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. 22TTtxα∂∂=∂∂We now have a length scale to work with so non-dimensionalize: 0ssTTTTθ−=−→Note that this θis a little different than before we want boundary conditions to equal zero. 22/pkttxLCLLαητρ==So we have 22θθτη∂∂=∂∂Boundary conditions: 0at0for all 1for all for 00for all 1 θητθητθτηη====∂==∂Assume that a solution of the form ( )()(),YXτηθ 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 at0;0 atdLdxθθηη====We’ll find out why these are the right type soon. Insert our new variables into the heat conduction equation: 22YXXYτη∂∂=∂∂. Divide both sides by XYto get: 2λ−≡1dYYdτ=221d XXdη}can replace ∂by d because Y and Xare 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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