Numerical Simulation of Palm Biodiesel Droplet Evaporation at Various Temperatures and Pressures

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Introduction
Biodiesel is one of the most promising fossil fuel replacements for automotive engines, furnaces, and turbines due to its sustainability, energy savings, and reduced carbon emissions; it can be a drop-in replacement or blended-in without structural modification [1].In Southeast Asia, palm biodiesel is of particular interest due to its predominant production capacity and global demand [2].It is a multicomponent methyl ester thoroughly tested with internal combustion engines where the quality of combustion is the primary determinant of engine torque, emissions, and fuel economy [3].The fuel injection and atomisation processes create numerous droplets from the jet prior to combustion [4].A greater viscosity of biodiesel compared to diesel results in larger atomised droplets, and as such, improving the evaporation of biodiesel is crucial [5].Hence, understanding the behaviour of droplets' evaporation is critical for investigating fundamental processes and the influence of operating conditions inside the combustion chamber.However, the complexity of its combustion makes both numerically and experimentally quantifying and observing the process itself difficult [6].
In the literature, the hallmark studies of Godsave and Spalding established the D 2 -law and the principal methodologies of droplet combustion and observation [7].The models presented a one-dimensional analysis of a spherical droplet in a quiescent environment with transient diffusion and diffusive convection in the liquid and gaseous phases, respectively.The experimental validation for these models, namely, suspended or free-falling droplet evaporation, formed the basis for all current experimental methods [8].A few experimental studies covered biodiesel, particularly palm methyl ester (PME).For instance, Morin et al. [9] investigated rapeseed and sunflower oil methyl ester vaporisation characteristics at elevated temperatures ranging from 473 to 1020 K. Hashimoto et al. [10] studied PME evaporation at 473 to 873 K at atmospheric pressure, and it was found that PME has a greater preheat period than diesel and subsequently a greater lifetime without a significant difference in evaporation coefficient.Polymerisation reactions, residue, and the influence of constituent fatty acids were observed at the latter stages of evaporation, particularly at lower temperatures.Manjunath et al. [11,12] monitored PME evaporation in a weakly convective flow at ambient temperatures of 423-573 K.The long-chain saturation factor is found to influence the vaporisation constants with larger-chain hydrocarbons exhibiting lower evaporation rates.Additive studies with ceria [13] and carbon nanotubes [14] further elucidated the influence of the multicomponent nature of PME with variance both in additive concentration and temperature.These studies revealed that a droplet experienced a significant expansion period during the initial preheating period, followed by a steady, linear evaporation period prior to extinction.Each study utilised various suspension and heating methodologies with varied inherent influences on evaporation.
Experimental research has utilised three droplet suspension methods: cross-fibres, silica rods, and thermocouples to support the droplets where the thermal transfer contribution of the support material to the evaporation process is a vital parameter.Han et al. [15] validated a theoretical model for heat transfer influenced by support type by comparing quartz fibres and thermocouples.They revealed that heat input through the support is proportional to the diameter of the support and the temperature differences between the droplet and the support.A nonlinear regressive relationship for evaporation was observed for supports ranging from 10-240 μm in diameter, where the evaporation rate increased and then decreased along the range.Thermocouples were found to have greater heat transfer than quartz supports due to thermal diffusivity.Chauveau et al. [16] experimentally investigated the influence of cross-support quartz fibres on the evaporation process.A significant increase in evaporation rate was observed with fibres larger than 14 μm.Greater conduction through fibres is suggested, and a correlation is presented to quantify the rate increase as a function of ambient temperature and fibre cross-sectional area.All such factors are taken into consideration in the numerical modelling of the evaporation process to ascertain properties which are difficult to observe experimentally such as temperatures, vapour flow, velocity, and density at the droplet interface.
