Dispersion of soluble matter in tube

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Discussion Overview

The discussion revolves around the article "Dispersion of soluble matter in solvent flowing slowly through a tube" by Sir Geoffrey Taylor, focusing on cases A2 and B2. Participants explore the implications of these cases regarding the concentration profiles of a solution injected into a tube filled with solvent, particularly the differences between models that disregard diffusion and those that include radial diffusion.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Jens expresses confusion over why case B2, which includes radial diffusion, extends infinitely in both directions while case A2, which disregards diffusion, does not.
  • Another participant argues that equation (29) does not provide a continuum between cases A and B, suggesting that minimal diffusion cannot be reconciled with the assumptions of case B.
  • Jens notes that both solutions have simplifying assumptions and questions which assumptions lead to the differences in concentration profiles.
  • One participant explains that the assumption of negligible radial variation in concentration in case B2 implies very fast fluid velocity or being far downstream, contributing to the infinite extension of the solution.
  • Jens seeks a method to determine when case A2 is sufficient for practical purposes and when to resort to the more complex B2 calculations, considering the Peclet number as a potential criterion.
  • Jens acknowledges the challenges of solving the general case and contemplates using specific concentration drop distances for practical illustration.

Areas of Agreement / Disagreement

Participants express varying degrees of understanding and agreement regarding the implications of the models, with no consensus on the assumptions leading to the differences in solutions or the criteria for when to use each case.

Contextual Notes

Participants note that both cases involve simplifying assumptions, and there is uncertainty regarding the conditions under which case A2 can be considered adequate compared to case B2.

Who May Find This Useful

Readers interested in fluid dynamics, dispersion phenomena, or those working on related engineering projects may find this discussion relevant.

jencam
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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
 
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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?
 
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
 
jencam said:
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.

jencam said:
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.
 
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
 

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