Abstract
Determining the thermal profile of ignition is important because the desired ignition behavior varies with the objective. For example, extended ignition prolongs the time that the engine runs; however, fast ignition offers a higher power gain. The pollution caused by undesirable chemical reactions, as determined by the ignition profile, is another important aspect. Based on a previously developed method, we examined the impact of different theoretical particle size distributions (PSDs) on the thermal ignition profile. We compared different PSDs of polydispersed fuel spray with normal distributions with various means, each corresponding to the same fuel volume. Our results revealed a significant dependence of thermal ignition on the PSD. Systems that comprised only lowradius droplets did not reach ignition, whereas systems with only highradius droplets required a long time to establish ignition. Moreover, the change in the mean droplet radius unexpectedly resulted in a double hump in the maximum temperature of the combustion process.
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Introduction
In general, spray combustion involves many particles, which allows only a statistical description of their behavior. When studying a model that describes spray combustion, various processes should be considered, such as the injection and atomization of the fuel liquid, mixing of droplets with oxidizing gas, heat transfer to droplets producing evaporation of liquid, size distribution and velocities of the droplets, evaporation of and drag on the droplets, and mass burning rate of the droplets. Therefore, to investigate combustion from the mathematical aspect, sufficient variables and parameters related to the combustion process of the spray should be considered. However, a complicated mathematical model could be extremely difficult to investigate and may allow only numerical analyses, as asymptotic and analytical approaches may not be feasible^{1,2,3,4,5,6,7,8}.
In most cases, mathematical models describing the spray combustion process include a system of nonlinear partial differential equations, which complicates their study^{9,10,11,12,13,14,15}. Additionally, researchers have developed mathematical models for the spray combustion process described by a system of nonlinear ordinary differential equations. Despite the complicated nature of these mathematical models, researchers have developed asymptotic and analytical methods to study thems^{16,17,18,19,20}.
As previously mentioned, an efficient way to model the spray combustion process when considering as many parameters and variables as possible but avoiding developing a long and complex mathematical model is to consider the droplet sizes as a distribution function or parcels of droplets. This is the aim of this research^{21,22,23}.
Obtaining the temperature profile of fuel during combustion is essential. Fuel profiles with long ignition times prolong engine operation over a given fuel volume. This characteristic is required for machines that operate over long combustion periods. For example, an unmanned aerial vehicle (UAV) or drone must run on a single tank of fuel for as long as possible ^{24}. However, there is a fast ignition profile designed for machines that require high power. A race car is a popular example, in which engineers attempt to run as fast as possible^{25}. Knowing the temperature graph during the ignition process can help reduce air pollution because temperature is considered a significant factor affecting the rate of a chemical reaction^{26}. Profiles of combustion processes that do not favor polluting chemical reactions are desirable.
The fuel distribution changes after the first cycle because of the residue from the previous cycle. Thus, we can accurately determine the first PSD. Since the efficiency and highpower of the motor are considered over many engine cycles. The effect of the initial distribution decreases over time. Nevertheless, its role is significant during acceleration as the engine first starts. Most air pollution occurs during the first ignition, which correlates with the combustion profile of the temperature. Owing to the importance of reducing air pollution, the design of fuel distributions to reduce air pollution is the major practical and beneficial aspect of the present study.
This study investigated the dynamics of several basic theoretical formations of PSDs. Singlehump PSDs with different peak locations and widths were evaluated over time. The results indicated a high dependence of the combustion profile on the initial PSD. Furthermore, this study provides initial suggestions on designing fuel distributions to account for the purpose of the engine and reduce toxic pollution. In a future study, we will examine the distribution of the PSD profile over cycles and the behavior of PSD, which the researcher changes. In addition, our research presents some novel results regarding the relationship between the PSD and the characteristics of the ignition. The behavior of the ignition, particularly the maximum temperature of the combustion process, does not depend trivially on the PSD. Moreover, the model results show that another hidden parameter is needed to understand the ignition process better.
Methods
A simplified thermal explosion model was selected for this study because of its advantages. Despite the simplicity of the model, it contains nontrivial information, as shown later. Additionally, the model is applicable to any realistic, accurate, or theoretical droplet PSD or its smooth approximation, thereby allowing studies using any PSD. Furthermore, unlike other methods, the computation time is low for any resolution; for example, the parcel method can be implemented on a smooth approximation of a PSD, which allows an analytical approach.
Below, we explain the key features of the model in detail and briefly introduce these assumptions. The analytical derivation of the model is not discussed.
