# NOx Formation Mechanism and Emission Prediction in Turbulent Combustion: A Review
## 3 NOx Formation-Kinetics Approach
### Simplified Kinetics  
The first method is to estimate NOx formation using a steady-state hypothesis. By classifying the formation mechanism of nitrogen oxides, the formation rate of nitrogen oxides is expressed in the form of compound concentrations product. By using the quasi-steady-state hypothesis and the steady-state hypothesis, the concentration of radicals in the elementary reaction is expressed by the concentration of compounds, thus reducing the difficulty of calculation. The concentration field distribution of NOx was obtained by solving the transport equation of species. Generally, this method mainly was involved in the post-treatment of CFD, and it is believed that the generation of nitrogen oxides is small and has a weak influence on the generation process of fuel combustion.  
Taking the thermal mechanism as an example, the nitrogen oxide generation rate, synthesized by the reaction of the three elementary elements, can be expressed as  
\[\frac{\mathrm{d[NO]}}{\mathrm{dt}}=\mathrm{k_{1}[N_{2}][O\cdot]+k_{2}[N\cdot][O _{2}]+k_{3}[N\cdot][OH\cdot]}, \tag{10}\]  
Combined on the assumption of the quasi-equilibrium of [N-]  
\[\frac{\mathrm{d[N\cdot]}}{\mathrm{dt}}=\mathrm{k_{1}[N_{2}][O\cdot]-k_{2}[N \cdot][O_{2}]-k_{3}[N\cdot][OH\cdot]}=0, \tag{11}\]  
Combining these formulas, the thermal NO production rate is [36]  
\[\frac{\mathrm{d[NO]}}{\mathrm{dt}}=2\mathrm{k_{1}[N_{2}][O\cdot]}, \tag{12}\]  
Because the [O-] is hard to calculate or measure, in practice, the concentration of radicals usually transfers to the concentrations of compounds, and this transfer can be performed via a calculation or measurement. Hence, the [O-] usually can be expresseddifferently based on various compounds. For an equilibrium system of combustion, the [O-] could be expressed as follows:  
\[[\text{O}\cdot]=\text{K}_{1}\frac{[H_{2}][O_{2}]}{[H_{2}\text{O}]}, \tag{13}\]  
For an oxygen-rich environment, the [O-] could be expressed as follows:  
\[[\text{O}\cdot]=\text{K}_{2}[\text{O}_{2}]^{0.5}. \tag{14}\]  
Similarly, the prompt NO formation rate could be expressed as follows [37]:  
\[\frac{\text{d}[\text{NO}]}{\text{dt}}=\text{f}_{\text{C}}\times\text{k}_{ \text{prompt}}\times[\text{O}_{2}]^{\text{a}}[\text{N}_{2}][\text{F}]\exp(- \text{Ea}/\text{R}/\text{T}). \tag{15}\]  
where a is the order of reaction, and f\({}_{\text{C}}\) is a correction factor dependent on the air-to-fuel ratio and fuel type. Also, k\({}_{\text{prompt}}\) equals 6.4 \(\times\) 10\({}^{6}\), and Ea equals 72,500 cal/g mol.  
This simplified mechanism is commonly used in the CFD simulation, despite the fact that this method is relatively simple compared with the detailed mechanism. The possible error can be summarized at two points. Firstly, the simplified mechanism could not include all pathways of NOx generation but only express the average NOx formation, and the pathways of NOx generation are complex and affect each other based on different reaction environments. Secondly, in the combustion process, there is a super-equilibrium state which occurs after the fuel decomposes. Although the time period is relatively short, the concentrations of radicals would exceed the equilibrium ones by hundreds of times.