Abstract
A new numerical approach for solving coagulation equation, TEMOM model, is first presented. In this model, the closure of the moment equations is approached using the Taylor-series expansion technique. Through constructing a system of three first-order ordinary differential equations, the most important indexes for describing aerosol dynamics, including particle number density, particle mass and geometric standard deviation, are easily obtained. This approach has no prior requirement for particle size spectrum, and the limitation existing in the log-normal distribution theory automatically disappears. This new approach is tested by comparing it with known accurate solutions both in the free molecular and the continuum regime. The results show that this new approach can be used to solve the particle general dynamic equation undergoing Brownian coagulation with sufficient accuracy, while less computational cost is needed.
1. INTRODUCTION
In theory, the evolution of aerosol size distribution in the flow field can be highly traced through solving the particle general dynamic equation (PGDE). This PGDE equation has an ability to describe the combined physical processes for very fine particles, such as the internal processes of nucleation, condensation, growth, and coagulation as well as the external transport processes of diffusion, convection, and thermophoresis. Unfortunately, the general dynamic equation is none other than Boltzmann's transport equation, in which only a limited number of known analytical solutions exist due to its own non-linear integro-differential structure. Hence, an alternative method, the numerical technique, has to be used to obtain approximate solution for this problem. However, the numerical calculations often become impractical, even with a modern super-computer, due to the requirement of large computational cost. In order to break the limitation in computational cost, three prominent methods were used, i.e., the moment method (MM) (CitationPratsinis 1988; Hulbert and Katz 1994, CitationChan et al. 2006, CitationLin et al. 2007), the sectional method (SM) (CitationGelbard and Seinfeld 1980; CitationTalukdar and Swihart 2004), and the stochastic particle method (SPM) (CitationWells and Kraft 2005; CitationMorgan et al. 2006). These methods have both advantages and disadvantages in accuracy and efficiency, and now they are used in different fields in terms of the particular requirement.
Because of the relative simplicity of implementation and low computational cost, the MM has been extensively used to solve most particulate problems, and has become a powerful tool for investigating aerosol microphysical processes in most cases (CitationWright et al. 2001; CitationYu et al. 2008b). As a key issue for the MM, the closure of the moment evolution equations has been achieved by making a prior assumption for the shape of the aerosol size distribution (CitationPratsinis 1988), approximating the integral moment by an n-point Gaussian quadrature (CitationMcGraw 1997) and assuming the pth-order polynomial form for the moments (CitationBarrett and Jheeta 1996). Typically, the moment method proposed by Pratsinis (PMM 1988) and the quadrature method of moments (QMOM) proposed by CitationMcGraw (1997) are two widely used moment methods now. The PMM is the early form of the MM, and has been developed to obtain the representative properties of the aerosol size distribution such as total particle concentration, mean size, and polydispersity only through solving the first three moments. However, the introduction of prior assumption with log-normal distribution in the PMM will lead to the method invalid at early stage of particle formation (CitationYu et al. 2008a, Citation2008b). The QMOM, to some extent, is an improved form of the PMM and has been proved to be applicable to any form of the growth laws and coagulation kernels (CitationUpadhyay and Ezekoye 2003). When the population balance is written in terms of one internal coordinate (e.g., particle length or size), the closure problem can be successfully solved with the use of a quadrature approximation without prior assumption. However, the weights and abscissas of the quadrature approximation must be additionally obtained using the product-difference algorithm (CitationGordon 1968), which greatly increases the computational cost in contrast to the PMM, especially in the case where the PGDE is required to be incorporated into the computational fluid dynamics (CFD) codes. In recent years, the particle formation occurring in a chemically reacting flow has been increasingly received attention, and accordingly it is possible to simultaneously capture the details of the fluid flow and transport, the evolution of the particle size distribution and complex chemical kinetics (CitationYu et al. 2008a, Citation2008b). For these particulate systems, consuming large computational cost will be inevitable for further investigations. Although some researchers such as CitationFriedlander (2000) and CitationFrenklach (2002) have shown the moment method without requiring a priori knowledge of the size spectrum is not new, it is still necessary to construct a new approach with respect to moment equation, which is easy to implement with low computational cost like the PMM and has not the prior requirement for particle size spectrum like the QMOM, to adapt to the requirement of modern complicated particulate industries.
