Part A: 1 pts each, 10 pts total, no partial credit. 1) (Correct: 1 pt/ Wrong: -3 pts). The sum of static, dynamic, and hydrostatic pressures is constant when flow is steady, irrotational, incompressible, and when frictional effects are negligible. 2) (Correct: 1 pt/ Wrong: -3 pts). A streamline cannot terminate or start in the middle of the flow unless it is at a stagnation point. 3) (Correct: 1 pt/ Wrong: -3 pts). If the flow is inviscid then the Reynolds number is equal to infinity 4) Dynamic pressure ρv 2 /2 is the pressure rise when the fluid in motion is brought to stop (or to zero velocity) isentropically. 5) Hydrostatic pressure ρgz accounts for the effects of the fluid weight on pressure acting on a surface. 6) The pressure head P/ρg is the elevation (or height) of a fluid column that produces the static pressure P. 7) The velocity head V 2 /2 is the elevation (or height) needed for a fluid to reach the velocity V during frictionless free fall. 8) The linear momentum equation is obtained by setting the intensive property n=_ velocity (U) and thus the extensive property N= mass x velocity (mu) in the Reynolds transport theorem. 9) (Correct: 1 pt/ Wrong: -3 pts). If a body moves with a Mach number of M=0.25 then the speed of sound of the fluid is equal to 4 times the speed of the body. 10) (Correct: 1 pt/ Wrong: -3 pts). Pathlines, streaklines and streamlines coincide only if the flow is steady
Part B: 3 pts each, 15 pts total, no partial credit. 1) Helicopter A hovers above sea level and a similar helicopter B, carrying the same weight, hovers above a mountain top at 1 mile altitude. Select the correct answer: a. Power of helicopter A is higher than power for helicopter B b. Power of helicopter B is higher than power for helicopter A c. Power of helicopter A is equal to the power for helicopter B because the lift the same weight d. Cannot be determined because we do not know the rotation speed of the blades 2) A flat plate is held stationary while a jet is impinging normal to its surface. If the velocity of the jet increases by a factor of 2 then the force required to hold the plate will: a. Increase by a factor of 2 b. Increase by a factor of 4 c. Stay the same d. Decrease by a factor of 2 e. Decrease by a factor of 4 3) A water jet of velocity V impinges on a plate moving toward the water jet with velocity ½V. If the force required to hold the plate stationary is F, then how much is the force required to move the plate towards the jet in terms of F. a. 2.25 F b. 1.50 F c. 1.25 F d. 1.0 F (Stay the same) e. 0.5 F
4) Water enters a centrifugal pump axially at a specified volume flow rate of 0.09 m 3 /s and velocity 5 m/s, and leaves in the normal direction along the pump casing. The area of the intake is 0.018 m 2. Determine the force acting on the shaft in the axial direction. Density of water is 1000 kg/m 3. a) 45 N b) 90 N c) 450 N d) 900 N e) 4500 N 5) Water from a storm drain flows onto a porous bed that absorbs the water with uniform vertical velocity of 8mm/s as shown in the figure below. The system is 5m deep into the page. Find the length L of the bed needed in order to completely absorb the storm water. a. 5 m b. 10 m c. 50 m d. 100 m e. 150 m
Part C: 10 pts total The drag coefficient as a function of the Reynolds number, for flow over a sphere and flow over a smooth cylinder is shown in the figure below. The Reynolds number for both cases is defined based on the free stream velocity and the diameter. 1) Using this figure and only the curve for the smooth cylinder a. Identify the approximate Reynolds number for which the flow on the cylinder becomes turbulent. Answer: (1pts) The drastic reduction in drag coefficient indicates the transition to turbulent flow at Re~300,000 b. Identify the range of Reynolds numbers for which the drag is not a function of the Reynolds number. Answer: (3pts) The almost constant drag between 1000< Re < 200,000
2) (2 pts) Draw the velocity profile at the point of separation for a turbulent flow over a smooth cylinder for a Reynolds number of Re = 1,000,000. Re = 1,000,000 will the typical critical velocity profile with zero shear stress 3) (4 pts total) Draw the streamlines for a flow over a smooth cylinder for a Reynolds number of Re=1,000,000. Show the streamlines for the vortices, the stagnation and the separation streamlines.
