Modern Engineering Thermodynamics

270 CHAPTER 8: Second Law Closed System Applications
EXAMPLE 8.13 (Continued )
production rate _S P vs. shaft angular velocity ω as ω varies from 0 to 10,000. rpm (Figure 8.16).
Answer: _S P = 133W/K:
FIGURE 8.16
Example 8.13, Exercise 25.
Bearing entropy production rate, W/K
250.
200.
150.
100.
50.
0
0
2,000. 4,000. 6,000. 8,000.
Engine rpm
10,000.
EXAMPLE 8.14
An electrical circuit board contains a variety of digital logic elements. When operating, the board draws 10.0 mA at 5.00 V
dc and it has a steady state surface temperature of 30.0°C. Estimate the entropy production rate of the circuit board.
Solution
First, draw a sketch of the system (Figure 8.17).
T = 30.0°C
Circuit
board
(S P ) W = ?
i = 10.0 mA
v = 5.00 V dc
FIGURE 8.17
Example 8.14.
The unknown is entropy production rate of the circuit board. The system is the circuit board.
Since nearly all the electrical energy entering the circuit board is being converted into heat, we could calculate the entropy
production rate of the board from its heat loss characteristics (convective heat transfer coefficient, surface temperature, environmental
temperature, surface area, etc.); however, none of this information is supplied in the problem statement. We
could also calculate the entropy production rate for electrical work mode dissipation directly from Eq. (7.73) if we knew
how the current density J e , electrical resistivity ρ e , and local internal temperature T are distributed throughout the circuit
board. But we do not know this information either. However, the problem statement asks for only an estimate of the
entropy production rate, and we can obtain this from the special electrical work mode dissipation Eq. (7.74) if we lump all
the components on the board into one uniform, isothermal, constant current density system. From Ohm’s law, ϕ = IR e ,we
can write Eq. (7.74) as
_S P
W = I2 R e
T
= ϕI
T = ð5:00
VÞ
ð 10:0 × 10−3 AÞ
30:0 + 273:15 K
= 1:65 × 10 −4 W/K
This is only a lumped parameter estimate of the entropy production rate for this system. The actual entropy production rate
is somewhat larger due to the nonuniform distribution of entropy-producing electrical components within the system
volume.