2.2. Numerical Model

Hereafter, the model used to simulate heating, evaporation, and combustion of spherical droplets in 1D geometry is briefly summarized. In this 1D model,¹⁷ already validated in previous papers^34,35 for droplet evaporation and combustion, the following assumptions are made:

spherically symmetric droplet

constant pressure

equilibrium conditions at the liquid/gas interface

absence of reactions in the liquid phase

Conservation equations for species, energy, and velocity in the droplet in the liquid phase are solved. For the liquid phase, equations are formulated as

where the subscript L refers to liquid-phase properties. ρ_L is the density, v_L is the convective velocity, Y_i,L is the mass fraction of species i, j_L,i is its diffusion flux (calculated according to the Stefan-Maxwell theory³⁶), and k_L is the thermal conductivity. c_L,i and c_L are the heat capacities of species i and of the mixture, respectively; r is the radial coordinate, and N_L is the total number of species in the liquid phase.

Similar equations are solved for the gas phase, although further contributions must be included to account for chemical reactions leading to autoignition and combustion and radiative heat transfer:

where the subscript G indicates gas-phase properties. With reference to the gas-phase species i, Y_i,G is its mass fraction, j_G,i is its mass diffusion flux (calculated according to the Fick’s law), j_soret,i is its flux due to the Soret effect, is its formation rate, and H_i is its mass enthalpy; q_R is the radiative heat flux, and N_G is the total number of species in the gas phase.

Boundary conditions at the droplet center require zero velocity for the liquid, and symmetry conditions are prescribed for temperature and mass fractions. The external flow of heated air induces an intense recirculation inside the suspended droplet. This effect has been discussed in detail in the case of nonreactive evaporation of droplets of acetic acid and ethylene glycol.²⁷ The predictions of a CFD model of droplet evaporation, in the same experimental apparatus discussed in this work, showed that the droplet is highly homogenized by liquid motions.²⁷ This effect is reproduced in the 1D model by adopting an enhancing factor, which is applied to the mass diffusion coefficients of species and to the thermal diffusion coefficient. The value of the enhancing factor was derived by a comparison of the results of 2D CFD simulations of bicomponent droplet evaporation²⁷ and the 1D model for the same mixture. The comparison with CFD results is presented in the Supporting Information. Interface properties are calculated from thermodynamic equilibrium using the Raoult law because of the low pressure at stake (at higher-pressure conditions, the use of a cubic EoS might instead be necessary).³⁷ Flux continuity was finally considered for mass and energy. The resulting set of equations is discretized using an adaptive grid, more refined in proximity of the interface from the liquid side, as well as in the whole flame region in the gas phase. Further details on this model and its capability to describe complex multicomponent mixtures are available in the literature.²⁴ Such model has been adopted to predict the experimental measurements discussed in this work.

It is important to underline that the 1D model cannot account for buoyancy, because of the assumption of spherical symmetry. In the experiments, the droplet is heated by a coil, which is placed below the droplet. Since the experiments are performed at normal gravity, the coil induces a buoyant flow heating the droplet.²⁷ The heating produced by the coil is experimentally characterized using the thermocouple in an experiment performed without a suspended droplet. To model this device using a 1D approach, some simplifications are needed. The time evolution of the temperature, measured by the thermocouple (i.e., during the experimental test without a suspended droplet) is assigned as a boundary condition for the gas-phase computational domain surrounding the droplet. The time-resolved temperature increase measured by the thermocouple is imposed at the boundary of the computational domain. In this way, the temperature of the gas phase around the droplet also increases due to the radial diffusion of heat, which progressively reaches the liquid droplet surface and triggers evaporation. However, because of this simplification, there is a delay in the numerical computations due to the time needed for the heat diffusion in the gas phase. Therefore, in order to compare model predictions and experimental measurements, it is necessary to apply a time shift. For the computational domain used in this example (i.e., a gas phase which extends up to a maximum radial distance equal to ∼30 initial droplet radii) the required time shift is 0.82 s.

Figure ​Figure33 shows an example of crude FPBO droplet heating and evaporation, which is followed by autoignition and droplet combustion (not of interest for this paper, since the model does not take into account the reactions in the liquid phase, which are expected to be significant only for liquid temperatures above 200–230 °C). It is possible to observe that a time of 0.82 s corresponds to the time needed for the temperature increase applied at the outer gas-phase boundary to reach the droplet surface. The same time shift (0.82 s) is applied in all the simulations discussed in this paper. In the numerical model, the autoignition time was quantified as the time in which the maximum heat release rate occurs.

Effect of initial diameter on crude FPBO droplet heating, evaporation, autoignition, and combustion. The experimentally measured heating rate experienced by the droplets (cf., Figure ​Figure11) is applied at the outer gas phase boundary.

Figure ​Figure33 and Figure ​Figure44 also show that, although the time shift is the same due to the common boundary condition, more time is required to heat the larger droplets because of the higher thermal inertia. As a result, the evaporation of the volatile components is delayed, and the gas-phase ignition requires significantly more time for the larger droplet. Finally, after ignition, the additional heat transfer from the flame to the droplet enhances the heating of the liquid phase.

Effect of initial diameter on pure FPBO droplet evaporation and liquid phase temperature (droplet center).