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I'm solving a heat conduction problem in cylindrical coordinates with a composite cylinder made of two different materials. Essentially the cylinder is split into a central cylinder of material A, surrounded by an annulus of material B. My mesh is set up so that a control volume boundary aligns with the interface between material A and material B. When discretizing the conduction equation, I need to come up with an "interface conductivity" for this control volume edge, since the two materials have different thermal conductivities. The method normally employed in Cartesian coordinates is the harmonic mean conductivity, which basically ensures that the heat flux is the same on either side of the control volume surface.

However, as I am working in cylindrical coordinates, I am somewhat unsure of the expression I came up with.

Using the standard notation of P being the node of interest and N being the adjacent node radially outward from P, this is what I came up with:

$k_{n}=(\frac{{\Delta}r}{k_{P}{\delta}r_{n}(1+r_{P}/r_{n})}+\frac{{\Delta}r}{k_N{\delta}r_{n}(1+r_{N}/r_{n})})^{-1}$

In the above equation, ${\Delta}r$ is the radial width of a cell, ${\delta}r_n$ is the distance between nodes P and N, $r_P$ is the radial coordinate of node P, $r_N$ is the radial coordinate of node N, and $r_n$ is the radial coordinate of the interface between nodes P and N.

Can anyone confirm my expression? I can walk through my derivation if that would help.

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This is the classical harmonic mean interface conductivity expression - See "Numerical Heat Transfer and Fluid Flow" by S.V. Patankar

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