AE6050/Seitzman
Summer 2013
HW#1
Due Tue., May 28
Review Problems:
Ideal Gas Dynamics, Kinetic Theory, Statistical Mechanics
 Homework solutions should be neat and logically presented, see format requirements
(www.ae.gatech.edu/people/jseitzma/classes/ae6050/homeworkformat.html). Please
note the requirement to draw some IMPLICATIONS/CONCLUSIONS from the results
of each problem. This could be implications for a practical device, a comparison of the
results of different parts of a problem, a physical interpretation for a mathematical
expression, etc.
To receive credit, you must show ALL work in the format described above. If you use
equations from the notes, the class textbook or another book, please cite the reference.
1. Isentropic Nozzle Expansion
a) Write a complete set of expressions (you DO NOT need to derive them, just
cite your source) for a thermally and calorically perfect, nonreacting gas as it
isentropically expands through a supersonic nozzle that relate the temperature
(T), pressure (p), density () and Mach number (M) of the flow to the local
cross-sectional area (A), the throat area (At), and any necessary upstream
reservoir conditions. Make sure to include any additional assumptions that
these expressions use (i.e., assumptions that are not specifically given to you
in this problem – those assumptions belong in the Givens).
b) Plot (not by hand!) T, p,  and M vs. A/At for a mixture that is 70% He and 30%
O2 by mole through such a nozzle as it expands from a reservoir (M~0) to an
exit Mach number of 6 with reservoir conditions of 800 K and 30 atm.
2. Normal Shocks
a) Write expressions (again, no need to derive them, just cite your source) that
give the temperature ratio, T2 T1 , pressure ratio, p2 p1 , density ratio  2 1 ,
and normalized entropy increase, s2  s1  R , for a normal shock in a thermally
and calorically perfect, nonreacting gas as a function of the shock Mach
number, M 1 . Again, be sure to include any additional assumptions used.
b) Plot the 1) temperature ratio, 2) pressure ratio, 3) density ratio, 4) normalized
entropy increase and 5) velocity ratio for normal shocks with Mach numbers up
to at least 10 using the gas mixture described in Problem 1.
3. Collision Frequency
If the rate at which a single carbon dioxide (CO2) molecule experiences elastic
collisions (i.e., collisions where only momentum or translational energy is
exchanged) with all other molecules in a pure CO2 gas is 1011 sec-1 at 300 K and
1 atm, what will the collision rate be if you keep the density constant, but triple
the pressure? (Don’t forget to include assumptions made in your analysis.)
ALSO, determine the collision cross-section (in units of cm2) based on the given
collision frequency.
4. Translational Energy
Determine the fraction of molecules with a translational kinetic energy greater
than 5 times the average translational kinetic energy for a thermally perfect gas.
5. Equilibrium Enthalpy
Consider a diatomic gas that can be modeled as a rigid rotor (with r=2.5K) and
harmonic oscillator (with v=2500K), and with three electronic energy levels
(g0=1; g 1=3 1=600 K; g2=5, 2=10,000 K).
a) Plot the normalized enthalpy (h/R) as a function of temperature (from 200 K to
4000 K) for this gas.
b) Using the standard thermodynamic definition of specific heat at constant
pressure (cp) for a thermally perfect gas AND a numerical approximation for cp
obtained from the results of part (a) of this problem, plot cp for this gas (again
from 200 K to 4000 K).
6. Specific Heat and Fully-Excited Conditions
a) Derive an expression for the normalized specific heat (cv/R) of a perfect gas
using the most general form of the partition function, i.e.,
n
Q  g 0   g i e i
T
i 1
where the lowest energy level (0) has zero energy, and i is the characteristic
temperature of the energy level (=i/k). Your answer will contain various
series summations.
SUGGESTION: you can start with the expression given in class for E as a
function of (lnQ)/T OR you can start with simpler expression:
E   Ni i  N  f i i
b) Use the result from (a) to plot cv,rot/R versus T/r for a linear rigid rotor starting
from ~0K and extending at least until the rotational energy mode is “fullyexcited”. Assume J=0 is the ground rotational energy level.
Also plot the value of Q as a function of T/r. (NOTE: you CAN NOT use the
assumption that Q=T/r.)
c) At what temperatures would you say that the rotational mode is fully excited
for the following gases: 1) H2, 2) He and 3) CO2.