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.