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:

: $\backslash boldsymbol\{u\}\; =\; w\backslash hat\{\backslash boldsymbol\{z\}\}\; =\; w\_0\; (1-r^2/a^2)\; \backslash hat\{\backslash boldsymbol\{z\}\}$

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:

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

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:

: $w(r)\; =\; \backslash bar\{w\}\; +\; w\text{'}(r)$

: $c(r,z)\; =\; \backslash bar\{c\}(z)\; +\; c\text{'}(r,z)$

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

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

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:

: $D\_\{\backslash mathrm\{eff\}\}\; =\; D\; \backslash left(\; 1\; +\; \backslash frac\{\backslash mathit\{Pe\}^\{2\}\}\{48\}\; \backslash right)\backslash ,\; ,$

where $\backslash mathit\{Pe\}=a\backslash bar\{w\}/D$ is the Péclet number, based on the channel radius $a$. 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 $\backslash xi=z-\backslash bar\; w\; t$, the dispersion process becomes a purely diffusion process,

: $\backslash frac\{\backslash partial\; \backslash bar\{c\}\}\{\backslash partial\; t\}\; =\; D\_\{\backslash mathrm\{eff\}\}\backslash frac\{\backslash partial^2\; \backslash bar\{c\}\}\{\backslash partial\; \backslash xi\; ^2\}$

with diffusivity given by the effective diffusivity.

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

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

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.{{cite journal | title= Shear dispersion in a porous medium. Part 1. An intrusion with a steady shape| year=2020 |journal=Journal of Fluid Mechanics| volume=899 | doi=10.1017/jfm.2020.478| bibcode=2020JFM...899A..38H| doi-access=free | last1=Hinton | first1=Edward M. | last2=Woods | first2=Andrew W. | hdl=11343/274191 | hdl-access=free }}

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 $(x\text{'},r\text{'},\backslash theta)$, consider the fully-developed Poiseuille flow $u=2\; U\; [1-(r\text{'}/a)^2]$ flowing inside a pipe of radius $a$, where $U$ is the average velocity of the fluid. A species of concentration $c$ with some arbitrary distribution is to be released at somewhere inside the pipe at time $t\text{'}=0$. 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 $U$. In a frame moving with the mean velocity and scaled with following non-dimensional scales

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

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

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

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

Suppose $t\backslash sim\; 1/\backslash epsilon\; \backslash gg\; 1$ (i.e., times large in comparison with the radial diffusion time $a^2/D$), where $\backslash epsilon\; \backslash ll\; 1$ is a small number. Then at these times, the concentration would spread to an axial extent $x\backslash sim\; \backslash sqrt\; t\backslash sim\; \backslash sqrt\{1/\backslash epsilon\}\backslash gg\; 1$. To quantify large-time behavior, the following rescalings{{cite journal | doi=10.1080/13647830.2020.1813335 | title=Taylor-diffusion-controlled combustion in ducts | year=2020 | last1=Liñán | first1=Amable | last2=Rajamanickam | first2=Prabakaran | last3=Weiss | first3=Adam D. | last4=Sánchez | first4=Antonio L. | journal=Combustion Theory and Modelling | volume=24 | issue=6 | pages=1054–1069 | bibcode=2020CTM....24.1054L | s2cid=225262971 }}

:$\backslash tau\; =\; \backslash epsilon\; t,\; \backslash quad\; \backslash xi\; =\; \backslash sqrt\backslash epsilon\; x$

can be introduced. The equation then becomes

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

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

Since $\backslash epsilon\; \backslash ll\; 1$, the solution can be expanded in an asymptotic series, $c=c\_0\; +\; \backslash sqrt\backslash epsilon\; c\_1\; +\backslash epsilon\; c\_2\; +\; \backslash cdots$ 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

:$\backslash frac\{1\}\{r\}\backslash frac\{\backslash partial\; \}\{\backslash partial\; r\}\backslash left(r\backslash frac\{\backslash partial\; c\_0\}\{\backslash partial\; r\}\backslash right)=0.$

Integrating this equation with boundary conditions defined before, one finds $c\_0=c\_0(\backslash xi,\backslash tau)$. At this order, $c\_0$ is still an unknown function. This fact that $c\_0$ is independent of $r$ is an expected result since as already said, at times $t\text{'}\backslash gg\; a^2/D$, the radial diffusion will dominate first and make the concentration uniform across the pipe.

