With the definition of the mass vaporization rate m˙(1)m˙=−dmdt=−ddt(43πrs3ρl),where m and t denote the droplet mass and the time, respectively, the index ‘s’ shows conditions at the droplet surface, rs is the actual droplet radius, and the subscript ’l’ denotes liquid properties. ρl is the liquid density, the mass vaporization rate of the droplet is given as [12] (2)m˙=ρ(u−drsdt)4πrs2.

In the above equation, ρ is the gas density, and u denotes the gas velocity.

Droplet heating is described considering the heat conduction inside the droplet as (3)∂∂t(ρlcplTl)=1r2∂∂r(r2λl∂Tl∂r),where cpl is the specific heat capacity of the liquid at constant pressure, Tl is the liquid temperature, and r denotes the radial direction. The liquid thermal conductivity is λl.

Under the present conditions, the momentum equation is trivially fulfilled [6] and the continuity equation includes the Stefan velocity u. The continuity equation is given by (4)∂ρ∂t+1r2∂∂r(r2ρu)=0,where ρ is gas density and r is radial direction.

The energy equation is written as (5)∂∂t(ρcpT)=−1r2∂∂r(r2ρucpT)+1r2∂∂r(r2λ∂T∂r)+ρ∂T∂r(∑k=1Ncp,kDk∂Yk∂r)−∑k=1NMkω˙khk,where Dk is the diffusion coefficient of species k into the gas mixture, cp,k is the specific heat capacity at constant pressure of species k, and hk and Mk are the specific enthalpy and the molecular mass of species k, respectively. The chemical reaction rate of species k is denoted by ω˙k,k=1,…,N for the N species.

The conservation of species mass fractions Yk,k=1,…,N is given by (6)∂∂t(ρYk)=−1r2∂∂r(r2ρuYk)+1r2∂∂r(r2ρDk∂Yk∂r)+Mkω˙k.

The chemical reaction scheme derived by Nanjaiah et al. [20] with 25 chemical species and 93 chemical reactions is used, containing both detailed and complex reactions; the mechanism was reduced from the detailed chemical reaction mechanism of Ranzi et al. [21]. Soot formation is not included in that chemical reaction scheme. However, soot precursors are considered, which will allow for future predictions of soot formation. Moreover, radiation is not considered since it is found to have a major influence only for large droplet burning [12].

The present droplet vaporization model was validated in the previous studies of Narasu et al. [22] who studied p-xylene vaporization in an infinite ambience of air. The mass evaporation model of Cuoci et al. [12] is used for the resolution of the ambience of the droplet, and it was validated for the autoignition of n-heptane droplets in air. The chemical reaction scheme derived by Nanjaiah et al. [20] has been successfully used for the simulation of the combustion of p-xylene sprays in the counterflow configuration [23].

All liquid- and gas-phase properties are variable, and they are evaluated following earlier work [22,24]. The boundary conditions are as follows.

The system has three different boundaries, one is at the droplet interior at r=0, another one at the droplet surface, r=rs, and in the ‘infinite’ ambience of the gas phase, r=∞.

The liquid and the gas phase equations are solved simultaneously. An in-house computer code in the programming language C has been developed for the single droplet combustion simulations. The numerical scheme is an extension of earlier work [22] where the gas phase was not resolved.

Explicit time marching is adopted, and central differencing is used to discretize the governing equations for the liquid and the gas phases. The numerical time step size for the liquid and gas phases is 10−10 s to fulfill the Courant criterion. The explicit time marching scheme requires such small time steps but it captures the highly transient ignition and flame dynamics and resolves the small time-scales during the ignition.

The droplet vaporization is assumed to be completed when the relative droplet radius rs/rs,0 has reached a value of 10−3, which refers to 10−9 of the initial droplet mass. The simulations are continued in the gas phase until the flame extinguishes to predict the droplet burnout stage.

The spatial grid inside the droplet consists of 10 uniform grid points for the droplet interior, where simulations with 20 grid points showed no significant difference compared to the 10 grid points used in the present study [22]. The computational domain of the gas phase is taken approximately 100 times the initial droplet radius, rs,0 [14]. The numerical grid consists of about M = 250 grid nodes. Different non-uniform meshes with increasing grid spacing from the droplet surface following Δri+1=(1+ε)Δri,i=1…,K are investigated to ensure grid independence. The grid size rg of the gas phase normalized by the initial droplet size rs,0 yields (16)rgrs,0=Δr1(1+ε)M−1−1ε.

