Explanation Write the expression for Energy balance equation for the controlled volume. E ˙ in + E ˙ g − E ˙ out = E ˙ st .......(1) Here, E ˙ in is the energy inflow, E ˙ out is the energy out flow, E ˙ g is the energy generated inside the controlled volume and E ˙ st is the rate of energy storage. Write the expression for heat inflow for controlled volume. E ˙ in = q r + q ϕ + q θ .......(2) Here, q r is the rate of heat conduction inflow in r direction, q ϕ is the rate of heat conduction inflow in ϕ direction and q θ is the rate of heat conduction inflow in θ direction. Write the expression for heat out flow for controlled volume. E ˙ out = q r + d r + q ϕ + d ϕ + q θ + d θ .......(3) Here, q r + d r is the rate of heat conduction outflow in r direction, q ϕ + d ϕ is the rate of heat conduction outflow in ϕ direction and q θ + d θ is the rate of heat conduction outflow in θ direction. Write the expression for rate of thermal energy generation. E ˙ g = q ˙ d V .......(4) Here, q ˙ is the heat generation per unit volume and d V is the elemental volume. Write the expression for energy stored in controlled volume. E ˙ st = ρ c p ∂ T d t d V .......(5) Here, ρ is the density, c p is the specific heat and ∂ T d t is the rate of change of temperature. Substitute q r + q ϕ + q θ for E ˙ in , q r + d r + q ϕ + d ϕ + q θ + d θ for E ˙ out , q ˙ d V for E ˙ g and ρ c p ∂ T d t d V for E ˙ st in equation (1). q r + q ϕ + q θ + q ˙ d V − ( q r + d r + q ϕ + d ϕ + q θ + d θ ) = ρ c p ∂ T d t d V .......(6) Write the expression for heat conduction rate in r direction. q r = − k ( d A ) r ∂ T ∂ r Substitute r d θ ( r sin θ d ϕ ) for ( d A ) r in above expression. q r = − k r d θ ( r sin θ d ϕ ) ∂ T ∂ r Here, k is the thermal conductivity, d ϕ is the length of elemental length in ϕ direction and d θ is the elemental length in θ direction. Differentiate above expression with respect to r . − ∂ q r d r = sin θ d ϕ d θ ∂ ∂ r ( k r 2 ∂ T ∂ r ) Write the expression for heat conduction rate in θ direction. q θ = − k ( d A ) θ ∂ T ∂ ( r d θ ) Substitute sin θ d ϕ d r for ( d A ) θ in above expression. q θ = − k sin θ d ϕ d r ∂ T ∂ ϕ Differentiate above expression with respect to θ . − ∂ q θ d θ = d ϕ d r ∂ ∂ θ ( k sin θ ∂ T ∂ θ ) Write the expression for heat conduction rate in θ direction. q ϕ = − k ( d A ) ϕ ∂ T ∂ ( ϕ r sin θ ) Substitute r d θ d r for ( d A ) ϕ in above expression. q ϕ = − k d θ d r sin θ ∂ T ∂ ϕ Differentiate above expression with respect to ϕ . − ∂ q ϕ d ϕ = d r d θ ∂ ∂ ϕ ( k ∂ T ∂ ϕ ) Write the expression for differential volume

6th Edition

ISBN: 9780470501962

Author: Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine

Publisher: Wiley, John & Sons, Incorporated

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Introduction to Heat Transfer

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The heat diffusion equation for spherical coordinates.

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