26 50 94 260 480 750 1,300 2,300 5,400 10,000 16,000 24,000 31,400 43,000 57,000 73,000 KTM Ball Valves Cv Values80

70

420 770 1,800 2,500 4,500 8,000 12,000 14,000 18,000 22,000

67.69 66.52 66.67 75.00 71.88 66.67 61.78 67.44 68.42 69.86

60

% of Maximum Cv

50

10 1.5 30 6 40 10 50 15 60 22 70 32.5 80 50 90 75

40

30

20

KTM Ball Valves have inherent equal percentage flow characteristics and are ideally suited for control applications over the widest possible range of flow. Equal percentage flow characteristics, in simple terms, means that a given percentage change in the valve opening will produce the same percentage change in flow.

10

0 0 10 20 30 40 50 60 70 80 90 100 % of Ball Opening

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Electronic File: 83236863.xls WorkSheet: KTM Ball Valves

Art Montemayor



KITZ Valves --- Cv valuesTYPICAL CV FACTORS FOR KITZ VALVES (FOR ALL VALVES IN FULL-OPEN POSITION) From Zidell Valve Corp.-Stafford, TX & KITZ catalog No. 510-E Trunnion Ball Reduced Bore ANSI CLASS 1500 Trunnion Ball Reduced Bore ANSI CLASS 2500 150 410 590 1,400 3,050 5,350 Trunnion Ball Reduced Bore ANSI CLASS 150 Trunnion Ball Reduced Bore ANSI CLASS 300 Trunnion Ball Reduced Bore ANSI CLASS 900 Trunnion Ball Reduced Bore ANSI CLASS 600

Nominal Valve Size inches

Gate

Globe

2.0 3.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 24.0 30.0 36.0

230 530 930 2,380 4,180 6,840 9,690 13,000 17,380 22,800 28,500 41,800 68,000 96,000

44 105 180 446

75 175 320 740 1,350 2,200 3,600 5,200 7,000 8,500 9,800 11,600

2 500 180 1,350 545 2,500 790 5,300 1,945 10,500 4,050 17,500 6,900 26,300 13,100 31,850 14,600 43,300 19,750 57,300 27,750 74,500 27,100 112,300 39,700 179,300 63,100 258,300

460 195 1,150 535 2,200 765 5,290 1,945 9,600 4,040 16,750 7,100 25,500 13,200 30,050 14,580 41,700 19,800 55,370 28,050 72,300 27,130 109,150 39,800 171,200 63,100 243,050

400 180 1,050 550 1,850 745 4,460 2,220 8,730 4,065 14,250 7,050 22,550 13,350 28,400 14,300 38,150 20,350 50,950 28,300 65,600 27,250 98,150 40,750 158,900 68,950 226,300

187 510 740 2,035 4,050 7,025 13,300 14,200 19,750 27,300 26,900

330 935 1,760 4,405 8,475 14,205 21,430 26,800 36,700 48,700 62,500 94,050

187 510 740 1,930 3,860 6,670 12,630 13,490 18,760 20,470 25,500

330 830 1,660 4,100 8,010 13,310 17,070 24,275 33,215 43,400 55,930 84,025

Cv VALUES FOR KITZ LOW-COST, ONE-PIECE, THREADED, CLASS 600, REDUCED BORE BALL VALVES ARE: Size, in. Cv 0.25 1.0 0.38 2.5 0.5 5.5 0.75 10.0 1 15.0 1.25 20.0 1.5 37.0 2 60.0

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Electronic File: 83236863.xls WorkSheet: Kitz Valves

Trunnion Ball Full Bore ANSI CLASS 2500 300 740 1,460 2,600 5,370 8,630 12,500

Trunnion Ball Full Bore ANSI CLASS 150





Swing Check

Art Montemayor



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Electronic File: 83236863.xls WorkSheet: Kitz Valves

Art Montemayor

Marlin/Crane Check Valves

Feb 16, 1999 Rev: 0

Marlin Style M High Performance Dual Disc Check ValvesSource: Marlin Catalog # MC - 196 Valve Size, In. Liquid Cv Value 2 2.5 3 4 5 6 8 10 12 14 16 18 20 24 30 36 42 48 54 60 66 72 54.3 82.1 139.3 271.3 482.5 725 1509 2640 4075 5303 7340 9940 12960 20500 37200 59000 92000 126000 186000 217000 280000 325000

Marlin Check Valves350000 300000

Cv Value for Liquid

250000 200000 150000 100000 50000 0 0 10 20 30 40 50 60 70 80

Note: These are very low pressure drop, wafer type of valves that require little "cracking" pressure to start to open.

Nominal Valve Size, Inches

02/16/99 As per Carlos Davila, Marlin/Crane Valves; Montgomery, TX: The "Cracking Pressure" required for Marlin Valves with standard springs is 6" WC and for valves with special, low torque it is 3.5" WC.

Page 14 of 148

Electronic FileName: 83236863.xls WorkSheet: Marlin Check Valves

Art Montemayor

Velan Valves Cv Flow Coefficients

July 27, 1998 Rev: 0

Cv FOR VELAN VALVESSources of Flow Coefficient (Cv) values: Velan Catalogs #VEL-SFV-96 & #VEL-BV-97 Gate Valve Reduced Port 150-800 1/2 3/4 1 1-1/2 2 2-1/2 3 4 6 8 10 12 14 16 18 20 24 7 14 30 100 160 Gate Valve Globe Valves Full Port St'd Design 150-800 3 4 6 14 33 900-1500 3 4 8 19 29 Globe Valves Bellows Seal Globe Valves Y - Pattern 4500 4 7 9 23 47 Piston Check Valves (Vertical) Ball Valves Split Body SB-150/300/600 Ball Valves Top-Entry Body SB-150/300/600 Ball Valves Unibody UB-150/300

Size Inches

Swing Check Valves 150-800 9 10 22 55 82 600-2680 7 8 12 25 60

900-1500 150-1500 14 14 30 100 160 14 22 34 110 200

150-800 1500-2500 1500-2680 2.5 3.5 6 12 28 5 7 9 34 38 7 8 12 25 60

150-800 900-1500 Reduced Port Full Port Reduced Port Full Port Reduced Port Only 2 3 6 14 25 2 4 9 21 25 12 50 100 250 430 720 1,020 2,000 5,500 9,800 16,400 23,800 27,500 36,000 46,000 57,000 75,000 8 13.5 34 65 104 200 720 26 75 103 206 322 760 9 15 38 125 165 250 540

130 250 540 770 1,900 3,900 6,700 5,200 8,050 12,500 15,500 27,000

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Electronic FileName: 83236863.xls WorkSheet: Velan Valves

Art Montemayor

Fittings' Losses FRICTION LOSS IN STANDARD VALVES AND FITTINGS

December 1, 1997 REV 0

The values given are equivalent lengths in feet of straight pipe at the referenced diameter sizeTYPE OF FITTING I.D. L/D 12 340 145 30 NOMINAL PIPE DIAMETER, INCHES 1/2 3/4 1 1 1/4 1 1/2 2 2 1/2 3 4 5 6 8 10 0.0518 0.0687 0.0874 0.1150 0.1342 0.1722 0.2057 0.2557 0.3355 0.4206 0.5054 0.6651 0.8350 0.35 17.0 8.0 1.5 1.3 1.0 1.0 3.2 3.5 0.5 22.0 12.0 2.2 1.8 1.3 1.3 4.5 5.0 0.6 27.0 14.0 2.7 2.3 1.7 1.7 5.7 6.0 0.8 38.0 18.0 3.6 3.0 2.3 2.3 7.5 8.5 1.2 44.0 22.0 4.5 3.6 2.8 2.8 9.0 10.0 1.2 53.0 28.0 5.2 4.6 3.5 3.5 12.0 13.0 1.4 68.0 33.0 6.5 5.5 4.3 4.3 14.0 15.0 1.7 80.0 42.0 8.0 7.0 5.2 5.2 16.0 18.0 2.3 120.0 53.0 11.0 9.0 7.0 7.0 22.0 24.0 2.8 140.0 70.0 14.0 12.0 9.0 9.0 27.0 30.0 3.5 170.0 84.0 16.0 14.0 11.0 11.0 33.0 37.0 4.5 220.0 120.0 21.0 18.0 14.0 14.0 43.0 50.0 5.7 280.0 140.0 26.0 22.0 17.0 17.0 53.0 63.0

Gate Valve, 100% open Globe Valve, 100% open Angle Valve, 100% open Standard Elbow Medium Sweep Elbow Long Sweep Elbow Tee (straight run) Tee (branch run) Return Bend

20 60 50

Source of data: Viking Pump Catalog Section 510 - Engineering Data Page 510.12; Issue C; Figure 11 NOTE: It is unclear what is meant by "Standard", "Medium Sweep" & "Long Sweep" Elbows. The pipe fitting industry supplies two basic Elbow geometries: 1) Short Radius Elbow: R/D = 1.0 (based on nominal, not specific, dimensions) 2) Long Radius Elbow: R/D = 1.5 (based on nominal, not specific, dimensions) Where, R = Radius of the Elbow curvature (nominal size) D = Nominal diameter of the Elbow (nominal size)

Checking out other fittings' resistance data, it is established that Standard Elbow = Short Radius Elbow Long Sweep Elbow = Long Radius Elbow Medium Sweep Elbow = ? unknown ?

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Electronic File: 83236863.xls WorkSheet: Fittings' Losses

Art Montemayor

Fittings' Losses FRICTION LOSS IN STANDARD VALVES AND FITTINGS

December 1, 1997 REV 0

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Electronic File: 83236863.xls WorkSheet: Fittings' Losses

Art Montemayor

EQUIVALENT LENGTH OF VALVES AND FITTINGS IN FEET Source: Rules of Thumb for Chem. Engrs; C.R. Branan; Gulf Publ.; p. 3

March 11, 1998 Rev: 0

Globe Valve or Ball Check Valve

90o Ell, Short Radius

90o Ell, Long Radius

Swing Check Valve

Gate or Ball Valve

Nominal Pipe size, in.

Through Tee

Angle valve

Expander 90o Miter Elbows

Reducer

Branch Tee

Plug Cock

45o Ell

Sudden Std Redu Sudden Std Redu Equivalent Length in terms of small diameter 2 Miters 3 Miters 4 Miters d/D=1/4 d/D=1/2 d/D=3/4 d/D=1/2 d/D=3/4 d/D=1/4 d/D=1/2 d/D=3/4 d/D=1/2 d/D=3/4

1 1/2 2 2 1/2 3 4 6 8 10 12 14 16 18 20 22 24 30 36 42 48 54 60

L/D = 407 55.0 70.0 80.0 100.0 130.0 200.0 260.0 330.0 400.0 450.0 500.0 550.0 650.0 688.0 750.0

L/D = 192 26.0 33.0 40.0 50.0 65.0 100.0 125.0 160.0 190.0 210.0 240.0 280.0 300.0 335.0 370.0

L/D = 99 13.0 17.0 20.0 25.0 32.0 48.0 64.0 80.0 95.0 105.0 120.0 140.0 155.0 170.0 185.0

L/D = 81 7.0 14.0 11.0 17.0 30.0 70.0 120.0 170.0 170.0 80.0 145.0 160.0 210.0 225.0 254.0 312.0

L/D = 12 1.0 2.0 2.0 2.0 3.0 4.0 6.0 7.0 9.0 10.0 11.0 12.0 14.0 15.0 16.0 21.0 25.0 30.0 35.0 40.0 45.0

L/D = 12 1.0 2.0 2.0 2.0 3.0 4.0 6.0 7.0 9.0 10.0 11.0 12.0 14.0 15.0 16.0 21.0 25.0 30.0 35.0 40.0 45.0

L/D = 17 2.0 3.0 3.0 4.0 5.0 8.0 9.0 12.0 14.0 16.0 18.0 20.0 23.0 25.0 27.0 40.0 47.0 55.0 65.0 70.0 80.0

L/D = 23 3.0 4.0 5.0 6.0 7.0 11.0 15.0 18.0 22.0 26.0 29.0 33.0 36.0 40.0 44.0 55.0 66.0 77.0 88.0 99.0 110.0

L/D = 58 8.0 10.0 12.0 14.0 19.0 28.0 37.0 47.0 55.0 62.0 72.0 82.0 90.0 100.0 110.0 140.0 170.0 200.0 220.0 250.0 260.0

L/D = L/D = L/D = L/D = L/D = L/D = L/D = L/D = L/D = L/D = L/D = L/D = L/D = L/D = 17 28 21 20 41 23 6 29 6 17 17 6 6 2 5 3 1 4 1 3 2 1 1 2.0 7 4 1 5 1 3 3 1 1 3.0 8 5 2 6 2 4 3 2 2 3.0 10 6 2 8 2 5 4 2 2 4.0 12 8 3 10 3 6 5 3 3 5.0 18 12 4 14 4 9 7 4 4 1 8.0 25 16 5 19 5 12 9 5 5 2 9.0 31 20 7 24 7 15 12 6 6 2 12.0 28 21 20 37 24 8 28 8 18 14 7 7 2 14.0 32 24 22 42 26 9 20 16 8 16.0 38 27 24 47 30 10 24 18 9 18.0 42 30 28 53 35 11 26 20 10 20.0 46 33 32 60 38 13 30 23 11 23.0 52 36 34 65 42 14 32 25 12 25.0 56 39 36 70 46 15 35 27 13 27.0 70 51 44 40.0 84 60 52 47.0 98 69 64 55.0 112 81 72 65.0 126 90 80 70.0 190 99 92 80.0