Several assumptions have been made to reduce model complexity and computation time, particularly targeting the effects of finite thermal conductivity and species diffusivity.Sazhin [17] developed the discrete component model (DC) and subsequent refinements, quasi-discrete (QD) and multidimensional quasidiscrete (MDQD) models, to account for recirculating flows, temperature gradients, and species diffusion inside the droplet.When applied to biodiesel [7], the evaporation rates of single-component and multiple-component droplets differed by less than 5.5%.The lower droplet temperatures were observed in multiple component droplets and accredited to small differences between the molar masses and thermal properties of component species.This behaviour was also reported by Ray and Raghavan [18] in their numerical study of biodiesel of Indian origin.The dropletSMOKE++ computational fluid dynamics (CFD) model [19] utilises the volume of fluid (VOF) methodology to describe the nonideality of multicomponent mixtures, two-phase velocity fields, fibre thermal interaction, interface advection, phase change, and combustion chemistry.It avoids parasitic current errors caused by numerical errors in surface tension calculation by implementing a centripetal force along with multiregion extensions for fibre modelling [20].Rybdylova et al. [21] implemented the heat conduction model presented in the DC model in single-component droplet evaporation with the commercial Ansys Fluent software.The Ansys model underpredicted evaporation rates compared to the MDQD model [22].Coarser meshes in ANSYS were observed to overpredict the internal temperatures and showed significant error and deviation near the end of evaporation [21].Finer meshes resulted in a larger variance in internal temperatures but more accurate thermal expansion and evaporation coefficients prior to the end of the evaporation period.Ghata and Shaw [23] modelled n-heptane evaporation at elevated pressures and temperatures with ANSYS Fluent, in which Marangoni stresses in an axisymmetric domain with thermal conductivities were taken into consideration.
Most of the literature reviewed in this study investigated the evaporation behaviours of droplets in an inert environment, except for the works by Manjunath et al. [11,12] and Amsal et al. [14], as these studies investigated droplet behaviours at temperature ranges below the ignition temperature of biodiesel.The turbulence induced by the injection process is more significant than the environmental composition in influencing the evaporation process under high pressure and temperature conditions.Amsal et al. [14] observed a small difference in the evaporation rate between inert and oxidizing environments.In engine operation, the fuel evaporation process often fluctuates with changes in pressure and oxidant concentration, as is the case when exhaust gas recirculation (EGR) is reused in an engine cycle [24].Although the influence of temperature, oxygen concentration, and pressure is well established in engine combustion studies, their influence on the fuel vaporisation process has not been thoroughly investigated.Therefore, this study is aimed at filling that knowledge gap by developing a numerical approach to model and observe the fundamental evaporation process of a single palm biodiesel droplet, considering three variables: temperature, pressure, and oxygen composition.The numerical results generated by this approach are then compared with the experimental results of Hashimoto et al. [10], Manjunath et al. [11,12], and our previous study [14].

Numerical Model Setup and Methodology
The numerical approach employed in this study simulates a spherical single-component fuel droplet.An axisymmetric where α is defined from the following transport equation [26] The transport equation of species i is given by where Y i is the mass fraction of species i.The diffusivity D i is derived from [27,28] where X i denotes the mole fraction of species i and D ij is the binary diffusion coefficient of species i with respect to species j.The mixture density, ρ, is defined as where Y j and ρ j indicate mass fraction and density of species j.Volumetric evaporation at the interface introduces the source term _ m i ′′′ in Equation (3).Besides the interface, it is zero in the entirety of the computational domain.

Mass, Momentum, and Energy Conservations. The mass conservation is
The source term S c in the continuity equation arises from the evaporation of the liquid phase at the interface.Like the mass source term in Equation (3), this value only exists at the interface.
The momentum conservation equation is where g ! is the gravitational acceleration.A momentum source f Γ at the interface is introduced to account for surface tension and only exists at the interface.This source term is an implementation of the volumetric source term into the surface force caused by surface tension, as shown as follows: where σ is the fluid surface tension coefficient and is a function of temperature and κ is the local curvature.The energy equation is given by where the energy density, U, is defined as The sensible enthalpy, H, in the gas phase is defined as The specific enthalpy, h j , for species/component j is computed as where T ref is 298.15K and c p and α j are specific heat and volume fraction of species j.The variable J ! j is the diffusion flux of species j and the term τ: v ! in the energy equation is the contribution of viscous dissipation.The energy source 3 International Journal of Energy Research term S e in Equations ( 9) exists due to energy associated with evaporation at the interface and does not occur anywhere else and is defined as where h f g is the latent heat of evaporation.
