# Computer Fluid Dynamics E181107

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1 Computer Fluid Dynamics E Combustion, multiphase flows Remark: foils with black background could be skipped, they are aimed to the more advanced courses Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

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

3 COMBUSTION aims Primary purpose of CFD analysis is to evaluate Temperature field (therefore thermal power, heat fluxes through wall ) Composition of flue gas (environmental requirements, efficiency of burning) To do this it is necessary to calculate Velocities and turbulent characteristics (mixing intensity) NS equations Transport of individual components (mass balances of species) Chemical reactions (reaction rates) Energy balances (with special emphasis to radiation energy transfer)

4 COMBUSTION balances m i mass fraction of specie i in mixture [kg of i]/[kg of mixture] ρm i mass concentration of specie [kg of i]/[m 3 ] Mass balance of species (for each specie one transport equation) Production of species is controlled by ( ρm ) + i( ρm u) = i( Γ m ) + S i i i i i Diffusion of reactants (micromixing) t diffusion (diffusion time constant) Rate of production of specie i [kg/m 3 s] Chemistry (rate equation for perfectly mixed reactants) t reaction (reaction constant) tdiffusion Damkohler number Da = t Da<<1 Da>>1 E Si = ρaexp( mi RT ε Si = ρcm i k ) a Because only micromixed reactants can react Reaction controlled by kinetics (Arrhenius) Turbulent diffusion controlled combustion reaction

5 COMBUSTION problems Specific problems related to combustion (two examples A and B) A) It is not correct to use mean concentrations in reaction rate equation Rotating turbulent eddy A B Bimolecular reaction A+B C Production rate S = km m locally m A m B S kmam A B B globally t m > 0, m > 0 but m m = 0 A B A B Actual reaction rate is zero even if the mean concentrations are positive

6 COMBUSTION problems Specific problems related to combustion (example B) B) It is not correct to use mean temperatures in reaction rate equation S NOx T min Example NOx production T mean Actual production rate of NOx Bimolecular reaction A+B C S E = AmAmB exp( ) RT E > AmAmB exp( ) RT Production rate locally T max S exp( E E ) >> exp( ) RT RT globally Underestimated reaction rate follows from nonlinear temperature dependence of reaction constant T[K]

7 COMBUSTION enthalpy Enthalpy balance is written for mixture of all species (result-temperature field) ( ρh) + i( ρhu) = i( Γ h) + Sh It holds only for reaction without phase changes h ~ c p T Sum of all reaction enthalpies of all reactions S = S h h i ri i Energy transport must be solved together with the fluid flow equations (usually using turbulent models, k-ε, RSM, ). Special attention must be paid to radiative energy transport (not discussed here, see e.g. P1-model, DTRMdiscrete transfer radiation, ). For modeling of chemistry and transport of species there exist many different methods and only one - mixture fraction method will be discussed in more details.

8 MIXTURE Fraction method Non-premixed combusion, and assumed fast chemical reactions (paraphrased as What is mixed is burned (or is at equilibrium) ) m fuel m oxidiser Flue gases Calculation of fuel and oxidiser consumption is the most difficult part. Mixture fraction method is the way, how to avoid it Mass balance of fuel ( ρm ) + i( ρm u) = i( Γ m ) + S Mass balance of oxidant fuel fuel fuel fuel ( ρ m ) + i( ρ m u ) = i( Γ m ) + S ox ox ox ox Mass fraction of oxidiser (e.g.air) Mass fraction of fuel (e.g.methane) kg of produced fuel [ ] 3 s m

9 MIXTURE Fraction method Stoichiometry 1 kg of fuel + s kg of oxidiser (1+s) kg of product Introducing new variable Φ = sm fuel m ox and subtracting previous equations ( ρsm ) + i( ρsm u) = i( Γ sm ) + ss ( ρmox ) + i( ρmoxu) = i( Γ mox ) + Sox fuel fuel fuel fuel ( ρφ ) + i( ρφ u ) = i( Γ Φ ) + ss fuel S ox This term is ZERO due to stoichimetry

10 MIXTURE Fraction method Mixture fraction f is defined as linear function of Φ normalized in such a way that f=0 at oxidising stream and f=1 in the fuel stream f Φ Φ 0 = = Resulting transport equation for the mixture fraction f is without any source term sm m + m fuel ox ox,0 Φ Φ sm + m 1 0 fuel,1 ox,0 ( ρ f ) + i( ρ fu ) = i( Γ f ) m ox is the mass fraction of oxidiser at an arbitrary point x,y,z, while m ox,0 at inlet (at the stream 0) Mixture fraction is 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.