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Electronic FileName: 83236863.xls WorkSheet: Equiv. Lengths #1

Art Montemayor

Equivalent Length of Fittings Source: Compressed Air Gas Data; Ingersoll-Rand; 1969; p. 34-78 Angle Valve L/D = 145 7.5 10.0 12.7 19.5 25.0 29.8 37.1 48.6 61 73 96 121 Gate Valve L/D = 13 0.67 0.89 1.14 1.74 2.24 2.67 3.32 4.36 5.5 6.6 8.6 10.9 12.9 14.4 16.5 18.6 20.9 25.2 Swing Check Valve L/D = 135 7.0 9.3 11.8 18.1 23.3 27.8 34.5 45.3 57 68 90 113 134 149 172 193 217 262 Plug 45o Std. 90o Std Cock Elbow Elbow L/D = 18 0.93 1.24 1.57 2.42 3.10 3.70 4.60 6.0 7.6 9.1 12.0 15.0 17.9 L/D = 16 0.8 1.10 1.40 2.15 2.76 3.29 4.09 5.4 6.7 8.1 10.6 13.4 15.9 17.7 20.3 22.8 25.7 31.0 L/D = 30 1.56 2.06 2.62 4.03 5.2 6.2 7.7 10.1 12.6 15.2 20.0 25.1 29.8 33.1 38.1 42.8 48.1 58 90o Long Radius Elbow L/D = 20 1.04 1.37 1.75 2.68 3.45 4.12 5.1 6.7 8.4 10.1 13.3 16.7 19.9 22.1 25.4 28.5 32.1 38.8 Standard Tee Through-Flow Branch Flow L/D = 20 1.04 1.37 1.75 2.68 3.45 4.12 5.1 6.7 8.4 10.1 13.3 16.7 19.9 22.1 25.4 28.5 32.1 38.8 L/D = 60 3.11 4.12 5.2 8.1 10.3 12.3 15.3 20.1 25.2 30.3 39.9 50.1 60 66 76 86 96 116 Close Return Bend L/D = 50 2.59 3.43 4.4 6.7 8.6 10.3 12.8 16.8 21.0 25.3 33.3 41.8 50 55 64 71 80 97 90o Welding Elbow Short Radius Long Radius L/D = 16 0.83 1.10 1.4 2.1 2.8 3.3 4.1 5.4 6.7 8.1 11 13 16 18 20 23 26 31 L/D = 12 0.6 0.8 1.0 1.6 2.1 2.5 3.1 4.0 5.0 6.1 8.0 10 12 13 15 17 19 23

March 16, 1998 Rev: 0

Globe Nominal Pipe's Inside Valve Pipe Sched. Diameter Size, Number Inches L/D = Inches Feet 1/2 3/4 1 1 1/2 2 2 1/2 3 4 5 6 8 10 12 14 16 18 20 24 40 40 40 40 40 40 40 40 40 40 40 40 40 30 30 30 20 20 0.622 0.824 1.049 1.610 2.067 2.469 3.068 4.026 5.047 6.065 7.981 10.020 11.938 13.250 15.250 17.124 0.052 0.069 0.087 0.134 0.172 0.206 0.256 0.336 0.421 0.505 0.665 0.835 0.995 1.104 1.271 1.427 340 17.6 23.3 29.7 45.6 59 70 87 114 143 172 226 284

19.250 1.604 23.250 1.938

Source: "Compressed Air and Gas Data"; Ingersoll-Rand Company; Charles W. Gibbs, Editor; (1969); page 34-78 The L/D values are cited as originating from Crane Co. Technical Paper No. 410 (1957) All valves and cocks are rated as fully open Check valves require 0.5 psi pressure loss to open fully Welding elbow data is from Midwest Piping Catalog 61 (1961) Crane Technical Paper No. 410 states: "The ratio L/D is the equivalent length, in pipe diameters of straight pipe, that will cause the same pressure drop as the obstruction under the same flow conditions." Page 19 of 148 Electronic FileName: 83236863.xls WorkSheet: Equiv. Lengths #2

Art Montemayor

Equivalent Length of Fittings Source: Compressed Air Gas Data; Ingersoll-Rand; 1969; p. 34-78

March 16, 1998 Rev: 0

The L/D, as defined by Technical Paper No. 410, is the pipe length divided by the pipe's Internal Diameter (ID).

Page 20 of 148


Art Montemayor Reference: Dobie Pump Spreadsheet Dobie Engineering (1992)

EQUIVALENT LENGTHS OF FITTINGS

April 21, 1998 Rev: 0

BASIS: All equivalent lengths are calculated using Crane Tech Paper #410. (1976) Pipe walls typical for 150 ASA carbon steel with 1/8 in. corrosion allow. Bends are based on using screwed or SO fittings < 3 in.; long radius weld elbows for 3 to 24 in; miter elbows for > 24in. Valves (except for Ball type) are based on full port types and swing check valves. Reducers assume a 30o included angle. Equivalent feet of the larger size pipe is indicated. The entrance losses are based on a sharp edge with no inward projection. Ball valves are assumed as Jamesbury Type 5150, ANSI CLASS 150, with reduced ball CvE E LO NT S RA S N C

ER

U

N D

C

ER

EQUIVALENT LENGTHS OF TYPICAL FITTINGS, IN FEET Nom. BENDS TEES Size, In. 90o ELL 45o ELL LINEAR BRANCH GATE GLOBE 1/2 3/4 1 1 1/2 2 3 4 6 8 10 12 14 16 18 20 24 30 36 42 48 54 1.4 1.9 2.4 3.8 4.8 3.4 4.4 6.7 8.7 11.0 13.2 14.6 16.8 19.0 21.2 25.6 73 88 103 118 132 0.7 1.0 1.3 2.0 2.6 2.2 2.9 4.3 5.6 7.1 8.5 9.4 10.8 12.2 13.6 16.5 37 44 52 59 66 0.9 1.2 1.6 2.5 3.2 5.1 6.7 10.1 13.2 16.7 20.0 22.1 25.4 28.8 32 39 49 59 69 78 88 2.7 3.7 4.8 7.5 9.7 15.3 20.1 30 39 50 60 66 76 86 96 116 146 176 206 235 265 0.4 0.5 0.6 1.0 1.3 2.0 2.7 4.0 5.3 6.7 8.0 8.8 10.2 11.5 12.8 15.5 19.5 23.5 27.5 31.3 35 15.5 21.0 27.1 42 55 87 114 172 224 284 340 n/a n/a n/a n/a n/a n/a n/a n/a n/a n/aR ED

BALL 1.7 1.9 1.3 1.7 5.4 9.2 9.7 69.6 48.2 35.3 n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a

BUTTERFLY CHECK one size one size n/a n/a n/a n/a n/a 11.5 15.1 22.7 29.6 29.2 35 39 32 36 40 48 61 73 86 98 110 2.3 3.1 4 6.2 8.1 12.8 16.8 25.3 33 42 50 55 64 72 80 97 122 147 172 196 221 n/a 0.8 0.8 4.3 2.0 11.5 5.4 20.4 11.6 12.8 10.1 4.9 15.3 9.0 8.7 23.5 42.2 37.5 35.1 31.8 33.7 n/a 1.2 1.0 8.1 2.6 22.1 7.2 36.4 15.1 15.5 9.7 2.8 7.3 6.3 5.5 23.6 49.5 37.3 30.2 23.3 23.0

EX

VALVES

Loss 0.8 1.2 1.7 2.8 4.3 7.3 10.4 16.8 23.5 31 38 44 52 60 68 84 109 136 165 196 232

Page 21 of 148


E LO XIT S S

P

A

Loss 1.7 2.5 3.5 5.7 8.5 14.6 20.7 34 47 62 77 88 103 120 136 167 218 272 331 392 465

Art Montemayor Reference: Dobie Pump Spreadsheet Dobie Engineering (1992) 60 148 74 98 295 39

EQUIVALENT LENGTHS OF FITTINGS n/a n/a 123 246 33.3 20.5 267

April 21, 1998 Rev: 0 534

Page 22 of 148


Art Montemayor

Equivalent Feet of Pipe Fittings and Valves

May 20, 1998 Rev: 0

EQUIVALENT LENGTH WORKSHEETFITTINGS & VALVES 90o ELL 45o ELL Through-Flow TEE Branch-Flow TEE GATE VALVE GLOBE VALVE BUTTERFLY VALVE BALL VALVE SWING CHECK VALVE REDUCTION-ONE SIZE EXPANSION-ONE SIZE ENTRANCE LOSS EXIT LOSS OTHER Straight Length PIPE QUANTITY 1 1 1 1 1 1 1 1 1 1 1 1 1 1 SIZE 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 EQ FT 4.8 2.6 3.2 9.7 1.3 55.0 n/a 5.4 8.1 2.0 2.6 4.3 8.5 0.0 FEET 4.80 2.60 3.20 9.70 1.30 55.00 #VALUE! 5.40 8.10 2.00 2.60 4.30 8.50 0.00 1250 #VALUE!

