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Lecture 7 - x x x T s q” o T ∞ h Case(1 T(x,0 = T i...

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Lecture 7 Dr. Srinath V. Ekkad
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Transient Conduction Methods: Lumped Capacitance Model Charts (graphic results) Numerical techniques Finite difference Finite element BEM Analytical Separation of variables Fourier series Etc.
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Lumped Capacitance Method Consider a solid, initially at T i and suddenly exposed to a fluid at T Solid T(t) V A s Fluid T , h If the solid is hotter, then quenching began at time, t=0. For t>0, the temperature of the solid will decrease until it reaches T . The essence of the lumped capacitance model is the assumption that the temperature of the solid is spatially uniform at any instant during this transient cooling process. Consider an energy balance for the control volume around the solid lump
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Time, t 1 0 θ θ i τ t
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Validity of lumped capacity model q conv q cond T s, 2 T s, 2 T s, 2 T s, 1 Bi >> 1 Bi << 1 T x L T , h Bi 1
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T , h x -L L T , h -L L -L L -L L T(x,0)=T i T(x,0)=T i Bi<<1 T T(t) Bi 1 T=T(x,t) Bi>>1 T=T(x,t) T T T
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Spatial Effects
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Semi-infinite solid
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Unformatted text preview: x x x T s q” o T ∞, h Case (1) T(x,0) = T i T(0,t) = T s Case (2) T(x,0) = T i-k dT/dx| x=0 = q” o Case (3) T(x,0) = T i-k dT/dx| x=0 = h(T ∞-T(0,t)) T s T i t T i T ∞ t t T i Case 1 Constant Surface Temperature T t T T x t T T T erf x t q t k T x k T T t s s i s s x s i ( , ) ( , ) ( ) ( ) " 2 =--= = -=-= α ∂ ∂ πα Case 2 Constant Surface Heat Flux q q T(x t T q t k x t q x k erfc x t s o i o o " " " / " , ) ( / ) exp =-=- - 2 4 2 1 2 2 α π α α Case 3 Surface Convection - = ---= - + + = ∞ ∞ k T x h T T t T x t T T T erfc x t hx k h t k erfc x t h t k x i i ∂ ∂ α α α α 2 2 2 2 [ ( , )] ( , ) exp...
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