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

2. 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

3. 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

4. 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

5. 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

6. 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

7. D2 LAW RESULTS

8. 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 = K
hfg = 256 kJ/kg
MW = 170
Dn-dodecane-N2 = 8.1x10-6 m2/s at 400 K and 1 atmosphere