Modern Engineering Thermodynamics

142 CHAPTER 4: The First Law of Thermodynamics and Energy Transport Mechanisms
27. Show that the first law of thermodynamics requires that, for an
ideal gas with a constant specific heat ratio c p /c v = k undergoing
a polytropic process (i.e., pv n = constant),
a. n must be greater than k for T 2 < T 1 when there is heat
transfer from the gas.
b. n must be less than k for T 2 < T 1 when there is a heat
transfer to the gas.
28. Find the moving boundary work done on a gas in compressing
it from V 1
= 10:0ft 3 , p 1 = 10:0 psia to V 2
= 1:000 ft 3 according
to the relation p V 3 = constant (Figure 4.30).
Piston
Gas
FIGURE 4.30
Problem 28.
Force
State 1
V 1 = 10.0 ft 3
p 1 = 10.0 psia
pV 3 = constant
Force
State 2
V 2 = 1.000 ft 3
29.* A brilliant young engineer claims to have invented an engine
that runs on the following thermodynamic cycle:
a. An isochoric pressurization from p 1 to p 2 = ∠p 1 .
b. An isobaric expansion from V 2
to V 3
= 2V 2
:
c. An isochoric depressurization from p 3 to p 4 = p 1 .
d. An isobaric compression back to the initial state, p 1 , V 1
:
Determine the net moving boundary work done during this
cycle if p 1 = 25.0 kPa and V 1
= 0:0300 m 3 : Sketch this cycle
on a p − V diagram.
30.* A balloon filled with air at 0.100 MPa-absolute is heated in
sunlight. As the balloon is heated, it expands according to the
following pressure-volume relation:
p = 0:1 + 0:15V + 0:06V 2
where p is in MPa and V is in m 3 (Figure 4.31). Determine the
moving boundary work transport of energy as the balloon
expands from 1.00 to 2.00 m 3 .
31. One lbm of an ideal gas with molecular weight 6.44 lbm/
lbmole is compressed in a closed system from 100. psia, 600.
R to a final specific volume of 8.00 ft 3 /lbm. At all points
during the compression, the pressure and specific volume are
related by
p = 50 + 4v + 0:1v 2
where p is in psia and v is in ft 3 /lbm. Determine the moving
boundary work required and the heat transfer during this
compression if the gas has a constant volume specific heat of
0.200 Btu/(lbm · R).
32. Three lbm of a substance is made to undergo a reversible
expansion process within a piston-cylinder device, starting from
an initial pressure of 100 psia and an initial volume of 2.00 ft 3 .
The final volume is 4.00 ft 3 . Determine the moving boundary
work produced by this expansion for each of the following
process paths. Note which process produces the maximum work
and which produces the minimum.
a. Pressure remains constant (p = K)
b. Pressure times volume remains constant ðpV = KÞ:
c. Pressure is proportional to volume ðp = KVÞ:
d. Pressure is proportional to the square of volume ðp = KV 2 Þ:
e. Pressure is proportional to the square root of volume
q
ðp = K
ffiffiffi
V Þ, where K is a constant in each case.
33.* The magnitude of the torque T on a shaft is given in
N ·m by
T = 6:3 cos θ
where θ is the angular displacement. If the torque and
displacement vectors are parallel, determine the work required to
rotate the shaft through one complete revolution.
34. The magnitude of the torque vector normal to the axis of a shaft
is given in ft ·lbf by
T = 21:7 sin θ for 0 < θ ≤ π
= 0 for π < θ ≤ 3π/2
= 50:4 for 3π/2 < θ ≤ 2π
Determine the work done in one complete revolution of the
shaft.
35. When the torque and angular displacement vectors are parallel,
the torque displacement relation for the drive shaft of a 1909
American Underslung automobile is given by
Tθ n = K
FIGURE 4.31
Problem 30.
State 1
p 1 = 0.100 MN/m 2
V 1 = 1.00 m 3
p = f(V)
State 2
V 2 = 2.00 m 3
where K and n are constants. Determine a general formula for
the shaft work when (a) n = 1.0, and (b) n ≠ 1.0.
36. How much elastic work is done in uniaxially stretching an
initially unstrained elastic steel bar (Young’s modulus = 3.0 ×
10 7 psi = constant) whose volume (also a constant) is 5.00 in 3
to a total strain of 0.00200 in/in?
37. When a rubber band is stretched, it exerts a restoring force (F)
that is a function of its initial length (L) and displacement (x).
For a certain rubber band this relation is
F = K x
L + x 2
,
L