MAE 5310: COMBUSTION FUNDAMENTALS Mass Diffusion and Droplet Evaporation November 28, 2007 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

1-D STEFAN PROBLEM Governing equations Solution To see effects of concentrations at interface and at top of tube, let mass fraction of A in freestream be zero, while arbitrarily varying YA,i, interface mass fraction, from zero to unity In terms of an experiment, this would be like blowing dry N2 across top of tube and interface mass fraction is controlled by partial pressure of liquid, which in turn is varied by changing temperature At small values of YA,i, dimensionless mass flux is basically proportional to YA,i, but for values greater than about 0.5, mass flux increases very rapidly

LIQUID VAPOR INTERFACE BOUNDARY CONDITIONS Key question: How can we assess the value of YA,i? Equilibrium exists between liquid and vapor phase of species A Partial pressure of species A on gas side of interface must equal saturation pressure associated with temperature of liquid: PA,i = Psat(Tliq,i) Partial pressure, PA,i, can be related to mole and mass fraction of species A Molecular weight of mixture also depends on cA,i, and hence on psat Analysis has transformed problem of finding vapor mass fraction at interface to finding temperature at the interface Must be found by writing an energy balance for liquid and gas phases and solving them with appropriate boundary conditions, including that at interface In cross liquid-vapor boundary, temperature continuity is maintained: Tliq,i(x = 0-) = Tvap,i(x = 0+) = T(0) Energy is conserved at interface: Heat is transferred from gas to liquid with some of energy going into heating liquid and remainder causing phase change

LATENT HEAT OF VAPORIZATION, hfg In many combustion systems a liquid ↔ vapor phase change is important Example: Liquid fuel droplet must first vaporize before it can burn Example: If cooled sufficiently, water vapor can condense from combustion products Latent Heat of Vaporization (also called enthalpy of valorization), hfg: Heat required in a constant P process to completely vaporize a unit mass of liquid at a given T hfg(T,P) ≡ hvapor(T,P)-hliquid(T,P) T and P correspond to saturation conditions Latent heat of vaporization is frequently used with Clausius-Clapeyron Equation to estimate Psat variation with T Assumptions: Specific volume of liquid phase is negligible compared to vapor Vapor behaves as an ideal gas If hfg is constant integrate to find Psat,2 if Tsat,1 Tsat2, and Psat1 are known We will do this for droplet evaporation and combustion, e.x. D2 law

SIMPLE EXAMPLE Liquid benzene (C6H6) at 298 K is contained in a 1 cm diameter glass tube and maintained at a level 10 cm below the top of the tube, which is open to the atmosphere Determine mass evaporation rate (kg/s) of benzene How long does it take to evaporate 1 cm3 of benzene? Compare evaporation rate of benzene with water Given data: Tboil = 353 K at 1 atm hfg = 393 kJ/kg at Tboil MWC6H6 = 78 kg/kmol Density of liquid benzene = 879 kg/m3 DC6H6-Air = 0.99x10-5 m2/s at 298 K DH20-Air = 2.6x10-5 m2/s at 298 K

DROPLET EVAPORATION Evaporation of a single liquid droplet in a quiescent atmosphere is the Stephan problem for spherical symmetry Key question: How long does it take for a liquid drop to evaporate? Assumptions for simplified treatment Evaporation process is quasi-steady Droplet temperature is uniform and below the boiling point of the liquid In many actual problems transient heating of liquid does not affect droplet lifetime substantially Mass fraction of vapor at droplet surface is determined by liquid-vapor equilibrium at the droplet temperature All thermodynamic properties are constant In reality properties vary greatly in gas phase Constant properties allow a closed form solution

D2 LAW RESULTS

SIMPLE EXAMPLE: DROPLET EVAPORATION In mass-diffusion-controlled evaporation of a fuel droplet, the droplet surface temperature is an important parameter Estimate the droplet lifetime, td, of a 100 mm diameter n-dodecane droplet evaporating in dry nitrogen at 1 atmosphere if the droplet temperature is 10 K below the n-dodecane boiling point. Repeat the calculation for a temperature 20 K below the boiling point and compare Assume that in both cases the mean gas density of nitrogen is that of nitrogen at 800 K and use the same temperature to estimate the fuel vapor diffusivity Given properties Density of n-dodecane is 749 kg/m3 Tboil = 489.5 K hfg = 256 kJ/kg MW = 170 Dn-dodecane-N2 = 8.1x10-6 m2/s at 400 K and 1 atmosphere