Lecture 7 Fin analysis student

Lecture 7 Fin analysis student - 1 MECE 4364 Heat Transfer...

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MECE 4364 Heat Transfer Prof. Dong Liu Department of Mechanical Engineering University of Houston 1 Lecture 7 – Sep 14, 2010 Lecture 7 Fin Analysis 2
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Lecture 7 Fin Equation ± Conservation of Energy ± Assume uniform cross-sectional area ± Alternative form ± General solution dT 2 dx 2 hP kA c T T () = 0 ( ) local fin temperature perimeter cross-sectional area surface area measured from the base to x T T c s b Tx P A A base tempearture fluid tempearture d θ 2 dx 2 m 2 = 0 2 excess temperature c hP m kA TT ≡− ⇒ ( ) 12 mx mx x Ce =+ 3 cs A const A Px == 2 2 11 0 cc dT dA dT h dA dx A dx dx A k dx ⎛⎞ +− = ⎜⎟ ⎝⎠ 0 s c dd T d A kA hT T dx dx dx −= x x dx conv qq d q + Lecture 7 Fin Equation ± General solution ± Boundary conditions ² At the base of the fin (x = 0) ² At the fin tip (x = L) ± Final solution ( ) mx mx x 4 ( ) 0 bb = −≡ x L dT hA T L T kA dx = ⎡⎤ ⎣⎦ xL d hL k dx = =− or ( ) ( ) cosh / sinh cosh / sinh b x mL x h m k mL h mk mL = + 1 sinh 2 1 cosh 2 x x x x y ee y where (where not include the side surface?) Case 1 Note: 1. Trend of temperature; 2. Axial heat flux
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Lecture 7 Fin Equation ± Total fin heat rate ± Alternatively ² It will yield the same result!! ± For a different 2 nd boundary condition: insulated tip (adiabatic) 00 fb c c x x dT d q q kA kA dx dx θ = = == = 5 () ff f ss AA qh T x T d A h x d A =− = ⎡⎤ ⎣⎦ ∫∫ ( ) sinh / cosh cosh / sinh fc b mL h mk mL P k A mL h mk mL + = + 0 xL d dx = = ( ) ( ) cosh cosh b x mL x mL = tanh b P k A m L = Case 2 (A f
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Lecture 7 Fin analysis student - 1 MECE 4364 Heat Transfer...

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