2.1.3.Interface Species/Mass Conservation.The evaporation rate at the interface via species conservation is given as [23] _ The following equations compute the unit normal vector and the curvature: The Wagner equation estimates the pressure of saturated vapour at the interface [29] where T r = T/T c , T c is the critical temperature and τ = 1 − T r and A, B, C, D, E are material constants which, for PME, are obtained from Rasulov and Abdulagatov [30].
The saturated fuel vapour mole fraction is given by where P is total gas pressure.The interface saturated vapour mass fraction is calculated by where M v and M g are the molar masses of vapour and ambient gas, respectively.The source terms for the liquid and gas phases are computed by where _ m′′′ is computed from the species balance (Equation ( 14)).The variables ρ l and ρ g are the liquid and gas densities, respectively.
The volume-averaged density ρ is obtained by 2.1.4.Thermocouple/Wall Interface Conditions.The wall temperature (T w ) is computed by the energy balance at the solid-fluid interface: where λ s and T s are thermal conductivity of solid and local solid temperature, respectively, Δn is the distance between the wall surface and solid cell centre.T f , λ f , and n are the local fluid temperature, the thermal conductivity of the fluid, and the local coordinate normal to the wall, respectively.The radiative heat flux is assumed to be negligible as the ambient temperature is relatively low (below 1000 K) [31].
The contact angle of the fuel is modelled by where θ is a user-defined contact angle and n !wall is the unit vector normal to the wall drawn outwardly from the flow region where an advection equation extrapolates the liquid interface [27].
where the extension velocity, v !ext , is given by [27] v where and the thermophysical properties are combined via the mixing law [32].Table 2 summarises the expressions for these properties used in the current model and the sources.
2.3.Solution Method, Meshing, and Validation.The finite volume method (FVM) is employed to solve the equations, which can be done with the commercial software ANSYS Fluent 20.1 R2. Figure 1 shows the domain's model geometry, boundary conditions, and mesh refinements.A droplet diameter of 1.1 mm is chosen to model the experimental setup used in [14].Due to its axisymmetric geometry, only a fourth of the droplet is modelled in a domain of 50 mm diameter.There is a symmetry plane on one side, an axis on the other side, one open boundary, and a pressure outlet, as the current model has no inlet velocities and targets thermal diffusion in and out of the domain in the gas phase.The implemented VOF model has three species: liquid fuel, gaseous fuel, and air.Synthesised air is modelled as a twocomponent fluid with oxygen and nitrogen such that it can be varied for the parametric study of the environmental composition.The thermocouple support is modelled as a wall at the respective ambient gas phase temperature with Marangoni stresses enabled.However, no heat generation rate is input to exclude the influence of heat transfer from the thermocouple; instead, only the evaporation behaviour of the droplet due to its thermophysical properties is modelled.The thermocouple material is set as nickel with the specific heat, density, and thermal conductivity of 460 J/kg, 8800 kg/m 3 , and 91 W/m⋅K, respectively.The transport equations are discretised with secondorder upwind and gauss-green cell-based spatial discretisation.The pressure and velocity coupling are achieved by the pressure-implicit with splitting operators (PISO) scheme, while pressure is discretised by the PRESTO!Scheme.Under-relaxation factors of 0.7, 0.9, 0.3, and 0.5 are used for momentum, energy, pressure, and gas phase species, respectively.A timestep of 1 μs is used with a second-order implicit transient formulation such that the Courant-Friedrichs-Levy (CFL) stability conditions are satisfied.While the flow could be assumed to be laminar, the komega shear stress transport (SST) turbulence model is selected for a low Reynolds number model, and it is able to predict flow separation and stability in adverse pressure gradient boundary layers.As no large normal strain regions exist in the physical model, the hybrid k-epsilon free stream model is appropriate for this problem [38].The evaporation rate ( _ m ′ ′′), mole/mass fraction of vapour at the interface, and source terms for energy (S e ) and continuity (S c ) are computed using UDF in ANSYS Fluent.This numerical study was carried out on an AMD Ryzen Threadripper PRO 3975WX 64-thread workstation with an average computation time of 27 hours per case.
