CIVE 401 - HYDRAULIC ENGINEERING PART I 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 HYDROSTATICS 1. A student weighs 135 lb, determine the weight in N, the mass in slugs and in kg. Ans. m = 4.19 slugs 2. A container weighs 450 lb when filled with 4 ft 3 of fluid. If the empty container weighs 50 lb, determine the specific weight, mass density and specific gravity of this fluid. Ans. G = 1.60 3. A fish tank is 2 ft long, 1 ft wide and 1 ft deep. What are the pressure and the force at the bottom of the tank? Ans. p = γh = 62.4 psf, W = 124.8 lb 4. A transducer measures a relative pressure of 65 psi. Determine (1) the value of the pressure in Pascals and (2) the absolute pressure in psi and kpa. Ans. P rel = 448 kpa, P abs = 79.7 psia = 550 kpa
5. What is the absolute vapor pressure of water (1) at 30 C in kpa and (2) at 180 F in psia and psfa. Ans. (2) p = 7.51 psia 6. A circular plate has a 2 m diameter. What is its area moment of inertia about the center of gravity? Ans. I XC = 0.785 m 4 7. A triangular plate has a base of 4 ft and a height of 3 ft. What are: (a) Surface area A? (b) The position of the center of gravity? (c) The area moment of inertia about the horizontal axis through the center of gravity? (d) The area moment of inertia about the base? 2 Ixo IXC A y 3 6ⅹ(1) 2 = 9 ft 4 Ans. A = 6 ft 2, y c = 1 ft, I xc = 3 ft 4, I xo = 9 ft 4
Pressure, piezometric head and forces 8. Find the relative and absolute pressure at the pipe centerline at A in kpa, psi and psf. What are the piezometric heads at A and B? C B Ans. p abs_a = 151.3 kpa 9. A circular plate 2m in diameter serves as a submarine window. Determine the average pressure and force on the vertical plate when the center of gravity of the plate is 500 m below the free surface. Also calculate the distance separating the center of gravity and the center of pressure. Ans. h = 0.5 mm
10. The Z m = 2 m circular gate centered at O can only rotate clockwise around a horizontal pivot at C located 5 cm below the center O of the plate. Determine the range of flow depths h for which the gate opens. Ans. Gate opens when h > 4 m 11. A rectangular plate is 4 ft wide and 8 ft long is submerged at an angle of 45 with the vertical. If the center of the plate is located 10 ft vertically below the free water surface, what is the magnitude of the hydrostatic force on the plate and locate the center of pressure. Ans. F = 19,968 lb = 10 tons, CP at a depth of 10.3 ft
Dam stability 12. For the dam sketched below, consider a unit width and determine the following: 6ft Concrete 16ft o 24ft A (a) the hydrostatic force on the dam 6ft F V Concrete F H 16ft o 24ft A F C1 F C2 (b) locate the center of pressure (c) the weight of concrete of the dam
(d) find the resultant force F R = ( F x ) 2 +( F y ) 2 = (7, 987) 2 +(2, 995 + 28, 800) 2 = 32,783 lb (e) sum the moments about O (f) find the point of application of the resultant force on the base of the dam OA (g) is the resultant passing through the central third of the base? Ans. y cp = 11.4 ft, W c = 28,800 lb, x = 10.6 ft in central third
13. Optimization problem! For the dam sketched below, determine the following per unit width: h Concrete O L A (a) calculate the horizontal force on the dam (b) find the vertical force from the weight of concrete (c) find the expression for the sum of moments about O (d) find the expression for the point application of the resultant force on the base of the dam (e) determine the ratio L/h for which the resultant will pass through the midpoint of the base OA (f) what is the range of L/h for which the resultant force would pass through the central portion of the base. Ans. (e) L = 0.91h, (f) no tension cracks when x R < 2L/3, or L > 0.645h
14. For the dam sketched below, determine the following per unit width: Y = kx 2 24 m A Concrete O A 12 m dx 4 m A (a) the constant k of the parabolic equation y = kx 2 (b) the horizontal hydrostatic force (c) the weight of water above the dam (d) divide the concrete part into three segments (e) determine the weight and CG for each segment (f) calculate the sum of horizontal and vertical forces (g) determine the resultant force (h) calculate the sum of moments about O 15. Where is Bonneville Dam located? What type of dam? 16. What is the main purpose of Tarbela Dam? 17. Why do we need a clay core in an earth-fill dam? 18. Google photos for Bartlett Dam and Daniel-Johnson Dam. What are the main differences between these two dams? 19. What happened to the 21 mile Dam in Nevada? What type of dam?
