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 n infinite medium. aASSUMPTIONS:(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. ()()ttTqkS T 4.20 , q3.19 , S1/ kRRΔ=Δ==(4.21) Using the thermal resistance relations developed in Chapter 3, their corresponding shape factors are: Plane wall:tR=<LAS.kAL=()Cylindrical shell:21t21ln r / r2LRSlnr / r .==π<2Lkπ(L into the page) Spherical shell:tR12121114S.4 krrl/rl/r⎡⎤=−=⎢⎥−⎣⎦ππ<(b) The shape factor for the sphere of diameter D in an
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