As far as I know, time-dependent heat conduction/diffusion in the solid sphere has never been solved mathematically, neither analytically nor numerically.
My personal view is that the solution does not exist.
The question arises, can we prove the existence or non- existence of such a solution?
Carslaw and Jaegers excellent old book "Conduction of Heat in Solids" covers this topic in detail. It is available free on the internet. There are a number of cases dependent upon the Biot number of the sphere.
See also Chapter Four "Heat Conduction in the Spherical Coordinate System" of the M. Nekati Ozisik's book ""Boundary Values Problems of Heat Conduction" Dover Publications, New York, 1968
This is only a preliminary introductory review to shed some light on the question and its proposed answer.
As far as I know, time-dependent heat conduction/diffusion in the solid sphere has never been solved mathematically, neither analytically nor numerically.
Neither in specific articles nor in respectable reference books suggested by Researchgate contributors.
My personal view is that the mathematical solution does not exist.
OR-In other words, the solution exists but is not mathematically accessible due to the existence of an irremovable singularity at r=0 .
There is no exact analytical integral solution and the approximate numerical solution satisfying an arbitrary Dirichlet BC cannot be found mathematically in 4D space (r,t) for the solid sphere.
The question arises, can we prove the existence or non-existence of such a solution?
However, Einstein and many of the greatest scientists of the time once said that nature did not care about our mathematical difficulties.
It integrates experimentally.
Briefly, we propose the simple form of the center-of-mass cooling curve c of a material object of any shape [including solid spheres]:
Tc(t)-T(S)=[Tc(0)-T(S)] . Exp(-C1.t.A/V) . . . (1)
T(S) is the ambient temperature assumed to be uniform in space and constant in time (t).
The presupposed condition of T(S) corresponds to the Dirichlet boundary conditions for the considered object.
C1 is a constant function of the body's diffusivity (alpha) where A and V are its area and volume, respectively.
(For simplicity, thermal conductivity and hence diffusivity is assumed to be an isotropic scalar quantity.)
It is surprising that Eq 1 describes exactly the experimental (and statistical) dependence of temperature on time which will be the subject of the following answer.
I'm not clear about your concern about solutions at R=0. The obvious case is as time tends to infinity the centre temperature = surface temperature. Solutions to the Fourier equation in spherical coordinates have existed for years usually with only simple boundary conditions at the surface of either constant heat flux or constant temperature and set temperature at time = 0. Is your concern about the mathematical or physical aspects of the problem?
II- Answer II continued.
Equation 1, which we call the BC equation for distinction, is an elegant time-dependent solution to the heat diffusion/conduction equation for a three-dimensional object of any shape.
The BC equation produces some interesting rules:
i- two 3D bodies of different shapes cannot have the same volume to area ratio unless both have exactly the same volume and area,
This physical rule produced by the laws of nature has exceptions, but only when applied outside its scope.
However, these exceptions confirm the rule rather than deny it.
ii- The minimum value of the V/A ratio of a body of 3D geometry is equal to 1/6 and is satisfied by cubic geometric shapes.
iii-Two three-dimensional bodies of different shapes and of the same materials can only have the same mass m if they both have exactly the same volume V and the same area A.
iv- Sabine's imperial formula for the reverberation time Tr in sound rooms:
Tr= constant * V/S. A.
is deduced in a regular way via the equation BC which is a product of the chains of the matrix B
The BC equation itself can be validated through transient cooling curve experiments as well as through statistical techniques.
The experimental results validating the BC equation are presented via the following four tables:
(The initial temperature T0=76°C and the cooling tank temperature is kept constant Tc=4°C.)
Table I-a , Cooling curve for a 2.5 kg aluminum cube with a side of 10 cm
t(s) 0 30 60 90 120 150 180 210 240 300 360 420 480 540 600
temp(c) 76 45 33 26.5 23 20 17.2 15.6 13.8 11.7 9.9 8.8 7.9 7.4 6.95
Table I-b, Cooling curve for a 2.5 kg aluminum cylinder with a diameter of 14.0 cm and a length of 6.5 cm
t(s) 0 30 60 90 120 150 180 210 240 300 360 420 480 540 600
temp(c) 76 44 32 26.23 20 17.2 15.6 13.8 11.6 9.8 8.6
7.9 7.4 6.
Table I-c, Cooling curve for a 6.2 kg aluminum pyramid with a 20 cm square base and a height of 19 cm.
t(s) 0 30 60 90 120 150 180 210 240 300 360 420 480 540 600
temp(c) 76 49 37.5 28 26 22.6 18.3 16.8 15, 13.5 12.6
--- 12 11.3 9
Table I-d, Cooling curve for a 2.9 kg solid aluminum sphere, 13 cm in diameter.
t(s) 0 30 60 90 120 150 180 210 240 300 360 420 480 540 600
temp(c) 76 49 37. 32 28.5 25 22.2 20 18.6 16 14.3 13.3 12.3 11.7 11.2
It should be noted that the temporal evolution of the temperature in the center of the solid sphere "mathematical singularity" is smooth, continuous and regular as in the case of the other geometrical forms.
In a following answer, we explain the numerical statistical basis of the derivation of the BC equation.
Interesting comparison of shape effects on heat transfer. Analytically, this would have to be approximate as the external natural convection heat transfer coefficient is not known for some of these shapes: that is what is being evaluated in fact. Transient heat transfer is a simple way of getting heat transfer coefficients. The internal conduction must obey the fourier equation: the boundary conditions are problematical.
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