Combustion

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What is combustion -- Physics versus modelling

Combustion phenomena consists of many physical and chemical processes with broad range of time scales. Mathematical description of combustion is not always trivial. Analytical solutions exists only for basic situations of laminar flame and because of its assumptions it is often restricted to few problems solved usually in zero or one-dimensional space.

Problems solved today concern mainly turbulent flows, gas as well as liquid fuels, pollution issues (products of combustion as well as for example noise pollution). These problems require not only extensive experimental work, but also numerical modelling. All combustion models must be validated against the experiments as each one has its own drawbacks and limits. However here the modelling part will be mainly addressed.

Reaction mechanisms

The combustion is mainly chemical process and although we can, to some extend, describe flame without any chemistry informations, for modelling of flame propagation we need to know the speed of reactions, product concentrations, temperature and other parameters. Therefore more or less detailed information about reaction kinetics is essential for any combustion model. Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to maintain until all of the mixture is burned into products. If the reaction is too slow, the flame will extinguish, if too fast, explosion or even detonation will occur. The reaction rate of typical combustion reaction is influenced mainly by concentration of reactants, temperature and pressure.

A stoichiometric equation of an arbitrary equation can be written as:

$\sum_{j=1}^{n}\nu' (M_j) = \sum_{j=1}^{n}\nu'' (M_j),$

where $\nu$ is the stoichiometric coefficient, $M_j$ is arbitrary species. One prime specifies the reactants and double prime products of the reaction.

Reaction rate, expressing the rate of disappearance of reactant i of such a reaction, is defined as:

$RR_i = k \, \prod_{j=1}^{n}(M_j)^{\nu'},$

in which k is the specific reaction rate constant. Arrhenius found that this constant is a function only of temperature and this function is defined as:

$k= A T^{\beta} \, exp \left( \frac{-E}{RT}\right)$

where A is pre--exponential factor, E is activation energy and $\beta$ is temperature exponent. These constants for given reactions can be found in literature. The reaction mechanism can be given from experiments for every reaction resolved, it could be also constructed numerically by automatic generation method (see [Griffiths (1994)] for review on reaction mechanisms). For simple hydrocarbon tens to hundreds of reactions are involved. By analysis and systematic reduction of reaction mechanisms global reaction (from one to five step reactions) can be found (see [Westbrook (1984)]).

Governing Equations for Reacting Flows

Together with the usual Navier-Stokes for compresible flows (See Governing Equations), additional equations are needed in reacting flows. The mass fraction transport equation for k-th species $Y_k$ is:

$\frac{\partial}{\partial t} \left( \rho Y_k \right) + \frac{\partial}{\partial x_j} \left( \rho u_j Y_k\right) = \frac{\partial}{\partial x_j} \left( \rho D_k \frac{\partial Y_k}{\partial x_j}\right)+ \dot \omega_k$

where Ficks law is assumed for scalar diffusion with $D_k$ , the species difussion coefficient and $\dot \omega_k$ is the species reaction rate. A non-reactive scalar (like the mixture fraction $Z$ ) had the following transport equation:

$\frac{\partial}{\partial t} \left( \rho Z \right) + \frac{\partial}{\partial x_j} \left( \rho u_j Z \right) = \frac{\partial}{\partial x_j} \left( \rho D \frac{\partial Z}{\partial x_j}\right)$

where $D$ is the diffusion coefficient of the passive scalar.

In turbulent flows, Favre averaging is often used and the mass fraction transport equation is transformed to

$\frac{\partial \overline{\rho} \widetilde{Y}_k }{\partial t} + \frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Y}_k}{\partial x_j}= \frac{\partial} {\partial x_j} \left( \overline{\rho D_k \frac{\partial Y_k} {\partial x_j} } - \overline{\rho} \widetilde{u''_i Y''_k } \right) + \overline{\dot \omega_k}$

where the turbulent fluxes $\widetilde{u''_i Y''_k}$ and reaction terms $\overline{\dot \omega_k}$ needs to be closed.