11 MIXTURE Fraction method Knowing f we can calculate mass fraction of fuel and oxidiser at any place x,y,z f stoichio = sm m ox,0 + m fuel,1 ox,0 At the point x,y,z where f=f stoichio are all reactants consumed (therefore m ox =m fuel =0) For example the mass fraction of fuel is calculated as f fstoich io < = 0 < f < (fuel lean region) m = 0 f 1 (fuel rich region, oxidiser is consumed m =0) m m f stoichio fuel stoichio ox fuel fuel,1 1 fstoichio f The concept can be generalized assuming that chemical reactions are at equlibrium f m i mass fraction of species is calculated from equilibrium constants (evaluated from Gibbs energies)

12 MIXTURE Fraction method 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 p 1 m m ( f ) p( f ) df fuel 0 fmean 1 = 0 fuel Frequently used β distribution Probability density function, defined in terms of mean and variance of f Variance of f is calculated from another transport equation Mass fraction corresponding to an arbitrary value of mixture fraction is calculated from equlibrium constant p( f ) = ε ( ρ f ' ) + i( ρ f ' u) = i( Γ f ' ) + C µ ( f ) C ρ f ' k f g t d 1 0 f (1 f ) p 1 q 1 p 1 q 1 f (1 f ) df

13 MIXTURE Fraction method Final remark: In the case, that m fuel is a linear function of f, the mean value of mass fraction m 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 m fuel (f) is usually highly nonlinear. 1 m = m ( f ) p( f ) df m ( f ) fuel fuel fuel 0

14 COMBUSTION of liquid fuel 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 du by solution of NS equations) m = F m fuel dt Drag force 1 FD = ρcd A u v ( u v) 2 Relative velocity (fluid-particle) c Drag coefficient c D depends upon Reynolds number D 24 3 = (1 + Re) Re < 5 Oseen Re 16 Effect of cloud (α c volume fraction of dispersed phase-gas) c D c c α / 3.7 D = D0 c c c D D 24 ( Re ) Re 800 Schiller Nauman = + < Re = < < Re 3.10 Newton Newton s region c D = Re

15 COMBUSTION of liquid fuel Evaporation of fuel droplet Diffusion from droplet surface to gas: Sherwood number Mass fraction of fuel at surface dm d π D = = dt dt 6 Ranz Marshall correlation for mass transport Sh = Re 2 2 ( ρ p ) π D Shρg Ddif ( m fg ms ) Sc Schmidt number =ν/d dif

16 MULTIPHASE flows examples Fluidised bed reactor or combustor Mixer (draft tube) and others Hydrotransport, cyclones, free surfaces, breakup of liquid jets, expanding foams, aerated reactors, cavitation, mold filling Spray dryer Flow boiling Annular flow Phases Gas-liquid Slug flow Gas-solid Liquid-liquid Bubble flow

17 MULTIPHASE flows methods Methods Lagrange (see liquid fuel burners, suitable for low concentration of particles) Mixture (not significant difference between phases, e.g. sedimentation) Euler (the most frequently used technique for any combination of phases) VOF (Volume Of Fluid) (evolution of continuous interface, e.g. shape of free surface modeling, moving front of melted solid )

18 MULTIPHASE EULER For each phase q are separately solved Continuity equation (mass balance of phase) ( ) ( v ) t α ρ + α ρ i = mɺ q q q q q pq p Volumetric fraction of phase q Velocity of phase q Mass transfer from phase p to phase q Momentum balance (each phase is moving with its own velocity, only pressure is common for all phases) ( v ) ( v v ) p ( ) R t α ρ α ρ + = α + α τ i i + q q q q q q q q q q pq p Stresses are calculated in the same way like in one phase flows Interphase forces

19 MULTIPHASE EULER Specific semiemprical correlations describe interaction terms Mass transfer for example Ranz Marschall correlation for Sh=2+ Momentum exchange R = k ( v v ) pq pq q p Special models for k pq are available for liquid-liquid, liquidsolid, and also for solid-solid combinations

20 MULTIPHASE MIXTURE method Mixture model solves in principle one-phase flow with mean density ρ m, mean velocity v m Continutity equation for mixture ρ m + i( ρ v ) = 0 m m Momentum balance for mixture (with corrections to drift velocities) ( ρ ) ( ) ( ( ( ) T mv m + i ρ mv mvm = p+ i µ m v m + v m ) + i( α pρpv dr, pv dr, p) p v = v v Drift velocities are evaluated from algebraic models (mixture acceleration determines for example centrifugal forces applied to phases with different density) dr, p p m Volumetric fraction of secondary phase (p) ( p p) ( p pvm) ( p pvdr, p) t α ρ α ρ + = α ρ i i

21 MULTIPHASE VOF Evolution of clearly discernible interface between immiscible fluids (examples: jet breakup, motion of large bubbles, free surface flow) There exist many different methods in this category, Level set method, Marker and cell, Lagrangian method tracking motion of particles at interface. ε ε=1 ε=0 ε ε=0 ε ε=1 ε=1 ε=1 ε=0 ε=0 ε Fluent Donor acceptor Geometric reconstruction ε + u i ε = Dissadvantage: initially sharp interface is blurred due to numerical diffusion 0

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