The parameter ε is varied between 0.020 and 0.025 and the initial grid cell Δr1 between 0.3 and 0.75 µm. It is found that the values ε = 0.025 and Δr1 = 0.5 µm are appropriate for an initial droplet radius of 100 µm and an ambient gas temperature, T∞ of 1500 K, cf. Table 1, as shown in Figs. 1 and 2.

Table 1 shows the ignition time τig, the droplet lifetime τd, and the total process time τt in dimensional units and divided by the square of the initial droplet diameter d02. From Figs. 1 and 2, which show the profiles of the gas temperature at different times for the four cases, it can be seen that case 3 seems to be the best choice for the parameters shown in Table 1.

For smaller initial droplet sizes studied in the remainder of the paper, the value of Δr1 is reduced accordingly and that of ε is reduced to achieve a computational domain of about 100 rs,0 while the other parameters are fixed.

Fig. 3 displays the characteristics of the single droplet vaporization and combustion for the different conditions considered in the present study, cf. Table 2. The temporal evolution of the normalized droplet surface and the mass vaporization rate during the droplet lifetime at different ambient temperature values are shown in Fig. 3a. Fig. 3b displays both the center and the surface temperatures Tcen and Ts, respectively. The time t is shown in terms of time divided by the square of the initial droplet diameter, t/d02, to better visualize the results for different initial droplet sizes.

The droplet is subjected to hot ambient air, and initially, droplet heating occurs, which results in droplet expansion-this reflects the variable physical properties of the liquid used in the present model. Droplet expansion occurs faster for the highest ambient air temperature and persists over a shorter time compared to the lower ambient air temperatures, see Fig. 3a. The period of major droplet vaporization constitutes the longest period during the droplet lifetime. As expected, the droplet lifetime increases with a reduction of the initial ambient gas temperature and with an increase in the initial droplet diameter.

The present result of τd/d02 = 0.69 µs/µm² or τd = 5.10 ms (case A) for the droplet lifetime at an ambient temperature of 1500 K and an initial droplet radius of 43 µm may be compared to the numerical result of Kunstmann et al. [4], who also followed the experimental conditions of Li et al. [2], however, they used a constant ambient temperature and did not resolve the gas phase. Their result is approximately 1.2 µs/µm² for the ambient oxygen, whereas the experiment gives a value of about 0.54 µs/µm²; however, in the experiment, spark ignition was used rather than autoignition, so that the experimental value lies below the present computations, which concern autoignition. Moreover, droplet combustion in oxygen is considerably faster than in air, so that the present droplet lifetime lies well in the range of what can be expected considering the different conditions.

Chen et al. [16] suggested the use of a revised d2 law with an exponent n, dn law. The initial heating period with droplet expansion, if variable transport properties of the liquid are considered, however, can never be captured by such a simplified law, but may be beneficial for the overall process. In the present study, the strong influence of the Stefan flow and the variable transport properties in the liquid phase, which allow for the prediction of droplet expansion during initial droplet heating, are responsible for the deviation of the results from the d2 law or any dn law, as will be further elaborated below.

The profiles of the mass vaporization rate m˙ are also shown in Fig. 3a. During droplet heating, the profiles of m˙ increase, and they peak at the time when droplet heating is completed. The negative slope in the profile of the mass vaporization rate shows the dominant period of droplet vaporization, where the variable physical properties are reflected in the deviation of a constant slope, which would present the d2-law. Towards the end of the droplet lifetime, there is an abrupt vaporization which is typical for this process [11]. Roughly between 0.07 and 0.2 µs/µm², all profiles of the mass evaporation rate show a non-monotonic behavior which is due to the interaction of the droplet vaporization with the gas phase, which will be addressed in Section 3.3 below.

The profiles of droplet surface temperature, Ts and the temperature at the droplet center, Tcen with time are displayed in Fig. 3b. They reflect the characteristics that are expected when the finite thermal conductivity inside the droplet is considered. The boiling temperature of the p-xylene is 420 K, and the figure reveals that the wet-bulb temperature depends on the conditions under consideration. It increases with higher ambient gas temperature and for larger droplet sizes.

The gas-phase characteristics will be studied next.

The gas-phase characteristics of the droplet heating and vaporization is displayed exemplarily for the situation of T∞ = 1500 K, and rs,0 = 43 µm (case A in Table 2). Figs. 4 through 8 show the radial profiles of the major chemical species mass fractions Yi, where p-xylene is in the vapor phase, and of gas temperature T. Note that r/rs=1 denotes the position of the droplet surface. In these figures, the left ordinate plotted in black labels shows the mass fraction of p-xylene and O2, the profiles of which are also in black color, whereas the mass fractions of CO, CO2, and H2O are shown on the first ordinate on the right-hand side in blue.

