D 2 Law For Liquid Droplet Vaporization References: Combustion and Mass Transfer, by D.B. Spalding (1979, Pergamon Press). “Recent advances in droplet vaporization and combustion”, C.K. Law, Progress in Energy and Combustion Science, Vol. 8, pp , Fluid Dynamics of Droplets and Sprays, by W.A. Sirignano (1999, Cambridge University Press).

Gas-Phase Streamlines Droplet With Internal Circulation Gas-Phase Streamlines Droplet With Internal Circulation Heat Fuel Vapour Buoyancy and droplet size destroy spherical symmetry evaporation Spherical Symmetry Model

Without/micro gravity With gravity

Mass Transfer I DEFINITIONS IN USE: density – mass of mixture per unit volume ρ [kg/m 3 ] species - chemically distinct substances, H 2 O, H 2, H, O 2, etc. partial density of A – mass of chemical compound (species) A per unit volume ρ A [kg/m 3 ] mass fraction of A – ρ A /ρ = m A note: ρ A + ρ B + ρ C + … = ρ m A + m B + m C + … = 1

DEFINITIONS IN USE: total mass velocity of mixture in the specified direction (mass flux density) – mass of mixture crossing unit area normal to this direction in unit time G TOT [kg/m 2 s], G TOT =  u (density x velocity) total mass velocity of A in the specified direction = G TOT,A [kg/m 2 s] note: G TOT,A + G TOT,B + G TOT,C …= G TOT convective mass velocity of A in the specified direction m A G TOT = ( ρ A /ρ) G TOT = G CONV,A note: G CONV,A + G CONV,B + G CONV,C …= G TOT but generally, G CONV,A ≠ G TOT,A diffusive mass velocity of A in the specified direction G TOT,A – G CONV,A = G DIFF,A note: G DIFF,A + G DIFF,B + G DIFF,C + … = 0

DEFINITIONS IN USE: velocity of mixture in the specified direction = G TOT /  [m/s] concentration – a word used for partial density or for mass fraction (or for mole fraction, partial pressure, etc.) composition of mixture – set of mass fractions

mass flux in mass flux out mass accumulated - =

The d 2 Law - assumptions (i)Spherical symmetry: forced and natural convection are neglected. This reduces the analysis to one-dimension. (ii) No spray effect: the droplet is an isolated one immersed in an infinite environment. (iii) Diffusion being rate controlling. The liquid does not move relative to the droplet center. Rather, the surface regresses into the liquid as vaporization occurs. Therefore heat and mass transfer in the liquid occur only because of diffusion with a moving boundary (droplet surface) but without convection. (iv) Isobaric processes – constant pressure. (v) Constant gas-phase transport properties. This causes the major uncertainty in estimation the evaporation rate (can vary by a factor of two to three by using different, but reasonable, averaged property value – specific heats, thermal conductivity, diffusion coefficient, vapour density, etc). (vi) Gas-phase quasi-steadiness. Because of the significant density disparity between liquid and gas. Liquid properties at the droplet surface (regression rate, temperature, species concentration) changes at rates much slower than those of gas phase transport processes. This assumption breaks down far away from the droplet surface where the characteristic diffusion time is of the same order as the surface regression time.

Gas-phase QUASI-steadiness – characteristic times analysis. In standard environment the gas-phase heat and mass diffusivities,  g and  g are of the same order of 10 0 cm 2 s -1, whereas the droplet surface regression rate, K = -d(D 0 2 )/dt is of the order of cm 2 s -1 for conventional hydrocarbon droplet vaporizing in standard atmosphere. Thus, there ratio is of the same order as the ratio of the liquid-to-gas densities,. It means that gas mass and heat diffusion occurs much faster than droplet surface regression time. If we further assume that properties of the environment also change very slowly, then during the characteristic gas-phase diffusion time the boundary locations and conditions can be considered to be constant. Thus the gas-phase processes can be treated as steady (time independent because the surface almost “freezes”), with the boundary variations occurring at longer time scales. When (at which value of D ∞ ) this assumption breaks down, i.e. when the diffusion time is equal to the surface regression time? When surface regression characteristic time becomes equal to gas mass/heat diffusion time, i.e. when D ∞ 2 /  g ≈ D 0 2 /K? Remembering that must still be valid (it doesn’t depend on the distance from the droplet), we can conclude that the steady assumption breaks down when. For standard atmospheric conditions it breaks down at It means that our model will be valid for the distances less than this one.

