1 CIVE 401 - HYDRAULIC ENGINEERING PART II Pierre Julien Colorado State University Problems with and are considered moderate and those with are the longest and most difficult. In 2018 solve the problems with PUMPS 1. 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]
2 2. 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. h p = 115m, P = 475 Hp]
3 3. At a flow rate of 0.25 m 3 /s, and head losses such that fl/d = 3.5, determine the power required at the pump and plot the EGL and HGL. [Ans. h p = 56 ft, h f = 7.3 ft, P= 56 Hp] Solution First, write the energy equation from the water surface to the water surface: γ + V 2 1 2g + z 1 + h p = p 2 γ + V 2 2 2g + z 2 + h L p 1 The velocity on the right side is V 2 = V 2 = 4Q = 4 0.25 (πd 2 1 ) = 3.54 m/s, and V 2 2 (3.14 0.3 2 ) 2g = 3.542 2 9.81 = 0.64 m With the datum passing at the level of the pump, we get: 4 + h p = 100 103 + 0.64 + 8 + 3.5 0.64 1,000 9.81 h p = 17.1 m = 56 ft P = γh pq = 62.4 56 0.25 35.32 = 56.3 Hp 550 550
4 4. 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] 5. For the pump curves #1 to #3 in the additional notes, look at conditions near maximum efficiency. Estimate the specific speed and locate each pump on the classification charts. [Ans: #2, N s = 3,243] 6. For a pump with N = 1,000, 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. [hint: look at the definition of constant values of C H and C Q ] [Ans. 50% increase in Q]
5 7. English Units - For the system below and the pump curve given, neglect minor losses and assume f = 0.02. 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 X max for this pump. Repeat with a pipe diameter of 24 in. [Ans. h f = 0.24Q 2, and X max = 4.4 ft in the 16 inch line]
6 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. h f = 18Q 2, and H = 2.5 m] Solution
7 D = 35.6cm = 0.356m, and A = πd2 = 3.14 0.3562 = 0.1 m 2 4 4 Assuming f = 0.02, we get h f = fl D V2 2g = fl D Q 2 0.02 (60 + 3) = [0.4 + 2gA2 0.356 ] Q 2 2 9.81 (0.1) 2 = 20Q2 Q h f H= h f +1.5 m 3 /s m m 0.05 0.05 1.55 0.10 0.20 1.70 0.15 0.45 1.95 0.20 0.80 2.30 0.25 1.25 2.75 0.30 1.80 3.30 We get the discharge in the plot: Q = 0.24 m3 s, and H = 2.5m, or N s = NQ1/2 H 3/4 = 690 0.24 35.32 (2.5 3.28) 3/4 = 415 X max = (14.7 0.12) 62.4 lb lb 144in2 in2 ft3 ft 2 X max = p atm p v γ h f NPSH 3.28 20 0.21 2 ft NPSH from the propeller level
8 9. Pump design experience! English units In the following set-up, the pump brings Q = 0.125m 3 /s of water to an elevation H = 120 ft above sea level in a L = 2,500 ft long pipe. 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.
9 10. 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-kQ 2. 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? [Ans. k = 0.084, Q = 155 h, and t = 16.9 hours] 0.097
10 TURBINES 11. 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] 12. A shaft produces 200 Hp at 600 rpm. Calculate the torque. [Answer: 1,760 lb-ft] 13. 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] 14. 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. 15. A Pelton wheel is 24 in 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? [Answer: H =109 ft, p = 18]
11 16. (advanced) For the Pelton wheel in example 8.8 of the handout, you need to synchronize the wheel speed. How many pairs of poles would you need to keep 300 rpm < N < 360 rpm. Calculate the Ns values for these conditions and 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 = 20, D = 2.7 m, and P = 22,794 Hp] Solution Calculate the number of poles: If p = 20. N = 120f p = 7200 p, or 7,200 360 H = 670m 3.28 ft = 2,198 ft m 7,200 < p <, and 20 < p < 24 300 N = 349 rpm P = 16,760 kw = 22,794 Hp V B = ω D 2 = 7,200 2π 2.77m 20 60 2 N s = NP1/2 349 22,7941/2 = H5/4 2,198 5/4 = 3.5 D d = 54 = 54 N s 3.5 = 15.4 D = 15.4 18 cm = 2.77m = 52.2 m/s which is close to 54.8 m/s
12 17. 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] 18. 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%, (d) the radial velocity in ft/s, and (e) the maximum tangential velocity of the runner? [Answer: Ns = 33 and ω = 12.6 rad/s] 19. A Francis turbine is designed with the following conditions: discharge 113 m 3 /s and N = 120 rpm, β 1 = 45, r 1 = 2.5 m and B = 2.9 m. Determine the angle α 1 that would avoid separation at the runner inlet. [Answer: α 1 = 4.2 ]
13 20. Improved design problem! A Francis turbine is designed with the following conditions: β 1 = 60, β 2 = 90, r 1 = 5 m, r 2 = 3 m and B = 1m. When the discharge is 126 m 3 /s and N = 60 rpm, calculate the entrance angle α 1 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. Finally, for the same discharge and runner size, can you suggest an improvement to the design of the runner blades. [Answer: α 1 = 6.8, P = 89 MW and with β 2 = 160, 133 MW]
14 21. (advanced) 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 V u2 =0. Determine the blade angles β 2 for r = 0.75, 1, 1.5 and 2 ft. [Answer: β 2 = 125 at r = 0.75 and β 2 = 133.5 at r = 1.0 ] β 2 = 90+35.5 = 125 at r = 0.75, β 2 = 90+55.1 = 145 at r = 1.5, and β 2 = 90+62.3 = 152 at r = 2.0 at r = 1, V u = 22.6 1, u = 8π 1 = 25 ft/s, V a = 26.3 ft/s, and β 2 = 90+ tan 1 ( 25 26.3 ) = 133.5 22. 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: N s = 151 and p = 72]
15 UNSTEADY FLOW and WATERHAMMER 23. A cast iron 18 diameter pipeline carries water at 70 F over a distance of 1,000 ft. If the flow velocity is 5 ft/s, the pressure is 46 psi and the pipe thickness is 0.5 inch, 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? 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 = 1.45 and Δp = 98 psi] f) A 10 ft diameter surge tank is built half-way between the power house and the reservoir. 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] 24. 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: ω d = 1.25 rad, ζ = 0.457 s and V max = 0.35 m/s] 25. 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. Would it make any difference if the roughness is increased to 0.08? [Ans: z 1 = 4 m, z 2 = 2.75 m, V m = 2.98 m/s when z = 0.9 ft]
16 26. The City of Thornton may build a pipeline with some details in the 2015 article. 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? c) what is the continuous equivalent discharge for 14,000 acre-ft per year? d) Assuming p 0 = 40 psi, what is the pressure head in the pipe? 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) given f = 0.02 and p 0 = 40 psi, what would be the pressure generated from a sudden closure? h) what if a valve closure requires 10 sec? 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?