Exam #2: Fluid Kinematics and Conservation Laws April 13, 2016, 7:00 p.m. 8:40 p.m. in CE 118


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1 CVEN (Socolofsky) Fluid Dynamics Exam #2: Fluid Kinematics and Conservation Laws April 13, 2016, 7:00 p.m. 8:40 p.m. in CE 118 Name: : UIN: : Instructions: Fill in your name and UIN in the space above. The exam is closed book, and only two doublesided sheets of notes are permitted. No collaboration with others! For multiple choice questions, choose the single, best answer. For short answer and workout problems, write down all steps necessary to solve the problem: show all your work. Failure to do so will result in a lower score. Be sure to answer all parts of all problems. Do not leave any problems blank. Certification: An Aggie does not lie, cheat, or steal or tolerate those who do. By my signature below, I certify that the work contained in this exam is my own and that I did not receive help from other students. Signature: Date: CIVIL 3136 TAMU College Station, Texas (979) FAX (979)
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3 A. True or False (25 points) For each of the following statements, check the box with the most appropriate response. 1. The total acceleration of a fluid particle as we follow the fluid is given by ~a = 2. Streamlines are everywhere tanget to the velocity vector. 3. A material system is allowed to exchange fluid across its material surface. 4. A control volume is allowed to exchange fluid across its control surface. 5. A control volume must be rigid and fixed in space. 6. Pumps do work on the fluid passing through the pump. 7. For steady flow through a constantdiameter pipe, friction losses act to increase the pressure with distance along the pipe. For Problems 8 10, assume the control volume in Figure 1 has been selected to solve a fluid flow problem using the conservation of momentum equation. Section 1 Section 2 Figure 1: Control volume for problems For the control volume in the figure, ~v ˆn at Section 1 is greater than zero. 9. For the control volume in the figure, p 1 A 1 is a force that acts to the right. 10. After solving the momentum equation, the values of R x and R y will be equal and opposite to the force required to hold the control volume in place. 2
4 B. Multiple Choice (45 points) For each of the following questions, circle the answer that is most appropriate or closest numerically to your answer. Be sure to clearly mark only one answer. Multiple selections will be graded as zero. 1. A fluid has velocity components u =(6y + t) and v =2tx, whereu and v are the velocity components in the x and ydirections, respectively. The xcomponent of the acceleration of a particle passing the point (1, 2) at t = 1 is closest to a. 11 b. 12 c. 13 d. 28 The unsteady discharge from a cistern is shown in Figure 2. Let A t be crosssectional area of the tank and A o be the area of the outlet. Figure 2: Schematic of the discharge from a cistern. 2. A solution using conservation of mass for the unsteady water depth h in the tank is given by a. dh/dt =(A o /A t ) p 2gh b. dh/dt = (A o /A t ) p 2gh c. dh/dt =(A t /A o ) p 2gh d. dh/dt = (A t /A o ) p 2gh 3
5 3. If a vertical pipe of length 3 m is attached to the outlet, then, ignoring friction, a. The tank will empty faster than without the pipe. b. The tank will empty slower than without the pipe. c. The tank will empty at the same rate as without the pipe. d. The tank will empty faster than without the pipe until half of the water is drained, and then it will empty slower than without the pipe. Figure 3 shows the steady flow through a pipe with a gate valve. The flow rate through the pipe is 0.2 m 3 /s and the gauge pressure at the upstream section is 600 kpa. The crosssectional area at the outlet is m 2. Figure 3: Schematic of a pipe discharge through a gate valve. 4. The velocity of the water at the outlet is a m/s b m/s c m/s d m/s 5. The force exerted by the water on the valve is a kn b kn c kn d kn 4
6 6. The head loss (in meters of head) through the valve is a m b m c m d m e m f. None of the above For Problems 9 10, a pump is used to deliver water at a flow rate of 0.8 m 3 /s from a large reservoir to another large reservoir that is 20 m higher in elevation. The friction head loss in the 200 mm diameter, 4 km long pipeline is 2.5 m for every 500 m of pipe at this flow rate. The ends of the pipes are submerged in the reservoirs. 7. What is the total pump head h p (in meters of head) that the pump must deliver to achieve the specified pumpage? a. 0 m b. 20 m c. 40 m d. 40 m 8. At a flow rate of 1.0 m 3 /s (which has a di erent amount of head loss), the pump head required is 75 m. The required power output of the pump for this case is a. 310 kw b. 740 kw c. 980 kw d MW 5
7 Crude oil flows through the horizontal tapered 45 elbow shown in Figure 4 at 0.02 m 3 /s. The pressure at A is 300 kpa, the crude oil is incompressible, with density o = 880 kg/m 3, and the velocities at A and B are v A = m/s and v B = m/s. Figure 4: Schematic of crude oil flow ( o = 880 kg/m 3 ) through a pipe bend and contraction. 9. Assuming ideal fluid flow and noting that the pipe continues beyond B, the pressure at B is a kpa b. 6.5 kpa c. 95 kpa d. 95 kpa 10. The force in the xdirection R x required to hold the elbow in place is a N b. 360 N c N d. 420 N 6
8 C. Workout Problem (30 points) 1. Water flows through the pipe in Figure 5 at 5 m/s. Between sections A and B, an orifice plate is installed, which has a round hole in its center. The gauge pressure at A is 230 kpa and that at B is 180 kpa. Figure 5: Schematic of pipe flow across an orifice plate. a. Sketch the control volume you would use to determine the force the water exerts on the plate. Show the fluxes of water into and out of the control volume and all forces acting on the control volume. b. Apply the energy equation to compute the head loss (in meters of head) between Sections A and B. 7
9 2. A pump draws water from a large tank and discharges to a reservoir at C, as shown in Figure 6. The pump supplies m of head, and the diameter of the pipe is 200 mm. Find the force required to hold the pump in place. Neglect friction losses. Figure 6: Schematic of pumped flow through a pipeline. 8
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11 D. Fluid Properties and Formulas Properties of Fluids: Gravitational acceleration g =9.81 m/s 2 = ft/s 2 (1) Weight, specific weight, and specific gravity W = gv = g SG = H2 O (2) Density and specific weight of water w = 1000 kg/m 3,1.94 slug/ft 3 (3) w =9, 810 kg/(m 2 s), 62.4 lb/ft 3 (4) Atmospheric pressure p atm = 101, 325 Pa = 14.7 psia (5) General Formulas: Area of a circle A = r 2 = 4 D2 (6) Volume of a cylinder V = r 2 h (7) Material derivative D (8) Twodimensional stream lines dy dx = v u (9) Bernoulli equation along a streamline p + v 2 2g + z = c (10) 9
12 Conservation of mass for a control volume Z dv + (~v ˆn)dA = 0 CV CS Conservation of linear momentum for a control volume Z ~vdv + ~v(~v ˆn)dA = X F ~ CV CS Energy Equation p 2 g + v2 2 2g + z 2 = p 1 g + v2 1 2g + z 1 + h S h L (13) Shaft work to flow rate relationship h S = Ẇ Q (14) 10
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