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Derivationofheatequation

# Derivationofheatequation - ,equation(2.3)inlecturenotes...

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Derivation of heat equation, equation (2.3) in lecture notes. Start from energy balance, equation (2.1): st st out g in E dt dE E E E & & & & = = - + (2.1) where dz z dy y dx x out z y x in q q q E q q q E + + + + + = + + = & & (1) By using a Taylor’s series expansion, neglecting the higher order terms: dz z q q q dy y q q q dx x q q q z z dz z y y dy y x x dx x + = + = + = + + + (2) Thermal generation due to an energy source: ) ( dV dz dy dx q q E g & & & = = (3) Energy storage: ) ( dz dy dx t T c E p st oe û ø OE º Ø r = & (4) Combining (2.1) with (1­4): dxdydz t T c dxdydz q dz z q dy y q dx x q p z y x r = + - - - & (5) From Fourier’s law: z T dxdy k z T kA q y T dxdz k y T kA q x T dydz k x T kA q z z y y x x - = - = - = - = - = - = ) ( ) ( ) ( (6)

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Differentiating these expressions: oe û ø OE º Ø ° l ± ² L ³ - =
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Derivationofheatequation - ,equation(2.3)inlecturenotes...

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