AE 2010 Thermodynamics (Problem Set #4: Mass Conservation: Control Volume
Analysis) NEW UPDATE Georgia Institute Of Technology
• Always indicate any assumptions you make. If you use any results or equations from
the class notes or text in you solutions, please note and reference them (but you
better be sure they are applicable to the problem at hand).
• Show all your work, no credit for just answers. When applicable, try to solve the
problem algebraically first. Only use numbers/values in the final steps of your
solution – and be sure to include units when you insert numbers.
• If the problem statement is given in ENGLISH units, the answer must also be in
English units; if the problem statement is in SI units, the answer must be in SI units.
1. Low Speed Wind Tunnel
A low speed, open-return wind tunnel is being designed with a fan at the inlet, and a
test section having a rectangular cross-section of 40in. 42in. The designers want to
achieve a 55 mph wind speed, with the air in the test-section nominally at 72 ºF and
0.95 atm. How much air (mass flow rate) will the fan that runs the wind tunnel have to
supply to meet these conditions? If the pressure of the air at the exit of the wind tunnel
was 0.2 psi lower than the pressure in the test section, what would be the air velocity
at the exit, assuming the air temperature remained at 72 ºF?
2. Steady-State Aircraft Engine
In a jet engine, air enters from the front, fuel is added farther downstream in the
engine, and the fuel and air burn. Thrust is produced when the gases leave the engine
through a nozzle at high velocity. Consider a case where a jet engine is being tested
on the ground. It is operating at steady-state and burning 0.25 kg/s of jet fuel. The
velocity (v) and temperature (T) profiles of the gases exiting the round nozzle have
been measured, and are found to depend only on the radial distance (r) from the
center of the nozzle – they are not dependent on the azimuthal position around the
nozzle. The measured data can be described approximated by the following
expressions: r
() r 2
T (r) = 1350K − 675K and v r = 420.m s − 105m s
R R
where R is the radius of the nozzle exit (=0.25 m for our engine). In addition, the
pressure and molecular weight of the exiting gas are 1.0 bar and 28.1 (and these
values are uniform at the exit). From the measured profiles, determine the fuel-air
ratio of the engine, i.e., the mass flow rate of fuel entering the engine divided by the
mass flow rate of air entering the engine.
, Solution for Problem Set #4: Mass Conserv.: Control Volume Analysis
Problem 1. Low Speed Wind Tunnel Test Section Exit
Given: Wind tunnel with test section
conditions shown in figure. m fan 42”
test 55 mph
72 F pexit=0.95 atm-0.2psi
0.95 atm 40”
Find: a) mass flow rate supplied by fan
b) air velocity at tunnel exit
Assume: 1) air is
ideal gas at room T and p conditions,
2) steady flow
3) uniform flow
4) exit has same area as test section
Analysis:
a) Mass Flowrate
Assuming steady conditions, 0
0= (
+ mfan − m test )
uniform flow in test section mfan = mtest = test vtest Atest
ptest
Using air is ideal gas mfan = vtest Atest
RTtest
mfan = 0.95atm 55mph (40in)(42in) 1 ft 2
3
0.7302 ft atm lbmol (460 + 72)R 60mph 144in 2
lbmol R 28.8lbm
lbm
m fan = 0.0704 3
(80.7 ft s)11.7 ft2 = 66.3lb m s
ft
b) vexit
At exit mexit = mtest
Or exit vexit Aexit = test vtest Atest
ptest RTtest p
Same T’s vexit = vtest = test v test
pexit RTexit pexit
0.95atm
vexit = 55mph = 55.8mph = 81.8 ft s
atm
0.95atm − 0.2 psi
14.7 psi
Implications: Large wind tunnels require lots of air, but if the numbers here are
accurate, the fan running them does not need to produce much of a
pressure increase.
Analysis) NEW UPDATE Georgia Institute Of Technology
• Always indicate any assumptions you make. If you use any results or equations from
the class notes or text in you solutions, please note and reference them (but you
better be sure they are applicable to the problem at hand).
• Show all your work, no credit for just answers. When applicable, try to solve the
problem algebraically first. Only use numbers/values in the final steps of your
solution – and be sure to include units when you insert numbers.
• If the problem statement is given in ENGLISH units, the answer must also be in
English units; if the problem statement is in SI units, the answer must be in SI units.
1. Low Speed Wind Tunnel
A low speed, open-return wind tunnel is being designed with a fan at the inlet, and a
test section having a rectangular cross-section of 40in. 42in. The designers want to
achieve a 55 mph wind speed, with the air in the test-section nominally at 72 ºF and
0.95 atm. How much air (mass flow rate) will the fan that runs the wind tunnel have to
supply to meet these conditions? If the pressure of the air at the exit of the wind tunnel
was 0.2 psi lower than the pressure in the test section, what would be the air velocity
at the exit, assuming the air temperature remained at 72 ºF?
2. Steady-State Aircraft Engine
In a jet engine, air enters from the front, fuel is added farther downstream in the
engine, and the fuel and air burn. Thrust is produced when the gases leave the engine
through a nozzle at high velocity. Consider a case where a jet engine is being tested
on the ground. It is operating at steady-state and burning 0.25 kg/s of jet fuel. The
velocity (v) and temperature (T) profiles of the gases exiting the round nozzle have
been measured, and are found to depend only on the radial distance (r) from the
center of the nozzle – they are not dependent on the azimuthal position around the
nozzle. The measured data can be described approximated by the following
expressions: r
() r 2
T (r) = 1350K − 675K and v r = 420.m s − 105m s
R R
where R is the radius of the nozzle exit (=0.25 m for our engine). In addition, the
pressure and molecular weight of the exiting gas are 1.0 bar and 28.1 (and these
values are uniform at the exit). From the measured profiles, determine the fuel-air
ratio of the engine, i.e., the mass flow rate of fuel entering the engine divided by the
mass flow rate of air entering the engine.
, Solution for Problem Set #4: Mass Conserv.: Control Volume Analysis
Problem 1. Low Speed Wind Tunnel Test Section Exit
Given: Wind tunnel with test section
conditions shown in figure. m fan 42”
test 55 mph
72 F pexit=0.95 atm-0.2psi
0.95 atm 40”
Find: a) mass flow rate supplied by fan
b) air velocity at tunnel exit
Assume: 1) air is
ideal gas at room T and p conditions,
2) steady flow
3) uniform flow
4) exit has same area as test section
Analysis:
a) Mass Flowrate
Assuming steady conditions, 0
0= (
+ mfan − m test )
uniform flow in test section mfan = mtest = test vtest Atest
ptest
Using air is ideal gas mfan = vtest Atest
RTtest
mfan = 0.95atm 55mph (40in)(42in) 1 ft 2
3
0.7302 ft atm lbmol (460 + 72)R 60mph 144in 2
lbmol R 28.8lbm
lbm
m fan = 0.0704 3
(80.7 ft s)11.7 ft2 = 66.3lb m s
ft
b) vexit
At exit mexit = mtest
Or exit vexit Aexit = test vtest Atest
ptest RTtest p
Same T’s vexit = vtest = test v test
pexit RTexit pexit
0.95atm
vexit = 55mph = 55.8mph = 81.8 ft s
atm
0.95atm − 0.2 psi
14.7 psi
Implications: Large wind tunnels require lots of air, but if the numbers here are
accurate, the fan running them does not need to produce much of a
pressure increase.
