ME 152 B – 2020 – Final Exam
Student name (if not using a separate sheet):
Please read these instructions carefully
This is an open-book exam: you may use course notes and textbook. You may not discuss it with other
students.
You must write the following statement on your exam, and sign underneath it: “I confirm that the work I
submit is my own and my own only, and that I did not use resources beside course notes and the
textbook.” If this statement is not written and signed, you will get a score of zero.
If you will print the exam or use a tablet, you may use the spaces provided to develop your answers.
You may also use a separate sheet of paper to develop your answers. If doing so, please scan your
solutions and upload them to Gauchospace by the posted time.
If you need to ask a question, please email the instructor.
1
Multiple choice
Note: you do not need to justify your answers to the multiple-choice questions on this page.
Select one answer per question.
1. The Buckingham-Pi theorem states that a problem with n independent variables and r different
dimensions can be reduced to a problem with ...
(a) r − n dimensions;
(b) r − n dimensionless numbers (or Pi-groups);
(c) n− r dimensions;
(d) n− r dimensionless numbers (or Pi-groups).
2. When the flow in a pipe is laminar, the velocity is ...
(a) Maximum at the wall;
(b) Minimum at the center of the pipe;
(c) A linear function of the distance to the center of the pipe;
(d) A parabolic function of the distance to the center of the pipe.
3. The momentum integral equation for a boundary layer...
(a) assumes irrotational flow everywhere;
(b) can be applied only to laminar flow;
(c) applies equally to laminar and turbulent flow;
(d) can be applied to time-varying flows.
4. Which of these assumptions is not required to derive the Euler turbomachinery equation for a cen-
trifugal pump:
(a) Incompressible and steady flow;
(b) Uniform flow at inlet and outlet;
(c) No swirl at the inlet;
(d) Neglecting torques not associated with shaft.
5. The ‘best efficiency point’ (BEP) corresponds to the point...
(a) Where the flow rate is maximum;
(b) Where the efficiency is maximum for any impeller diameter;
(c) Where the pressure head is maximum;
(d) Where the efficiency is maximum for a given impeller diameter.
6. The speed of sound in an ideal gas is...
(a) faster than in a solid;
(b) only dependent on the density of the gas;
(c) only dependent on the temperature of the gas;
(d) the same for all gases.
7. The flow process through a shock wave...
(a) is adiabatic and is reversible;
(b) is adiabatic and is irreversible;
(c) is not adiabatic and is reversible;
(d) is not adiabatic and is irreversible.
2
8. A small aerosol particle of diameter d and density ρp = 2, 500 kg/m
3 falls in air (density ρair =
1.2 kg/m3, and dynamic viscosity µair = 1.8×10−5 Pa · s). For small Reynolds numbers, Re 1, the
terminal settling speed of the particle V depends only on d, µair, the density difference (ρp − ρair)
and the gravitational acceleration g.
(a) List all the relevant variables and the dimensions associated with this problem. Using the
Buckingham Pi theorem, determine how many dimensionless groups govern this problem.
(b) Form the dimensionless groups.
It is often difficult to visualize the settling of small aerosol particles of diameter d = 10µm.
Therefore, a model system is built in the lab using water as a fluid (density ρw = 1, 000 kg/m
3,
and dynamic viscosity µw = 10
−3 Pa · s) and particle with the same density ρp.
(c) What should be the size of the particle settling in the model system, dmodel to ensure dynamical
similarity?
(d) The model particle of diameter dmodel falls at a velocity 3.4 cm/s in water. What is the velocity
at the small aerosol particle of diameter d = 10µm will fall in the air?
(e) What time will it takes for the small aerosol particles to fall over a distance of 2 m?
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4
9. Water flows from an upper reservoir into a lower reservoir through a pipe of diameter D = 35 cm,
roughness ε = 0.7 mm and length L = 150 m. The height between the water surface of the top and
bottom reservoirs is H = 45 m. The minor losses are captured by the entrance and exit coefficients are
K = 0.8 and K = 1, respectively. The situation is sketched below. Recall, for water, ρ = 1, 000 kg/m3
and µ = 10−3 kg/(m s).
Upper reservoir
Lower reservoir
H
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Pipe
L
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(a) We first assume that the major losses are negligible compared to the minor losses. Determine
the flow rate in the pipe.
(b) Calculate the value of the Reynolds number in the pipe.
(c) We now consider the major losses. As a first approximation, we can assume that the Reynolds
number remains the same to determine the friction factor. Estimate the flow rate in the pipe.
(d) Using the new flow rate obtained from the previous question, estimate the new Reynolds number
and comment on the approximation.
(e) Given the Reynolds number that you estimated in the previous question, what type of flow do
you expect in the pipe? Provide a schematic of the velocity profile in the pipe.
5
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10. Consider flow with velocity U over a flat plate, in a zero pressure gradient. An approximation to a
laminar boundary layer is given by
u
U
= a
(y
δ
)3
+ b
y
δ
+ c = a η3 + b η + c.
(a) Using appropriate boundary conditions, find the constants a, b, and c;
(b) Calculate the nondimensional momentum thickness, given by
β =
θ
δ
=
∫ 1
0
u
U
(
1− u
U
)
dη
(c) Using τw = µ(∂u/∂y)y=0, find an expression for the wall shear stress τw in terms of δ.
