# 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.

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