1. HEAT PROCESSES Combustion, Flames and CFD HP11
CFD analysis of combustion. Transport equations and modeling of chemical reactions. Non-premix combustion and mixture fraction methods. Lagrangian and Eulerian methods.
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

2. Combustion and Flames HP11
Warnatz J.: Combustion. Springer, Berlin 1996
Mark Bradford

3. Combustion HP11 Laminar flame
Use EBU (Eddy Break Up models)
Laminar flame
Premixed (only one inlet stream of mixed fuel and oxidiser)
Homogeneous reaction in gases
Turbulent flame
Use mixture fraction method (PDF)
Laminar flame
Non premixed (separate fuel and oxidiser inlets)
Liquid fuels (spray combustion)
Turbulent flame
Combustion of particles (coal)
Lagrangian method-trajectories of a representative set of droplets/particles in a continuous media
Lagrangian method-trajectories of a representative set of droplets/particles in a continuous media

4. Combustion HP11 Premixed combustion
Fuel and oxidiser are mixed first and even then burn
Bunsen laminar flame
uL burning velocity (=umsin)
flamelets
Turbulent premixed flame  um
flame front
Flat flame um
Bunsen burner fully premixed. Colour corresponds to radicals CH (photo Wikipedia)
porous plate
R.N. Dahms et al. / Combustion and Flame (2011)
Laminar premixed flames are characterised by a narrow flame front (front width depends upon the rate of chemical reaction and diffusion coefficient). Laminar burning velocity (uL) must be greater than the velocity of flow (um) in case of a flat flame front, otherwise the flame will be extinguished (in fact this is the way how to determine the burning velocity experimentally). Example: flat laminar flame in porous ceramic burner. Nonuniform transversal velocity profile is manifested by a conical flame front, and the flame front cone angle depends upon burning velocity uL=um sin . Example: Bunsen burner.
In case of turbulent flow the flame front is wavy and fluctuating (flamelets, the turbulent flame can be viewed as an ensemble of premixed laminar flames). Example: Combustion in a cylinder of spark ignition engine, gas turbines in power plants.
Advantage of the premixed combustion: In a gas turbine operating at a fuel-lean condition high temperatures and associated NOx formation are avoided. Soot formation is also suppressed. Disadvantage: Risk of uncontrolled explosion of premixed reactants.

5. Combustion HP11 Nonpremixed combustion
Fuel and oxidiser enter combustion chamber as separated streams. Mixing and burning take place simultaneously (nonpremix is called diffusive burning).
Laminar nonpremixed flame
Flame front
fuel
air
Burned gases
Turbulent nonpremixed flame
fuel+flue gas
air+flue gas
air
stoichiometric line
Laminar coflow (left) counter flow (right).
Nonpremixed flames unlike premixed flames do not propagate. Flame simply cannot propagate into the fuel side because there is no oxygen, neither to the oxidising region with lack of fuel. Only burned products diffuse to both sides of the flame front. Flame front position corresponds to stoichiometric ratio fuel/air – there is also the highest temperature.
In turbulent nonpremixed flames the flamelet concept can be used again. Chemistry of nonpremixed combustion is more complicated because on one side of the flame front is the fuel-rich burning accompanied by formation of sooth and on the air side a fuel lean combustion occurs. Existence of glowing sooth is manifested by yellow luminiscence.
Examples: Industrial burners, combustion engines.

6. Combustion-flame HP11 Flame propagation
Previous lecture HP10 was devoted to statics (fuel consumption, overall mass and enthalpy balances,…). Questions like: What is the distance of the flame front disc levitating above a porous ceramic plate? What is the thickness of the flame front? What is the burning velocity? Which species are generated inside the flame? Concentration profiles? Temperature profiles? What is the height and width of the flame?... should be addressed to kinetics.
Flame propagation
An idea about processes related to burning velocity and flame propagation can be obtained from analysis of the simplest 1D cases. There is a similarity with drying: process is again controlled by diffusion (now diffusion of fuel and products of burning), and heat fluxes are consumed at a moving interface (flame front, now the enthalpy changes of chemical reaction substitute the enthalpy of phase changes).
On the other hand the problem is more difficult, because there are much more transported species and chemical reactions strongly depend upon temperature. Transport equations are therefore nonlinear and numerical solution is necessary.

