ANSYS FLUENT 12.0 Theory Guide - 13.1.9 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.

In turbulent combustion calculations, ANSYS 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 [ 380] or probability density function (PDF) techniques [ 148]. ANSYS FLUENT uses the PDF approach.

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 non-premixed combustion model. For details on these models, refer to Chapters 7 and 8.

The PDF method has proven very useful in the theoretical description of turbulent flow [ 149]. In the ANSYS 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 non-premixed or partially premixed 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 mean turbulent reaction rate $\overline{w}$ can be described in terms of the instantaneous rate

and a single or joint PDF of various variables. In general,

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

(13.1-103)

where $V_{1}, V_{2}$ ,... are temperature and/or the various species concentrations present.

is the probability density function (PDF).

The PDF is used for weighting against the instantaneous rates of production of NO (e.g., Equation 13.1-15) and subsequent integration over suitable ranges to obtain the mean turbulent reaction rate. Hence we have

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

(13.1-104)

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

(13.1-105)

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 13.1-15, and

and

are the PDFs of the variables $V_{1}$ and, if relevant, $V_{2}$ . The same treatment applies for the HCN or NH

source terms.

Equation 13.1-104 or 13.1-105 must be integrated at every node and at every iteration. For a PDF in terms of temperature, the limits of integration are determined from the minimum and maximum values of temperature in the combustion solution. For a PDF in terms of mixture fraction, the limits of the integrations in Equation 13.1-104 or 13.1-105 are determined from the values stored in the look-up tables.

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

$P (V_{1}, V_{2}) = P_{1} (V_{1}) P_{2} (V_{2})$

(13.1-106)

ANSYS FLUENT can assume

to be a two-moment beta function that is appropriate for combustion calculations [ 123, 231]. The equation for the beta function is

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

(13.1-107)

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

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

(13.1-108)

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

(13.1-109)

The beta function requires that the independent variable

assume values between 0 and 1. Thus, field variables such as temperature must be normalized. See this section in the separate User's Guide for information on using the beta PDF when using single-mixture fraction models and two-mixture fraction models.

ANSYS FLUENT can also assume

to exhibit a clipped Gaussian form with delta functions at the tails.

The cumulative density function for a Gaussian PDF ( $G_{\rm CDF}$ ) may be expressed in terms of the error function as follows:

$G_{\rm CDF} = \frac{1}{2} \left(1 + {\it erf} \left(\left(m - \overline{m} \right) / \sqrt{2 \sigma^2} \right) \right)$

(13.1-110)

where ${\it erf}(\;\;)$ is the error function,

is the quantity in question, and $\overline{m}$ and $\sigma^2$ are the mean and variance values of

, respectively. The error function may be expressed in terms of the incomplete gamma function ( $gammp(\;\;)$ ):

$\displaystyle \mbox{for} \; m < 0: \; erf (m )$	$\textstyle =$	$\displaystyle - gammp (\mbox{0.5, } m^2 )$
$\displaystyle \mbox{for} \; m \geq 0: \; erf (m )$	$\textstyle =$	$\displaystyle gammp (\mbox{0.5, } m^2 )$	(13.1-111)

The variance, $\sigma^2$ , can be computed by solving the following transport equation during the combustion calculation or pollutant postprocessing stage:

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

(13.1-112)

where the constants $\sigma_t$ ,

and

take the values 0.85, 2.86, and 2.0, respectively.

Note that the previous equation may only be solved for temperature. This solution may be computationally intensive, and therefore may not always be applicable for a postprocessing treatment of NOx prediction. When this is the case or when solving for species, the calculation of $\sigma^2$ is instead based on an approximate form of the variance transport equation (also referred to as the algebraic form). The approximate form assumes equal production and dissipation of variance, and is as follows:

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

(13.1-113)

For a PDF in terms of 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.