1. Droplet evaporation Liquid fuel combustion
Combustion of liquid fuels requires their previous atomization, i.e., breaking the liquid fuel into a spray of small drops (droplets). The droplets often vaporize before combustion begins.
A simplified model of droplet evaporation is described below. The interaction between droplets is neglected, and the droplet surface temperature is assumed to be close to the boiling temperature of the liquid, so that the evaporation rate is controlled by heat transfer from the ambient to the droplet.
These assumptions are reasonable if the spray is not dense and the temperature of the medium is high.
Heat transferred from the ambient supplies the energy necessary to vaporize the liquid fuel, and the fuel vapor then diffuses from the droplet surface into the ambient. The mass loss causes the droplet radius to shrink with time, until the droplet is completed evaporated.
We wish to determine
Liquid fuel combustion
Combustion

2. Droplet evaporation Liquid fuel combustion Simplifying assumptions:
The droplet evaporates in a quiescent, infinite medium.
The evaporation process is quasi-steady, i.e., at any instant in time the process can be described as if it were in steady state.
The fuel is a single-component liquid with zero solubility for gases.
The droplet temperature is uniform and equal to the boiling point of the fuel (Ts = TBP).
Binary diffusion with Le=1.
All the thermophysics properties of the gaseous phase (lg, r e cp,g,) are constant. Even though they may vary greatly from the droplet surface to the ambient, a judicious choice of mean values allows reasonably accurate predictions.
Mass conservation equation in gaseous phase
Liquid fuel combustion
Combustion

3. Droplet evaporation Liquid fuel combustion
Energy conservation equation in gaseous phase
From mass conservation, and for constant
properties, this yields
with T = Ts for r = rs and T = T for r  .
The solution of this equation is
with Z = cp,g/(4π lg)
Liquid fuel combustion
Combustion

4. Droplet evaporation Liquid fuel combustion
Energy balance at the droplet surface
Calculating the temperature derivative at the surface from the temperature profile determined in previous slide, and solving for the evaporation rate, , leads to
where the non-dimensional parameter Bq is known as transfer number or Spalding number (subscript q indicates that it is only based on heat transfer) given by
Liquid fuel combustion
Combustion

5. Droplet evaporation Liquid fuel combustion
Mass balance for the droplet
The mass of the droplet is given by Inserting the mass of the droplet and the evaporation rate in the above equation
or, after some algebra,
Liquid fuel combustion
Combustion

6. Droplet evaporation Liquid fuel combustion
Integration in time yield the D2 law:
and the droplet lifetime
The D2 law holds after an initial transient period associated with the heating of the droplet to near boiling point.
The following approximations may be used to evaluate the gas phase properties:
with
Liquid fuel combustion
Combustion

7. Droplet burning Liquid fuel combustion
The previous model is now extended to include a spherically symmetric diffusion flame that surrounds the droplet. The restriction that the droplet is at the boiling point is removed.
Simplifying assumptions
The burning droplet, surrounded by a spherically symmetric flame, exists in a quiescent, infinite medium. There are no interactions with other droplets, and the effects of convection are ignored.
The burning process is quasi-steady.
The fuel is a single-component liquid with zero solubility for gases. Phase equilibrium prevails at the liquid-vapour interface.
The pressure is uniform and constant.
The gas phase consists only of fuel vapor, oxidizer and combustion products, and is divided in two zones. The inner zone, between the droplet surface and the flame, contains only fuel vapor and products, while the outer zone consists of oxidizer and products. Binary diffusion prevails in each zone.
Liquid fuel combustion
Combustion

8. Droplet burning Liquid fuel combustion
The fuel and the oxidizer react in stoichiometric proportions at the flame. Chemical reaction is assumed to be infinitely fast, resulting in an infinitely thin flame front.
The Lewis number is unity
Radiative heat transfer is negligible.
The thermophysical properties (lg, cp,g , rDM ) are constant.
The liquid fuel is the only condensed phase. No soot or liquid water is present.
We may write five equations:
Mass species conservation in the inner and outer zones
Energy conservation in the inner and outer zones
Phase equilibrium at the liquid-vapour interface
These allow us to determine the five unknowns:
Liquid fuel combustion
Combustion

9. Droplet burning Liquid fuel combustion Mass conservation still holds:
Fuel mass conservation in the inner zone
These equations lead, after some algebra, to
Liquid fuel combustion
Combustion

10. Droplet burning Liquid fuel combustion
Oxidizer mass conservation in the outer zone:
This equation and the global mass conservation lead, after some algebra, to
s =(mO2/mfu)stoich
Noting that yO2  yO2,∞ when r  ∞, then
Liquid fuel combustion
Combustion

11. Droplet burning Liquid fuel combustion
Energy conservation equation (the equation for droplet evaporation remains valid):
In the inner zone, and
In the outer zone, and
This yields
Liquid fuel combustion
Combustion