7. Combustion-flame
HP11
… and numerical solution is necessary. There are few exceptions: Zeldovic and Frank-Kamenetskii (1938) described the flame propagation of a steady premixed planar flame by a transport equation of fuel (mass fraction F(z)) and by transport equation for enthalpy (temperature T(z)) assuming first-order chemical reaction (fuelproduct) and similarity of transport equations (assuming that the diffusion coefficient D is approximately the same as the temperature diffusion coefficient a). Analysis of these equations reveals the fact that a steady solution (stationary position of the flame front) exists only as soon as the flow velocity u (=uF flame velocity) is uF
T-temperature
F - fuel z
[mm]
E is activation energy and A preexponential factor characterising rate of the chemical reaction (Arrhenius term).
Result: The flame velocity increases when the diffusion coefficient or the chemical reaction rate increases.

8. all fuel is consumed in the thin flame front
Combustion-flame
HP11
Some information about the flame front can be obtained from the solution of diffusion equation. For example the flame front position around the burning fuel particle in air can be estimated from the radial mass fraction profiles obtained by solution of the diffusion equation R  r
fueĺ=1
ox=1
all fuel is consumed in the thin flame front
…and now: position of front (distance ) is determined by stoichiometry of chemical reaction (1 kg of fuel consumes s kg of oxidiser), therefore and assuming the same diffusion coefficient in fuel and in air

9. Combustion-flame HP11 Flame size
The size (length) of a nonpremixed burning jet can be estimated from the velocity of jet and from the diffusion time necessary for mixing in the transversal direction. Burke and Schumann (1928), Ind.Eng.Chem 20:998 h
r-mixing length
Velocity of circular jets decreases with the distance from nozzle (u~1/x) however this approximation assumes a constant velocity at axis and also cylindrical form of jet, having radius r of fuel nozzle. Mixing length (r) is associated with the mixing time by theory of penetration depth
and the height of flame h is estimated as the product of the mixing time and axial velocity
This is qualitatively correct conclusion stating that the flame length is proportional to flowrate and indirectly proportional to diffusion coefficient
(Warnatz 1996 documents the diffusion coefficient effect by comparing the height of flame at hydrogen combustion that is 2.5 shorter than the flame height at carbon monoxide combustion)

10. Combustion-flame-explosion
HP11
Previous examples concern steady combustion processes (time independent solutions). Ignition and explosions are always time dependent processes.
Explosion is characterized by a positive feedback, when a temperature increase releases more heat than could be removed. The simplest demonstration is a perfectly mixed chamber filled by reaction mixture (Semenov 1928)
Qremoved(T)
Qreaction(T)
There exists a critical temperature Tcrit when Qreaction(Tcrit)=Qremoved(Tcrit). For T>Tcrit begins the so called run-away and temperature increases without any control (this is explosion).
Remark: the term deflagration is used for combustion and flame velocity determined by diffusion (this velocity is rather small, of the order 1 m/s). The term detonation concerns explosions, driven be pressure gradients (propagation velocities are very high of the order 1000 m/s, because speed of sound is high in burnt gases).

11. Chemical reactions
HP11

12. Rate of chemical reactions
HP11
When discussing combustion statics a detailed knowledge of actual chemical reactions is not needed – released heat can be evaluated only from the composition of fuel and burned gas (this composition can be obtained for example by analyzers of flue gas). This is because according to the Hess’s law the enthalpy change of a reaction (enthalpy of products minus enthalpy of reactants) is independent of a specific reaction mechanism:
where are stoichiometric coefficients of reactants (R) and products (P), is the standard enthalpy of formation corresponding to the formation of compound from elements in their standard state (at temperature 298 K and atmospheric pressure).
What is substantial: It is sufficient to evaluate only the overall chemical reaction and it is not necessary to sum up reaction enthalpies of many actual elementary reactions.

