Chapter 2 Heat Conduction Equation2-23 We consider a thin spherical shell element of thickness Δrin a sphere (see Fig. 2-17 in the text).. The density of the sphere is ρ, the specific heat is C, and the length is L. The area of the sphere normal to the direction of heat transfer at any locationis where r is the value of the radius at that location. Note that the heat transfer area A depends on r in this case, and thus it varies with location. When there is no heat generation, an energy balanceon this thin spherical shell element of thickness Δrduring a small time interval Δt can be expressed as Ar=42π&&QQEtrrr−=+ΔΔΔelementwhere ΔΔΔΔEEEmCTTCArTttttt telement=−=ΔT−=−+++()ρSubstituting, &gAr CArTTtr−+=−++ΔΔΔwhere . Dividing the equation above by AΔrgives =42−−=−1ArCtrTaking the limit as and yields Δr→0Δt→0tTCrTkArA∂=⎟⎠⎞⎜⎝⎛1
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This note was uploaded on 01/14/2012 for the course PHY 4803 taught by Professor Dr.danielarenas during the Fall '10 term at UNF.