1. 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 Combustion, multiphase flows reduced… Computer Fluid Dynamics E181107 CFD8r 2181106

2. COMBUSTION CFD8r Homogeneous reaction in gases Premixed (only one inlet stream of mixed fuel and oxidiser) Non premixed (separate fuel and oxidiser inlets) Laminar flame Turbulent flame Laminar flame Turbulent flame Use mixture fraction method (PDF) Use EBU (Eddy Breakup models) Liquid fuels (spray combustion) Combustion of particles (coal) Lagrangian method-trajectories of a representative set of droplets/particles in a continuous media

3. COMBUSTION aims CFD8r 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 CFD8r m i mass fraction of specie i in mixture [kg of i]/[kg of mixture]  m i mass concentration of specie [kg of i]/[m 3 ] Mass balance of species ( for each specie one transport equation) Rate of production of specie i [kg/m 3 s] Production of species is controlled by  Diffusion of reactants (micromixing) – t diffusion (diffusion time constant)  Chemistry (rate equation for perfectly mixed reactants) – t reaction (reaction constant) Damkohler number

5. COMBUSTION enthalpy CFD8r 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 ~ c p T 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 and only one - mixture fraction method will be discussed in more details.P1-modelDTRM

6. MIXTURE Fraction method CFD8r Bacon

7. MIXTURE Fraction method CFD8r 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 Mass balance of fuel Mass balance of oxidant Calculation of fuel and oxidiser consumption is the most difficult part. Mixture fraction method is the way, how to avoid it Mass fraction of oxidiser (e.g.air) Mass fraction of fuel (e.g.methane)

8. MIXTURE Fraction method CFD8r Stoichiometry 1 kg of fuel + s kg of oxidiser  (1+s) kg of product and subtracting previous equations Introducing new variable This term is ZERO due to stoichimetry

9. MIXTURE Fraction method CFD8r 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 Resulting transport equation for the mixture fraction f is without any source term 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. 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)

10. MIXTURE Fraction method CFD8r Knowing f we can calculate mass fraction of fuel and oxidiser at any place x,y,z For example the mass fraction of fuel is calculated as 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) At the point x,y,z where f=f stoichio are all reactants consumed (therefore m ox =m fuel =0)

11. MIXTURE Fraction method CFD8r 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 Probability density function, defined in terms of mean and variance of f 0 fmean 1 p Frequently used  distribution Variance of f is calculated from another transport equation

12. MIXTURE Fraction method CFD8r 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.

13. COMBUSTION of liquid fuel CFD8r m 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 by solution of NS equations) Relative velocity (fluid-particle) Drag force Drag coefficient c D depends upon Reynolds number 1 10 4 10 5 Re cDcD Newton’s region c D =0.44 Effect of cloud (  c volume fraction of dispersed phase-gas)

14. COMBUSTION of liquid fuel CFD8r Evaporation of fuel droplet Diffusion from droplet surface to gas: Mass fraction of fuel at surface Sherwood number Schmidt number = /D dif Ranz Marshall correlation for mass transport

15. MULTIPHASE flows examples CFD8r Fluidised bed reactor or combustor Mixer (draft tube) Spray dryer Annular flow Slug flow Bubble flow Flow boiling …and others Hydrotransport, cyclones, free surfaces, breakup of liquid jets, expanding foams, aerated reactors, cavitation, mold filling… Phases Gas-liquid Gas-solid Liquid-liquid visualisation THERMOPEDIA THERMOPEDIA

16. MULTIPHASE flows methods CFD8r 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…)

17. MULTIPHASE EULER CFD8r For each phase q are separately solved  Continuity equation (mass balance of phase)  Momentum balance (each phase is moving with its own velocity, only pressure is common for all phases) Volumetric fraction of phase q Velocity of phase q Mass transfer from phase p to phase q Interphas e forces Stresses are calculated in the same way like in one phase flows

18. MULTIPHASE EULER CFD8r Specific semiemprical correlations describe interaction terms Mass transfer for example Ranz Marschall correlation for Sh=2+… Momentum exchange Special models for k pq are available for liquid-liquid, liquid- solid, and also for solid-solid combinations

19. MULTIPHASE MIXTURE method CFD8 Mixture model solves in principle one-phase flow with mean density  m, mean velocity v m Continutity equation for mixture Momentum balance for mixture (with corrections to drift velocities) Volumetric fraction of secondary phase (p) Drift velocities are evaluated from algebraic models (mixture acceleration determines for example centrifugal forces applied to phases with different density)

20. MULTIPHASE VOFVOF CFD8 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.Level set method  =0  =1     Fluent  Donor acceptor  Geometric reconstruction Dissadvantage: initially sharp interface is blurred due to numerical diffusion

21. Level Set method CFD8

22. RADIATION CFD8 Pollock

23. RADIATION CFD8 Heat flow (W) between gas and wall Emissivity of gas corresponding to temperature of gas T s Absorptivity of gas corresponding to wall temperature T w Hottel’s diagram for emissivity of CO 2 and H 2 O as a function of temperature and pL (pressure x length) Kirchhoff’s law (  = a ) Emissivity=Absorptivity at the same wavelength

24. RADIATION Fluent models CFD8 Radiation models are selected according to optical thickness of media (flue gas) a.L aL <1 DTRM (discrete transf.radiation modelling) DO (discrete ordinates) 1< aL <3 P-1 model (transport equation for radiation temperature) 3< aL Roseland model (simplified P-1 model) Absorption and scatter