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Equipartition Theorem

A hollow tube...

Question

A hollow tube has a length l, inner radius R₁ and outer radius R₂. The material has a thermal conductivity K. Find the heat flowing through the walls of the tube if (a) the flat ends are maintained at temperature T₁ and T₂ (T₂ > T₁) (b) the inside of the tube is maintained at temperature T₁ and the outside is maintained at T₂.

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Solution

(a) When the flat ends are maintained at temperatures T₁ and T₂ (where T₂ > T₁):

Area of cross section through which heat is flowing, $A = π (R_{2}^{2} - R_{1}^{2})$

Rate of flow of heat $= \frac{d θ}{d t}$

$= \frac{K A . ∆ T}{l} = \frac{K π (R_{2}^{2} - R_{1}^{2}) (T_{2} - T_{1})}{l}$

(b)

When the inside of the tube is maintained at temperature T₁ and the outside is maintained at T₂_:

Let us consider a cylindrical shell of radius r and thickness dr.
Rate of flow of heat, q $= K A . \frac{d T}{d r}$
$q = K A . \frac{dT}{d r} q = K (2 π r l) \frac{dT}{d r} \int_{R_{1}}^{R_{2}} \frac{d r}{r} = \frac{2 π K l}{q} \int_{T_{1}}^{T_{2}} d T {[\ln (r)]}_{R_{1}}^{R_{2}} = \frac{2 π K l}{q} [T_{2} - T_{1}] \ln (\frac{R_{2}}{R_{1}}) = \frac{2 π K l}{q} (T_{2} - T_{1}) q = \frac{2 π K l (T_{2} - T_{1})}{\ln (\frac{R_{2}}{R_{1}})}$

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