1

Optimum Pipe Flow

Department of

Mechanical and Aerospace Engineering

University of California: Irvine

by

Nicholas Cordero

MAE 108

Student ID: 80377983

December 14, 2014

2

Table of Contents I. Summary…………………………………………………………………………..3

II. Solution…………………………………………………………………...……..4-9

III. Table of Results……………………………………………………….……….9-12

IV. Plots………………………………………………………………………...……13

V. Conclusion………………………………………………………...……………..14

List of Figures

1. Figure1. Schematic of CW Cycle for UCI Central Plant Figure 2

2. Figure2. Schematic of HTW Cycle for UCI Central Plant

3. Figure3. Viscosity of Water vs. Temperature from Munson Fundamentals of Fluid

Mechanics 7th

edition textbook

4. Figure4. Density of Water vs. Temperature from Munson Fundamentals of Fluid

Mechanics 7th edition textbook

5. Figure5. Total Annual Cost vs. Pipe ID for HTW cycle

6. Figure6. Total Annual Cost vs. Pipe ID for CW Cycle

List of Tables

1. Table1. HTW Cycle Fluid Mechanics Calculations

2. Table2. HTW Cycle Mortgage Calculations

3. Table3. CW Cycle Fluid Mechanics Calculations

4. Table4. CW Cycle Mortgage Calculations

5. Table5. HTW Cycle Mortgage Calculations for Optimum Pipe Diameter of 14 inches

6. Table6. CW Cycle Mortgage Calculations for Optimum Pipe Diameter of 36.5 inches

3

Summary

The purpose of this design project is to determine the optimum pipe inner diameter (ID)

for the Cold Water (CW) and High Temperature Water (HTW) supply cycles for the UCI Central

Plant which will yield the lowest total cost annually. The total annual cost accounts for the cost

of electricity required to per year to power the pump, the cost of the pipe, the cost for

installation, pumps, valves, etc. , and the mortgage cost on the total capital cost, which is the sum

of the cost of the pipe and cost of installation, pumps, valves, etc. For the amortization, a 30-year

unsecured mortgage at 3.4% APR with payment yearly is assumed while the electricity cost is

assumed to be $.17/kW*Hour. Each cycle are assumed to have level pipes of schedule 40 steel of

each 10,000ft in length and thirteen pipe inner diameters to choose from for the optimal

diameter, ranging from 2.5 inches to 36.5 inches. The temperature and flow rate is assumed to be

4.111˚C and 50,000 gallons per minute (gpm) for the CW cycle and 186.667˚C and 4,000gpm

for the HTW cycle. The HTW cycle is always assumed to be kept in the liquid phase by the

nitrogen blanket and both cycles are assumed to be run for 365 days per year, 24 hours per day

with no shut downs.

In general, pumping power decreases with increasing pipe diameter and the cost of the

pipes and installation increase as well, nonlinearly. The pumping power can be determined by

first by finding the pressure drop across the pipes with the various diameter sizes and then

multiplying the pressure drop by the flow rate for the particular cycle being analyzed. The

pressure drop is determined by various parameters such as the area of the pipe, the velocity of the

flow, Reynolds’s number, surface roughness of the pipe, and friction factor. These properties are

determined based on the properties of water, namely density and kinematic viscosity, as a

function of temperature which is obtained the Munson Fundamentals of Fluid Mechanics 7th

edition textbook and extrapolated in order to obtain the properties of water at the temperature for

each cycle. The pumping power determined from the pressure drop and flow rate can now be

multiplied by the electricity cost to determine the amount it costs to run each year. The mortgage

and electricity cost are added together to yield the final total cost per year for each pipe diameter.

The calculations showed the optimum pipe diameter for the CW cycle was 36.5 inches at

a total annual cost of $3,831,537.60 while the optimum pipe diameter for the HTW cycle was

14.0 inches at a total cost of $315,261.32. The annual electricity cost for the CW cycle will be

$1,935,723.06 while the annual mortgage cost will be $1,895,814.54. For the HTW cycle, the

annual electricity cost will be $114,511.34 while the annual mortgage cost will be $200,749.98.