13. Combustion of Methane
HP11
Example: Gas burner operating at atmospheric pressure consumes mass flowrate of CH kg/s. Calculate mass flowrate of flue gas. Assuming temperature of fuel and air 298K and temperature of flue gases 500 K calculate heating power.
Remark: methane combustion is described by more than 200 elementary chemical reactions. Because we need only overall heat it is sufficient to use only one overall chemical reaction.
Air
CH4 Q
fg-flue gas
Flue gas production 1kg CH4 needs 4 kg O2. Mass fraction of oxygen in air is
Heat removed
cp0.001 MJ/kgK

14. Rate of chemical reactions
HP11
Released heat calculated from the equation
corresponds to a certain number of reacting moles of reactants [J/mole].
So that we could calculate the power of heat released in a unit volume [W/m3] it is necessary to know number of moles of reactants consumed by each chemical reaction in unit volume and in unit time, therefore it is necessary to know the time rate of concentration
where cA[A] is the molar concentration of specie A [kmol/m3], MA is the molecular mass [kg/kmol], A is the mass concentration [kg/m3] and A is the mass fraction of the specie A in a mixture.

15. Rate of chemical reactions
HP11
If we need to calculate the composition inside a flame and the composition of flue gas or if it is necessary to evaluate temperature distribution inside a combustion chamber, the rate of chemical reactions must be known.
Reaction rate of a general reaction A+B+…  D+E+… can be expressed as k
where [A] is molar concentration of the specie A [kmol/m3], exponents a,b,… are reaction orders with respect to the species A,B,… (only reactants, not products!) and the sum of all exponents a+b+…=n is the overall reaction order. K is the rate coefficient of reaction that depends upon temperature and pressure.
Example:
Combustion of hydrogen with chlorine H2+Cl22HCl
Reaction is of the first order with respect to hydrogen and of order 0.5 with respect to chlorine (overall reaction order n=1.5).

16. Rate of chemical reactions
HP11
Previous example demonstrates that the reaction orders need not be integers, they are usually rational, sometimes even negative, numbers. Even worse: the rate equation of overall chemical reaction has a limited validity, reaction orders and reaction coefficients depend upon temperature and pressure.
The reason for this complicated behavior is in complexity of chemical reactions. Overall reactions should be decomposed to several elementary reactions, and in the previous case instead of H2+Cl22HCl the following set of four elementary reactions should be considered.
Elementary reactions can be either unimolecular (for example the reaction 1, decomposition of Cl2 molecule to radicals Cl) or bimolecular (reactions 2,3,4). Unimolecular elementary reactions are always of the first order (reaction exponent a=1) and bimolecular are of the second order (exponents a=b=1). Actual rate of overall chemical reaction is evaluated by simultaneous solution of the four differential equations with the rate coefficients k1,k2,k3,k4. Solution of these differential equations can be simplified by assuming that the radicals Cl and H are at chemical equilibrium and the time derivatives of their concentration can be neglected. Then the differential equations describing the time changes of radicals reduce to algebraic equations and the concentration of [H] and [Cl] can be eliminated from the set of elementary reactions. Rate equations are therefore expressed only in terms of [Cl2] and [H2], giving previously presented result
Remark: Not all bimolecular reactions are elementary reactions (H2+Cl22HCl is an example). Elementary bimolecular reactions are characterized by presence of highly reactive radicals as reactants.

17. Elementary reactions HP11
Rate equations of uni- and bi-molecular elementary reactions are characterized by reaction orders that are identical with the stoichiometric coefficients 0,1, or 2
Kinetics of a general system of R elementary reactions (r=1,2,…,R) of S species (s=1,2,…,S) can be described as
i=1,2,…,S
stoichiometric coefficient of the specie s in the reaction r (negative for reactant, positive for product)
Explanation will be shown in the following example (H2+Cl22HCl)

18. matrix of stoichiometric coefficients
Elementary reactions
HP11
There is 5 species S1(Cl2), S2(H2), S3(Cl), S4(H), S5(HCl) and four reactions
matrix of stoichiometric coefficients
Let us apply rate equations for the concentration of chlorine Cl2 (i=1)
I hope that it is comprehensible… all other rate equations are similar.

