CVEN 311-501 (Socolofsky)

Fluid DynamicsExam #2: Bernoulli Equation and Conservation of Mass and MomentumMarch 31, 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 two double-sided 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 College Station, Texas 77843-31363136 TAMU (979) 845-4517 FAX (979) 862-8162

A. True or False (25 points)

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

to the following figure.

Figure 1: A free jet issuing from a pipeline attached to a large tank and impinging on a plate.

1. Points A and B cannot be connected by a streamline because the velocity at A is zero.

2 True 2 False

2. The pressure at point A is atmospheric.

2 True 2 False

3. The pressure at points B and C are identical because they are at the same height in a

connected fluid.

2 True 2 False

4. The elevation head at point A is h.

2 True 2 False

5. The gage pressure a point B is zero.

2 True 2 False

6. The only requirement to be able to use the Bernoulli equation between points A and B is

that the fluid is incompressible.

2 True 2 False

7. At steady state, the velocity at point B of an ideal fluid is√

2gh.

2 True 2 False

8. To use the control volume shown in the figure with the momentum equation, the control

volume must be moved to touch the water in the jet

2 True 2 False

9. Neglecting gravity, the vertical force on the block from the water jet would be ρv2BAB upward.

2 True 2 False

10. At steady state, the total acceleration of water in the free jet where the streamlines are

expanding at point D is zero.

2 True 2 False

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B. 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. In fluid kinematics, a line that is everywhere tangent to the velocity vector is called a

a. Path line

b. Streak line

c. Stream line

d. Material line

e. None of the above

2. Considering the flow of an ideal fluid through a horizontal pipe, which of the following is a

correct statement?

a. The fluid density is constant

b. As the cross section of the pipe reduces, the velocity increases

c. As the cross section of the pipe reduces, the pressure reduces

d. All of the above

e. Only a. and b.

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3. Consider the free jet of an incompressible fluid flowing through an orifice fitted to a constant-

level tank as shown in Figure 2. Ignoring all losses, which of the following is a correct

statement about the magnitude of the initial velocity U of the jet?

Figure 2: Free flow of water through an orifice plate fitted to a large tank.

a. U is directly proportional to the orifice diameter

b. U is inversely proportional to the fluid’s density

c. U is proportional to the square root of the depth h

d. U is proportional to the square of the depth h

e. None of the above

4. Figure 3 shows a horizontal pipeline with a sudden enlargement. The energy grade line

and the hydraulic grade line under certain flow of an incompressible fluid of specific weight

γ = 10 kN/m3 are also shown. The pressure change due to the enlargement is most nearly

Figure 3: Schematic of a pipe expansion, showing the Energy Line and the Hydraulic Grade Line.

a. an increase of 3 kPa

b. a decrease of 3 kPa

c. an increase of 30 kPa

d. a decrease of 30 kPa

e. None of the above

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5. At a certain section in a pipeline, a reducer is used to reduce the diameter from 2D gradually

to diameter D (refer to Figure 4). When an incompressible fluid flows through this pipeline,

the velocity is U1 in the first section and U2 in the second section. Which of the following is

a true conclusion?

Figure 4: Schematic of a pipe section with a reducer between Points 1 and 2.

a. U2 = 4U1

b. U2 = 2U1

c. U2 = U1/2

d. U2 = U1/4

e. None of the above

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6. A 0.3 m diameter pipeline terminates in a nozzle of outlet diameter = 0.15 m. The free jet

from the nozzle is deflected through 90◦ by a flat plate as shown in Figure 5. When water

flows through the pipe at a rate of 0.25 m3/s, the magnitude of the horizontal force required

to hold the plate (N) is most nearly

Figure 5: Schematic of a free jet exiting a nozzle and impinging on a flate plate.

a. 880

b. 1,760

c. 2,640

d. 3,530

e. None of the above

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C. Workout Problem (45 points)

An engineer has proposed to use a jet mounted on the bottom of a reservoir to prevent sediment

accumulation near the inlet to a water treatment plant. A schematic of the jet design is shown in

Figure 6. Water is drawn into the pump housing at a rate of 5 m3/s through a 0.5 m diameter intake

at A. An impeller accelerates the water and forces it out horizontally through a 0.1 m diameter

nozzle at B. Refer to this figure for each of the following problems.

Figure 6: Schematic of a jet designed to prevent sediment accumulation in the bottom of a drinkingwater reservoir.

1. Sketch a control volume (CV) directly in Figure 6 that you can use to find the force necessary

to anchor the jet housing to the reservoir bed. Draw the CV as a free-body diagram in the

space below. Show the flows at the inlets and outlets, any external forces acting on the control

volume, and the reaction forces of the jet housing on the water.

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2. Compute the magnitude of the velocity of the water at the inlet and outlet.

3. Find the horizontal component of the force necessary to hold the pump housing in place.

Report both the magnitude and direction of this force.

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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 · n̂)dA = 0 (12)

• Conservation of linear momentum for a control volume

∂

∂t

∫CV

ρ~vdV +

∫CS

ρ~v(~v · n̂)dA =∑

~F (13)

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

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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