2009 International Conference on Signal Processing Systems
Mathematical Modelling of Droplet Atomization Using the Population Balance Equation
Hossam S. Aly Tholudin M. Lazim
Yehia A. Eldrainy
Mohammad Nazri Mohd Jaafar
Department of Aeronautical Engineering Faculty of Mechanical Engineering Universiti Teknologi Malaysia Skudai, JB - Malaysia
[email protected] [email protected]
Department of Aeronautical Engineering Faculty of Mechanical Engineering Universiti Teknologi Malaysia Skudai, JB - Malaysia
[email protected] [email protected]
physics can lead themselves to a two dimensional description. This drawback is overcome in the Eulerian approach [6-10] in which the Navier-Stokes equations are solved for both the continuous and dispersed phase simultaneously. Away from Eulerian and Lagrangian models, the volume of fluid (VOF) model has gained some attention in simulating spray dispersion and liquid sheet disintegration [11-13]. Recent studies proposed methodologies to extend the VOF for predicting atomization characteristics [14-15]. Recently, the population balance model was introduced to the CFD community as an alternative method for modelling multiphase flow problems in which coalescence and break up are of great importance [16]. The model solves a transport equation for a number density function and thus providing a description for the discrete phase SMD and its distribution function. Furthermore, the model can be easily coupled with the Eulerian multi-fluid framework which eliminates the need for semi empirical models employed in Lagrangian tracking models. That is, the population balance model gains the advantages of both the Lagrangian and Eulerian methods while keeping the computational complexity at a minimum level. Since it was introduced to the CFD community, the population balance model was extensively used in various multiphase problems such as bubble columns [17-25], and aerosol [26]. However, only few studies can be found that discusses the implementation of the various population balance models in droplet atomization problems [26-30]. Consequently, this paper aims to introduce the basics of the population balance equation, the numerical procedure for its solution, the CFD framework that couples the model with the Navier-Stokes solver, and preliminary results for simulating a plain jet airblast atomizer. The numerical results are compared with the experimental measurements of Liu et. al. [31]. Finally, recommendations on the implementation of the population balance model in this type of problems are presented for future improvements of the model.
Abstract—This study investigates numerically the atomization process occurring in a plain jet airblast atomizer. The population balance equation is solved for the dispersed phase coupled with a CFD Eulerian multi-fluid model. The Sauter mean diameter values obtained numerically compare favorably with previous experimental data only at certain flow conditions. Finally, this study proposes some enhancements on using the numerical model which were revealed from the model formulation and the results obtained. Keywords-population balance equation; Eulerian approach; multiphase flow modelling; atomization; CFD; droplet break-up.
I.
INTRODUCTION
Atomization is a vital process in most combustion systems. It can be described as a procedure in which bulk liquid is converted into small droplets. The resulting droplets are characterized by a mean diameter called the Sauter mean diameter (SMD). Values of SMD impact the combustion efficiency significantly, for instance, droplets with very low SMD values will tend to evaporate instantly and contribute to the increase of NOx emissions. On the other hand, High SMD values will perform as a source for particulate formation. The intensive research work aiming to understand and characterize the atomization process is motivated by its significant impact on combustion efficiency as well as its fundamental obscurity. Lefebvre [1] describes many empirical equations derived from experimental measurements for various types of atomizers. On the other hand, many researchers are interested in predicting atomization characteristics numerically. The Lagrangian particle tracking method [2-4] was found to be very efficient in predicting SMD values especially when droplet evaporation and combustion are considered. Moreover, the Lagrangian model has the advantage of including the effects of droplet coalescence and break up without dividing the dispersed phase into a number of separate classes and thus reducing the computational expenses significantly [5]. However, one major drawback of the model is that the problem must be solved in three dimensions even if the flow 978-0-7695-3654-5/09 $25.00 © 2009 IEEE DOI 10.1109/ICSPS.2009.190
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II.
III.
THE POPULATION BALANCE EQUATION
The population balance equation is a transport equation for the number density function, assuming that coalescence is negligible for dilute sprays the equation reduces to:
THE NUMERICAL DETAILS
The CFD code Fluent 6.3 was used in this study to simulate water droplets atomization through a plain jet airblast atomizer using the discrete population balance model. A grid consisting of 35188 volume cells was constructed. The solution domain and grid are shown in figure 1 and figure 2 respectively.
