Lecture 12 Droplet Combustion Spray Modeling Spray combustion: Many practical applications liquid fuel is injected into the combustion chamber resulting in fuel spray. Spray combustion involves many physical processes, including atomization, oplet collision and agglomeration, vaporization, heat and mass transfer, oplet-air and vapor-air mixing, ignition, turbulence, premixed and/or di usion flames, pollutant production and flame extinction. The liquid fuel is atomized by the combined action of aerodynamical shear, strain and surface tension producing a large number of small oplets which increase the overall surface area exposed to the oxidizer and enhance the rates of heat and mass transfer. The atomization of a 1cm diameter oplet of liquid into oplets of 100 µm diameter, for example, produces a million oplets and increases the overall surface area by a factor of 10 4. University of Illinois at Urbana- Champaign 1
The oplets distribution range from few microns to around 500µm, primarily because practical nozzles cannot produce sprays of uniform op size at typical operating conditions. Furthermore, many of the larger oplets produced in the initial liquid disintegration undergo further breakup into smaller oplets. Depending on the oplet distribution, the mode of operation and the state of the ambience, di erent regimes of burning occur, including a di usion flame lies outside the cloud of oplets burning the ambient oxygen with the fuel that originates from the vaporizing oplets oplet burning occurring all throughout the cloud or in a layer on the edge of the cloud, internal group combustion with an internal flame that separates a group of vaporizing oplets from a group of individually burning oplets individual oplet combustion where individual oplets burn with oxygen di using across the resulting cloud of burning oplets oxidizer entrainment atomizer vaporization oplet burning flame Liquid jet liquid S P R A Y atomization oxidizer entrainment cloud burning University of Illinois at Urbana- Champaign 2
Droplet Vaporization and Combustion: Assumptions: oplet is a sphere spherical symmetry single component fuel uniform properties inside the oplet Fuel vapor op no fluid motion inside the oplet quiescent ambience no gravity quasi-steady approximation one-step overall chemical reaction (F + O! Products) Overall mass conservation of the oplet d dt 4 3 lr 3 s = ṁ mass flow rate mass per unit time leaving the oplet 4 r 2 s l s dt =4 r2 s g (v s dt ) Since g / l 1 (typically 10 2 10 3 ) the oplet recedes very slowly compared to the gas-phase processes, and s dt g l v a direct consequence of the jump relation [ (v n V I ) ]] = 0 across the interface r = r s, namely l ṙ s = g (v ṙ s ) after using g / l 1. quasi-steady approximation ṁ or g v can be determined from the steady equations in the gas phase, with the oplet history determined a-posteriori from the relation above University of Illinois at Urbana- Champaign 3
Conditions for the mass fractions at the liquid-gas interface can be derived by integrating the species equations across the moving liquid-gas interface, r = r s, or using the previously derived jump condition [[ Y i ((v+v i ) n V I )]]=0 where we have assumed that no reaction occurs along the interface Using the approximation ṙ s ( g / l )v g Y i (v ṙ s ) g D i dy i = ly i ṙ s g vy F g vy O g D F dy F = gv g D O dy O =0 Similarly, integrating the energy (expressed in terms of enthalpy) across the moving liquid-gas interface, r = r s, or using the previously derived jump condition [[ h(v n V I )+pv I + q n ]] = 0 g h g (v ṙ s ) g dt dt = l h l ṙ s l a + a where we have assumed that no reaction occurs along the interface and neglected kinetic energy and viscous dissipation in the energy equation (the small Mach number approximation). Then, using the approximation ṙ s ( g / l )v g v (h g h l ) {z } Q v latent heat of vaporization dt a + dt a =0 University of Illinois at Urbana- Champaign 4
g vq v dt a + dt a =0 assuming all the heat conducted inwards goes to heating the liquid fuel, dt 4 a 2 4 dt 3 a3 l c pl dt dt = vq v + a 3 dt lc pl dt {z } l often neglected. a Alternatively it may be accounted for by replacing Q v with an e ective value that accounts for internal heating. dt = v Q v In the following, the gas density will be denoted by (with no subscript) and the oplet radius a. The steady conservation equations are d v dy F v dy O (r2 v) =0 d r 2 dy F D F r 2 D O r 2 d v dt 1 c p r 2 r 2 dy O d r 2 dt = F W F! = O W O! = Q c p! a liquid Conditions far away, as r!1 T = P 0 R/W YF YO! = B e E/RT W F W O T = T 1, = 1 Y F =0 Y O = Y O 1 University of Illinois at Urbana- Champaign 5
