CVEN 311-501 (Socolofsky)

Fluid DynamicsExam #3: Energy Equation and Dimensional AnalysisApril 21, 2017, 10:20 a.m. – 11:10 a.m. in CE 118

Name: :

UIN: :

Instructions:

Fill in your name and UIN in the space above.

The exam is closed book, and only three double-sided sheets of notes are permitted. No collabora-

tion 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 College Station, Texas 77843-31363136 TAMU (979) 845-4517 FAX (979) 862-8162

Answers

True / False

1. 2 True 2 False

2. 2 True 2 False

3. 2 True 2 False

4. 2 True 2 False

5. 2 True 2 False

6. 2 True 2 False

7. 2 True 2 False

8. 2 True 2 False

9. 2 True 2 False

10. 2 True 2 False

Multiple Choice

1. 2 a. 2 b. 2 c. 2 d. 2 e.

2. 2 a. 2 b. 2 c. 2 d. 2 e.

3. 2 a. 2 b. 2 c. 2 d. 2 e.

4. 2 a. 2 b. 2 c. 2 d. 2 e.

5. 2 a. 2 b. 2 c. 2 d. 2 e.

6. 2 a. 2 b. 2 c. 2 d. 2 e.

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A. Workout Problem (45 points)

Answer each question carefully, showing all of your work.

1. (15 points) Consider a boundary layer growing along a thin flat plate, as shown in Figure 1.

This problem involves the following parameters: boundary layer thickness δ, downstream

distance x, free stream velocity V , fluid density ρ, and fluid viscosity µ. The dependent

parameter is δ. If we choose three repeating parameters as x, ρ, and V , find a complete set

of non-dimensional Π -groups to describe this problem.

Figure 1: Diagram of a boundary layer velocity profile above a flat plate.

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2. (30 points) As shown in Figure 2, water is drawn from the well at B through the 3 in diameter

suction pipe and discharged through the pipe of the same size at A. If the pump supplies

20 ft of head to the water, determine the flow rate of water into the tank at A. Assume the

frictional head loss in the pipe system is 1.5V 2/(2g).

Figure 2: Schematic of a pipe network and pump delivering water from a pit to a storage tank.

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B. True or False (25 points)

For each of the following statements, check the box with the most appropriate response and refer

to the pipe network and pump in Figure 3. The pipe diameter is constant.

Figure 3: Schematic of a pump in a water distribution system.

1. The velocity at B is greater than the velocity at A.

2 True 2 False

2. The pressures at C and B are the same because they are at the same height in a connected

fluid.

2 True 2 False

3. The pressure at C is the lowest pressure among the pressures at A, B, and C.

2 True 2 False

4. We can solve this problem to determine the energy input of the pump using the Bernoulli

equation.

2 True 2 False

5. The pump does shaft work on the fluid in the pipe.

2 True 2 False

6. For the flow direction shown in the figure, the pressures must be wrong.

2 True 2 False

7. If the pump head is 15 m, then the head loss in the system is about 5 m.

2 True 2 False

8. Head losses in the pipe network include the effects of friction.

2 True 2 False

9. Real pumps can operate at 100% efficiency.

2 True 2 False

10. If the pump head is 10 m when the flow rate is 1 m3/s, then the pump adds 100 kW of power

to the water in the pipe

2 True 2 False

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C. Multiple Choice (30 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. The drag coefficient CD is a non-dimensional parameter and is a function of drag force FD,

density ρ, velocity V , and area A. The drag coefficient can be expressed as CD =

a. FDV2/(2ρA)

b. 2FD/(ρV A)

c. ρV A2/FD

d. FDA/(ρV )

e. 2FD/(ρV2A)

2. Figure 4 shows the energy line when lubricating oil is flowing through a horizontal pipe. The

head loss due to friction per meter of pipe (m/m) is most nearly

Figure 4: Schematic of a horizontal pipe showing the energy line with friction loss.

a. 0.04

b. 0.4

c. 0.12

d. 1.2

e. None of the above

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3. The schematic of a pumping system to pump water from a canal to an overhead storage tank

is shown in Figure 5. At the design pumping rate of 0.0083 m3/s, the total head loss of the

system is 10% of the total static head (pressure head when the water is not moving). The

power added by the pump (kW) is most nearly

Figure 5: Schematic of a pipe network and pump to move water from a canal to an overhead storagetank.

