Verification of Expansion Orders of the Taylor-Series Expansion Method of Moment Model for Solving Population Balance Equations

We verified the effect of the Taylor expansion order on the accuracy of the Taylor-series expansion method of moment (TEMOM) model in both the free molecular and continuum-slip regimes. The ordinary differential equations for moments with fourth-order Taylor expansion were first derived for fractal-like agglomerates, which were further compared to the existing TEMOM model with third-order Taylor expansion. We confirmed that the TEMOM model with a fourth-order Taylor expansion is less accurate than that with a third-order Taylor expansion. Moreover, the scope of application of the TEMOM model with a fourth-order Taylor expansion is limited. The existing TEMOM model with a third-order Taylor expansion was verified as the most reliable model for solving population balance equations for agglomerates undergoing Brownian coagulation.


INTRODUCTION
Compared with extant studies on spherical particles, the study of the dynamic behavior of aerosols consisting of agglomerates is a relatively new field (Friedlander, 2000).The dynamics of these two types of aerosol, however, differ considerably.A recent study showed that the evolution of size distributions of fractal-like agglomerates over time can be traced by solving the Smoluchowski equation or the simplified population balance equation (PBE).The PBE, however, is an integral-differential equation, which cannot be solved analytically without any assumption (Lee et al., 1984).Particularly for agglomerates, the fractal theory must be employed to construct the governing equations; this inevitably increases the complexity of the equations.Therefore, numerical methods must be introduced in this research field.
Two main development directions exist in current research on PBEs.The first one is the application of PBEs in engineering (Buesser and Pratsinis, 2012;Buffo and Marchisio, 2014;Solsvik and Jakobsen, 2014).In the present study, the form of a governing PBE must be constructed on the basis of specific dynamic processes, which typically comprise internal processes such as coagulation, nucleation, condensation, breakage, and coalescence, and external processes including convection and diffusion transport and thermophoresis (Friedlander, 2000).Currently, the effect of microscale turbulence on the dynamic process in PBEs has received increasing attention (Clement, 2011;Garrick, 2011;Murfield and Garrick, 2013;Ramkrishna and Singh, 2014) because it is crucial for successfully coupling various turbulent models to resolve Navier-Stokes and PBEs.The second direction is the studying of the solution of PBEs by implementing various mathematical techniques.In the past 30 years, numerous scholars have attempted to solve the PBE.Three main numerical methods have been proposed and evaluated: the method of moments (Hulburt and Katz, 1964;Lee et al., 1984;McGraw, 1997;Frenklach, 2002;Marchisio et al., 2003;Park, 2003;Yu et al., 2008a), the sectional method (Landgrebe and Pratsinis, 1990), and the stochastic particle method or Monte Carlo method (Kraft, 2005;Morgan et al., 2006;Kruis et al., 2012;Zhao and Zheng, 2013).Both the advantages and disadvantages of these three methods were compared by Kraft (2005).Although some studies have sought to use analytical methods to solve the PBE (Otto et al., 1998;Yu et al., 2015), their application was considerably limited compared with numerical solutions.In particular, applying them to couple with computational fluid dynamic (CFD) models is inconvenient.
The method of moments has been extensively applied to solve most particulate problems because of its relative simplicity in implementation and low computational cost (Swift and Friedlander, 1964).One of the moment methodologies, the Taylor-series expansion method of moments (TEMOM) was proposed as a novel method in 2008.Subsequently, this method has shown potential in solving the PBE (Yu et al., 2008a).Since the groundbreaking work of Smoluchowski (1917), the PBE has become a basic equation for aerosol dynamics modeling.To date, the TEMOM has been widely used in research on aerosol dynamics and nanoparticle formation (Yu and Lin, 2010;Lin et al., 2012;Xie and He, 2013).In addition, numerous recent studies have applied the analytical solution for the PBE based on the TEMOM governing equation (Chen et al., 2014;Xie, 2014;Yu et al., 2014bYu et al., , 2015)).The TEMOM has no prior requirement regarding the particle size spectrum, which is similar to the quadrature method of moments (McGraw, 1997), and the number of governing moment equations is equal to the order of the Taylor-series expansion.Previous studies have revealed that maintaining three terms of the Taylor series is preferable when considering both precision and efficiency simultaneously (Yu et al., 2008b).However, whether the reliability of the model increases with the number of Taylor-series terms remains unclear.
To address this research gap, we derived a new TEMOM with a fourth-order Taylor expansion and compared its accuracy with that of the third-order Taylor expansion.To validate the TEMOM model, a sectional method was selected as a reference.In addition, in this work, the application scope of the new model was studied.

