Page 418 - Bird R.B. Transport phenomena

400 Chapter 12 Temperature Distributions with More Than One Independent Variable

(b) A typical timetable for roasting turkey at 350°F is 4
Mass of turkey Time required per unit mass
(min/lb m)

6-10 20-25
10-16 18-20
18-25 15-18
Compare this empirically determined cooking schedule with the results of part (a), for geo-
metrically similar turkeys at an initial temperature T o , cooked with a given surface tempera-
ture Tj to the same dimensionless temperature distribution S = ®(f, 77, £).
12B.8. Use of asymptotic boundary layer solution. Use the results of Ex. 12.4-2 to obtain 8 r and q 0
for the system in Problem 12D.4. By comparing 8 T with D, estimate the range of applicability
of the solution obtained in Problem 12D.4.
12B.9. Non-Newtonian heat transfer with constant wall heat flux (asymptotic solution for small axial
distances). Rework Example 12.2-2 for a fluid whose non-Newtonian behavior is described ade-
quately by the power law model. Show that the solution given in Eq. 12.2-2 may be taken over
for the power law model simply by an appropriate modification in the definition of v 0.
12C.1. Product solutions for unsteady heat conduction in solids.
(a) In Example 12.1-2 the unsteady state heat conduction equation is solved for a slab of
thickness 2b. Show that the solution to Eq. 12.1-2 for the analogous problem for a rectangular
block of finite dimensions 2a, 2b, and 2c may be written as the product of the solutions for
three slabs of corresponding dimensions:
Т, - Т(х, у, z, О
(12C.1-1)

2
in which 0(у/Ь, at/b ) is the right side of Eq. 12.1-31.
(b) Prove a similar result for cylinders of finite length; then rework Problem 12A.4 without
the assumption that the cylinder is infinitely long.

12C.2. Heating of a semi-infinite slab with variable thermal conductivity. Rework Example 12.1-1
for a solid whose thermal conductivity varies with temperature as follows:

(12C.2-1)
— T
1
in which k is the thermal conductivity at temperature T , and /3 is a constant. Use the follow-
0
o
ing approximate procedure:
(a) Let 0 = (T — T )/(Ti - T ) and 77 = y/8(t), where 8(t) is a boundary layer thickness that
o
Q
changes with time. Then assume that
0(y, t) = (12C.2-2)
in which the function Ф(т}) gives the shapes of the "similar" profiles. This is tantamount to as-
suming that the temperature profiles have the same shape for all values of /3, which, of
course, is not really true.
(b) Substitute the above approximate profiles into the heat conduction equation and obtain
the following differential equation for the boundary layer thickness:
, сч d 8 л т
л (12C.2-3)
M8 — = a N
0
Woman's Home Companion Cook Book, Garden City Publishing Co., (1946), courtesy of Jean Stewart.