(a) Explanation Given: The inner radius of the cylindrical shell is r i . The outer radius of the cylindrical shell is r o . The volumetric heat generation rate is q ˙ . The temperature at the outer surface of the shell is T ∞ . The heat transfer coefficient is h . Formula used: The one dimensional radial heat conduction equation for the cylindrical shell with uniform heat generation is given as, 1 r d d r [ r d T d r ] + q ˙ k = 0 Here, q ˙ is the volumetric heat generation rate and k is the thermal conductivity of the material. The steady-state conduction equation is given as, q ι cond . = k A c ( d T d r ) Here, A c is the area conducting heat. The steady state convection equation is given as, q ι conv . = h A c ( T − T ∞ ) Here, T is the temperature at any radius and h is the convection heat transfer coefficient. The Boundary conditions are given as, d T d r | r = r i = 0 and ( q ι cond . ) r = r o = ( q ι conv . ) r = r o Calculation: The one dimensional radial heat conduction equation for the cylindrical shell with uniform heat generation is further expanded as, 1 r d d r [ r d T d r ] + q ˙ k = 0 d d r [ r d T d r ] = − q ˙ k r Integrate the above expression, r d t d r = − q ˙ r 2 2 k + C 1 d t d r = − q ˙ r 2 k + C 1 r Integrate the above expression, T ( r ) = − q ˙ r 2 4 k + C 1 ln ( r ) + C 2 ...... (1) Apply the Boundary condition d T d r | r = r i = 0 on the above expression, d d r ( − q ˙ r 2 4 k + C 1 ln ( r ) + C 2 ) r = r i = 0 − q ˙ r i 2 k + C 1 r i + 0 = 0 C 1 = q ˙ r i 2 k Apply the Boundary condition ( q ι cond . ) r = r o = ( q ι conv (b) To determine The heat transfer rate at the outer radius of the shell.

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ISBN: 9780470501962

Author: Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine

Publisher: Wiley, John & Sons, Incorporated

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Chapter 3, Problem 3.96P

(a)

To determine

The expression for the steady-state temperature distribution.

(b)

To determine

The heat transfer rate at the outer radius of the shell.

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Chapter 3, Problem 3.95P

Chapter 3, Problem 3.97P

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Chapter 3 Solutions

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The expression for the steady-state temperature distribution.

Solution Summary: The author explains the one dimensional radial heat conduction equation for the cylindrical shell with uniform heat generation.

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The expression for the steady-state temperature distribution.

The heat transfer rate at the outer radius of the shell.

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Chapter 3 Solutions