CFD modeling of combustion

Transcription

1 CFD modeling of combustion Rixin Yu 1

2 Lecture 8 CFD modeling of combustion 8.a Basic combustion concepts 8.b Governing equations for reacting flow Reference books An introduction to computational fluid dynamics, the finite volume method, H.K. versteeg, W. Malalasekera Chapter 12 Theoretical and numerical combustion, (2 nd edition), T. Poinsot, D. Veynante, Chapter 1 2

3 A few examples of combustion Keywords: Heat, burning, light, Power, emission pollution, Chemical reactions, multi-component mixture, radicals Flame, unstable combustion, detonation, etc. 3

4 A combustionmixturecontains multiple species ( 1) The mass fractions, the mole fractions for each species (1,., ),, Molecular weight for species : [,,,] = [,,, ] Mole fractionfor species : [,,,] = [X,,, ] = 1; = / = = / = Massfractionfor species : [,,,] = [Y,,, ] = 1 Mixture-averaged mean molecular weight = 1/ = 4

5 Total pressure, partial pressure, equation of state for a mixture containing multiple species (1,., ) Total pressure: Partial pressure: = = = = = = = : universal gas const. Equation of state = = / = Mean molecular weight = 1/ 5

6 Total enthalpy, sensible enthalpy and enthalpy of formation for a specie h : Enthalpy [ ] of a species (k) with respect to reference enthalpy at standard pressure (1ATM) and temperature ( =298.15K) Total = sensible +chemical h = h, + Δh, =, + Δh, Sensible enthalpy: h, chemical, enthalpy of formation( ),, : specific heat capacity at constant pressure for species k Enthalpy of formation Δh, : increase in enthalpy when a compound is formed from its constitute elements in their nature formsat standard conditions, for H2, O2, N2, C (graphite) it is zero, for CO2 it is KJ/kmol, because the exothermic reaction(heat release): (h) + 6

7 Sensible energy and chemical energy for a single specie Sensible+chemical energy = h = h, =, + Δh, + Δh, =, + Δh, Sensible energy, Chemical, enthalpy of formation ( ), : specific heat capacity at constant volume for species 7

8 Enthalpy and Energy in a multi-component mixture Enthalpy of Mixture: Energy of Mixture: h = h =, + Δh, =, + Δh, = = =, + Δh, + Δh, =, Enthalpy of formation for mixture + Δh, = / + Δh, & :Mixture-averagedheat capacity at constant volume and pressure respectively, &, : Heat capacities for a single spices 8

9 Relation between Energy and enthalpy for the entire mixture and for a single specie = h = = h = h 9

10 Apply first law of thermodynamics to a combustion system Assume homogenous (no spatial gradient), zero velocity (no kineticenergy) Given Unburned (fresh) state,,with mass fraction = [Y,, To find Burned (product) state, with mass fraction = Y,,, ],? constant pressure = h = h + Δh, = + Δh,? constant volume = = + Δh, = / + Δh, = / 10

11 Apply first law of thermodynamics to a 0-D combustion system Example: Assume a single-step reaction, lets determine the final burned state mass fraction of unburned state = [Y,,, mole fraction of unburned state ] = [X,,, ] () Left coeff Right coeff (1) 1 0 (2) 2 0 (3) 0 1 (4) Δ 2Δ 1Δ 2Δ = ( ) Δ = + Δ = min (, ) Either fuel or oxydizerare completely consumed! Note: = 1 + just normalize to get mole fractionof burned state 1, = = mole fractionof burned state! 11

12 The chemical-kinetic equation Different ways of preparing the reactant-mixture with a same fuel ( ) ( ) ( ) ( ) ( ) ( ) stoichiometry stoichiometry ( ) () Left Right (1) 1 ¾ (2) 0 0 (3) 0 ¼ (4) 0 ½ (5) /2 (6) Air ½ 0 () Left Right (1) 1 ¾ (2) ½ 0 (3) 0 ¼ (4) 0 ½ (5) 3.76/2 3.76/2 Elements conservation : [,,,] = [,,,] [] : number of a element [C] contained within the molecular of species 12