Fig. 4a displays the initial conditions at t=0 µs and Fig. 4b at 0.05 µs/µm² which is within the droplet heating zone, cf. Fig. 3. At 0.10 µs/µm², see Fig. 5a, the peak gas temperature starts rising to a value of 1758 K at r/rs=1.85. At this time, an inflection point appears in the temperature profile, and thus, this can be considered as the start of ignition. Note that the ignition time includes the droplet heating and is different from the ignition delay time used in chemical kinetics. The premixed mixture of p-xylene vapor and the O2 results in chemically controlled reactions at this stage. The species H2O, CO, and CO2 start to build up in the chemical reaction zone at ignition.

At 0.15 µs/µm², see Fig. 5b, the location of the peak temperature has moved away from the droplet surface to about r/rs=3.57 and reaches 2406 K. The species mass fractions of the reaction products increase, where is dominating.

At 0.2 µs/µm², see Fig. 6a, the peak temperature rises to 2609 K and its location has moved to about r/rs=6.27. A transition from the chemically controlled combustion to diffusion-controlled combustion occurs. As the gas temperature increases, CO2 is formed and dominates in the high-temperature region where the forward step of the chemical reaction CO + OH ⇌ CO2 + H prevails.

In Fig. 6b, at 0.50 µs/µm², quasi-steady combustion prevails with a rise in peak temperature to 2678 K and a relocation of the flame to r/rs=11.58. At 0.69 µs/µm², i.e., at the end of the droplet lifetime, see Fig. 7a, the peak temperature drops to 2625 K. This drop in the peak gas temperature is associated with a reduced fuel mass evaporation rate, cf. Fig. 3a, towards the end of droplet lifetime. At this time, the droplet burnout regime is initiated since the flame is no longer fed with fuel vapor. This phase in single droplet combustion has not yet been studied in the literature. The location of the peak temperature has now moved to r/rs=23. Fig. 7b shows the species and gas temperature profiles at times after the droplet has completely evaporated, but the chemical reactions still proceed, which is the droplet burnout region. At 0.85 µs/µm², the peak temperatures drops to 2003 K. A continuous reduction in the species mass fractions of CO and CO2 is also observed at these time instants where the burnout occurs.

The droplet combustion simulations are continued after the droplet lifetime until the complete breakdown of the chemical reactions. Fig. 8 displays the radial variation of species mass fractions and gas temperature at time instants after the droplet lifetime. At 1.0 µs/µm², see Fig. 8a, the peak temperatures drops to 1691 K and the location of the peak temperature has moved to r/rs=9. Further, at 1.5 µs/µm², the peak temperature reaches the ambient conditions with complete breakdown of the chemical reactions, terminating the process.

Fig. 9 shows an overview of the entire process. Fig. 9a displays the temperatures inside the droplet at r/rs≤1 (shown in the inset) and the gas temperature in the ambience of the droplet at r/rs>1. During the droplet heating and before ignition at about 0.05 µs/µm², near the droplet surface, the gas temperature decreases from its initial value of 1500 K, and at the ignition instant at 0.10 µs/µm², it increases to 1758 K. Thereafter, at 0.15 µs/µm² the flame temperature reaches 2406 K and raises to its peak flame temperature of 2678 K at 0.50 µs/µm². This is accompanied by an increase in droplet surface temperature and vaporization rate, cf. Fig. 3. At 0.20 µs/µm², a transition to the diffusion-controlled combustion occurs, followed by a quasi-steady combustion state. The transition from chemically-controlled to diffusion-controlled combustion occurs when there is a considerable increase in Stefan velocity, (see discussion in Section 3.3), which marks the initiation of remarkable diffusion, which is associated with the local minimum in the profile of the Stefan velocity after autoignition has occurred. At 0.69 µs/µm², with reduced mass vaporization rate, see Fig. 3a, a drop in the flame temperature occurs. From 1 µs/µm², which corresponds to the post-droplet burnout regime, the chemical reactions slow down, and the ambient temperature approaches the initial gas temperature of 1500 K.

In Fig. 9b, the radial variations of the fuel vapor mass fractions, Yxyl and of YO2 are shown. The mass fraction of fuel vapor at the droplet surface ranges from 0.6 and is clipped at 0.25 for a better resolution of the small mass fractions. During droplet heating and initial vaporization, the vapor fraction of p-xylene increases drastically near the droplet and distributes from the droplet surface into the ambient due to diffusion effects and the Stefan flow. At the ignition time of 0.10 µs/µm², the fuel vapor and O2 are consumed and reduced in this region. At t/d02 = 0.50 µs/µm², the region in the reaction zone where both fuel vapor and O2 coexist is wider, and the maximum flame temperature prevails as seen in Fig. 9a.

