heat_2_27 - PROBLEM 2.27 KNOWN Temperature distribution and...

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PROBLEM 2.27 KNOWN: Temperature distribution and distribution of heat generation in central layer of a solar pond. FIND: (a) Heat fluxes at lower and upper surfaces of the central layer, (b) Whether conditions are steady or transient, (c) Rate of thermal energy generation for the entire central layer. SCHEMATIC: ASSUMPTIONS: (1) Central layer is stagnant, (2) One-dimensional conduction, (3) Constant properties ANALYSIS: (a) The desired fluxes correspond to conduction fluxes in the central layer at the lower and upper surfaces. A general form for the conduction flux is ʹʹ = = + ± ! " $ # qk T x k A ka eB cond -ax . Hence, = = + ± ! " $ # = = + ± ! " $ # qq k A ka e B q q k A ka B l cond x=L -aL u cond x=0 16 . < (b) Conditions are steady if T/ t = 0. Applying the heat equation, α ∂α 2 11 T x q k T t - A k e A k e T t 2 -ax -ax
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