# NOx Formation Mechanism and Emission Prediction in Turbulent Combustion: A Review
## 4 The Empirical Formula Approach  
At the early stage, the computer was not able to support the computational fluid dynamics of the NOx generation. Hence, making use of a large amount of experiment results could predict the tendency of NOx production. Aiming at evaluating several gas turbine combustors, [PERSON] [40] proposed a method of evaluation that incorporates the product of reaction rate, mean residence time, and mixing coefficient into the following calculation:  
\[NOx=A\left(\frac{PV_{\text{C}}}{\dot{m}_{A}T}\right)\left(\frac{\Delta P}{P} \right)^{x}\exp(zT) \tag{16}\]where \(A\), \(x\), and \(z\) are constants representing the empirical coefficients of NOx generation; \(\Delta P\) represents the pressure differential; and \(T\) and \(P\) represent the temperature and pressure in the combustion chamber, respectively. The reaction rate could be represented as a function comprising the exponential of temperature and pressure. The mean residence time can be determined by considering the mass flow rate, density, and volume of the reactor. The fitting results showed that the NOx generation depended on the operating conditions and residence time and had little relationship with the fuel properties. The subsequent research followed this idea to calculate the NOx amount by correcting parameters in the empirical formula. [PERSON] and [PERSON] [41] pointed out that fuel type, steam and water, and fuel vaporization affected the accuracy of the empirical formula for the gas turbine. And the empirical formula could be expressed as follows:  
\[EI_{NOx}=29\exp(-21,670/T)P_{3}^{0.66}(1-\exp(-250\tau)) \tag{17}\]  
where \(\tau\) represents the residence times; \(P_{3}\) represents the combustor inlet pressure; and \(T\) represents the effective flame temperature, which is relative to the combustor inlet pressure and temperature. The revised empirical formula could be within the margin of error of 20%. At the same time, to overcome the shortcomings of the empirical formula, they designed a method based on the simplest chemical reaction rate model to predict NOx emission as follows, which has a stronger theoretical basis and is preliminarily considered more satisfactory than the empirical formula. \[EI_{NOx}=\exp(A-B/T)\text{tanh}(C\times P_{3}^{D}\times T_{3}^{E}\times\phi^{ F}\times\tau) \tag{18}\]  
where \(A\), \(B\), \(C\), \(D\),\(E\), and \(F\) are the empirical coefficients; T represents the effective flame temperature, which is relative to the combustor inlet pressure and temperature; \(P_{3}\) represents the combustor inlet pressure; \(T_{3}\) represents the combustor inlet temperature; \(\phi\) represents the equivalence ratios; and \(\tau\) represents residence times. Based on the traditional empirical formula for predicting NOx emissions, [PERSON] and [PERSON] [42] took into account the influence of droplet evaporation and droplet size and combustor type, including the RQL combustor, diffusion flame combustor, and premixed combustor. By improving the prediction accuracy to modify the empirical formula, the simulation results were verified in the 3D model, and good matching results were obtained. The error between calculation and measurement is around 10%, which provides support for the subsequent utilization in advanced combustion systems. [PERSON] and [PERSON] [43] used mathematical methods that introduced a large range of experimental data to optimize the coefficients of [PERSON]'s empirical formula and [PERSON] and [PERSON]'s modified formula:  
\[EI_{NOx}=1.902\times 10^{5}\frac{P_{3}^{0.4048}(1-\exp(-480\tau))}{\exp(37,902/T)} \tag{19}\]  
where \(\tau\) represents residence times; \(P_{3}\) represents the combustor inlet pressure; and \(T\) represents the effective flame temperature, which is relative to the combustor inlet pressure and temperature. By increasing the coefficient of determination and smaller standard deviation, the experiment results can match the prediction results well. [PERSON] et al.