a. 1.0

b. 1.5

c. 2.0

d. 3.0

e. None of the above

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4. A hydropower turbine fed with water from a reservoir is shown in Figure 6. If the total

head losses in the system at a flow rate of Q = 0.5 m3/s are hL = 1.983 m, then the power

generation potential of this sytem (kW) is most nearly

Figure 6: Schematic of the flow through a turbine between two reservoirs.

a. 80

b. 100

c. 140

d. 160

e. None of the above

5. A dam spillway is 40 ft long and has a fluid velocity of 10 ft/s. Considering a Froude scale

model (Fr = V/√gh), the corresponding model fluid velocity (ft/s) for a model length of 5 ft

is most nearly

a. 1.25

b. 3.50

c. 5.25

d. 8.00

e. None of the above

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6. A screw propellor has the following relevant dimensional parameters: axial thrust F , propellor

diameter D, fluid kinematic viscosity ν, fluid density ρ, gravitational acceleration g, ship

speed V , and propellor rotational speed, N . Which of the following statements is true when

determining the non-dimensional variables for this problem.

a. There are a total of 7 variables in this problem

b. There are 4 linearly independent non-dimensional groups

c. When using the repeated variables method, one acceptable set of repeated variables

would be N , D, and ρ.

d. All of the above

e. (a) and (b) only

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D. Formulas and Fluid Properties

• Gravitational acceleration

g = 9.81 m/s2, 32.17 ft/s2

• Absolute zero

Tabs = -273.15 ◦C, -459.67 ◦F

• Atmospheric pressure

patm = 101,325 Pa, 14.7 psia

• Properties of water

ρw = 1000 kg/m3, 1.94 slug/ft3

γw = 9,810 kg/(m2s), 62.4 lb/ft3

µ = 1 · 10−3 N·s/m2, 20 · 10−6 lb·s/ft2

ν = 1 · 10−6 m2/s, 10 · 10−6 ft2/s

σ = 0.072 N/m, 0.005 lb/ft

• Vapor pressure of water at 20 ◦C (70 ◦F)

pv = 2,340 Pa, 0.363 psia

• Ideal gas law

PV = mRT , ρ =P

RT(1)

• Hydrostatic pressure

p = p0 + ρgh (2)

• Forces on a plane surface

FR = γ sin θ

∫AydA = γAhc

yR =IxcycA

+ yc (3)

• Weight, specific weight, and specific gravity

W = ρgV , γ = ρg, SG =ρ

ρH2O(4)

• Viscosity and shear stress

τ = µdu

dy(5)

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• Area of a circle

A = πr2 =π

4D2 (6)

• Volume of a cylinder

V = πr2h (7)

• Material derivative

Dα

Dt=∂α

∂t+ u

∂α

∂x+ v

∂α

∂y+ w

∂α

∂z(8)

• Two-dimensional stream lines

dy

dx=v

u(9)

• Bernoulli equation along a streamline

p

γ+v2

2g+ z = c (10)

• Bernoulli equation normal to a streamline

p

γ+

∫ r

r0

v2

gRdn+ z = c (11)

• Conservation of mass for a control volume

∂

∂t

∫CV

ρdV +

∫CS

ρ(~v · n)dA = 0 (12)

• Conservation of linear momentum for a control volume

∂

∂t

∫CV

ρ~vdV +

∫CS

ρ~v(~v · n)dA =∑

~F (13)

• Energy Equation

p2ρg

+v222g

+ z2 =p1ρg

+v212g

+ z1 + hS − hL (14)

• Shaft work to flow rate relationship

hS =W

γQ(15)

• Major losses in a pipe

hL =fl

D

v2

2g(16)

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• Minor losses in a pipe

hL = KLv2

2g(17)

• Drag force

D =1

2ρAU2CD (18)

Figure 7: Areas and moments of inertia about the centroid of different plane shapes.

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Table 1: Properties of common gases

Gas Molecular Weight Gas Constant (SI) Gas Constant (BG)

M (g/mol) R (J/(kg·K)) R (ft·lb/(slug·◦R))

Air 35.34 286.9 1716

Oxygen 31.9988 259.8 1554

Nitrogen 28.0134 296.8 1775

Carbon dioxide 44.01 188.9 1130

Methane 16.043 518.3 3099

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