Page 310 - Bird R.B. Transport phenomena

294 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

Once the temperature and heat flux distributions are known, various information
about the system may be obtained:

(i) Maximum temperature rise (at r = 0)

T m a x - T o = ^ (10.2-14)

(ii) Average temperature rise
27Г [R
- т

2 rdrdO
Jo

Thus the temperature rise, averaged over the cross section, is half the maximum temper-
ature rise.
(iii) Heat outflow at the surface (for a length L of wire)

Q\ r=R = lirRL • q \ =R = 2irRL ш ^ п = irRl L ' e (10.2-16)
S
'
r r
This result is not surprising, since, at steady state, all the heat produced by electrical dis-
2
sipation in the volume TTR L must leave through the surface r = R.
The reader, while going through this development, may well have had the feeling of
deja vu. There is, after all, a pronounced similarity between the heated wire problem and
the viscous flow in a circular tube. Only the notation is different:

Tube flow Heated wire
First integration gives
Second integration gives v (r) T(r) -Т„
z
Boundary condition at r = 0 T = finite finite
rz
Boundary condition at r = R v = 0 T- - U
z i 0
Transport property k
Source term (% - 9> )/L Se
L
Assumptions /л = constant k,k = constant
e
That is, when the quantities are properly chosen, the differential equations and the
boundary conditions for the two problems are identical, and the physical processes are
said to be "analogous." Not all problems in momentum transfer have analogs in energy
and mass transport. However, when such analogies can be found, they may be useful in
taking over known results from one field and applying them in another. For example,
the reader should have no trouble in finding a heat conduction analog for the viscous
flow in a liquid film on an inclined plane.
There are many examples of heat conduction problems in the electrical industry. 1
The minimizing of temperature rises inside electrical machinery prolongs insulation life.
One example is the use of internally liquid-cooled stator conductors in very large
(500,000 kw) AC generators.

1 M. Jakob, Heat Transfer, Vol. 1, Wiley, New York (1949), Chapter 10, pp. 167-199.