19. Temperature dependence
HP11
Reaction constants k depend upon temperature according to Arrhenius law
Preexpontial factor (the same unit as k)
Activation energy of chem.reaction J/mol
Collision theory: only the molecules having energy greater than Ea react at mutual collision.
Relative amount of molecules having kinetic energy greater than Ea is given by Maxwell distribution of energies
The greater is temperature the more molecules have energy greater than E
Reactions having high activation energy proceed at high temperatures and vice versa (thermal NOx reaction with Ea~300 kJ/mol starts at about 1500 K, while prompt NOx reaction with Ea~100 kJ/mol is important at relatively low temperature about 1000 K)

20. Simplified reaction mechanism
HP11
Reality is such that even the simple looking reaction like hydrogen combustion 2H2+O22H2O requires about 40 elementary reactions for a satisfactory reaction mechanism. Combustion of hydrocarbons is even more complicated and is represented by hundreds and thousands of elementary reactions (oxidation of the simplest hydrocarbon CH4 is described by 277 reactions among 49 chemical species). Simultaneous solution of thousands differential equations is too expensive for simulation of practical systems with 3 dimensional inhomogeneous turbulent flows. There exist several techniques how to reduce the complicated system, generally based upon exclusion of reactions that are very fast and are in partial equilibrium (QSSA Quasi-Steady-State-Assumption, ILDM Intrinsic Low Dimensional Method, see Peters 2000). Methane oxidation can be in this way reduced to global mechanism consisting in only four or even two steps

21. Simplified reaction mechanism
HP11
Simplified reaction mechanisms are applied also for description of NOx formation (important for design of burners from the point of view of emissions). While the concentrations of CO2, H2O can be predicted by assuming equilibrium, NO reactions are slow and far from equilibrium at short residence times in burning zone (some ms, time scales of reactions are thousand times longer).
Thermal NOx (Zeldovic mechanism)
Prompt NOx (Fenimore)
Fuel NOx
Source of NO from atmospheric nitrogen formed by the following reactions:
The first reaction (attack of oxygen radical to N2) has very high activation energy (319 kJ/mol) and proceeds only at very high temperatures.
Prompt NO is formed also from atmospheric N2 but at the flame front from radicals CH (fuel rich flame)
Fuel-bound nitrogen is converted into NO mainly in coal combustion (there is approximately 1% chemically bound N in coal)

22. CFD models HP11 CFD - TURBULENT COMBUSTION CFD - TURBULENT FLAMES
Peters N.: Turbulent combustion. Cambridge Univ.Press, 2000
CFD - TURBULENT FLAMES
Habisreuter P. et al. Mathematical modeling of turbulent swirling flames in Cox M.et al: Combustion Residues: Current, Novel and Renewable Applications. J.Wiley 2008
CFD - THERMOCHEMISTRY
Kubota, N. (2007) Combustion Wave Propagation, in Propellants and Explosives: Thermochemical Aspects of Combustion, Wiley Weinheim
Alexander

23. Flow modelling – Navier Stokes
HP11
Combustion modelling is always based upon the solution of Navier Stokes equations for vector of velocity field (even if it is combustion of liquid or solid fuels). There are 4 unknowns: 3 components of velocity ux,uy,uz and pressure. Therefore 4 partial differential equations must be solved simultaneously
Continuity equation, stating that the accumulation of mass is the divergence of mass flux
Momentum balance, stating that the mass x acceleration equals the sum of forces acting upon a fluid element (pressure forces + viscous forces + volumetric forces)
Solution of these equations (velocity field) describes convective transport of fuel, oxidiser as well as products of burning inside a combustion chamber. These equations are quite general, describe incompressible/compressible fluids, laminar as well as turbulent flow regime, but…

24. Flow modelling – Navier Stokes
HP11
…laminar as well as turbulent flows, but in the turbulent regime it would be necessary to model all details of chaotic motion of turbulent eddies. Large and slowly rotating energetic eddies decay to smaller and smaller ones down to tiny Kolmogorov eddies so small that viscous effects prevail and dissipate all kinetic energy to heat. Extremely fine mesh is necessary for resolution of Kolmogorov eddies and number of nodes Nnodes rapidly increases with the Reynolds number. Also the the range of time scales is very broad due to high frequency of turbulent fluctuations of small eddies (10 kHz).
Very small time step of Direct Numerical Simulation and number of nodes 109 make standard numerical methods of finite volumes or elements prohibitively expensive for the solution of engineering problems (including combustion).

