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In this study, a model for the Joint droplet size and velocity distribution has been developed based on the thermodynamically consistent concept-the maximization of entropy generation during the liquid atomization process. The present model prediction compares favorably with the previous model of droplet size distribution produced by an air-blast annular nozzle and near the liquid bulk breakup region and provides velocity distribution for the same. A parametric study describes the model's prediction under different operating conditions.
Atomization of liquids is a process used in a wide range of industrial operations such as combustion in furnaces, gas turbines, diesel engines, and rocket engines; process industry utilizing spray drying, evaporative cooling, powdered metallurgy, and spray painting; and agriculture usage for herbicide and insecticide spraying, amongst others. During atomization, the surface area for a given amount of liquid can be increased considerably by which the mass and heat transfer processes associated with surface phenomena, can be relatively enhanced. In many applications of heat and mass transfer as well as flow processes, it is convenient to work with mean or average diameters. Hence, spray drop size distributions can be conveniently expresed in terms of some representative mean diameters, such as the surface, volume, or sauter mean diameter. The mean diameters appear in empirical correlations for predicting atomizer performance and some researchers have devoted much effort to the evaluation of mean diameters1,2. However, the drops generated by atomizers vary widely in their sizes because of the random nature of the atomization process. Detailed information regarding droplet size and velocity distributions in sprays is of ultimate importance for the design, operation, and optimization of spray systems. In recent years, computational fluid dynamics (CFD) analysis has been playing an increasingly important role, and the combustion model in CFD codes for practical spray combustion systems requires the initial droplet size and velocity distribution in the vicinity of the nozzle (known as the initial condition). All of these lead to the increasing interest in the joint drop size and velocity distribution prediction in the past two decades.
Over the years, considerable investigations have been made to describe the breakup of the liquid sheets into droplets and also predict the nature of the size and velocity distributions. The droplet size and velocity distributions are typically expressed as probability distribution function (PDF). Attempts were done since 1930's, when empirical correlations were used with an objective to find the best fit to the measured data for a wide range of atomizers or nozzles and operating conditions. In 1933, Rossin and Rammler3 published an equation for coal particle size distribution, which is still popular till date for its mathematical simplicity and the possibility of being extrapolated into range of very fine droplets. Following this, in 1939 Nukiyama and Tanasawa4 developed an exponential function to represent the droplet size distribution in sprays from pneumatic atomizers. In 1951, Mugele and Evans5 developed an upper limit function as a modified expression to log normal distribution. Later, in 1985, Rizk and Lefebvre6 used a modified expression of the Rosin-Rammler formula to provide a better data fit for larger drop sizes. Using this modified Rosin-Rammler formula, Han et al.7 obtained a reasonable agreement for a pressure-swirl atomizer and Rizk and his colleagues also achieved reasonable success in a number of investigations8,9. Bhatia et al.10 first applied the log hyperbolic distribution in their work. Xu et al.11 presented a stable model as compared to the log hyperbolic distribution, but it was slightly less accurate fit.
As an alternative to the empirical approach, primarily two analytical methods for obtaining the pdfs have been developed in the past two decades: the maximum entropy principle (MEP) and the discrete probability function (DPF). The DPF method was first employed by Sovani et al.12 to model the droplet size distributions in sprays. On the other hand, the MEP was developed by Jaynnes13 based on Shannon's concept14 of information entropy, which is a measure of uncertainty of probability distribution. The application of MEP to spray modeling was pioneered by Sellens and Brzustowski15-17, and Li and Tankin 18-23, and is able to predict the least biased pdf that duly satisfies a set of constraints expressing the available information related to distribution sought. Chin et al.24, 25 discussed that this is physically realistic because in the Rayleigh regime surface tension plays predominant role for droplet formation and the effects of liquid kinetic energy as well as the surrounding gas medium are negligible.