13 A global Fuel/Oxydizer reaction: stoichiometry and equivalence ratio ( ) ( ) ( ) Δ 2Δ 1Δ 2Δ Δ = = 1 2 = = 1 2(W W ) No fuel or oxidizer on the product side Both fuel and oxidizer are completely consumed! Δ = = Equivalence ratio: = / > 1: h < 1: = 1: h ( ) = 1 3 / 1 2 = 2 3 < 1: 13

14 Simple estimate of the adiabatic flame temperature If a fuel/oxidizer mixture is burned completely (assume under constant pressure), and if noexternal heat or worktransfer takes place, then all energy liberated by chemical reaction will be used to heat the product, achieving max (adiabatic ) flame temperature! stoichiometry ( ) Δ 2Δ Δ 1Δ 2Δ Δ Left Right () (1) 1 0 (2) 2 0 (3) 0 1 (4) 0 2 (5) [Y,,, Reaction Δ = mi (, ) = ( ) Δ = + normalize = 0, = 0] Note: for non-stoichiometry mixture (i.e. 1),the product mixture may contain unburned fuel or oxidizer (i.e. 0or 0) + Δh, = + Δh, 14

15 Chemical equilibrium and reverse reaction In practice, some reactions occur in the reversedirection (more prominent at high temperature), which is sometime called disassociation Equilibrium maximize Gibbs function Gibb function [ ]: = h specific entropy : Condition for equilibrium: Δ = log Equilibriums constant. =.. =... =.. 15

16 Mechanisms of combustion and chemical kinetics Example for hydrogen oxidization The approximated one-step global reaction The detailed reaction mechanism contain multiple steps of elementary reactions involving multiple intermediate species ,,,,intermediate species (radicals), denotes third body ( or, arbitrary atom/radical/molecureswhich can increase the collision chance, high pressure means more concentration third body to collide) 16

17 Detailed chemistry, Intermediate species Another example for methane oxidization 17

18 Detailed chemistry, Intermediate species Example for methane oxidization The detail (not fully complete) GRI-mechanism contains: 325 elementary reactions, 53 species, which is optimized for certain ranges of temperature and pressure conditions Different chemical reaction pathway or subsystem. 18

19 Chemical reaction does not happen in an instant, it takes time Elementary reactions and the reaction rate Molecularity Unimolecular Bimolecular Termolecular Elementary Step Rate Law for Elementary step [ ] = [] []: = [][] = = [] = = [][][] = ate = [] Note: forward/backward reaction can also be related through equilibrium condition 19

20 Reaction rate constant and Arrhenius law Reaction rate constant : (Arrhenius law) () = exp ( ) 0 when 0 when : pre-exponential constant : temperature exponent : Activation energy. Just a note: has different unit for different order of elementary reaction Unimolecular, = [] Bimolecular, = [][].. exp ( ) 20

21 Determine the reaction rate of a specie involved in multiple elementary reactions All species All elementary reactions (all rewritten as forward reaction) 1: 2: : : 1: : : rate in mole unit rate in mass unit, = (, =,, = ),, k = 1, Total mole concentration = exp ( ) 21

22 Governing equations for the time-evolution of a homogenous, zerovelocity stationary, reacting mixture (0D) = 1 =, = 1, 1 = / Const. pressure or + + h = 0 = 0 Adiabatic + 2Unknowns for the above + 2equations: = [ (), (),, = 1, ] Initial conditions: = [ ( ), ( ),, = 1, ]

23 Chemical reaction can be viewed a (nonlinear) dynamic system problem Typical features in terms of trajectory and attractors for gas phase combustion system Let s examine a simplified reaction system being reduced to contain only three unknowns, the solution for this nonlinear ordinary differential equations (ODES) are trajectory moving in a 3D phase space spanned by (,, ), say ( Y ). Attractingcurveformed by all chemical equilibrium states rapid change of state after being activated = (,, ) = (,, ) = (,, ) Initially slow incubation = Δ (,, ) = Δ (,, ) = Δ (,, ) Catalyst drill a tunnel 23