Fig. 10 displays a summary of the four cases studied in the present paper, see Table 2. Fig. 10a gives the peak gas temperature Tmax plotted against time t/d02 for all conditions under consideration. For all cases, ignition occurs after the initial heating and vaporization of the droplet. A transition from chemically controlled combustion to diffusion-controlled combustion occurs, resulting in quasi-steady combustion. The predictions of the flame temperature are in line with the predictions of n-heptane droplet combustion for different droplet diameters and ambient temperature values [27]. Also, the larger initial droplet size and higher ambient temperature result in the highest flame temperature among the different cases considered. Towards the end of droplet lifetime, a reduction in the flame temperature occurs, which is more pronounced in the smaller droplet due to the lower amount of fuel vapor from the smaller droplet. After complete evaporation of the droplet, the peak temperature drops for all the cases, gradually leading to the complete breakdown of the chemical reactions.

The profiles of the gas temperatures reflect the characteristics of the time scales provided in Table 2. The strong increase in gas temperature marks the ignition time τig. The lowest ambient gas temperature shows a strong delay in ignition time, which results in a local maximum of the flame temperature at ignition. The quasi-steady vaporization period is characterized by an almost constant flame temperature before it dramatically decreases due to a lack of fuel vapor that results from the end of the droplet vaporization marked by the droplet lifetime τd. The values τd/d02 in Table 2 confirm that the initial droplet size has a significant influence on the droplet lifetime, in contrast to the ambient gas temperature. The total process time τt is considerably longer for lower ambient temperatures and larger droplets. However, τt/d02 is shortest for the largest initial droplet, which is associated with the shortest values of τig/d02, τd/d02, and τt/d02. The droplet surface temperature for the largest droplet increases fastest, see Fig. 3, causing a short ignition time which is associated with a fast rise in gas temperature seen in Fig. 10a.

This also affects the flame standoff distance, which is taken as the position of the maximum flame temperature rf normalized by the instantaneous droplet radius rs, see Fig. 10b. Ignition occurs near the droplet surface at r/rs=1. The flame standoff distance of p-xylene droplet increases throughout the droplet lifetime, which is also reported for n-heptane droplets [27]. At later times beyond the droplet lifetime, the flame moves closer to the droplet surface due to a shortage of fuel vapor, resulting in a decrease in peak temperature, see Fig. 10a. The flame standoff distance is largest for the biggest droplet and reduces with ambient gas temperature.

The non-monotonic behavior of the mass evaporation rate shown in Fig. 3a is addressed in this section. For this purpose, the vaporization rate constant K=−dds2dt=−8rsdrsdt with time t/d02, where ds=2rs is the instantaneous droplet diameter, is shown in Fig. 11a. For all cases, K initially decreases slightly due to droplet expansion. Most often, this phase is not shown in literature [28] since droplet expansion is only visible when variable liquid properties are used in simulations. After droplet expansion, see Fig. 3a, the vaporization rate constant increases during droplet heating, reaching a maximum where a quasi-steady phase is seen during which the vaporization rate constant attains a plateau-like behavior. At the end of the droplet lifetime, the vaporization rate constant abruptly decreases to zero. A small oscillatory change in K occurs after the ignition and is most significant for the larger droplet diameter. The strong variation of the vaporization rate constant K confirms that a constant value as used in approximations with the d2 law is not appropriate, particularly during droplet expansion and heating.

Fig. 11b shows the temporal variation of Stefan velocity for the cases under consideration. An initial rise in the Stefan velocity is followed by an almost constant period during quasi-steady droplet vaporization, and towards the end of droplet lifetime, a sudden rise occurs. As expected, the Stefan velocity is lower for larger droplet sizes and increases with higher ambient gas temperature.