The MEP approach has also been used by other researchers to obtain drop size distribution. Van der Geld and Vermeer26 have obtained a size distribution function by considering primary and satellite droplet formation separately with an assumed Gaussian distribution without substantiating by experiments. Cousin et al.27, derived the drop size distribution for pressure swirl atomizers by maximizing a more general expression of Shannon entropy, which is equal to Bayesian entropy 28. Their approach needed a priori knowledge about mean droplet diameter in a particular spray, which cannot be known without experimental measurements. This work is further extended by Dumouchel et al.29, 30, to obtain the parameters to fit their results for practical sprays. According to Kapur29, the MEP method does not necessarily give the right distribution under some kinds of constraints. Ahmadi and Sellens31 suggested the simplified model in order to obtain only the droplet size distribution. They reached the conclusion that prediction of the droplet size distribution is independent of the velocity distribution and the constraints on momentum and kinetic energy carry only velocity information. Hence, the set of constraints is the conservation of mass, and of surface energy, and the partition constraint.
Almost all of the models mentioned above are semi-empirical in nature and need experimental results as inputs. Mitra and Li32 combined a linear and nonlinear instability model to predict the mass mean diameter and a nonlinear model to predict the breakup length. Boundary layer theory was used to evaluate the momentum and energy source terms. Shannon's entropy is maximized to obtain the distributions sought. The completely predictive model incorporates both the deterministic and stochastic aspect of spray droplet formation processes. Mitra33 further modified this model by replacing Shannon's entropy with Bayesian entropy. Hence, the effect of the unstable wave growths on atomization can be represented by a priori distribution, which is extracted from the first-order growth rate curves, and the nonlinear instability is incorporated in the droplet distribution through the breakup length present in the source terms and the breakup wavelength in the mass mean diameters. The agreement between the model prediction and experimental data was satisfactory37. Babinsky and Sojka34 provided a comprehensive review on the modeling of droplet size distributions in sprays.
The MEP method, which had been popular during the two decades since 1980, was adequate only for isolated systems in thermodynamic equilibrium and, hence, is not physically consistent with real conditions of the atomization process. Recently, Li et al.36 formulated a new model on the prediction of droplet size distribution based on the thermodynamically consistent concept-the maximization of entropy generation (MEG) during the spray formation, which is actually a nonisolated and irreversible process.
The MEG method described by Li et al.36, was limited to only droplet size distribution and did not provide information about the droplet velocity distribution. Further in his study there were requirement of the mean diameters, and . can however be determined from theoretical considerations described in33, but and have to be provided from the experimental measurements. For a given flow condition, different values of and represent the different degrees of departure from the thermodynamic equilibrium and, hence, the different extents of irreversibilities during the atomization process. Irreversibilities are to be induced by the liquid flow, nozzle configurations, liquid-gas interactions, etc., and vary from spray to spray; hence, and also vary from spray to spray, and so does the droplet size distribution.
The objective of the present study is to predict joint probability distribution function providing information on both droplet diameter and velocity using MEG. The present work is the extension of the work done by Li et al.36, 37 applicable only for droplet sixe distribution. The present study formulates modelling for a new pdf that provides joint velocity and diameter distribution. The current model is applicable for non isolated spray systems in thermodynamic equilibrium, hence physically consistent with realistic condition of the atomization process. A comparative study has also been done between the existing Li's model36, and the present joint droplet diameter and velocity distribution model based on MEG principle.
The subsequent section will explain the formulation of the thermodynamic entropy, which is an extension of the entropy expressed in36. The principle of formulation is the second law of thermodynamics considering the system to be non-isolated and irreversible. The constraints given in 36,37 are imposed on the newly formulated entropy, by using the Jaynnes concepts of maximization of entropy13.
