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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 = pi (R_2^2 - R_1^2)`
Rate of flow of heat = `(d theta)/dt`
`= (KA ( Rpi - 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= KA. {aT}/{dr}``
`q = KA. (dt)/(dr)`
`q = K (2pirl)dt/(dr)`
\[\int\limits_{R1}^{R2}\] `(dr)/r = 2piKl` \[\int\limits_{T1}^{T2}\] `dT`
`[In (r)]_{R1 }^{R2} (dr)/(r) =( 2pirL)/q [T_2 - T_1]`
In `((R_2)/(R_1)) = "2piKl"/ q [T_2 - T_1]`
`q = (2piKl(T_2-T_1))/"in" (R_2/R^1)`
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