Page 490 - Advanced thermodynamics for engineers

482 CHAPTER 20 IRREVERSIBLE THERMODYNAMICS

l

Q
(J ) T (J ) T+ T J I
Q

J' Q
FIGURE 20.6
Small element of wire subject to current flow.
Let the required heat transfer, to a series of reservoirs, be J . Under these conditions, the heat flow due
0
Q
to heat conduction is the same as previously.
Then
0
J ¼ J Q TþDT J Q T þ J I Dε (20.61)
Q
|fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflffl{zfflffl} |ffl{zffl}
Heat flow rate; dQ=dt; Heat flow rate; Ohmic heating
at TþDT dQ=dt; at T
At any given temperature, T, the heat of transport, Q*,is(Eqn (20.47))

J Q

Q ¼ ¼ TS
J I
T
and hence from Eqn (20.47)

J Q ¼ J I TS (20.62)
T
At a temperature, T þ DT

dS

J Q ¼ J I ðT þ DTÞ S þ DT
TþDT dT
(20.63)
TdS

¼ J I TS þ DT þ S DT if multiples of small terms are neglected
dT
Now, for the element
dε
Dε ¼ D‘ (20.64)
d‘
and
dT
DT ¼ D‘ (20.65)
d‘
Equation (20.38) can be rearranged to give
dε J I dT
¼ S (20.66)
d‘ l d‘