Initially, the most significant effects were considered based on the rapid nature of the ignition process. The system characteristics imply minor heat loss and pressure changes. Consequently, the system was considered adiabatic and isothermic for the liquid phase (i.e., all the droplets are at the same temperature at any given time)^{27}. Moreover, the heat transfer coefficient was assumed to be a function of the gas phase^{28}. The application of the quasisteadystate approximation allowed the assumption that the liquid and saturated liquid temperatures were equal^{1}. A system of ODEs was used to describe the process under the aforementioned approximation. The system comprises one equation each for the temperature and concentration changes, as well as equations for changes in the radius of each droplet^{20}.
The initial conditions are as follows:
Previous studies demonstrated the reduction of the ODE system via mathematical manipulation^{18,29}. First, the summation of the radii exchanged by integration over the droplets allows the definition of a probability density function. The natural candidate for the integrand is a piecewise continuous function that describes the PSD accurately times the radius, whereas the limit of the integral ranges from zero to the maximum radius \(R_m\) (5). Alternatively, it may be approximated using a smooth function that holds equality. We denote the PSD or its approximation, as \(\hat{P}_0\). The droplet distribution can be normalized to obtain a proper probability distribution function (PDF) \(P_{0}()\) (6).
Second, any radius is replaced by a function of the current maximum radius and maximum radius at the beginning of the combustion process (7). Note that \(R_m\) is the current maximums radius while \(R_{m,0}\) is the maximum radius in the beginning of the process. The latter allows rewriting the summation of the radii as an integral function \(F(R_{m})\) (8).
The function G(R) (7) can be viewed as the effective radius of the PSD system. We refer to this function as the effective radius function:
where
Thus, the system is reduced to three equations that represent the system of the original ODEs.
Finally, introducing nondimensional variables are introduced to obtain the final simplified form of the system.
We rewrite \(F(R_{m})\) as a function of \(r_{m}\).
Notice:
Thus, we define a new function for the summation:
\(F(R_{m})\) can be considered as the mean effective radius, and thus, it is termed as the mean effective radius function.
A simple system that describes this process is obtained as follows:
The nondimensional initial conditions were as follows:
Eq. (17) can be replaced by (20).
We used Python to numerically solve the system of ODE. Additionally, a program was used to plot the system solutions.
Results
We investigated the dynamics of the PSD considering a maximum radius of \(R_{m0}\). To compare different PSDs unbiasedly, the volumes of all PSDs were considered to be equal to a monodisperdsed fuel volume. The reference volume was considered as \(\frac{4\pi }{3}NR^{3}_{m0}\) where N is the number of droplets, and \(R_{m0}\) is the radius of the droplet. Any PDF is equivalent to a PSD up to a certain factor or normalization. Thus, for any distribution, the PDF function f(x) can be considered as a PSD of the reference volume by multiplication of factor A to maintain the equality (21).
The normal distribution (22) was first chosen for the calculation because it is wellknown and has been used to approximate a realistic PSD. The normal distribution has two parameters that hold the mathematical data: mean \(\mu\) and the standard deviation \(\sigma\). \(\mu\) is the center of the curve while \(\sigma\) is linked with the curve’s width as \(2\sqrt{2ln2}\sigma\) is the width of the curve in half of the maximal curve value. A factor replaces the distribution coefficient \(\frac{1}{\sqrt{2\pi }\sigma }\) to account for the PSD volume.
We evaluated three PSDs corresponding to the normal PDF with a cutoff in the maximum width of \(R_{m0}\), seen at Figs. 1, 2 and 3. The PSDs are of equal width (\(\sigma =\frac{R_{m0}}{8}\)), they are centered at \(\mu =\frac{R_{m0}}{4}\), \(\mu =\frac{R_{m0}}{2}\), and \(\mu =\frac{3R_{m0}}{4}\) respectively. The first PSD attains combustion the fastest, followed by the second PSD, and finally the third PSD. The final PSD \(\theta\) value at the end of the combustion process is approximately \(25\%\) lower than those of the other PSDs. Moreover, the combustion graph shows a decline in the slope near the end of the combustion process. Additionally, the two first \(r_m\) versus \(\theta\) graphs are concave, while the last graph transitions from concavity to convexity at the end of the graph. The change in combustion temperature is indicated by the variable \(\theta\) and the change in ignition time can be attributed to the droplet size. A smaller droplet has larger surface area and thus, burns more rapidly. Nevertheless, the changes in the combustion temperature between the first two PSDs were negligible, suggesting that both PSDs reached thermal combustion. Moreover, the third PSD did not reach thermal combustion. Thus, we can deduce that until a specific value of the center of the PSDs, complete combustion occurs, and the radii of the droplets affects the combustion time. The second anomaly in the \(r_m\) versus \(\theta\) graph requires a higher resolution for better investigations.
Following the previous results, we evaluated several PSDs that corresponded to the normal PDF with smaller widths. Each PSD had an equal small width (\(\sigma =\frac{R_{m0}}{32}\)), but was centered at various points. For a comfortable visualization, the results are presented in three sets of graphs based on the center of the PDF.
First, we evaluate the PDFs, which correspond to small drops of radii, centered at \(\mu =\frac{R_{m0}}{8}\), \(\mu =\frac{3R_{m0}}{16}\), \(\mu =\frac{R_{m0}}{4}\) and \(\mu =\frac{3R_{m0}}{8}\) as shown in Figs. 4, 5, 6 and 7. The normal PDF with a small width centered at low radii (\(\mu =\frac{R_{m0}}{8}\)) is linear; however, the characteristics of combustion process does not occur. The graph of \(\theta\) versus \(\tau\) show linear behavior and don’t have the characteristic of a combustion (high \(\theta\) values and a steep incline) are absent. Because the droplets are small, they evaporate rapidly, which does not provide sufficient time for maintaining the temperature. However, larger droplets maintain the temperature. The \(r_m\) versus \(\theta\) graph is convex function. As the center of the PDF advances to higher values, the values of \(\theta\) increase and reach the combustion profile at \(\mu =\frac{R_{m0}}{4}\) whereas the graph of $\rm$ vs. \(\theta\) becomes concave with an incline at \(\theta \sim 8\). Next when \(\mu =\frac{3R_{m0}}{8}\), the combustion process with the highest temperature decreases with an incline at \(\theta \sim 5\). The temperature decreases with increasing droplet size, which can be attributed to the absence of small droplets that burn rapidly. The final natural assumption was proven to be faulty, as shown below.