25. Temporary fluctuation
Turbulent flow RANS/LES
HP11
The most frequently used technique for modeling of turbulent flows consists in averaging of fluctuating quantities (velocities, density, pressure, temperature and concentrations). Filtration of time fluctuations (substituting actual noisy time course at a point x,y,z by average) is the principle of RANS (Reynolds Averaging Navier Stokes) family of numerical methods while the spatial fluctuations filtering is a characteristic of LES (Large Eddy Simulation) class.
It is assumed that filtering the time fluctuations smooth out the spatial fluctuations and vice versa (because turbulent fluctuations are correlated). Δ t u
Temporary fluctuation
Mean value

26. Turbulent flow RANS
HP11
Applying time average operator to each term of continuity equation results in equation which is exactly the same as the original equation just only instead of actual velocities the mean (filtered) velocities are substituted. Time averaging applied to the momentum equation also results to a similar equation for the mean velocities but a new term, Reynolds turbulent stresses t, appears on the right hand side
Boussinesque hypothesis suggests that the Reynolds turbulent stresses are again proportional to the gradient of mean velocity as in the case of laminar flow but with a different proportionality coefficient - turbulent viscosity
Resulting equation can be solved by the same numerical techniques as in the laminar flows, but using variable viscosity ef(k,).

27. Turbulent flow RANS k-
HP11
Effective viscosity can be expressed as the sum of actual (molecular) viscosity  and turbulent viscosity t representing momentum transport by turbulent eddies. This contribution (t) depends upon the magnitude of velocity fluctuation (or square of this magnitude, expressed by the kinetic energy of turbulence k [m2/s2]) and also upon the rate of how the kinetic energy of turbulent eddies is converted to smaller eddies and in the final instance to heat. This second characteristic is the dissipation of energy of turbulence  [m2/s3] (the unit of  follows from the unit of k, because  is the change of k per second).
Turbulent viscosity can be expressed by the model
where C is a constant (usually 0.09). There is no mystery in the correlation: it can be derived easily by dimensional analysis (unit of viscosity is Pa.s) assuming power law formula and comparing exponents at kg, m, s

28. Production term (generator of turbulence)
Turbulent flow RANS k-
HP11
It is assumed that k and  are scalar properties, that can be calculated from
Transport equation for the kinetic energy of turbulence k [m2/s2]
Transport equation for dissipation of energy of turbulence  [m2/s3]
Production term (generator of turbulence)
Dissipation term

29. Turbulent flow RANS
HP11
The choice k- and the two transport equations for k and  is not by far the only way how to characterize a state of turbulence.
Two equation models calculate turbulent viscosity from the pairs of turbulent characteristics k-, or k-, or k-l (Rotta 1986)
k (kinetic energy of turbulent fluctuations) [m2/s2]
(dissipation of kinetic energy) [m2/s3]
(specific dissipation energy) [1/s]
l (length scale) [m]
Wilcox (1998) k-omega model, Kolmogorov (1942)
Jones,Launder (1972), Launder Spalding (1974) k-epsilon model
These pairs of turbulent characteristics can describe not only the turbulent viscosity but also transport coefficients (diffusivity, thermal conductivity) and the rate of micromixing Rm (kg of component mixed in unit volume per second) that controls the rate of combustion, e.g.
as follows from dimensional analysis (the same way as turbulent viscosity)

30. Turbulent flow RANS
HP11
All the presented CFD RANS models (and many others), are available in commercial programs like Fluent.
Knowing velocity field and scalar characteristics of turbulence it is possible to analyzed transport of individual species, chemical reactions, and temperatures…