A non-uniform, structured mesh consisting of 4371 elements with a greater density near the droplet's location, as seen in Figure 1, is chosen to model the diffusion at the interface and droplet thermal properties.Table 3 provides details of the mesh independence test, where the target is the droplet diameter at a time of 3 s and an ambient temperature of 873 K.It is revealed that results obtained from the selected mesh are within a 1% difference from that obtained from finer mesh sizes.In addition, the results from the simulation model are validated by comparing them with the experimental results of Hashimoto et al. [10].In these validation cases, the ambient temperature varies in a nitrogen environment at a pressure of 1 atm.As shown in Figure 2, the two results show minimal differences in evaporation rate constant of 9.1% (473 K), 13.8% (673 K), and 5.9% (873 K), which could be attributed to the difference in support geometry between the two studies.It is also noted that the simulated initial thermal expansion is slightly greater than the experiment's, which could be attributed to the differences in compositions of biodiesel used in [10] and the present study, as presented in Table 1.   3 presents normalised droplet lifetimes with normalised diameters D 2 /D 2 0 at an ambient temperature ranging from 473 to 873 K.The results exhibit distinct initial expansion, unsteady evaporation, and steady evaporation periods for all tested temperatures.It is observed that the droplet behaviour in the steady evapora-tion period could comply with the D 2 -law [39].D 2 = D 2 0 − k t, from which the evaporation rate constant (or evaporation coefficient) k (mm 2 /s) can be obtained.The main observation in Figure 3 is that the droplet lifetime gets shorter with an increase in ambient temperature.The increase in droplet diameter in the initial thermal expansion period is greatest at 623 K, with subsequent higher temperatures having lower peak normalised diameters.This thermal expansion behaviour was also observed in the experimental cases.It could be caused by the heat transfer rate at higher temperatures being significantly greater than the thermal expansion rate initiating the evaporation process [14].Figure 4 presents the fuel mass fraction of the droplet along selected points of the droplet lifetime at an ambient temperature of 673 K. Figure 4(b) shows the droplet in the thermal expansion period where the droplet diameter is larger than its initial value (Figure 4(a)).Figure 4(c) presents the droplet in the steady evaporation period, while Figure 4(d) shows the droplet near the end of the evaporation process, highlighting the droplet boundary changes.It should be noted that droplets having D 2 /D 2 0 < 0:2 are neglected in evaporation coefficient analysis because the steady evaporation period ends at D 2 /D 2 0 = 0:2 for most of the cases.Figure 5 compares the evaporation coefficient obtained from this numerical study and the three experimental studies [10,12,14].The results show that the evaporation coefficient increases nonlinearly with the ambient temperature.However, there are discrepancies between the results obtained from different studies.In detail, Manjunath et al. [12] reported significantly greater evaporation rates at a low temperature range (423-573 K), while the numerical results lie between the experimental results of Amsal et al. [14] and Hashimoto et al. [10].It should be noted that the standard deviations of the experiments are 3.98% [14] and 3.21% [10], respectively.Besides the differences in initial droplet diameter, which is neglected via normalisation of the droplet diameter (D 2 /D 2 0 ), the primary reason could be the differences in experimental setups as they used different droplet support systems: silica rods [12], thermocouples   [14], and quartz cross-fibres [10].The thermal conductivity of quartz decreases with temperature, while that of the Ntype thermocouple increases.Without geometric consideration, a greater thermal contribution of the thermocouples would therefore be observed in Amsal et al. [14], which results in a higher evaporation rate than in Hashimoto et al. [10].Another factor that could also contribute to the differences between the experimental palm biodiesel results could be the variance in thermal conductivity from the two legs of the thermocouple.Nicrosil in the positive leg has a greater thermal conductivity increase with temperature compared to nisil in the negative leg.This difference could introduce greater internal convective flows as a thermal gradient is produced with heat transfer from the thermocouple.This phenomenon would not be observed in silica rods or fibre supports as they are the same materials, and their thermal conductivities do not change significantly with temperature.Therefore, considering the geometric variance, the  7 International Journal of Energy Research significantly larger support used by Manjunath et al. [12] results in a greater heat transfer.

