# Diffusion distance (microfluidics)

1. Mar 3, 2015

### Soren

The diffusion distance for a Brownian particle is given as x^2 = 2Dt
D: diffusion coeficient, t: time
However, not all textbooks include the factor 2 (i.e. they just write x^2=DT). Is there a difference or are they just beeing sloppy?

I run into trouble when I want to calculate the lenght of a channel required for mixing two fluids in a laminar flow based on the Péclet number
Simple explanation here: http://en.wikibooks.org/wiki/Microfluidics/Mixing
Here the time it takes for two fluids to mix (diffuse across the channel width) is only given as t = x^2/D
The factor 2 seems to be missing.

2. Mar 3, 2015

### Staff: Mentor

I can help you with this, but please understand that there is not going to be an exact number for x. There will be one distance to get 90% of the way there, a longer distance to get 99% of the way there, and still longer distance to get 99.9 % of the way there, but, mathematically the distance to get 100% of the way there will be infinite. Also understand that the distances involved are going to be functions of the specific problem.

Now please tell me in detail your laminar flow diffusional mixing problem: geometry, placement of concentrations, etc. If you can, also write out the differential equation and boundary conditions.

Chet

3. Mar 3, 2015

### Soren

I'm not looking for a numerical solution, I'm just trying to understand.

I was looking at a case with a strictly laminar pipeflow (i.e. circular geometry). For such a flow the mean velocity (u) =Q/ pi*x^2, with x being the diameter and Q the flowrate = u/pi*x^2.
Now if I have two fluid inlets that mix (like the classic T-mixer) mixing occurs per diffusion across the diameter of the pipe and the mixing time (td) =x^2/D.
Now I also know that the mixing length (length of the channel required for mixing) ld =td*u. By simple insertion of u and ld it is easy to see that ld=u*x^2/2D. So far so good.

However the Péclet number (Pe) can also be used to find the mixing length, as ld = Pe*x. Now, the subsition of the Péclet number as defined in original post I get
ld = ux^/D

So there is a factor of 2 missing in the Péclet definition?

4. Mar 3, 2015

### Staff: Mentor

These are only very crude order of magnitude approximations (rules of thumb), so a factor of 2 difference is nothing to make a big deal about. If you want to know a more accurate answer, you need to solve the partial differential equations for the diffusional mass transfer in the pipe. This can be done either analytically or numerically for either a flat velocity profile or a parabolic velocity profile.

Chet

5. Mar 5, 2015

### Soren

Thank you Chet - this answers my question.

On a further note I am actually looking into the differential equations (unsteady diffusion in finite medium actually) and may have further qestions, do you have experience with these?

6. Mar 5, 2015

### Staff: Mentor

Yes. This same mathematical framework also applies to transient heat transfer situations involving either conduction or conduction and convection. So lots of information and techniques are applicable from that area.

Chet

7. Mar 10, 2015

### Soren

So I have read up a bit on the subject. I want to find a solution for fick's second law of diffusion in my described problem (see attached pdf). In brief it is diffusion from the walls of a tube into the bulk solution. First step is to make sure that I have the proper initial and boundary conditions. I think I have them sorted out, but am really relying on some textbook examples, that I think are comparable. Supposing I've got things right, what do I do from here? (separation of variables??)

#### Attached Files:

• ###### Diffusion problem.pdf
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8. Mar 10, 2015

### Staff: Mentor

You really have two separate problems here. First you have a transient convection/diffusion problem for the concentration profile as a function of r, z, and t when fluid of concentration A is displacing fluid of concentration B in your tube. Whenever you shut off this flow, the concentration is not going to be just a function of r, but also a function of z. However, you are assuming that it is just a function of r. This then provides the initial condition for your second problem. If you really want to do this right, you need to do a better job of establishing the initial concentration profile for the second problem.

The second problem, as you have described it, is for transient radial diffusion of a material away from a wall, given a small slug of higher concentration immediately adjacent to the wall. Depending on what you are trying to determine specifically, there are approximations that can be made that can greatly simplify the solution to this problem. Do you want to follow the development of the concentration profile all the way to the point where the concentration is fully homogenized, or would it be adequate to follow it for a shorter amount of time, say until the concentration effectively begins changing at maybe 2/3 of the way out from the centerline?

Chet

9. Mar 10, 2015

### Soren

Yes, I am aware that setting the initial concentration to be evenly distributed over r and z, and ignoring the convective dispersion is a big simplification (I am aware of taylor-aris dispersion) But because of the dimensions, the velocity and time that the flow is switched on, I believe that at least the concentration gradient in the z direction (direction of flow) is well approximated dc/dz = 0. I suppose that finding the concentration gradient dc/dr (or concentration profile over r) would be of some interest.

My primary concern in regards to the second problem, is to calculate the time required for the concentration to be equalized over r. At least to the point where the concentration at r= 0 is 95% of the slab concentration. I have some experimental data that I wish to compare my results with, and see if things are occurring on at least the same timescale.

I am aware that I can simplify matters to some extend by treating this a as diffusion from a point source, and using a semi-infinite medium approximation would be adequate for a short time abd I think the result would somewhat like the propability density function. If there is an approximation that could at least point towards what kind of function (square, errorfunction, etc.) the concentration as a function of time in the slab occurs as, then that would be a good start.

10. Mar 10, 2015

### Staff: Mentor

I have a number of ideas on how to get at this, if your main objective is an order of magnitude estimate.

If you are willing to ease up on the cylindrical geometry temporarily, then you can use an infinite sequence of dirac delta functions for the initial concentration (in flat geometry), spaced a diameter distance apart, and add up the contributions of all the spikes. This will simulate the condition of a finite region, because the fluxes half way between the spikes will be zero. So it's a kind of method of images.

Another thing you can do is to look up the problem where you have a cylinder, and, at time zero, you suddenly change the concentration at r = R to a new value. The time for the concentration at the center to get to 95% of the concentration at the wall will give you a good order of magnitude estimate for your problem.

Another thing you can do is to look up the problem of applying a constant flux of material to the wall and holding it at that value for all time. Then you can take this solution and, by first applying a constant positive flux for a short time, and then superimposing the same solution with a negative flux after the time interval, you will obtain the solution for a pulse over the time interval followed by zero wall flux after that. This is pretty much identical to what you have. You just add the two solutions at the centerline, and compare that with the sum of the two solutions at the wall (of course, with the two solutions separated by the time delay).

Chet

11. Mar 10, 2015

### Staff: Mentor

This is a continuation of my previous post. If f(r,t) is the solution to the problem of constant mass flux to a cylinder, then the solution for the pulse problem is $Δt_p\frac{\partial f}{\partial t}$, where Δtp is the duration of the pulse.

Chet

12. Mar 11, 2015

### Soren

This is exactly what I have been trying to "look up" and it does not exist, at least I cannot find it.

13. Mar 11, 2015

### Staff: Mentor

Transport Phenomena, Bird, Stewart, and Lightfoot, Fig. 12.1-2, p. 378

Conduction of Heat in Solids, Carslaw and Jaeger, p. 200

Chet

14. Mar 11, 2015

### Soren

Ah, yes. I have similar example in my textbook, however, as I understand this then the temperature (or concentration) imposed on the radius =R does not vary with time, i.e. it is constant for t>0.

I think your latter suggestion that simulates a pulse is interesting, although I am unsure how to go about it as my experienced with PDE's is just about nill.

15. Mar 11, 2015

### Staff: Mentor

I'm pretty sure Carslaw and Jaeger has the solution to this problem in its compilation.

Chet