In this article we present a new approach to solve the PGDE undergoing Brownian coagulation. The underlying idea of the approach is that the closure of the moment evolution equations is approached using the Taylor-series expansion technique. The Taylor-series expansion method of moments (TEMOM) has no prior requirement for particle size spectrum, and the number of moment equations needed is equal to the order of the Taylor-series expansion. Theoretically, the precision of solution and the computational cost increase with increasing the terms of Taylor-series expansion. Based on the trial numerical simulation, keeping 3 terms of Taylor-series is preferable if both precision and efficiency are simultaneously considered. The present investigation is concerned with the Brownian coagulation both in the free molecule and the continuum regime. Additionally, it should be pointed out that the TEMOM is easy to be extended to investigate other aerosol dynamics such as coalescence, surface growth and nucleation, and the derivation of Bivariate extension of the TEMOM simultaneously tracking the particle size and surface is also easy to be obtained, which is similar to the QMOM method (CitationWright et al. 2001).
2. THEORY
The scientific description for ultrafine particle dynamics starts from Schmoluchowski's discrete coagulation equation (CitationSmoluchowski 1917) and follows the integro-differential equation governing the continuous size distribution for the number concentration function developed by CitationMuller (1928). It is believed that the Muller's equation remains the absolute predominated status for investigating aerosol Brownian coagulation problems since its appearance in 1928, which is written as:
In the past, some efforts have been made to achieve the closure of Equation (Equation2). Three prominent methods, i.e., making a prior assumption for the shape of the aerosol size distribution (CitationPratsinis 1988; CitationYu et al. 2006), approximating the integral moment by an n-point Gaussian quadrature (CitationMcGraw 1997; CitationYu et al. 2007), and assuming the pth-order polynomial form for the moments (CitationBarrett and Jheeta 1996), have been evaluated and compared (CitationBarrett and Webb 1998). In this article, we develop a new approach, Taylor expansion technique, to solve this problem.
2.1. Brownian Coagulation in the Free Molecule Regime
In this size regime the collision frequency function βFM is (CitationPratsinis 1988):
Let us first concentrate on (1/v+1/v 1)1/2 in Equation (Equation4) and then rewrite it in the form:
For Equation (Equation8), we take the first three terms of Taylor-series expansion, which is consistent with the following treatments for the fractional moment. Disposing Equation (Equation7) with Equation (Equation8) and substituting it into Equation (Equation2) results in the following complicated equation with respect to the first three moments m 0, m 1, and m 2:
. Using the transformed Equation (Equation3), the right terms of Equation (Equation9) can be integrated. At the case, their detailed expressions take the following form:
For all the coefficients from ξ*1 to ς*3, it is clear that all the same terms are not grouped and all the terms completely correspond to the coefficients from ξ1 to ς3 in Equation (Equation9). In particular, Equation (Equation9) denoted by these coefficients is absolutely composed with fractional-order moments, not integral moments. In order to further close the moment equation, it is necessary to dispose these fractional moments. Therefore, we continue to use the Taylor-series expansion technique to dispose Equation (Equation3) with respect to fractional kth moment. In the Equation (Equation3), vk can be expanded with Taylor-series about point v=u (this is consistent with the expansion point u in (v+v 1)1/2):
Similar to the disposition for (v+v 1)1/2, we take the first three terms of Taylor-series. In order to meet the requirement of moment transformation, it is necessary to group the terms of truncated Equation (Equation11):
Substituting Equation (Equation12) into Equation (Equation3), we have
Here, the expansion point of Taylor-series expansion is denoted by the mean particle size v (= m 1 / m 0). At the case, Equation (Equation14) can be written in the following form:
2.2. Brownian Coagulation in the Continuum Regime
The disposition for aerosol Brownian coagulation equation in the free molecule regime can be also expanded to be in the continuum regime. In this regime the collision frequency function is (CitationBarrett and Webb 1998):
3. COMPUTATIONS
The numerical computations are all performed on an Intel (R) Pentium 4 CPU 3.00 GHz computer with memory 1 GB. The 4th-order Runge-Kutta method with fixed time step is used to solve the system of ordinary differential equations, and the Jacobi method is applied to obtain the eigenvalues and the eigenvectors of Jacobi matrix, which occurs in the computation of the QMOM. In the computations, all the numerical simulations are based on dimensionless time, and the time step is supposed to be small enough, 0.001, in order to allow an accurate result. The detailed dimensionless time is defined by τ=B 1 t in the free molecular regime and τ=B 2 t in the continuum regime. All the programs are written using the C Programming language and are performed on Microsoft Visual C++ 6.0 compiler.