31. Combustion – mass balances
HP11
i mass fraction of specie i in mixture [kg of i]/[kg of mixture]
  i mass concentration of specie [kg of i]/[m3]
Mass balance of species (for each specie one transport equation)
Because only micromixed reactants can react
Rate of production of specie i [kg/m3s]
Production of species is controlled by
Diffusion of reactants (micromixing) – tdiffusion (diffusion time constant)
Chemistry (rate equation for perfectly mixed reactants) – treaction (reaction constant)
Damkohler number
Da<<1
Reaction controlled by kinetics (Arrhenius)
Da>>1
Turbulent diffusion controlled combustion

32. Combustion – mass balances
HP11
The source term Si [kg/(m3s)] is the sum of contributions of all chemical reactions where the specie i exists either as a reactant or a product (when stoichiometric coefficient ri of specie i in reaction r is nonzero)
S-is number of species (i=1,2,…,S) and R is number of chemical reactions.

33. Combustion - enthalpy HP11
Enthalpy balance is written for mixture of all species (result-temperature field)
Sum of all reaction enthalpies of all reactions
It holds only for reaction without phase changes h ~ cpT
Energy transport must be solved together with the fluid flow equations (usually using turbulent models, k-, RSM,…). Special attention must be paid to radiative energy transport (not discussed here, see e.g. P1-model, DTRM-discrete transfer radiation,…). For modeling of chemistry and transport of species there exist many different methods but only one - mixture fraction method for the non-premixed combustion will be discussed in more details.

34. Combustion - enthalpy HP11
The production term Sh [W/m3] is the sum of reaction enthalpies over all exo- or endothermic reactions.
It can be calculated from a simplified reaction scheme (from reduced number of overall reactions, reaction enthalpies and reaction progresses of these reactions) but also from the previously calculated Si values (kg of specie i generated in unit volume and unit time due to all chemical reactions) and corresponding enthalpies of formation

35. Mixture Fraction method
HP11
Bingham

36. Mixture Fraction method
HP11
Let us consider non-premixed combusion, and fast chemical reactions (paraphrased as “What is mixed is burned (or at equilibrium)”)
Calculation of fuel and oxidiser consumption is the most difficult part. Mixture fraction method is the way, how to avoid it
mfuel
moxidiser
Flue gases
Mass fraction of fuel (e.g.methane)
Mass balance of fuel
Mass balance of oxidant
Mass fraction of oxidiser (e.g.air)

37. This term is ZERO due to stoichimetry
Mixture Fraction method
HP11
Stoichiometry
1 kg of fuel s kg of oxidiser  (1+s) kg of product
Introducing new variable
and subtracting previous equations
-Sfuel is kg of fuel consumed in unit volume per one second (at a given place x,y,z and time t).
This term is ZERO due to stoichimetry

38. Mixture Fraction method
HP11
Mixture fraction f is defined as linear function of  normalized in such a way that f=0 at oxidising stream and f=1 in the fuel stream
ox is the mass fraction of oxidiser at an arbitrary point x,y,z, while ox,0 at inlet (at the stream 0)
Resulting transport equation for the mixture fraction f is without any source term
Mixture fraction is a property that is CONSERVED, only dispersed and transported by convection. f can be interpreted as a concentration of a key element (for example carbon). And because it was assumed that „what is mixed is burned“ the information about the carbon concentration at a place x,y,z bears information about all other participating species, see next slide…

39. Mixture Fraction method
HP11
Knowing f we can calculate mass fraction of fuel and oxidiser at any place x,y,z
At the point x,y,z where f=fstoichio are all reactants consumed (therefore ox=fuel=0)
For example the mass fraction of fuel is calculated as
The concept can be generalized assuming that chemical reactions are at equlibrium
f → i mass fraction of species is calculated from equilibrium constants (evaluated from Gibbs energies)

40. Mixture Fraction method
HP11
Equilibrium depends upon concentration of the key component (upon f) and temperature. Mixture fraction f undergoes turbulent fluctuations and these fluctuations are characterized by probability density function p(f). Mean value of mass fraction, for example the mass fraction of fuel is to be calculated from this distribution
Mass fraction corresponding to an arbitrary value of mixture fraction is calculated from equlibrium constant p
Probability density function, defined in terms of mean and variance of f
Frequently used  distribution
fmean
Variance of f is calculated from another transport equation

41. Mixture Fraction method
HP11
Final remark: In the case, that fuel is a linear function of f, the mean value of mass fraction  fuel can be evaluated directly from the mean value of f (and it is not necessary to identify probability density function p(f), that is to solve the transport equation for variation of f). Unfortunately the relationship  fuel(f) is usually highly nonlinear.