4

Solution Figure1. Schematic of CW Cycle for UCI Central Plant

5

Figure2. Schematic of HTW Cycle for UCI Central Plant

UCI’s Central plant provides the UCI campus with cold and hot water for 24 hours a day,

7 days a week, and 365 days out of the year. The cold and hot water are supplied from the

Central plant to campus through underground pipes and the cost for this process can be

optimized by varying the pipe diameters for the HTW cycle and CW cycle.

Problem

Due to the HTW cycle and CW cycle having different flow rates and operating

temperatures, each cycle has an optimum pipe diameter which has the lowest annual cost and

ultimately achieves the goal of delivering the HTW and CW to campus.

Key Parameters

1. Inner Diameter (D)

2. Temperature (T)

3. Area (A)

4. Flow Rate (Q)

5. Friction Factor (f)

6. Pipe Surface Rougness (ε)

7. Velocity (V)

8. Pressure Drop (∆P)

9. Power (P)

6

10. Pipe Length (L)

11. Density (ρ)

12. Kinematic Viscosity (υ)

13. Reynolds number (Re)

Assumptions

1. One of the assumptions made for this design is that the pipe run is level, neglecting any

pressure drop variation due to gravitational effects from either increasing or decreasing

sloped pipe lines. In reality, there will be different configurations for the pipe lines in

which they may have a decreasing slope which will result in less power need to drive the

flow or an increasing slope which will result in more power needed to drive the flow.

Either way, they both affect the amount of power needed and may vary from each pipe

length segment.

2. The design will assume maximum expected flow rates for HTW of 4,000 gpm and 50,000

gpm however, in reality the flow rates may vary depending on variation in the pumping

power of the motor.

3. The HTW is also assumed to be kept in the liquid phase by the nitrogen blanket.

4. The flows are assumed to be maintained 365 days per year, 24 hours per day however, in

reality there might be technical difficulties in certain areas meaning the central plant may

have to shut down temporarily for maintenance repair. This means that more pumping

power is needed to make up for this time interval of no operation.

5. The fluid properties, mainly density and viscosity, as a function of temperature are

obtained from the Munson Fundamentals of Fluid Mechanics 7th edition textbook and

then extrapolated using a best fit curve meaning. This best fit curve is used to obtain the

fluid properties at our desired temperatures which leads to error, and can be seen in

Figures 3 and 4.

Sample Calculation

This calculation will be done for the CW cycle (T=4.111˚C and Q=5,000gpm) using the

2.5 inch (.0635 meter) pipe ID

1. Find A

( )

2. Find V

3. Find

7

The data for viscosity as a function of temperature was obtained from the Munson

Fundamentals of Fluid Mechanics 7th edition textbook and extrapolated using a best fit

exponential curve as shown in Figure 3 below. For the CW at T=4.111˚C, the corresponding υ=

0.000001687

which is hardly different from υ at standard air conditions but for the HTW

cycle, the υ=.00000000087302

at and cannot be assumed to be at standard air conditions.

Figure3. Viscosity of Water vs. Temperature from Munson Fundamentals of Fluid Mechanics 7th

edition textbook

4. Find Re

5. Find f

Colebrook Equation which is only valid for Re>4,000 and obtained from the Munson

Fundamentals of Fluid Mechanics 7th edition textbook.

( (( ⁄

)

))

Where =.000045m for schedule 40 steel pipe

y = 2E-06e-0.018x

0.000E+00

2.000E-07

4.000E-07

6.000E-07

8.000E-07

1.000E-06

1.200E-06

1.400E-06

1.600E-06

1.800E-06

2.000E-06

0 20 40 60 80 100 120

Vis

cosi

ty [

m2

/s]

Temperature [°C]

Kinematic Viscosity of Water vs. Temperature

8

( (( ⁄

)

))

6. Find ρ

The data for density as a function of temperature was obtained from the Munson

Fundamentals of Fluid Mechanics 7th edition textbook and extrapolated using a best fit

polynomial curve as shown in Figure 4 below. For the CW at T=4.111˚C, the corresponding ρ =

1000.15755

which again is hardly different from ρ at standard air conditions but for the HTW

cycle, the ρ=862.2728625

at 186.667˚C and cannot be assumed to be at standard air

conditions.

