simulations - Reaction-diffusion. From kinetic parameters in 3D, can we infer their equivalent in 1D?

I am studying a reaction-diffusion system : $A+B ⇌_{k_{-}}^{k_{+}} C$. From experimental data I have all the kinetic parameters : diffusion coefficients $d$ and reaction rates $k_+$, $k_-$. Beside I am solving numerically the reaction diffusion equations. For example for A:

$ \frac{\partial A}{\partial t}=-k_+ A B + k_+C + d_A \nabla^2 A \\ $

First I wish to simulate a system in only one dimension. From the coefficients found experimentally, is there any way to find what should be the 1D coefficients? If that makes any sense at all.

Answer

There is no difference based on dimensionality.

The rate parameters $k$ are a way of saying "When an $A$ molecule collides with a $B$ molecule with enough energy, a $C$ molecule is produced." In other words, it's based only on the number of collisions per unit time, which is an isotropic property (at least in gases). It is a point operation occurring in an infinitesimal volume -- it is a zero dimensional (0D) property.

Likewise your diffusion coefficients are based on things like the cross-sectional area of the colliding molecule pairs $(A, B)$, $(A, C)$ and $(B, C)$ which is independent of the number of dimensions. It may also depend on the forces between molecules like VDW forces if those happen to exist, which again are not dependent on the number of dimensions.

So -- your 3D coefficients should match your 2D which should match your 1D. For that matter, it should match in a 0D reactor as well (where you would drop the diffusion obviously since there are no more spatial gradients).

Where you might see differences -- 3D tends to have "relieving" effects, in the sense that a change in pressure in 1D only has 2 directions to spread while one in 3D has 6 directions to spread. This shows up in things like turbulence too. So your 1D results may be different from the 3D results when using the same numbers.

If this occurs, you have two choices:

1. 
   

   If the differences are small you can say they are negligible and that 1D is a good approximation to the system. If they are large, then the correct choice is to say that 1D is not a suitable model for the system and you must simulate it in higher dimensions to capture the relevant physics of the problem.

   
   



2. 
   

   Re-calibrate your coefficients for 1D. You would basically start with what you have and tweak $k$ and $d_i$ until your results matched the 3D. This can be an acceptable choice under some limited circumstances, but to be valid you would need to show that your new values of $k$ and $d_i$ reproduced the same physics as the original set. So for instance, change equivalence ratio and you should still match the 3D results without having to tweak your numbers again. This may only work for a narrow range of $\phi$ but if that range is what you are studying then maybe it's fine. On the other hand, if you have to tweak the numbers for every case then that will never be acceptable as a rigorous scientific approach. So proceed with caution in that direction.

   
   

Then again, it may depend on what you are looking at. Maybe peak temperature is too high in 1D but flame speed matches exactly. If you are interested in flame speed, you can say that the relieving effects don't matter. If you are looking at peak $T$ then you have a problem.

All of this is to say that you have to very clearly and carefully construct both the assumptions and goals of you analysis and frame all of your simulations within those. Document everything, justify everything, and if you find that your model just isn't useful then you'll have to fix it.