Transient convective burning of interactive fuel droplets in single-layer arrays

The transient convective burning of n-octane droplets interacting within single-layer arrays in a hot gas flow perpendicular to the layer is studied numerically, with considerations of droplet surface regression, deceleration due to the drag of the droplets, internal liquid motion, variable properties, non-uniform liquid temperature and surface tension. Infinite periodic arrays, semi-infinite periodic arrays with one row of droplets (linear array) or two rows of droplets, and finite arrays with nine droplets with centers in a plane are investigated. All arrays are aligned orthogonal to the free stream direction. This paper compares the behavior of semi-infinite periodic arrays and finite arrays with the behavior of previously studied infinite periodic arrays. Furthermore, it identifies the critical values of the initial Damköhler number for bifurcations in flame behavior at various initial droplet spacing for all these arrays. The initial flame shape is either an envelope flame or a wake flame as determined by the initial Damköhler number, the array configuration and the initial droplet spacing. The critical initial Damköhler number separating initial wake flames from initial envelope flames decreases with increasing interaction amongst droplets at intermediate droplet spacing (when the number of rows in the array increases or the initial droplet spacing decreases for a specific number of rows in the array). In the transient process, an initial wake flame has a tendency to develop from a wake flame to an envelope flame, with the moment of wake-to-envelope transition advanced for the increasing interaction amongst droplets at intermediate droplet spacing. For the array with nine droplets with centers in a plane, the droplets at different types of positions have different critical initial Damköhler number and different wake-to-envelope transition time for initial wake flame.     

Investigation of interaction between methanol fed tandem porous spheres burning in a mixed convective environment

Flame interaction during the burning of two porous spheres in tandem arrangement fed with methanol and subjected to a mixed convective environment, has been studied experimentally and numerically. Porous sphere technique is employed for experimentally simulating the burning characteristics of methanol transpired spheres of different sizes, separated by fixed distances. The mass burning rates from both the spheres and visible flame stand-off distances from the sphere surfaces have been measured in the experiments. In the numerical simulations, transient, axisymmetric, mass, momentum, species and energy conservation equations are solved using a finite volume technique based on non-orthogonal semi-collocated grids. Features of the numerical model include finite rate chemistry and temperature and mixture composition dependent thermo-physical properties. Burning of tandem porous spheres in an air stream flowing vertically upwards, at atmospheric pressure has been simulated for different sphere sizes, separation distances and free stream velocities. The numerical predictions have been compared with experimental results. Results reveal that when two spheres burn one over the other, the transition from envelope to wake flame is delayed when compared with that of an isolated sphere. For two spheres of same diameter burning one over the other, depending on the separation distance, flame blows-off after the occurrence of transition from envelope to wake flame in the bottom sphere. For the case of larger sphere at the top, either the flame stabilises in the recirculation zone formed in between the spheres or the flame from the smaller sphere lifts off and stabilises near the front portion of the larger sphere, depending on the separation distance. At higher separation distances, around four times the diameter of the sphere, both the spheres burn independently. The burning rate undergoes complex variations with air stream velocity depending on the sphere sizes and separation distances.     

Autoignition and early flame behavior of a spherical cluster of 49 monodispersed droplets

Autoignition and early flame behavior of a spherical cluster of 49 monodispersed droplets in a high-temperature air were examined in microgravity. The monodispersed suspended-droplet cluster (MSDC) model with which both droplet spacing and initial droplet diameter were well-controlled was developed, and the solidified-fuel fiber-suspension technique was utilized for making the MSDC model. The tested 3D MSDC models had the HCP (hexagonal closest packing) structure. Individual flames, which enveloped each droplet, or group flame, which enveloped the whole droplet cluster, were formed immediately after ignition. The flame changed from the group flame to a cluster of the individual flames either with increasing the droplet spacing or decreasing the initial droplet diameter. Each of the individual flames merged into the group flame with the lapse of time. Burning sphere diameter decreased at the beginning, and then increased. The transition from the individual flames to the group flame occurred around the time period at which the burning sphere diameter reached its minimum. The time period at which the burning sphere diameter reached its maximum was delayed and the expansion rate of the burning sphere was enhanced with decreasing the droplet spacing or with increasing the initial droplet diameter.     

