Thermodynamics HW Solutions 86

Thermodynamics HW - Chapter 2 Heat Conduction Equation 2-23 We consider a thin spherical shell element of thickness r in a sphere(see Fig 2-17 in

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Chapter 2 Heat Conduction Equation 2-23 We consider a thin spherical shell element of thickness Δ r in 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 location is 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 balance on this thin spherical shell element of thickness Δ r during a small time interval Δ t can be expressed as Ar = 4 2 π && QQ E t rr r −= Δ Δ element where Δ Δ ΔΔ EE E m C T T C A r T tt t tt t element =− = Δ T = ++ + () ρ Substituting, & g A r C A r TT t r −+= + + Δ Δ Δ where . Dividing the equation above by A Δ r gives = 4 2 = 1 A r C t r Taking the limit as and yields Δ r 0 Δ t 0 t T C r T kA r A = 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.

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