Page 372 - Modelling in Transport Phenomena A Conceptual Approach

352 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

Macroscopic equation

The integration of the governing equation, Eq. (9.2-26) over the volume of the
system gives the macroscopic energy balance as

-LL1'"1 --z (rk$)rdrdOdz= 1"
d
R1
1" 12= Rr drdedz (9.2-46)
Integration of Eq. (9.2-46) yields

(9.2-47)

Rate of energy out Rate of energy generation

Equation (9.2-47) is the macroscopic energy balance under steady conditions by
considering the solid cylinder as a system. It is also possible to make use of New-
ton's law of cooling to express the rate of heat loss from the system to the sur-
roundings at T, with an average heat transfer coefficient (h). In this case &.
(9.2-47) reduces to
R
R(h) (TR -T-) = 1 Srdr (9.2-48)

Example 9.3 Rate of heat generation per unit volume, ?Re, during the transmis-
sion of an electric current through wires is given by

where I is the current, ke is the electrical conductivity, and R is the mdius of the
Wire.
a) Obtain an expression for the digerence between the maximum and the surface
temperatures of the wire.
b) Develop a correlation that will pennit the selection of the electric current and the
wire diameter if the dierence between the maximum and the surface temperatures
is specified. If the wire mwt carry a larger current, should the wire have a larger
or smaller diameter?
Solution

Assumption

1. The thennal conductivity and the electrical conductivity of the wire are
constant.