If one assumes that the phase change at the interface occurs very fast, equilibrium conditions prevail and a definite relation exists between the vapor partial pressure and the oplet surface temperature. This relation is of the form py F = k exp Qv R 1 1 T B T known as the Claussius Clapeyron relation. Instead we shall use for simplicity that the surface temperature is constant, and at r = a T = T B Boundary conditions at r = a vy F vy O D F dy F = v D O dy O =0 dt = vq v T = T B We write dimensionless equations using a as a unit of distance, D th /a as a unit of velocity and normalize the density and temperature with respect to their ambient values 1 and T 1. We also use M = ṁ/4 a( /c p ) to denote the dimensionless mass flux. M = r 2 v T =1 M dy F r 2 M dy O r 2 M dt r 2 Le 1 F r 2 Le 1 O r 2 d d 1 d r 2 r 2 dy F r 2 dy O r 2 dt! = D 2 Y F Y O e 0/T =! =! = q! at r = 1: MY F Le 1 dy F F = M MY O Le 1 dy O O =0 as r!1 dt = ML v, T = T s T = =1, Y O = Y O 1 Y F =0 We have a 6th-order system, with 7 BCs; the extra one for the determination of M. University of Illinois at Urbana- Champaign 6
The parameters are = O W O / F W F 0 = E/RT 1 q = Q/ F W F c p T 1 D = 2 1a 2 F B ( /c p )W O L v = Q v /c p T 1 T s = T B /T 1 We note in, particular, the dependence on the Damköhler number on the oplet radius D a 2 and on the ambient pressure D P 0. We also assume unity Lewis numbers. Pure vaporization: D =0 Y F =1 e M/r Y O = Y O 1 e M/r M(1 1/r) T = T s L v + L v e M =ln 1+ 1 T s L v In dimensional form, the vaporization rate is ṁ = 4 a c p ln 1+ c p(t 1 T B ) Q v transfer number B v = c p(t 1 T B) Q v = impetus for transfer resistance to transfer Clearly B v > 0, which implies that, for vaporization, T 1 >T B. University of Illinois at Urbana- Champaign 7
T 1 Y F =1 T T B Y F 1 r Combustion: We will consider the Burke-Schumann limit D!1and make use of the coupling functions. T + qy F =1+(T s 1+q L v )(1 e M/r ) T + q 1 Y O =(T s L v )(1 e M/r )+(1+q 1 Y O 1 )e M/r from which we can also write Y F 1 Y O =1 (1 + 1 Y O 1 )e M/r The profiles are readily available 8 < 1 (1 + 1 Y O 1 )e M/r (r<r f ) Y F = : 0 (r>r f ) 8 < 0 (r<r f ) Y O = : ( + Y O 1 )e M/r (r >r f ) University of Illinois at Urbana- Champaign 8
8 < T = : (T s L v )(1 e M/r )+(1+q 1 Y O 1 )e M/r (r<r f ) 1+(T s 1+q L v )(1 e M/r ) (r>r f ) There are few conditions that remain to be satisfied; these are T = T s at r = 1, and continuity of all variables at r = r f. The first provides and expression for M and the others determine the flame stando distance and the flame temperature. M =ln 1+ 1+q 1 Y O 1 T s L v In dimensional form, the burning rate is ṁ = 4 a c p ln 1+ c p(t 1 T B )+QY O 1 / O W O Q v B c = c p(t 1 T B )+QY O 1 / O W O Q v = impetus for transfer resistance to transfer transfer number The flame stando distance r f = a ln(1 + B c ) ln(1 + F W F Y O 1 / O W O ) Note that r f > 1 implies that c p (T 1 T B ) > (Q v Q/ F W F ) 1 Y O 1 which is not restrictive because typically Q>Q v. The flame temperature T f = T 1 + (Q/ F W F Q v )/c p (T 1 T B ) 1+ O W O / F W F Y O 1 University of Illinois at Urbana- Champaign 9
Y F =1 T Y O 1 T B Y O Y F T 1 1 r f r If the activation energy parameter is considered large, the asymptotic analysis provides a more complete picture spanning the entire range of Damköhler number D. M c BS limit T a Burning rate M v Vaporization limit D ext D ign Damköhler number D D ~ a 2, P 0 Extinction may occur as the oplet size diminishes below a critical value; auto-ignition may occur if the ambient pressure is increased significantly. University of Illinois at Urbana- Champaign 10
Droplet history: ( use back the notation a = r s and = g ) The vaporization and burning rates were found to have the form ṁ =4 r s ( /c p )ln(1+b) with the appropriate transfer number. ṁ =4 r 2 s g v ) g v = cp ln(1 + B) 1 r s s dt = g l v = k k = 2 ln(1 + B) with 2r s c p l 2r s s dt = k ) r2 s = r 2 s(0) kt d 2 -law d 2 = d 2 (0) Kt K = 8( /c p) l ln(1 + B) d 2 t life = l g d 2 0 8D th ln(1 + B) 1/K t Nuruzzaman et al. PCI, 1971 The d 2 -law prediction is verified experimentally for most of the oplet lifetime with the exception, perhaps, of a short initial period following ignition University of Illinois at Urbana- Champaign 11
