Taylor dispersion

We use z as an axial coordinate and r as the radial coordinate, and assume axisymmetry. The pipe has radius a, and the fluid velocity is:

${\displaystyle {\boldsymbol {u}}=w{\hat {\boldsymbol {z}}}=w_{0}(1-r^{2}/a^{2}){\hat {\boldsymbol {z}}}}$

The concentration of the diffusing species is denoted c and its diffusivity is D. The concentration is assumed to be governed by the linear advection–diffusion equation:

${\displaystyle {\frac {\partial c}{\partial t}}+{\boldsymbol {w}}\cdot {\boldsymbol {\nabla }}c=D\nabla ^{2}c}$

The concentration and velocity are written as the sum of a cross-sectional average (indicated by an overbar) and a deviation (indicated by a prime), thus:

${\displaystyle w(r)={\bar {w}}+w'(r)}$

${\displaystyle c(r,z)={\bar {c}}(z)+c'(r,z)}$

Under some assumptions (see below), it is possible to derive an equation just involving the average quantities:

${\displaystyle {\frac {\partial {\bar {c}}}{\partial t}}+{\bar {w}}{\frac {\partial {\bar {c}}}{\partial z}}=D\left(1+{\frac {a^{2}{\bar {w}}^{2}}{48D^{2}}}\right){\frac {\partial ^{2}{\bar {c}}}{\partial z^{2}}}}$

Observe how the effective diffusivity multiplying the derivative on the right hand side is greater than the original value of diffusion coefficient, D. The effective diffusivity is often written as:

${\displaystyle D_{\mathrm {eff} }=D\left(1+{\frac {{\mathit {Pe}}^{2}}{48}}\right)\,,}$

where ${\displaystyle {\mathit {Pe}}=a{\bar {w}}/D}$ is the Péclet number, based on the channel radius ${\displaystyle a}$. The interesting result is that for large values of the Péclet number, the effective diffusivity is inversely proportional to the molecular diffusivity. The effect of Taylor dispersion is therefore more pronounced at higher Péclet numbers.

In a frame moving with the mean velocity, i.e., by introducing ${\displaystyle \xi =z-{\bar {w}}t}$, the dispersion process becomes a purely diffusion process,

${\displaystyle {\frac {\partial {\bar {c}}}{\partial t}}=D_{\mathrm {eff} }{\frac {\partial ^{2}{\bar {c}}}{\partial \xi ^{2}}}}$

with diffusivity given by the effective diffusivity.

The assumption is that ${\displaystyle c'\ll {\bar {c}}}$ for given ${\displaystyle z}$, which is the case if the length scale in the ${\displaystyle z}$ direction is long enough to smooth the gradient in the ${\displaystyle r}$ direction. This can be translated into the requirement that the length scale ${\displaystyle L}$ in the ${\displaystyle z}$ direction satisfies:

${\displaystyle L\gg {\frac {a^{2}}{D}}{\bar {w}}=a{\mathit {Pe}}}$.

Dispersion is also a function of channel geometry. An interesting phenomenon for example is that the dispersion of a flow between two infinite flat plates and a rectangular channel, which is infinitely thin, differs approximately 8.75 times. Here the very small side walls of the rectangular channel have an enormous influence on the dispersion.

While the exact formula will not hold in more general circumstances, the mechanism still applies, and the effect is stronger at higher Péclet numbers. Taylor dispersion is of particular relevance for flows in porous media modelled by Darcy's law.^[6]

One may derive the Taylor equation using method of averages, first introduced by Aris. The result can also be derived from large-time asymptotics, which is more intuitively clear. In the dimensional coordinate system ${\displaystyle (x',r',\theta )}$, consider the fully-developed Poiseuille flow ${\displaystyle u=2U[1-(r'/a)^{2}]}$ flowing inside a pipe of radius ${\displaystyle a}$, where ${\displaystyle U}$ is the average velocity of the fluid. A species of concentration ${\displaystyle c}$ with some arbitrary distribution is to be released at somewhere inside the pipe at time ${\displaystyle t'=0}$. As long as this initial distribution is compact, for instance the species/solute is not released everywhere with finite concentration level, the species will be convected along the pipe with the mean velocity ${\displaystyle U}$. In a frame moving with the mean velocity and scaled with following non-dimensional scales

