Droplets diameter distribution using maximum entropy formulation combined with a new energy-based sub-model

Seyed Mostafa Hosseinalipour ,Hadiseh Karimaei,*,Ehsan Movahednejad

1 Energy and Environmental Lab,Dpt.Mech.Eng.,Iran University of Science and Technology,16846-13114,Narmak,Tehran,Iran

2 University of California,Berkeley,USA

1.Introduction

The droplet size distribution in sprays is one of the vital parameters required for fundamental analysis of practical sprays.Detailed information about the droplet size distribution is consequently important for the design,performance and optimization of spray systems[1].Information of droplets size distribution in immediate downstream the breakup is necessary as the boundary condition for computational fluid dynamics(CFD)and two-phase spray computations[2].The classic models to predict Sauter Mean Diameter(SMD)were emanated principally from the experimental information.In this method,a distribution curve is fitted on different experimental data resulted from different operational conditions of an injector.This procedure is the basic foundation of available probability distribution functions such as Rosin–Rambler,Nukiyama–Tanasawa,Log–Kernel etc.[2,3].Several studies were carried out to obtain a more general droplet distribution based on statistical approaches.Since 1985,the maximum entropy principle(MEP)has been applied in atomization and spray fields in order to estimate droplet size and velocity distributions and gained a lot of success.This approach can estimate the most probable droplet size and velocity distributions under a set of constraints.Sellens and Brzustowski[4]and Li and Tankin[5]were pioneers of applying the MEP approach for modeling of spray and atomization.In this approach besides the conservation of mass,momentum and energy,maximum entropy principle must be satisfied by the droplet size distribution function.The most probable size distribution could be determined from the conservation equations if the system entropy is maximized.Liand Tankin[5]utilized a single constraint of energy but Sellens et al.[6]used separate constraints for the liquid–gas surface energy and kinetic energy of the system.After solving the equation system by the Newton–Raphs on method,Sellens et al.found that the probability size and velocity distribution functions(PDF)are in a way that the probability of droplets with a zero diameter does not tend to zero.Li and Tankin[7]used from the volumetric distribution function instead of the diametrical distribution function and therefore used droplets volume instead of droplets size in the equations.The results were in close agreement with the experimental data.However,the method of Li and Tankin[7]for very small droplets predicted zero frequency.Dumouchel[8]published a review paper about droplet size distributions of sprays using the maximum entropy method in which he investigated different types of ME models with emphasizing on similarities and differences of them and discussed about the expectations that can be accessed by these models.The maximum entropy formulation needs a mean diameter as an input that was earlier provided from the experimental measurements.In the present paper,it is attempted to provide the input of the ME model by a theoretical model rather than experimental measurement.

In the past,there have been many attempts to predict the mean droplets diameter.Concerning the calculation methods for mean droplets diameter,empirical correlations have been proposed for different types of nozzles[8,9].As an engineering tool to calculate the mean droplets diameter,convenient models which do not require the experimental parameters and much time to compute are preferred.In the present paper,a simple estimation method,which is entitled the energy-based model,is proposed based on the energy conservation law for the liquid sheet atomization to predict the mean droplets diameter.This method is based on the deterministic aspects of liquid atomization process using the energy approach.The other advantage of this method is that it does not need any experimental data or coefficient.

In this study,the maximum entropy formulation(MEF)combined with the energy-based method(EBM)was used for simulation of droplets formation.The two models have been connected together using the mean droplets diameter.An industrial swirl injector(Miser)with hollow cone spray characterized by previous researches[10,11]is investigated as a case study for the present model.The new results show a close agreement with the available experimental data[12].The major objective of this paper is investigation of the ability of maximum entropy formulation combined with the energy-based model to predict the mean droplets diameter of a spray.

2.The Maximum Entropy Formulation

Droplets formation is a random process;therefore,it can be modeled using a statistical method in which PDFs are utilized for droplet size and velocity distributions[12–14].Governing equations are written for a control volume which extends from the injector exit plane towards the zone of primary droplets formation at starting point of the breakup process.The control volume is illustrated in Fig.1.The control volume length is identical to the breakup length obtained from experimental data.The conservation equations of liquid mass,momentum,and energy must be satisfied through the atomization process.Concerning the entropy maximization,the conservation equations can be represented based on probability density function pijwhich is the probability of being present a droplet with volume Viand velocity uj,where i and j are related to i th and j th intervals of volume and velocity of droplets.Therefore,the conservation equations can be written as follows[13–16]:

where,˙n is the droplet production rate and also˙mo,˙JO,and˙EOare respectively the mass flow rate,momentum and energy that enter to the control volume from the injector outlet.Sm,Smuand Seare the source terms of mass,momentum and energy equations,respectively.

Since the sum of probabilities has to be equal to one,the following equation has to be considered besides the above equations.

