Modern Engineering Thermodynamics

266 CHAPTER 8: Second Law Closed System Applications
EXAMPLE 8.11 (Continued )
Exercises
20. Rework part c in Example 8.11 with an immersion time of 1.00 s rather than 5.00 s. Note that T 2 now becomes 130.5°C
rather than 32.5°C. Keep the values of all the other variables the same as in the example. Answer: 1 (S P ) 2 = 0.0192 J/K.
21. Determine 1 (S P ) 2 in Example 8.11 when T 4 = 30.0°C rather than 15.0°C. With this value of T 4 , the value of T 2 changes
from 32.5°C to 136°C. Use this new value of T 2 but keep the values of all of the other variables the same as in the
example. Answer: 1 (S P ) 2 = 0.0845 J/K.
22. Plot 1 (S P ) 2 vs. immersion time t over the range 0 ≤ t ≤ 30.0 s for the process described in Example 8.11. Note that
T 2 varies between 15.0°C ≤ T 2 ≤ 200.°C overthisrangeoft. Do this by using commercial equation solver software or by
choosing several values of t in the range 0 ≤ t ≤ 30.0 s and carry out the calculations and plot the results with a spreadsheet.
Answer: Your results should look like Figure 8.12.
Entropy production, J/K
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
0
Entropy production
Temperature T 2
5 10 15 20
Time, seconds
200
175
150
125
100
75
50
25
0
25 30
T 2 ,°C
FIGURE 8.12
Example 8.11, Exercise 22.
The next three examples illustrate the use of the direct method of determining the entropy production. This
method is usually more difficult than the entropy balance or indirect method, because it requires detailed information
about local property values (i.e., properties at each point inside the system boundary) and the integration
of Eq. (8.6) or (8.7) is often quite difficult. The formulae for the work mode entropy production per unit time
per unit volume (σ W ) are given in Chapter 7 for viscous and electrical work mode irreversibilities.
EXAMPLE 8.12
The heat transfer rate from a very long fin of constant cross-section is given by
p
_Q =
ffiffiffiffiffiffiffiffiffiffiffiffi
hPk t A T f − T ∞
where h is the convective heat transfer coefficient, P is the perimeter of the fin in a plane normal to its axis, k t is the thermal
conductivity of the fin, and A is the cross-sectional area of the fin (again in a plane normal to its axis). T ∞ is the temperature
of the fin’s surrounding (measured far from the fin itself) and T f is the temperature of the foot (or base) of the fin. The temperature
profile along the fin is given by
where
Tx ðÞ= T ∞ + T f − T ∞ e
−mx
m =
hP
1/2
k t A
The fin is attached to an engine whose surface temperature is 95.0°C. Determine the entropy production rate for the fin if
it is a very long square aluminum fin, 0.0100 m on a side, in air at 20.0°C. The thermal conductivity of aluminum is
204 W/(m ·K) and the convective heat transfer coefficient of the fin is 3.50 W/(m 2·K).