4. RESULTS AND DISCUSSION
We now test our new approach by comparing it with known accurate models (CitationBarrett and Jheeta 1996; CitationBarrett and Webb 1998) both in the free molecular and the continuum regime. In order to make better comparison, we have written the codes based on the QMOM model, the PMM model and the TEMOM model. For QMOM, five different cases are included, i.e., quadrature approximation with 2, 3, 4, 5, and 6 nodes. The PMM used here is based on the log-normal distribution approximation developed by CitationPratsinis (1988). The initial distribution of all cases is assumed to be log-normal, i.e., n(v, t) = 
with N = 1, v g = 
and w g = 
, which is consistent with the initial distribution conducted by CitationBarrett and Jheeta (1996), CitationBarrett and Webb (1998), and CitationWilliams (1986). Under the case with log-normal initial distribution, the initial kth moments m k are obtained through solving m k = 
. The calculation of relative error for any variables follows the definition:
4.1. Brownian Coagulation in the Free Molecular Regime
In this regime the computation time is scaled by the definition τ=B 1 t, which is consistent with the disposition by CitationBarrett and Webb (1998). Here, we follow the study of CitationBarrett and Webb (1998) in which they compared various models at several special time points using tabulation approach. shows the values of zero moment m 0 and second moment m 2 given by various methods (the first moment m 1 is a constant when only Brownian coagulation is considered) at time τ= 1.0, 5.0, and 10.0. In this table, the values are all from Table 3 of CitationBarrett and Webb (1998) except for the QMOM6 and the TEMOM. For QMOM2, QMOM3, and PMM models, it should be noted that our own codes give the same values with those provided by CitationBarrett and Webb (1998). The last column in shows the consumed CPU time up to τ= 10. Since some studies have shown that the QMOM model with 2 or 3 nodes produces the accurate result, and the precision can be increased with increasing the number of nodes (CitationMcGraw 1997; CitationUpadhyay and Ezekoye 2003), it is feasible to take the QMOM6 as a reference to validate other models. In fact, all the approximate methods in give reasonable agreement with the reference results. For the zero moment, the results obtained by TEMOM are all identical to those by the QMOM6 except at τ= 1.0. At τ= 1.0, the result of TEMOM is closer to QMOM6 than Gamma model, QMOM2, and QMOM3 models. Although different models provide different results for the second moment, the TEMOM model produces the same precision as QMOM3 and PMM model. In addition, it can be easily seen that the consumed CPU time for the TEMOM is the shortest among all the investigated models.
TABLE 1 Moments m 0 and m 2 in the free molecular regime at different dimensionless time and the consumed dimensionless time for various methods
In order to further validate this new method, it is necessary to have the comparisons between our TEMOM model and other known models over large evolution time. The evolutions of m 0 with scaled time τ together with the relative error in various methods are shown in . The relative error denotes the ratio of m 0 from the various methods to the QMOM6. In fact, all the errors mentioned in this paper, including the zero and second moment, are all the relative values of TEMOM, PMM, or QMOM2-QMOM6 to QMOM6. In , all the curves overlap each other, and it is difficult to distinguish from one to the other. Hence, the differences between them must be evaluated using the relative error, Error%, which is shown in . From this figure, it can be found that the error for the TEMOM is always less than 1% from τ= 0 to τ= 100. In addition, it is clear that the curves of error exhibit the same trend for all the models, and more importantly, the TEMOM gives the least relative error beyond τ= 6. For all the investigated models, the relative error reaches constant values at τ>6, indicating that these models can give the reasonable results for zero moment.
FIG. 1 The zero moment m 0 and the relative error for various methods in the free molecular regime. The relative error denotes the ratio of m 0 from various methods to the QMOM with 6 points.
The second moment m 2 as well as the relative error are shown in . Like the zero moment, the relative error is still denoted by the ratio of various methods to the QMOM6. Similar to , all the curves overlap in . In , the curves of relative error are very similar for the TEMOM and QMOM3, except that values are negative for the former but positive for the latter. For the TEMOM, the error initially increases in the negative direction and then holds nearly constant at –1.65%. For the PMM, the error starts to show its largest value at τ= 0.01, and then decreases until it attains its nearly constant value at about τ=1. However, the absolute values of relative error of the PMM are less than that of the TEMOM in the entire time range. This indicates the TEMOM can achieve the same precision as the QMOM3 for solving the second moment, although the precision is little less than that produced by the PMM.
FIG. 2 The second moment m 2 and the relative error for various methods in the free molecular regime. The relative error denotes the ratio of m 2 from various methods to the QMOM with 6 points.
There exists an asymptotic solution for the distribution of reduced particle size, the so-called self-preserving size distribution (SPSD) (CitationFrenklach 1985), which has become an important tool to explore the aerosol coagulation mechanisms. The self-preserving form is usually approximated by a lognormal distribution with a geometric standard deviation σ g which is obtained by solving the following equation (CitationPratsinis 1988):
Although Equation (20) was originally derivated from log-normal theory, it may be also applied in QMOM and TEMOM models. shows the variations of σ g with time in the free molecular regime for various methods. We can see that the SPSD attains for all methods at about τ= 5. For the QMOM, the asymptotic values of σ g decrease from 1.378 to 1.346 with an increase of nodes from 2 to 6. The asymptotic value of 1.345 for the TEMOM is the closest to the QMOM6 among all the investigated models. In addition, the asymptotic value of 1.355 for the PMM is identical to the value given by CitationLee et al. (1984) who used log-normal functions for particle size distribution. The above comparisons suggest that the TEMOM has a higher precision in describing the asymptotic aerosol size distribution than other methods when compared to QMOM6.