Surface area per unit mass, m2/kg
Gas to liquid velocity ratio
Parameters in formulation
Gas velocity, m/s
Coefficient of drag for flow over flat plate
Droplet velocity, m/s
Droplet diameter, m
Liquid sheet velocity, m/s
Dimensionless droplet diameter
Dimensionless droplet velocity
Arithmetic mean diameter, m
Specific volume, m3/kg
Surface area mean diameter, m
Mass or volume mean diameter, m
Mean volume of droplet diameter , m3
Specific internal energy, kJ/kg
Joint droplet size and velocity probability distribution function
Specific entropy generation due to particle nature of spray, kJ/kg
Parameter in formulation
A constant, kJ/K
Weber number based on
Specific kinetic energy, kJ/kg
Dimensionless liquid sheet breakup length
Modified Lagrange's multipliers
Mass flow rate of liquid, kg/s
Isothermal compressibility of liquid, m2/N
Mass of drop, kg
Gas to liquid density ratio
Number of droplet of diameter and velocity per unit time
Gas density, kg/m3
Total number of droplets per unit time
Liquid density, kg/m3
Surface tension, N/m
Specific potential energy, kJ/kg
Discrete probability function for drop size and velocity distribution
Specific entropy, kJ/K.kg
Entropy flow rate, kJ/K.s
Entropy generation per unit time, kJ/K.s
Dimensionless entropy generation
Dimensionless mass source term
Index identifying droplet diameter
Dimensionless momentum source term
Index indentifying droplet velocity
Dimensionless energy source term
Specific internal energy, kJ/kg
Consider an air blast atomizer producing a conical sheet of liquid at the nozzle exit (Figure 1.). As the liquid proceeds further downstream the thickness decreases and the instabilities within the sheet sets in, breaking it into ligaments and finally into the droplets. The system under study involves the region bounded by the planes at the nozzle exit and the plane where droplets are formed. The study proposes a model for the droplet size and velocity distribution at the later plane.
As explained the control volume starts from the atomizer exit (location 1) and ends at the plane where droplets are formed (location 2). The spray is assumed to be in steady flow and isothermal condition. According to second law of thermodynamics,
where is mass flow rate of liquid from the atomizer, demotes the entropy flow rate and s denotes the specific entropy. For steady flow process and neglecting mass exchange between system and surrounding,
At the nozzle exit, the flow at state 1 is in bulk liquid form with free surface produced. Therefore, the entropy can be obtained deterministically since the temperature is known; at state 1, the entropy at the nozzle exit is due to the liquid bulk and the surface tension effect. A multitude of droplets form in the breakup region (state 2), and the total surface area is increased drastically. A liquid phase exists inside each of the droplets. The total rate of entropy flow is composed of two parts. One part is associated with the liquid bulk with free surface and is similar to that at state 1 and can therefore be denoted as. The other part is due to the existence of the numerous individual droplets, represented as, which is directly associated with the particle nature of the droplets. Hence,
Entropy associated with particle nature of droplet [ ]
For quantitative evaluation of entropy quantitatively, the present case is made analogous to Gibbs ensemble in statistical thermodynamics. Therefore, the entropy at state 2 is associated with the probability distribution representing droplet size and velocity and a different distribution results in different amount of entropy.
Let at location 2, is the total number of droplets per unit time and is the number of droplets of diameter and velocity. Then the total number of possible states is expressed as,
The entropy due to particle nature can therefore be written as,
where , is a constant and the Sterling's Approximation provides .
It is assumed that the droplets are spherical in shape due to surface tension effects. However there may be droplets of non spherical shape due to large size or due to oscillations, but they can be neglected under statistically stationary assumption.
Entropy change for the bulk liquid,
According to the Gibbs equation for a simple compressible substance with free interfaces, the liquid bulk entropy change can be expressed as,
and the internal energy can be expanded to , where the terms have their usual meanings. Since the atomization process is considered to be isothermal, the internal energy, which is the function of temperature only, will remain unchanged. The potential energy term is taken to be zero for a constant level spray. Therefore, the entropy equation is
where is the isothermal compressibility of the liquid.
