Page 308 - Bird R.B. Transport phenomena

292 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow
d. At a solid-fluid interface, the normal heat flux component may be related to the
difference between the solid surface temperature T o and the "bulk" fluid temper-
ature T b:
q = h(T 0 - T b) (10.1-2)

This relation is referred to as Newton's law of cooling. It is not really a "law" but
rather the defining equation for h, which is called the heat transfer coefficient.
Chapter 14 deals with methods for estimating heat-transfer coefficients.

All four types of boundary conditions are encountered in this chapter. Still other kinds
of boundary conditions are possible, and they will be introduced as needed.

§10.2 HEAT CONDUCTION WITH AN
ELECTRICAL HEAT SOURCE
The first system we consider is an electric wire of circular cross section with radius JR and
1 1
electrical conductivity k e ohm" cm" . Through this wire there is an electric current with
2
current density / amp/cm . The transmission of an electric current is an irreversible
process, and some electrical energy is converted into heat (thermal energy). The rate of
heat production per unit volume is given by the expression

S e = £ (10.2-1)

The quantity S e is the heat source resulting from electrical dissipation. We assume here
that the temperature rise in the wire is not so large that the temperature dependence of
either the thermal or electrical conductivity need be considered. The surface of the wire
is maintained at temperature T o. We now show how to find the radial temperature distri-
bution within the wire.
For the energy balance we take the system to be a cylindrical shell of thickness Ar
and length L (see Fig. 10.2-1). Since v = 0 in this system, the only contributions to the en-
ergy balance are
Rate of heat in
across cylindrical (27rrL)q r\,) = (2irrLqX (10.2-2)
surface at r
Rate of heat out
across cylindrical (2тг(г + kr)L){q\ ) = (2irrLq)\ Ar (10.2-3)
r
r r+Ar
r+
surface at r + Ar
Rate of thermal
energy production by (2irr^rL)S e (10.2-4)
electrical dissipation
в
The notation q means "heat flux in the r direction/' and (• • )|,+д means "evaluated at
г
r
r + Ar." Note that we take "in" and "out" to be in the positive r direction.
We now substitute these quantities into the energy balance of Eq. 9.1-1. Division by
2vrLAr and taking the limit as Ar goes to zero gives

Ar
The expression on the left side is the first derivative of rq with respect to r, so that Eq.
r
10.2-5 becomes
j{rq ) = S r (10.2-6)
r e