Results and Discussion
Figures 6(a)-6(c) compare simulated droplet lifetimes with those presented in Hashimoto et al. [10] at 473, 673, and 873 K, respectively.The simulation's initial expansion periods show a greater droplet expansion magnitude than the experimental results, followed by a steady expansion period with a lower gradient, i.e., a greater evaporation coefficient.A possible cause of the expansion and rate variance with ambient temperature is the influence of liquid phase thermal conductivity on the surface temperature of the droplet [17].Given gas-phase quasisteadiness, the D 2 -law is expressed as [39] where λ g is the thermal conductivity of gas [W/(m•K)], ρ l is the density of liquid [kg/m 3 ], c p,g is the specific heat of gas [J/ (kg•K)], and B is the Spalding transfer number which is defined in the steady evaporation process as where h f g is the heat of evaporation [J/kg] and T s and T a are droplet surface and ambient temperatures [K], respectively.The B number represents the ratio of resistance to gasification by a driving force.The greater thermal conductivity of the model results in a greater surface temperature immediately after the expansion phase.Ambient temperature similarly affects the rate, as the increased temperature difference increases B even more significantly.The choice to model biodiesel, as a composite fuel, as single-component fuel could also affect the surface temperature.In the multicomponent model, each PME fuel component has its own set of thermophysical properties and therefore evaporates at individual rate.The thermophysical properties of larger PMEs and unsaturated PMEs result in slower overall evaporation with time-varying internal flows.While unsaturated PMEs are more volatile and evaporate faster at higher temperatures, their lower overall composition results in varying surface temperatures in the multicomponent model.Therefore, the surface temperatures are greater, as single-component models cannot account for this temporal temperature variance [18].From Table 1, it can be surmised that the primary comparison of Hashimoto et al. [10] presented in Figure 6 has a greater composition of unsaturated methyl esters than this study's modelled biodiesel blend.This difference was evident in the experimental results where residue from polymerisation reactions was observed in [10] but not in [14] nor [12].Droplet studies for pure methyl esters showed that a greater level of unsaturation in the ester resulted in greater residue and lower evaporation rates compared to that of unsaturated methyl esters [10].
Considering the initial diameter of the droplets in the experiments, a trend can be observed with the larger droplets having greater evaporation rates.This is because larger droplets have a greater mass transfer area [12].This behaviour was also observed using discrete component models with kerosene in which increasing droplet sizes yielded greater droplet evaporation coefficients [40].Thermal expansion characteristics, as discussed earlier, do not change with environmental conditions.Likewise, the normalisation with initial droplet diameter removes the influence of observation time in obtaining the evaporation coefficient, i.e., when the droplet evaporation rate is determined at elevated ambient temperatures.Besides the influences of suspension, droplet sizes, ambient velocity, and polymerisation reactions, a comparison of experimental and numerical results suggests an additional influence on increasing evaporation rates: the environmental composition.Both [12,14] conducted their experiments in an oxidising environment compared to the inert nitrogen environment of [8,10], opening a possible parameter of control, the oxygen composition.

Effect of Air/Nitrogen Composition in the Environment.
Varying oxygen concentration in diesel fuel engines through exhaust gas recirculation is a well-established implementation for emission mitigation, particularly for diesel engines, as they can function with up to 50% recirculated air [24].While injection rate, pressure, and mixing play key roles in combustion, the evaporation of atomised droplets is the key effect on fuel consumption.Therefore, this section of the study investigates the impact on evaporation rate of changing oxygen concentration in the environment in which oxygen percentages are 15%, 21%, and 27%, with the rest for nitrogen.Current exhaust gas recirculation percentage upper limits for biodiesels range from 20% to 50% [41]; however, these biodiesels have higher viscosities than palm biodiesel [11] and are less favourable as a direct replacement in engines [31].Therefore, a lower limit of 15% oxygen, which is equivalent to 30% recirculation [21], was selected in this simulation study.