Figure4. Density of Water vs. Temperature from Munson Fundamentals of Fluid Mechanics 7

th

edition textbook

7. Find ∆P

Where L= 10,000ft (or 3048m) for both the HTW and CW cycles

(

)

(

)

8. Find P

y = -0.0036x2 - 0.0685x + 1000.5

955

960

965

970

975

980

985

990

995

1000

1005

0 20 40 60 80 100 120

De

nsi

ty (

kg/m

3 )

Temperature [°C]

Density of Water vs. Temperature

9

Pipe ID

(in)

Pipe ID

(m)

Cost of Pipe

[$/ft]

Cross Sectional Area

[m2]

Q(flow rate)

[gpm]

Q(flow rate)

[m3/s]V[m/s] Re f ΔP [Pa] Power [W] Power [HP] Power [kW]

Cost to Run Annually

[$]

2.5 0.0635 2.99 0.003166922 4000 0.252360667 79.6864233 5.796E+09 0.01812371 2419613457 610615265 818848.5582 610615.265 $909,931,357.39

3.0 0.0762 3.99 0.004560367 4000 0.252360667 55.33779396 4.830E+09 0.01737572 932256373.7 235264840 315495.3471 235264.84 $350,588,770.76

4.0 0.1016 4.99 0.00810732 4000 0.252360667 31.1275091 3.623E+09 0.01628634 207358805.4 52329206.37 70174.62163 52329.20637 $77,980,339.68

6.0 0.1524 7.99 0.018241469 4000 0.252360667 13.83444849 2.415E+09 0.01491691 25010458.88 6311656.076 8464.070215 6311.656076 $9,405,552.25

8.0 0.2032 12.99 0.032429279 4000 0.252360667 7.781877275 1.811E+09 0.01404752 5589187.822 1410491.165 1891.499808 1410.491165 $2,101,896.58

10.0 0.254 17.99 0.050670748 4000 0.252360667 4.980401456 1.449E+09 0.01342457 1750247.384 441693.5968 592.3208698 441.6935968 $658,206.37

12.0 0.3048 24.99 0.072965877 4000 0.252360667 3.458612122 1.208E+09 0.01294595 678307.6118 171178.1611 229.5536952 171.1781611 $255,087.59

14.0 0.3556 33.99 0.099314666 4000 0.252360667 2.541021151 1.035E+09 0.01256108 304498.9804 76843.5657 103.048919 76.8435657 $114,511.34

16.0 0.4064 43.99 0.129717115 4000 0.252360667 1.945469319 9.056E+08 0.01224151 152206.762 38410.99994 51.50999937 38.41099994 $57,239.60

20.0 0.508 54.99 0.202682992 4000 0.252360667 1.245100364 7.245E+08 0.01173435 47808.81049 12065.06329 16.17951637 12.06506329 $17,979.21

24.0 0.6096 66.99 0.291863508 4000 0.252360667 0.864653031 6.038E+08 0.0113434 18573.16064 4687.1352 6.285551837 4.6871352 $6,984.71

30.0 0.762 146.99 0.456036731 4000 0.252360667 0.55337794 4.830E+08 0.01089164 5843.671186 1474.712756 1.977622381 1.474712756 $2,197.60

36.5 0.9271 320.99 0.675061039 4000 0.252360667 0.373833849 3.970E+08 0.01051743 2116.627475 534.1535206 0.71631167 0.534153521 $795.99

Pipe ID (in) Cost to Run Annually [$] Cost of Pipe [$] Cost to Install [$] Annual Mortgage Payment [$] Annual Interest Payment [$] Annual Principle Payment [$] Total Annual Cost [$]

2.5 $909,931,357.39 $29,900.00 $299,000.00 ($17,659.38) ($580.68) ($17,078.71) $909,949,016.77

3.0 $350,588,770.76 $39,900.00 $399,000.00 ($23,565.53) ($774.88) ($22,790.65) $350,612,336.29

4.0 $77,980,339.68 $49,900.00 $499,000.00 ($29,471.68) ($969.09) ($28,502.59) $78,009,811.36

6.0 $9,405,552.25 $79,900.00 $799,000.00 ($47,190.12) ($1,551.71) ($45,638.42) $9,452,742.38

8.0 $2,101,896.58 $129,900.00 $1,299,000.00 ($76,720.87) ($2,522.74) ($74,198.13) $2,178,617.45

10.0 $658,206.37 $179,900.00 $1,799,000.00 ($106,251.61) ($3,493.77) ($102,757.84) $764,457.97