PIPE FLOW Losses in pipes 20. For the pipe shown below, neglect all minor losses and calculate the velocity of the jet and the discharge in the pipe. Plot the HGL and EGL. Check f on the Moody diagram. Ans. V jet = 36.5 ft/s, Q= 7.15 ft 3 /s, and f = 0.015 21. Water flows from reservoir A to B at a water temperature of 10 C. Given the cast iron pipe length 300 m and diameter 1 m, use the Moody diagram to determine the discharge when H = 18 m and h = 3 m. Plot the HGL and the EGL and find the pressure at point P half way between the two reservoirs. Ans. V = 8.24 m/s, Q= 6.5 m 3 /s, and pressure at P = 101 kpa
Three-reservoirs 22. For the system shown assume f = 0.02 and determine the discharges in the three pipes and the elevation of the EGL at the junction. K = 8fL gπ 2 D 5 K BD = 169, K AD = 24,788 Ans. Q AD = 0.038 m 3 /s, Q BD = 0.18 m 3 /s, Q DC = 0.217 m 3 /s
23. For the three reservoirs below, assume f = 0.02 and neglect minor losses. Determine the discharge in each pipe and the energy grade line at node B. K = 8fL gπ 2 D 5 K AB = 12.53, K CB = 15.29 Ans. Q AB = 2.79 ft 3 /s, Q CB = 1.77 ft 3 /s, Q BD = 4.56 ft 3 /s
Pipe networks 24. Write a computer program to solve the pipe network below. All pipe diameters are 150 cm and all pipe lengths are indicated. Assume f = 0.02 and neglect all minor losses. A head of 40m is given at point A. Determine the discharges in all pipes and the EGL at all nodes. Keep printed evidence of your program. Once your model works, recalculate the conditions for the following two cases: (A) replace the outflow at D with an outflow of 0.5 m 3 /s at both B and F; and (2) based on the initial condition sketched below, if you increase the diameter of a single pipe to 250 cm, which pipe would give the maximum head at point F? Ans. Q AB = 0.787 m 3 /s, Q BC = 0.787 m 3 /s, Q CD = 0.187 m 3 /s, Q AD = 0.123 m 3 /s, Q AE = 0.986 m 3 /s, Q FE = 0.014 m 3 /s, Q DF = 0.415 m 3 /s, Q CG = 0.60 m 3 /s, Q FG = 0.4 m 3 /s. Case A.Q AB = 0.938 m 3 /s, Q BC = 0.437 m 3 /s, Q DC = 0.285 m 3 /s, Q AD = 1.066 m 3 /s, Q AE = 0.997 m 3 /s, Q EF = 0.003 m 3 /s, Q DF = 0.782 m 3 /s, Q CG = 0.721 m 3 /s, Q FG = 0.279 m 3 /s.
25. For the system shown, assume f = 0.02 and a piezometric head of 100 ft at point A. Determine the discharges in all pipes and the elevation of the EGL at all nodes. Note that CI designates cast iron and WI wrought iron. Ans. Q AB = 2 ft 3 /s, Q BC = 0.4 ft 3 /s, Q AD = 1.0 ft 3 /s, Q DC = 1.0 ft 3 /s, Q BE = 1.6 ft 3 /s, Q CF = 1.4 ft 3 /s, Q EF = 0.6 ft 3 /s 26. In this example Part I is solved, and you then have to calculate Part II. Part I. - For the system shown all pipes are 12 in diameter. Assume f = 0.02 and a head of 200 ft at point A. Write a computer program to solve this pipe network. Determine the discharges in all pipes and the elevation of the EGL at all nodes.
Solution to Problem 26 Part II. - Once you have solved Part I, replace the outflow of 10 cfs at F with 5 cfs outflows at both C and H. Which of the two cases gives a higher pressure at point E?