There is an unlimited number of probability distribution functions(pij)which satisfy Eqs.(1)–(3),but the most probable and proper distribution is the one that can maximize Shannon entropy[17]:

where K is the Boltzmann's constant.If volume and velocity of droplets are transformed to diameter and velocity,then the formalism can be rewritten according to the probability of being present the droplets whose diameters are betweenand whose velocities are betweenand[7].Eqs.(1)–(3)can be written in a nondimensional and integral form as Eq.(7)[3].The Lagrangian multipliers method is used for maximizing of Shannon entropy(Eq.(5)).By using the method of Lagrange multipliers,a probability distribution,that should maximize the Shannon entropy,can be obtained.Therefore,probability function will be as follows:

The non-dimensional form of probability function is represented in Eq.(8).The set of λiis a set of Lagrangian multipliers which must be computed.Consequently,to obtain the Lagrange coefficients(λi)in the probability function(f),it is essential to solve the following normalized set of equations[18].

Fig.1.Control volume which extends from the injector exit plane towards the zone of primary droplets formation.

In the present study,the mass source term is set to zero implying that the evaporation during the atomization is disregarded.It should be noted that,any energy conversion inside the control volume is not regarded as a source term.In the control volume,there is a momentum exchange between the gas and liquid flow because of drag force acting on liquid bulk.It should be considered as a momentum source term.There is a source term of energy in the ME formulation for the purpose of including turbulence effect in estimation of the dropletsize distribution that can be estimated and considered in the model.The MEF needs a mean droplets diameter as an input provided in the next section.The Newton–Raphson method is used to solve equation set(7)and the probability function is determined by Eq.(8).The range of variation for both the nondimensional diameter and velocity were considered from 0 to 3.

3.The Energy-Based Model

In the following,the energy conservation law for the atomization process is implemented with considering some assumptions.Then,a calculation method to predict the mean droplets diameter is proposed.According to Fig.1,a control volume extends from the injector outlet towards the primary breakup region.This control volume covers the distance between the nozzle outlet and breakup region.Eq.(10)presents the energy conservation law for the mentioned control volume.It is assumed that there are no energy input,work output and energy source.Therefore the fluid and gas energies including the total enthalpy,kinetic and potential energy,and also the liquid surface free energy are considered in the control volume.

The pressure inside the liquid at both the input and output of control volume is equal to the sum of ambient gas pressure and Laplace pressure as Eq.(11)shows.

By substituting Eq.(11)into Eq.(10)and considering some further assumptions,Eq.(12)can be derived.In this equation,the potential energy is neglected.The change in the internal energy is adequately lower than the change in the kinetic energy,therefore,it is neglected,and also there is no change in the output gas energy rather than the input gas.

Eq.(12)shows that an increase in the Laplace pressure and surface free energy is equivalent to a decrease in the kinetic energy.It means that the energy is converted to the Laplace pressure and surface free energy through the atomization.Eqs.(13)and(14)represent the relations between the mass flow rate and flux of surface area of liquid for spherical droplets and liquid sheet,respectively.It is assumed that the liquid sheet enters into the control volume and the spherical droplets exit from that.

Based on the equation derived from the energy conservation law,the mean droplets diameter(d)can be estimated.Eq.(15)is obtained by substituting Eqs.(13)and(14)into Eq.(12)and manipulating that as follows:

To obtain an appropriate formula,one parameter is defined as atomization efficiency(η).The atomization efficiency shows the ratio of the decrease in the kinetic energy to the in flow energy.Eq.(16)represents the atomization efficiency.

By employing Eq.(16)and the Weber number into Eq.(15),the following formula is derived to obtain the mean droplets diameter of spray as follows.

The formulation proposed has the ability to calculate the mean droplets diameter with using no experimental parameter or coefficient.As can be seen the mean droplets diameter is proportional to the reverse of Weber number and atomization efficiency.

4.Results and Discussions

Fig.2.Mean droplets diameter versus atomization efficiency drawn based on Eq.(17).

Mean droplets diameter versus atomization efficiency in terms of the mean liquid velocity is drawn as Fig.2 based on Eq.(16).Based on the amount of atomization efficiency,the mean droplets diameter can be estimated.With considering Eq.(16),the maximum atomization efficiency(ηmax)is obtained when the decrease in the kinetic energy becomes the maximum value,however,with assuming that there are no other energy losses.As a case study an industrial swirl injector with the geometrical and performance characteristics presented in Table 1 is investigated.The Weber and Reynolds numbers of the liquid sheet,drag force on the liquid sheet and atomization efficiency are calculated and presented in Table 2.Regarding Fig.2,the mean droplets size is estimated equal to 73 μm for this case study.The mean droplets diameter estimated using the energy-based model furthermore the available experimental data[12]are represented in Table 3.As can be seen,this model as a simple,time-saving and reliable enough model can provide a very good estimation of the mean droplets size at the primary breakup stage rather than using complicated methods like the linear instability theory to use as an input in the MEF.

