Simplifying Heat Diffusion Problems for Small Heat Transfer

The temperature distribution in a sphere for heat transfer coefficient h equal 0 is presented. The main point is that the functional form is quite different from that for h not equal zero. This is not in the literature. It is a useful approximation for small h and small times and it makes the solution much clearer intuitively. For example recent research in cryobiology thaws cells at ultra-fast rates by suspending the cell in a medium containing carbon black which absorbs near IR laser light, generating a heat source surrounding the cell. It is necessary to know the rate (and uniformity) of warming which the cell experiences due to this indirect heating. Specific results are given. keywords: heat diffusion, small heat transfer, cryobiology Statement of Problem Find the Temperature Distribution T (r, t) for a homogenous sphere of radius b subject to a timeindependent heat source and boundary condition ~ ∇ T (b, t) = 0 at the surface. The initial distribution T (r, 0) and the heat source are given. The functional form of this solution for heat transfer coefficient h = 0 is different from that for h > 0 [1-3]. The general solution is applied to a specific research problem in cryobiology [4]. General Solution The standard approach would be to find a steady-state solution TS(r) which generates the source and satisfies the boundary condition. Here there is no such solution because the temperature increases indefinitely as long as the heat source exists. We find the asymptotic behavior and subtract this from the desired solution. The difference is amenable to the standard approach. Specifically: cP ρ ∂T (r, t) ∂t − k ∇ T (r, t) = q(r) T (r, 0) = F (r) and ∂T (b, t) ∂r = 0 (1) The heat source q(r) and the initial temperature distribution F (r) are given. We reduce the problems to its essentials by using dimensionless variables r → rb and t→ tτ where τ = cP ρ b 2 k and a heat source Q(r) = b k q(r). Then ∗e-mail: [email protected]