## Radioactive Waste

Consider radioactive waste that is stored in a spherical container, buried 9 m below the earth’s surface. The thermal conductivity of the soil can be assumed as 0.52 W/m∙K. The container’s outside diameter is 2 m (D), and 1500 W of heat (q) is released as a result of radioactive decay. If the soil surface temperature is kept at 20°C, what is the outside surface temperature of the container under steady state conditions?

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__**Hint**

**Hint**

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__**Hint 2**

**Hint 2**

Thermal resistance:

$$$q=Sk(T_1-T_2)$$$

where
$$S$$
is the shape factor,
$$k$$
is the thermal conductivity, and
$$T$$
is the temperature.

Thermal resistance:

$$$q=Sk(T_1-T_2)$$$

where
$$S$$
is the shape factor,
$$k$$
is the thermal conductivity, and
$$T$$
is the temperature. To determine shape factor, referencing a heat transfer equation table. The problem scenario qualifies as an isothermal sphere buried in a semi-infinite medium by meeting the below conditions:

As a result, the heat transfer equation is:

$$$q=\frac{2\pi D}{1-\frac{D}{4z}}k(T_1-T_2)$$$

$$$=\frac{2\pi (2m)}{1-\frac{2m}{4(10m)}}(0.52W/mK)(T_1-20^{\circ}C)$$$

$$$q=(6.87\:W/K)(T_1-20^{\circ}C)=1500\:W$$$

Thus:

$$$T_1=\frac{1500W}{6.87\frac{W}{K}}+20^{\circ}C=238^{\circ}C$$$

238°C