The surface tension term and the kinetic energy term are not negligible in the Eq. (8) since the process of atomization increases the net surface area thereby contributing to surface tension effect and there will be overall change in kinetic energy.
Entropy, a thermodynamic property, depends only on the state of the system. Therefore, integrating the
At the atomizer exit, a thin liquid sheet forms from the annular air-blast atomizer, and the liquid pressure is almost the same as the surrounding air pressure since the curvature effect is small and negligible. However, a circular liquid jet forms solid-cone spray produced by small-orifice atomizers, and the liquid pressure is larger than the ambient air pressure due to the surface tension effect. However, the specific value of p1 will not affect the determination of the droplet size distribution, as shown later. Therefore, for simplicity, we assume
For liquid pressure at state 2, let us consider only one liquid droplet for the moment. For a liquid droplet in the air, the pressure difference across the free interface is related to the surface tension effect as follows,
At the atomizer exit the kinetic energy is dependent on the liquid sheet velocity while at state 2 the kinetic energy is dependent of the droplet velocity, therefore,
The total entropy flow rate at state 2 by considering all the droplets is
Therefore, the entropy generation in the atomization process is,
The non dimensional form is obtained by dividing the Eq. (14) by, which yields,
By substituting and , and we get, the simplified non dimensional form as
Since the term A is also non dimensional, therefore,
where and .
Joint Droplet Size and Velocity Distribution Model
The objective of the present study is the determination of the joint droplet size and velocity distribution at state 2, which is in reality the initial droplet size distribution for droplets just formed in a spray. There are infinite sets of the probability distribution that can satisfy the global constraints on the atomization process, such as the conservation laws. For an isolated system reaching thermodynamic equilibrium, the entropy of the system reaches a maximum. Thus, the most appropriate distribution for such a system can be determined by maximizing the system entropy. However, for liquid atomization, which is a non-isolated system undergoing a non-equilibrium process, according to irreversible thermodynamics, the least biased (or the most realistic) distribution is the one that maximizes the amount of entropy generation during the naturally occurring atomization process, under a given set of constraints.
The constraints imposed on the physical problem for determination of the droplet size and velocity distribution are the conservation of mass, momentum and energy. These conservation principles are applied to the control volume enclosing the bulk liquid and the extended from the atomizer exit to the plane where the droplets are produced immediately after the breakup of the liquid sheet. The conservation of mass can be written as,
where is the mass source term.
For the conservation of the liquid momentum, the total momentum of all the droplets must be equal to the momentum of the bulk liquid at the atomizer exit, plus a momentum source term, ,which accounts for the momentum exchange between the bulk liquid and the surrounding gas medium. Therefore the momentum conservation can be written as,
Similarly the conservation of the liquid energy requires that the total energy of all the droplets formed in the spray; i.e. the sum of their kinetic and their surface energies, be equal to the kinetic energy of the liquid at the atomizer exit, plus the source term,, which represent the exchange of energy between the two phases. Hence, the energy conservation can be written as,
The non dimensional form the Eqs. (18), (19), and (20) are as follows,
where , , , , and .
In addition to the conservation equation there are three more constraints that are required for completing defining the distribution function, . From the definition of the probability, the sum of individual probabilities in a universe is unity, provides the following normalization relation,
The other two constraints are based on the definition of the droplet's mean diameters, viz. D10 and D20, which can either be provided by experimental data or other theoretical consideration . These are as follows,
where and .
The Joint Droplet Size and Velocity Distribution Function
To maximize the entropy generated, , under the set of constraints, Eqs. (21 -24), the Lagrange's method is adopted. That is,
where are the Lagrange's multipliers. Upon simplification of the Eq. (27), we obtain the
If the constant is known then the terms A, B and C can be evaluated and there will be no requirement of the constraints, Eq. (25) and Eq. (26), and the remaining can be determined by numerical techniques. But if is unknown then we assuming total of six Lagrange's multipliers and not four as shown below,
Now for the above, the six constraints can be utilized to determine the six multipliers .
