Particle Deposition Mechanisms: Mass Diffusion

Diffusion in a Stream in a Tube

The equation governing the convective diffusion in a tube is given as

(54)

with

(55)

where U is the mean velocity in the tube. In a coordinate system moving with the mean fluid velocity U, Equation (54) may be restated as

(56)

where the axial diffusion  is neglected.

For zero flux to the wall, the boundary condition at the tube surface is given as

(57)

As a first approximation,  in the moving frame is negligibly small and

(58)

Now solving Equation (56) for c, it follows that

(59)

where

(60)

Using (59) in (60), the value of  may be evaluated and then

(61)

The total flow of substance across the pipe then is given by

(62)

The flux

(63)

has the same form as Fick's law with an effective diffusivity

(64)

In the next approximation we drop the assumption that  thus

(65)

Equation (65) is applicable if the Peclet number, satisfy

(66)

If a certain amount N of substance is introduced at , that is

(67)

Then the solution to Equation (65) is given as

(68)

Variation of concentration as a function of space and time are shown in Figure 5. It is seen that the concentration travels like a wave but also dispersed along it path.