Part IV 1. Which of the following pump curves can lead to an unstable operation? Why? (2 points) Head Head Flow rate (A) Flow rate (B) Since two operating points are possible 2. Identify (mark X ) the best efficiency operating point for the three pumps shown in the figure below. (3 points) 3. A gasoline (specific gravity 0.72) pump requires P 1 watts of power at a certain flow rate. If the pump is now used to pump water at the same flow rate what would be the power requirement? Assume that the pump speed is the same in both cases. (2 points) A. P 1 B. 0.72 P 1 C. P 1 /0.72 D. (0.72) 2 P 1
4. Select one answer. Consider the flow through a pipe as shown below. The minor loss for case 1 would be greater than / less than / equal to the minor loss for case 2. (2 points) Q Q D 1.5D 2D D 2D Case 1 Case 2 5. Water flows through a circular pipe of length L and diameter D at a certain flow rate. If the circular pipe is replaced with a square duct having the same cross- sectional area, the major loss would increase / decrease / stay the same. Assume flow is in the fully rough regime. (2 points) 6. Consider viscous flow through a pipe bend. Draw the streamlines and identify the source of minor loss. (2 points)
7. The system curve for a pipe system is as shown in the figure. If a gate valve is added to the pipe system, draw the new system curve. (2 points) Head System curve starts at the same point but gets shifted up due to increase in minor loss Flow rate Problem 8 (15 points) Pitch (density = 1100 kg/m 3 ) is a highly viscous fluid. The viscosity of pitch can be estimated using data from the famous pitch drop experiment. It is estimated that a volume V = 4.7 10-5 m 3 of pitch drained out from the funnel (dimensions as shown in the figure) over a period of 582 months (17708 days). Estimate the viscosity of pitch. Neglect minor losses. Also neglect the kinetic energy of the fluid. Hint: Assume laminar flow and check the validity of that assumption at the end. Solution:
Assumptions: 1. Laminar flow 2. Neglect minor losses 3. Neglect kinetic energy of fluid Apply extended Bernoulli s equation (EBE) from 1- >2 2 2 2 p V p V 1 1 2 2 l + α V 1 + z1 + α2 + z2 = f ρg 2g ρg 2g d 2g p 1 = p 2 = p atm (2 points) V 1 = V 2 0 since kinetic energy can be neglected (2 points) z 1 = h+l, z 2 = 0 (2 points) For laminar flow 64 64µ f = = (2 points) Re ρvd Substitute into EBE 2 64µ lv h+ l = ρvd d 2g Solve for µ 2 ρgd h µ = 1+ (2 points) 32V l The average velocity can be obtained from the flow rate Q 4Q V = 2 A = πd 4 πρgd h µ = 1+ 128Q l The flow rate can be obtained using the data provided 3 V 4.7e-5 m 14 3 Q= = = 3.07 10 m / s (2 points) T 17708*3600*24s Plug in all numbers 4 π (1100)(9.81)(0.0094) 2.9 µ = 1+ (128)(3.07e-14) 7.5 = 9.3e7 Pa s Check for Reynolds number (1 point) ρvd 4ρQ Re = = = 4.9e-17 Laminar flow assumption OK! (2 points) µ µπd
Problem 1 (3 pts) 1) Consider the two nozzle in figure. They both work between two environments at given pressure with air. Assume 1D flow and neglect both friction and heat transfer effects. How is the mass flow rate of nozzle a compared with the nozzle b? Nozzle A Nozzle B 1 m 2 0.5 m 2 1 m 2 0.5 m 2 p 1 = 200kPa T 1 = 100 C p2 = 150kPa p 1 = 200kPa T 1 = 100 C p 2 = 150kPa LA LB = 2 LA a) m! = 0.75 m! b) m! = m! c) m! = 2 m! d) m! =!!!!!!! m!
Problem 2 (11 pts) 2A (6 pts) For which exit pressure sonic conditions are reached for the flow inside this Convergent- Divergent nozzle operating with air? Assume 1D isentropic flow conditions. P1 = 100 kpa A2 = 6 mm 2 A1 = 6 mm 2 Amin = 2 mm 2 a) Sonic condition cannot be established for any value of exit pressure b) Sonic condition are always established for any value of exit pressure c) For any outlet pressure below 97.3 kpa d) Only for a pressure between 4.7 kpa and 97.3 kpa e) Only below 4.7 kpa
2B (5 pts) Represent realistic trends for the pressure and the Mach number inside the nozze for A. p 2 = 99 kpa; B. p 2 =97 kpa; C. p 2 =50 kpa; D. p 2 =1 kpa. Clearly indicate the curves A,B,C,D in the diagrams below p 100 kpa 50 kpa 0 kpa x A1 x Amin x A2 x M 1 0 x A1 x Amin x A2 x
Problem 3 (3 pts) The coupling in figure can be considered as a minor loss in a pipe flow problem. Since the minor loss coefficient k is not given, an experiment on a smaller model (1:5 geometry scale) is performed. Determine which input velocity you would use for the model case. 3 m/s A. 0.6 m/s B. 6.71 m/s C. 15 m/s D. 75 m/s
Problem 4 (18 pts) A rigid tank of volume 10-5 m 3 contains fluid with k = 1.2 initially at 700 kpa and 100 C. At t = 0 s, a valve connected to the tank is opened. The valve can be assumed as nozzle with minimum area 10-7 m 2. The pressure outside the tank is 100 kpa. Ambient 100 kpa 25 C 0.1 m3 RIGID TANK k=1.2; R=290 Nm/kgK 700kPa 100 C 10-5 m 2 Assuming isentropic flow conditions: a) Determine the mach number at nozzle exit, at t = 0 s (3 pts) b) Determine the temperature of the fluid at the exit section of the nozzle, at t = 0 s (2 pts) c) Determine the flow rate at nozzle exit, at t = 0 s (2 pts) d) If the temperature in the tank remains constant, what is the pressure in the tank after 10 s? (10 pts) e) What is the M number after 10 s? (1 pt)