TOTAL EQUIVALENT FEET = INSTRUCTIONS:

107.5

1) Enter the known quantities and sizes of each fitting type in your hydraulic circuit in the designated YELLOW cells as well as the amount of straight pipe. Note: The sizes of the fittings to be keyed in are the NOMINAL pipe size, not the I. D.. You must furnish both the quantities and sizes to generate a calculated answer. 2) The resultant calculated equivalent pipe length(s) appear in RED numbers. Art Montemayor

Page 23 of 148


Art Montemayor


May 20, 1998 Rev: 0

0

0

0

0

0

0

0

0

0

0.5 0.75 1 1.5 2 3 4 6 8 10 12 14 16 18 20 24 30 36 42 48 54 60

1.4 1.9 2.4 3.8 4.8 3.4 4.4 6.7 8.7 11 13.2 14.6 16.8 19 21.2 25.6 73 88 103 118 132 148

0.7 1 1.3 2 2.6 2.2 2.9 4.3 5.6 7.1 8.5 9.4 10.8 12.2 13.6 16.5 37 44 52 59 66 74

0.9 1.2 1.6 2.5 3.2 5.1 6.7 10.1 13.2 16.7 20 22.1 25.4 28.8 32 39 49 59 69 78 88 98

2.7 3.7 4.8 7.5 9.7 15.3 20.1 30 39 50 60 66 76 86 96 116 146 176 206 235 265 295

0.4 0.5 0.6 1 1.3 2 2.7 4 5.3 6.7 8 8.8 10.2 11.5 12.8 15.5 19.5 23.5 27.5 31.3 35 39

15.5 21 27.1 42 55 87 114 172 224 284 340 n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a

n/a n/a n/a n/a n/a 11.5 15.1 22.7 29.6 29.2 35 39 32 36 40 48 61 73 86 98 110 123

2.3 3.1 4 6.2 8.1 12.8 16.8 25.3 33 42 50 55 64 72 80 97 122 147 172 196 221 246

Page 24 of 148


Art Montemayor


May 20, 1998 Rev: 0

0

0

0

0

0

n/a 0.8 0.8 4.3 2 11.5 5.4 20.4 11.6 12.8 10.1 4.9 15.3 9 8.7 23.5 42.2 37.5 35.1 31.8 33.7 33.3

n/a 1.2 1 8.1 2.6 22.1 7.2 36.4 15.1 15.5 9.7 2.8 7.3 6.3 5.5 23.6 49.5 37.3 30.2 23.3 23 20.5

0.8 1.2 1.7 2.8 4.3 7.3 10.4 16.8 23.5 31 38 44 52 60 68 84 109 136 165 196 232 267

1.7 2.5 3.5 5.7 8.5 14.6 20.7 34 47 62 77 88 103 120 136 167 218 272 331 392 465 534

1.7 1.9 1.3 1.7 5.4 9.2 9.7 69.6 48.2 35.3 n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a

Page 25 of 148


Art Montemayor

Typical Resistance Coefficients Fluid Flow in Piping Systems

March 12, 1998 Rev: 0

TYPICAL K VALUES FOR PIPE FITTINGS This table is taken from API Recommended Practice 521, 2nd Edition, Sep.1982, which obtained it in turn from: Tube-Turn Catalogue and Engineering Data Book No. 211, Chemetron Corp., Louisville, Kentucky.

K VALVES: Globe, Open Typical Depressuring, Open Angle, Open Swing Check, Open Gate, Open SCREWED FITTINGS: 180o close return Branch-Flow Tee 90o Standard Elbow Through-Flow Tee 45o Elbow MITERED FITTINGS: 90o Single-Miter Elbow 60o Single-Miter Elbow 90o Double-Miter Elbow 90o Triple-Miter Elbow 45o Single-Miter Elbow FABRICATED FITTINGS: Through-Flow Tee WELDED FITTINGS: Branch-Flow Tee 45o Branch-Flow Lateral 90o Long-Sweep Elbow Through-Flow Lateral 180o Return Through-Flow Tee 90o Elbow 45o Elbow CONTRACTION OR EXPANSION: Contraction (ANSI) Contraction (Sudden) Expansion (ANSI) Expansion (Sudden) 1.37 0.76 0.59 0.50 0.43 0.38 0.32 0.21 0.000 --0.500 --1.000 0.200 --0.460 --0.950 0.50 1.72 0.93 0.59 0.46 0.46 1.95 1.72 0.93 0.50 0.43 9.70 8.50 4.60 2.30 0.21

Art's Notes 100% wide open position ---' ???? 100% wide open position 100% wide open position 100% wide open position

w/ flow in all 3 branches w/ flow in all 3 branches

w/ flow in all 3 branches

w/ flow in all 3 branches w/ flow in all 3 branches w/ flow in all 3 branches w/ flow in all 3 branches

d/d' 0.400 0.210 0.380 0.900 0.740

0.600 0.135 0.290 0.500 0.410

0.800 0.039 0.120 0.110 0.110

Page 26 of 148

Electronic FileName: 83236863.xls WorkSheet: Typical K values

Art Montemayor

Typical Pipe Friction Factors

March 12, 1998 Rev: 0

Typical Friction Factors and Conversion Factors for Clean Steel Pipe (Based on Equivalent Roughness of 0.00015 Feet) Source: API Recommended Practice 521; 2nd Ed., Sep. 1982 (p.49) Conversion Factor for Equivalent Length per Unit of K Feet Meters 8.8 14.3 20.5 33.6 48.2 63.0 78.8 89.6 104.6 136.0 169.6 222.6 273.4 2.7 4.4 6.3 10.2 14.7 19.2 24.0 27.3 31.9 41.5 56.7 67.9 83.3

Nominal Pipe Size, Inches NPS 2, Schedule 40 NPS 3, Schedule 40 NPS 4, Schedule 40 NPS 6, Schedule 40 NPS 8 x 1/4-inch wall NPS 10 x 1/4-inch wall NPS 12 x 1/4-inch wall NPS 14 x 1/4-inch wall NPS 16 x 1/4-inch wall NPS 20 x 1/4-inch wall NPS 24 x 1/4-inch wall NPS 30 x 1/4-inch wall NPS 36 x 1/4-inch wall

Moody Friction Factor (f) 0.0195 0.0178 0.0165 0.0150 0.0140 0.0135 0.0129 0.0126 0.0123 0.0119 0.0115 0.0110 0.0107

Note: NPS = Nominal Pipe Size. The above friction factors and conversion factors apply at high Reynolds numbers, namely above 1 x 106 for NPS 24 and larger. For smaller pipe, down to NPS 2", the applicable Reynolds number is 2 x 105.

Page 27 of 148

Electronic FileName: 83236863.xls WorkSheet: Typical Friction Factors

Art Montemayor

Pipe Pro

May 07, 1998 Revision: 0

This information is taken from the PipePro computer program sold by: Professional Designers & Engineers, Inc. P.O. Box 11380 Boulder, CO 80301 Phone & Fax: (303) 530-1551

VALUES OF ABSOLUTE ROUGHNESS, , FOR VARIOUS MATERIALS

Type of Pipe Material Asphalted Cast Iron Brass & Copper Concrete Cast Iron Galvanized Iron Wrought Iron Plastic Steel Riveted Steel* Wood Stave*

Range of values (x 10-6), Ft Normal Design value (x 10-6), Ft 400 5 1,000 to 10,000 850 500 150 5 150 3,000 to 30,000 600 to 3,000 400 5 4,000 850 500 150 5 150 6,000 2,000

* Note: These two types of material have been obsolete and outdated for approximately 50 years. They are kept here as a reminder to young engineers of the danger and stupidity that some data represents when the authors do not review, proof and edit their technical product(s). Art Montemayor

Page 28 of 148

Electronic FileName: 83236863.xls WorkSheet: Pipe Pro

Art Montemayor

Pipe Pro


REASONABLE VELOCITIES FOR WATER FLOW THROUGH PIPE Application & Service Condition Municipal Water Distribution System General Plant Service Pump Suction Drain Lines Boiler Feed System Range of Typical Velocities, Ft/sec 2 to 7 4 to 10 4 to 7 4 to 7 8 to 15

Note: Long piping systems may require lower velocities to prevent an undesirable pressure drop. Higher velocities may be used or exceeded where pressure drop is not important.

Page 29 of 148


Art Montemayor

Pipe Pro


RESISTANCE COEFFICIENTS "K" FOR PIPE ENTRANCE AND EXITS The K resistance coefficient represents the resistance to flow in pipes for various configurations of entrances, exits, contraction or enlargement. The resistance coefficient method is based on the assumption that the fluid's pressure loss is all due to pressure drag and is independent of the Reynolds Number.

Description of Pipe Entrance or Exit Inlet - Sharp edge flush to the tank or reservoir Inlet - Slightly rounded at connection to tank, r/d=0.02 Inlet - Slightly rounded at connection to tank, r/d=0.04 Inlet - Medium rounded at connection to tank, r/d=0.06 Inlet - Medium rounded at connection to tank, r/d=0.10 Inlet - Well rounded at pipe connection, r/d=0.15 & up Inlet - Pipe projecting into tank or reservoir Exit - Open discharge (complete loss of velocity head) Exit - Submerged gradual enlargement discharge (complete recovery of velocity head)

Coefficient K 0.50 0.28 0.24 0.15 0.09 0.04 0.78 1.00 0.00

Description of Enlargement or Contraction: Flow may be directed through pipes of differing sizes by using especially designed enlargements or contractions to achieve the desired coefficient "K". These designs are generally required when a minimum "K" value is desired to limit the pressure drop in the system. The input cells below are provided to calculate the "K" value based on the dimensions of the enlargement or contraction.

Where, Large Diameter, D Small Diameter, d Length, L = = = Coefficient K Gradual contraction from large diam. to small diam. Gradual enlargement from small diam. to large diam. Sudden square edge contraction from D to d Sudden square edge enlargement from d to D

Page 30 of 148


Art Montemayor

Pipe Pro


RESISTANCE COEFFICIENTS "K" FOR VALVES AND FITTINGS The resistance coefficient K represents the reduction in static head for flow in pipes caused by valves and fittings. The K value is the number of velocity heads and is valid for all flow conditons. K/f = Constant where, f = turbulent Darcy friction factor DESCRIPTION OF VALVES AND FITTINGS 15 degree Corner Miter 30 degree Corner Miter 45 degree Standard Elbow 45 degree Street Elbow 45 degree Corner Miter 60 degree Corner Miter 90 degree Short Radius Elbow 90 degree Long Radius Elbow 90 degree Street Elbow 90 degree Square Corner Miter 180 degree Close Pattern Return Bend Tee Flow-through Run Tee Flow-through Branch Globe valve; perdendicular stem with unguided disc Globe valve; perdendicular stem with guided disc Globe valve; 60o stem to run with unguided disc Globe valve; 45o stem to run with unguided disc Angle valve; 90o with unguided disc Angle valve; 90o with guided disc Gate valve; full size, fully open Gate valve; 3/4 open Gate valve; 1/2 open Gate valve; 1/4 open Ball valve; round opening with full bore Plug valve; round opening with full bore Plug valve; rectangular opening with full bore, fully open Plug valve; 3-way rectangular opening with full bore, straignt flow Plug valve; 3-way 80% bore, fully open, branch flow Butterfly valve; 2" to 8", fully open Butterfly valve; 10" to 14", fully open Butterfly valve; 16" to 24", fully open Check valve; conventional swing, angle seat, fully open Check valve; conventional swing, perpendicular seat, fully open Check valve; tilting disc at 5o to flow (2" to 8") Check valve; globe type Foot valve; hinged disc with strainer Foot valve; poppet disc with strainer RECOMMENDED K/f 4 8 16 26 15 25 20 14 50 60 50 20 60 340 600 175 145 150 200 8 35 160 900 3 3 18 30 90 45 35 25 100 50 40 400 75 420

Page 31 of 148


Art Montemayor

Pipe Pro


FLOW COEFFICIENT, Cv, FOR CONTROL VALVES

The Cv flow coefficient for valves expresses the flow rate of 60 oF water in gallon per minute with 1.0 psi pressure drop across the valve. Cv varies for other fluids according to the square root of the weight density of the fluid. For Control Valves:

Cv = Q x (density/62.37/pressure drop)0.5Where, Q = liquid flowrate in gpm density = liquid density in lb/ft3 pressure drop = pressure drop in psi Values for Cv are normally provided by the control valve manufacturer. However, desired values for Cv may be calculated below as follows:

For Control Valves: Flowrate, gpm = Density, lb/ft3 = Pressure drop, psi = Cv = 100.00 62.37 5.00 44.72

Page 32 of 148


Art Montemayor

Pipe Pro


The Cv sizing coefficient must be obtained from the valved manufacturer; however, a sampling of some typical valve coefficients is listed below. Size, inches Control Valve Description 1/2 3/4 1 1 1/4 1 1/2 2 2 1/2 3 4 6 8 10 12 1 1 1/2 2 2 1/2 3 4 6 8 10 12 14 16 18 20 24 30 36 48 60 72 Plug; top guided Plug; top guided Plug; cage guided Plug; top guided Plug; cage guided Plug; cage guided Plug; cage guided Plug; cage guided Plug; cage guided Plug; cage guided Plug; cage guided Plug; port guided Plug; port guided Ball; V-notched Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Butterfly; disc Flow Characteristic Liquid coefficient, Cv Equal percentage Equal percentage Equal percentage Equal percentage Equal percentage Equal percentage Equal percentage Equal percentage Equal percentage Equal percentage Equal percentage Quick opening Quick opening 90o Opening 60o Opening 60o Opening 60o Opening 60o Opening 60o Opening 60o Opening 60 Opening 60o Opening 60o Opening 60o Opening 60o Opening 60o Opening 60o Opening 60o Opening 60o Opening 60o Opening 60o Opening 60o Opening 60o Openingo

