Explanation Consider the steady state, one dimensional heat conduction in rod with constant thermal properties. Write the general form of steady state radial heat conduction equation for rod. 1 r ∂ ∂ r ( k r r ∂ T r ∂ t ) + q ˙ = 0 Here, r is the radius of the rod, k r is the thermal conductivity of the rod, T r is the temperature of the rod at radius r and q ˙ is the volumetric heating rate. Simplify the above expression. k r r ∂ ∂ r ( r ∂ T r ∂ t ) + q ˙ = 0 The appropriate form of heat equation of rod is k r r ∂ ∂ r ( r ∂ T r ∂ t ) + q ˙ = 0 . Write the expression for the steady state radial heat conduction for the cladding material. 1 r ∂ ∂ r ( k c r ∂ T c ∂ t ) + q ˙ = 0 Here, r is the radius of the cladding, k c is the thermal conductivity of the cladding material, T c is the temperature of the cladding at radius r and q ˙ is the volumetric heating rate in the cladding. Substitute 0 for q ˙ in above expression. k c r ∂ ∂ r ( r ∂ T c ∂ t ) + 0 = 0 Simplify above expression. ∂ ∂ r ( r ∂ T c ∂ t ) = 0 The appropriate form of heat equation for the cladding is ∂ ∂ r ( r ∂ T c ∂ t ) = 0 . The boundary condition corresponding to symmetry at the centerline is d T r d r | r = 0 = 0 . The boundary condition corresponding to the thermal equilibrium at the interface between the conducting rod and the cladding material can be identified by equating the temperature of the conducting rod at radius r i to the temperature of the cladding material at radius r i by considering there is no contact resistance and hence no temperature drop at the interface between the rod and the cladding as T r ( r i ) = T c ( r i )

6th Edition

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 2, Problem 2.48P

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The appropriate forms of heat equation for rod and cladding for steady state and the boundary conditions.

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