Modern Engineering Thermodynamics

Problems 165
where v, T, and r are the specific volume, temperature, and
radius of the star.
a. Show that the collapse process is a polytropic process with
n = 4/3.
b. Beginning with the per unit mass differential form of the
first law, find an expression for the star’s heat transfer as a
function of its specific heats and temperature. Note that the
star does p V work on itself as it collapses.
c. Using the expression found in part b, along with c v =
0.200 Btu/(lbm · R) and k = 1.4, calculate the amount of
heat transfer per unit mass of star as its temperature changes
from 5000. to 10,000. R.
d. Explain the critical condition that exists when the specific
heat ratio k takes on the value of 4/3.
Computer Problems
The following computer problems are designed to be completed
using a spreadsheet or equation solver. They may be used as part of a
weekly homework assignment.
57. Develop a computer program that performs an energy balance on
a closed system containing an incompressible substance (either a
liquid or a solid). Include the following input (in proper units):
the heat and work transports of energy, the system volume, the
initial temperature of the system, and the density and specific
heat of the incompressible material contained in the system.
Output to the screen the system mass and final temperature.
58. Develop a computer program that performs an energy balance on a
closed system containing an ideal gas with constant specific heats.
Include the following input (in proper units): the heat and work
transports of energy, the system volume, the initial temperature
and pressure of the system, and the constant volume specific heat
and gas constant of the gas contained in the system. Output to the
screen the system mass and the final pressure and temperature.
59. Repeat Problem 58, except allow the user to choose the system
ideal gas from a screen menu, and omit the prompts for gas
properties. Use the data in Table C.13 of Thermodynamic Tables to
accompany Modern Engineering Thermodynamics for the properties
of the gases in your menu.
60. Develop a computer program that generates data to allow you
to plot on the computer the explosive energy contained in a
1000. ft 3 pressure vessel containing compressed air vs.
a. The vessel’s initial temperature when the initial pressure is
held constant at 100. psia.
b. The vessel’s initial pressure when the initial temperature is
held constant at 80.0°F.
c. Create a three-dimensional plot with the explosive energy on
the vertical axis and the initial pressure and initial
temperature on the horizontal axes.
Assume the final temperature and pressure of the air after the
vessel has ruptured are 70.0°F and14.7psiaineachcaseand
that the air undergoes a polytropic decompression process
with n = 1.25 when the vessel ruptures. Also assume that
the air behaves as a constant specific heat ideal gas with
k =1.40.
61. A white dwarf is a spherical mass of gas in outer space. Its radial
pressure gradient must always be in equilibrium with its own
gravitational force field, or
dp
dr = − Gmρ/r2
where G is the gravitational constant, ρ is the density of the gas
at radius r (i.e., ρ = ρ(r)), and m is the mass of gas inside a
sphere of radius r,
m = 4π
Z r
0
ρr 2 dr
During its formation, the gas of a white dwarf obeys the
polytropic equation
pv −5/3 = α = constant
These relations can be combined to yield a differential equation
for the density field ρ = ρ(r) inside a white dwarf of the form
d 2 ϕ
dx 2 + 2
dϕ
+ ϕ 2/3 = 0
x dx
where ϕ = ðρ/ρ 0 Þ 2/3 , ρ 0 = ρ(r = 0), and x = r/r*, where
r = 5α/ 8πGρ 1/3
0 .
a. Solve the preceding differential equation for ϕ(x) using a
computer numerical solution with the boundary conditions
ϕ = 1 and dϕ/dx =0atx = 0 to show that ϕ(x) =0at
x = 3.6537.
b. Show that ϕ(x) = 0 corresponds to the radius R of the white
dwarf and a white dwarf therefore has a mass m given by
m = − 45:91ρ 0 ðr Þ 3 ðdϕ/dxÞj x=R .