Question

(1) A sphere of decaying radioactive material of radius ro produces heat at a rate of q"" (W/m3). The sphere is contained in a spherical shell of graphite of outside radius r1. The outside surface of the graphite is cooled uniformly by flowing air of temperature To. The heat transfer coefficient at the outside surface is h. The constant thermal conductivities of the radioactive material and the graphite are ko and ki, respectively. Densities and heat capacities are ρο, co and ρι,G, respectively, following this subscript convention p1, ki, C1 Po, ko, Co The one-dimensional, transient heat equation with heat generation, for a region of constant k, in a spherical coordinate system, is (a) Write one-dimensional heat equations governing the radioactive material and governing the graphite shell, for steady-state conditions. (One heat equation for each region). (b) Define the boundary conditions required to solve for the steady-state temperature distribution throughout the composite

Answer 1

a) for the radioactive material, consider a spherical shell of thickness dr at a radial distance of r (0<r<ro) from the center

Applying heat balance on this particular shell:

In- Out GenerationAccumulation

Assuming steady state, Accumulation =0

OT Or

$Out =-K_{o}A\frac{\partial T}{\partial r}|_{r+dr}=-K_{o}(4\pi r^{2})\frac{\partial T}{\partial r}|_{r+dr}$

" , Generation-q(4π , 2.Sr )

от ar Эт

Dividing by $\Delta r$ and $\Delta r\rightarrow 0$

or

where q''' is the heat generated per unit volume of radioactive material

For the graphite shell, the equation remains same except the generation term changes. Since there is no generation inside this shell,

Generation-0

Hence the energy balance for ro<r<r1,

ar ar

Dividing by $\Delta r$ and $\Delta r\rightarrow 0$

1HTr Эт

b) The boundary conditions,

By Energy transferred at the interface should be balanced, at r=ro

1. at r=ro

от o ar

2. At r=r1, $T=T\infty$ at steady state