Dispersion of soluble matter in tube

[jencam]

Hi

I am reading and trying to comprehend the article "Dispersion of soluble matter in solvent flowing slowly through a tube", Sir Geoffrey Taylor, Proceedings of the Royal Society of London, 1953.

I am particularly interested in cases A2 and B2, where a concentrated solution is injected into one end of a tube filled with solvent only.

Case A2 handles the (average) concentration at different positions in the tube disregarding diffusion, which happens to become a linear decrease in concentration.

Case B2 generalizes the model to include radial diffusion still ignoring axial diffusion. The solution in this case becomes a symmetric erf function, which principally extends infinitely in both directions.

What bothers me is that when flow increases or coefficient of diffusion decreases, case B2 should in my opinion asymptotically fall back to case A2. Principally I would think the concentration profile should be limited in length - we have no axial diffusion so the concentrated part shouldn't be able to extend beyond case A2.

I don't know if I am plain stupid or if this is due to some assumptions that are not fulfilled at low-D.

Can anyone please give me a hint?

Regards

--Jens

 

[Mapes]

But equation (29) doesn't describe a continuum between cases A and B, it only describes case B, where convective effects are already assumed to be minimal. That being assumed, we can't then consider the case of minimal diffusion and hope to recover equation (6). We would need a general solution of equation (11) to be able to recover the extreme cases A and B. Does this make sense?

 

[jencam]

I understand neither solution is complete and simpliying assumptions have been made in both cases. I don't see which of the assumptions that causes the elongation of the slope to (-infinity,+infinity) instead of (0,u0). If there were axial diffusion I would understand. I did the experiment with syrup (colored/colorless) and understand that the diffusion-less theory is pretty accurate.

What I really need for my project is a means to determine when case A2 is "good enough", when I can estimate the concentrated part as a plain "edge" and when I need to use the complicated B2 calculations. I can do this with the equations in the article so principally I don't "need" the generic result that works in all cases. I am just a bit unsatisfied with it. A "full" solution would be better but may of course not be possible. An alternative would be for me to understand *why* the solutions are so different.

As an engineer I could of course invent an interpolation model between the cases. This isn't a very scientific approach though ;-).

Regards

--Jens

 

[Mapes]

I understand neither solution is complete and simpliying assumptions have been made in both cases. I don't see which of the assumptions that causes the elongation of the slope to (-infinity,+infinity) instead of (0,u0). If there were axial diffusion I would understand. I did the experiment with syrup (colored/colorless) and understand that the diffusion-less theory is pretty accurate.

At the top of p191 Taylor assumes that any radial variation in concentration is negligible. This is equivalent to assuming that the fluid velocity is very fast, or equivalently that we are very far downstream of where the solute was introduced. That's why the B2 solution extends essentially to infinity.

You may already see this, but the way I think about Taylor dispersion is that the parabolic profile of fully developed flow creates relatively large concentration gradients as high-solute fluid is continually positioned next to low-solute fluid. These gradients promote fast mixing that smears out radial nonuniformities.

What I really need for my project is a means to determine when case A2 is "good enough", when I can estimate the concentrated part as a plain "edge" and when I need to use the complicated B2 calculations.

This may be the Peclet number, but check this.

 

[jencam]

OK, I see now, and appreciate the difficulties solving the general case. I wish I had more experience with partial differential equations...

I'll look into the Peclet number but I think just estimating the distance to fall from e.g. 90 to 10% concentration (or whatever I am interested in) as in equation (30) will do. This will have an illustrative meaning to people that haven't studied physics.

Thanks for helping me out.

--Jens