24 Chemical reaction can be viewed a (nonlinear) dynamic system problem In additional to the gas-phase combustion problem, there are plenty of complex phenomena for more generalnonlinear dynamics system (Examples: pendulum system, three-body problem, there are lots of math ) The famous 3D butterfly trajectory with chaotic attractor. 24

25 For a more general 0-D reacting system Neglect transport (i.e. zero spatial gradient) A set of equations solved for (),, (); k=1,..,n), starting at = 0. The solution to the ODEs is a trajectoryin high dimensional phase space spanned by N+2unknowns variables. A few simple algebraic constraints such as conservation of elements and also total masscan reduce the number of unknowns. A note from theoretical chemistry: chemical reactions do not have to be attracted to some equilibriumthermodynamic behavior! (It may have some type of limit-cycle or chaotic orbit). The Belousov-zhabotinsky reaction! YouTube showing Belousov-zhabotinsky reaction! 25

26 Theoretic and numerical aspects for 0-D reacting system A set of ODE equations solved for (),, (), k=1,,,n), starting at = 0. 1) For most gas-phase combustion, there often exists fast and slow elementary reactions, the time scales may differ by several order of magnitude. It is a mathematical stiff system with multiple drastically different time-scales, an expensiveadaptive-time-step ODE solver must be used to perform numerical time-integration. 1) Such calculation will usually be performed by popular software package: such as Chemkin(free before, not any more), Cantera(free) and Flamemaster. Note, evolutions of all thermodynamic and transport coefficients (,, Δh,,.., ) are usually based on the NASA polynomials, the chemical kinetic mechanism including all elementary reactions and the reacting constants can be downloaded together with a published journal article. 2) For common gas combustion reaction, there often exist certain intrinsic lower-dimentional manifolds (ILDM) in the phase space, towards which a trajectory will be quickly attracted. When the trajectory come close to the vicinity of such manifold region, the solution along trajectory then stay parallel and move slowly within such manifold. 3) Very expensive calculations of stiff-ode solver for every CFD-cells. Ideal: Tabulation The In-situ adaptive-tabulation (ISAT), by S.B. Pope. 26

27 CFD modeling of combustion Governing equations for reacting flow 27

28 Governing equation for reacting flow Combustion does not create new mass, it just redistributes mass among different species. Burning liberated heat causes flow dilatation Global Mass + = 0 = Momentum + = + +, = ( + ) Typical combustion causes = = 5 10 large variation in dynamic viscosity () and large dilatation term 28

29 Conservation of species mass Mass conservation for species k + ( +, ) =, = 1,, + = (, ) +, = 1,,, : the diffusion velocity Gobla mass eq. 1 + =, + 1 = = 1, = 0 = 0 29

30 Compute the diffusion velocity An less accurate simple gradient model (Fick law ) Fick law, = violate:, = 0 + ( + ) = (, ) +, = 1,, Assume const for all species, i.e. = = D = D, =, = = 0 Note: some CFD code does not use this strategy of correctionveloclity, the inconsistence error will be pumped into an abundant diluting gas such as N2 30

31 Compute the diffusion velocity Solve the more accurate full equations mole = + +, for = 1,.. = is binary mass diffusion of species diffuse into, = / is the mole fraction of, Neglect Soret effect (mass diffusion due to temperature gradient). 31

32 Diffusion velocity Binary diffusion in a two-species system + = 1: Assume: is mall, neglect volume force: = + +, for = 1,.. Binary diffusion: = = + = 0 = = / Fick law is exact for binary diffusion 32

33 Diffusion velocity Multi-species diffusion: Hirschfelder-Curtiss approximation = + +, for = 1,.. Multi-species diffusion: A complicated inversion problem, Hirschfelder-Curtiss approximation is a best first-order approximation of exact system. not Fick law anymore = = species diffuse into the "mixture" = = / / 33

34 Species mass equations with different models of diffusion velocity = = / Hirschfelder-Curtiss approx. (more accurate) = + ( + ) = ( ) +, = 1,, = Fick approx. (not accurate, but easy for numerical implementation) + ( + ) = + 34

35 Various definition of Energy and enthalpy Kinetic energy : Chemical energy: Δh,, h, enthalpy of formation 35

36 Derive the kinetic energy equation from mass and momentum eq.s Useful indentiy: material-derivative + = + + = + +, Momentum eq = 0 = + +, = + = ( +, ) 36 viscous-stress contributes to reversible mechanical work!

