A S ingle Cell with One-Directional Reaction

1. Dissociation reaction between nitrogen molecules, (N₂ +N₂ →N +N+N₂);

For the 1^st simulation, the initial gas is 100% nitrogen molecules (N₂). The dissociation reactions will occur when two nitrogen molecules collide with each other with enough collision energy. The probability of dissociation uses the total collision energy model (TCE). Table 6.2 is the simulation and theoretical results. Figure 6.2 is the average probability and rate constant of simulation and theoretical data as a function of temperature. In Figure 6.2, the data of simulation are in agreement with the theoretical data except the data at 5,000 K. The reason is that very few reactions occur due to lower reaction probability. Only ten reactions occur in this situation, which leads to inaccuracy by insufficient sampling.

2. Dissociation reaction between nitrogen molecules, (N₂+N →N+N+N );

The 2^nd test case is the dissociation reaction involving the collision between nitrogen molecule (N₂) and atom (N). The initial number densities of both species are the same. Only collision between N₂ and N will be accumulated and processed. Table 6.3 and Figure 6.3 are the comparison of simulation and theoretical results. As mentioned previously, the data of lower temperature is not perfect due to fewer chemical reactions.

3. Recombination reaction between nitrogen molecule and atoms (N+N+N₂ →N₂+N₂);

The 3^rd test case is the recombination reaction of nitrogen atoms (N), and the third body must be nitrogen molecule (N₂). The initial particle densities of N₂ and N are the same. Comparisons between the simulation and the theoretical results are shown in Table 6.4 and Figure 6.4. The agreement is remarkable. But the data of higher temperature is not very close to the theoretical data. The value is overestimated because a continuous reaction probability, i.e. TCE model, is used. The TCE model is constructed in such a manner that it reproduced the rate constant given by the Arrhenius

equation for an equilibrium continuous distribution. This model will has some questions with using discrete vibrational energy model [70].

4. Recombination reaction between nitrogen molecule atoms (N +N+N → N₂+N);

The 4^th test case is the recombination reaction with all nitrogen atoms (N). All the particles are nitrogen atoms. The results are shown in Table 6.5 and Figure 6.5. This simulation has the same trend with the third test case, that is, the error increased with the temperature. The reason has been mentioned in previous section.

5. Dissociation reaction between oxygen molecules, (O₂ +O₂ →O+O+O₂);

After simulating the dissociation and recombination of nitrogen gas, we repeat all types of analysis for oxygen gas as the cases 5~8. The range of temperatures of these simulations is from 3,000 to 7,000 K. Table 6.6 and Figure 6.6 are the simulation and theoretical results. In Figure 6.6, the data of simulation are in agreement with the theoretical data except the data at 3,000 K, which has the same trend as nitrogen cases.

6. Dissociation reaction between oxygen molecule and oxygen atom (O₂ +O→O+O+O);

The 6^th test case is the dissociation reaction involving the collision between oxygen molecule (O₂) and atom (O). The initial number densities of both species are the same.

Table 6.7 and Figure 6.7 are the comparisons of simulation and theoretical results. The agreement is remarkable.

7. Recombination reaction between oxygen molecule and atoms (O+O+O₂ →O₂+O₂);

The 7^th test case is the recombination reaction of oxygen atoms (O), and the third body is oxygen molecule. The initial particle densities of O₂ and O are the same.

Comparisons between the simulation simulation and the theoretical results are shown in Table 6.8 and Figure 6.8. The results are not very close to the theoretical results due to low reaction probability, but it should be acceptable.

8. Recombination reaction between oxygen molecule atoms (O+O+O→O₂+O);

The 8^th test case is the recombination reaction with all oxygen atoms (O) and the third body particle is oxygen atom. All the particles are O in the beginning. Table 6.9 and Figure 6.9 are the comparisons of simulation and theoretical results. By comparing this reaction with case 7, this reaction has higher reaction probability when the third body is O. And the simulated results agree with theoretical results.

9. Dissociation of nitric oxide with nitrogen molecule, (NO+N₂ →N+O+ N₂);

Now, we are interested in the chemical reactions of nitric oxide (NO). Cases 9~13 are the dissociation reaction of nitric oxide (NO) with nitrogen molecule (N₂), oxygen molecule (O₂), nitric oxide (NO), nitrogen atom (N) and oxygen atom (O), respectively.

When nitric oxide collides with diatomic particles, which are cases 9~11, the setting of variables of rate constants are the same. The setting of chem..dat of cases 12 and 13 also are the same for monatomic particles. In case 9, the initial number density of NO and N₂ are the same. Table 10 and Figure 6.10 are the average probability and rate constant of simulation and theoretical data as a function of temperature. The data of simulation are in agreement with the theoretical data.

10. Dissociation of nitric oxide with oxygen molecule, (NO+O₂ → N+O+O₂);

The 10^th test case is the dissociation reaction involving the collision between nitric oxide and oxygen molecules. Table 6.11 and Figure 6.11 are the comparison of simulation and theoretical results. The results are reasonable.

11. Dissociation of nitric oxide with nitric oxide molecule, (NO+NO→ N+O+NO);

The 11^th test case is the dissociation reaction between nitric oxide molecules (NO).

