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2 Problem 1 SCORE: (1 pts) Part A: 3 pts each, no partial credit. 1) Two identical tanks A and B have holes with round corners at the bottom. The hole on tank B is fitted with a pipe with the same cross-sectional area. If viscous effects are negligible, which tank will have a larger flow rate (Q)? a. Q A > Q B b. Q A < Q B c. Q A = Q B d. Insufficient data for an answer ) In the previous problem, what would be the answer if viscous effects are not negligible? a. Q A > Q B b. Q A < Q B c. Q A = Q B d. Insufficient data for an answer Flow rate of tank B depends on the friction losses inside the pipe, so we need more information. 3) Air flows out of a nozzle from a pressurized tank at room temperature. The air comes out cold. What is the jet s density relative to ambient? a. Higher b. Same c. Lower d. Cannot determine. P/RT, and the pressure is the same and temperature of the jet is lower, so its density is higher.

3 4) Air flows over a flat plate as shown below. The flow is laminar and a boundary layer forms on the plate. Which profile best represents the shear stress inside the boundary layer at the position 1? There is a finite shear stress on the wall and it decreases with distance from the wall. Outside the BL the shear stress is zero. Free stream velocity U Boundary layer thickness Boundary layer Position 1 Flat plate A B C D E Height above plate 0 Shear stress 0 Shear stress 0 Shear stress 0 Shear stress 0 Shear stress 5) Water flows over two flat plates as shown below. The free stream velocity is the same for both plates and the flow remains laminar throughout the plates. The second plate is twice as long as the first plate, but both plates have the same width. Circle the letter of the correct statement. A. The drag force F is greater than times the drag force F 1 B. The drag force F equals times the drag force F 1 C. The drag force F equals the drag force F 1 D. The drag force F is less than the drag force F 1 E. The drag force F is greater than F 1 but less than times the drag force F 1 Plate 1 Length = L V freestream Plate Length = L V freestream Drag force F 1 Drag force F The shear stress decreases along the plate. Therefore, the drag force will be larger but less than twice of F 1

4 6) Water flows through two pipes with the same diameter, length, and friction factor as shown below. The flow rate through the second pipe is twice that through the first pipe. Both flows are turbulent and fully developed. Which statement is correct about the pressure drop over the pipe length, p, for two pipes? a. p =0.5 p 1 b. p =0.5 p 1 c. p = p 1 d. p = p 1 e. p =4 p 1 p=f(l/d)( V /) V =V 1 Flow rate 1 = m 1 Pipe 1 Flow rate = m 1 Pipe 7) Two manometers are connected to a pipe as shown below. Which statement is correct about the flow inside the pipe? a. There is no flow in the pipe. b. The flow is toward right. c. The flow is toward left. d. Cannot determine the direction of the flow.

5 Problem SCORE: (5 pts) A fluid of constant density and viscosity passes over a flat plate at a velocity U. Two stagnation tubes are placed inside the boundary layer at distances of L and L from the leading edge of the plate and at a height of a above the plate. The stagnation tubes are connected to each other through a manometer that shows a pressure difference of p. a) (10 pts) The pressure difference p is expected to depend on, U, L, and a. Perform a dimensional analysis to find the dimensionless groups in the problem using (, U, L) as repeating parameters. b) (10 pts) Assume a laminar boundary layer with a linear velocity profile and a constant outer flow pressure. For this flow, the velocity and boundary layer thickness are u/u=y/ and /x=3.46/re x 0.5, respectively, where x is measured from the leading edge of the plate. What is the stagnation (gage) pressure at the location of each stagnation tube? c) (3 pts) What is the pressure difference p measured by the manometer? d) ( pts) Rearrange the solution you obtained in part (c) and write it in terms of a relationship between the dimensionless groups that you found in part (a).

