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problem4-06 - PROBLEM 4.6 KNOWN Uniform media of prescribed...

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PROBLEM 4.6 K NOWN: 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 n infinite medium. a A SSUMPTIONS: (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 t T q kS T 4.20 , q 3.19 , S 1/ kR R Δ = Δ = = (4.21) Using the thermal resistance relations developed in Chapter 3, their corresponding shape factors are: Plane wall: t R = < L A S . kA L = ( ) Cylindrical shell: 2 1 t 2 1 ln r / r 2 L R S lnr / r . = = π < 2 Lk π ( L into the page ) Spherical shell : t R 1 2 1 2 1 1 1 4 S . 4 k r r l/r l/r = = π π < (b) The shape factor for the sphere of diameter D in an
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