Page 480 - Advanced thermodynamics for engineers

472 CHAPTER 20 IRREVERSIBLE THERMODYNAMICS

20.5 THE CALCULATION OF ENTROPY PRODUCTION OR ENTROPY FLOW
In Section 20.2, the concept of entropy flow was introduced. At any point in the bar, [, the entropy flux,
J S may be defined as the entropy flow rate per unit cross-sectional area. At [, the entropy flow rate will
0
be the following:
Q
dS d 1 dQ
¼ ¼ (20.15)
dt dt T T dt
and the entropy flux is
1 dS 1 1 dQ
J S ¼ ¼ (20.16)
A dt A T dt
The heat flow rate J Q is defined as
dQ=dt
J Q ¼ (20.17)
A
Hence

J Q
J S ¼ (20.18)
T
The rate of production of entropy per unit volume, q,is
d dS d ðdS=dtÞ d 1 dS
q ¼ ¼ ¼
dV dt d‘ A d‘ A dt
i.e.

dJ S
q ¼ ðfrom Eqn ð20:16ÞÞ (20.19)
d‘
Substituting from Eqn (20.18) for J S gives
d J Q J Q dT
q ¼ ¼ (20.20)
2
d‘ T T d‘
Equation (20.20) is the rate of production of entropy per unit volume at point [ in the rod, where the
temperature is T. This equation was derived for thermal conduction only.
dT
If the rod was at a uniform temperature, i.e. in equilibrium, ¼ 0and q ¼ 0. For conduction
d‘
dT
< 0thus q is positive, i.e. entropy is produced not dissipated.
d‘
A similar calculation maybe madefortheflow of electricityalongawire. Thewire maybe considered
to be in contact along its length with a reservoir at temperature T. If an electric current density, J I (¼ I/A),
flows due to a potential difference dε, and since the wire is at constant temperature T because of its
contact with the reservoir, then the electrical work must be equal to the heat transferred from the wire.
_
The rate of doing electrical work (power) is J I A dε, and the total rate of heat production is Q.
Hence
_
Q ¼ J I A dε (20.21)