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Question
A hole of radius r1 is made centrally in a uniform circular disc of thickness d and radius r2. The inner surface (a cylinder a length d and radius r1) is maintained at a temperature θ1 and the outer surface (a cylinder of length d and radius r2) is maintained at a temperature θ2 (θ1 > θ2). The thermal conductivity of the material of the disc is K. Calculate the heat flowing per unit time through the disc.
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Solution
Let `(d theta)/(dt)` be the rate of flow of heat.
Consider an annular ring of radius r and thickness dr.
Rate of flow of heat is given by
`(d theta)/(dt) = K (2pird)`
Rate of flow of heat is constant.
∴ `(d theta)/(dt) = i`
`i = - k ( 2pir.d) (d theta)/(dr)`
`int_{r_1}^{r_2} dr/r = (2piKd)/l int_{theta_1}^(theta_2) d theta`
`["ln" (r) ]_{r1}^{r2} = (2pikd)/l [ theta_2 - theta 1]`
`i =(2pikd(theta_2 -theta_1))/("ln"(r_2/r_1)`
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