Figures 7(a)-7(c) present the droplet lifetimes of palm biodiesel at three O 2 /N 2 compositions (15%, 21%, and 27% oxygen content) with the variation of ambient temperature.At a given ambient temperature, the initial expansion period shows little difference between compositions, where the little change does not influence the liquid phase in the composition of the gaseous phase.The primary trend observed is the increase in evaporation rate with an increase in oxygen concentration.The evaporation rate differences observed between compositions in Figure 7(a) (473 K) are not very large, with percentage differences of 1.5%, 8.6%, and 8.7%, respectively, compared to the baseline nitrogen environment.From Figures 7(b) and 7(c) at 673 and 873 K, respectively, it can be qualitatively observed that the differences between the evaporation rates at each ambient temperature increase progressively with a steeper gradient at higher O 2 percentages.At 673 K, the evaporation rates increasing from the baseline N 2 environment are 3.4%, 14.6%, and 16.1%, respectively, for 15%, 21%, and 27% O 2 , while at 873 K, these are 6.4%, 20.8%, and 27.8%, respectively.As shown in Figure 7, the difference in evaporation behaviour exists only in the steady evaporation period and not in the initial expansion period; the resultant evaporation rate could primarily be caused by viscosity, thermal diffusivity, and conductivity changes in the gaseous environment.The Euken formula with Hirschfelder corrections presents the relationship between these properties [42] where f tr is the translational Euken factor (= 5/2), c v,tr and c v,int are the constant volume-specific heats for translational and internal degrees of freedom, and D i is the self-diffusion coefficient of the gas.Thermal conductivity is, therefore, a property directly influenced by the ability of a gaseous molecule to move its mass through translation and rotation.The degrees of freedom in each direction influence one term of the above definition.Both density and diffusion are affected by internal or rotational degrees of freedom, while viscosity is influenced by translational degrees [42].Thermal diffusivity is a function of thermal conductivity and the volumetric heat capacity at constant pressure; hence, the specific heat capacities are relative to both thermal diffusivity and conductivity.While nitrogen has a greater heat capacity at constant volume, its viscosity, density, and diffusivity are 10 International Journal of Energy Research lower than oxygen.Hence, a mixture with increasing oxygen composition has greater thermal transfer capability due to the combined effects of greater viscosity, density, and diffusion coefficient.Consequently, as stated previously, the B number increases with T a , resulting in a greater evaporation constant for mixtures at higher ambient temperatures.

Pressure Effects at Various Ambient Temperatures.
Figure 8 presents the effect of ambient pressure on the evaporation rate of palm biodiesel.A nonlinear reduction in evaporation rate with pressure is observed for all temperatures tested.A decrease in the evaporation rate is more noticeable in the pressure range of 0.2-1 bar; however, the evaporation rate decreases more leniently in the 1-5 bar pressure range.Among the ambient temperatures tested, evaporations of droplets at 473 K show the most influence on pressure variation with an overall reduction of 219% from 0.2 to 5 bar compared to 213% and 196% reductions for 673 K and 873 K, respectively.It can be observed that the change in ambient temperature has a greater influence on the evaporation process than the change in pressure.Figure 9 illustrates the impact of pressure on the droplet lifetimes at 673 K.The magnitude of droplet expansion remains unchanged as expected, with liquid fuel being incompressible, while the duration of 11 International Journal of Energy Research the unsteady evaporation period increases with increasing pressure.Steady evaporation periods are observed at all pressures, with the evaporation rate increasing at lower pressures.This primary behaviour of droplet lifetime reduction via decreased initial unsteady evaporation time and decreased evaporation rates with increasing pressure is also observed for 473 and 873 K (not shown).
Figure 10 presents the liquid fuel mass fraction and streamlines of droplets of D 2 /D 2 0 = 0:554 at 673 K and various pressures.The underlying process can be observed and compared as all droplets are in the steady evaporation region of their lifetime regardless of the normalised time, as can be seen in Figure 9.The internal droplet flow is similar for all droplets, and it shows a Hill's spherical vortex [43] confined in the annular region between the droplet surface and thermocouple wall regardless of pressure.This behaviour is caused by Marangoni stress and viscous stress generated through the relative motion of liquid and vapour at the droplet interface, as reported in several studies of singlecomponent models [7,18,23].It is stipulated to be the cause of an overestimated surface temperature and, consequently, a greater evaporation rate than multicomponent models [17].These vortices are counterclockwise due to the influence of mass transfer interactions at the interface and external flows in the environment.The external flows of Figures 10(a)-10(c) reveal that larger external vortices are observed at lower pressure.While Figure 10 presents a truncated view of the simulation domain to focus on the droplet itself, the external streamlines extend outwards to the pressure inlet with mass and velocity transfer present at that boundary.As this system does not model any external velocities or forced convection, the primary driving forces for the streamlines are thermal buoyancy, mass diffusion, pressure gradients, and gravity [18].As convective flows from thermal and mass gradients are initiated at the droplet interface; hence, the evaporation process is thereby limited by the vapour layers generated at the interface.
