Consider steady - state diffusion through two sheets of material stacked togethe

Question

Consider steady - state diffusion through two sheets of material stacked together in a series configuration. The thickness and diffusion coefficient of the first and second layer are x1 , D1 and x2 , D2 respectively.

a) show that the diffusion through the stacked configuration is the same as through a shingle material of thickness (x1+x2) and single diffusion coefficient.

D=(D1D2(x1 + x2)/(x1D2 + x2D1)

b) generalize this result for "N" substances staked in series and, separately, "M" substances arrange side - by side in parallel.

Explanation / Answer

The flux of the diffusing atoms is given by J = D(C/x) Where D is the diffusion constant             C is the change in concentration             x is the thickness For the first sheet J1 = D1(C/x1) For the second sheet J2 = D2(C/x2) a) As the sheets are connected in series, 1/J = 1/J1 + 1/J2 1/J = 1/D1(C/x1) + 1/D2(C/x2) If the sheet is replaced by another sheet of thickness x1 + x2 and having difusion constant D' then J = D'C/(x1 + x2 ) Therefore 1/D'C/(x1 + x2 ) = 1/D1(C/x1) + 1/D2(C/x2) (x1 + x2 )/D' = x1/D1 + x2/D2 D' = (x1 + x2 )/[x1/D1 + x2/D2] D' = D1D2(x1 + x2 )/[x1D2 + x2D1] D' = (x1 + x2 )/[x1/D1 + x2/D2] D' = D1D2(x1 + x2 )/[x1D2 + x2D1] b) If there are n number of sheets connected whose thickness is x1 + x2 + x3 +....+ xn Then D' = (x1 + x2 + x3 +....+ xn)/[x1/D1 + x2/D2  + x3/D3 + ... + xn/Dn ] Then D' = (x1 + x2 + x3 +....+ xn)/[x1/D1 + x2/D2  + x3/D3 + ... + xn/Dn ]

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