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CIVE 401 - HYDRAULIC ENGINEERING – PART II

HYDROMACHINERY

Pierre Julien

Colorado State University

Problems with ♦ and ♦♦ are considered moderate and those with ♦♦♦ are the longest and/or most difficult.

In 2021, you solve the odd-numbered problems with ☺

PUMPS

☺ 1. ♦ English Units - Determine the power required to pump 2.5 cfs at 68°F from the lower to

upper reservoir. Assume that the friction factor is f = 0.015, and sketch the HGL and EGL.

[Ans. H = 104 ft, P = 29.5 HP]

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2. ♦ SI Units -What power must be supplied by the pump if water (T = 20°C) is pumped through a

200 mm diameter steel pipe from the lower tank to the upper one at a rate of 0.314 cms? Draw the

EGL and HGL. [hint: find f from the Moody diagram].

[Ans. ℎ𝑝 = 108𝑚, 𝑃 = 445 𝐻𝑃]

☺ 3. ♦♦ SI Units - At a flow rate of 0.25 m3/s, and head losses such that fL/D = 3.5, determine the

power required at the pump and plot the EGL and HGL.

[Ans. ℎ𝑝 = 56 𝑓𝑡, ℎ𝑓 = 7.3 𝑓𝑡, P= 56 HP]

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4. ♦♦ English Units - For the pump curves #1 to #3 in the class notes, look at conditions near

maximum efficiency. Estimate the specific speed and locate each pump on the classification

charts.

[Ans: #2, 𝑁𝑠 = 2,800]

☺ 5. ♦ English Units - What type of pump at N = 1,500 rpm should be used for a discharge of 12

cfs and head of 25 ft?

[Ans: mixed flow pump]

6. ♦♦♦ For the pump below, you burned the motor and want to replace it with a faster motor. What

is the maximum head that can be generated if the speed is increased to 1500 rpm? Also determine

the increase in discharge at a given head for the same condition. Finally, would this increase or

decrease the potential for cavitation, or NPSH? [hint: look at the definition of constant values of CH

and CQ]

[Ans. 50% increase in Q, and V]

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☺ 7. ♦♦ English Units - For the system below and the pump curve given, assume f = 0.02, and

neglect other minor losses. Determine the head losses as a function of discharge and plot on the

performance on the diagram below. Determine the discharge and head at the pump when the pipe

diameter is 16 inches. What type of pump would you recommend? If the NPSH is 8’ for the 16”

line, what is the maximum elevation Xmax for this pump. Repeat with a pipe diameter of 24 in.

[Ans. ℎ𝑓 = 0.24𝑄2, and Xmax = 4.4 ft in the 16-inch line]

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8. ♦♦ SI Units - Find the discharge under the conditions shown and the given pump characteristics.

If the motor is located at an elevation of 23 m, which elevation (propeller or motor) should be

considered for the NPSH calculation?

[Ans. ℎ𝑓 = 18𝑄2, and H = 2.5 m]

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PROBLEMS 9 and 10 MUST BE SOLVED

☺ 9. ♦♦♦ English units - Optimization Problem

In the following set-up, the pump brings Q =

0.125m3/s of water to an elevation H = 120 ft above

sea level in a L = 2,500 ft long pipe. The pump is set

to run 16 hours per day.

Your company (JJ Engineering) is asked to design

the pipe diameter (single value) and pump size. Your supervisor Jan tells you to assume f = 0.02

and neglect minor losses. Determine the flow velocity for various pipe diameters of 8”, 10”, 12”, etc.

A) Jan asks you to estimate the cost for each alternative based on a 2014 table of the cost for

galvanized iron pipe and pumps. The cost of pipes is 2” (6$/ft), 3” (10$/ft), 4” (25$/ft), 6” (35$/ft), 8”

(60$/ft), 10” (90$/ft), 12” (125 $/ft), 15” (200 $/ft), 18” (320$/ft), 20” (360$/ft), 24” (500$/ft), 30” (720

$/ft), 36” (900$/ft), 42” (1,250$/ft), and 48” (1,600$/ft). The cost of pumps is ~$200/Hp, and there is

a fixed base cost of $150,000 for the construction.

First meeting with your client - Jan has to travel and asks you to make the recommendation to your

very important client and elected representative in your District named Jill.

♦♦♦ Prepare a 2 page report with your analysis and recommendation for your meeting.