It is generally regarded that the droplet size and velocity in sprays varies continuously rather than discretely. Therefore, to obtain a continuous probability function, the probability of finding the drop between the volume Vn and Vn+1 and between the velocity Un and Un+1 is determined.
where is the non dimensional volume.
Substitute which then yields
The Eq. (31) represents the final continuous joint droplet size and velocity distribution function. The multipliers can be determined by solving the following set of six equations,
In order to solve for joint droplet size and velocity distribution, the source terms, and , need to be evaluated. The source terms forms a link between the deterministic and stochastic sub models. The following values of source terms33 are taken in the present formulation,
where is the drag coefficient for flow over flat plate, is the dimensionless breakup length, and .
The set of six equations, Eqs. (32) to (37) are highly non linear and very sensitive to the Lagrange's multipliers. But since the set of equations are continuous and their derivatives are also continuous in the region, Newton-Raphson is often the techniques employed to solve such equations. Details of solving the equation up to second order accuracy are given in35.
In order to simplify the complexities of finding the solution of six unknowns, only five multipliers are solved and the sixth value is evaluated by enforcing the normalization condition. This means that the following equation is computed at each iteration step,
Using the Eq. (41) the set of six equations can now be reduced to five equations as shown below,
In this way solving the five equations rather than six will result in faster convergence.
The number based droplet size distribution can be obtained by integrating Eq. (31), over the velocity space from minimum to maximum value. Similarly the number based droplet velocity distribution can be obtained by integrating the Eq. (31) over the diameter space from minimum to maximum. The minimum and maximum limiting values of the diameter and velocity space are taken according to36.
A program was developed in Matlab to solve the above set of equations. Standard routines like fsolve and dblquad have been used to evaluate the results. The initial guess is an important parameter that controls the convergence of the solution. The program flow chart describing the Matlab program code is given in Fig. 2.
RESULTS AND DISCUSSIONS
The MEG model calculates the joint droplet size and velocity distribution for sprays when the nozzle condition and also the measured parameters are provided. The nozzle exit conditions are (a) density ratio, ρ, (b) velocity ratio, U, (c) liquid sheet breakup length, (d) half sheet thickness, (e) physical properties of
gas and liquid. The measured parameters are (a) arithmetic mean diameter, (b) surface mean diameter, (c) mass mean diameter. Table 1 gives the values of the mean diameters and the liquid sheet breakup length and Table 2 provides the operating conditions; are taken from .
Fig. 3 (a) shows a three dimensional plot of the joint pdf for a typical spray resulting from the breakup of the liquid sheet with half sheet thickness of 10 µm, Weber number of 50, density ratio of 0.001, and velocity ratio of 4. The corresponding isoprobability curves are shown in Fig. 3 (b). It is observed that there exists a global maximum for the distribution which is located at dimensionless diameter slightly close to 0.5 and dimensionless velocity slightly close to 0.5. The probability of drops of large size having high velocity is low while the probability is high for drops having lower velocity.
The effect of Weber number on the number based droplet diameter distribution and velocity distribution is shown in Fig. 4 (a) and (b) for the case II from Table 1. It is observed that with the increase in Weber number, while other conditions are kept fixed, there is no significant difference in distribution of droplet diameter except that at near peak. The deviation in distribution is very less as compared to work done by Mitra33, due to the fact that the droplet distribution is completely defined by the three mean drop diameters under MEG analysis36. In the velocity distribution, as the Weber number increases the peak shifts lower and towards larger droplet velocities. At very high Weber number (We = 300) the distribution is wider and the drops of wider velocity range will occur (0 to 12.5 m/s). However, at large droplet velocities, the distribution curves tend to collapse onto each other.