4 9 17 22 33 56 87 121 203 357 808 1,280 1,700 22 26 55 91 136 271 768 1,340 2,170 3,180 3,880 5,210 6,510 8,210 12,200 19,900 29,400 54,200 85,300 123,000

Page 33 of 148


Art Montemayor

RESISTANCE TO FLUID FLOW Source: Crane Tech Paper #410 D

May 08, 1998 Rev: 0

CONCENTRIC REDUCERSfor included angle less than 45o l Steps: 1) Enter the required information in the YELLOW cells 2) The calculated results appear in RED numbers.

d D L

= small diameter, in. = large diameter, in. = diameter ratio = length of fitting, in. = included concentric angle, degrees = included concentric angle, radians

= = = = = =

3.07 6.07 0.51 5.5 30 0.52 = =

d

K1 = Resistance Coefficient (based on small diameter) K2 = Resistance Coefficient (based on large diameter)

0.2 2.4

CONCENTRIC EXPANDERSfor included angle less than 45o d D L = small diameter, in. = large diameter, in. = diameter ratio = length of fitting,in = included concentric angle, degrees = included concentric angle, radians = = = = = = 3.07 6.07 0.51 5.5 30 0.52 = = 0.0 1.5

K1 = Resistance Coefficient (based on small diameter) K2 = Resistance Coefficient (based on large diameter)

NOTE: Resistance to fluid flow due to pipe reducers, pipe expanders, entrances and exits to pipes is independent of the pipe and fluid's friction factor. These pressure losses are caused by local disturbances, such as changes in pipe cross section. Although considered "minor", these may actually be major losses in the case of a short pipe run. Whenever the velocity of a flowing stream is altered either in direction or in magnitude, eddy currents are set up and a loss of energy in excess of the pipe friction in that same length is created. Such losses may be expressed as velocity heads (KV2/2g) --- where K must be determined for each case. For concentric reducers, a minimum K value of about 0.10 is obtained with an included angle of 20o to 40o. Smaller or larger angles result in higher values of K.

Page 34 of 148

Electronic FileName: 83236863.xls WorkSheet: Reducers & Expanders

Art Montemayor

RESISTANCE TO FLUID FLOW Source: Crane Tech Paper #410

May 08, 1998 Rev: 0

This table calculates the included angle for Stainless Steel, schedule 40s, Concentric Reducers. It should also apply to Carbon Steel fittings as well as to Concentric Expanders. Data source: Flowline Fittings Catalog; 1965; p.48 D d

L

Fitting Size Large Diameter, in. Small Diameter, in. Length, in, Included Angle, Degrees 2x1 2 x 1-1/2 3x1 3 x 1-1/2 3x2 4x2 4x3 6x3 6x4 8x6 10 x 6 12 x 6 2.067 2.067 3.068 3.068 3.068 4.026 4.026 6.065 6 8 10 12 1.049 1.610 1.049 1.610 2.067 2.067 3.068 3.068 4 6 6 6 3.000 3.000 3.500 3.500 3.500 4.000 4.000 5.500 6 6 7 8 19 9 32 24 16 28 14 30 21 18 32 41

Page 35 of 148

Electronic FileName: 83236863.xls WorkSheet: Reducers & Expanders

Art Montemayor VOLUMETRIC CAPACITY FOR BUTT-WELDING FITTINGS All volumes expressed in cubic inches Reference: Piping Engineering; Tube Turns Division of Chemetron Corp.; Nov. 1971; p.47

June 02, 1999 Rev: 0

90o Elbows 180o Returns 45o Elbows Tees Nominal Long Radius Short Radius Long Radius Short Radius Long Radius Full-size outlets Pipe Size, in. Standard X-Strong Standard X-Strong Standard X-Strong Standard X-Strong Standard X-Strong Standard X-Strong 1/2 3/4 1 1 1/4 1 1/2 2 3 4 6 8 10 12 14 16 18 20 22 24 26 30 34 36 42 48 0.7 0.6 0.9 0.8 2.0 1.7 4.4 3.7 7.2 6.2 15.8 13.9 52.2 46.7 119.8 108.3 408.1 368.3 942.3 860.3 1856.9 1758.2 3195.9 3064.2 4545.9 4376.1 6882.2 6658.4 9906.5 9621.7 13707.5 13353.8 18365.2 17935.8 23995.6 23482.4 30644.8 30041.0 47449.7 46642.1 69490.1 68449.0 82695.4 81526.6 132116.7 130520.2 198322.0 196203.0 1.4 1.9 4.1 8.7 14.4 31.6 104.4 239.7 816.1 1884.7 3713.8 6391.8 9091.8 13764.5 19813.1 27415.0 36730.9 47991.1 61289.6 94899.4 110260.5 176155.6 1.1 1.5 3.4 7.4 12.5 27.8 93.3 216.6 736.6 1720.6 3516.5 6128.5 8752.2 13316.9 19243.4 26707.6 23914.5 46964.7 40054.6 62189.4 0.4 0.5 1.0 2.2 3.6 7.9 26.1 59.9 204.0 471.1 928.5 1598.0 2273.0 3441.1 4953.3 6853.8 9182.7 11997.8 15322.4 34745.0 41347.7 66054.8 40971.0 0.2 0.4 0.8 1.8 3.1 7.0 23.3 54.1 184.2 430.1 879.1 1532.1 2188.1 3329.2 4810.8 6676.9 8967.9 11471.2 15020.5 34224.5 40763.3 65260.1 40538.0 0.8 0.6 1.6 1.3 3.5 2.9 7.5 6.5 12.4 10.8 22.2 19.7 65.2 58.7 135.8 123.6 413.2 367.1 881.1 811.2 1675.4 1594.9 2816.9 2712.3 3809.3 3681.0 5804.3 5634.3 8396.5 8179.2 11701.3 11429.4 14348.0 14049.2 20647.2 20249.7 23912.3 23493.2 35985.0 35442.5 52836.7 52135.9 62945.1 62157.5 94207.0 93209.0 145443.0 144092.0

1.4 2.9 4.8 10.5 34.8 79.9 272.1 628.3 1238.0 2130.7 3030.7 4301.4 6054.0 9366.8 14664.0 31648.8 55130.3 88077.8

4.2 9.2 31.1 72.0 245.4 572.7 1172.2 2042.8 2917.4 4161.5 5879.9 9125.1 14350.3 31110.5 54351.0 87013.4

2.7 5.9 9.6 21.1 69.6 159.9 544.3 1256.7 2476.0 4261.5 6061.4 8602.8 12108.0 18733.6 29327.9 63297.7

8.3 18.3 62.2 143.9 490.8 1145.5 2344.3 4085.6 5834.8 8323.0 11759.8 18250.2 28700.7 62221.0

110260.5 108702.1

Page 36 of 148

Electronic FileName: 83236863.xls WorkSheet: Fittings' Volumes

Art Montemayor 90o Elbows: V = 2D2A/8 V = Volume D = Inside diameter A = Center to face distance

June 02, 1999 Rev: 0

180o Returns: V = 2D2O/8 V = Volume D = Inside diameter A = Center to center distance

45o Elbows: V = 2D2A/8 V = Volume D = Inside diameter A = Center to face distance

Full Size Outlet Tees: V = (D2/2) (C + M/2) - D3/3 V = Volume D = Inside diameter

Page 37 of 148


Art Montemayor C = Center to end of run M = center to end of branch

June 02, 1999 Rev: 0

Pipe Caps: V = (D2/4) (E - t - D/12) V = Volume D = Inside diameter E = length t = wall thickness

Crosses: V = (D2/2) (C + M) - (2/3)D3 V = Volume D = Inside diameter C = Center to end of run M = center to end of branch

Concentric & eccentric reducers:

Page 38 of 148


Art Montemayor

June 02, 1999 Rev: 0

Page 39 of 148


Art Montemayor

June 02, 1999 Rev: 0

Crosses Stub Ends Full-size outlets Lap Joint Standard X-Strong Standard X-Strong Standard X-Strong 0.3 0.6 1.1 1.8 2.5 3.9 11.1 24.2 77.3 148.5 295.6 517.0 684.6 967.6 1432.6 2026.4 2784.1 3451.0 4014.3 5163.1 7010.8 10936.3 13480.0 0.2 0.4 0.9 1.5 2.0 3.2 9.4 20.8 65.7 122.3 264.4 475.0 640.0 911.0 1363.0 1938.0 2682.9 3313.0 3884.1 5006.4 6811.5 10666.7 13157.0 0.9 1.6 3.5 6.0 8.1 20.1 44.4 76.4 231.1 400.2 788.5 1131.0 1654.6 2191.8 2804.5 3492.5 5094.7 0.7 1.3 2.9 5.1 7.1 17.8 39.6 69.0 208.5 365.3 746.6 1084.3 1592.8 2120.6 2723.8 3402.4 4985.7

Caps

Reducers Nominal Pipe Size, in. Concentric & Eccentric Large end Small end Standard X-Strong 3/8 1/2 3/4 1/2 3/4 1 1/2 3/4 1 1 1/4 3/4 1 1 1/4 1 1/2 1 1 1/4 1 1/2 2 2 1/2 1 1/2 2 2 1/2 3 3 1/2 2 1/2 3 3 1/2 4 1.5 1.8 2.1 2.6 3.1 3.7 4.0 4.6 5.3 6.5 7.6 8.5 10.0 11.4 20.2 21.9 25.5 29.5 37.3 41.8 46.6 54.4 62.1 11.9 123.7 134.8 147.1 1.3 1.4 1.8 2.2 2.6 3.1 3.4 3.9 4.5 5.6 6.6 7.4 8.8 10.0 17.9 19.4 22.7 26.3 33.4 37.5 41.8 48.9 55.9 100.6 111.3 121.5 132.8

Tees with Reducing Outlet Standard 2.7 2.8 3.0 5.8 6.0 6.3 9.4 9.6 9.9 10.8 16.7 16.9 17.6 18.4 50.9 51.8 52.9 55.1 58.6 108.0 110.4 113.9 119.7 125.9 334.1 340.0 346.3 354.1

1

9.5 15.5 27.7 80.5 166.5 501.3 1061.9 2010.4 3371.9 4171.5 6311.7 9081.3 12634.1 22189.4

8.2 13.7 24.7 72.9 152.4 441.0 983.0 1920.6 3255.9 4043.7 6144.0 8868.1 12368.2 21802.9

1 1/4

1 1/2

2

3

4

6

Page 40 of 148


Art Montemayor

6 5 3 3 1/2 4 5 6 4 5 6 8 5 6 8 10 6 8 10 12 6 8 10 12 14 8 10 12 14 16 8 10 12 14 16 18 10 175.3 221.7 235.6 269.6 309.2 385.9 428.3 476.8 586.0 639.4 697.7 827.0 993.8 1,496 1,738 2,041 2,382 2,321 2,656 3,029 3,289 3,413 3,821 4,104 4,598 158.7 201.8 215.0 245.8 280.9 362.0 401.4 444.8 546.7 606.4 658.8 779.8 947.7 1,419 1,646 1,952 2,288 2,212 2,552 2,920 3,175 3,291 3,695 3,973 4,458

June 02, 1999 Rev: 0 375.7 716.9 722.6 730.6 753.0 791.1 1,373.0 1,396.0 1,432.0 1,506.0 2,318.0 2,348.0 2,430.0 2,567.0 1,992 2,587 2,922 2,976 2,621 3,396 3,803 4,891 5,054 4,318 4,849 4,993 6,147 7,180 5,360 6,010 6,204 7,606 8,877 10,163 12,027

8

10

12

14

16

18

20

6,248 6,922 7,974 9,404

6,059 6,717 7,747 9,150

22 Page 41 of 148


Art Montemayor 12 14 16 18 20 10 12 14 16 18 20 22 12 14 16 18 20 22 24 14 16 18 20 22 24 26 28 16 18 20 22 24 26 28 30 32

