Page 311 - Bird R.B. Transport phenomena

§10.2 Heat Conduction with an Electrical Heat Source 295

To illustrate further problems in electrical heating, we give two examples concern-
ing the temperature rise in wires: the first indicates the order of magnitude of the heating
effect, and the second shows how to handle different boundary conditions. In addition,
in Problem 10C.2 we show how to take into account the temperature dependence of the
thermal and electrical conductivities.

EXAMPLE 10.2-1 A copper wire has a radius of 2 mm and a length of 5 m. For what voltage drop would the
temperature rise at the wire axis be 10°C, if the surface temperature of the wire is 20°C?
Voltage Required for a
Given Temperature Rise QQJJJJIQJSI
in a Wire Heated by an
Electric Current Combining Eq. 10.2-14 and 10.2-1 gives

max (10.2-17)
° 4kk e
The current density is related to the voltage drop E over a length L by

(10.2-18)
I = K
L
Hence
2
_ T _ E R 2 (K (10.2-19)
° 4L U
2
from which

(10.2-20)
2
2
8
For copper, the Lorenz number of §9.5 is k/k T = 2.23 X 10~ volt /K . Therefore, the voltage
0
e
drop needed to cause a 10°C temperature rise is
E = ( 5 0 0 0 m m \ - vglt 93)(10)K
2 V 2 2 3 x l 0 8 V(2
\ 2 mm / К
4
= (5000X1.49 X 10~)(54.1) = 40 volts (10.2-21)
EXAMPLE 10.2.2 Repeat the analysis in §10.2, assuming that T is not known, but that instead the heat flux at
o
the wall is given by Newton's "law of cooling" (Eq. 10.1-2). Assume that the heat transfer co-
Heated Wire with efficient h and the ambient air temperature T are known.
air
Specified Heat Transfer
Coefficient and SOLUTION I
Ambient Air e
Temperature ^ solution proceeds as before through Eq. 10.2-11, but the second integration constant is de-
termined from Eq. 10.1-2:

B.C. 2': (10.2-22)

2
Substituting Eq. 10.2-11 into Eq. 10.2-22 gives C = (S R/2h) + {S R /4k) + T , and the tem-
e
air
2
e
perature profile is then
(10.2-23)
4k
From this the surface temperature of the wire is found to be T + S R/2h.
air e