${\displaystyle t={\frac {t'}{a^{2}/D}},\quad x={\frac {x'-Ut'}{a}},\quad r={\frac {r'}{a}},\quad Pe={\frac {Ua}{D}}}$

where ${\displaystyle a^{2}/D}$ is the time required for the species to diffuse in the radial direction, ${\displaystyle D}$ is the diffusion coefficient of the species and ${\displaystyle Pe}$ is the Peclet number, the governing equations are given by

${\displaystyle {\frac {\partial c}{\partial t}}+Pe(1-2r^{2}){\frac {\partial c}{\partial x}}={\frac {\partial ^{2}c}{\partial x^{2}}}+{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c}{\partial r}}\right).}$

Thus in this moving frame, at times ${\displaystyle t\sim 1}$ (in dimensional variables, ${\displaystyle t'\sim a^{2}/D}$), the species will diffuse radially. It is clear then that when ${\displaystyle t\gg 1}$ (in dimensional variables, ${\displaystyle t'\gg a^{2}/D}$), diffusion in the radial direction will make the concentration uniform across the pipe, although however the species is still diffusing in the ${\displaystyle x}$ direction. Taylor dispersion quantifies this axial diffusion process for large ${\displaystyle t}$.

Suppose ${\displaystyle t\sim 1/\epsilon \gg 1}$ (i.e., times large in comparison with the radial diffusion time ${\displaystyle a^{2}/D}$), where ${\displaystyle \epsilon \ll 1}$ is a small number. Then at these times, the concentration would spread to an axial extent ${\displaystyle x\sim {\sqrt {t}}\sim {\sqrt {1/\epsilon }}\gg 1}$. To quantify large-time behavior, the following rescalings^[7]

${\displaystyle \tau =\epsilon t,\quad \xi ={\sqrt {\epsilon }}x}$

can be introduced. The equation then becomes

${\displaystyle \epsilon {\frac {\partial c}{\partial \tau }}+{\sqrt {\epsilon }}Pe(1-2r^{2}){\frac {\partial c}{\partial \xi }}=\epsilon {\frac {\partial ^{2}c}{\partial \xi ^{2}}}+{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c}{\partial r}}\right).}$

If pipe walls do not absorb or react with the species, then the boundary condition ${\displaystyle \partial c/\partial r=0}$ must be satisfied at ${\displaystyle r=1}$. Due to symmetry, ${\displaystyle \partial c/\partial r=0}$ at ${\displaystyle r=0}$.

Since ${\displaystyle \epsilon \ll 1}$, the solution can be expanded in an asymptotic series, ${\displaystyle c=c_{0}+{\sqrt {\epsilon }}c_{1}+\epsilon c_{2}+\cdots }$ Substituting this series into the governing equation and collecting terms of different orders will lead to series of equations. At leading order, the equation obtained is

${\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c_{0}}{\partial r}}\right)=0.}$

Integrating this equation with boundary conditions defined before, one finds ${\displaystyle c_{0}=c_{0}(\xi ,\tau )}$. At this order, ${\displaystyle c_{0}}$ is still an unknown function. This fact that ${\displaystyle c_{0}}$ is independent of ${\displaystyle r}$ is an expected result since as already said, at times ${\displaystyle t'\gg a^{2}/D}$, the radial diffusion will dominate first and make the concentration uniform across the pipe.

Terms of order ${\displaystyle {\sqrt {\epsilon }}}$ leads to the equation

${\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c_{1}}{\partial r}}\right)=Pe(1-2r^{2}){\frac {\partial c_{0}}{\partial \xi }}.}$

Integrating this equation with respect to ${\displaystyle r}$ using the boundary conditions leads to

${\displaystyle c_{1}(\xi ,r,\tau )=c_{1a}(\xi ,\tau )+{\frac {Pe}{8}}(2r^{2}-r^{4}){\frac {\partial c_{0}}{\partial \xi }}}$

where ${\displaystyle c_{1a}}$ is the value of ${\displaystyle c_{1}}$ at ${\displaystyle r=0}$, an unknown function at this order.