The behavior of a single oplet in a combustion chamber depends on the Reynolds number Re = g d 0 u u p /µ g,whereuisacharacteristic velocity of the gas and u p the particle velocity. So far, the e ect of the relative velocity of the ambient gas on the oplet has been neglected, practically taking Re = 0. The spherical symmetry assumption cannot be justified when Re 6= 0. A general approach that has been often used to adess the problem of oplet combustion in a convective field has been to correct a-posteriori the results derived for Re = 0 with empirical correlations. Formulas that have been used are of the form K = K (0) 1+CRe 1/2 Pr 1/3 C 0.3 where K is the evaporation constant, K (0) the evaporation constant for a stationary burning oplet, Pr = µ g /D th g is the Prandtl number. Similarly, burning rate corrections for free convection has been proposed, in the form K = K (0) 1+CGr 0.52 C 0.5 where Gr is the Grashof number, representing the ratio of the buoyancy to viscous forces. Gr = 2 gd 3 0 g T µ 2 g T Here T may be taken as the di erence between the flame temperature and the ambient temperature. The assumption of spherical symmetry for suspended oplets can be justified either in micro-gravity conditions (g 1) or in reduced pressure environment (P 0 g 1) University of Illinois at Urbana- Champaign 12
Combustion of a solid particle: Similar to a fuel oplet a solid particle may be surrounded by a di usion flame, so that ṁ = 4 r s c p ln (1 + B) if the flame is close enough to the surface, r f 1 ln (1 + B) =ln 1+ 1 Y O r s solid ṁ =4 r s g D O ln 1+ 1 Y O This is the di usion-controlled limit, where the di usion of oxidizer towards the particle controls the burning. For Le = 1, we have /c p = D O. Using ṁ = d dt 4 3 srs 3 = 4 s rs 2 s dt ṁ =4 r s g D O ln 1+ 1 Y O we find in the di usion-controlled limit r 2 s = r 2 s(0) 2 g D O s ln (1 + 1 Y O ) t d 2 = d 2 0 8 g D O s ln (1 + 1 Y O ) t t life = s g d 2 0 8D O ln(1+ 1 Y O ) namely a d 2 -law. University of Illinois at Urbana- Champaign 13
If burning occurs on the surface (and possibly inside the pores) of the particle, the burning rate is controlled by the chemical kinetics. For an Arrhenius-type heterogenous surface chemical reaction rate, we have ṁ =4 r 2 sk s Y O r s where k s = B s e E/RTs solid kinetic-controlled burning This is the kinetic-controlled limit, where the chemical reaction rate controls the burning. Using ṁ = d 4 dt 3 srs 3 = 4 s r 2 s s dt k s r s = r s (0) Y O t s d = d 0 2k s g s Y O t namely a d 1 -law (linear). t life = s g d 0 2B s Y O e E/RT t life = 8 >< >: s g s g d 2 0 8D O ln(1+ 1 Y O ) d 0 2B s Y O e E/RTs di usion-controlled kinetic-controlled t independent of T t d 2 0 t e E/RT t d 0 The controlling mechanism is determined by a surface Damköhler number D s = t di usion t kinetic = d 0 B s Y O e E/RTs 4D O ln(1+ 1 Y O ) large particles, or higher ambient temperature ) D s large ) di usion-controlled small particles, or low ambient temperature ) D s small ) kinetic-controlled University of Illinois at Urbana- Champaign 14
Spray Modeling: The accurate description of spray combustion, say by DNS, requires identification of whether liquid or gas is present at every point in space, to employ appropriate equations of state in each of the separate phases and to determine the evolution of a large number of liquid-gas interfaces with boundary conditions that involve phase transformation. The large number of oplets required to simulate real dispersed flows in turbulent environments that are encountered in practical applications, make such calculations prohibiitve. There are a practically three approaches to spray combustion modeling. Continuum formulation Lagrangian formulation Probabilistic formulation We will not be discussing any of these in detail, but we will try to get a general idea of what each of these formulations involve. Continuum Formulation. Properties of both the particles and the gas are assumed to exist at every spatial point (the average value of a small parcel that includes both phases) regardless of whether that point is actually in the liquid, solid or gas at that instant. The spray contains N distinct chemical species and M di erent class of oplets identified by fuel content, size and/or velocity. Let m i be the mass of species i and ˆm (k) the mass of oplets of class k. The total volume of the mixture will be denoted by V T and will be given by V T = V g + V l,wherev g and V l are the volumes occupied by the gas and the liquid phases, respectively. ' = V g /V T is the volume fraction of the gas, namely the ratio of the volume occupied by the gas to the total volume of the mixture. 1 ' is the void fraction, or the ratio of the volume occupied by the condensed phase from the total volume V T. University of Illinois at Urbana- Champaign 15