The initial particles are 100% nitric oxide molecules. Comparisons between the simulation and the theoretical results are shown in Table 6.12 and Figure 6.12. The agreement is remarkable.

12. Dissociation of nitric oxide with nitrogen atom, (NO+N →N +O+N );

The 12^th test case is the dissociation reaction of nitric oxide (NO) and nitrogen atom (N). The initial particle number densities of the reactants are the same.

Comparisons between the simulation and the theoretical results are shown in Table 6.13 and Figure 6.13. The simulated results are very close to the theoretical results.

13. Dissociation of nitric oxide with oxygen atom, (NO+O→ N+O+O);

The 13^th test case is the dissociation reaction of nitric oxide (NO) and oxygen atom (O). The initial particle number densities of NO and O are the same. Comparisons between the simulation and the theoretical results are shown in Table 6.14 and Figure 6.14. The simulated results agree with the theoretical results.

14. Recombination of nitric oxide with nitrogen molecule, (N +O+N₂ →NO+ N₂) The following test cases are the reverse reactions of cases 9~13, which are the recombination reaction to form nitric oxide. The third body particles of case 14 to 18 are

N₂, O₂, NO, N and O in order. When the third particles are molecules (cases 14~16), the coefficients of rate constant in chem.dat are the same. For cases 17 and 18, in which the third body particles are monatomic particles, are the same. Case 14 is the recombination reaction with nitrogen molecules (third body particle). Table 6.15 and Figure 6.15 are the comparison of the simulation and theoretical results. The simulated results are scatter and different from the theoretical results due to low reaction probability. Only about 50 real reactions occur in these simulations, which lead to inaccurate results.

15. Recombination of nitric oxide with oxygen molecule, (N +O+O₂ →NO+O₂);

The 15^th test case is the recombination of nitric oxide (NO) and the third body particle are oxygen molecules (O₂). Table 6.16 and Figure 6.16 are the results of simulation and theory. The simulations do not agree perfectly because of the low reaction probability, especially for the case of higher temperature, in which only 13 real reactions occur.

16. Recombination of nitric oxide with nitric oxygen molecule, (N +O+NO→NO+NO);

The 16^th test case is recombination reaction of nitric oxide (NO), and the third body particles are nitric oxides (NO). Comparisons of simulation and theoretical results are shown in Table 6.17 and Figure 6.17. This case has the same situation as case 14 and 15.

The simulated results should be improved by running move time-steps to obtain enough sampling.

17. Recombination of nitric oxide with nitrogen atom, (N+O+N→NO+N );

The 17^th test case is the recombination to form nitric oxide (NO) by colliding with the third body particle, which is nitrogen atom (N). Table 6.18 and Figure 6.18 are the comparisons between simulation and theoretical results. From Table 6.18, the reaction probability is higher than the reaction with monatomic third body particle. But the samplings are still not enough to obtain accurate results.

18. Recombination of nitric oxide with oxygen atom, (N +O+O→NO+O);

The 18^th test case is the final simulation about the recombination of nitric oxide (NO), and the third body particle is oxygen atom (O). Table 6.19 and Figure 6.19 are the results of simulation and theory. In conclusion, the reaction probability of recombination is lower than the dissociation. The recombination reaction needs more time-steps to obtain enough sampling and accurate results.

19. Exchange reaction between nitrogen molecule and oxygen atom

(N₂+O→NO+N)

After verifying the dissociation and recombination reaction, the following benchmarks are about the exchange reaction between molecules and atoms. The 19^th test case is the exchange reaction (forward) between nitrogen molecule (N₂) and oxygen atom. In this case, nitrogen molecule will dissociate into two nitrogen atoms and exchange one of the nitrogen atoms with the oxygen atom, then forms one nitric oxide molecule and one nitrogen atom. The initial particle number densities of N₂ and O are the same. Only collisions between these two species will be accumulated to calculate the reaction probability. The results are shown in Table 6.20 and Figure 6.20. The simulated results agree perfectly with the theoretical data.

20. Exchange reaction between nitric oxide and nitrogen atom (NO+N →N₂ +O) The 20^th test case is the exchange reaction between nitric oxide (NO) and nitrogen atom (N), which is the reverse reaction of the 19^th test case. The initial particle number densities of NO and N are the same. The results are shown in Table 6.21 and Figure 6.21. Although the simulated results have some discrepancies, the simulations are acceptable.

21. Exchange reaction between nitric oxide and oxygen atom (NO+O→O₂+N) The 21^st test case is also the exchange reaction (forward), which is a reaction between nitric oxide (NO) and oxygen atom (O). The initial particle number densities of NO and O are the same. The results are shown in Table 6.22 and Figure 6.22. These simulated results are also remarkable by comparing to the theoretical data.

22. Exchange reaction between oxygen molecule and nitrogen atom (O₂ +N →NO+O)

The 22^nd test case is the exchange reaction between oxygen molecule (O₂) and nitrogen atom (N), which is the reverse reaction of 21^st test case. The initial particle number densities of O₂ and N are the same. The results are shown in Table 6.23 and Figure 6.23. The simulated results are in good agreement with theoretical data.