12. Droplet burning Liquid fuel combustion with ZT = cp,g/(4π lg)
Energy balance at the droplet surface
The heat conducted into the droplet interior can be handled in several ways:
A common approach is to model the droplet as consisting of two zones: an inner zone where the temperature is uniform and equal to its initial temperature, To; and a thin surface layer at the surface temperature, Ts. Hence,
Another approach is to assume that the droplet has a uniform temperature with a transient heat-up period
Liquid fuel combustion
Combustion

13. Droplet burning Liquid fuel combustion
The simplest approach is to assume that the droplet rapidly heats up to a steady state temperature Ts, so that
Expressing qg-i according to Fourier´s law and evaluating the temperature gradient at the flame surface from the previously determined temperature profile yields, after some algebra,
Energy balance at the flame sheet
(note that there is no flow of products inward from the flame to the droplet surface due to assumption 3)
Liquid fuel combustion
Combustion

14. Droplet burning Liquid fuel combustion Noting that
it follows that, for constant cp,g,
using Fourier’s law and the temperature profiles formerly obtained, we then have
Liquid fuel combustion
Combustion

15. Droplet burning Liquid fuel combustion
Liquid-vapor equilibrium at the surface of the droplet – Clausius-Clapeyron equation
A and B are constants that depend on the fuel and which may be determined from Clausius-Clapeyron equation.
Noting that
It follows that
and
Liquid fuel combustion
Combustion

16. Droplet burning Liquid fuel combustion
The model for droplet burning is constituted by the following five equations:
Liquid fuel combustion
Combustion

17. Droplet burning Liquid fuel combustion
The solution of the governing equations may be obtained as follows:
i) Assuming Ts, equations (2), (3) and (4) may be solved to obtain, successively,
where the transfer number Bo,q is defined as
Liquid fuel combustion
Combustion

18. Droplet burning Liquid fuel combustion
Then, yfu,s is calculated as follows from Eq. (1)
Eq. (5) is solved to obtain a new value for Ts:
Steps (i) to (iii) are repeated until the value of Ts guessed in step (i) is close enough to that determined in step (iii)
Liquid fuel combustion
Combustion

19. Droplet lifetime Liquid fuel combustion
If it assumed that the droplet is at the boiling temperature, as in the simplified model of droplet evaporation, the present droplet burning model greatly simplifies.
In such a case, the previous equations allow the direct calculation of
without iteration, and the equation for yfu,s becomes irrelevant, since yfu,s = 1 in that case. This simplification is reasonable when the droplet is burning vigorously after its initial heat-up transient.
Droplet lifetime
The equation for in terms of the transfer number Bo,q is similar to that derived for the evaporation of a droplet. Hence, a burning rate constant may be defined as
The burning rate constant is truly a constant only after a steady-state surface temperature is reached, since only then Bo,q is a constant
Liquid fuel combustion
Combustion

20. Droplet lifetime Liquid fuel combustion
Assuming that the transient heat-up period is small in comparison to the droplet lifetime, the D2 law is recovered for droplet burning:
The droplet lifetime is found by setting D to zero, yielding:
The D2 law is a good representation of experimental data after the heat-up transient.
Liquid fuel combustion
Combustion

21. Burning rate Liquid fuel combustion
The burning rate constant does not change much with the fuel. The droplet lifetime is mainly dependent on its initial diameter, and therefore on the atomization process.
The burning rate constant is approximately equal to10-6 m2/s and 2x10-6 m2/s for the burning of hydrocarbons in air and in oxygen, respectively, increasing with the temperature of the ambient.
cetane
gasoil
kerosene
benzene
n-heptane
Liquid fuel combustion
Combustion

22. Droplet lifetime Liquid fuel combustion
The thermophysical properties may be estimated as follows:
The spherical symmetry in the model arises from neglecting the relative velocity between the droplet and the medium, as well as the buoyancy.
The convective effects may be incorporated using the film theory, which replaces the boundary conditions prescribed at by similar boundary conditions prescribed at r = dM for the species and at r = dT for the temperature.
Liquid fuel combustion
Combustion

23. Extension to convective environments
The heat and mass transfer rates at the surface of the droplet increase due to convective effects.
The radius dM and dT are related to the Nusselt and Sherwood numbers as follows:
In a medium at rest, Nu=2
Assuming Le=1 for all species, then Nu=Sh and dM =dT
Liquid fuel combustion
Combustion

24. Extension to convective environments
The following correlation may be used to evaluate Nu for droplet burning in forced convection:
where the Reynolds number is defined from the diameter of the droplet and the relative velocity (physical properties may be determined at the average temperature of the medium)
Convection influences the temperature and species profiles on the outer side of the flame front. The 2nd and 4th equations of the system of equations are modified as follows:
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Combustion

25. Extension to convective environments
The burning rate is now given by
with the transfer number still given by
The expression for the burning rate reduces to the original one for Nu=2, i.e.,when the medium is quiescent.
Liquid fuel combustion
Combustion