Identification of droplet burning modes in lean, partially prevaporized swirl-stabilized spray flames

Experimental and numerical investigations of single droplet burning modes in a lean, partially prevaporized swirl-stabilized spray flame are reported. In the experiment single droplet flames have been visualized by CH-PLIF and simultaneous recording of the Mie signal. Two single droplet burning modes were identified: the envelope flame is a spherical diffusion flame burning at near-stoichiometric conditions. The wake flame is a potentially lean, partially premixed flame located downstream of the droplet. The droplet burning mode is of practical relevance, since it has significant impact on NO formation due to incomplete prevaporization.The droplet burning mode is determined by the ratio of chemical and convective time scales. The convective time scale is related to the droplet slip velocity. The impact of turbulent gas phase velocity fluctuations on droplet mechanics and droplet burning is discussed, based on a previous numerical investigation. In the present study the droplet slip velocity was measured with the 3D Phase Doppler (3D-PD) technique. For the measured slip velocities and ambient conditions in the hot gas region of the spray flame, simulations of single droplet burning were performed utilizing detailed models for chemical reaction, diffusive transport and vaporization. An agreement between the droplet burning modes predicted by the simulation and the droplet burning modes observed in the experiments was found.     

Combustion behaviors of isolated n-decane and ethanol droplets in carbon dioxide-rich ambience under microgravity

The burning and sooting behaviors of isolated fuel droplets for ethanol and n-decane are examined in high concentration of the ambient carbon dioxide under microgravity. A quartz fiber with the diameter of 50 μm maintains the droplet in the center of the combustion chamber and the range in the initial droplet diameter is from 0.30 to 0.80 mm. The ambience consists of oxygen, nitrogen and carbon dioxide. The concentration of oxygen is 21% in volume, and that of carbon dioxide is varied from 0% to 60% in volume. Detail measurements of the projected image of the droplet are conducted by using a high speed video camera and the effective droplet diameter squared are calculated from the surface area of the rotating body of the projected object. From evolutions of the droplet diameter squared, the instantaneous burning rates are calculated. Time history of the instantaneous burning rate clearly represents the droplet combustion events, such as the initial thermal expansion, ignition and following combustion. The instantaneous burning rate for n-decane shows an increasing trend during combustion, while that for non-sooting ethanol remains almost constant or shows a decreasing trend. A slight stepwise increase in the instantaneous burning rate is observed for larger n-decane droplets in air, which may be attributed to soot accumulation. However, this behavior of the burning rate disappears in higher concentration of carbon dioxide. Direct observation of the droplet flame indicates suppression of soot production in higher concentration of carbon dioxide and the suppression is enhanced for smaller droplet.     

Combustion characteristics and liquid-phase visualisation of single isolated diesel droplet with surface contaminated by soot particles

The combustion generated soot contamination effect on a single diesel droplet ignition and burning was investigated experimentally for the first time. Diesel droplet flame was used to contaminate the droplet to be investigated prior to ignition. Distinct differences in lifetime and stability of the burning of the neat and contaminated droplet samples were observed in their heating, boiling and disruptive phases. For a soot-contaminated droplet surface, the evaporation rate became weaker as a result of slower mass transfer thus contracted the flame formation. Contrary to the burning rate enhancement of droplet with stable and uniform suspension of particles observed by other researchers, the slightest contamination of soot particles in a fuel droplet surface can significantly reduce the burning rate. Denser agglomeration of soot can form a shell on the droplet surface which blocks the flow of gas escaping through the surface thus distort the droplet even further. At late combustion stage, bubbles are observed to rapture on the surface of the soot-contaminated droplet. Strong ejections of volatile liquid and vapour that would explode shortly after parting from the droplet are observed. It seems that the explosion and burning of ejected mixture have little interactions with the enveloped flame surrounding the primary droplet. Enhanced visualisation of droplet liquid-phase has clearly indicated the cause of declining trend in the burning rate and flame stand-off ratio of soot-contaminated diesel droplet. These insights are of significance for understanding the effect of fuel droplet contamination by combustion generated soot particles.     

对流速度对燃料液滴火焰形态及燃烧的影响

考虑气相非稳态及液滴内部环流,建立运动液滴非稳态蒸发燃烧模型.模型采用动网格方法精确追踪液滴表面位置,采用守恒方程组更新液滴表面边界条件.根据单步全局化学反应机理,仿真研究正庚烷燃料液滴在不同对流速度下的火焰形态及燃烧.结果表明:运动液滴内部环流使液滴内部低温区向环流中心移动.当液滴运动速度大于某临界值后,火焰形态由包覆火焰转变为尾迹火焰.包覆火焰的富燃区范围、高温区范围及燃烧速率明显较尾迹火焰大;包覆火焰的液滴表面温度及表面蒸发流率分布也明显不同于尾迹火焰.     