THEORY
In this section, we illustrate only the main derivation because the TEMOM with third order and fourth order have the same procedure for converting the PBE to the governing equations for moments.The integral-differential PBE was first proposed by (Müller, 1928) before it became the basic equation in aerosol research.The PBE can be expressed as where n(v,t)dv is the number of particles with volumes between v and v + dv at time t, and (v,v 1 ) is the collision frequency function between two particles of volumes v and v 1 .By multiplying v k in both sides of Eq. ( 1) simultaneously and then integrating them over the entire particle size distribution, the final transformed moment equation can be written as where the moment m k is defined as 0 ( ) The application of Eq. ( 3) is shown in APPENDIX A.

Coagulation in the Continuum-Slip Regime
The collision frequency function for agglomerates is shown as follows (Yu and Lin, 2009): where ϕ = λA/(3/4) 1/3 and f = 1/D f .Here, B 2 = 2k B T/3μ, k B is Boltzmann constant, T is the temperature, μ is the gas viscosity, λ is mean free path of the gas, A is a constant, 1.591, v p0 is the volume of primary particles, and D f is the mass fractal dimension characterizing particle morphology.Substituting Eq. ( 4) into Eq.( 2), the equations with respect to the first four moments m 0 , m 1 , m 2 , and m 3 can be derived as follows: where The right terms of Eq. ( 5) can be integrated using the transformed Eq. (3), as follows: In Eq. ( 6), some fractal moments such as m f , m -f , and m -2f can be replaced by functions of m 0 , m 1 , m 2 and m 3 .Accordingly, we must expand v k in Eq. ( 3) with a Taylor series around point v = u; finally, the closure model for moments with a fourth-order Taylor expansion is Consequently, the moment equation merely composed of integral moment variables can be easily obtained by substituting Eq. ( 7) into Eq.( 6): where  4 2 2 2  0 1 2 3  0 1 3  0 1  2   4  4  4 6  2 4 2  2 3  0 1  2  1  0 3  0 1 2 3   2 2 3  2 2 2 2  0 1 3  0 1 In the above derivation, the Taylor expansion point u is defined as u = m 1 /m 0 , which is the same as our previous model with third-order expansion (Yu et al., 2008).Eq. ( 8) is clearly a system of first-order ordinary differential equations, wherein the right terms are denoted by the first four moments m 0 , m 1 , m 2 , and m 3 .Therefore, we can easily solve the system.Obtaining the first three moments, which are crucial for describing aerosol dynamics, by solving the first-order ordinary differential system is reasonable.

Free Molecular Regime
Similar to nonspherical particles in the continuum-slip regime, the collision frequency function for agglomerates in the free molecular regime must be modified from the classic spherical collision frequency function, and can be expressed as follows: where , r p0 is the radius of the primary particle and ρ is the density of particles.The agglomerates would be coalescing spheres on the condition that f = 1/3, and thus Eq. ( 9) can be simplified to the equation for classical spherical particle coagulations.Substituting Eq. ( 9) into Eq.( 2) and performing a fourthorder Taylor-series expansion for those terms that cannot be integrated-that is, (1/v + 1/v 1 ) 1/2 -results in the following fourth-order set of moment equations (Yu et al., 2008): 12 2 9 2 3 2 2 7 2 5 2 2 5 7 2 2 3 2 1 2 9 2 3 2 7 2 5 2 5 2 6 2 4 3 6 24 12 30 where u is the expansion point of the Taylor-series expansion.Eq. ( 10) cannot be solved because the right terms comprise numerous fractional-order moments instead of integral moments.Consequently, the general closure model shown in Eq. ( 7) should be used to achieve closure of Eq. ( 10).In principle, the closure of Eq. ( 10) can be achieved easily by selecting the expansion point u = m 1 /m 0 and substituting the transformed equation into it.Here, the derivation of collision frequency function for nonspherical particles is absolutely identical to that for spheres in the previous works (Yu et al., 2008).