Second, the PDFs that correspond to largesize droplets are centered at \(\mu =\frac{R_{m0}}{2}\), \(\mu =\frac{5R_{m0}}{8}\), \(\mu =\frac{3R_{m0}}{4}\) and \(\mu =\frac{7R_{m0}}{8}\) as shown in Figs. 8, 9, 10 and 11 respectively. At \(\mu =\frac{R_{m0}}{2}\) and \(\mu =\frac{5R_{m0}}{8}\) an unexpected phenomenon occurs when the combustion temperature begins to increase with inclines at \(\theta \sim 6\) and \(\theta \sim 8\), respectively. As the centers of the PDFs continue to increase, \(\mu =\frac{3R_{m0}}{4}\) and \(\mu =\frac{7R_{m0}}{8}\) the combustion temperature decreases with incline at \(\theta \sim 6\) and \(\theta \sim 5\), respectively, as expected.
Because the droplet size decreased as the process continued, many smaller droplets were formed. This observation explains why combustion occurs only for droplets with large radii. As small droplets do not increase in size, larger droplets that hold heat cannot be formed. However, large droplets may reduce in size and form smaller droplets to increase heat. As a large droplet becomes smaller, some of the energy is used for evaporation. Hence, the combustion temperature decreases, but combustion continues to occur.
The unusual behavior of the change in combustion temperature, that is, the double maximum in incline Fig. 12, shows that a single parameter of droplet size cannot explain. There may be a mechanism that depends on the division of the droplets into two groups: small and large. At a certain point, depending on the parameters, the small and large droplets of the PSD behave differently.
For PDFs containing droplets of mostly small or mostly large radii, the system behaves as expected by the heat stored in droplets by size and surfacetovolume ratio, which depends only on their radii. However, the midsized normally distributed PSD exhibits a more complex phenomenon that must be explored.
The tendency to store heat or evaporate to increase the heat depends on the ratio of the surface area (\(4 \pi R^2\)) and volume (\(\frac{4\pi }{3} R^3\)) of the droplets, which is proportional to 1/R (or inversely proportional to R). As 1/R increases, the tendency to evaporate increases; therefore, this parameter influences the combustion profile and explains the evolution of the maximum temperature, as explained for the classic case of normal droplet width distribution, as shown in Figs. 1, 2 and 3. However, a closer examination of the maximum temperature of the smallerwidth distribution in Fig. 12, indicates that a single parameter cannot explain the profile. Thus, at least one hidden parameter must exist to account for the double hump phenomenon in the maximum temperature profile of the combustion process. The results suggest that this parameter can explain the interplay between droplets with smaller and larger radii, which account for building and storing heat, respectively. In future studies, we will attempt to justify our results analytically and extract these hidden parameters.
Discussion
The dependence on droplet distributions is a major problem of polydisperse fuel combustion. The problem is complicated, and we have only a very previous theoretical study on the ignition processes of such fuels. It is evident that no injection process can produce a monodisperse spray, and there exists a PDF that corresponds to the different injection processes. The unrealistic nature of a monodisperse injection can be discussed separately. The necessity of droplets with both large and small radii for proper thermal combustion was demonstrated.
In the previous section, the influence of the PDF on ignition processes was shown to be essential. However, for some PDFs, the ignition process was not possible. In contrast, the monodisperse injection does not appear "optimal" for fast ignition, which further depends on the engine type. In this study, we used a simple model to demonstrate the influence of the PDF on the ignition process. The plot of the maximum combustion temperature vs. the center of the PDF with a normal distribution exhibited a double hump with two local maxima, as shown in Fig. 12. The unpredictable shape of the graph demonstrates that more than one small parameter may exist in the system.
Unfortunately, we did not extract the main parameters of the PDF that influenced the ignition. Hence, this question remains unanswered, and these hidden parameters, which might differ for lean and rich fuels, could be obtained in future studies. This paper highlights the importance of the droplet radius distribution, and combined with the tools for assessing fuel dynamics^{30} it is expected to encourage studies employing more advanced PSDs.
Conclusions
In this study, we analyzed several normal PSDs. First, the combustion profile indicated a significant dependence on the PSD. No combustion was observed for PSDs comprising large amounts of tiny droplets. On the other hand, the system did not reach combustion for PSD which has many large droplets, unlike the common logic that would dictate the graph of the maximum temperature vs. the mean location of the normal distribution will be with one hump. A curve with a double hump was obtained. The nature of the plots was nontrivial because the PSDs had narrow widths. The results indicate the presence of at least one other small parameter that affects the system behavior. To obtain a better understanding of the combustion process, the basic model should be modified to explicitly include this parameter. Both the small and large droplets contributed to the combustion profile, and a monodisperse approximation must be more subtle to account for such phenomena. Our simple investigation shows that the combustion profile relies heavily on the PSD, which may aid in obtaining a combustion profile that does not support undesired chemical reactions, and thereby reduces air pollution.
Data availibility
All data generated or analysed during this study are included in this published article [and its supplementary information files].
Abbreviations
 A :