June 02, 1999 Rev: 0 12,226 7,129 12,394 7,841 12,620 8,622 12,995 9,474 1,345,813,172 8,701 14,972 10,979 8,419 12,795 9,221 14,628 10,094 16,477 16,606 20,062 20,233 20,465 20,846 21,316 21,887 22,565 30,283 30,520 30,908 31,386 31,964 32,652 33,458 34,102 44,385 44,779 45,265 45,851 46,548 47,364 48,018 49,058 50,242 Electronic FileName: 83236863.xls WorkSheet: Fittings' Volumes

22

7,333 8,055 8,848 9,711

24

8,637 9,451 10,334

26

30

34

Page 42 of 148

Art Montemayor 16 18 20 22 24 26 28 30 32 34 20 22 24 26 28 30 32 34 36 22 24 26 28 30 32 34 36 38 40 42 44 46

June 02, 1999 Rev: 0 52,701 53,098 53,587 54,177 54,878 55,700 56,359 57,404 58,594 59,940 76,746 77,172 77,639 78,899 79,574 80,636 81,462 82,341 83,276 125,186 125,667 126,693 127,344 128,044 129,561 130,462 131,419 133,521 134,710 135,960 138,742 140,256

36

30,539 32,314 34,176

30,071 31,831 33,698

32,031 33,617 37,053 38,902 40,840 42,866 85,143 89,354 98,305 103,044 107,959 113,050 118,317 123,760 129,378 135,173 141,144

31,561 33,132 36,540 38,375 40,229 42,311 83,984 88,163 97,047 101,753 106,635 111,693 116,927 122,337 127,923 133,685 139,622

42

48

Page 43 of 148


Art Montemayor

June 02, 1999 Rev: 0

Tees with Reducing Outlet X-Strong 2.3 2.4 2.6 5.1 5.4 5.7 8.3 8.6 9.1 9.9 15.2 15.6 16.5 17.4 45.5 46.4 47.3 49.5 52.6 97.7 99.9 103.2 108.6 114.5 301.8 307.4 313.7 321.2 Electronic FileName: 83236863.xls WorkSheet: Fittings' Volumes

Page 44 of 148

Art Montemayor 361.8 655.5 661.0 668.7 690.5 719.4 1,300.0 1,323.0 1,350.0 1,426.0 2,224.0 2,250.0 2,329.0 2,468.0 1,915 2,488 2,816 3,022 2,532 3,283 3,701 4,741 4,902 4,191 4,711 5,013 5,976 6,985 5,214 5,855 6,221 7,414 8,657 9,916 11,745

June 02, 1999 Rev: 0

Page 45 of 148


Art Montemayor 11,944 12,110 12,337 12,710 8,519 14,603 10,745 12,526 14,325 16,141 16,287 19,668 19,837 20,070 20,449 20,919 21,487 22,164 29,768 30,006 30,392 30,869 31,445 32,131 32,936 33,583 43,724 44,116 44,600 45,184 45,879 46,693 47,351 48,389 49,571

June 02, 1999 Rev: 0

Page 46 of 148


Art Montemayor 51,958 52,353 52,841 53,429 54,128 54,947 55,610 56,653 57,842 59,185 75,825 76,253 76,723 77,972 78,650 79,710 80,540 81,425 82,365 123,874 124,358 125,377 126,031 126,736 128,245 129,151 130,113 132,207 133,402 134,660 137,432 138,954

June 02, 1999 Rev: 0

Page 47 of 148


Art Montemayor Nominal Outside Wall Pipe Diameter Thickness Size Inches Inches Inches

U.S.A. Pipe Dimensions Pipe Inside Diameter Inches Feet d2 Inside Diameter Functions d3 (in Inches) d4 d5

September 30, 2003 Rev: 0 Transverse Internal ("Flow") Area in2 ft2

Commercial Wrought Steel Pipe DataSchedule Wall Thickness - Per ASA B36.10 - 1950 14 Schedule 10 16 18 20 24 30 8 10 Schedule 20 12 14 16 18 20 24 30 8 10 Schedule 30 12 14 16 18 20 24 30 1/8 1/4 3/8 1/2 3/4 1 1.25 1.5 Schedule 40 2 2.5 3 3.5 4 5 6 8 10 14 16 18 20 24 30 8.625 10.750 12.750 14 16 18 20 24 30 8.625 10.750 12.750 14 16 18 20 24 30 0.405 0.540 0.675 0.840 1.050 1.315 1.660 1.900 2.375 2.875 3.500 4.000 4.500 5.563 6.625 8.625 10.750 0.250 0.250 0.250 0.250 0.250 0.312 0.250 0.250 0.250 0.312 0.312 0.312 0.375 0.375 0.500 0.277 0.307 0.330 0.375 0.375 0.438 0.500 0.562 0.625 0.068 0.088 0.091 0.109 0.113 0.133 0.140 0.145 0.154 0.203 0.216 0.226 0.237 0.258 0.280 0.322 0.365 13.500 1.1250 15.500 1.2917 17.500 1.4583 19.500 1.6250 23.500 1.9583 29.376 2.4480 8.13 0.6771 10.25 0.8542 12.25 1.0208 13.38 1.1147 15.38 1.2813 17.38 1.4480 19.25 1.6042 23.25 1.9375 29 2.4167 8.07 0.6726 10.14 0.8447 12.09 1.0075 13.25 1.1042 15.25 1.2708 17.12 1.4270 19 1.5833 22.88 1.9063 28.75 2.3958 0.27 0.0224 0.36 0.0303 0.49 0.0411 0.62 0.0518 0.82 0.0687 1.05 0.0874 1.38 0.1150 1.61 0.1342 2.07 0.1723 2.47 0.2058 3.07 0.2557 3.55 0.2957 4.03 5.05 6.07 7.98 10.02 0.3355 0.4206 0.5054 0.6651 0.8350 182.25 240.25 306.25 380.25 552.25 862.95 66.02 105.06 150.06 178.92 236.42 301.93 370.56 540.56 841.00 65.14 102.74 146.17 175.56 232.56 293.23 361.00 523.31 826.56 0.07 0.13 0.24 0.39 0.68 1.10 1.90 2.59 4.27 6.10 9.41 12.59 16.21 25.47 36.78 63.70 100.40 2,460.4 3,723.9 5,359.4 7,414.9 12,977.9 25,350.0 536.4 1,076.9 1,838.3 2,393.2 3,635.2 5,246.3 7,133.3 12,568.1 24,389.0 525.8 1,041.4 1,767.2 2,326.2 3,546.6 5,021.3 6,859.0 11,971.3 23,763.7 0.0 0.0 0.1 0.2 0.6 1.2 2.6 4.2 8.8 15.1 28.9 44.7 65.3 128.6 223.1 508.4 1,006.0 33,215.1 57,720.1 93,789.1 144,590.1 304,980.1 448,403.3 894,661.0 1,641,308.6 2,819,506.2 7,167,031.5 143.14 188.69 240.53 298.65 433.74 677.76 51.85 82.52 117.86 140.52 185.68 237.13 291.04 424.56 660.52 51.16 80.69 114.80 137.89 182.65 230.30 283.53 411.01 649.18 0.06 0.10 0.19 0.30 0.53 0.86 1.50 2.04 3.36 4.79 7.39 9.89 12.73 20.01 28.89 50.03 78.85 0.994 1.310 1.670 2.074 3.012 4.707 0.360 0.573 0.818 0.976 1.289 1.647 2.021 2.948 4.587 0.355 0.560 0.797 0.958 1.268 1.599 1.969 2.854 4.508 0.000 0.001 0.001 0.002 0.004 0.006 0.010 0.014 0.023 0.033 0.051 0.069 0.088 0.139 0.201 0.347 0.548

744,681.6 21,875,767.4 4,358.1 11,038.1 22,518.8 32,011.4 55,895.1 91,158.9 137,316.6 292,207.8 35,409.3 113,140.8 275,854.7 428,184.9 859,442.6 1,583,977.6 2,643,343.9 6,793,831.7

707,281.0 20,511,149.0 4,243.4 10,555.2 21,365.1 30,822.2 54,085.3 85,984.6 130,321.0 273,854.8 34,248.1 106,987.5 258,304.2 408,394.0 824,801.1 1,472,401.0 2,476,099.0 6,264,702.3

683,205.6 19,642,160.0 0.0 0.0 0.1 0.1 0.5 1.2 3.6 6.7 18.3 37.2 88.6 158.5 262.7 648.8 1,353.1 4,057.2 10,080.2 0.0 0.0 0.0 0.1 0.4 1.3 5.0 10.8 37.7 91.7 271.8 562.2 1,057.7 3,274.7 8,206.4 32,380.7 101,004.0

Page 48 of 148

FileName: 83236863.xls WorkSheet: Pipe Tables

Art Montemayor 12 14 16 18 20 24 12.750 14 16 18 20 24 0.406 0.438 0.500 0.562 0.593 0.687 11.94 13.12 15 16.88 18.81 22.63

U.S.A. Pipe Dimensions 0.9948 1.0937 1.2500 1.4063 1.5678 1.8855 142.52 172.24 225.00 284.80 353.97 511.94 1,701.4 2,260.5 3,375.0 4,806.3 6,659.5 11,583.1 20,310.8 29,666.4 50,625.0 81,110.7 125,292.4 262,078.3

September 30, 2003 Rev: 0 242,469.9 111.93 0.777 389,341.9 135.28 0.939 759,375.0 176.71 1.227 1,368,823.9 223.68 1.553 2,357,250.3 278.00 1.931 5,929,784.5 402.07 2.792

Page 49 of 148


Art Montemayor 8 Schedule 60 10 12 14 16 18 20 24 1/8 1/4 3/8 1/2 3/4 1 1.25 1.5 2 2.5 3 3.5 4 5 6 8 10 12 14 16 18 20 24 8 10 12 14 16 18 20 24 4 5 6 8 10 12 14 16 18 20 24 8.625 10.750 12.750 14 16 18 20 24 0.405 0.540 0.675 0.840 1.050 1.315 1.660 1.900 2.375 2.875 3.500 4.000 4.500 5.563 6.625 8.625 10.750 12.750 14 16 18 20 24 8.625 10.750 12.750 14 16 18 20 24 4.500 5.563 6.625 8.625 10.750 12.750 14 16 18 20 24 0.406 0.500 0.562 0.593 0.656 0.750 0.812 0.968 0.095 0.119 0.126 0.147 0.154 0.179 0.191 0.200 0.218 0.276 0.300 0.318 0.337 0.375 0.432 0.500 0.593 0.687 0.750 0.843 0.937 1.031 1.218 0.593 0.718 0.843 0.937 1.031 1.156 1.281 1.531 0.438 0.500 0.562 0.718 0.843 1.000 1.093 1.218 1.375 1.500 1.812

U.S.A. Pipe Dimensions 7.81 0.6511 9.75 0.8125 11.63 12.81 14.69 16.5 18.38 22.06 0.22 0.3 0.42 0.55 0.74 0.96 1.28 1.5 1.94 2.32 2.9 3.36 3.83 4.81 5.76 7.63 9.56 11.38 12.5 14.31 16.13 17.94 21.56 0.9688 1.0678 1.2240 1.3750 1.5313 1.8387 0.0179 0.0252 0.0353 0.0455 0.0618 0.0798 0.1065 0.1250 0.1616 0.1936 0.2417 0.2803 0.3188 0.4011 0.4801 0.6354 0.7970 0.9480 1.0417 1.1928 1.3438 1.4948 1.7970 61.04 95.06 135.16 164.20 215.74 272.25 337.68 486.82 0.0462 0.0912 0.1789 0.2981 0.5506 0.9158 1.6333 2.2500 3.7597 5.3963 8.4100 11.32 14.64 23.16 33.19 58.14 91.47 129.41 156.25 204.89 260.05 321.77 465.01 55.34 86.75 122.41 147.04 194.27 246.11 304.08 438.4 13.13 20.82 30.26 51.68 82.16 115.56 139.57 183.98 232.56 289.00 415.18 476.9 926.9 1,571.4 2,104.0 3,168.8 4,492.1 6,205.2 10,741.2 0.0099 0.0275 0.0757 0.1628 0.4085 0.8765 2.0873 3.3750 7.2901 12.5357 24.3890 38.1 56.0 111.5 191.2 443.3 874.8 1,472.2 1,953.1 2,932.8 4,193.5 5,771.9 10,027.4 411.7 808 1354.37 1783.01 2707.71 3861.03 5302.61 9179.22 47.6 95.0 166.5 371.5 744.7 1,242.3 1,648.9 2,495.5 3,546.6 4,913.0 8,459.7 3,726.2 9,036.9 18,269.3 26,961.2 46,542.6 74,120.1 114,026.0 236,993.8 0.0021 0.0083 0.0320 0.0889 0.3031 0.8388 2.6676 5.0625 14.1355 29.1204 70.7281 128.1 214.3 536.6 1,101.5 3,380.3 8,366.8 16,747.8 24,414.1 41,980.2 67,624.9 103,537.1 216,230.7 3,062.4 7525.67 14984.72 21620.73 37740 60571.78 92466.98 192194.42 172.5 433.5 915.7 2,671.0 6,749.6 13,354.7 19,480.0 33,849.4 54,085.3 83,521.0 172,375.6