COMPUTATIONS
In the present study, we applied the same characteristic variables m k = M k m k0 with m k0 = Nv g0 k to normalize all of the aforementioned TEMOM equations.Here, N and v g0 are the initial total particle number and geometric mean volume, respectively.In addition, v p0 is specified as v g0 in the continuum-slip regime.The geometric mean volume is defined as (Pratsinis, 1988).The fourth-order Runge-Kutta method with a fixed time step was used to solve the set of ordinary differential equations.In this work, we used the SM model proposed by Landgrebe and Pratsinis (1990) as reference.The error function in the SM model was computed using the incomplete Gamma function method.For the sake of accuracy, the dimensionless time step selected for the TEMOM and SM models was 0.001.In all the cases, the surrounding air temperature and pressure were assumed to be 300 K and 1.103 × 10 5 Pa, respectively.The relative error for any variable was calculated using the following definition: where ϕ MM is the variable obtained from the TEMOM, and ϕ SM is the referenced SM variable.In the calculation, the initial dimensionless moments are set to 00 10 , and σ g0 is the initial aerosol geometric standard deviation (GSD).The program code of the SM used in this work was verified in a previous study, and is generally considered an accurate solution for solving the PBE (Yu and Lin, 2009;Yu et al., 2015).To enable the TEMOM to be compared with our SM model, the mass fractal dimension in our TEMOM code was assumed to be 3.It needs to note in this kind of specification of initial values, three key parameters including the total particle number concentration, the geometric standard deviation and geometric mean volume should be firstly specified, which exactly correspond to log-normal distributions.

RESULTS
In principle, the GSD of aerosols can be an arbitrary value that is equal to or larger than 1.0000.However, the existing TEMOM with a third-order Taylor expansion is only valid from 1.0000 to 1.6583 for a GSD in the free molecular regime, and from 1.0000 to 1.6990 for a GSD in the continuum regime (Yu et al., 2014a).Therefore, the application scope of the TEMOM with a higher order, such as the fourth order, as well as the accuracy of the TEMOM with a fourth-order Taylor expansion should be evaluated.The accuracy of the TEMOM can be characterized by the relative errors between it and the referenced SM solution for three essential quantities, namely, M 0 , σ g , and v g , representing the total particle number concentration, GSD of the number distribution, and geometric mean volume, respectively.

