Question

Am I correct in using volumetric properties when simulating the heat diffusion equation with FiPy?

Context

I am solving the transient heat diffusion equation with an explicit source term.

Where:

• T is temperature [K]
• C is volumetric heat capacity [J/m³ K]
• k is conductivity [W/m K]
• S is volumetric heat source [W/m³]

I represent this in FiPy as follows:

eq = TransientTerm(C) == DiffusionTerm(k) + source

Research

• For starters, the units in the above PDE all match. The transient term has units [J/m³ s], which simplify to [W/m³]. The product of k and T in the diffusion term has units [W/m], which is then divided by [m] twice when we apply the two nablas. So the diffusion term has units [W/m³]. And then the source term is given in [W/m³].
• The diffusion.mesh1D example shows how to solve this PDE in FiPy. They use ρCp as their transition coefficient. I assume that their Cp denotes specific (gravimetric) heat capacity, so ρCp (ie: the product of specific heat capacity & density) is volumetric heat capacity.
• This thread from ~1 million years ago mentions a heat diffusion error caused by using a volumetric source term. This initially threw me, but I think it is because the author had written their PDE in terms of thermal diffusivity, instead of having separate transient and diffusion coefficients.

If the units are consistent between terms, then FiPy doesn't care.

Yes, in [examples.diffusion.mesh1D](https://pages.nist.gov/fipy/en/latest/generated/examples.diffusion.mesh1D.html#module-examples.diffusion.mesh1D), Cp is specific heat capacity and rho is mass density.