# 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 .

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