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☺ 10. ♦♦♦ English Units – Analytical Solution

A pump delivers water from a large reservoir to the tank through a 800 ft long 2 ft diameter cast

iron pipe. The pump curve can be approximated with the function Δh = 155-kQ2. Assume f = 0.02

and an efficiency of 70% to determine the following: (a) what is the constant k of the pump curve?;

(b) how long does it take to fill the reservoir to an elevation of 255ft?; and (c) what kind of problem

may emerge between elevation 250 and 255 ft? [hint: analytically solve to find the operating point

and write Q as a function of h, and then analytically solve dt = AT dh/Q – with math tables …]

[Ans. 𝑘 = 0.084 , 𝑄 = √155−ℎ

0.097, and t = 16.9 hours]

Time to search the web

☺ 11. ♦ Examine photos of positive displacement pumps

12. ♦ Examine photos of deep well pumps

☺ 13. ♦ Examine photos of axial pumps

14. ♦Examine photos of centrifugal pumps

☺ 15. ♦ Find an axial pump curve

16. ♦ Find a centrifugal pump curve

☺ 17. ♦ Examine photos of pump cavitation

18. ♦Examine photos of a multi-stage pump

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TURBINES

☺ 19. ♦ English Units - A turbine discharges 1,200 cfs under a head of 26 ft at an efficiency of

86%. How many horsepower can be generated under those conditions?

[Answer: 3,040 HP]

20. ♦ English Units - A shaft produces 200 HP at 600 rpm. Calculate the torque.

[Answer: 1,760 lb-ft]

☺ 21. ♦ English Units - The moment of momentum of water is reduced by 20,000 lb-ft in a turbine

moving at 400 rpm. Determine the power generated by this turbine.

[Answer: 1,523 HP]

22. ♦ English Units - Determine the power and select the type of turbine for the following

conditions:

a) the head is 600 ft and the discharge is 10 cfs.

b) the head is 200 ft and the discharge is 200 cfs [Answer: Francis].

c) the head is 50 ft and the discharge is 4,000 cfs.

☺ 23. ♦ English Units - A Pelton wheel is 24 inches in diameter and rotates at 400 rpm. What is the

head that is best suited for this wheel? How many pairs of poles are needed? [hint: can you find

the velocity of the wheel, and from it the velocity of the jet?]

[Answer: H =109 ft, 9 pairs of poles in the US]

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24. ♦♦(advanced) SI Units - For the Pelton wheel in the lecture notes, you need to synchronize the

wheel speed. Find out what happens with 22 poles. Find N and calculate the Ns value for these

conditions. Select the best design if the maximum efficiency is reached when D/d = 54/Ns where D is

the diameter of the wheel, d is the diameter of the jet and Ns is the specific speed.

[Answer: p = 22, optimum D = 2.7 m, and 𝑉𝐵 = 46.3 𝑚/𝑠]

☺ 25. ♦ English Units - For the Francis turbine at Hoover Dam (P = 115,000 HP, N = 180 rpm and

H = 480 ft), find Ns and the discharge if e = 0.85.

[Answer: Q = 2,484 cfs]

26. ♦♦ English Units - For the Francis turbine at Grand Coulee (P = 150,000 HP, H = 330 ft, and N

= 120 rpm), determine: (a) the angular velocity in rad/s, (b) the specific speed, (c) the discharge if

the efficiency is 90%, and search the appropriate information in the notes to determine (d) the

radial velocity in ft/s, and (e) the maximum tangential velocity of the runner?

[Answer: Ns = 33, and 𝑉𝑡 = 103𝑓𝑡

𝑠]

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☺ 27. ♦♦ SI Units

A Francis turbine is designed with the following conditions: β1 = 60°, β2 = 90°, r1 = 5 m, r2 = 3 m and

B = 1m. When the discharge is 126 m3/s and N = 60 rpm, calculate the entrance angle α1 that would

prevent separation of the streamlines at the entrance of the runner. Determine the maximum power

and torque that can be generated under these conditions.

♦♦♦ Improved design problem! Finally, for the same discharge and runner size, can you suggest an

improvement to the design of the runner blades.

[Answer: α1 = 6.8°, P = 89 MW and with β2 = 160°, P = 133 MW]

28. ♦♦ SI Units - A Francis turbine is designed with the following conditions: discharge 113 m3/s and

N = 120 rpm, β1 = 45°, r1 = 2.5 m and B = 2.9 m. Determine the angle α1 that would avoid separation

at the runner inlet.

[Answer: α1 = 4.2°]

☺ 29. ♦ English Units - A Kaplan Turbine is designed to generate 24,500 HP at N = 100 rpm under

41 ft of head. If the efficiency is 85%, what are the flow discharge, the specific speed and the

number of poles for the generator. [Answer: 𝑁𝑠 = 151 and p = 72]

30. ♦♦♦ (advanced) English Units - For the Kaplan turbine from the handout, draw the velocity

diagrams at the trailing edge of the propeller turbine blade such that the tangential velocity Vu2 =0.

Determine the blade angles β2 for r = 0.75, 1, 1.5 and 2 ft.

[Answer: β2 = 125° at r = 0.75 and β2 = 133.5° at r = 1.0 ]

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Time to search the web!