September 30, 2003 Rev: 0 29,113.1 88,109.6 212,398.6 345,480.5 683,617.7 1,222,981.0 2,095,342.0 5,229,031.3 0.0005 0.0025 0.0135 0.0485 0.2249 0.8027 3.4092 7.5938 27.4087 67.6466 205.1115 430.8 819.8 2,582.7 6,345.8 25,775.0 80,019.9 190,523.2 305,175.8 600,904.0 1,090,519.1 1,857,248.9 4,662,798.2 22,781.0 70094.05 165790.96 262172.91 526020.05 950250.05 1612439.27 4024166.83 625.1 1,978.1 5,037.4 19,201.8 61,178.6 143,562.9 230,136.1 459,133.4 824,801.1 1,419,857.0 3,512,324.7 47.94 74.66 106.16 128.96 169.44 213.82 265.21 382.35 0.036 0.072 0.141 0.234 0.432 0.719 1.283 1.767 2.953 4.238 6.605 8.888 11.497 18.194 26.07 45.66 71.84 101.64 122.72 160.92 204.24 252.72 365.21 43.46 68.13 96.14 115.48 152.58 193.3 238.83 344.32 10.31 16.35 23.77 40.59 64.53 90.76 109.62 144.50 182.65 226.98 326.08 0.333 0.518 0.737 0.896 1.177 1.485 1.842 2.655 0.0003 0.0005 0.0010 0.0016 0.0030 0.0050 0.0089 0.0123 0.0205 0.0294 0.0459 0.0617 0.0798 0.1263 0.1810 0.3171 0.4989 0.7058 0.8522 1.118 1.418 1.755 2.536 0.302 0.47 0.67 0.8 1.06 1.34 1.66 2.39 0.072 0.114 0.165 0.282 0.448 0.630 0.761 1.003 1.268 1.576 2.264

Schedule 80

7.44 0.6199 9.31 0.78 11.06 0.92 12.13 1.01 13.94 1.16 15.69 1.31 17.44 1.45 20.94 1.74 3.62 4.56 5.5 7.19 9.06 10.75 11.81 13.56 15.25 17 20.38 0.3020 0.3803 0.4584 0.5991 0.7553 0.8958 0.9845 1.1303 1.2708 1.4167 1.6980

Schedule 120

Schedule 100

Page 50 of 148


Art Montemayor 8 10 12 14 16 18 20 24 1/2 3/4 1 1.25 1.5 2 2.5 3 4 5 6 8 10 12 14 16 18 20 24 8.625 10.750 12.750 14 16 18 20 24 0.840 1.050 1.315 1.660 1.900 2.375 2.875 3.500 4.500 5.563 6.625 8.625 10.750 12.750 14 16 18 20 24 0.812 1.000 1.125 1.250 1.438 1.562 1.750 2.062 0.187 0.218 0.250 0.250 0.281 0.343 0.375 0.438 0.531 0.625 0.718 0.906 1.125 1.312 1.406 1.593 1.781 1.968 2.343 7 8.75 10.5 11.5 13.12 14.88 16.5 19.88 0.47 0.61 0.82 1.16 1.34 1.69 2.13 2.62 3.44 4.31 5.19 6.81 8.5 10.13 11.19 12.81 14.44 16.06 19.31

U.S.A. Pipe Dimensions 0.5834 0.7292 0.8750 0.9583 1.0937 1.2397 1.3750 1.6563 0.0388 0.0512 0.0679 0.0967 0.1115 0.1408 0.1771 0.2187 0.2865 0.3594 0.4324 0.5678 0.7083 0.8438 0.9323 1.0678 1.2032 1.3387 1.6095 49.01 76.56 110.25 132.25 172.24 221.30 272.25 395.06 0.22 0.38 0.66 1.35 1.79 2.85 4.52 6.89 11.82 18.60 26.93 46.42 72.25 102.54 125.17 164.20 208.46 258.05 373.03 343.1 669.9 1,157.6 1,520.9 2,260.5 3,292.0 4,492.1 7,852.1 0.1 0.2 0.5 1.6 2.4 4.8 9.6 18.1 40.6 80.2 139.7 316.2 614.1 1,038.3 1,400.4 2,104.0 3,009.7 4,145.3 7,204.7 2,402.4 5,861.8 12,155.1 17,490.1 29,666.4 48,971.6 74,120.1 156,068.8 0.0 0.1 0.4 1.8 3.2 8.1 20.4 47.4 139.7 346.0 725.0 2,154.5 5,220.1 10,513.6 15,667.9 26,961.2 43,453.8 66,590.9 139,151.8

September 30, 2003 Rev: 0 16,819.0 51,290.9 127,628.2 201,135.7 389,341.9 728,502.2 1,222,981.0 3,102,022.5 0.0 0.1 0.4 2.1 4.3 13.7 43.3 124.4 480.3 1,492.4 3,762.0 14,678.8 44,370.5 106,460.8 175,292.1 345,480.5 627,386.5 1,069,716.0 2,687,578.4 38.50 60.13 86.59 103.87 135.28 173.80 213.82 310.28 0.17 0.30 0.52 1.06 1.41 2.24 3.55 5.41 9.28 14.61 21.15 36.46 56.75 80.53 98.31 128.96 163.72 202.67 292.98 0.267 0.418 0.601 0.721 0.939 1.207 1.485 2.155 0.001 0.002 0.004 0.007 0.010 0.016 0.025 0.038 0.064 0.101 0.147 0.253 0.394 0.559 0.683 0.896 1.137 1.407 2.035

Schedule 160

Schedule 140

Standard Wall ("Std") Pipe (Similar to Schedule 40)1/8 1/4 3/8 1/2 3/4 1 1.25 1.5 2 2.5 3 3.5 4 5 6 8 10 12 0.405 0.540 0.675 0.840 1.050 1.315 1.660 1.900 2.375 2.875 3.500 4.000 4.500 5.563 6.625 8.625 8.625S 10.75 10.75 10.75S 12.75 12.75S 0.068 0.088 0.091 0.109 0.113 0.133 0.140 0.145 0.154 0.203 0.216 0.226 0.237 0.258 0.280 0.277 0.322 0.279 0.307 0.365 0.330 0.375 0.269 0.364 0.493 0.622 0.824 1.049 1.380 1.610 2.067 2.469 3.068 3.548 4.026 5.047 6.065 8.071 7.981 10.192 10.136 10.020 12.090 12.000 0.0224 0.0303 0.0411 0.0518 0.0687 0.0874 0.1150 0.1342 0.1723 0.2058 0.2557 0.2957 0.3355 0.4206 0.5054 0.6726 0.6651 0.8493 0.8447 0.8350 1.0075 1.0000 0.0724 0.1325 0.2430 0.3869 0.6790 1.100 1.904 2.592 4.272 6.096 9.41 12.59 16.21 25.47 36.78 65.14 63.70 103.88 102.74 100.40 146.17 144.00 0.0195 0.0482 0.1198 0.2406 0.5595 1.154 2.628 4.173 8.831 15.051 28.9 44.7 65.3 128.6 223.1 525.8 508.4 1,058.7 1,041.4 1,006.0 1,767.2 1,728.0 0.0052 0.0176 0.0591 0.1497 0.4610 1.211 3.627 6.719 18.254 37.161 88.6 158.5 262.7 648.8 1,353.1 4,243.4 4,057.2 10,790.4 10,555.2 10,080.2 21,365.1 20,736.0 0.0014 0.0064 0.0291 0.0931 0.3799 1.270 5.005 10.818 37.731 91.750 271.8 562.2 1,057.7 3,274.7 8,206.4 34,248.1 32,380.7 109,975.8 106,987.5 101,004.0 258,304.2 248,832.0 0.0568 0.1041 0.1909 0.3039 0.5333 0.864 1.496 2.036 3.356 4.788 7.393 9.887 12.730 20.006 28.890 51.162 50.027 81.585 80.691 78.854 114.80 113.10 0.0004 0.0007 0.0013 0.0021 0.0037 0.0060 0.0104 0.0141 0.0233 0.0332 0.0513 0.0687 0.0884 0.1389 0.2006 0.3553 0.3474 0.5666 0.5604 0.5476 0.7972 0.7854

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U.S.A. Pipe Dimensions

September 30, 2003 Rev: 0

Extra Strong ("XS") Wall Pipe (Similar to Schedule 80)1/8 1/4 3/8 1/2 3/4 1 1.25 1.5 2 2.5 3 3.5 4 5 6 8 10 12 0.405 0.540 0.675 0.840 1.050 1.315 1.660 1.900 2.375 2.875 3.500 4.000 4.500 5.563 6.625 8.625 10.75 12.75 0.095 0.119 0.126 0.147 0.154 0.179 0.191 0.200 0.218 0.276 0.300 0.318 0.337 0.375 0.432 0.500 0.500 0.500 0.215 0.302 0.423 0.546 0.742 0.957 1.278 1.500 1.939 2.323 2.900 3.364 3.826 4.813 5.761 7.625 9.750 11.750 0.0179 0.0252 0.0353 0.0455 0.0618 0.0798 0.1065 0.1250 0.1616 0.1936 0.2417 0.2803 0.3188 0.4011 0.4801 0.6354 0.8125 0.9792 0.0462 0.0912 0.1789 0.2981 0.5506 0.916 1.633 2.250 3.760 5.396 8.41 11.32 14.64 23.16 33.19 58.14 95.06 138.06 0.0099 0.0275 0.0757 0.1628 0.4085 0.876 2.087 3.375 7.290 12.536 24.4 38.1 56.0 111.5 191.2 443.3 926.9 1,622.2 0.0021 0.0083 0.0320 0.0889 0.3031 0.839 2.668 5.063 14.136 29.120 70.7 128.1 214.3 536.6 1,101.5 3,380.3 9,036.9 19,061.3 0.0005 0.0025 0.0135 0.0485 0.2249 0.803 3.409 7.594 27.409 67.647 205.1 430.8 819.8 2,582.7 6,345.8 25,775.0 88,109.6 223,969.7 0.0363 0.0716 0.1405 0.2341 0.4324 0.719 1.283 1.767 2.953 4.238 6.605 8.888 11.497 18.194 26.067 45.664 74.662 108.43 0.0003 0.0005 0.0010 0.0016 0.0030 0.0050 0.0089 0.0123 0.0205 0.0294 0.0459 0.0617 0.0798 0.1263 0.1810 0.3171 0.5185 0.7530

