Consider steady one$-$dimensional diffusion through a funnel of varying cross section

$($Figure below$).$ Unlike a tube of a constant cross section, the concentration varies

nonlinearly with distance.

$longrightarro{w}^{\text{'}}$

$($a$)$ Neglect chemical reactions and convection and derive the following materials

balance for steady one$-$dimensional diffusion through a region of varying cross section

$($i$.$e$.,A=A(x)$

$\frac{d(N_{ix}A(x))}{dx}=0$

Integrate this equation and explain briefly what the result indicates.

$($b$)$ The radius varies linearly with distance along the funnel according to this

formula:

$r(x)=r_0(1+\frac{x}{L})$

Use Fick's law, the result from Part $($a$)$ and equation $r(x)=r_0(1+\frac{x}{L})$ to show that the

steady state flux is:

$N_{ix}=\frac{2D_{ij}(C_0-C_L)}{L(1+\frac{x}{L}{)}^2}$

The boundary conditions are:

$x=0,C_i=C_0$

$x=L,C_i=C_L$