The presence of a second mechanism leading to NO_x formation was first identified by Fenimore [171] and was termed “prompt NO_x”. There is good evidence that prompt NO_x can be formed in a significant quantity in some combustion environments, such as in low-temperature, fuel-rich conditions and where residence times are short. Surface burners, staged combustion systems, and gas turbines can create such conditions [41].

At present, the prompt NO_x contribution to total NO_x from stationary combustors is small. However, as NO_x emissions are reduced to very low levels by employing new strategies (burner design or furnace geometry modification), the relative importance of the prompt NO_x can be expected to increase.

Prompt NO_x is most prevalent in rich flames. The actual formation involves a complex series of reactions and many possible intermediate species. The route now accepted is as follows:

(9–16)

(9–17)

(9–18)

(9–19)

A number of species resulting from fuel fragmentation have been suggested as the source of prompt NO_x in hydrocarbon flames (for example, CH, CH₂, C, C₂H), but the major contribution is from CH (Equation 9–16) and CH₂, via

(9–20)

The products of these reactions could lead to formation of amines and cyano compounds that subsequently react to form NO by reactions similar to those occurring in oxidation of fuel nitrogen, for example:

(9–21)

Prompt NO_x formation is proportional to the number of carbon atoms present per unit volume and is independent of the parent hydrocarbon identity. The quantity of HCN formed increases with the concentration of hydrocarbon radicals, which in turn increases with equivalence ratio. As the equivalence ratio increases, prompt NO_x production increases at first, then passes a peak, and finally decreases due to a deficiency in oxygen.

The reaction described by Equation 9–16 is of primary importance. In recent studies [576], comparison of probability density distributions for the location of the peak NO_x with those obtained for the peak CH have shown close correspondence, indicating that the majority of the NO_x at the flame base is prompt NO_x formed by the CH reaction. Assuming that the reaction described by Equation 9–16 controls the prompt NO_x formation rate,

(9–22)

There are, however, uncertainties about the rate data for the above reaction. From the reactions described by Equation 9–16 – Equation 9–20, it can be concluded that the prediction of prompt NO_x formation within the flame requires coupling of the NO_x kinetics to an actual hydrocarbon combustion mechanism. Hydrocarbon combustion mechanisms involve many steps and, as mentioned previously, are extremely complex and costly to compute. In the present NO_x model, a global kinetic parameter derived by De Soete [139] is used. De Soete compared the experimental values of total NO_x formation rate with the rate of formation calculated by numerical integration of the empirical overall reaction rates of NO_x and N₂ formation. He showed that overall prompt formation rate can be predicted from the expression

(9–23)

In the early stages of the flame, where prompt NO_x is formed under fuel-rich conditions, the O concentration is high and the N radical almost exclusively forms NO_x rather than nitrogen. Therefore, the prompt NO_x formation rate will be approximately equal to the overall prompt NO_x formation rate:

(9–24)

For C₂H₄ (ethylene)-air flames,

(9–25)

is 251151 , is the oxygen reaction order, is the universal gas constant, and is pressure (all in SI units). The rate of prompt NO_x formation is found to be of the first order with respect to nitrogen and fuel concentration, but the oxygen reaction order, , depends on experimental conditions.

Equation 9–24 was tested against the experimental data obtained by Backmier et al.  [35] for different mixture and fuel types. The predicted results indicated that the model performance declined significantly under fuel-rich conditions and for higher hydrocarbon fuels. To reduce this error and predict the prompt NO_x adequately in all conditions, the De Soete model was modified using the available experimental data. A correction factor, , was developed, which incorporates the effect of fuel type, that is, number of carbon atoms, and the air-to-fuel ratio for gaseous aliphatic hydrocarbons.   Equation 9–24 now becomes

(9–26)

so that the source term due to prompt NO_x mechanism is

(9–27)

In the above equations,

(9–28)

(9–29)

is 303474.125 , is the number of carbon atoms per molecule for the hydrocarbon fuel, and is the equivalence ratio. The correction factor is a curve fit for experimental data, valid for aliphatic alkane hydrocarbon fuels (C_nH_2n+2) and for equivalence ratios between 0.6 and 1.6. For values outside the range, the appropriate limit should be used. Values of and were developed at the Department of Fuel and Energy at The University of Leeds in England.

Here, the concept of equivalence ratio refers to an overall equivalence ratio for the flame, rather than any spatially varying quantity in the flow domain. In complex geometries with multiple burners this may lead to some uncertainty in the specification of . However, since the contribution of prompt NO_x to the total NO_x emission is often very small, results are not likely to be significantly biased.

Oxygen reaction order depends on flame conditions. According to De Soete [139], oxygen reaction order is uniquely related to oxygen mole fraction in the flame:

(9–30)