CVE 341-Assignement 8_CHAPTER 13_SOLUTION

CVE 341-Assignement 8_CHAPTER 13_SOLUTION - American...

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Unformatted text preview: American University of Sharjah Civil Engineering Department CVE 341 – WATER RESOURCES ASSIGNMENT 7 Q.1) Water flowing in a wide channel encounters a 22cm high bump at the bottom of the channel. If the flow depth is 1.2 m and the velocity is 2.5 m/s before the bump, determine if the flow is chocked over the bump, and discuss. flow (Answers yc=0.97m & Ec=1.46m3/s) 97m Assumptions 1 The flow is steady. 2 Frictional effects are negligible so that there is no dissipation of mechanical Frictional energy. 3 The channel is sufficiently wide so that the end effects are negligible. Analysis The upstream Froude number and the critical depth are The flow is subcritical since Fr < 1, and the flow depth decreases over the bump. The upstream, over the bump, and critical specific energy is We have an interesting situation: The calculations show that Es2 < Ec. That is, the specific energy of the fluid That decreases below the level of energy at the critical point, which is the minimum energy, and this is impossible. Therefore, the flow at specified conditions cannot exist. The flow is choked when the specific energy drops to the Therefore, minimum value of 1.46 m, which occurs at a bump-height of Discussion A bump-height over 6 cm results in a reduction in the flow rate of water, or a rise of upstream water level. Therefore, a 22-cm high bump alters the upstream flow. On the other hand, a bump less than 6 cm high will not affect the upstream flow Q.2) Water flow in a wide rectangular channel approaches a 10 cm-high bump at 1.5 m/s and a depth of 1m. Estimate (a) The water depth y2 over the bump (b) The bump height that will cause the crest flow to be critical. V2 1 .5 Fr = = ⇒ Fr = 0.479 gy c 9.81 * 1 (Answers y2=0.859m & 0.197m) 2 E = y+ Subcritical flow Q2 1 .5 2 = 1 .0 + = 1.115m 2 * 9.81 2gA 2 E2=E1-∆h=1.015 y 3 − 1.015 y 2 + 0.115 = 0 2 2 There are three real roots y2=0.859m, 0.451m and -0.296. The negative value is physically impossible. The second solution (0.451m) is supercritical condition for E2 and it is not possible for this subcritical bump. The first solution is correct. Q.3) A trapezoidal channel with a bottom width of 5 m and side slopes t = 2 runs on a slope of 0.0005 and carries a discharge of 50 m3/s at normal depth. Take Manning’s n as 0.021 (metric). a) What is the specific energy of the flow in the channel? b) Is the flow subcritical or supercritical? c) What is the alternate depth to the normal depth? d) What is the critical depth in the channel? (Answers E=3.07m, Fr=0.359, y=1.12m, yc=1.72m) Y(m) 0.4 0.6 0.8 1 1.116 1.2 1.4 1.6 1.72 1.8 2 2.2 2.4 2.6 2.8 2.95 3 3.2 3.4 3.6 3.8 4 4.2 4.4 A(m^2) 2.32 3.72 5.28 7 8.0709 8.88 10.92 13.12 14.28 15.48 18 20.68 23.52 26.52 29.68 32.155 33 36.48 40.12 43.92 47.88 52 56.28 60.72 Q^2 /(2*g*A^2) 23.674 9.208 4.571 2.600 1.956 1.616 1.069 0.740 0.625 0.532 0.393 0.298 0.230 0.181 0.145 0.123 0.117 0.096 0.079 0.066 0.056 0.047 0.040 0.035 E(m) 24.074 9.808 5.371 3.600 3.072 2.816 2.469 2.340 2.325 2.332 2.393 2.498 2.630 2.781 2.945 3.073 3.117 3.296 3.479 3.666 3.856 4.047 4.240 4.435 y Using Excel 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 E Q.4) Water flows with a velocity of 3.0 m/s and a depth of 3.0 m in a rectangular channel. (a) What is the change in water surface elevation produced by a gradual upward change in bed elevation produced by a gradual upward change in bed elevation (hump) of 30 cm? (b) What would be the water surface if there were a gradual drop of 30 cm? (Answers 0.205m drop, 0.1m rise) 1 9 Y(m) 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.495 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 A(m^2) Q^2 /(2*g*A^2) 0.4 25.803 0.6 11.468 0.8 6.451 1 4.128 1.2 2.867 1.4 2.106 1.6 1.613 1.8 1.274 2 1.032 2.2 0.853 2.4 0.717 2.495 0.663 2.6 0.611 2.8 0.527 3 0.459 3.2 0.403 3.4 0.357 3.6 0.319 3.8 0.286 4 0.258 4.2 0.234 4.4 0.213 4.6 0.195 t g 0 9.81 E(m) 26.203 12.068 7.251 5.128 4.067 3.506 3.213 3.074 3.032 3.053 3.117 3.158 3.211 3.327 3.459 3.603 3.757 3.919 4.086 4.258 4.434 4.613 4.795 y b Q SBM: E1=0.3+E2 E2=3.459-0.30 E2=3.158m y2=2.495m 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 E SBM: E1+0.3=E2 E2=3.459+0.30 E2=3.758m y2=3.4m Q.5) Referring to Problem 4, what is the maximum height of the setup that would not affect the upstream flow condition? What is the minimum width that could exist with a 0.3-m step-up without changing the upstream condition, knowing that the upstream width is 6.0 m? (Answers 0.428m, 2.105m, 5.64m) c) ...
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