Page 665 - Bird R.B. Transport phenomena

§20.5 'Taylor Dispersion" in Laminar Tube Flow 645

The total mass flow of A through a plane of constant z (that is, the flow relative to
the average velocity (v )) is
z
[н

«-!?«« (205-13.
482) dz
Next we note that, with the assumption of p = constant, p(co (v )) = (p )(v ) and
A
z
z
A
n
V
p(co v ) ~ (PA AZ) = ( Az)- (Replacing v by v is allowed here because, with axial mol-
A z z Az
ecular diffusion neglected, species A and В are moving with the same axial speed).
Therefore when Eq. 20.5-13 is divided by TTR, 2 we obtain the averaged mass flux
expression
(n ) = (p )(v ) ~ K ^ = (p )(v ) - K ^ (20.5-14)
A
z
A
Az
z
relative to stationary coordinates. Here К is an axial dispersion coefficient, given by Tay-
lor's analysis as
This formula indicates that axial dispersion (in the range Pe > > 1 considered so far) is
enhanced by the radial variation of v and reduced by radial molecular diffusion.
z
Although Eq. 20.5-14 has the form of Fick's law in Eq. (C) of Table 17.8-2, the present
equation does not include any axial molecular diffusion. Also it should be emphasized
that К is not a property of the fluid mixture, but depends on R and (v ) as well as on Я1 .
АВ
z
Next we write the equation of continuity of Eq. 19.1-6, averaged over the tube cross
section, as
j (pA)= -Ji^Az) (20.5-16)
t
When the expression for the mass flux of A from Eq. 20.5-14 is inserted, we get the fol-
lowing axial dispersion equation'.

^(PA) + (V )-J-(PA) = KA (PA) (20.5-17)
Z
dt dZ $2T
This equation can be solved to get the shape of the traveling pulse resulting from a 8-
function input of a mass m A of solute A into a stream of otherwise pure B:

(20.5-18)

This can be used along with Eq. 20.5-15 to extract 4L from data on the concentrations in
AB
the traveling pulse. In fact, this is probably the best method for reasonably quick mea-
surements of liquid diffusivities.
Taylor's development laid the foundation for an extensive literature on convective dis-
persion. However, it remained to study the approximations made and to determine their
5
range of validity. Aris gave a detailed treatment of dispersion in tubes and ducts, covering
the full range of t and including diffusion in the z and в directions. His long-time asymptote

5
R. Aris, Proc. Roy. Soc, A235 r 67-77 (1956).