w >n V , t @ >u&n V , t @ wt (1) c c c c Q g V E , t V V n V , t d V g V n V ³
40
:v
The terms on the left hand side represent the rate of change of the number density function and its convective derivative respectively; the terms on the right hand side represent the rates of birth and death of droplets due to breakage respectively. There are several methods for solving the population balance equation, such as the monte carlo method [16], the discrete method [16], the quadrature method of moments [32], and the direct quadrature method of moments [27,28]. This work will be restricted to the application of the discrete method only, other methods will be considered in future work. The discrete method is based on representing the continuous particle distribution function PDF in terms of discrete classes or bins. The major advantages of the discrete method is that the number density is calculated directly and that the droplet distribution function can be calculated, while in the method of moments the solver calculate only the moments of the number density function. However, when applying the discrete method, the size bins of the droplets must be determined priori to the solution, moreover, a large number of size bins may be required in order to obtain an accurate solution which may lead to computational complexity. In the discrete method the population balance equation is written in terms of volume fraction of particle size i:
w U s D i U s u i D i wt ³Qg V c E V V c n V c, t dV c g V n V , t
100
Fig.1. Solution domain (dimensions in mm) (2)
:v
Air Inlet Water Inlet
Where U s is the density of the dispersed phase and D i is the volume fraction of particle size i. In order to obtain a solution for the number density function, the population balance equation is coupled with the Eulerian multi-fluid model via a two-way coupling procedure, i.e. the velocity ui is computed from the Navier-Stokes equations and substituted into the population balance equation. The population balance model in turn computes the SMD of the droplets from the obtained size distribution function. The calculated SMD values are then returned to the Eulerian solver in order to compute the momentum exchange between The continuous and dispersed phases i.e. the drag acting on the droplets.
Fig.2. Grid and inlet boundaries
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An unsteady pressure based solver was used with the Eulerian multiphase model and turbulence was modeled using the mixture k İ turbulence model [33]. A time step of 1e-5 was chosen in order to ensure that the solution converges within each time step. The velocity boundary conditions for the water and air are 1 and 60 m/s respectively. The population balance equation was descritized into 7 bins, the size of the bins are given in table 1.
accurate breakage kernel would be sensitized to the Weber and Reynolds number of the flow field
Table 1. Bins size Bin Size Pm
0 1024
1 322
2 102
3 32
4 10
5 3
6 1
The breakage kernel was computed as follows:
E V V c =
2 Vc
(3)
This represents a case where equal droplet fragments are distributed to all daughter size bins. Equation 3 was implemented into the Fluent solver using a User Defined Function (UDF). The frequency by which break up occurs 1
was chosen to be 2000 s which is approximately equal to the reciprocal of the mean characteristic time scale of turbulence eddies. Fig. 3 Contours of SMD in mm
IV. RESULTS AND DISCUSSION 1200
Contours of SMD are shown in figure 3. The figure shows that SMD values continue to decrease as the droplets depart from the injector until they reach a minimum value near the outlet. The figure also reveals the impact of droplets diameter on the spray dispersion. The spreading rate of the spray increases as the droplets diameter decrease. This finding is consistent with the two way coupling between the discrete population balance model and the Eulerian multifluid model. The distribution of the SMD values along the centerline is shown in figure 4. It is clear that the SMD near the outlet reaches a constant value. The value of SMD at the outlet plane was found to be 77 Pm while the SMD value obtained experimentally by Liu et. al. [31] for the same geometry and boundary conditions is 58 Pm . However, this good agreement ceases to exist at different inlet air velocities. The inconsistency of the results at different air velocities results from the constant frequency of the breakage kernel. While this value described the break up process reasonably at air velocity of 60 m/s, it failed to represent the frequency of break up at different velocities. Another drawback in the breakage kernel is the probability density function describing the break up process. A more
1000
Sauter diameter, Pm
800
600
400
200
0 0
20
40 60 Axial distance, mm
80
Fig.4. SMD variation along the centerline
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V. CONCLUSION AND FUTURE WORK A preliminary numerical simulation of water droplet atomization through a plain jet airblast atomizer was performed using the recently introduced population balance model. The model obtained good agreement with the experimental data as long as the frequency of the break up process was close to the reciprocal of the mean characteristic time scale of turbulence eddies. In order to improve the performance of the model the following recommendations are proposed: x The frequency of the break up process should be a variable and calculated as a function of the turbulence time scale. x The breakage kernel probability density function should be sensitized to the non dimensional parameters controlling the break up process such as the Weber and Reynolds number. x Increasing the number of bins would improve the solution accuracy but at the expenses of further computational complexities. x The performance of other solution methods for the population balance model such as the quadrature method of moments should be tested and compared to results obtained from the discrete method. VI. ACKNOWLEDGMENT Partial support was provided by the Malaysian Ministry of Science, Technology and Innovation (MOSTI) under Science Fund grant 79253. References [1] [2] [3]
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