sec6-1 - ChE 120B Conduction Separation of Variable...

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ChE 120B 6 - 1 Conduction Separation of Variable Consider a rod of finite length insulated at one end Same basic equation, but quite a different physical problem. 2 2 TT tx α ∂∂ = We now have a length scale to work with so non-dimensionalize: 0 s s θ = Note that this θ is a little different than before we want boundary conditions to equal zero. 22 / p kt t xL CL L ητ ρ == So we have 2 2 τ η = Boundary conditions: 0 at 0 for all 1 for all for 0 0 for all 1 θη τη Assume that a solution of the form () () ( ) , YX ηθ = exits. This is called a separation of variables approach. This will only work if the boundary conditions are of a certain type. Check our boundary conditions. 0a t 0 ; t d L dx = = We’ll find out why these are the right type soon. Insert our new variables into the heat conduction equation: 2 2 XY = . Divide both sides by XY to get: 2 λ −≡ 1 dY Yd = 2 2 1 dX X d } can replace by d because Y and X are functions of only one variable can only be true if equal to a constant = function of time only = function of position
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ChE 120B 6 - 2 Hence, we can divide the partial differential equation into two ordinary differential equations. 2 22 2 dY d X YX dd λ τη =− and we wind up with 2 linear ODE’s 2 12 3 cos sin YC e XC C λτ ηλ η == + 2 is called the separation constant (we chose a negative sign on 2 so that the solution would remain finite with time) Hence, () [ ] 2 3 ,c o s s i n Ce C C θτη λη =+ Now, we impose the boundary conditions, but we have to pick them in the best order (easiest to hardest to apply): 0 for 0 for all (easy to apply) θ ητ →= = [ ] 2 ,0 0 C θτ Unless 2 0 C = , we have a trivial solution. So 21 3 0a n d c a l l CC C C = This means our solution reduces to: [ ] 2 3 o s s i n C C 2 conditions left (remember, both C and are unknown constants) Next, apply the 2nd B.C.: 2 0 for or 1 cos 0 d xL d d d γτ λλ = We know that C can’t equal zero (the solution would be trivial) So we need to look at a value of that will make this come true or cos = 0. We know this is true pretty often: if 35 , , ; for any integer 0 cos 0!
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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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sec6-1 - ChE 120B Conduction Separation of Variable...

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