37 Energy equation for total energy (sensible + chemistry+ kinetic energy) Useful indentiy: material-derivative + = + Total energy = + + +, ( +, ) = + h, = Fourier s law Diffusion of multispecies with different enthalpy : external heat source (not burning released heat), ( +, ), power produced by volume force. Buoyance, etc. 37

38 Energy equation for total enthalpy (sensible + chemistry+ kinetic energy) Total Enthalpy: h = + / = h = + + +, ( +, ) h = + + +, ( +, ) = h = + + +, ( +, ) 38

39 Energy equation for enthalpy (sensible + chemistry+ kinetic energy) Enthalpy: h=h h = + + +, ( +, ) 1 2 = + +, h = + + +,, 39

40 Energy equation for sensible enthalpy (sensible + chemistry+ kinetic energy) Sensible Enthalpy: h = h Δh, Δh, h = + + +,, =, +, = 1,, h = + + Δh,, Δh, + +,, = + h, h, = h Δh, h = + + h,, Δh, + +,, 40

41 Energy equation in temperature form h = +, (, ), [,] h h = + +, h,, Δh, - - =, + + +,, (,) h,, h = h, + Δh, = + +,, h + +,, 41

42 Various form of energy eq. 42

43 Summary of equations no body force, no external heat Global Mass + = 0 Species conservation + =, +, = 1,, 1 Momentum + = + Energy h = + = +,, + Δh, 43

44 Other simplification of the governing equations Low Mach number assumption (, ) = () + (, ) and p Thermodymamic + hydrodynamic Transport coeff. ( such as Heat capacity ) Equal (among k) for all species Const(t) for mixture Non-dimentional number. Lewis number (the ratio of thermal diffusivity to mass diffusivity.) Schmidt number (the ratio of momentum diffusivity (kinematic viscosity) and mass diffusivity ) Prandtl number (ratio of momentum diffusivity to thermal diffusivity) 44

45 Let s consider a toy reacting system involving only two species and a single step reaction Mass fraction of: Product: Fuel : 1 Fuel Product (For example 3 2 ) Assume: (1)1D (2)Same molecular weight: = = = ) (3) Δh = 0, Δh < 0 (heat release, exothermal reaction) (4) Constant thermodynamic/transport properties for fuel/product and perfect ideal gas, heat capacity:, mass diffusivity: = = (to be used later) (more) = Δh, h = + p = ; = 45

46 Let obtain a greatly reduced equations for the toy system (1) species mass equation: + ( + ) =, +, = 1,, 3D1D Fick law + = +, = 1,2 Only two species Assume const. 46

47 Let obtain the reduced equations for the toy system (2) momentum equation: + = + +, 3D1D = ( + ) + ( + ) = 4 3 Assume const. 47

48 Let obtain the reduced equations for the toy system (3) energy equation: Assume const. Neglect viscous heating h = + + +, ( +, ) Compressible Conservative form: + h = = +, = 0,, + Δh, Low Mach number assumption: (, ) = () + (, ), Nonconservative form: = + Δh, 48

49 Mass fraction of: Product: Fuel : 1 Conservation law for: Specie mass: product total mass: Momentum: Energy: Simple 1D Toy reacting system Summary of compressiblegoverning equations + Fuel Product = + + = = + h = Arrhenius reaction = 1 (1 ) = Δh, h = + = Compressible flow = 49

50 Mass fraction of: Product: Fuel : 1 Simple 1D Toy reacting system Summary of governing equations with low Mach assumption Fuel Product Conservation law for: Specie mass: product total mass: Momentum: + = + + = = Arrhenius reaction = 1 (1 ) Energy: = + Δh, Low Mach assumption ( incompressible ): =, (, ), (, ) () 50