12.0 $255,087.59 $249,900.00 $2,499,000.00 ($147,594.65) ($4,853.21) ($142,741.44) $402,682.24

14.0 $114,511.34 $339,900.00 $3,399,000.00 ($200,749.98) ($6,601.06) ($194,148.92) $315,261.32

16.0 $57,239.60 $439,900.00 $4,399,000.00 ($259,811.46) ($8,543.12) ($251,268.34) $317,051.06

20.0 $17,979.21 $549,900.00 $5,499,000.00 ($324,779.09) ($10,679.39) ($314,099.70) $342,758.30

24.0 $6,984.71 $669,900.00 $6,699,000.00 ($395,652.87) ($13,009.86) ($382,643.01) $402,637.59

30.0 $2,197.60 $1,469,900.00 $14,699,000.00 ($868,144.74) ($28,546.35) ($839,598.39) $870,342.34

36.5 $795.99 $3,209,900.00 $32,099,000.00 ($1,895,814.54) ($62,338.20) ($1,833,476.34) $1,896,610.53

9. Cost to Run Annually

10. The cost of the pipe and installation are assumed to be mortgaged at a 3.4% APR for

30 years. Once this annual payment is found, it is added to the total cost to run

annually (CRA) and this will be the total cost per year. All these calculations are

down on excel using the PPMT, IPMT, and PMT functions.

Table of Results

Table1. HTW Cycle Fluid Mechanics Calculations

Table2. HTW Cycle Mortgage Calculations

10

Pipe ID

(in)

Pipe ID

(m)

Cost of Pipe

[$/ft]

Cross Sectional Area

[m2]

Q(flow rate)

[gpm]

Q(flow rate)

[m3/s]V[m/s] Re f ΔP [Pa] Power [W] Power [HP] Power [kW]

Cost to Run Annually

[$]

2.5 0.0635 2.99 0.003166922 50000 3.15450981 996.0807575 3.749E+07 0.01813304 4.31856E+11 1.3623E+12 1826868096 1362295305 $2,030,075,707,659.10

3.0 0.0762 3.99 0.004560367 50000 3.15450981 691.7227482 3.124E+07 0.017388578 1.66428E+11 5.24999E+11 704035262.9 524999005.4 $782,347,060,354.88

4.0 0.1016 4.99 0.00810732 50000 3.15450981 389.0940459 2.343E+07 0.016307734 37039274555 1.16841E+11 156686033.2 116840754.9 $174,114,655,867.16

6.0 0.1524 7.99 0.018241469 50000 3.15450981 172.9306871 1.562E+07 0.014960889 4474762206 14115681275 18929440.39 14115681.27 $21,035,014,613.00

8.0 0.2032 12.99 0.032429279 50000 3.15450981 97.27351147 1.172E+07 0.014120914 1002262952 3161648314 4239840.226 3161648.314 $4,711,449,428.84

10.0 0.254 17.99 0.050670748 50000 3.15450981 62.25504734 9.373E+06 0.013533684 314763850.1 992925652.9 1331535.233 992925.6529 $1,479,645,595.02

12.0 0.3048 24.99 0.072965877 50000 3.15450981 43.23267177 7.811E+06 0.013096587 122411068.8 386146917.4 517831.5457 386146.9174 $575,431,386.64

14.0 0.3556 33.99 0.099314666 50000 3.15450981 31.76277926 6.695E+06 0.012758563 55173480.08 174045284.1 233398.5705 174045.2841 $259,360,141.68

16.0 0.4064 43.99 0.129717115 50000 3.15450981 24.31837787 5.858E+06 0.012490672 27704778.47 87394995.47 117198.6194 87394.99547 $130,234,947.29

20.0 0.508 54.99 0.202682992 50000 3.15450981 15.56376184 4.687E+06 0.012099485 8793984.448 27740710.21 37200.90515 27740.71021 $41,338,865.14

24.0 0.6096 66.99 0.291863508 50000 3.15450981 10.80816794 3.906E+06 0.011838218 3457792.551 10907640.52 14627.38688 10907.64052 $16,254,431.74

30.0 0.762 146.99 0.456036731 50000 3.15450981 6.917227482 3.124E+06 0.011599213 1110173.963 3502054.658 4696.332653 3502.054658 $5,218,718.78