The passive scalar turbulent transport equation is

$\frac{\partial \overline{\rho} \widetilde{Z} }{\partial t} + \frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z} }{\partial x_j}= \frac{\partial} {\partial x_j} \left( \overline{\rho D \frac{\partial Z} {\partial x_j} } - \overline{\rho} \widetilde{u''_i Z'' } \right)$

where similar to the mass fraction equation, $\widetilde{u''_i Z''}$ needs modelling.

Infinitely fast chemistry

All combustion models can be divided into two main groups according to the assumptions on the reaction kinetics. We can either assume the reactions to be infinitely fast - compared to e.g. mixing of the species, or of the comparable time scale of the mixing process. The simpler approach assuming chemistry fast enough, that the limiting process is mixing of the species is historically older approach and even today can be appropriate approach. It is simpler to solve then #Finite rate chemistry models, but introduces errors to the solution which may or may not be important.

Premixed Combustion

Premixed flame occurs in mixtures of fuel and oxidiser, homogeneously premixed prior to the flame. These flames are not limited only to gas fuels, but also to the pre-vaporised fuels. Typical example of premixed laminar flame is bunsen burner, where the air enters the fuel stream. The mixture burns in the wake of the riser tube walls forming nice stable flame. The premixed flames has many advantages in terms of control of temperature and products and pollution concentration, but introduce also some dangers like the autoignition (in the supply system).

Turbulent flame speed model

Eddy Break-Up model

The Eddy Break-Up model is the typical example of mixed-is-burnt combustion model. It is based on the work of Magnussen and Hjertager, and Spalding and can be found in all CFD packages. The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. The combustion is described by a single step global chemical reaction:

$F + \nu_s O \rightarrow (1+\nu_s) P$

in which F stands for fuel, O for oxidiser and P for products of the reaction. Alternativelly we can have multistep scheme, where each reaction has its own mean reaction rate. The mean reaction rate is given by:

$\bar{\dot\omega}_F=A_{EB} \frac{\epsilon}{k} min\left[\bar{C}_F,\frac{\bar{C}_O}{\nu}, B_{EB}\frac{\bar{C}_P}{(1+\nu)}\right]$

$\bar{C}$ denotes mean concentrations for fuel, oxidiser and products respectively, A and B are model constants with typical values of 0.5 and 4.0 respectively. The values of these constants are fitted according to the experimental results and they are suitable for most of the general cases. Still they are just constants based on experimental fitting and they need not be suitable for all the situations. Care must be taken especially in highly strained regions, where the ratio of $k$ to $\epsilon$ is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed. CFD codes usually has some remedies to overcome this problem.

This model largely over-predicts temperatures and concentrations of species like CO and other species. Still this model is quite popular for its simplicity and relatively easy convergence and implementation.

Bray-Moss-Libby Model

Non premixed combustion

Conserved scalar equilibrium models

Finite rate chemistry

Premixed Combustion

Coherent Flame Model

Flamelets based on G equation

Non-premixed Combustion

Flamelets based on conserved scalar

Peters (2000) define Flamelets as "thin diffusion layers embedded in a turbulent non-reactive flow field". If the chemistry is fast enough, the chemistry is active within a thin region where the chemistry conditions are in (or close to) stoichiometric conditions, the "flame" surface. This thin region is assumed to be smaller than Kolmogorov length scale and therefore the region is locally laminar. The flame surface is defined as an iso-surface of a certain scalar $Z$ , mixture fraction in non-premixed combustion.

The reactive problems is therefore split into two parts: First, the mixing , which consists of the location of the flame surface which is a non-reactive problem concerning the propagation of a passive scalar. And second, the flame structure , which deals with the distribution of the reactive species inside the flamelet.

To obtain the distribution inside the flame front we assume it is locally one-dimensional and depends only on time and the scalar coodinate.