Double Extra Strong ("XXS") Wall Pipe (Similar to Schedules > 80)1/2 3/4 1 1.25 1.5 2 2.5 3 3.5 4 5 6 8 0.840 1.050 1.315 1.660 1.900 2.375 2.875 3.500 4.000 4.500 5.563 6.625 8.625 0.294 0.308 0.358 0.382 0.400 0.436 0.552 0.600 0.636 0.674 0.750 0.864 0.875 0.252 0.434 0.599 0.896 1.100 1.503 1.771 2.300 2.728 3.152 4.063 4.897 6.875 0.0210 0.0362 0.0499 0.0747 0.0917 0.1253 0.1476 0.1917 0.2273 0.2627 0.3386 0.4081 0.5729 0.0635 0.1884 0.359 0.803 1.210 2.259 3.136 5.29 7.44 9.94 16.51 23.98 47.27 0.0160 0.0817 0.215 0.719 1.331 3.395 5.555 12.2 20.3 31.3 67.1 117.4 325.0 0.0040 0.0355 0.129 0.645 1.464 5.103 9.837 28.0 55.4 98.7 272.5 575.1 2,234.0 0.0010 0.0154 0.077 0.577 1.611 7.670 17.422 64.4 151.1 311.1 1,107.2 2,816.1 15,359.0 0.0499 0.1479 0.282 0.631 0.950 1.774 2.463 4.155 5.845 7.803 12.965 18.834 37.122 0.0003 0.0010 0.0020 0.0044 0.0066 0.0123 0.0171 0.0289 0.0406 0.0542 0.0900 0.1308 0.2578

Stainless Steel Pipe (Schedule 5S)1/2 3/4 1 1.25 1.5 2 2.5 3 3.5 4 5 6 8 10 12 0.840 1.050 1.315 1.660 1.900 2.375 2.875 3.500 4.000 4.500 5.563 6.625 8.625 10.75 12.75 0.065 0.065 0.065 0.065 0.065 0.065 0.083 0.083 0.083 0.083 0.109 0.109 0.109 0.134 0.156 0.710 0.920 1.185 1.530 1.770 2.245 2.709 3.334 3.834 4.334 5.345 6.407 8.407 10.482 12.438 0.0592 0.0767 0.0988 0.1275 0.1475 0.1871 0.2258 0.2778 0.3195 0.3612 0.4454 0.5339 0.7006 0.8735 1.0365 0.5041 0.8464 1.404 2.341 3.133 5.040 7.339 11.12 14.70 18.78 28.57 41.05 70.68 109.87 154.70 0.3579 0.7787 1.664 3.582 5.545 11.315 19.880 37.1 56.4 81.4 152.7 263.0 594.2 1,151.7 1,924.2 0.2541 0.7164 1.972 5.480 9.815 25.402 53.856 123.6 216.1 352.8 816.2 1,685.1 4,995.3 12,071.9 23,933.3 0.1804 0.6591 2.337 8.384 17.373 57.027 145.897 411.9 828.4 1,529.1 4,362.5 10,796.3 41,995.7 126,537.9 297,682.1 0.3959 0.6648 1.103 1.839 2.461 3.958 5.764 8.730 11.545 14.753 22.438 32.240 55.510 86.294 121.50 0.0027 0.0046 0.0077 0.0128 0.0171 0.0275 0.0400 0.0606 0.0802 0.1024 0.1558 0.2239 0.3855 0.5993 0.8438

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Stainless Steel Pipe (Schedule 10S)1/8 1/4 3/8 1/2 3/4 1 1.25 1.5 2 2.5 3 3.5 4 5 6 8 10 12 0.405 0.540 0.675 0.840 1.050 1.315 1.660 1.900 2.375 2.875 3.500 4.000 4.500 5.563 6.625 8.625 10.75 12.75 0.049 0.065 0.065 0.083 0.083 0.109 0.109 0.109 0.109 0.120 0.120 0.120 0.120 0.134 0.134 0.148 0.165 0.180 0.307 0.410 0.545 0.674 0.884 1.097 1.442 1.682 2.157 2.635 3.260 3.760 4.260 5.295 6.357 8.329 10.420 12.390 0.0256 0.0342 0.0454 0.0562 0.0737 0.0914 0.1202 0.1402 0.1798 0.2196 0.2717 0.3133 0.3550 0.4413 0.5298 0.6941 0.8683 1.0325 0.0942 0.1681 0.2970 0.4543 0.7815 1.203 2.079 2.829 4.653 6.943 10.63 14.14 18.15 28.04 40.41 69.37 108.58 153.51 0.0289 0.0689 0.1619 0.3062 0.6908 1.320 2.998 4.759 10.036 18.295 34.6 53.2 77.3 148.5 256.9 577.8 1,131.4 1,902.0 0.0089 0.0283 0.0882 0.2064 0.6107 1.448 4.324 8.004 21.647 48.208 112.9 199.9 329.3 786.1 1,633.1 4,812.5 11,788.8 23,566.0 0.0027 0.0116 0.0481 0.1391 0.5398 1.589 6.235 13.463 46.693 127.029 368.2 751.5 1,403.0 4,162.3 10,381.5 40,083.4 122,839.7 291,982.3 0.0740 0.1320 0.2333 0.3568 0.6138 0.945 1.633 2.222 3.654 5.453 8.347 11.104 14.253 22.020 31.739 54.485 85.276 120.57 0.0005 0.0009 0.0016 0.0025 0.0043 0.0066 0.0113 0.0154 0.0254 0.0379 0.0580 0.0771 0.0990 0.1529 0.2204 0.3784 0.5922 0.8373

Note:Stainless Steel Pipe Schedule 40S values are the same, size for size, as those shown above on the Standard Wall Pipe Table (heaviest weight on 8, 10, and 12-inch sizes). Stainless Steel Pipe Schedule 80S values are the same, size for size, as those shown above on the Extra Strong Pipe Table.

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May 8, 2003

Although pipe classification is common knowledge that is taken for granted among a lot of us old engineers, I have found that young engineers are lacking this information because both academic professors and we experienced engineers are both guilty of not passing on the information which used to be common and available when piping and fitting catalogs like Vogt, Tube Turns, Walworth, etc. used to be freely available to us. Now, these valuable free catalogs have become a thing of the past Because I regard this subject as very basic and important for all engineers to dominate, some years back I prepared the following explanation for young engineers working under me and with me in plant projects. I would like to share it with any one else who hasn't had the opportunity to find out this logical explanation of how pipe is classified. Industrial pipe thicknesses follow a set formula, expressed as the schedule number as established by the American Standards Association (ASA) now re-organized as ANSI - the American National Standards Institute. Eleven schedule numbers are available for use: 5, 10, 20, 30, 40, 60, 80, 100, 120, 140, & 160. The most popular schedule, by far, is 40. Sch 5, 60, 100, 120, & 140 have rarely, if ever been employed by myself in over 40 years as a practicing engineer. The schedule number is defined as the approximate value of the expression: Schedule Number = (1,000)(P/S) Where, P = the internal working pressure, psig S = the allowable stress (psi) for the material of construction at the conditions of use. For example, the schedule number of ordinary steel pipe having an allowable stress of 10,000 psi for use at a working pressure of 350 psig would be: Schedule Number = (1,000)(350/10,000) = 35 (approx. 40) This would be the proper schedule for welded joints and steel fittings but not for threaded connections and cast-iron or malleable-iron fittings. In practice, schedule 40 would be used for welded construction and Sch 80 (about 2x the computed value) for iron fittings. The higher schedule is required because of weaknesses in the iron fittings and the metal lost in cutting the threads. For all pipe sizes below 10", Sch 40 pipe is identical with what was once called standard pipe, and Sch 80 is identical with the former extra-strong pipe. There is no equivalent schedule number for double-extra-strong pipe, and Sch 160 is the only other weight in which pipe smaller than 4" is available. Temperature has no direct bearing on the schedule, except as it either weakens (or strengthens) the material's allowable stress. Stainless steels (304ELC & 316ELC), for example, yield a stronger allowable stress at the low temperatures near the cryogenic zone (-50 to -150 oF). Copper and Brass also exhibit the same behavior. I've used the rule of thumb that the softer the metal, the stronger it is at the lower temperatures. I hope this has helped you in explaining how pipe is classified. Art Montemayor

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In actual, industrial practice, every operating company (organizations such as DuPont, Dow, Monsanto, Huntsman, Union Carbide, etc.) as well as all the major Engineering and Construction companies (such as Fluor, Bechtel, etc.) have a listing of their standard pipe sizes which they apply to all their projects. These companies make it a point to reduce the quantity of different sizes employed in any one project or plant. Obviously, when confronted with a

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September 30, 2003 Rev: 0 customer's list of different standard sizes, the E&C contractor will accede to his customer's desires and wishes. U.S.A. Pipe Dimensions Certain pipe sizes are labeled as "Bastard" sizes in the USA, and they are not incorporated nor used in applications. Some of these Bastard sizes are 1-1/4", 2-1/2", 3-1/2", and 5". The reasons for not using these sizes are usually: 1. Traditionally, the concept of stepping up the size of piping by a magnitude of doubling the capacity is what has been predominant thinking - at least in the USA. In other words, an attempt is made to have pipe sizes that are separated by roughly a factor of 2 x the cross-sectional area. This concept is applied to pipes sizes up to 10" in nominal diameter. 2. The tradition of establishing pipe sizes by the method described above results in a series of pipe sizes for Schedule 40 as follows: Nominal Internal pipe size, flow area, in2 in. 1 1 1/2 2 3 4 6 8 10 0.864 2.036 3.355 7.393 12.73 28.89 50.03 78.86

3. The reduction of pipe sizes actually employed forces the pipe fabricator to increase the price of those sizes (such as 1-1/4, 2-1/2, 3-1/2, and 5") due to their diminished market demand. 4. Reducing the variety of pipe sizes also allows a user to reduce his pipe plant inventory, together with the many required fittings. This results in increased economy in many ways and simplifies maintenance. The applications or project engineer should always strive to adhere to a standard of pipe sizes in order to simplify the subsequent operation of the equipment and installation that he installs.

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gineers, I have

when piping and

ack I prepared ould like to share

ns and cast-iron 0 (about 2x the fittings and the

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March 12, 2003 Rev: 0

Insert Basic Data in Input Cells and Obtain Results in Bold Red Numbers Inputs Nominal Pipe Diameter 2 inches Pipe's Inside Diameter Pipe's Schedule No. 40 Flow Rate 100 GPM Click this Button Fluid's Viscosity 5 cPs to see a list of Fluid's Density 8.45 lbs/gal acceptable pipe diameters Pipe's Abs. Roughness 0.00015 feet and pipe schedules Calculation Methods Churchill Equation (1977)Chen Equation (1979) P Per 100 feet of pipe P Per 100 feet of pipe Reynolds Number 30,979 9.31 psi 9.28 psi 21.18 feet 21.12 feet Linear Velocity Darcy Friction Factor Darcy Friction Factor 9.55 ft/sec 0.0257 0.0256

2.067 inches

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FileName: 83236863.xls WorkSheet: Liq Pressure Drop

This sheet gives the value of the Fanning Friction Factor, f, using the Colebrook-White, Moody and Barr formulas Enter the known basic data in the YELLOW cells as input for the calculated solution. The Colebrook - White Equation The Moody Equation

1 1.26 =4 log1 0 3.71d R e f f 1.26 f n1=1/ 4log1 0 3.71d Re f nf n1 = 1

[

]

200 106 f =0. 001375 1 d ReThe Barr Equation

[[

]1/3

2

[4 log 10 ab ]

2

1 5. 1286 =4 log10 0 .89 3. 71d Re ff =1/ 4log10

]

Enter the basic data input here (mm) 0.03 Absolute Pipe Roughness d (mm) 300 Pipe Internal diameter Re 664,000 Reynolds Number Solution f 0.01207 0.01406 0.01398

[

5. 1286 0 . 89 3. 71 d Re

]

2

Moody 0.003018 Barr 0.003515 Colebrook-White 0.003495 Colebrook-White iteration f(n) a b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(f = Fanning Friction Factor) ( = Darcy Friction Factor)

1 2 3 4 5 6 7 8 9 ### ### ### ###

0 0 0 0 0 0 0 0 0 0 0 0 0

LHS 18.2 16.84 16.92 16.91 16.92 16.92 16.92 16.92 16.92 16.92 16.92 16.92 16.92

RHS 16.84 16.92 16.91 16.92 16.92 16.92 16.92 16.92 16.92 16.92 16.92 16.92 16.92

Error 1.36 -0.07 0 0 0 0 0 0 0 0 0 0 0

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

This spreadsheet was downloaded from a British WebSite on the InterNet. The author is identified as Andrew Sleigh. -Art Montemayor (02/15/2003) 0.7500 inches = 19.05 mm

From: "Fluid Mechanics With Engineering Applications"; R. L. Daugherty & J. B. Franzini; McGraw-Hill Book Co.; 7th Ed.; 1977: Values of Absolute Roughness, , For New Pipe Pipe Type Drawn tubing, brass, lead, glass, centrifugally spun cement, bituminous lining, transite Commercial steel or wrought iron Welded-steel pipe Asphalt-dipped cast iron Galvanized iron Cast iron, average Concrete Wood stave Riveted steel Units of Feet Millimeters 0 0.00015 0.00015 0.00040 0.00050 0 0.046 0.046 0.122 0.152

0.00085 0.259 0.001 to 001 0.3 to 3.0 0.0006 to 0.003 0.18 to 0.9 0.003 to 0.03 0.9 to 9.0

Note: Daugherty & Franzini state that the Colebook Equation is:

1 d 2 . 51 =2 Log 3 . 7 Re f for, expressed in Natural Logarithms:

1 d 2 . 51 =0 . 8686 Ln 3 . 7 Re f f

The Friction Factor used in the above equation is the Darcy Friction Factor. The Darcy Friction Factor is four times (4x) the Fanning Fricition Factor.

