Chapter 2 Heat Conduction Equation2-22 We consider a thin cylindrical shell element of thickness Δrin a long cylinder (see Fig. 2-15 in the text). The density of the cylinder is ρ, the specific heat is C, and the length is L. The area of the cylinder normal to the direction of heat transfer at any locationis ArL=2πwhere 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. An energy balanceon this thin cylindrical shell element of thickness Δrduring a small time interval Δt can be expressed as &&&QQGEtrrr−+=+ΔΔΔelementelementwhere ΔΔΔΔEEEmCTTCArTttttt telement=−=ΔT−=−+++()ρrAgVgGΔ==&&&elementelementSubstituting, tTTrCArAgQQtttrrrΔ−Δρ=Δ+−Δ+Δ+&&&where =2L. Dividing the equation above by AΔrgives −−+=
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