Question 481: Thermodynamics & KTG

Question

The figure shows a system of two concentric spheres of radii ${r_1}$ and ${r_2}$ are kept at temperatures ${T_1}$ and ${T_2}$, respectively. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to
![](https://imagex.cdn.examgoal.net/q0NWGCiLMsCb79lCF/WSeEDxwJ95FpxoyUYXTC6Ya5dPvTe/wHis2k2cNpzML7NMEXxZAf/image.png)

Accepted Answer

Consider a shell of thickness $(dr)$ and of radius $(r)$ and the temperature of inner and outer surfaces of this shell be $T,(T-dT)$
$H = {{KA\left[ {\left( {T - dT} \right) - T} \right]} \over {dr}} = {{ - KAdT} \over {dr}}$
$H =  - 4\pi K{r^2}{{dT} \over {dr}}$ ( as $A = 4\pi {r^2}$ )
Then, $\left( H \right)\int\limits_{{r_1}}^{{r^2}} {{1 \over {{r^2}}}} dr =  - 4\pi K\int\limits_{{T_1}}^{{T_2}} {dT} $
$H\left[ {{1 \over {{r_1}}} - {1 \over {{r_2}}}} \right] =  - 4\pi K\left[ {{T_2} - {T_1}} \right]$
or $H = {{ - 4\pi K{r_1}{r_2}\left( {{T_2} - {T_1}} \right)} \over {\left( {{r_2} - {r_1}} \right)}}$
![](https://imagex.cdn.examgoal.net/5f2Sn8CWo8UlPXQYi/WFW7mXUNheB3yTKB2qIHKOt8mvXz3/FB0HCOZGadEqtdDP0dliM3/image.svg)

Answer

$In\left( {{{{r_2}} \over {{r_1}}}} \right)$

Answer

${{\left( {{r_2} - {r_1}} \right)} \over {\left( {{r_1}{r_2}} \right)}}$

Answer

${\left( {{r_2} - {r_1}} \right)}$

Answer

${{{r_1}{r_2}} \over {\left( {{r_2} - {r_1}} \right)}}$

Thermodynamics & KTG

The figure shows a system of two concentric spheres of radii {r_1} and {r_2} are kept at temperatures {T_1} and {T_2}, respectively. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to

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