6 Problem Six dimensional parameters and three repeating parameters result in three dimensionless groups which can be identified by inspection: Π 1 = p ρu Π = a L Π 3 = ρul µ The pitot tubes measure the difference between the stagnation pressures, however the static pressure is constant in the boundary layer. Therefore, p = 1 ρu 1 1 ρu = 1 ( ) Ua ρ 1 ( Ua δ 1 ρ δ For a boundary layer with linear velocity profile In trems of dimensionless parameters: δ x = 3.46 δ = 1µx Rex ρu p = 1 ρu a ρu ( 1 1µ L 1 ) = 1 ρ U 3 a L 48 µl p ρu = 1 48 ( ρul µ ) ( a ) L )

7 Problem 3 SCORE: (30 pts) Water is to be pumped through 600 m of pipe from reservoir 1 to at a rate of 0.1m 3 /s as shown below. The pipe is cast iron of diameter 15 cm and the pump is 75% efficient. Neglect minor losses in the pipe kg/m kg/m s (a) (8 pts) What is the friction factor of the pipes? (b) (10 pts) What is the head produced by the pump (in meters)? (c) (5 pts) What is the power consumption of the pump (in watts)? (d) (7 pts) What is the NPSHA at the pump (in meters)?

8 Problem 3: For cast iron, take =0.6 mm, or /d= 0.6/(150)= s (a) V Q m ; A ( /4)(0.15) s Moody chart Vd 1000(5.66)(0.15) Re , / d , f E 3 (b) The extended Bernoulli s equation, with p 1 = p and V 1 =V =0, yields an expression for pump head: L V 600 (5.66) Hp z f m d g 0.15 (9.81) (c) gqhp 1000(9.81)(0.1)(184) Power: P 41 kw 0.75 (d) Extended Bernoulli s equation between points 1 and s: L V V 300 (5.66) (5.66) Hs z f m d g g 0.15 (9.81) (9.81) Ps Patm Pv V (5.66) NPSHA m g g 1000(9.81) (9.81)

9 Problem 4 SCORE: (4 pts) Air flows isentropically through a converging-diverging duct. At section 1, A 1 = 0 cm, p 1 = 300 kpa (abs), 1 = 1.75 kg/m 3, and Ma 1 = 0.6. Assume that the throat is choked. Determine (a) (5pts) the throat area; (b) (5pts) the stagnation temperature; (c) (6pts) the mass flow rate. (d) (5pts) At section, the area is exactly the same, but the flow is much faster. At this location, find the Mach number, Ma. (e) (3pts) Which one of the following statements is correct about the relative location of the throat and sections 1 and? 1) Section must be downstream of section 1. ) Section must be upstream of section 1. 3) There must be a throat between section 1 and section. 4) Section 1 must be downstream of the throat. 5) None of the above.

10 Problem 4: (a) We can find the throat area A* right away. For k = 1.4 and Ma 1 = 0.6, 3 A1 0cm Ma 1.64, solve A* Athroat 7.58 cm A* A* Ma 1 1. (b) Temperature and stagnation temperature: T R 1 87(1.75) 597 K T T(1 0. Ma ) (597 K)[1 0.(0.6) ] 603K o p 1 1 (c) Compute V 1 and then we can find the mass flow: m V1 Ma1c (87)(597) s kg m kg m 1AV 1 1 (1.75 )(0.000 m )(110.7 ) m s s (d) Now go over to section and compute those properties. Section has the same cross section as section 1 but the flow is much faster. The only way that this could happen is that the flow must be supersonic in section. We have the same area ratio: A A1.64 ; Isentropic flow table: Ma.50 A* A* (e) If the areas are the same but the velocities different, there must be a sonic throat between two sections (3).

### Signature: (Note that unsigned exams will be given a score of zero.)

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### Signature: (Note that unsigned exams will be given a score of zero.)

Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.) Circle your lecture section (-1 point if not circled, or circled incorrectly): Prof. Dabiri Prof. Wassgren Prof.

### Prof. Scalo Prof. Vlachos Prof. Ardekani Prof. Dabiri 08:30 09:20 A.M 10:30 11:20 A.M. 1:30 2:20 P.M. 3:30 4:20 P.M.

Page 1 Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.) Circle your lecture section (-1 point if not circled, or circled incorrectly): Prof. Scalo Prof. Vlachos

### Page 1. Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.)

Page 1 Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.) Circle your lecture section (-1 point if not circled, or circled incorrectly): Prof. Vlachos Prof. Ardekani

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