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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 - Physics

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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.

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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Solution 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 Concept: Thermal Expansion of Solids.
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