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 T T t x α = We now have a length scale to work with so non-dimensionalize: 0 s s T T T T θ = Note that this θ is a little different than before we want boundary conditions to equal zero. 2 2 / p kt t x L C L L α η τ ρ = = So we have 2 2 θ θ τ η = Boundary conditions: 0 at 0 for all 1 for all for 0 0 for all 1 θ η τ θ η τ θ τ η η = = = = = = Assume that a solution of the form ( ) ( ) ( ) , Y X τ η θ τ η = 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. 0 at 0; 0 at d L dx θ θ η η = = = = We’ll find out why these are the right type soon. Insert our new variables into the heat conduction equation: 2 2 Y X X Y τ η = . Divide both sides by XY to get: 2 λ 1 dY Y d τ = 2 2 1 d X X d η } 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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