Derivationofheatequation

Derivationofheatequation - Derivation of heat equation,...

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Unformatted text preview: Derivation of heat equation, equation (2.3) in lecture notes. Start from energy balance, equation (2.1): dE t & s & & & E n + E g - E = = E st i out dt where & E in = q + q + q x y z & E = q +dx + q +dy + q +dz out x y z (2.1) (1) By using a Taylor’s series expansio n, neglecting the higher order terms: ¶q x dx ¶x ¶q y q y +dy = q + dy y ¶y ¶q q +dz = q + z dz z z ¶z q x + dx = q x + Thermal generat ion due to an energy source: & & & ( E g = q dV = q dx dy dz ) Energy storage: ¶T ù é & E t = êrc ( x dy dz d ) s p ¶t ú ë û Combining (2.1) with (1­4): ¶q ¶q x ¶q ¶T y & dx dy - z dz + q dydz = rc p dx dxdydz ¶x ¶y ¶z ¶t (5) (3) (2) (4) From Fourier’s law: ¶T ¶T = - k dydz ( ) ¶x ¶x ¶T ¶T q = - kA = -k dxdz ) ( y y ¶y ¶y ¶T ¶T q = -kA = -k dy ) (dx z z ¶z ¶z q x = -kA x (6) Different iat ing these expressio ns: ¶q x é ¶ æ ¶T ö ù = -dydz ç k ÷ ú ê ¶x ¶x ¶x øû ëè ¶q y é ¶ æ ¶T öù = -dxdz ê ç k ÷ú ç ÷ ¶y ë ¶y è ¶y øû ¶q é ¶ æ ¶T öù z = -dydz ê ç k ÷ú ¶z ë ¶z è ¶z øû Subst ituting into (5): é ¶ æ ¶T öù é ¶ æ ¶T öù é ¶ æ ¶T öù ¶T & d dxdydz ê ç k ÷ú + dxdydz ê ç k ÷ú + dxdydz ê ç k ÷ú + q xdydz = rC p dxdydz ç ¶y ÷ ¶ t ë ¶x è ¶x øû øû ë ¶z è ¶z øû ë ¶y è (7) Dividing by (dxdydz) results in the heat equation ¶ æ ¶T ö ¶ æ ¶T ö ¶ æ ¶T ö & ¶T ç k ÷ + ç k ÷ + q = rc ç k ÷ + p ç ¶y ÷ ¶z ¶z ¶x è ¶x ø ¶y è ¶t è ø ø ...
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This note was uploaded on 09/01/2010 for the course CHEE 318 taught by Professor Ku during the Spring '10 term at University of Washington.

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