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sec4-1 - ChE 120B Transient Heat Conduction and Combining...

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ChE 120B 4 - 1 Transient Heat Conduction and Combining Variables (Set #4) Examples of the general equations: one-D transient heating of a semi-infinite solid bar Boundary conditions T = T s for all T (0, t ) At time t = 0, one end of the bar is raised to temp T s and kept there. 0 x TT →∞ ( ) 0 ,0 at 0 Tx T t = = Can simplify the general equations for a solid: 2 2 tx α = Define new variable 0 0 s θ = so as to make ( ) () 0 0 2 2 0 0, 1 and s s x t t θθ = ∞= == ∂∂ =

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ChE 120B 4 - 2 Combination of Variables When no obvious length or time scale exists in the problem, it is often necessary to create one by combining variables. Define a new variable 4 x n t α = Note that n is dimensionless. Applying the chain rule for derivatives: 2 n tt n θ −∂ = 22 1 4 x tn = so that our original partial differential equation becomes ordinary. 2 2 20 . dd n dn dn θθ + = Let p dn = integrating this expression gives dp np dn + = 2 dp ndn p =− 2 np n C = −+ A 2 1 n p Ce = 2 1 n d dn =
ChE 120B 4 - 3 and a second integration gives () 2 12 0 . n n nC ed n C θ =+ Examine the boundary conditions. Note that the first two conditions: ,0 0 0 4 x x n t α =⎫ ⎪⎪ = ⎬⎬ ∞= combine into one boundary condition when 0 or , t xn ⎛⎞ ⎜⎟ →∞ ⎝⎠ ( ) 0 n →∞ = This is an essential condition for combination of variables to work. For an ordinary second order differential equation, we can have only 2 boundary conditions. An inside tip: Combination of variables — The general approach 1) To reduce the partial differential equation into an ordinary differential equation, we need to combine 2 variables into one. A. Check B.C.'s — if two of the B.C.'s cannot be made equivalent, forget it, cannot be done. B. Often used if no length or timescale available. Start by assuming that , bc ax t η = and use chain rule for derivatives: dd d dt d dt θη = 2 22 2 2 d d dx d x d dx ηη

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ChE 120B 4 - 4 and substitute into original equation: () 2 11 2 2 1 bc b c dd d ax ct abx t ab b x t d θθ θ αα ηη η −− =+ reorganize to get: ( ) 22 1 2 1 01 .
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sec4-1 - ChE 120B Transient Heat Conduction and Combining...

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