Conduction: quick check

How does the rate of heat transfer by conduction change when all spatial dimensions are doubled?

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Answer
Because area is the product of two spatial dimensions, it increases by a factor of four when each dimension is doubled $(A_{final} = (2 d)^{2} = 4 d^{2} = 4 A_{initial})$ . The distance, however, simply doubles. Because the temperature difference and the coefficient of thermal conductivity are independent of the spatial dimensions, the rate of heat transfer by conduction increases by a factor of four divided by two, or two:
${(\frac{Q}{t})}_{final} = \frac{{kA}_{final} (T_{2} - T_{1})}{d_{final}} = \frac{k ({4A}_{initial}) (T_{2} - T_{1})}{{2d}_{initial}} = 2 \frac{{kA}_{initial} (T_{2} - T_{1})}{d_{initial}} = 2 {(\frac{Q}{t})}_{initial} .$
14.36

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