ME331-430 Advanced Fluid Mechanics

Prof. A. Banerjee
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Solutions to Home Work Problem Set 3

1. (3 points) A mountain community is connected to a large city

in the valley by a single highway. The rate at which autos
enter the highway from the community’s single on-ramp is
non . There is one off-ramp between the mountain community
and the city, and autos are exiting the highway at half the rate
1
that autos enter from the on-ramp, i.e. noff  non . On bad
2
weather days when visibility is severely limited, the rate at
dN (non  noff ) 2

which accidents occur is  , where N is the

dt 
number of cars originating from the mountain community and
α is a constant equal to 100 autos per hour. Assume that
accidents remove the autos involved from the flow of traffic.
In a severe storm, because of a large number of accidents, the
number of autos moving along the highway is constant. At
what rate must autos be entering the highway from the on-ramp?
 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
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2. A subway train has a cross-sectional area, Acs = 0.1A, where
A is the area of the tunnel through which it moves. The train
is traveling at a constant velocity U. What is the average
velocity, u, between the train and the tunnel walls in the
indicated direction?
a. Use a control volume moving with the subway train
to solve (2 points)
b. Use a control volume fixed in space to solve (2 points)
[Hint for (b): Use a Galilean transformation so that the subway train is at rest →The fluid and tunnel
walls move to the left with velocity U. The flow is steady in this coordinate frame.]

Part (a):
 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
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Part (b):
 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
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3. A hover craft has two air compressors, neither of which draws any air from the surroundings. One
compressor supplies a large number of small hover jets with vertical velocity downward of
magnitude we that provide a mass-flow rate m 1   we Ah as shown, where ρ is air density and Ah is
the total area of the hover jets. The other compressor supplies a horizontal jet with velocity relative
to the hover craft, ue = −uei and mass-flow rate m 2   ue Ae , where Ae is the area of the jet. The flow
is incompressible and flow through the compressors is steady as observed from within the hover
craft. Ignoring both pressure variations around the hover craft’s surface and buoyancy effects on the
air flow, determine the equations of motion for mass and x momentum. Express your answers for
dM/dt and MdU/dt in terms of m 1 , m 2 and any pertinent velocities, where M is the mass of the hover
craft and the air remaining in the compressors. (4 points)
 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
-------------------------------------------------------------------------------------------------------------------------------
 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
-------------------------------------------------------------------------------------------------------------------------------
4. (Conservation of Momentum with CV accelerating) Starting from rest, the cart shown in the
figure is propelled by a hydraulic catapult (liquid jet). The jet strikes the curved surface and makes a
180 turn, leaving horizontally. Air and rolling resistance
may be neglected. If the mass of the cart is 100 kg and the
jet of water leaves the nozzle (of area 0.001 m2) with a
speed of 35 m/s, determine the speed of the cart 5s after the
jet is directed against the cart. (3 points)
 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
-------------------------------------------------------------------------------------------------------------------------------
5. (Conservation of Energy) Your yard-drainage system is backing up. To relieve the problem, you
install a pump to help drain the water from the primary pipe whose diameter is d. Fluid moves from
right to left and its temperature
increases by ΔT. The outlet pipe has
diameter d1 = 2d and the inlet pipe
has diameter d2 = d. The pump
supplies energy to the flow such that
the upstream pressure is p2 = p and the downstream pressure is p1 = 4p. The steady mass-flow rate is
m . Neglecting effects of gravity, determine the power, P, delivered by the pump to the flow.
Express your answer in terms of m , p, d, ΔT, water density, ρ, and specific-heat coefficient, cv.
(3 points)
 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
-------------------------------------------------------------------------------------------------------------------------------
 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
-------------------------------------------------------------------------------------------------------------------------------
6. (Conservation of Energy) A power plant uses a river to discharge waste heat. Heat is transferred
from the plant to the river through a heat exchanger at a rate ˙Q. The river’s volume-flow rate is
V and the product of its density and specific-heat coefficient is ρcp. The flow is steady, the river’s
cross-sectional area is constant, and the difference between the temperature downstream and
upstream of the power plant is ΔT = Tf − Ti. Frictional work from the river, ground and atmosphere
can be neglected. Determine the flow rate as a function of ρcp, V and ΔT. (3 points)