sec2-1 - ChE 120B Shell Balances Spherical Geometry S.S...

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ChE 120B 2 - 1 Shell Balances — Spherical Geometry S.S. Heat conduction with source term: Try spherical geometry using a shell balance: Input — Output + Source = 0 2 || 4 | 0 rr rr r r r QQ Sr r π −+ Δ = Remember, we can substitute Fourier's Laws anytime to get: r dT Qk A dr =− A = 4 π r 2 () 22 2 4| 4|4 | 0 r r r dT dT kr S r r dr dr ππ ⎛⎞ + Δ = ⎜⎟ ⎝⎠ , rearranging we get: 2 0 lim r r r dT dT S dr dr r rk Δ→ ⎛⎞⎛⎞ +− ⎜⎟⎜⎟ ⎝⎠⎝⎠ = Δ or r S dT dr r k Integrating once we get 23 1 3 r S dT C dr k + or 1 2 3 r rS C dT dr k r + We can use the obvious boundary condition that nothing real goes to : as 0 dT r dr →→ unless 1 0 C = 3 r rS dT dr k Alternative: We can also say that the center of a sphere is a point of symmetry, so 0 0 | r T r = = ; we get the same result.
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ChE 120B 2 - 2 Integrating again () 2 2 6 r rS Tr C k =− + If the surface temperature is 1 T then 2 12 6 r RS TTR C k == + 2 6 r TC k += and rearranging 2 2 1 1 6 r r Tr T kR ⎛⎞ =+ ⎜⎟ ⎝⎠ The flux leaving the sphere r dT Qk A dr is 2 4 3 r R S Rk k π ⎡⎤ ⎢⎥ ⎣⎦ 3 4 3 r R S = or simply the total heat generated by the sphere! which makes a lot of sense.
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ChE 120B 2 - 3 Conduction with Convection Example 2 Heat Conduction in a Cooling Fin: How to simplify a multi - dimensional problem with judicious approximations. T True Situation Model 1) T is a function of x,y,z 1) For 2B « L,W T = T(z) only 2) A small amount of heat is lost from edges 2) No heat is lost from edges 2LW » (2BW + 4 BL) 3) Heat transfer coefficient is a function of temperature, position 3) Heat flux from fin is given by Newton's Law of cooling q = h(T-T ) h = constant T=T(z) only Starting from these assumptions, we can model the fin as a 1-D problem: A balance across an area of a segment Δ z of the fin gives Input by conduction — (Output by conduction + Output by convection) = 0 () | | h 2 W z 0 zz zz z QQ T T −− Δ =
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ChE 120B 2 - 4 Inserting Fourier's Law: 2 BW zz dT dT Qk A k dz dz =− . Inserting this into the previous equation: () 2 BW | 2 BW | h 2 BW z 0 z dT dT kk T T dz dz −+ Δ = dividing by 2 W z KB Δ and taking limit as z0 Δ || 0 z dT dT h dz dz TT zk B
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This note was uploaded on 03/22/2011 for the course CHE 120B taught by Professor Zasadinski during the Winter '10 term at UCSB.

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sec2-1 - ChE 120B Shell Balances Spherical Geometry S.S...

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