Unformatted text preview: PROBLEM 4.6
KNOWN: Uniform media of prescribed geometry. FIND: (a) Shape factor expressions from thermal resistance relations for the plane wall, cylindrical shell and spherical shell, (b) Shape factor expression for the isothermal sphere of diameter D buried in an infinite medium. ASSUMPTIONS: (1) Steady-state conditions, (2) Uniform properties. ANALYSIS: (a) The relationship between the shape factor and thermal resistance of a shape follows from their definitions in terms of heat rates and overall temperature differences. T q = kST q= S = 1/ kR t (4.21) ( 4.20 ) , ( 3.19 ) , Rt Using the thermal resistance relations developed in Chapter 3, their corresponding shape factors are: Plane wall: Rt = L kA
ln ( r2 / r1 ) 2 Lk S= A . L
2 L lnr2 / r1. < < < Cylindrical shell: (L into the page) Spherical shell: Rt = S= Rt = 1 1 1 - 4 k r1 r2 S= 4 . l/r1 - l/r2 (b) The shape factor for the sphere of diameter D in an
infinite medium can be derived using the alternative conduction analysis of Section 3.2. For this situation, qr is a constant and Fourier's law has the form dT q r = -k 4 r 2 . dr Separate variables, identify limits and integrate. ( ) T2 dr q - r D / 2 2 = T1 dT 4 k r D q r = 4 k ( T1 - T2 ) 2 q 1 q 2 - r - = - r 0 - = ( T2 - T1 ) 4 k r D/2 4 k D or S = 2 D. < COMMENTS: Note that the result for the buried sphere, S = 2D, can be obtained from the expression for the spherical shell with r2 = . Also, the shape factor expression for the "isothermal sphere buried in a semi-infinite medium" presented in Table 4.1 provides the same result with z . ...
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