Question

Radioactive waste is packed in a long, thin-walled cylindrical container. The waste generates thermal energy non-uniformly, according to the relationship Qgen=Qo[1-(r-ro)^2], where Qo is a constant and ro is the radius of the container. Steady-state conditions are maintained by submerging the container in a liquid is at T(infinity) and provides a uniform convection coefficient h. Obtain an expression for the temperature Ts of the container wall under steady conditions.

Answer 1

Answer:

$T_s=T_\infty +\dfrac{Q_or_o}{4h}$

Explanation:

Heat generated per unit volume

$Q_{gen}=Qo\left [{1-\dfrac{r^2}{r_o^2}}\right]$

Total heat generation through the volume

Eg= Q(gen) .dV

$E_g=\int_{0}^{r_o}Qo\left [{1-\dfrac{r^2}{r_o^2}}\right]\ 2\pi rLdr$

$E_g=2\pi LQ_o\int_{0}^{r_o}\left [{r-\dfrac{r^3}{r_o^2}}\right]dr$

$E_g=2\pi LQ_o\left [\dfrac{r^2}{2}-\dfrac{r^4}{4r_o^2}\right]^{r_0}_0$

$E_g=\dfrac{1}{2}Q_0\pi L\ r_0^2$    -------1

Heat transfer due to convection

q= h A (Ts- T ∞)              ----------2

By equating equation 1 and 2

$h 2\pi r_0 L(T_s-T_\infty ) =\dfrac{1}{2}Q_0\pi L\ r_0^2$

$T_s=T_\infty +\dfrac{Q_or_o}{4h}$