The experimental work by Chauveau et al. [29] on n-decane droplet vaporization in a hot atmospheric environment also predicts non-monotonicity in the mass vaporization rate m˙ for the larger droplet size, and they attributed this to the lower Stefan flow velocity that causes vapor accumulation and thus reduces the mass evaporation rate. Awasthi et al. [28] performed numerical simulations for the auto-ignition of methanol droplet combustion, and they predicted a significant variation of vaporization rate constant K during the droplet lifetime. An oscillatory change in K is observed after the droplet ignition, and they attributed it to the changes in the surface composition arising from internal circulation of the droplet. Fang et al. [30] carried out experimental and theoretical studies to demonstrate the deviation of d2 law during droplet combustion for various liquid hydrocarbon fuels under the influence of gravity. They attributed their deviation to the coupled effects of momentum, heat, and mass transfer that result in the non-quadratic shrinkage law. Chen et al. [16] reported the deviation of the d2 law in their droplet combustion studies under gravity conditions, and the deviation of their results was argued to result from the simultaneous momentum, heat, and mass transfer from the buoyant convection set up by the flame around the droplet. In the present study, the rapid droplet vaporization in hot air conditions is associated with a significant Stefan flow velocity, which influences the vaporization rate constant. Moreover, it will be shown below that the chemical reactions also contribute to a deviation of the vaporization rate constant from the d2 law.

All droplet characteristics, including the Stefan velocity, the vaporization rate constant, and the mass vaporization rate, show a non-monotonic, somewhat oscillatory behavior around the time when ignition occurs. To analyze the effect of chemical kinetics on these profiles, the p-xylene consumption rate of the chemical reactions ω˙C8H10 and heat release rate Q˙=−∑k=1NMkω˙khk during ignition are discussed further for case D, cf. Table 2, where this effect is most significant.

The chemical reactions for the consumption of p-xylene are:(R1)C8H10+O→0.5C2H4+C7H7+OH (R2)C8H10+OH→0.5C2H4+C7H7+H2O (R3)C8H10+H→0.5C2H4+C7H7+H2 (R4)C8H10+O2→0.5C2H4+C7H7+HO2 (R5)C8H10+M⇌CH3+C2H4+M

From these reactions, (R1), (R2), and (R3) in which the fuel reacts with the radicals O, OH, and H are most significant and are plotted against radial position at six different times, see Fig. 12. (R3) with its lowest impact increases during the first time steps, whereas (R1) and (R2) show an initial increase and decrease thereafter. The reaction of the fuel vapor with OH is most important, followed by that with the O-radical. At later times, the reaction with H becomes more important.

At 0.06 µs/µm², i.e., prior to the droplet ignition, Fig. 12a shows that the chemical kinetics control the process. At this time, the chemical reactions occur near the droplet surface. Further, at time 0.085 µs/µm² and beyond, see Figs. 12b and 13a, the chemical reaction zone moves away from the droplet surface and a transition to diffusion-controlled combustion occurs with a sharp rise in the gas phase temperature. This transition behavior can influence the droplet vaporization at these times and the oscillatory change in the profiles of K, m˙, and the Stefan velocity u occurs, see Figs. 3a and 11a,b.

Fig. 13b displays the profile of the heat release rate Q˙ at different times for case D in Table 2. At 0.06 µs/µm², i.e., prior to ignition, a significant positive Q˙ value exists near the droplet surface. Further, at the ignition time of 0.07 µs/µm², a sharp rise in the heat release rate occurs, indicating an intense chemical activity with energy-releasing reactions near the droplet surface. After ignition, a drop in Q˙ within the positive region and at certain radial locations, the Q˙ takes negative values. The negative values of Q˙ are associated with the energy-absorbing chemical reactions after ignition.

Thus, the initial chemical reactions consuming the p-xylene vapor have a pronounced impact on the droplet vaporization characteristics, leading to some oscillation in the profiles of the mass vaporization rate, the Stefan velocity, and the vaporization constant. In reverse, the vaporization of the droplet enables the ignition through the formation of fuel vapor, which feeds the flame. After the completion of droplet vaporization, the combustion is sustained, retards, and eventually breaks down due to a lack of fuel vapor.

The present numerical simulations of pure p-xylene droplet auto-ignition and combustion with detailed chemical kinetics are novel. The model predictions include the temporal variation of the Stefan velocity during the droplet lifetime. Moreover, a new analysis of the interaction between the droplet ignition and vaporization, the temporal variation of the peak temperature, and flame standoff ratio is presented all the way until the chemical reactions break down completely due to a lack of combustible fuel vapor.

The one-dimensional model assumed in the present study has some limitations in neglecting the effects of possible asymmetric droplet behavior, which may occur under gravity or buoyancy. The detailed chemical reaction scheme does not include soot formation even though its precursors are included. Also, the assumption of a lack of radiation may lead to too high flame temperature for the larger droplet sizes. These differences may affect the standoff distance of the flame as well.

The present study builds the ground for further simulations considering the detailed chemistry in the droplet heating, vaporization, and combustion of p-xylene in air, which will be used in future studies of soot formation and nanoparticle formation in the bi-component flame spray pyrolysis of the precursor solution of TTIP/p-xylene.