Particle Deposition Mechanisms: Mass Diffusion
Diffusion in a Stream in a Tube
The equation governing the convective diffusion in a tube is given as
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(54)
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with
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(55)
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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
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(56)
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where the axial diffusion
is neglected.
For zero flux to the wall, the boundary condition at the tube surface is given as
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(57)
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As a first approximation,
in the moving
frame is negligibly small and
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(58)
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Now solving Equation (56) for c, it follows that
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(59)
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where
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(60)
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Using (59) in (60), the value of
may be evaluated and then
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(61)
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The total flow of substance across the pipe then is given by
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(62)
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The flux
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(63)
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has the same form as Fick's law with an effective diffusivity
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(64)
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In the next approximation we drop the assumption that
thus
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(65)
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Equation (65) is applicable if the Peclet number, satisfy
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(66)
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If a certain amount N of substance is introduced at , that is
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(67)
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Then the solution to Equation (65) is given as
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(68)
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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.
![](formulas_massdiffusion/image246.gif)
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