☺ 31. ♦ Examine photos of a 42 pole electric generator

32. ♦ Examine photos of a Pelton wheel

☺ 33. ♦ Examine photos of a Francis Turbine

34. ♦Examine photos of a Kaplan Turbine

☺ 35. ♦ Examine photos of a Deriaz turbine

36. ♦ Examine photos of a bulb turbine

☺ 37. ♦ Examine photos of turbine cavitation

38. ♦Examine photos of tidal power hydro turbines

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UNSTEADY FLOW and WATERHAMMER

☺ 39. ♦♦ English Units - A cast iron 18” diameter pipeline carries water at 70°F over 1,000 ft from

a reservoir to a powerhouse. The flow velocity is 5 ft/s and the initial pressure is 46 psi. If the pipe is

1 inch thick, refer to Section 1 for cast iron pipe properties and determine the following:

a) The wave celerity in this pipe.

b) The added pressure generated by a sudden closure.

c) Would the sudden closure cause cavitation or burst the pipe open?

d) How long would the closure time have to be to reduce the maximum pressure?

e) What is the increased pressure if the time of closure is 1 s? [Answer: N = 2.1 and Δp = 130 psi]

f) A 10 ft diameter surge tank is built half-way in the line. What is the maximum surge height?

[Answer: S = 3 ft]

g) What is the period of oscillations in the surge tank? [Answer: T = 165 s]

40. ♦♦ English Units - The City of Thornton is considering building a pipeline with the specs given

in this 2015 newspaper article.

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a) what is the cost of the pipeline per linear foot?

b) if the population starts at 138,000 residents, what is the annual population increase?

138,000 (1 + 𝑖)10 = 242,000, 𝑖 = (242,000

138,000)

0.1

− 1 = 5,78% 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟

c) what is the continuous equivalent discharge: (a) in cfs for 14,000 acre-ft per year? and (b) in

gallons per resident per day?

d) assuming p0 = 40 psi, what is the pressure head in the pipe?

𝐻 =𝑝

𝛾=

40 × 144

62.4= 93 𝑓𝑡

e) how long would it take to establish 95% of the equilibrium flow velocity after a sudden opening?

f) assuming that friction is negligible, what is the period of oscillations in this pipe?

g) assuming a rigid pipe with f = 0.02 and p0 = 40 psi, what would be the pressure generated from a

sudden closure?

h) what if a valve closure requires 10 sec?

𝑡 =2𝐿

𝑐=

2×60×5,280

4,756= 133𝑠, same unless closure takes more than 2 minutes

i) based on the value of water from the second article in 2017, do you think this project has a high

or low benefit to cost ratio?

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☺ 41. ♦♦ Consider that the cost to build a hydropower plant is $50,000,000. The annual energy

generation is equal to 52,000 MWh and the value of electricity is constant at $70 per MWh. The

period of contract for energy supply is 20 years and the annual interest rate is 5% annual.

a) Is this contract lucrative? (Tip: use present worth analysis).

This contract is not lucrative.

b) What should be the minimum value of electricity to break even in this contract?

c) Suppose that it is not possible to raise the value of electricity, but you can sell the

hydropower plant at the end of contract. What should be the minimum sale value?

The hydropower plant sale price should be at least ≅ $12,500,000

42. ♦♦ Consider the penstock design problem in the Case Study (Example 9) of the lecture notes.

You currently have a 7.5m diameter penstock, but you could increase your revenue by $250,000 per

year over a period of 30 years by constructing a 9.5 m diameter penstock. Consider interest rates of

0% and 5% and compare the results.

a) What is the incremental construction cost?

b) Is this contract lucrative over 30 years if the interest rate is 0%?

The cost would be $4,000,000 and the revenue would be $250,000 × 30 yrs = $7,500,000, thus

profitable.

c) Is this expansion project valuable if the interest rate is 5%?

d) Is this contract profitable if the interest rate is 4% but you lose the first year of revenue

because of the project construction?

Break even with the construction cost.

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Time to search the web!

☺ 43. ♦ Examine photos of burst pipes

44. ♦ Examine photos of surge tanks

Problems for graduate/honor students

45. ♦♦ A 10 m long 5 mm diameter U-shaped plastic tube is holding water at 20°C. If there is a 1 m

head difference between both ends when the pressure is suddenly released at t = 0. Determine the

following: (a) the natural circular frequency of the oscillations; (b) the damping factor; (c) the

circular frequency of the damped oscillations; (d) the period of the damped oscillations; (e) the

lowest water level; (f) the maximum flow velocity; and (g) is the flow laminar?

[Answer: 𝜔𝑑 = 1.25𝑟𝑎𝑑

𝑠, 𝜁 = 0.457 and 𝑉𝑚𝑎𝑥 = 0.31 𝑚/𝑠]

46. ♦♦ A 25 m long 25cm diameter U-shaped pipe has a 8 m difference between both ends as the

system is released from rest. If f = 0.04, calculate the successive maxima during the oscillations and

find the maximum velocity in the pipe. Compare the calculations with f = 0.08.

[Ans: 𝑧1 = 4 𝑚, 𝑧2 = −2.75 𝑚, 𝑉𝑚 = −2.98 𝑚/𝑠 when 𝑧′ = 0.9 𝑓𝑡, and 𝑧2 = −2.12 𝑚 when 𝑓 = 0.08