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 .