Modern Engineering Thermodynamics

354 CHAPTER 10: Availability Analysis
Problems (*indicates problems in SI units)
1. Determine the potential for the following vector field:
! ! ! !
A = yi + xj + 0k :
2. Compute the vector field for the following potential:
P = ax + by + cz.
3. Compute the vector field for the following potential:
P = 3x 3 +2y 2 − z +7.
4. Compute the vector field for the following potential:
P = 5x 2 y/z + sin(x).
5. Coulomb’s law of the electrostatic force of attraction between
two isolated charges, Q 1 and Q 2 , separated by a distance r is
!
F = ðQ1 Q 2 /4πεr 2 ! !
Þ ir, where ir is the unit vector pointing along
the line of centers of the charges and ε is the dielectric constant
of the medium containing the charges. Show that this force has a
potential given by P = Q 1 Q 2 /4πεr:
6. Newton’s law for the gravitational force between to bodies of
mass m 1 and m 2 separated by a distance r is ! F =ðGm 1 m 2 /r 2 !
Þ i r,
!
where G is the gravitational constant and i r is a unit vector
pointing along the line of centers of the masses. Determine the
gravitational potential for this force and show that it satisfies
the Laplace equation in cylindrical coordinates,
∇ 2! !
F = ∂2 F
∂r 2 + 1 ∂ ! F
r ∂r + 1 ∂ 2!
F !
r 2 ∂θ 2 + ∂2 F
∂z 2
7.* The total energy contained in a closed rigid system is 550. kJ
and its total entropy is 2.7521 kJ/K. The ground state total
energy, total entropy, and absolute temperature of the system
are 50.0 kJ, 1.0000 kJ/K, and 273 K, respectively. Determine
the maximum reversible work this system can produce.
8.* The total energy contained in a closed, sealed, rigid can of
tuna is 3000. J with a total entropy of 2.8330 J/K. The ground
state total energy, total entropy, and temperature of the system
are 100.0 J, 0.0275 J/K, and 20.0°C, respectively. Determine
the maximum reversible work available from the can of tuna.
9. The total energy contained in a closed, sealed, rigid can of
carbonated soda is 10.0 Btu with a total entropy of 0.0739
Btu/R. The ground state total energy, total entropy, and
temperature of the system are 1.00 Btu, 0.0619 Btu/K, and
70.0°F, respectively. Determine the maximum reversible work
available from the can of soda.
10.* An inventor claims to have developed a new closed, sealed
battery that can produce a maximum reversible work of 3.80
kJ. The total energy contained in the battery is 2.00 kJ with a
total entropy of 7.5150 J/K. If the ground state total energy
and total entropy are 100. J and 0.5660 J/K, respectively, what
ground state temperature is required to meet the inventor’s
maximum reversible work claim?
11. As a designer, you are required to develop a new closed, sealed
thermal energy storage cell that has a maximum reversible work
output of 500. Btu. The ground state total energy, total entropy,
and temperature are 5.00 Btu, 0.11690 Btu/R, and 50.0°F,
respectively. If the total entropy of the cell must be ten times its
ground state value, what should the total energy of the cell be?
12. An open bucket containing 30.0 lbm of liquid water at 70.0°F
is sitting 6 ft above the floor on a ladder. Determine the total
availability of the water in the bucket relative to the floor. The
local environment is at 14.7 psia and 70.0°F.
13.* An open bucket containing 14.0 kg of liquid water at 20.0°C is
spun on a rope in a horizontal plane 2.00 m above the floor
with a tangential velocity of 5.00 m/s. Determine the total
availability of the water in the bucket relative to the floor. The
local environment is at 0.101 MPa and 20.0°C.
14.* Determine the total availability of a 7.00 × 10 −3 kg
incompressible lead bullet traveling vertically at 1000. m/s at a
height of 50 m above the ground. The temperature of the bullet
is 150.°C and its specific heat is 0.167 kJ/kg·K. The local
environment is at 0.101 MPa and 20.0°C.
15.* A stationary tethered balloon contains helium gas (an ideal
gas here) at 0.00°C and 0.0700 MPa at a height of 1000. m.
Determine the specific availability of the helium in the
balloon relative to the ground, where the local environment is
at p 0 = 0.101 MPa and T 0 = 20.0°C.
16.* The air (an ideal gas here) in the ballast tanks of a submarine is
at 10.0°C and 1.50 MPa when the submarine is cruising at 3.00
m/s, 100. m below sea level. Determine the specific availability
of the air in the ballast tanks relative to sea level where the local
environment is at 0.101 MPa and 20.0°C.
17. Integrate Eq. (7.52) and use Eqs. (10.13a) and (10.14a) to
determine the irreversibility and total availability change for
an aergonic closed system in which the temperature increases
from T 1 = 70.0°F toT 2 = 200.°F for the cases where the heat
transfer varies with the system absolute temperature according
to the relationships
a. Q = K 1 T (convection).
b. Q = K 2 T 4 (radiation)
where K 1 = 3.70 Btu/R and K 2 = 5.40 × 10 −4 Btu/R 4 .The
system boundary is maintained isothermal at 350.°F and the
local environment is at 14.7 psia and 70.0°F.
18. The temperature distribution due to conduction heat transfer
inside a flat plate with an internal heat generation is given by
T = T 0 +(T s − T 0 )(x/L) 2 , where T s is the surface temperature at
x = L,andT 0 is the centerline temperature at x = 0 (Figure 10.23).
Use Eqs. (7.66) and (10.13b) to determine a formula for the
steady state irreversibility rate for this system.
T s
FIGURE 10.23
Problem 18.
19. A current of 100. A is passed through a 6.00 ft long stainless
steel wire 0.100 inch in diameter. The electrical resistivity of
the wire is 1.97 × 10 −5 Ω ·in, and its thermal conductivity is
x
L
T s