# Combustion models. How does one determine the most relevant reactions?

I am not entirely sure if this is better suited to Chemistry.SE or Matter modelling, but hey, you guys are nicer.

I am fascinated by just how complicated combustion processes are.

I was watching an old SpaceX video on how they model the combustion processes in their Raptor engines. (Here is a really relevant video with timestamp, which I recommend people watch in full.)

They give the example of hydrogen burning with oxygen, which requires 23 separate chemical equations to model, and that is with the reduced mechanism.

(Incidentally, I would like to know what column 'b' refers to in the table above.)

The situation with methane combustion is even worse. 53 species and 325 reactions are involved.

If one were to attempt to model simplified versions of these processes, how would they go about it?

Is there an a-priori way of telling which individual reactions contribute little to the overall combustion, just by the identity of the species involved and the other entries in the table? Intuitively, one expects the interactions of two different rare high-energy intermediates to not contribute greatly to the overall combustion, but what if it does?

While highly nonlinear, I don't believe these equations are chaotic, at least in a situation with perfect mixing. Even so, I suspect changes in the proportion of trace intermediates could very easily have large effects on subsequent reactions.

Does it come down to running the full simulation, then successively removing reactions until the accuracy drops below an acceptable threshold? Or are there tricks one can use to simplify these equations?

• I deleted my comment after I watched the video, since it is about kinetic modeling. Aug 15, 2021 at 2:27
• However, for chemical reaction equilibria, you choose the set of reactions that works best for your numerical solver. Some folks change the reaction sets on the fly in their solver. Aug 15, 2021 at 2:28
• I don't know much about this so I hesitate to answer, but one approach is to use a master equation: en.wikipedia.org/wiki/Master_equation . In that approach, you have to have some idea of the rate of moving between any two of the states described by the reactions you've listed. This rate is then related to a probability that enters a transition matrix, and the entire system is modelled via a Markov process. This allows you to find out how various initial conditions will behave over time and what possible equilibria can be achieved. Aug 16, 2021 at 21:32
• @ingolifs would you be interested in my putting how to determine a set of reactions for chemical reaction equilibria, how there are infinite possibilities, and how people more in the business than myself choose their sets of reactions? It wouldn't answer your question directly, but I would include a monologue about how SpaceX is probably doing it the overly hard way and how that doesn't surprise me :) Aug 19, 2021 at 13:54
• +1 (I gave it earlier, but didn't get a chance to comment at that time). Welcome to our new community and thank you so much for contributing your question here! We hope to see much more of you in the future !!! Here's a related (though not the same) question that I answered on this site: mattermodeling.stackexchange.com/a/6584/5. I think your question could be answered by this community if we wait long enough, but a bounty would probably get people more motivated to put the effort in! Aug 21, 2021 at 18:06

## 1 Answer

One way to model a multistep complex mechanism is to built a microkinetic model from the rate constants for of all relevant elementary reactions (forward and reverse).

For example, for reaction 1 you have

$$\ce{H2 + O2 ->[k_1] 2OH}$$

with $$k_1(T)$$ given by some parameters in a modified Arrhenius expression like $$k(T) = AT^b e^{-E_\mathrm{A}/RT}$$ or $$k(T) = A(T/T_0)^b e^{-E_\mathrm{A}/RT}$$. It is hard to say which one is used, or if it is some other form, or to be sure of the units without more information.

The net reaction rate, $$r$$, can be expressed as the rate of formation of any chosen product of the consumption of any chosen reactant and it is generally a complex function of the rate constants for all of its elementary steps and the thermodynamics of all of its reaction intermediates.

For example: $$r = -\frac{d\ce{[H2]}}{dt}$$

And the concentration of $$\ce{H2}$$ is affected by every elementary reaction that produces or consumes it.

$$\frac{d\ce{[H2]}}{dt} = -k_1 \ce{[H2][O2]} + k_{-1}\ce{[OH]^2} - k_2\ce{[OH][H2]} + k_{-2}\ce{[H2O][H]} - k_4\ce{[O][H2]} + k_{-4}\ce{[OH][H]} + k_{10}\ce{[H]^2[M]} - k_{-10}\ce{[H2][M]} +k_{11}\ce{[H]^2[H2]} - k_{-11}\ce{[H2]^2} + k_{12}\ce{[H]^2[H2O]} - k_{-12}\ce{[H2][H2O]} + k_{16}\ce{[H][HO2]} - k_{-16}\ce{[H2][O2]} + k_{19}\ce{[H][H2O2]} - k_{-19}\ce{[HO2][H2]}$$

where $$k_i$$ are the rate constant for reaction $$i$$ and $$k_{-i}$$ is the rate for the reverse of $$i$$. Both rate constants are related to each other by the equilibrium constant $$K_i = \frac{k_i}{k_{-i}} = \exp(-\Delta_\mathrm{rxn}G_i/RT)$$ and $$\Delta_\mathrm{rxn}G_i$$ is the Gibbs free energy change for reaction $$i$$.

One can integrate the concentrations of all species involved for a given set of initial conditions.

To identify which elementary steps in the complex mechanisms are most important to the overall reaction rate one can use the degree of rate control $$X_{\mathrm{RC} i}$$ for elementary step $$i$$:

$$X_{\mathrm{RC}, i}=\frac{k_{i}}{r}\left(\frac{\partial r}{\partial k_{i}}\right)_{k_{j \neq i} K_{i}}=\left(\frac{\partial \ln r}{\partial \ln k_{i}}\right)_{k_{j \neq i}, K_{i}}$$

The larger the numeric value of $$X_{\mathrm{RC}, i}$$ is for a given step, the bigger is the influence of its rate constant on the overall reaction rate $$r$$.

Of course that one can do more complicated things such as including the heat and mass transfer across the system.

## References

1. D. A. McQuarrie, Physical chemistry: A molecular approach (University Science Books, Sausalito, CA, 1997).

2. C. Stegelmann, A. Andreasen, C. T. Campbell, Degree of rate control: how much the energies of intermediates and transition states control rates. J. Am. Chem. Soc. 131, 8077–8082 (2009).

3. https://kinetics.nist.gov/kinetics/

• this is a nice answer Apr 11 at 12:55