(d) The momentum integral equation for a flat plate is τ = ρU2 dδdxβ. Using the results you obtained
so far in this question, obtain an expression for δ(x)x , as a function of Rex =
ρUx
µ .
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9
11. An autonomous electric aircraft has a mass of 650 kg and has a wing with planform area A = 25 m2.
The lift and drag coefficients CL and CD are given by the figure below (note that these curves already
includes finite span effects, so you do not need to calculate those). The lift and drag L and D are
given by
L =
1
2
ρU2CL(α)A,
D =
1
2
ρU2CD(α)A.
0 5 10 15 20
, (deg)
0
0.2
0.4
0.6
0.8
1
1.2
(C
L) w
in
g
0 5 10 15 20
, (deg)
0
0.05
0.1
0.15
0.2
0.25
(C
D
) wi
ng
The aircraft needs to fly as slowly as possible in order to perform an imaging survey. Steady horizontal
flight needs to be maintained. Assume ρ = 1.19 kg/m3.
(a) At what angle of attack should the aircraft fly?
(b) What is the corresponding slowest aircraft velocity U?
(c) What is the propulsive power requirement in this slow-flying condition?
A momentary error in the flight controls causes α to increase to 22◦. The velocity U is unchanged.
(d) Is steady horizontal flight still possible? If not, what is the vertical acceleration? Is the aircraft
climbing or descending?
(e) Assuming that the horizontal engine thrust is unchanged, what is the horizontal acceleration?
Is the aircraft accelerating or decelerating?
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11
12. A wind turbine has diameter D = 82.4 m. Its power coefficient has been determined empirically
as (Cp)actual = 0.38, such that the power extracted is P =
1
2(Cp)actualρV
3
0 Arotor. The air density
is ρ = 1.22 kg/m3. The turbine operates in a wind V0 = 12 m/s. Recall also that the ideal power
coefficient is (Cp)ideal = 4a(1 − a)2, where a is the axial induction factor relating the wind velocity
V0 to the air velocity through the rotor V1 = V0(1− a).
(a) What is the power extracted by the turbine in these conditions?
(b) For a system extracting energy from a moving fluid, argue, on physical grounds, that a maximum
in power coefficient Cp must exist as the axial induction factor a is varied. (Hint: consider the
limiting cases a = 0, a = 1, and a case in between).
(c) An inventor claims to have designed a turbine that, with D = 82.4 m and in the same wind
conditions, would give P = 4.5 MW of power. Is this possible?
(d) What is the ideal maximum of power that could be produced by a turbine with D = 82.4 m and
in the same wind conditions?
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Blank page for developing problem 12.
13
13. Properties for a pump are given by the following table:
Q (gpm) 50 100 150 200 250 300 350 400 450
H (ft) 205 202 198 194 185 177 167 156 139
The pump needs to deliver water across a vertical distance ∆z, using a pipe with diameter D = 4 in.
The pump performance data are fitted by the parabolic curve H = H0 − AQ2, where H0 = 205.8 ft
and A = 3.22 × 10−4 ft/(gpm)2 Since the flow is expected to be in the fully rough regime, you may
assume that the friction factor is a constant, f = 0.016. Minor losses are negligible. By the definition
of the friction factor, the pressure drop due to major losses is
∆p =
8ρQ2
pi2D4
f
L
D
,
and you may find useful 1 gal = 0.003785 m3, 1 ft = 0.3048 m, 1 in = 2.54 cm. Assume ρ = 1000 kg/m3.
(a) The vertical distance across which water needs to be delivered is ∆z = 180 ft. At what flow rate
is the water delivered?
(b) Sketch the pump head capacity versus Q for the given pump. On the same axes, sketch the
combined system head versus Q. Label the operating point.
(c) Would you install the pump at the bottom or at the top of the vertical pipe? Why?
(d) A second pump, with the same performance data as the first pump, is added in series. What
would be the equation governing the total pump head, and what would be the values of the
overall H0, A? Please sketch the change how the operating point would change, in a diagram
of H versus Q.
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15
14. A new supersonic passenger aircraft is being designed. At the nose of the aircraft, air is brought to
rest (relative to the aircraft) inside a sensor.
Shortly after takeoff, the aircraft is flying at V = 265 m/s, in air at T = 270 K and ρ = 0.909 kg/m3.
(a) What is the Mach number M?
(b) Could this flow be approximated as incompressible?
(c) In the sensor, is the flow brought to rest isentropically?
(d) What is the temperature inside the sensor?
(e) What is the pressure inside the sensor?
During cruise, the aircraft flies at V = 790 m/s, in air with T = 216 K and ρ = 0.0889 kg/m3.
(f) In the sensor, is the flow brought to rest isentropically?
(g) What is the temperature inside the sensor?
(h) What is the pressure inside the sensor?
You may find useful the relations (listed here without corresponding assumptions) :
c2 = kRT, p = ρRT, p/ρk = constant,
T0 = T
(
1 +
k − 1
2
M2
)
, p0 = p
(
1 +
k − 1
2
M2
) k
k−1
k = 1.4, R = 287 J/(kg K), sin(α) = 1/M, (1)
as well as the normal shock relations
p2
p1
=
2k
k + 1
M21 −
k − 1
k + 1
T2
T1
=
(
1 + k−12 M
2
1
) (
kM21 − k−12
)(
k+1
2
)2
M21
M22 =
M21 +
2
k−1
2k
k−1M
2
1 − 1
Please justify the applicability of an equation if using it.
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