36.5 0.9271 320.99 0.675061039 50000 3.15450981 4.672925302 2.568E+06 0.011470065 411784.8527 1298979.357 1741.960011 1298.979357 $1,935,723.06

Pipe ID (in) Cost to Run Annually [$] Cost of Pipe [$] Cost to Install [$] Annual Mortgage Payment [$] Annual Interest Payment [$] Annual Principle Payment [$] Total Annual Cost [$]

2.5 $2,030,075,707,659.10 $29,900.00 $299,000.00 ($17,659.38) ($580.68) ($17,078.71) $2,030,075,725,318.48

3.0 $782,347,060,354.88 $39,900.00 $399,000.00 ($23,565.53) ($774.88) ($22,790.65) $782,347,083,920.42

4.0 $174,114,655,867.16 $49,900.00 $499,000.00 ($29,471.68) ($969.09) ($28,502.59) $174,114,685,338.84

6.0 $21,035,014,613.00 $79,900.00 $799,000.00 ($47,190.12) ($1,551.71) ($45,638.42) $21,035,061,803.13

8.0 $4,711,449,428.84 $129,900.00 $1,299,000.00 ($76,720.87) ($2,522.74) ($74,198.13) $4,711,526,149.70

10.0 $1,479,645,595.02 $179,900.00 $1,799,000.00 ($106,251.61) ($3,493.77) ($102,757.84) $1,479,751,846.62

12.0 $575,431,386.64 $249,900.00 $2,499,000.00 ($147,594.65) ($4,853.21) ($142,741.44) $575,578,981.29

14.0 $259,360,141.68 $339,900.00 $3,399,000.00 ($200,749.98) ($6,601.06) ($194,148.92) $259,560,891.66

16.0 $130,234,947.29 $439,900.00 $4,399,000.00 ($259,811.46) ($8,543.12) ($251,268.34) $130,494,758.75

20.0 $41,338,865.14 $549,900.00 $5,499,000.00 ($324,779.09) ($10,679.39) ($314,099.70) $41,663,644.24

24.0 $16,254,431.74 $669,900.00 $6,699,000.00 ($395,652.87) ($13,009.86) ($382,643.01) $16,650,084.62

30.0 $5,218,718.78 $1,469,900.00 $14,699,000.00 ($868,144.74) ($28,546.35) ($839,598.39) $6,086,863.51

36.5 $1,935,723.06 $3,209,900.00 $32,099,000.00 ($1,895,814.54) ($62,338.20) ($1,833,476.34) $3,831,537.60