A molecular dynamics study of fuel droplet evaporation in sub- and supercritical conditions

Evaporation processes of a fuel droplet under sub- and supercritical ambient conditions have been studied using molecular dynamics (MD) simulations. Suspended n-dodecane droplets of various initial diameters evaporating into a nitrogen environment are considered. Both ambient pressure and temperature are varied from sub- to supercritical values, crossing the critical condition of the chosen fuel. Temporal variation in the droplet diameter is obtained and the droplet lifetime is recorded. The time at which supercritical transition happens is determined by calculating the temperature and concentration distributions of the system and comparing with the critical mixing point of the n-dodecane/nitrogen binary system. The dependence of evaporation characteristics on ambient conditions and droplet size is quantified. It is found that the droplet lifetime decreases with increasing ambient pressure and/or temperature. Supercritical transition time decreases with increasing ambient pressure and temperature as well. The droplet heat-up time as well as subcritical to supercritical transition time increases linearly with the initial droplet size d₀, while the droplet lifetime increases linearly with ${\mathrm{d}}_0^2$. A regime diagram is obtained, which indicates the subcritical and supercritical regions as a function of ambient temperature and pressure as well as the initial droplet size.     

An analytical study of droplet combustion under microgravity: Quasi-steady transient approach

An analytical model based on an assumption of combined quasi-steady and transient behavior of the process is presented to exemplify the unsteady, sphero-symmetric single droplet combustion under microgravity. The model used in the present study includes an alternative approach of describing the droplet combustion as a process where the diffusion of fuel vapor residing inside the region between the droplet surface and the flame interface experiences quasi-steadiness while the diffusion of oxidizer inside the region between the flame interface and the ambient surrounding experiences unsteadiness. The modeling approach especially focuses on predicting; the variations of droplet and flame diameters with burning time, the effect of vaporization enthalpy on burning behavior, the average burning rates and the effect of change in ambient oxygen concentration on flame structure. The modeling results are compared with a wide range of experimental data available in the literature. It is shown that this simplified quasi-steady transient approach towards droplet combustion yields behavior similar to the classical droplet theory.     

Influence of water mist on propagation and suppression of laminar premixed flame

The combustion of premixed gas mixtures containing micro droplets of water was studied using one-dimensional approximation. The dependencies of the burning velocity and flammability limits on the initial conditions and on the properties of liquid droplets were analyzed. Effects of droplet size and concentration of added liquid were studied. It was demonstrated that the droplets with smaller diameters are more effective in reducing the flame velocity. For droplets vaporizing in the reaction zone, the burning velocity is independent of droplet size, and it depends only on the concentration of added liquid. With further increase of the droplet diameter the droplets are passing through the reaction zone with completion of vaporization in the combustion products. It was demonstrated that for droplets above a certain size there are two stable stationary modes of flame propagation with transition of hysteresis type. The critical conditions of the transition are due to the appearance of the temperature maximum at the flame front and the temperature gradient with heat losses from the reaction zone to the products, as a result of droplet vaporization passing through the reaction zone. The critical conditions are similar to the critical conditions of the classical flammability limits of flame with the thermal mechanism of flame propagation. The maximum decrease in the burning velocity and decrease in the combustion temperature at the critical turning point corresponds to predictions of the classical theories of flammability limits of Zel'dovich and Spalding. The stability analysis of stationary modes of flame propagation in the presence of water mist showed the lack of oscillatory processes in the frames of the assumed model.     