Using the following chain rules for the time

$\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z}$

and spatial coordinate

$\frac{\partial Y_k}{\partial x_j} = \frac{\partial Z}{\partial x_j}\frac{\partial Y_k}{\partial Z}$

to the species transport equation (see #Governing Equations for Reacting Flows) and re-arranging, we obtain

$\rho \frac{\partial Y_k}{\partial t} + Y_k \left[ \frac{\partial \rho}{\partial t} + \frac{\partial \rho u_j}{\partial x_j} \right] + \frac{\partial Y_k}{\partial Z} \left[ \rho \frac{\partial Z}{\partial t} + \rho u_j \frac{\partial Z}{\partial x_j} - \frac{\partial}{\partial x_j}\left( \rho D \frac{\partial Z}{\partial x_j} \right) \right] = \rho D \left( \frac{\partial Z}{\partial x_j} \frac{\partial Z}{\partial x_j} \right) + \dot \omega_k$

The second and third term in the LHS cancel due to continuity and mixture fraction transport, the equation therefore boils down to

$\frac{\partial Y_k}{\partial t} = \frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot \omega_k$

where $\chi = 2 D \left( \frac{\partial Z}{\partial x_j} \right)^2$ is called the scalar dissipation and controls the mixing, providing the interaction between the flow and the chemistry.

If the flame dependence on time is dropped, even though he field $Z$ still depends on it.

$\dot \omega_k= -\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2}$

This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of $Y_k=f(Z,\chi)$ can be pre-computed and stored in look-up tables with all the required complex chemistry.

Flamelet Computation and Flamelet Libraries

The computation of non-premixed turbulent flames based on laminar-flamelet models is generally based on two-dimensional or three-dimensional CFD codes that employ standard models for fluid-mechanical closure of the govening equations. In many cases, for that purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.

Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on

$\frac{\partial Y_k}{\partial t} = \frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,$

then the variable parameter is the scalar dissipation rate $\chi$ . The flamelet profiles for the various parameter values are stored in a dataset or file which is called a "flamelet library". For the generation of such libraries ready to use software is avalable such as Softpredict's Combustion Simulation Laboratory COSILAB [1] with its relevant solver RUN1DL, which can be used for a variety of relevant geometries; see various publications that are available for download.

Flamelets in turbulent combustion

In turbulent flames the interest is $\widetilde{Y}_k$ . In flamelets, the flame thickness is assumed to be much smaller than Kolmogorov scale and obviously is much smaller than the grid size. It is therefore needed a distribution of the passive scalar within the cell. $\widetilde{Y}_k$ cannot be obtained directly from the flamelets library $\widetilde{Y}_k \neq Y_F(Z,\chi)$ , where $Y_F(Z,\chi)$ corresponds to the value obtained from the flamelets libraries. A generic solution can be expressed as

$\widetilde{Y}_k= \int Y_F( \widetilde{Z},\widetilde{\chi}) P(Z,\chi) dZ d\chi$

where $P(Z,\chi)$ is the joint Probability Density Function (PDF) of the mixture fraction and scalar dissipation which account for the scalar distribution inside the cell and "a priori" depends on time and space.

The most simple assumption is to use a constant distribution of the scalar dissipation within the cell and the above equation reduces to

$\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) dZ d\chi$

$P(Z)$ is the PDF of the mixture fraction scalar and simple models (such as Gaussian or beta PDFs) can be build depending only on two moments of the scalar mean and variance, $\widetilde{Z},Z''$ .

If the mixture fraction and scalar dissipation are consider independent variables, $P(Z,\chi)$ can be written as $P(Z) P(\chi)$ . The PDF of the scalar dissipation is assumed to be log-normal with variance unity.

$\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) P(\chi) dZ d\chi$

Conditional Moment Closure (CMC)

In Conditional Moment Closure (CMC) methods we assume that the species mass fractions are all correlated with the mixture fraction (in non premixed combustion).