42. Relative velocity (fluid-particle)
Combustion – liquid fuel
HP11
Lagrangian method: trajectories, heating and evaporation of droplets injected from a nozzle are calculated.
Sum of all forces acting to liquid droplet moving in continuous fluid (fluid velocity v is calculated by solution of NS equations)
mfuel
Relative velocity (fluid-particle)
Drag force
Drag coefficient cD depends upon Reynolds number
Effect of cloud (c volume fraction of dispersed phase-gas) Re cD
Newton’s region cD=0.44

43. Combustion – liquid fuel
HP11
Evaporation of fuel droplet
Diffusion from droplet surface to gas:
Sherwood number
Mass fraction of fuel at surface
Ranz Marshall correlation for mass transport
Schmidt number =/Ddif

44. CFD examples
Vít Kermes, Petr Belohradský, Jaroslav Oral, Petr Stehlík: Testing of gas and liquid fuel burners for power and process industries. Energy, Volume 33, Issue 10, October 2008, Pages
Lempicka

45. CFD – gas burner
HP11

46. CFD – gas burner
HP11

47. CFD – gas burner
HP11
Zeldovic NOx – thermal NOx

48. CFD – gas burner
HP11

49. CFD thermoacoustic burner
HP11
Low computational cost CFD analysis of thermoacoustic oscillations Applied Thermal Engineering, Volume 30, Issues 6-7, May 2010, Pages Andrea Toffolo, Massimo Masi, Andrea Lazzaretto
Timing of the pressure fluctuation due to the acoustic mode at 36 Hz and of the heat released by the fuel injected through the main radial holes (if the heat is released in the gray zones, the necessary condition stated by Rayleigh criterion is satisfied and thermoacoustic oscillations at this frequency are likely to grow).

50. HP11

51. EXAM
HP11
Combustion chemistry and CFD

52. What is important (at least for exam)
HP11
Heat released by reaction of reactants with stoichiometric coefficients R
Kinetics of R- elementary chemical reactions (ci is molar concentration of specie i). This rate equation holds only for elementary reactions (mono and bimolecular reactions) and not for overall reactions.
stoichiometric coefficient of the specie s in the reaction r (negative for reactant, positive for product)
reaction constant depends upon temperature (Arrhenius term ~exp(-Er /RT), where Er is activation energy of reaction r)
Example: complicated reaction mechanism of NO formation from atmospheric nitrogen can be simplified to only 3 elementary bimolecular reactions (Zeldovic mechanism of NOx formation)

53. What is important (at least for exam)
HP11
Concentration of species ci (or mass fractions i) in the combustion chamber follows from the mass balance (transport equation for specie i)
The source term Si corresponds to all chemical reactions where the specie i exists either as a reactant or a product (when stoichiometric coefficient ri is nonzero)
Temperature field follows from enthalpy balance where all chemical equations (and their reaction enthalpies) should be included

54. What is important (at least for exam)
HP11
CFD (Computer Fluid Dynamics) solution is based upon numerical solution of previous transport equations for concentrations (or i) and temperatures (or enthalpies h) together with the Navier Stokes equations and continuity equation for velocity and pressure. In case of turbulent flow it is necessary to solve also additional transport equation for turbulent characteristics (for example k and ).
This complicated system of partial differential equations can be sometimes simplified: In the case of non-premixed combustion (mixing of fuel and air proceeds not in front of but inside the combustion chamber) and assuming practically instantaneous chemical reactions (concept “what is mixed is burned”), the transport equations for all species can be substituted by only one transport equation for “mixture fraction f” (mass fraction of a key element, usually carbon).
There is no source term in the transport equation for the mixture fraction f because elements on contrary to species are conserved. Concentrations of individual species i(x,y,z) can be derived from the calculated f(x,y,z) and from stoichiometry (or assuming immediately established chemical equilibrium).