Terms of order $\backslash sqrt\backslash epsilon$ leads to the equation

:$\backslash frac\{1\}\{r\}\backslash frac\{\backslash partial\; \}\{\backslash partial\; r\}\backslash left(r\backslash frac\{\backslash partial\; c\_1\}\{\backslash partial\; r\}\backslash right)=Pe\; (1-2r^2)\backslash frac\{\backslash partial\; c\_0\}\{\backslash partial\; \backslash xi\}.$

Integrating this equation with respect to $r$ using the boundary conditions leads to

:$c\_1(\backslash xi,r,\backslash tau)\; =\; c\_\{1a\}(\backslash xi,\backslash tau)\; +\; \backslash frac\{Pe\}\{8\}(2r^2-r^4)\backslash frac\{\backslash partial\; c\_0\}\{\backslash partial\; \backslash xi\}$

where $c\_\{1a\}$ is the value of $c\_1$ at $r=0$, an unknown function at this order.

Terms of order $\backslash epsilon$ leads to the equation

:$\backslash frac\{1\}\{r\}\backslash frac\{\backslash partial\; \}\{\backslash partial\; r\}\backslash left(r\backslash frac\{\backslash partial\; c\_2\}\{\backslash partial\; r\}\backslash right)=Pe\; (1-2r^2)\backslash frac\{\backslash partial\; c\_1\}\{\backslash partial\; \backslash xi\}\; +\; \backslash frac\{\backslash partial\; c\_0\}\{\backslash partial\; \backslash tau\}\; -\; \backslash frac\{\backslash partial^2\; c\_0\}\{\backslash partial\; \backslash xi^2\}.$

This equation can also be integrated with respect to $r$, but what is required is the solvability condition of the above equation. The solvability condition is obtained by multiplying the above equation by $2r\; dr$ and integrating the whole equation from $r=0$ to $r=1$. 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

:$\backslash frac\{\backslash partial\; c\_0\}\{\backslash partial\; \backslash tau\}\; =\backslash left(1+\backslash frac\{Pe^2\}\{48\}\backslash right)\; \backslash frac\{\backslash partial^2\; c\_0\}\{\backslash partial\; \backslash xi^2\}\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash frac\{\backslash partial\; c\_0\}\{\backslash partial\; t\}\; =\backslash left(1+\backslash frac\{Pe^2\}\{48\}\backslash right)\; \backslash frac\{\backslash partial^2\; c\_0\}\{\backslash partial\; x^2\}.$

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

:$\backslash frac\{\backslash partial\; c\_0\}\{\backslash partial\; t\text{'}\}\; +\; U\; \backslash frac\{\backslash partial\; c\_0\}\{\backslash partial\; x\text{'}\}\; =D\backslash left(1+\backslash frac\{U^2\; a^2\}\{48\; D^2\}\backslash right)\; \backslash frac\{\backslash partial^2\; c\_0\}\{\backslash partial\; x\text{'}^2\}.$

By the way in which this equation is derived, it can be seen that this is valid for $t\text{'}\backslash gg\; a^2/D$ in which $c\_0$ changes significantly over a length scale $x\text{'}\backslash gg\; a$ (or more precisely on a scale $x\backslash sim\; \backslash sqrt\{Dt\text{'}\})$. At the same time scale $t\text{'}\backslash gg\; a^2/D$, at any small length scale about some location that moves with the mean flow, say $x\text{'}-Ut\text{'}=x\_s\text{'}-Ut\text{'}$, i.e., on the length scale $x\text{'}-x\_s\text{'}\backslash sim\; a$, the concentration is no longer independent of $r$, but is given by $c=c\_0\; +\; \backslash sqrt\{\backslash epsilon\}\; c\_1.$

Integrating the equations obtained at the second order, we find

:$c\_2(\backslash xi,\backslash tau)\; =\; c\_\{2a\}(\backslash xi,\backslash tau)\; +\; \backslash frac\{Pe\}\{4\}\backslash left(r^2-\backslash frac\{r^4\}\{2\}\backslash right)\; \backslash frac\{\backslash partial\; c\_\{1a\}\}\{\backslash partial\backslash xi\}\; +\; \backslash frac\{Pe^2\}\{32\}\backslash left(\backslash frac\{r^2\}\{6\}+\backslash frac\{r^4\}\{2\}-\backslash frac\{5r^6\}\{8\}+\backslash frac\{r^8\}\{8\}\backslash right)\; \backslash frac\{\backslash partial^2c\_0\}\{\backslash partial\backslash xi^2\}$

where $c\_\{2a\}(\backslash xi,\backslash tau)$ is an unknown at this order.

