Explanation Given information: Inner radius of the container is r 1 , outer radius of the container is r 2 , temperature of outer surface is T ∞ , heat flux is q ˙ 1 and the coefficient of heat transfer is h . Write the expression for the differential equation for heat conduction in spherical coordinate. d d r ( r 2 d T d r ) = 0 ...... (I) Here, radius of the container is r . Write the expression for the Fourier’s law of heat conduction in the radial direction. q s = − k d T d r ...... (II) Here, thermal conductivity of material is k and the temperature gradient is d T d r . Write the expression for the heat loss due to convection. q = h [ T − T ∞ ] ...... (III) Here, the amount of heat lost due to heat transfer by convection is q . The heat lost due to conduction is same as that lost due to convection. Calculation: Integrate the Equation (I). ∫ d d r ( r 2 d T d r ) = ∫ 0 r 2 d T d r = C 1 d T d r = C 1 r 2 ...... (IV) Integrate the Equation (IV). ∫ d T d r = ∫ C 1 r 2 T ( r ) = − C 1 r + C 2 ...... (V) Apply the boundary conditions that is, at r = r 1 , T = T ( r 1 ) and q s = q ˙ 1 . Substitute T ( r 1 ) for T and r 1 for r in Equation (II). q ˙ 1 = − k d T ( r 1 ) d r ...... (VI) Substitute T ( r 1 ) for T and r 1 for r in Equation (IV). d T ( r 1 ) d r = C 1 r 1 2 Substitute C 1 r 1 2 for d T ( r 1 ) d r in Equation (VI). q ˙ 1 = − k ( C 1 r 1 2 ) C 1 = − q 1 r 1 2 k Apply the boundary conditions that is, at r = r 2 , T = T ( r 2 ) . Substitute T ( r 2 ) for T and r 2 for r in Equation (IV). d T ( r 2 ) d r = C 1 r 2 2 Substitute T ( r 2 ) for T and r 2 for r in Equation (V). T ( r 2 ) = − C 1 r 2 + C 2 Substitute h [ T − T ∞ ] for q s , T ( r 2 ) for T and r 2 for r in Equation (II). h [ T ( r 2 ) − T ∞ ] = − k d T ( r 2 ) d r

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Chapter 2, Problem 78P

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The expression for the variation of temperature in the shell wall.

The temperature of the outer surface of the container.

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Chapter 2, Problem 77P

Chapter 2, Problem 79P

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HEAT+MASS TRANSFER:FUND.+APPL.

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The expression for the variation of temperature in the shell wall. The temperature of the outer surface of the container.

The expression for the variation of temperature in the shell wall. The temperature of the outer surface of the container.

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Chapter 2 Solutions

The expression for the variation of temperature in the shell wall.

The temperature of the outer surface of the container.