Extended lattice Boltzmann scheme for droplet combustion

The available lattice Boltzmann (LB) models for combustion or phase change are focused on either single-phase flow combustion or two-phase flow with evaporation assuming a constant density for both liquid and gas phases. To pave the way towards simulation of spray combustion, we propose a two-phase LB method for modeling combustion of liquid fuel droplets. We develop an LB scheme to model phase change and combustion by taking into account the density variation in the gas phase and accounting for the chemical reaction based on the Cahn-Hilliard free-energy approach. Evaporation of liquid fuel is modeled by adding a source term, which is due to the divergence of the velocity field being nontrivial, in the continuity equation. The low-Mach-number approximation in the governing Navier-Stokes and energy equations is used to incorporate source terms due to heat release from chemical reactions, density variation, and nonluminous radiative heat loss. Additionally, the conservation equation for chemical species is formulated by including a source term due to chemical reaction. To validate the model, we consider the combustion of n-heptane and $n$-butanol droplets in stagnant air using overall single-step reactions. The diameter history and flame standoff ratio obtained from the proposed LB method are found to be in good agreement with available numerical and experimental data. The present LB scheme is believed to be a promising approach for modeling spray combustion.

Figure 1

The flow chart of the solution procedure for combustion.

Figure 2

Schematic of the computational domain. $r_d$ is the radius of the droplet and $r_f$ is the radius of the flame. Detailed explanation on implementing the boundary conditions is given in the Appendix.

Figure 3

The $d^2$-law results for a 100-μm droplet obtained using three different grid resolutions. The square of the drop diameter, which is rescaled by its value at $t=0$, is plotted versus normalized time (for convenience, time is divided by $d_0^2$).

Figure 4

The temperature profile, rescaled by the boiling point, versus the dimensionless distance, which is measured from the center of the droplet, for three different grids. The distance is nondimensionalized by the instantaneous radius of the droplet so that the location of maximum temperature stands for the flame standoff ratio (FSR).

Figure 5

The $L^2$-norm error in calculating the droplet diameter versus the number of grid points in the $x$ direction.

Figure 6

The diameter histories for evaporation of an $n$-heptane droplet in quiescent air at 1200 K. Comparison between (a) analytical solution [Eq. (49)]; and (b) finite-volume results [14]. The square of the drop diameter, which is rescaled by its value at $t=0$, is plotted versus normalized time (for convenience, time is divided by $d_0^2$).

Figure 7

Combustion of an $n$-butanol droplet of diameter 560 μm in ambient air at 298 K. Comparison between the numerical results and the experimental data by Alam et al. [15] for (a) the law, and (b) the FSR.

Figure 8

Combustion of $n$-heptane droplets of diamater 100 μm in ambient air at 1200 and 1600 K. The square of the droplet diameter, which is rescaled by its initial value at $t=0$, is plotted versus time.

Figure 9

The FSR for combustion of an $n$-heptane droplet of diameter 100 μm in ambient air at (a) 1200 K, and (b) 1600 K.

Figure 10

The dimensionless temperature profile at different times $(T_b=371\mathrm{K})$. The $x$ distance is nondimensionalized by instantaneous droplet radious so that the $x$ location where the temperature is maximum stands for the FSR.

Figure 11

Mass fraction profiles at different times. The $x$ distance is nondimensionalized by the instantaneous droplet radius so that the location where the fuel mass fraction is zero stands for the FSR.

Figure 12

Dimensionless temperature contours at different times during combustion. The flame radius, shown by maximum temperature, increases until it reaches a maximum value, after which it remains nearly constant.

Figure 13

Contours of density. Comparison between (a) constant-density, and (b) variable-density approaches.

Figure 14

A comparision between dimensionless velocity profiles for constant-density and variable-density approaches. The velocity is rescaled by the Stefan velocity (which is defined as the bulk velocity of evaporation, at the interface). In the variable-density approach, the velocity increases beyond the Stefan velocity outside the interface.