The analytic study of heat conduction near spheres is a relatively mature science. Even so, there are still a number of analytic conduction solutions being published. For example, Atefi and Moghimi (1) recently presented a Fourier series solution for heat transfer near a hollow sphere.

Probably the most comprehensive collection of classic analytic solutions to heat conduction problems is found in the text by Carslaw and Jaeger (2). One of the solutions in that text relates to the temperature profile about a spherical void where the surface of the void was exposed to a uniform heat flux (Eq. (4) of Section 9.10 of Ref. 2).

One modification of the above classical solution is to consider an encapsulated sphere with a uniform volumetric heat generation rate, $q$. If the temperature of the sphere is initially...
1.
Atefi
,
G.
, and
,
M.
, 2006, “
A Temperature Fourier Series Solution for a Hollow Sphere
,”
ASME J. Heat Transfer
0022-1481,
128
(
9
), pp.
963
968
.
2.
Carslaw
,
H. S.
, and
Jaeger
,
J. C.
, 1959,
Conduction of Heat in Solids
,
2nd ed.
,
Oxford
,
New York
.
3.
Necati Ozisik
,
M.
, 1993,
Heat Conduction
,
2nd ed.
,
Wiley-Interscience
,
New York
.
4.
Abramowitz
,
M.
, and
Stegun
,
I. A.
, 1964,
Handbook of Mathematical Functions
,
National Bureau of Standards
.
5.
Munoz
,
V.
, 2004, Octave-based function