Nonane droplet combustion with and without buoyant convection: Flame structure, burning rate and extinction in air and helium

The burning and extinction characteristics of isolated small nonane droplets are examined in a buoyant convective environment and in an environment with no external axial convection (as created by doing experiments at low gravity) to promote spherical droplet flames. The ambience is air and a mixture of 30%O₂/70%He to assess the influence of soot formation. The initial droplet diameter (D_o) ranges from 0.4 to 0.95 mm. Measurements are reported of the extinction diameter and time to extinction, and of the evolution of droplet diameter, flame diameter, soot shell diameter, burning rate, and broadband radiative emissions.In a buoyancy-free environment for air larger droplets burn slower than smaller droplets for the range of D_o examined, which is attributed to the influence of soot. In the presence of a buoyant flow in air, no influence of D_o is observed on the burning rate while the buoyant flames are still heavily sooting. The effect of D_o is believed to be due to a combination of dominance of the nonluminous, nonsooting, portion of the buoyant flame around the forward half of the droplet on heat transport and the secondary role of the luminous wake portion of the flame. In a non-sooting helium inert at low gravity, no effect of D_o is found on the evolution of droplet diameter.Flame extinction is observed only in the 30%O₂/70%He ambience. For all of the observations, extinction appears to occur before the disappearance of the droplet which is then followed by a period of evaporation. The extinction diameter and time to extinction increases with D_o and an empirical correlation is presented for these two variables.     

Flame-spread limit from interactive burning droplets in microgravity

This research conducted microgravity experiments on the flame spread over droplet-cloud elements with strong droplet interaction aboard Kibo on the ISS. The droplet-cloud element represents a local droplet pattern appearing in randomly distributed droplet clouds near the group-combustion-excitation limit and consists of small-droplet-spacing droplets and large-droplet-spacing droplets. As droplet-cloud elements, we used four n-decane droplets, Droplets C, B, A and L, placed at fiber intersections of two-dimensional SiC-fiber lattice with a 4-mm fiber interval in a combustion chamber. The flame spreads over the droplet-cloud element in order of Droplets C, B, A and L. The position of Droplet L relative to Droplet A was varied to investigate the flame-spread-limit distribution around burning Droplet A. The position of Droplet B relative to Droplet A was varied to investigate the effect of two-droplet interaction between Droplets B and A on the flame spread to Droplet L. The position of Droplet C relative to Droplet B was also varied to investigate the effect of three-droplet interaction among Droplets C, B and A. The results shows that in the case with the strong interaction by two or three interactive droplets, the high-temperature region is enlarged by the droplet interaction, centers near the center of mass of the interactive droplets and plays an important role in the flame-spread-limit distribution. Since the burning lifetime of Droplet A is finite, the flame-spread time from burning Droplet A to Droplet L is limited by burning lifetime of Droplet A and is less than 80% of the burning lifetime of Droplet A, which increases with the interactive effect. The flame-spread-limit distance from the center of mass of the interactive droplets increases with the burning lifetime.     

Combustion characteristics and detailed simulations of surrogates for a Tier II gasoline certification fuel

An experimental and numerical study of combustion of a gasoline certification fuel (‘indolene’), and four (S4) and five (S5) component surrogates for it, is reported for the configurations of an isolated droplet burning with near spherical symmetry in the standard atmosphere, and a single cylinder engine designed for advanced compression ignition of pre-vaporized fuel. The intent was to compare performance of the surrogate for these different combustion configurations and to assess the broader applicability of the kinetic mechanism and property database for the simulations. A kinetic mechanism comprised of 297 species and 16,797 reactions was used in the simulations that included soot formation and evolution, and accounted for unsteady transport, liquid diffusion inside the droplet, radiative heat transfer, and variable properties. The droplet data showed a clear preference for the S5 surrogate in terms of burning rate. The simulations showed generally very good agreement with measured droplet, flame, and soot shell diameters. Measurements of combustion timing, in-cylinder pressure, and mass-averaged gas temperature were also well predicted with a slight preference for the S5 surrogate. Preferential vaporization was not evidenced from the evolution of droplet diameter but was clearly revealed in simulations of the evolution of mixture fractions inside the droplets. The influence of initial droplet diameter (D_o) on droplet burning was strong, with S5 burning rates decreasing with increasing D_o due to increasing radiation losses from the flame. Flame extinction was predicted for D_o =3.0 mm as a radiative loss mechanism but not predicted for smaller D_o for the conditions of the simulations.     