FIG. 3 The geometric standard deviation σ g in the free molecular regime for various methods.
4.2. Brownian Coagulation in the Continuum Regime
Similar to the treatment for Brownian coagulation in the free molecular regime, we now compare various methods in the continuum regime. The results from the QMOM6 are considered to be exact and are used as references. The computational time in Equation (Equation18) is scaled by τ=B 2 t. The variations of the zero moment m 0 and the relative error for various methods in the continuum regime are shown in . In , all the curves overlap with each other and thus we cannot distinguish one from another. Hence, it is still necessary to use the relative error to distinguish the precisions produced by all the investigated models. In the whole range up to τ=100, shows the TEMOM model nearly exhibits the same curve with the PMM model, and in particular, the results of TEMOM are closer to the QMOM6. Moreover, like the PMM and QMOM, the relative error produced by TEMOM converges to a nearly constant value beyond τ=10, and more importantly, this error is always below 1.5%. The results clearly indicate that the TEMOM is capable of handling the evolution of zero moment with high precision in the continuum regime.
FIG. 4 The zero moment m 0 and the relative error for various methods in the continuum regime. The relative error denotes the ratio of m 0 from various methods to the QMOM with 6 points.
The variations of second moment and relative error for various methods are shown in . Similarly, all the curves exhibit the same trend and there are no visible differences with each other in . In the entire evolution time up to τ=100, it is clear that in all the investigated models, the QMOM2 model produces the largest error, but its value is always below 0.6%. These suggest all the investigated models can achieve very high precision for representing second moment in this regime. Although the error produced by the TEMOM is slightly larger than the QMOM3, their curves almost show the same trend. Like the zero moment, the error produced by the TEMOM converges to a nearly constant value of 0.4% beyond τ= 10. Typically, the curves of other models show the same trend except that there are negative values below τ=5 for the PMM. From the above comparisons, for the second moment, it can be easily found that the TEMOM gives relative low error close to the QMOM3.
FIG. 5 The second moment m 2 and the relative error for various methods in the continuum regime. The relative error denotes the ratio of m 2 from various methods to the QMOM with 6 points.
In the continuum regime, the particle size distribution can also achieve the self-preserving size spectrum. All the studies (CitationLee et al. 1997; CitationPratsinis 1988; CitationLee 1983; CitationLee et al. 1984) showed that the asymptotic value based on the geometric standard deviation σ g is near to 1.32 if the log-normal distribution theory is used. In , the values of the geometric standard deviation for various methods are presented and compared. It is obvious that all curves show the same trend and attain each asymptotic value at about τ=10. However, like in the free molecular regime, the diversity for these asymptotic values still exists in this regime. For the QMOM, the asymptotic value of σ g decreases from 1.325 to 1.315 with an increase of nodes from 2 to 6. In this figure, the curves for QMOM3–QMOM5 are not shown because their asymptotic values are always between QMOM2 and QMOM6. Here, it is worth pointing out that the TEMOM gives the same curve as the PMM model, and their asymptotic values are the same value of 1.319, which is consistent with the asymptotic value of 1.32 reported in the literatures (CitationLee et al. 1997; CitationLee 1983; CitationLee et al. 1984). This suggests that the TEMOM has an ability to handle the evolution of particle size spectrum with time based on geometric standard deviation, in particular with the same precision as the PMM model.
FIG. 6 The geometric standard deviation σ g in the continuum regime for various methods.
5. CONCLUSIONS
In this study we used the Taylor-series expansion technique to dispose the collision terms and the fractional moments to obtain a new form for the moment equations. This approach requires no prior requirements for particle size spectrum, and the number of moment equations to be solved simultaneously is equal to the reserved terms of Taylor-series. The present study was focused on Brownian coagulation in the free molecular regime as well as in the continuum regime, and the new approach was validated by comparing it with known methods. In order to validate TEMOM model in accuracy and efficiency, three important indexes of aerosol dynamics, i.e., the zero moment, the second moment and the geometric standard deviation were compared. The results show that the new approach can be used to solve particle general dynamic equation undergoing Brownian coagulation with sufficient accuracy, while less computational cost is needed.
Acknowledgments
We would like to acknowledge the financial supported by the Major Program of the National Natural Science Foundation of China (Grant No. 10632070).
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