The effect of density ratio on the distributions is shown in Fig. 5 (a) and (b). The peak of droplet distribution lowers, but there is marked change in the shape of the distribution curve, due to the reasons stated earlier. The peaks in the velocity distribution get lower as there is increase in the density ratio. The distribution widens and produce more drops of higher and lower velocity, for the case having ρ = 0.0014. But for the case of ρ = 0.001, there is less number of drops of higher velocity and more drops are produced having velocity equal to 4.3 m/s.
The effect of gas to liquid velocity ratio on the distribution curves is shown in Fig. 6 (a) and (b). In the droplet diameter distribution, there is a very slight decrease in peaks, but not any marked changes in the distribution. In the velocity distribution curve, the increase in the velocity ratio lowers the peak of the curve and there is slight shifting of the peak towards lower velocity. At U = 10, the peak lies slightly less than 4 m/s, but for the case U = 8, the peak lies around 4.3 m/s.
A comparison between the present model's, Eq. 31, predictions on droplet size distribution are compared with the distribution given by Li et al.36 model, are presented for some of the cases given in . The specific flow condition for each case is listed in Table 1, and Table 2 presents the corresponding arithmetic mean diameter, the surface area mean diameter , and mass mean diameter , at the spray centerline and at a distance 30 mm downstream of the nozzle exit, measured for each case and the liquid sheet breakup length estimated from the flow visualization technique. The corresponding values of αi for the model prediction, Eq. (31), are presented in Table 3. The model prediction is compared in Figs. 7 (a) to 10 (a) with the Li et al. model of droplet size distribution for each of the four cases shown in Tables 1 and 2. It is seen that the model prediction agrees very well with the Li et al. droplet size probability distribution, which was experimentally validated in .
The droplet velocity distributions are presented for each of the four cases in Fig 7 (b) to 10 (b). The nature of distribution conforms to the velocity distributions given in [33, 37]. The velocity distribution predicts that the probability of finding the drops with velocity is not zero but has certain finite value. The probability then increases as the velocity of drops increases and reaches a maximum, after which the probability starts decreasing if further droplet velocity is increased.
In this study, a new model for the joint droplet size and velocity distribution in sprays has been formulated based on a thermodynamically consistent concept-the maximization of entropy generation during the non-isolated and irreversible liquid atomization process. A parametric study has been provided, that is able to describe the behavior of the present model under different operating conditions; Weber number, gas to liquid velocity ratio, and gas to liquid density ratio. Also a comparison between the present model's prediction and the Li et al.36 model's prediction are provided. Therefore, the present model is relatively more physically consistent with the realistic nature of atomization in sprays. Further work is under way to experimentally verify the present model for the joint droplet size and velocity distribution.
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Table 1 Experimentally Measured Parameters36
Liquid Sheet Breakup Length (mm)
Table 2 Experimental Flow Conditions36
Water Velocity (m/s)
Inner air Velocity (m/s)
Outer Air Velocity (m/s)
Table 3 Lagrange's multipliers for the cases taken given in Tables 1 and 2
Figure 1 Schematic of the control volume chosen for the analysis. Location 1 represents the atomizer exit and location 2 represents the droplet formation plane36
Figure 2 Program Code Flow
Figure 3: (a) Joint 3D distribution (b) Joint Iso-probability curves for Case II
Figure 4 Effect of Weber number on (a) Droplet Diameter Distribution function and (b) Droplet Velocity Distribution for Case II
Figure 5: Effect of gas to liquid density ratio on (a) Droplet Diameter Distribution (b) Droplet Velocity Distribution for Case II
Figure 6: Effect of gas to liquid velocity ratio on (a) Droplet Diameter Distribution (b) Droplet Velocity Distribution for Case II
Figure 7 (a) Droplet diameter distribution and (b) Droplet Velocity Distribution for Case I
Figure 8: (a) Droplet Diameter Distribution (b) Droplet Velocity Distribution for Case II
Figure 9: (a) Droplet Diameter Distribution (b) Droplet Velocity Distribution for Case III
Figure 10 (a) Droplet Diameter Distribution (b) Droplet Velocity Distribution for Case IV