Computerized Moody Diagram -- Rick Sellens, Queen's Mechanical Engineering

0.05 e/D Re Smooth 3,000 0.04356 3,150 0.04291 3,308 0.04228 3,473 0.04166 3,647 0.04106 3,829 0.04046 4,020 0.03988 4,221 0.03931 4,432 0.03875 4,654 0.03821 4,887 0.03767 5,131 0.03714 5,388 0.03663 5,657 0.03612 5,940 0.03563 6,237 0.03514 6,549 0.03467 6,876 0.03420 7,220 0.03374 7,581 0.03329 7,960 0.03285 8,358 0.03242 8,776 0.03200 9,215 0.03158 9,675 0.03117 10,159 0.03077 10,667 0.03038 11,200 0.02999 11,760 0.02961 12,348 0.02924 12,966 0.02887 13,614 0.02851 14,295 0.02816 15,010 0.02782 15,760 0.02747 16,548 0.02714 17,375 0.02681 18,244 0.02649 19,156 0.02617 20,114 0.02586 21,120 0.02555 22,176 0.02525 23,285 0.02495 24,449 0.02466 25,671 0.02438 26,955 0.02409 28,303 0.02382 29,718 0.02354 31,204 0.02328 32,764 0.02301 34,402 0.02275 36,122 0.02250 37,928 0.02225 39,825 0.02200 41,816 0.02176 43,907 0.02152 46,102 0.02128 48,407 0.02105 50,828 0.02082 53,369 0.02060 56,038 0.02038 58,839 0.02016 61,781 0.01994 64,870 0.01973 68,114 0.01952 71,520 0.01932 75,096 0.01912 78,850 0.01892 82,793 0.01872 86,933 0.01853 91,279 0.01834 95,843 0.01815 100,635 0.01797 105,667 0.01779 110,951 0.01761 116,498 0.01743 122,323 0.01726 128,439 0.01709 134,861 0.01692 141,604 0.01675 148,684 0.01659 156,119 0.01643 163,924 0.01627 172,121 0.01611 180,727 0.01595 189,763 0.01580 199,251 0.01565 209,214 0.01550 219,674 0.01536 230,658 0.01521 242,191 0.01507 254,301 0.01493 267,016 0.01479 280,366 0.01465 294,385 0.01452 309,104 0.01438 324,559 0.01425 340,787 0.01412 357,827 0.01399 375,718 0.01387 394,504 0.01374 414,229 0.01362 434,940 0.01350 456,687 0.01338 479,522 0.01326 503,498 0.01314 528,673 0.01303 555,106 0.01291 582,862 0.01280 612,005 0.01269 642,605 0.01258 674,735 0.01247 708,472 0.01236 743,896 0.01226 781,090 0.01215 820,145 0.01205 861,152 0.01195 904,210 0.01185 949,420 0.01175 996,891 0.01165 1,046,736 0.01155 1,099,073 0.01146 1,154,026 0.01136 1,211,728 0.01127 1,272,314 0.01118 1,335,930 0.01109 1,402,726 0.01100 1,472,863 0.01091 1,546,506 0.01082 1,623,831 0.01073 1,705,023 0.01065 1,790,274 0.01056 1,879,787 0.01048 1,973,777 0.01039 2,072,466 0.01031 2,176,089 0.01023 2,284,893 0.01015 2,399,138 0.01007 2,519,095 0.00999 2,645,050 0.00992 2,777,302 0.00984 2,916,167 0.00976 3,061,976 0.00969 3,215,074 0.00961 3,375,828 0.00954 3,544,619 0.00947 3,721,850 0.00940 3,907,943 0.00933 4,103,340 0.00926 4,308,507 0.00919 4,523,932 0.00912 4,750,129 0.00905 4,987,636 0.00898 5,237,017 0.00892 5,498,868 0.00885 5,773,812 0.00879 6,062,502 0.00872 6,365,627 0.00866 6,683,909 0.00860 7,018,104 0.00853 7,369,009 0.00847 7,737,460 0.00841 8,124,333 0.00835 8,530,549 0.00829 8,957,077 0.00823 9,404,931 0.00817 9,875,177 0.00812 10,368,936 0.00806 10,887,383 0.00800 11,431,752 0.00795 12,003,340 0.00789 12,603,507 0.00784 13,233,682 0.00778 13,895,366 0.00773 14,590,134 0.00767 15,319,641 0.00762 16,085,623 0.00757 16,889,904 0.00752 17,734,400 0.00747 18,621,120 0.00742 19,552,176 0.00737 20,529,784 0.00732 21,556,274 0.00727 22,634,087 0.00722 23,765,792 0.00717 24,954,081 0.00712 26,201,785 0.00708 27,511,874 0.00703 28,887,468 0.00698 30,331,842 0.00694 31,848,434 0.00689 33,440,855 0.00685 35,112,898 0.00680 36,868,543 0.00676 38,711,970 0.00671 40,647,569 0.00667 42,679,947 0.00663 44,813,944 0.00658 47,054,642 0.00654 49,407,374 0.00650 51,877,742 0.00646 54,471,630 0.00642 57,195,211 0.00638 60,054,972 0.00634 63,057,720 0.00630 66,210,606 0.00626 69,521,136 0.00622 72,997,193 0.00618 76,647,053 0.00614 80,479,406 0.00610 84,503,376 0.00607 88,728,545 0.00603 93,164,972 0.00599 97,823,221 0.00596

e/d 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

Re 3,000 3,150 3,308 3,473 3,647 3,829 4,020 4,221 4,432 4,654 4,887 5,131 5,388 5,657 5,940 6,237 6,549 6,876 7,220 7,581 7,960 8,358 8,776 9,215 9,675 10,159 10,667 11,200 11,760 12,348 12,966 13,614 14,295 15,010 15,760 16,548 17,375 18,244 19,156 20,114 21,120 22,176 23,285 24,449 25,671 26,955 28,303 29,718 31,204 32,764 34,402 36,122 37,928 39,825 41,816 43,907 46,102 48,407 50,828 53,369 56,038 58,839 61,781 64,870 68,114 71,520 75,096 78,850 82,793 86,933 91,279 95,843 100,635 105,667 110,951 116,498 122,323 128,439 134,861 141,604 148,684 156,119 163,924 172,121 180,727 189,763 199,251 209,214 219,674 230,658 242,191 254,301 267,016 280,366 294,385 309,104 324,559 340,787 357,827 375,718 394,504 414,229 434,940 456,687 479,522 503,498 528,673 555,106 582,862 612,005 642,605 674,735 708,472 743,896 781,090 820,145 861,152 904,210 949,420 996,891 1,046,736 1,099,073 1,154,026 1,211,728 1,272,314 1,335,930 1,402,726 1,472,863 1,546,506 1,623,831 1,705,023 1,790,274 1,879,787 1,973,777 2,072,466 2,176,089 2,284,893 2,399,138 2,519,095 2,645,050 2,777,302 2,916,167 3,061,976 3,215,074 3,375,828 3,544,619 3,721,850 3,907,943 4,103,340 4,308,507 4,523,932 4,750,129 4,987,636 5,237,017 5,498,868 5,773,812 6,062,502 6,365,627 6,683,909 7,018,104 7,369,009 7,737,460 8,124,333 8,530,549 8,957,077 9,404,931 9,875,177 10,368,936 10,887,383 11,431,752 12,003,340 12,603,507 13,233,682 13,895,366 14,590,134 15,319,641 16,085,623 16,889,904 17,734,400 18,621,120 19,552,176 20,529,784 21,556,274 22,634,087 23,765,792 24,954,081 26,201,785 27,511,874 28,887,468 30,331,842 31,848,434 33,440,855 35,112,898 36,868,543 38,711,970 40,647,569 42,679,947 44,813,944 47,054,642 49,407,374 51,877,742 54,471,630 57,195,211 60,054,972 63,057,720 66,210,606 69,521,136 72,997,193 76,647,053 80,479,406 84,503,376 88,728,545 93,164,972 97,823,221

f 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

0.08533 0.08470 0.08410 0.08352 0.08297 0.08244 0.08194 0.08146 0.08100 0.08056 0.08014 0.07974 0.07936 0.07899 0.07865 0.07831 0.07800 0.07769 0.07741 0.07713 0.07687 0.07662 0.07638 0.07615 0.07593 0.07572 0.07552 0.07533 0.07515 0.07498 0.07482 0.07466 0.07451 0.07437 0.07423 0.07410 0.07398 0.07386 0.07375 0.07364 0.07354 0.07344 0.07335 0.07326 0.07318 0.07310 0.07302 0.07295 0.07288 0.07281 0.07275 0.07269 0.07263 0.07258 0.07253 0.07248 0.07243 0.07239 0.07234 0.07230 0.07226 0.07223 0.07219 0.07216 0.07212 0.07209 0.07207 0.07204 0.07201 0.07199 0.07196 0.07194 0.07192 0.07190 0.07188 0.07186 0.07184 0.07182 0.07181 0.07179 0.07178 0.07176 0.07175 0.07174 0.07173 0.07171 0.07170 0.07169 0.07168 0.07167 0.07166 0.07166 0.07165 0.07164 0.07163 0.07162 0.07162 0.07161 0.07161 0.07160 0.07159 0.07159 0.07158 0.07158 0.07157 0.07157 0.07157 0.07156 0.07156 0.07155 0.07155 0.07155 0.07154 0.07154 0.07154 0.07154 0.07153 0.07153 0.07153 0.07153 0.07152 0.07152 0.07152 0.07152 0.07152 0.07152 0.07151 0.07151 0.07151 0.07151 0.07151 0.07151 0.07151 0.07151 0.07150 0.07150 0.07150 0.07150 0.07150 0.07150 0.07150 0.07150 0.07150 0.07150 0.07150 0.07150 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07149 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148 0.07148

calculate to converge 0.07836 0.07865 0.07806 0.07833 0.07778 0.07802 0.07751 0.07773 0.07724 0.07745 0.07699 0.07718 0.07675 0.07692 0.07652 0.07668 0.07630 0.07644 0.07609 0.07622 0.07588 0.07600 0.07569 0.07579 0.07550 0.07560 0.07532 0.07541 0.07515 0.07523 0.07498 0.07506 0.07482 0.07489 0.07467 0.07474 0.07453 0.07459 0.07439 0.07444 0.07426 0.07431 0.07413 0.07418 0.07401 0.07405 0.07390 0.07393 0.07379 0.07382 0.07368 0.07371 0.07358 0.07361 0.07348 0.07351 0.07339 0.07341 0.07330 0.07332 0.07322 0.07324 0.07314 0.07315 0.07306 0.07308 0.07299 0.07300 0.07292 0.07293 0.07285 0.072

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