From Probability density function we have

$\overline{Y_k}= \int <Y_k|\eta> P(\eta) d\eta$

where $\eta$ is the sample space for $Z$ .

CMC consists of providing a set of transport equations for the conditional moments which define the flame structure.

Experimentally, it has been observed that temperature and chemical radicals are strong non-linear functions of mixture fraction. For a given species mass fraction we can decomposed it into a mean and a fluctuation:

$Y_k= \overline{Y_k} + Y'_k$

The fluctuations $Y_k'$ are usually very strong in time and space which makes the closure of $\overline{\omega_k}$ very difficult. However, the alternative decomposition

$Y_k= <Y_k|\eta> + y'_k$

where $y'_k$ is the fluctuation around the conditional mean or the "conditional fluctuation". Experimentally, it is observed that $y'_k<< Y'_k$ , which forms the basic assumption of the CMC method. Closures. Due to this property better closure methods can be used reducing the non-linearity of the mass fraction equations.

The Derivation of the CMC equations produces the following CMC transport equation where $Q \equiv <Y_k|\eta>$ for simplicity.

$\frac{ \partial Q}{\partial t} + <u_j|\eta> \frac{\partial Q}{\partial x_j} = \frac{<\chi|\eta> }{2} \frac{\partial ^2 Q}{\partial \eta^2} + < \dot \omega_k|\eta>$

Multiple Mapping Closure (MMC)

Linear Eddy Model

PDF transport models

PDF methods are based on the transport equation of the joint-PDF of the scalars. Denoting $P \equiv P(\underline{\psi}; x, t)$ where $\underline{\psi} = ( \psi_1,\psi_2 ... \psi_N)$ is the phase space for the reactive scalars $\underline{Y} = ( Y_1,Y_2 ... Y_N)$ . The transport equation of the joint PDF is:

$\frac{\partial <\rho | \underline{Y}=\underline{\psi}> P }{\partial t} + \frac{ \partial <\rho u_j | \underline{Y}=\underline{\psi}> P }{\partial x_j} = \sum^N_\alpha \frac{\partial}{\partial \psi_\alpha}\left[ \rho \dot{\omega}_\alpha P \right] - \sum^N_\alpha \sum^N_\beta \frac{\partial^2}{\partial \psi_\alpha \psi_\beta} \left[ <D \frac{\partial Y_\alpha}{\partial x_i} \frac{\partial Y_\beta}{\partial x_i} | \underline{Y}=\underline{\psi}> \right] P$

where the chemical source term is closed. Another term appeared on the right hand side which accounts for the effects of the molecular mixing on the PDF, is the so called "micro-mixing " term. Equal diffusivities are used for simplicity $D_k = D$

Lagrangian

Eulerian

References

Griffiths J.F. (1994), "Reduced Kinetic Models and Their Application to Practical Combustion Systems", Prog. in Energy and Combustion Science,Vol. 21, pp. 25-107.
Westbrook, Ch.K., Dryer,F.L., (1984), "Chemical Kinetic Modeling of Hydrocarbon Combustion", Prog. in Energy and Combustion Science,Vol. 10, pp. 1-57.

External links and sources

SoftPredict's COSILAB homepage

SoftPredict's flamelet paper download page

Combustion

From CFD-Wiki

Contents

What is combustion -- Physics versus modelling

Reaction mechanisms

Governing Equations for Reacting Flows

Infinitely fast chemistry

Premixed Combustion

Turbulent flame speed model

Eddy Break-Up model

Bray-Moss-Libby Model

Non premixed combustion

Conserved scalar equilibrium models

Finite rate chemistry

Premixed Combustion

Coherent Flame Model

Flamelets based on G equation

Non-premixed Combustion

Flamelets based on conserved scalar

Flamelet Computation and Flamelet Libraries

Flamelets in turbulent combustion

Conditional Moment Closure (CMC)

Multiple Mapping Closure (MMC)

Linear Eddy Model

PDF transport models

Lagrangian

Eulerian

References

External links and sources

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