Microexplosion of an emulsion droplet during Leidenfrost burning

An experimental study has been made of the microexplosion of an emulsion droplet on a hot surface during Leidenfrost burning. Photographic observation is used to study how the emulsion droplet behaves and what happens inside the droplet and to measure the waiting time for the onset of microexplosion. Weibull analysis was used to obtain the distribution function of the waiting time for the onset of microexplosion and to derive the formula for the rate of microexplosion as a function of water volume and emulsion temperature. The base fuels employed were n-decane, n-dodecane, n-tetradecane, and n-hexadecane. The results show that the increase in emulsion temperature with lapse of time results in the agglomeration and coalescence of microdroplets of base fuel dispersed in the continuous phase of water inside the emulsion droplet, terminated by the complete separation of the two phases. At the end of the phase separation process, an opaque water droplet is formed in the central core and is enveloped by the transparent shell of base fuel. Preferential evaporation of the base fuel occurs after the phase separation. The volume of the base fuel decreases while the water volume remains constant. The onset of the microexplosion of an emulsion droplet burning on the hot surface is classified by the wearout type of the Weibull distribution. The waiting time for the onset of the microexplosion decreases with increases in the normal boiling point of base fuel, initial water content, ambient pressure, and test surface temperature. The rate of microexplosion increases with the lapse of time and with increased normal boiling point of the base fuel. The rate of microexplosion increases linearly with increasing water volume in the emulsion droplet and decreases exponentially with the inverse of emulsion temperature.     

Vaporization and combustion in three-dimensional droplet arrays

The quasi-steady vaporization and combustion of multiple-droplet arrays is studied numerically. Utilizing the Shvab–Zeldovich formulation, a transformation of the governing equations to a three-dimensional Laplace’s equation is performed, and the solution to Laplace’s equation is obtained numerically to find the effects of droplet interactions in symmetric, multiple-droplet arrays. Vaporization rates, flame surface shapes, and flame locations are found for different droplet array configurations and fuels. The number of droplets, the droplet arrangement within the arrays, and the droplet spacing within the arrays are varied to determine the effects of these parameters. Computations are performed for uniformly spaced three-dimensional arrays of up to 216 droplets, with center-to-center spacing ranging from 3 to 25 droplet radii. As a result of the droplet interactions, the number of droplets and relative droplet spacing significantly affect the vaporization rate of individual droplets within the array, and consequently the flame shape and location. For small droplet spacing, the individual droplet vaporization rate decreases below that obtained for an isolated droplet by several orders of magnitude. A similarity parameter which correlates vaporization rates with array size and spacing is identified. Individual droplet flames, internal group combustion, and external group combustion can be observed depending on the droplet geometry and boundary conditions.     

Oscillatory cool flame combustion behavior of submillimeter sized n-alkane droplet under near limit conditions

This paper reports simulation results of oscillatory cool flame burning of an isolated, submillimeter sized n-heptane (n-C₇H₁₆) droplet in a selectively ozone (O₃) seeded nitrogen-oxygen (N₂-O₂) environments at atmospheric pressure. An evolutionary one-dimensional droplet combustion code encompassing relevant physics and detailed chemistry was employed to explore the roles of low-temperature chemistry, O₃ seeding, and dynamic flame structure on burning behaviors. For X_O2= 21% and a range of selective ozone seeding, near-quasi-steady cool flame burning is achieved directly (without requiring hot flame initiation and radiative extinction). Under low oxygen index conditions, but with significant O₃ seeding (X_O3 = 5%), a nearly quasi-steady cool flame is initially established that then transitions to a dynamically oscillating cool flame burning mode which continues until the droplet is completely consumed. It is found that the oscillation occurs as result of a initial depletion of fuel vapor-oxidizer layer evolving near the droplet surface and its dynamic re-establishment through liquid vaporization and vapor/oxidizer transport. A kinetic analysis indicates that the dynamic competition between the reaction classes- (a) degenerate chain branching and (b) chain termination/propagation - along with continuous fuel and oxygen leakage through the flame location contributes to an oscillatory burning phenomena of ever-increasing amplitude. Analysis based on single full-cycle of oscillatory burning shows that the reaction progression matrices (evolution of heat and species) for QOOH➔chain propagation/termination reactions (here, Q = C₇H₁₄-) directly scales with the gas phase temperature field. On the contrary, the QOOH➔degenerate branching reactions undergoes three distinct stages within the same oscillatory cycle. The coupled flame dynamics and kinetics suggest that in the oscillatory burning mode, kinetic processes dynamically cross through conditions characterizing the negative temperature coefficient (NTC) turnover temperature, separating low temperature and NTC kinetic regimes. In addition, a parametric study is conducted to determine the role of O₃ seeding level on the observed oscillation phenomena.     

