Page 382 - Bird R.B. Transport phenomena

364 Chapter 11 The Equations of Change for Nonisothermal Systems

Hole at top
(at "north pole")
Solid

Insulated
surfaces

Similar hole
at "south pole"

(a) (b)
Fig. 11B.4. Heat conduction in a spherical shell: (a) cross section
containing the z-axis; (b) view of the sphere from above.

11B.5. Axial heat conduction in a wire 2 (Fig. 11B.5). A wire of constant density p moves downward
with uniform speed v into a liquid metal bath at temperature T . It is desired to find the tem-
o
perature profile T(z). Assume that Г = T x at z = °°, and that resistance to radial heat conduc-
tion is negligible. Assume further that the wire temperature is T = T at z = 0.
o
(a) First solve the problem for constant physical properties C and k. Obtain
p
pC vz
v
(11B.5-1)
(b) Next solve the problem when C and к are known functions of the dimensionless temper-
p
ature 0: /с = /СоД(0) and C p = C L(0). Obtain the temperature profile,
poc
C vz
P px
(11B.5-2)
1 Г0 = ==
L(0W0
•'О
(c) Verify that the solution in (b) satisfies the differential equation from which it was derived.

4
Temperature of wire far
from liquid metal
surface is T^

Wire moves downward
with constant speed v

Liquid metal surface
/ at temperature T
o
Fig. 11B.5. Wire moving into a liquid metal bath.

2
Suggested by Prof. G. L. Borman, Mechanical Engineering Department, University of Wisconsin.