Page 348 - Bird R.B. Transport phenomena

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

T = ambient air Fig. 10D.1. Circular fin on a
a
temperature Temperature T = T at r = R 0
o
/

10D.1. Heat loss from a circular fin (Fig. 10D.1).
(a) Obtain the temperature profile T(r) for a circular fin of thickness IB on a pipe with outside
wall temperature T . Make the same assumptions that were made in the study of the rectan-
o
gular fin in §10.7.
(b) Derive an expression for the total heat loss from the fin.
10D.2. Duct flow with constant wall heat flux and arbitrary velocity distribution.
(a) Rework the problem in §10.8 for an arbitrary fully developed, axisymmetric flow velocity
distribution v /v zmax = ф(£), where f = r/R. verify that the temperature distribution is given by
z
(10D.2-1)

in which
1(0 = (10D.2-2)

1
Show that Q = 0 and C o = [/(I)]" . Then show that the remaining constant is
2
c = -umv (10D.2-3)
2
Verify that the above equations lead to Eqs. 10.8-27 to 30 when the velocity profile is parabolic.
These results can be used to compute the temperature profiles for the fully developed
tube flow of any kind of material as long as a reasonable estimation can be made for the ve-
locity distribution. As special cases, one can get results for Newtonian flow, plug flow, non-
Newtonian flow, and even, with some modifications, turbulent flow (see §13.4). 6
(b) Show that the dimensionless temperature difference driving force @ 0 - S b is

(10D.2-4)

(c) Verify that the dimensionless wall heat flux is

(10D.2-5)
- T b) 0 O -
k(T 0 ® b
and that, for the laminar flow of Newtonian fluids, this quantity has the value ff.
(d) What is the physical interpretation of Д1)?

6 R. N. Lyon, Chem. Engr. Prog., 47, 75-59 (1951); note that the definition of ф{£) used here is different
from that in Tables 14.2-1 and 2.