Terms of order ${\displaystyle \epsilon }$ leads to the equation

${\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c_{2}}{\partial r}}\right)=Pe(1-2r^{2}){\frac {\partial c_{1}}{\partial \xi }}+{\frac {\partial c_{0}}{\partial \tau }}-{\frac {\partial ^{2}c_{0}}{\partial \xi ^{2}}}.}$

This equation can also be integrated with respect to ${\displaystyle r}$, but what is required is the solvability condition of the above equation. The solvability condition is obtained by multiplying the above equation by ${\displaystyle 2rdr}$ and integrating the whole equation from ${\displaystyle r=0}$ to ${\displaystyle r=1}$. This is also the same as averaging the above equation over the radial direction. Using the boundary conditions and results obtained in the previous two orders, the solvability condition leads to

${\displaystyle {\frac {\partial c_{0}}{\partial \tau }}=\left(1+{\frac {Pe^{2}}{48}}\right){\frac {\partial ^{2}c_{0}}{\partial \xi ^{2}}}\quad \Rightarrow \quad {\frac {\partial c_{0}}{\partial t}}=\left(1+{\frac {Pe^{2}}{48}}\right){\frac {\partial ^{2}c_{0}}{\partial x^{2}}}.}$

This is the required diffusion equation. Going back to the laboratory frame and dimensional variables, the equation becomes

${\displaystyle {\frac {\partial c_{0}}{\partial t'}}+U{\frac {\partial c_{0}}{\partial x'}}=D\left(1+{\frac {U^{2}a^{2}}{48D^{2}}}\right){\frac {\partial ^{2}c_{0}}{\partial x'^{2}}}.}$

By the way in which this equation is derived, it can be seen that this is valid for ${\displaystyle t'\gg a^{2}/D}$ in which ${\displaystyle c_{0}}$ changes significantly over a length scale ${\displaystyle x'\gg a}$ (or more precisely on a scale ${\displaystyle x\sim {\sqrt {Dt'}})}$. At the same time scale ${\displaystyle t'\gg a^{2}/D}$, at any small length scale about some location that moves with the mean flow, say ${\displaystyle x'-Ut'=x_{s}'-Ut'}$, i.e., on the length scale ${\displaystyle x'-x_{s}'\sim a}$, the concentration is no longer independent of ${\displaystyle r}$, but is given by ${\displaystyle c=c_{0}+{\sqrt {\epsilon }}c_{1}.}$

Integrating the equations obtained at the second order, we find

${\displaystyle c_{2}(\xi ,\tau )=c_{2a}(\xi ,\tau )+{\frac {Pe}{4}}\left(r^{2}-{\frac {r^{4}}{2}}\right){\frac {\partial c_{1a}}{\partial \xi }}+{\frac {Pe^{2}}{32}}\left({\frac {r^{2}}{6}}+{\frac {r^{4}}{2}}-{\frac {5r^{6}}{8}}+{\frac {r^{8}}{8}}\right){\frac {\partial ^{2}c_{0}}{\partial \xi ^{2}}}}$

where ${\displaystyle c_{2a}(\xi ,\tau )}$ is an unknown at this order.

Now collecting terms of order ${\displaystyle \epsilon {\sqrt {\epsilon }}}$, we find

${\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c_{3}}{\partial r}}\right)=Pe(1-2r^{2}){\frac {\partial c_{2}}{\partial \xi }}+{\frac {\partial c_{1}}{\partial \tau }}-{\frac {\partial ^{2}c_{1}}{\partial \xi ^{2}}}.}$

The solvability condition of the above equation yields the governing equation for ${\displaystyle c_{1a}(\xi ,\tau )}$ as follows

${\displaystyle {\frac {\partial c_{1a}}{\partial \tau }}-\left(1+{\frac {Pe^{2}}{48}}\right){\frac {\partial ^{2}c_{1a}}{\partial \xi ^{2}}}=-{\frac {Pe^{3}}{2880}}{\frac {\partial ^{3}c_{0}}{\partial \xi ^{3}}}.}$

• Erdogan–Chatwin equation

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