Now collecting terms of order $\backslash epsilon\backslash sqrt\backslash epsilon$, we find

:$\backslash frac\{1\}\{r\}\backslash frac\{\backslash partial\; \}\{\backslash partial\; r\}\backslash left(r\backslash frac\{\backslash partial\; c\_3\}\{\backslash partial\; r\}\backslash right)=Pe\; (1-2r^2)\backslash frac\{\backslash partial\; c\_2\}\{\backslash partial\; \backslash xi\}\; +\; \backslash frac\{\backslash partial\; c\_1\}\{\backslash partial\; \backslash tau\}\; -\; \backslash frac\{\backslash partial^2\; c\_1\}\{\backslash partial\; \backslash xi^2\}.$

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

:$\backslash frac\{\backslash partial\; c\_\{1a\}\}\{\backslash partial\; \backslash tau\}\; -\backslash left(1+\backslash frac\{Pe^2\}\{48\}\backslash right)\; \backslash frac\{\backslash partial^2\; c\_\{1a\}\}\{\backslash partial\; \backslash xi^2\}\; =\; -\backslash frac\{Pe^3\}\{2880\}\backslash frac\{\backslash partial^3c\_0\}\{\backslash partial\backslash xi^3\}.$

{{reflist}}

• {{cite journal |doi=10.1098/rspa.1956.0065 |title=On the dispersion of a solute in a fluid flowing through a tube |journal=Proceedings of the Royal Society of London, Series A |year=1956 |volume=235 |issue=1200 |pages=67–77 |bibcode=1956RSPSA.235...67A |last1=Aris |first1=R. |s2cid=95229777 }}
• {{cite journal |doi=10.1017/S0022112089001679 |title=On the foundations of generalized Taylor dispersion theory |year=1989 |last1=Frankel |first1=I. |last2=Brenner |first2=H. |journal=Journal of Fluid Mechanics |volume=204 |page=97 |bibcode=1989JFM...204...97F |s2cid=123719494 }}
• {{cite journal |doi=10.1098/rspa.1953.0139 |title=Dispersion of soluble matter in solvent flowing slowly through a tube |journal=Proceedings of the Royal Society of London, Series A |year=1953 |volume=219 |issue=1137 |pages=186–203 |bibcode=1953RSPSA.219..186T |last1=Taylor |first1=Geoffrey |s2cid=97372019 }}
• {{cite journal |doi=10.1098/rspa.1954.0130 |title=The dispersion of matter in turbulent flow through a pipe |journal=Proceedings of the Royal Society of London, Series A |year=1954 |volume=223 |issue=1155 |pages=446–468 |bibcode=1954RSPSA.223..446T |last1=Taylor |first1=Geoffrey |s2cid=96182418 }}
• {{cite journal |doi=10.1098/rspa.1954.0216 |title=Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion |journal=Proceedings of the Royal Society of London, Series A |year=1954 |volume=225 |issue=1163 |pages=473–477 |bibcode=1954RSPSA.225..473T |last1=Taylor |first1=Geoffrey |s2cid=97701431 }}
• {{cite journal |doi=10.1098/rsta.1980.0205 |title=Dispersion resulting from flow through spatially periodic porous media |journal=Philosophical Transactions of the Royal Society of London, Series A |year=1980 |volume=297 |issue=1430 |pages=81–133 |bibcode=1980RSPTA.297...81B |last1=Brenner |first1=H. |s2cid=121853893 }}
• Mestel. J. [http://www.ma.ic.ac.uk/~ajm8/M4A33/taylor.pdf Taylor dispersion — shear augmented diffusion], Lecture Handout for Course M4A33, Imperial College.

Category:Fluid mechanics

Category:Fluid dynamics