Thermodynamics HW Solutions 85

Thermodynamics HW Solutions 85 - Chapter 2 Heat Conduction...

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Chapter 2 Heat Conduction Equation 2-22 We consider a thin cylindrical shell element of thickness Δ r in 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 location is Ar L = 2 π 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. An energy balance on this thin cylindrical shell element of thickness Δ r during a small time interval Δ t can be expressed as && & QQ G E t rr r −+ = Δ Δ element element where Δ Δ ΔΔ EE E m C T T C A r T tt t tt t element =− = Δ T = ++ + () ρ r A g V g G Δ = = & & & element element Substituting, t T T r CA r A g Q Q t t t r r r Δ Δ ρ = Δ + Δ + Δ + & & & where = 2 L . Dividing the equation above by A Δ r gives +=
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