Transient, One Dimensional Heat Conduction In A Convectively Cooled Sphere

3867 words - 16 pages

Transient, One-Dimensional Heat Conduction
in a Convectively Cooled Sphere
Gerald Recktenwald∗
March 16, 2006†

1

Overview

This article documents the numerical evaluation of a well-known analytical
model for transient, one-dimensional heat conduction. The physical situation
is depicted in Figure 1. A sphere of uniform material is initially at a uniform
temperature Ti . At time t = 0 the sphere is immersed in a stream of moving
fluid at some different temperature T∞ . The external surface of the sphere exchanges heat by convection. The local heat flux from the sphere to the fluid
is
q = h(Ts − T∞ )
(1)
where h is the heat transfer coefficient, and Ts is the local surface temperature.
The problem ...view middle of the document...

g. [1, 2]. A universal solution is obtained in terms of the dimensionless variables
θ∗ =

T − T∞
,
Ti − T∞

r∗ =

r
,
ro

Fo =

αt
.
2
ro

(5)

The dimensionless form of the boundary condition in Equation (3) is
∂θ∗
∂r∗


= Bi θs

(6)

r ∗ =1

where the Biot number is

hro
(7)
k

and θs = (Ts − T∞ )/(Ti − T∞ ). If Bi
1 the internal temperature gradient is
small compared to the scaled difference between the surface temperature and
the fluid. In that case the temperature in the sphere is spatially uniform.
The analytical solution is the infinite series
Bi =


2
Cn exp(−ζn Fo)

θ=
n=1

where
Cn =

1
sin(ζn r∗ )
ζn r ∗

4 [sin(ζn ) − ζn cos(ζn )]
.
2ζn − sin(2ζn )

Copyright c 2011, Gerald Recktenwald. All rights reserved.

(8)

(9)

3

20
f(ζ)

0

−20
0

5

10

15

20
f(ζ)

0

−20
0

10

20

ζ

30

40

50

Figure 2: Plots of f (ζ ) = 1 − ζ cot(ζ ) − Bi for Bi = 5 showing roots (*) and
singularities (o) of Equation (10). Upper plot shows the range 0 ≤ ζ ≤ 15.
Lower plot shows the range 0 ≤ ζ ≤ 50.
The ζn are the positive roots of
1 − ζn cot(ζn ) = Bi.

(10)

Equations (8) through (10) provide a compact representation of the solution.
Obtaining numerical values from these formulas is a nontrivial effort except in
the case where Fo
1 when a one term approximation is sufficient. In general,
at least the first few terms terms in the series are needed.

3

Evaluating the Solution in Matlab

In this section, a procedure for evaluating Equation (8) is presented. The Matlab programs, or (for long programs) the function prologues, are listed in the
appendix.

3.1

Finding the Roots of Equation (10)

The first task is to find the roots, ζn , of Equation (10) for a given Bi. This is a
standard root-finding problem, with two important complications. First, there
are an infinite number of ζn on the positive real line. To evaluate Equation (8)
a large number of roots must be available and sorted in order of increasing
ζn . A missing root will cause erroneous evaluation of the series. The second
complication is that Equation (10) has a singularity between successive roots.

Copyright c 2011, Gerald Recktenwald. All rights reserved.

3.1

Finding the Roots of Equation (10)

4

Figure 2 shows the function f (ζ ) = 1 − ζ cot(ζ ) − Bi for Bi = 5 in the range
0 ≤ ζ ≤ 50. Note that f (ζ ) = 0 when ζ is a root of Equation (10). The top
half of Figure 2 shows f (ζ ) (for Bi = 5) in the range 0 ≤ ζ ≤ 15. The roots
are identified with the * symbol, and the singularities are identified by vertical
dashed lines and the o symbol. The bottom half of Figure 2 shows f (ζ ) over the
larger range 0 < ζ ≤ 50. As ζ increases the spacing between roots approaches a
constant, and the roots are located midway between the singularities. At small
ζ the spacing between the roots is not equal.
The zRoots function in Listing 1 uses the zfun function (Listing 2), the
bracket function (Listing 3), and the built-in fzero function to find the roots
of Equation (10). fzero is a very efficient root-finding...

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