The movement of the vapour away from the droplet through generated flows outside the droplet is a fundamental driver of the evaporation rate in a quiescent environment [31].The Grashof and Prandtl numbers could be utilised to analyse the convection process in such an environment.For quiescent-free convection mass transfer, the Grashof number is expressed as where g is the gravitational acceleration (m/s 2 ), β is the coefficient of volumetric expansion (1/K), ν is the kinematic viscosity (m 2 /s), and L C is the characteristic length which for a spherical droplet is defined as D/2 [44].The Prandtl number is the ratio of momentum diffusivity to thermal diffusivity and is defined as As can be inferred from the definition of Gr, the increase in the kinematic viscosity of the fuel vapour and environment due to elevated pressure would result in the predominance of viscous rather than buoyancy forces.However, as the flow is laminar, the viscous forces do not generate eddies that enhance the vaporisation process, as observed for droplet evaporation in turbulent environments [45].In addition, momentum diffusivity takes precedence as a higher Pr number would be obtained at higher pressures due to increased dynamic viscosity.Mass diffusivity, a process dependent on the concentration gradient, is also affected by increased pressure as the fuel vapour movement is suppressed.Due to decreased convection, poor diffusivity results in the relative stagnation of vapour layers around the droplet in a high-  12 International Journal of Energy Research pressure environment.This stagnation with elevated pressures is also observed in droplets of n-heptane [19], ndecane [46], and kerosene [47] at high pressures.As such, the relative importance of convection, either free or forced, significantly influences evaporation enhancement at higher pressures [19].The decrease in density as pressure decreases from 5 to 0.2 bar also results in greater convection through buoyancy, and vice versa.Therefore, for palm biodiesel, the results suggest an inversely proportional relationship with ambient pressure and a proportional relationship with ambient temperature.Comparing the effect on evaporation rates through pressure reduction and temperature increase, higher temperatures yield greater enhancements as they result in increased free convention through increased mass and momentum diffusivity along with reduced viscosities for the fuel vapour.

Conclusion
A numerical study of palm biodiesel droplets is carried out under normal gravity with elevated temperatures, pressures, and varying oxygen contents.Temperatures range from 473 to 873 K, while pressures range from 0.2 to 5 bar.The environmental composition varies from 0 to 27% oxygen content in a blend with nitrogen.The resulting evaporation rates for each parameter are compared with results available from experimental studies.The developed numerical model provides a simplified CFD approach to determining the singledroplet evaporation behaviour under desirable engine conditions.With the availability of experimental data for palm biodiesel and diesel blends, the current model has the potential to be expanded to assess those blends.It could also provide a basis for modelling vapour flow prior to ignition in single-droplet combustion simulations.In summation, the key findings of the study are as follows: (i) The evaporation rate is significantly enhanced by increased ambient temperature at all pressures and environmental composition conditions due to a significantly higher thermal conductivity and droplet surface temperature (ii) The droplet diameter and support geometry are found to significantly impact the evaporation rate, in which increased initial droplet and support fibre diameters greatly enhance the evaporation coefficients (iii) Changes in oxygen concentration in the environment where evaporation occurs have a less significant impact on evaporation rate than temperature.Increasing oxygen composition from 0 to 27% results in a 6.4% increase in evaporation rate at 473 K, while a 27.8% increase is observed at 873 K (iv) Elevated pressures reduce the evaporation rate for palm biodiesel.However, the impact of pressure is less noticeable at high temperatures.Across the range of pressures tested (0.

3. 1 .
Evaporation Rate of PME Droplets.The first parametric study focuses on ambient temperature variation at

2 )Figure 5 :
Figure 5: Droplet evaporation rate coefficient comparison with the increase in ambient temperature.

Figure 9 :
Figure 9: Droplet lifetime at 673 K with pressures ranging from 0.2 to 5 bar.

Table 1 :
Chemical formula and compositions of major components of biodiesel.

Table 3 :
Mesh refinement details (note that the ambient environment is nitrogen).
2 to 5 bar), an overall reduction in the evaporation rate of 219%, 213%, and 196% is observed for temperatures of 473, 673, and 873 K, respectively