Constant preexponential rate factor
 C :

Molar concentration (kmol m\(^{3}\))
 c :

Specific heat capacity (J kg\(^{1}\) K\(^{1}\))
 E :

Activation energy (J kmol\(^{1}\))
 L :

Liquid evaporation energy (i.e., latent heat of evaporation, enthalpy of evaporation) (J kg\(^{1}\))
 \(n_{d_{i}}\) :

Number of droplets of size i per unit volume (m\(^{3}\))
 Q :

Combustion energy (J kg\(^{1}\))
 B :

Universal gas constant (J kmol\(^{1}\)K\(^{1}\))
 \(R_{d_{i}}\) :

Radius of size i drops (m)
 \(R_{max, 0}\) :

Maximal droplet radius at \(t=0\) (m)
 T :

Temperature (K)
 t :

Time (s)
 \(t_{react}\) :

\(A^{1}e^{1/\beta }\) (s)
 \(P(\cdot )\) :

Probability density function
 \(\hat{P}(\cdot )\) :

Probability distribution function
 \(U(\cdot )\) :

Unit step function
 \(\Pi (\cdot )\) :

Rectangle function
 \(\alpha\) :

Dimensionless volumetric phase content
 \(\nu\) :

The quantity equivalent to the volumetric phase content for the continuous model
 \(\beta\) :

Dimensionless reduced initial temperature (with respect to the socalled activation temperature E/B)
 \(\gamma\) :

Dimensionless parameter that represents the reciprocal of the final dimensionless adiabatic temperature of the thermally insulated system after the explosion has been completed
 \(\epsilon _{1},\hspace{2 pt}\epsilon _{2}\) :

Dimensionless parameters introduced in Eq. (13) and describing the interaction between gaseous and liquid phases
 \(\mu\) :

Molar mass (kg kmol\(^{1}\))
 \(\lambda\) :

Thermal conductivity (W m\(^{1}\) K\(^{1}\))
 \(\rho\) :

Density (kg m\(^{3}\))
 \(\tau\) :

Dimensionless time
 \(\Psi\) :

Represents the internal characteristics of the fuel (the ratio of the specific combustion energy and the latent heat of evaporation) and defined in Eq. (13) (dimensionless)
 \(\eta\) :

Dimensionless fuel concentration
 \(\theta\) :

Dimensionless temperature
 r :

Dimensionless radius
 d :

Liquid fuel droplets
 f :

Combustible gas component of the mixture
 g :

Gas mixture
 i :

Number of droplet sizes
 L :

Liquid phase
 p :

Under constant pressure
 s :

Saturation line (surface of droplets)
 0:

Initial state
 m :

Number of droplet sizes
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S.H.: Data analysis, numerical application. N.O.: Mathematical modeling, numerical analysis. V.G.: Results analysis and validation of the results.
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Hareli, S., Nave, O. & Gol’dshtein, V. Impact of a polysidpersed fuel distribution on the ignition characteristic. Sci Rep 14, 18304 (2024). https://doi.org/10.1038/s41598024654569
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DOI: https://doi.org/10.1038/s41598024654569