Table 1 Geometrical characteristics of the injector[9,10]

Table 2 The Weber and Reynolds numbers of the liquid sheet and the drag force on the liquid sheet

Table 3 the mean droplets diameter estimated from the combined model(MEF/EBM)and experiment

The source terms of the MEF including the energy and momentum are respectively as follows:

Cfis the drag coefficient of the air passing over a liquid flat plate with contact area A.It was suggested and used in some previous studies[10,12–14].All energy exchanges which happen inside the control volume are ignored.Just the energy which comes in or out of the control volume is considered as a source term.It should be noted that the heat transfer during the process is ignored.The source terms are calculated and presented in Table 4.

Table 4 The calculated source terms

Fig.3 presents the probability of size distribution calculated from the ME model and obtained from the experimental measurement for the under-studied injector.In this figure,the size distributions resulting from the ME model using the energy-based sub-model,ME model using the mean droplets diameter experimentally measured and the distribution experimentally measured are illustrated.As can be observed,the theoretical result of the ME model using the new sub model is in good agreement with the experimental data reported in reference[12].As can be seen,the new sub-model provides the reliable prediction close to the experimental data.Therefore,this model can be a good substitute for the experimental measurements or use of complicated methods such as the linear instability theory[20].

Fig.3.Comparison of the droplets size distributions obtained from the ME modelusing the energy-based sub-model and experimental data[12].

For the droplets diameter smaller than 100 μm,the theoretical model overestimates the frequency of them a little bit,but totally the prediction of theoretical model matches well with the experimental data in the entire range of droplets size.Therefore it can be concluded that the combined model(MEF/EBM)presented in the paper can well estimate the initial droplet size distribution.Also the result shows that the existence of the small droplets(smaller than 75 μm)is more probable than the larger ones.The existence probability of the droplets greater than 180 μm is obtained equal to zero from the MEF but the experimental data shows that the droplets with sizes up to 200 μm have the existence probability.A bit difference between the experimental data and theoretical study can be because of the probable errors in the experimental measurement and assumptions considered in the theoretical model.Dumouchel[8]believes that due to the interaction between the surface tension and aerodynamic forces,very small or very large droplets cannot form;therefore,the droplets smaller than 15 μm have not been observed among the experimental data.

Fig.4 presents the probability of size distribution calculated from the ME model as per the different mean droplet diameters in the range of 50–100 μm.Using Fig.4 it is possible to compare the predicted droplet diameters distribution in terms of atomization efficiency.This figure shows the importance of obtaining a good estimation of mean droplet size and its effect on the results of ME model.Therefore obtaining a mean droplets size close to the experimental data can lead to predict the droplet size distribution close to the experimental data.The objective is investigation of the effect of mean droplet diameter of a spray on the prediction of maximum entropy formulation and here,the other parameters of MEF are not considered.

Fig.4.Comparison of the droplets size distributions obtained from the MEF as per the different mean droplets diameters(50–100 μm).

5.Conclusions

In this paper,the stochastic process of the atomization was modeled using the maximum entropy principle(MEP)in order to estimate the distribution of the droplet size in the primary breakup zone.Moreover,one predictive modelas a simple and time-saving modelto estimate the mean droplets size of sprays based on the deterministic aspects of a liquid atomization process has been discussed.This model was proposed as a theoretical model based on the energy conservation law to calculate the droplets diameter produced by the liquid sheet atomization.This model shows that an increase in the Laplace pressure and surface free energy is equal to a decrease in the kinetic energy and therefore the kinetic energy loss causes the atomization process and droplets production.Therefore the droplets diameter can be estimated based on the kinetic energy loss evaluation.The mean droplet diameter estimated using the energy-based model is in good agreement with the available experimental data.

The MEmodeland energy-based sub-model were combined together by the mean droplets diameter of spray.The droplets diameter distribution estimated using the combined model(MEF/EBM)has been compared with the available experimental data.Although some assumptions were considered to simplify and solve the governing equations,but the results showed a very good agreement with the experimental data in terms of both the quantity and trend.The parametric study showed that the droplet size distribution predicted by the ME model is significantly sensitive to the mean droplets diameter.Since the experimental measurement of a spray is a hard and expensive job to do,therefore,replacing it with an analytical model to well predict the mean droplets diameter and a numerical model to well predict the droplets diameter distribution can be very noticeable.Consequently,it can be concluded that the energy-based model can provide a good prediction of the mean droplets size to use subsequently in the modeling of droplets size distribution using the MEF independent of the experimental data.

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