Table3. CW Cycle Fluid Mechanics Calculations

Table4. CW Cycle Mortgage Calculations

11

Table5. HTW Cycle Mortgage Calculations for Optimum Pipe Diameter of 14 inches

Year Total Interest Paid Total Principle Paid Total Paid

1 $6,825.50 $3,953.22 $10,778.72

2 $6,691.09 $4,087.63 $10,778.72

3 $6,552.11 $4,226.61 $10,778.72

4 $6,408.41 $4,370.31 $10,778.72

5 $6,259.82 $4,518.90 $10,778.72

6 $6,106.17 $4,672.55 $10,778.72

7 $5,947.31 $4,831.41 $10,778.72

8 $5,783.04 $4,995.68 $10,778.72

9 $5,613.18 $5,165.54 $10,778.72

10 $5,437.56 $5,341.16 $10,778.72

11 $5,255.96 $5,522.76 $10,778.72

12 $5,068.18 $5,710.54 $10,778.72

13 $4,874.02 $5,904.70 $10,778.72

14 $4,673.26 $6,105.45 $10,778.72

15 $4,465.68 $6,313.04 $10,778.72

16 $4,251.04 $6,527.68 $10,778.72

17 $4,029.09 $6,749.62 $10,778.72

18 $3,799.61 $6,979.11 $10,778.72

19 $3,562.32 $7,216.40 $10,778.72

20 $3,316.96 $7,461.76 $10,778.72

21 $3,063.26 $7,715.46 $10,778.72

22 $2,800.93 $7,977.78 $10,778.72

23 $2,529.69 $8,249.03 $10,778.72

24 $2,249.22 $8,529.50 $10,778.72

25 $1,959.22 $8,819.50 $10,778.72

26 $1,659.36 $9,119.36 $10,778.72

27 $1,349.30 $9,429.42 $10,778.72

28 $1,028.70 $9,750.02 $10,778.72

29 $697.20 $10,081.52 $10,778.72

30 $354.43 $10,424.29 $10,778.72

12

Table6. CW Cycle Mortgage Calculations for Optimum Pipe Diameter of 36.5 inches

Year Total Interest Paid Total Principle Paid Total Paid

1 $64,457.69 $37,332.87 $101,790.56

2 $63,188.38 $38,602.18 $101,790.56

3 $61,875.90 $39,914.66 $101,790.56

4 $60,518.80 $41,271.76 $101,790.56

5 $59,115.56 $42,675.00 $101,790.56

6 $57,664.61 $44,125.95 $101,790.56

7 $56,164.33 $45,626.23 $101,790.56

8 $54,613.04 $47,177.52 $101,790.56

9 $53,009.00 $48,781.56 $101,790.56

10 $51,350.43 $50,440.13 $101,790.56

11 $49,635.47 $52,155.09 $101,790.56

12 $47,862.19 $53,928.37 $101,790.56

13 $46,028.63 $55,761.93 $101,790.56

14 $44,132.72 $57,657.84 $101,790.56

15 $42,172.36 $59,618.20 $101,790.56

16 $40,145.34 $61,645.22 $101,790.56

17 $38,049.40 $63,741.16 $101,790.56

18 $35,882.20 $65,908.36 $101,790.56

19 $33,641.32 $68,149.24 $101,790.56

20 $31,324.24 $70,466.32 $101,790.56

21 $28,928.39 $72,862.17 $101,790.56

22 $26,451.07 $75,339.49 $101,790.56

23 $23,889.53 $77,901.03 $101,790.56

24 $21,240.90 $80,549.67 $101,790.56

25 $18,502.21 $83,288.35 $101,790.56

26 $15,670.40 $86,120.16 $101,790.56

27 $12,742.32 $89,048.24 $101,790.56

28 $9,714.68 $92,075.88 $101,790.56

29 $6,584.10 $95,206.46 $101,790.56

30 $3,347.08 $98,443.48 $101,790.56

13

Plots

Figure5. Total Annual Cost vs. Pipe ID for HTW cycle

Figure6. Total Annual Cost vs. Pipe ID for CW Cycle

2.5

3.0

4.0

6.0

8.0

10.0 12.0 14.0 16.0 20.0 24.0

30.0

36.5

$100,000.00

$1,000,000.00

$10,000,000.00

$100,000,000.00

$1,000,000,000.00

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0

Tota

l An

nu

al C

ost

[$

]

Diameter [in]

Total Annual Cost vs. Pipe ID for HTW Cycle

2.5 3.0

4.0

6.0

8.0 10.0

12.0 14.0

16.0 20.0

24.0 30.0 36.5

$1,000,000.00

$10,000,000.00

$100,000,000.00

$1,000,000,000.00

$10,000,000,000.00

$100,000,000,000.00

$1,000,000,000,000.00

$10,000,000,000,000.00

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0

Tota

l An

nu

al C

ost

[$

]

Diameter [in]

Total Annual Cost vs. Pipe ID for CW Cycle

14

Conclusion Based on the calculated results, it can be seen that the mortgage cost was the same for

each cycle with the same pipe diameters do to mortgage only being a function of total capital

cost however; the electricity cost was different for each cycle with the same pipe diameter do to

each cycle having different flow rates and operating temperatures. The HTW cycle has a much

lower annual cost at the optimum pipe diameter than that of the CW cycle with its optimum pipe

diameter mainly due to its low flow rate and high operating temperature.

There is error associated with extrapolating the plots for density and kinematic viscosity

as well as with using the Colebrook equation to get the friction factor, which in turn can lead to

an error with the pressure drop calculated. Since the pressure drop is proportional to the pumping

power and the pumping power determines the cost to run the cycle each year, it can affect the

actual total annual cost. These calculations however would give a good idea as to what pipe

diameter to choose and the pressure drop that would be expected to see. By actually performing

an experiment using the optimum pipe diameter of schedule 40 steel with the same temperature

and flow rate specifications, we could determine the actual pressure drop and compare it to the

calculated pressure drop and see the error. Experimentation would yield a more accurate pressure

drop and hence the actual pumping power needed.