On the behaviour of organic gel multi-size spray diffusion flames

The recently reported, experimentally observed, unusual behaviour of organic gellant-based fuel droplets which, under appropriate ambient thermal conditions, evaporate and burn in an oscillatory fashion is incorporated in a phenomenological manner in a model of a two-dimensional arbitrary multi-size spray diffusion flame. Non-unity Lewis numbers are permitted for the fuel vapour and oxidant. A combined analytical/numerical solution of the governing equations is presented and used to investigate how a spray's initial polydispersity and the frequency of oscillatory evaporation influence the combustion field. It is demonstrated that the initial droplet size distribution and the frequency of evaporation of the burning gel droplets can have an acute impact both on the homogeneous diffusion flame shape, height and width and on the thermal field downstream of the flame front. Hot spots of individual (or clusters of) burning droplets can be created and under certain operating conditions can lead to hotter temperatures than experienced in the main homogeneous flame. The intensity of these hotspots, their number and location are sensitive to spray related parameters. In realistic combustion chambers there is a danger inherent in the existence of hotspots in undesirable regions as they can damage the structural integrity. Other computed results demonstrate that, in relation to the spray diffusion flames obtained using an equivalent purely liquid fuel spray, the use of a gel fuel spray can lead, under certain operating conditions, to a reduction in flame height and temperature. The latter effect is critical when considering flame extinction.     

Numerical study on flame spread of an n-decane droplet array in different temperature environment under microgravity

A series of numerical calculations of flame spread of an n-decane droplet array was conducted at different ambient temperatures (T_a = 300 and 573 K) for S/d₀ from 1.5 to 10, where S is the droplet interval and d₀ is the initial droplet diameter. The authors compared these numerical results with experimental results under similar conditions at different ambient temperatures for the first time in this study. Good qualitative agreement in flame spread behavior between numerical results and microgravity experiments is obtained. Flame spread mode changed with an increase in S/d₀. Also, appearance of the flame spread mode in a stepping-stone manner (Mode III in [Jpn. Soc. Mech. Eng. 68 (672) (2002) 2423]) in a normal temperature environment was verified by numerical calculations and microgravity experiments, although it was not predicted in the theoretical analysis. In addition, good qualitative agreement of flame spread rate V_f versus S/d₀ was obtained between numerical and experimental results, although numerical results were at least twice as large as experimental results. V_f had a maximum peak at a specific S/d₀ for a different ambient temperature. Employment of improved reaction model and consideration for thermal radiation heat transfer are expected to produce quantitatively better results. An increase in surface temperature of unburned droplets and the development of a flammable gas layer around the droplets were promoted in a high-temperature environment, due to an increase in heat transfer from ambient air to the droplet. As a result, V_f was increased by the higher ambient temperature, suggesting that ambient temperature plays a significant role both in the flame spread mode and the flame spread rate through promotion of a flammable gas layer around unburned droplets.     

Microgravity experiments of single droplet combustion in oscillatory flow at elevated pressure

An experimental study for 1-butanol single droplet flames in constant and oscillatory flow fields was conducted under microgravity conditions at elevated pressure. In the constant flow experiments, flow velocities from 0 to 40 cm/s were tested. Using obtained data of d², the burning rate constants were evaluated. The burning rate constant in the quiescent condition was also calculated successfully at high pressure by the extrapolation method based on the Frössling relation. In the oscillatory flow experiments, the flow velocities were varied from 0 to 40 cm/s at the frequencies of 2–40 Hz. Results showed that the burning rate constant during the droplet lifetime varied following the quasi-steady relation at 0.1 MPa; however, in the conditions with higher frequencies at 0.4 MPa, the average burning velocity became larger than that for the constant flow case with the velocity equivalent to the maximum velocity in the oscillatory flow. Under the condition where the burning rate constant increased, it was observed that the flame did not sufficiently move back upstream, leading to enhancement of the heat transfer from the flame to the droplet surface. Therefore, the instantaneous burning rate constant increased. To investigate the mechanism of such flame behavior, the ratio of two characteristic times, τ_f/τ_D (τ_f: flow oscillation characteristic time, τ_D: diffusion characteristic time), were compared. As the flow oscillatory frequency increased, τ_f/τ_D becomes smaller. τ_f/τ_D also became smaller at high pressure. If τ_f/τ_D is small due to the small mass diffusion rate, the droplet flame could not move back to the appropriate position for the minimum velocity in steady flow, leading to an increase of the burning rate constant, especially in the case of higher frequency at high pressure.