Modern Engineering Thermodynamics

8.3 Systems Undergoing Irreversible Processes 267
Solution
First, draw a sketch of the system (Figure 8.13).
T f
Fin of thermal conductivity k t
Perimeter P
x
L
T ∞
Area A
FIGURE 8.13
Example 8.12.
The unknown is the entropy production rate for the fin. The material is aluminum.
Since there is no work mode entropy production here, Eq. (7.66) is used to find the fin’s entropy production rate by the direct
method (note that we have insufficient information to use the more convenient indirect method or entropy balance here) as
Z
_S P = _S P Q = − _q
dT
T 2 dx
0
V
dV
act
Since this is a one-dimensional heat transfer problem, we can substitute Adxfor dV: Then,
Z
L
_S P Q = − _q
Z ∞
dT
k
Adx= t A dT 2
T 2 dx
T 2 dx
dx
where we have used Fourier’s law, _q = −k t ðdT/dxÞ, and have let L → ∞ for a very long fin. We can differentiate the fin’s temperature
profile given previously to obtain
dT
dx = −m T
f − T ∞ e
−mx
then
pffiffiffiffiffiffiffiffiffiffiffiffi
A dT 2 hPk t AðmÞ T f − T 2
e
−2mx
k t
=
T 2
dx T ∞ + ðT f − T ∞ Þe −mx 2
and
2
pffiffiffiffiffiffiffiffiffiffiffiffi
ð Þ Q = mT f − T ∞ hPk t A
S P
Z ∞
0
0
e −2mx dx
T ∞ + T f − T ∞ e
−mx 2
This expression can be integrated using a table of integrals and a change of variables (e.g., let y =e –mx ) to obtain
_S P Q =
p ffiffiffiffiffiffiffiffiffiffiffiffi
hPk t A ln T f
+ T
∞
− 1
T ∞ T f
In this problem, we have
h = 3:50 W/ m 2 ⋅K A = 1:00 × 10 −4 m 2
P = 0:0400 m
T ∞ = 20:0°C = 293:15 K
k t = 204 W/ ðm⋅KÞ T f = 95:0°C = 368 K
Then,
_S P = _S P Q
p
= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½3:50 W/ ðm 2 ⋅KÞŠð0:0400 mÞ½204 W/ ðm⋅KÞŠð1:00 × 10 −4 m 2 Þ ln 368
293:15 + 293:15
368 − 1
= 0:00128 W/K
Note that the entropy production rate in this example is quite small.