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20.1.9 NOx Formation in Turbulent Flows

The kinetic mechanisms of NOx formation and destruction described in the preceding sections have all been obtained from laboratory experiments using either a laminar premixed flame or shock-tube studies where molecular diffusion conditions are well defined. In any practical combustion system, however, the flow is highly turbulent. The turbulent mixing process results in temporal fluctuations in temperature and species concentration that will influence the characteristics of the flame.

The relationships among NOx formation rate, temperature, and species concentration are highly nonlinear. Hence, if time-averaged composition and temperature are employed in any model to predict the mean NOx formation rate, significant errors will result. Temperature and composition fluctuations must be taken into account by considering the probability density functions which describe the time variation.



The Turbulence-Chemistry Interaction Model


In turbulent combustion calculations, FLUENT solves the density-weighted time-averaged Navier-Stokes equations for temperature, velocity, and species concentrations or mean mixture fraction and variance. To calculate NO concentration, a time-averaged NO formation rate must be computed at each point in the domain using the averaged flow-field information.

Methods of modeling the mean turbulent reaction rate can be based on either moment methods [ 404] or probability density function (PDF) techniques [ 163]. FLUENT uses the PDF approach.

figure   

The PDF method described here applies to the NOx transport equations only. The preceding combustion simulation can use either the generalized finite-rate chemistry model by Magnussen and Hjertager or the nonpremixed combustion model. For details on these models, refer to Chapters  14 and 15.



The PDF Approach


The PDF method has proven very useful in the theoretical description of turbulent flow [ 164]. In the FLUENT NOx model, a single- or joint-variable PDF in terms of a normalized temperature, species mass fraction, or the combination of both is used to predict the NOx emission. If the nonpremixed combustion model is used to model combustion, then a one- or two-variable PDF in terms of mixture fraction(s) is also available. The mean values of the independent variables needed for the PDF construction are obtained from the solution of the transport equations.



The General Expression for the Mean Reaction Rate


The mean turbulent reaction rate $\overline{w}$ can be described in terms of the instantaneous rate $w$ and a single or joint PDF of various variables. In general,


 \overline{w}=\int \cdot \cdot \cdot \int w (V_{1}, V_{2},\dots) P (V_{1}, V_{2},\dots) dV_{1} dV_{2} \dots (20.1-103)

where $V_{1}, V_{2}$,... are temperature and/or the various species concentrations present. $P$ is the probability density function (PDF).



The Mean Reaction Rate Used in FLUENT


The PDF is used for weighting against the instantaneous rates of production of NO (e.g., Equation  20.1-15) and subsequent integration over suitable ranges to obtain the mean turbulent reaction rate. Hence we have


 \overline{S}_{\rm NO} = \int S_{\rm NO}(V_1) P_1(V_1) dV_1 (20.1-104)

or, for two variables


 \overline{S}_{\rm NO} = \int\int S_{\rm NO} (V_1, V_2) P(V_1, V_2) dV_1 dV_2 (20.1-105)

where $\overline{S}_{\rm NO}$ is the mean turbulent rate of production of NO, S $_{\rm NO}$ is the instantaneous rate of production given by, for example, Equation  20.1-15, and $P_1(V_1)$ and $P(V_1, V_2)$ are the PDFs of the variables $V_{1}$ and, if relevant, $V_{2}$. The same treatment applies for the HCN or NH $_3$ source terms.

Equation  20.1-104 or 20.1-105 must be integrated at every node and at every iteration. For a PDF in temperature, the limits of integration are determined from the minimum and maximum values of temperature in the combustion solution. For a PDF in mixture fraction, the limits of the integrations in Equation  20.1-104 or 20.1-105 are determined from the values stored in the look-up tables.



Statistical Independence


In the case of the two-variable PDF, it is further assumed that the variables $V_{1}$ and $V_{2}$ are statistically independent so that $P(V_{1}, V_{2})$ can be expressed as


 P (V_{1}, V_{2}) = P_{1} (V_{1}) P_{2} (V_{2}) (20.1-106)



The Beta PDF Assumption


$P$ is assumed to be a two-moment beta function as appropriate for combustion calculations [ 135, 245]. The equation for the beta function is


 P(V) = \frac{{\Gamma}(\alpha +\beta)}{{\Gamma}(\alpha){\Gamm... ...ta-1}} {\displaystyle{\int_0^1 V^{\alpha-1}(1-V)^{\beta-1}dV}} (20.1-107)

where ${\Gamma}(\;\;)$ is the Gamma function, $\alpha$ and $\beta$ depend on $\overline{m}$, the mean value of the quantity in question, and its variance, $\sigma^2$:


 \alpha = \overline{m} \left( \frac{\overline{m}(1-\overline{m})}{\sigma^2} -1 \right) (20.1-108)


 \beta = (1-\overline{m}) \left( \frac{\overline{m}(1-\overline{m})}{\sigma^2} -1 \right) (20.1-109)

The beta function requires that the independent variable $V$ assume values between 0 and 1. Thus, field variables such as temperature must be normalized. See Section  20.1.10 for information on using the beta PDF when using single-mixture fraction models and two-mixture fraction models.



The Calculation Method for $\sigma^2$


The variance, $\sigma^2$, can be computed by solving a transport equation during the combustion calculation stage. This approach is computationally intensive and is not applicable for a postprocessing treatment of NOx prediction. Instead, calculation of $\sigma^2$ is based on an approximate form of the variance transport equation.

A transport equation for the variance $\sigma^2$ can be derived:


 \frac{\partial}{\partial t} \left(\rho \sigma^2 \right) + \n... ... (\nabla \overline{m})^2 - C_d \rho \frac{\epsilon}{k}\sigma^2 (20.1-110)

where the constants $\sigma_t$, $C_g$ and $C_d$ take the values 0.85, 2.86, and 2.0, respectively. Assuming equal production and dissipation of variance, one gets


 \sigma^2 = \frac{\mu_t}{\rho} \frac{k}{\epsilon} \frac{C_g}{... ...left(\frac{\partial \overline{m}}{\partial z}\right)^2 \right] (20.1-111)

The term in the brackets is the dissipation rate of the independent variable.

For a PDF in mixture fraction, the mixture fraction variance has already been solved as part of the basic combustion calculation, so no additional calculation for $\sigma^2$ is required.


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