The bulk density (mass per total volume V T ) of species i is i. NX The bulk density of the gaseous mixture is = i = 1 NX V T It di ers form the gas density g = 1 V g N X unit volume that includes only gas. Similarly, the density of oplets of class k is ˆ (k), so that the bulk density of oplets is MX ˆ = ˆ (k) = 1 MX ˆm (k) V T k=1 which di ers from the liquid density l = 1 V l M X k=1 i=1 k=1 ˆm (k) = V T V l ˆ = 1 1 ' ˆ i=1 m i = V T V g = 1 ' i=1 m i which is the mass per For a dilute spray the condensed phase volume fraction is small (1 that g,but ˆ =(1 ') l 1 ' = O(1) because g / l 1. g / l ˆ is often referred to as the spray density. ' 1) so Statements of mass conservation of the gas and condensed phases lead to @ @t + r v = Ṁ @ ˆ @t + r ˆ ˆv = Ṁ Here Ṁ is the total mass (per unit volume, per unit time) added to the gas phase due to vaporization and v, ˆv are the gas and particle velocities. This can be determined, for example, from the expressions we derived for the vaporization rate of isolated oplets. Note: @ ( +ˆ )+r ( v +ˆ ˆv) =0 @t i.e., the total mass is conserved. Using ˆ =(1 ') l, we can recast the mass conservation of the oplets in the following form @' + r (ˆv') =Ṁ + r ˆv, @t l ' is no longer conserved; the volume occupied by the oplets changes as a result of (i)vaporization, and (ii) distance between oplets University of Illinois at Urbana- Champaign 16
Similarly, separate statements of momentum and energy conservation are written for the gas and liquid phases with appropriate source/sink terms representing the total momentum and energy carried to the gas from the vaporizing oplets, the total force exerted on the particles from their surroundings and the internal heating of the oplets which, in general, could have a di erent temperature than the bulk gas. Probabilistic Formulation. In this approach the condensed phase is described by a distribution function, or probability density function (pdf) f(r, x, v,e,t) fdx dv de... is the probable number of particles in the radius range about r, in the spatial range dx about the position x, with velocities in the range dv about v and internal energy in the range e about de, etc..., at time t. The time evolution of the distribution function f is given by @f @t + r x (vf)+r @ v (af)+ @r (ṙf)+ @ (ėf)+ = Q, @e and is known as the spray equation. the terms on the l.h.s. of the spray equation represent op-carrier gas interactions, the source term Q on the r.h.s. represents particle-particle interactions (fragmentation, coalescence, etc..). University of Illinois at Urbana- Champaign 17
The quantities in the spray equation are determined by the laws governing the dynamics of individual oplets dx dt = v, dv dt = a, dt =ṙ, de dt =ė, etc... The conservation laws for the gas-oplet mixture will include additional source terms involving the integral of f. For example, the mass and momentum equations are Z f For example, the dependence of ṙ on the radius for liquid ops can be specified by adopting the d 2 -law /c p ln (1 + B) ṙ =, l r @u @t + u ru = @ f @t + r ( f u)= rp + f g 4 r 2 l ṙf dvde, Z 4 3 r3 l a +4 r 2 l ṙ(v u) fdvde, where f is the fluid density, namely the mass of the gas mixture per unit volume of space, which due to the presence of the particles di ers from the gas density g that corresponds to the mass per unit volume occupied by the gas. The main di culty in this approach is the high dimensionality of the distribution function f(r, x, v, e,t) which, under the simplest assumptions, contains nine variables. Lagrangian Discrete Particle Formulation. The oplets are considered as individual point-particles and their trajectories are solved for, separately from the flow field which exists only where there are no oplets. The main di culty is probably the inability to track a su ciently large number of particles in order to University of Illinois at Urbana- Champaign 18