Effective thermal conductivity for a solid with spherical inclusions
Input(s)
\(\mathrm{T}\): Initial Temperature \((\mathrm{K})\)
\(\boldsymbol{k}_{\text {eff }}\): Effective Thermal Conductivity ( \(\mathrm{J} / \mathrm{ft}\) s K) \(\boldsymbol{k}_{\boldsymbol{o}}\): Thermal Conductivity of Solid ( \(\mathrm{J} / \mathrm{ft}\) s K)
\(\mathrm{R}\): Effective Radius (ft)
\(\mathrm{r}\): Distance from Centre \((\mathrm{ft})\)
A: Temperature Gradient \((\mathrm{K} / \mathrm{ft})\)
\(\Theta\): Position of Substance from Centre (rad)
Output(s)
\(\boldsymbol{T}_{\boldsymbol{r}}\): Temperature at \(\mathrm{R}(\mathrm{K})\)
Formula(s)
\[
\mathrm{T}_{\mathrm{r}}=\mathrm{T}+\left(1-\left(\frac{\mathrm{k}_{\mathrm{eff}}-\mathrm{k}_{\mathrm{o}}}{\mathrm{k}_{\mathrm{eff}}+2 * \mathrm{k}_{\mathrm{o}}}\right) *\left(\left(\frac{\mathrm{R}}{\mathrm{r}}\right)^{3}\right)\right) * \mathrm{~A} * \mathrm{r} * \cos (\Theta)
\]
Reference(s)
Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley & Sons, Chapter: 11, Page: 371 .