The d 2 Law – assumptions (vii) Single fuel species. Thus it is unnecessary to analyze liquid- phase mass transport (no diffusion term). (viii) Constant and uniform droplet temperature. This implies that there is no droplet heating. Where all the heat goes? Combined with (vii), we see that liquid phase heat and mass transport processes are completely neglected. Therefore the d 2 Law is essentially a gas-phase model. (ix) Saturation vapour pressure at droplet surface. This is based on the assumption that the phase-change process between liquid and vapour occurs at a rate much faster than those for gas- phase transport. Thus, evaporation at the surface is at thermodynamic equilibrium, producing fuel vapour which is at its saturation pressure corresponding to the droplet surface temperature. (x) No Soret (mass flow because of the temperature gradient), Dufour (heat flow because of the concentration gradient) and radiation effects (how this effects the validity of the model?).

Rate of accumulation of mass of component j Mass flow rate of component j into the system Mass flow rate of component j out of system Rate of generation of mass of component j from reaction Rate of depletion of mass of component j from reaction

Droplet evaporation I (no energy concerns) The phenomenon considered: A small sphere of liquid in an infinite gaseous atmosphere vaporizes and finally disappears. What is to be predicted? Time of vaporization as a function of the properties of liquid, vapor and environment. Assumptions: spherical symmetry (non-radial motion is neglected) (quasi-) steady state in gas Γ VAP independent of radius large distance between droplets no chemical reaction

Vapor concentration distribution m VAP in the gas. roro r GoGo G = G TOT,VAP

m VAP m VAP,0 m VAP,∞ Droplet surface r = r 0 r 1 Why the curve doesn’t start at m VAP =1?

Example

m VAP,0 has a strong influence, but is not usually known, it depends on temperature. relative motion of droplet and air augments the evaporation rate (inner circulation of the liquid) by causing departures from spherical symmetry. the vapour field of neighboring droplets interact m VAP,0 and m VAP,∞ may both vary with time. Γ VAP usually depends on temperature and composition. Limitations

The Energy Flux DEFINITIONS IN USE:

E E +dE S xx

0

0, for the case of Stefan flow

Droplet evaporation II roro r GoGo G = G TOT,VAP E QoQo heat flow to gas phase close to liquid surface

y = -x 0 1 So, a positive G 0 reduces the rate of heat transfer at the liquid surface. It means that if the heat is transferred to some let us say solid surface, that we want to prevent from heating up, we should eject the liquid to the thermal boundary layer (possibly through little holes). This liquid jets will accommodate a great part of the heat on vaporization of the liquid. Thus, we’ll prevent the surface from heating – transpiration cooling. The smaller the holes the smaller a part of heat towards the liquid interior and, subsequently towards the solid surface.

Clausius-Clayperon equation for

Linkage of equations

Equilibrium vaporization – droplet is at such a temperature that the heat transfer to its surface from the gas is exactly equals the evaporation rate times the latent heat of vaporization: This implies: What if –Q 0 ≠ G 0 L

Unity Le number

Temperature and concentration profiles look exactly the same. !

Cases of interest: (i)When T ∞ is much greater than the boiling-point temperature T BOILING, m VAP,0 is close to 1 and T 0 is close to T BOILING. Then the vaporization rate is best calculated from: (ii)When T ∞ is low, and m VAP,∞ is close to zero, T 0 is close to T ∞. This implies T 0 ≈T ∞. Thus, m VAP,0 is approximately equal to the value given by setting T 0 =T ∞ in and the vaporization rate can be calculated by: As in example with water droplet evaporating at 10 0 C

Evaporation rate [m 2 /s] The choice depends on whether T 0 or m VAP,0 is easier to estimate

Qualitative results for D 2 -Law

Droplet heat up effect on temperature and lifetime We can divide the droplet evaporation process into two stages. At first, while the droplet is cold (evaporation is slow), all the heat from the hot environment will be used for droplet interior heat up. As the droplet temperature will approach its steady state value, droplet heat up will slow down, while evaporation will accelerate. r 0 =r 0 (t)

Droplet heat up effect on temperature and lifetime Slowest limit Fastest limit Distillation limit Diffusion limit D 2 Law Center Temperature Surface Temperature T (  LIQ /r 0,INITIAL 2 )t (r  /r 0,INITIAL ) 2 (  LIQ /r 0,INITIAL 2 )t Distillation limit Diffusion limit D 2 Law