Page 309 - Bird R.B. Transport phenomena

§10.2 Heat Conduction with an Electrical Heat Source 293

Fig. 10.2-1. An electrically heated wire, show-
ing the cylindrical shell over which the energy
Uniform heat balance is made,
I I production
by electrical
heating

4r\r I I
I Heat in by | Heat out by
I conduction I conduction

Ar-H h^-

This is a first-order differential equation for the energy flux, and it may be integrated to give

S/ C, (10.2-7)
~2 + T

The integration constant Q must be zero because of the boundary condition that
B.C. 1: at r = 0, q is not infinite (10.2-8)
r
Hence the final expression for the heat flux distribution is

S r
e
(10.2-9)
This states that the heat flux increases linearly with r.
We now substitute Fourier's law in the form q r = —k(dT/dr) (see Eq. B.2-4) into
Eq. 10.2-9 to obtain

(10.2-10)
dr~ 2
When к is assumed to be constant, this first-order differential equation can be integrated
to give
(10.2-11)

The integration constant is determined from
B.C. 2: at r = R, T = T o (10.2-12)

2
Hence C = (S R /4fc) + T and Eq. 10.2-11 becomes
e
o
2
(10.2-13)
Equation 10.2-13 gives the temperature rise as a parabolic function of the distance r from
the wire axis.