27. In the following network, Part I is solved, you can program this and solve Part II. The pressure at A is 60 psig, all pipes are horizontal, and f = 0.012. Now, solve it when Qc = 5 cfs and Qe = 10 cfs
Minor losses in pipes 28. The pipe system shown is open to the atmosphere at pt 2. Calculate the discharge given f = 0.02 and H = 30 ft. Ans. did you include all losses? Q = 1.66 ft 3 /s 29. For the pipe shown below, assume f = 0.02 and determine the discharge in the pipe when H = 50 ft and plot the EGL and HGL. Secondly, close the valve and find the value of K required to decrease the discharge by 50%? Ans.(1) k exp = 0.57, (K + 60. 1) V 2 2 and (2) K = 194 2g = 50 ft, V 2 = 7. 12 ft s, Q = 5. 6 ft3 /s;
30. For the pipe shown below, determine the pressure in psi half way between A-B. What is the energy loss at the valve B-C? Why would you place a nozzle at the downstream end of the pipe? 31. Ans. p @40 ft = 23.7 psi, h BC = 16. 6 ft, to keep high pressure inside the pipe Ans. k con = 0.37, H = 84. 3 ft, t = 140 s
Siphon and negative pressure 32. For the 4 pipe shown below, point C is the mid-point of the 200 pipe and determine the following: Neglect all friction losses (a) what is the velocity in the pipe? (b) plot the EGL and HGL (c) what is the energy level at point C? (d) what is the pressure at point C? (e) Assuming the project at sea level and water T at 65 F, how far above the current position could you elevate point C without cavitation Ans. V = 25.4 ft/s, EGL 10 ft above datum, at C, p = -749 psf, h 21.9 ft 33. Repeat Problem 32 with friction losses assuming f = 0.02 and determine the discharge in the pipe? Ans. Q = 0. 6 cfs
FORCES and MOMENTUM 34. A nozzle ejects water horizontally at 40 mi/h at a volumetric flow rate of 30 m 3 /s. The stream is deflected horizontally by a plate shown in plan view. Determine the force in kn exerted on the plate by the jet in cases (a), (b) and (c). Ans. (b) R x = 537kN to the left, (c) R x = 1,072kN to the left 35. Use continuity and the sum of forces along the incline plate to determine: (1) the flow distribution, Q 1 and Q 2 as a function of the angle, and (2) the horizontal force on the plate. Ans. Q 1 = 0.5 Q 0 (1 + cos )
36. The 1 ft wide vane shown in plan view has a discharge Q 0 = 3 cfs and a jet velocity of 300 ft/s, determine the following: Q 1 = 1.8 cfs 300 ft/s 300 ft/s Q = 3 cfs 300 ft/s Q 2 = 1.2 cfs A. What are the force components F x and F y required to hold the vane in place? B. Find the force components when the plate moves to the right at v p = 100 ft/s? C. What is the work done per unit time on the moving plate? Ans. (a) F x = - 0.9 Q 0 V 0, (b) Q = A(V-v p ), (c) P = 69,840 lb.ft/s
37. A discharge of 0.1 m 3 /s passes through the pipe bend shown in planview. If the pressure head on the left hand side is 10 m of water, calculate the force required to anchor this pipe. Ans. p 2 = 22,114 N, R x = 2,375 N to the left, R y = 1,022 N up,
38. From the pipe junction in a horizontal plane shown below, what is the velocity of section 3 if the diameter is 0.15m? If the pressure head at (2) is 10 m, determine the pressure head at (3) and find the net force required to hold this junction when = 60. Ans: V 3 = 5. 66 m/s, P 3 = 100, 082 Pa, R x = 348 N right, R y = 1,047 N up
39. In the horizontal pipe branch shown here, determine the pressure in each branch and the force required to hold the junction in place. (2) (3) (1) Ans: V 3 = 15. 3 ft/s, p 3 = 1339 psf, R x = 687 lb right, R y = 1,444 lb down 40. Water flows through a horizontal contraction at arate of 25 cfs. The contraction loss coefficient is 0.2 based on the higher velocity. The upstream pipe pressure is 30 psig. What is the anchor force to hold the system in place. Ans: V 1 = 7. 95 ft s, V 2 = 14. 15 ft s, P 1 γ = 69. 2 ft, P 2 γ = 66. 5 ft, p 1 A 1 = 13, 570 lb, ρqv 1 = 390lb, p 2 A 2 = 7, 330 lb, ρqv 2 = 690lb, The anchor force is R x = 5, 940 lb to the left.
41. A 1m diameter pipe carries 3 m 3 /s of water at 10 C under constant pressure at 75 kpa. A 30 horizontal pipe bend weighs 4 kn and contains 1.8 m 3 of water. What is the anchor force to hold the bend in place. Ans: p = 75, 000 Pa, V = 3. 82 m s, p 1 A 1 = 58, 875 N, ρqv 1 = 11, 460 N, R x = 9, 420 N to the left, R y = 35, 170N down, The vertical anchor force is R z = 21,658 N vertical upward.
Flow meters 42. Neglect friction in the Venturi shown and determine the discharge in the pipe. What is the Cv for this flow condition? What is the discharge difference when using the correction coefficient? Ans. Q = 1.41 ft 3 /s, Cv = 0.98, error less than 2% 43. Derive an expression relating the flow discharge as a function of the parameters indicated in the venturi shown below. Note that the flow results are independent of the angle of the pipe with the horizontal. A B D C Ans. ( V 2 V 1 ) 2 = ( D 1 D 2 ) 4, P γ = (ρ m ρ 1) R, and Q = πd 2 1 4 2gR (ρ m [( D 4 1) 1] D 2 ρ 1) 1
44. In Breckenridge Colorado, a Venturi with d = 15 cm is placed in a D = 30 cm pipe under a head H = 5m. If you neglect friction losses, what is the maximum discharge before cavitation occurs. Ans. Q = 0.272 m 3 /s 45. What is the main difference between a ball valve and a plug valve? 46. Why are the energy losses in a globe valve so high? 47. What is the main difference between pipes schedule 40 and schedule 80? 48. Why do you need a strainer for a foot valve? 49. What type of valve is the Howell-Bunger valve? 50. What error percentage would you have in taking 1.0 for the calculations of a nozzle when the diameter ratio is less than 0.5?