Temperature profile with a nuclear heat source

Input(s)

$S_{n o}$: Volume Rate Heat Production at the Center of the Source $\left(\mathrm{cal} / \mathrm{cm}^{3} \mathrm{~s}\right)$

$R^{F}$: Radius of Fissionable Substance (cm)

$K^{F}$: Thermal Conductivity of Fissionable Substance (W/cm K)

$\mathrm{r}$: Radius of Core $(\mathrm{cm})$

b: Constant of Fission (dimensionless)

$k^{c}$: Thermal Conductivity of Casing Substance (W/cm K)

$R^{c}$: Radius of Casing Substance $(\mathrm{cm})$

$T_{o}$: Initial Temperature (K)

Output(s)

$T_{f}$: Temperature of Fissionable Substance (K)

Formula(s)

$$ \mathrm{T}_{\mathrm{f}}=\left(\frac{\mathrm{S}_{\mathrm{no}} *\left(\mathrm{R}^{(\mathrm{F}) 2}\right)}{6 * \mathrm{~K}^{\mathrm{F}}}\right) *\left(\left(1-\left(\frac{\mathrm{r}}{\mathrm{R}^{\mathrm{F}}}\right)^{2}\right)+\frac{3}{10} * \mathrm{~b} *\left(1-\left(\frac{\mathrm{r}}{\mathrm{R}^{\mathrm{F}}}\right)^{4}\right)\right)+\left(\frac{\mathrm{S}_{\mathrm{no}} *\left(\mathrm{R}^{(\mathrm{F}) 2}\right)}{3 * \mathrm{k}^{\mathrm{c}}}\right) *\left(1+\frac{3}{5} * \mathrm{~b}\right) *\left(1-\frac{\mathrm{R}^{\mathrm{F}}}{\mathrm{R}^{\mathrm{c}}}\right)+\mathrm{T}_{\mathrm{o}} $$

Reference(s)

Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley \& Sons, Chapter: 10, Page: 298.