Continuum-Slip Regime
In the continuum-slip regime, we found that the application scope of the TEMOM with a fourth-order Taylor expansion is highly dependent on two crucial quantities, namely Kn and σ g0 .On the basis of a trial numerical simulation, we determined the application scope of the TEMOM with a fourth-order Taylor expansion (Table 1).Increasing Kn causes a decrease in the limit of σ g0 .Because Kn is 0.5000, the TEMOM with a fourth-order Taylor expansion fails to yield a reasonable result, indicating that it must be used when Kn is less than 0.5000.Here, it should be pointed out that the TEMOM is considered to be successful as the relative errors of it to the referenced SM for the zero-th moment, the GSD and geometric mean volume are all less than 5.0%.Regarding the TEMOM with a third-order Taylor expansion in the continuum regime, its application scope is confined from 1.0000 to 1.6583 (Yu et al., 2015), whereas that of the TEMOM with a fourth-order Taylor expansion proposed in this work is from 1.0000 to 1.3350.This indicates that compared with the initial TEMOM with a third-order Taylor expansion, the application of the newly derived TEMOM with a fourth-order Taylor expansion is considerably limited.It needs to be pointed out the valid range of the fourth-order model comes from a trial numerical simulation, whereas that of the third-order model comes from a theoretical analysis.
To evaluate the accuracy of the TEMOM as its orders are increased, we selected several representative cases for comparison.Four aerosols with initial (Kn, σ g0 ) values of (1.3350, 0.0001), (1.3314, 0.0110), (1.3190, 0.0500), and (1.3070, 0.0900) were selected and investigated as representative cases.The selection of these aerosols was based on their characteristic quantities (i.e., Kn and σ g0 , corresponding to the limit of application scope) shown in Table 1.The accuracy of the TEMOM is the lowest within these limits (Yu et al., 2015).Fig. 1 shows the results of the relative errors of M 0 , σ g , and v g of the TEMOM with a third-order and fourth-order Taylor expansion compared with the referenced SM model.The figure shows that the relative errors of M 0 of the TEMOM model with third-order Taylor expansion are always small.The TEMOM with a fourth-order Taylor expansion, however, yielded relatively large errors.Similar to M 0 , the TEMOM with a fourthorder Taylor expansion yielded relatively larger errors compared with the TEMOM with a third-order Taylor expansion, indicating that the initial TEMOM with a thirdorder Taylor expansion was considerably more accurate than the other models for solving the PBE.Moreover, the TEMOM with a third-order Taylor expansion has a larger application scope according to the GSD.Thus, including higher-order moments (i.e., M 3 in the governing equations for moments) results in large errors in the final results.To improve the accuracy of the three essential quantities investigated, constructing a general closure model composed of polynomials for moments with orders less than 2 would be an ideal approach (Yu and Lin, 2015).

Free Molecular Regime
Unlike in the continuum-slip regime, the property of initial size distribution is dependent only on the GSD in the free molecular regime.We found that both the accuracy and application scope of the newly derived TEMOM with a fourth-order Taylor expansion were strongly affected by this quantity in this regime.Similar to the study in the continuum-slip regime, we used the trial numerical method to determine the application scope of the TEMOM with a fourth-order Taylor expansion, and finally found that it should be (1.000,1.2660), beyond which the TEMOM with a fourth-order Taylor expansion would fail.
To verify the accuracy of the TEMOM with a fourthorder Taylor expansion in the free molecular regime, two cases with an initial GSD of 1.2000 and 1.2300 were selected and investigated.Fig. 2 shows the results of the relative errors of M 0 , σ g , and v g for the TEMOM with a fourthorder Taylor expansion and third-order Taylor expansion compared with the referenced SM model.The figure indicates that the variation range of the relative error of M 0 of the TEMOM model with a third-order Taylor expansion is extremely small and close to 0, but the relative error of M 0 of the TEMOM model with a fourth-order Taylor expansion is higher than 1%.For σ g , the relative errors of the TEMOM with a fourth-order Taylor expansion and the TEMOM with a third-order Taylor expansion approach 0.05%.Regarding v g , the relative error of the TEMOM with a third-order Taylor expansion reaches 0.8%, whereas the relative error of the TEMOM with a fourth-order Taylor expansion is close to 0. Overall, the TEMOM with a third-order Taylor expansion model was regarded as the more accurate model in this regime.

DISCUSSIONS
In this work, we only focused on a specific topic although it seems like unimportant as the model itself, i.e., whether the accuracy of the TEMOM increases with an increase of reserved orders of Taylor expansion series.This is because for the TEMOM model, two critical issues we have to concern, namely how many orders of Taylor expansion series is the best suitable for solving PBE, and what kind of Taylor expansion point should be chosen.In this work, we definitely confirmed the accuracy of the TEMOM cannot increase with an increase of reserved orders of Taylor expansion series.This is because higher order of Taylor expansion leads to much complicated mathematical form of ordinary differential equations, and also more numbers of equation.
The reconstruct of the size distribution from several lowerorder moments remains a challenging issue now.For the TEMOM, we verified the reconstruction of size distribution can be achieved if we assume the size distribution follows a log-normal size distribution (Yu et al., 2008).Our study reported several critical quantities exhibiting size distribution including geometric standard deviation and geometric mean volume are nearly the same for the TEMOM with threeorder Taylor expansion and the log normal method of moments proposed by Lee et al. (1984).In the valid range of the TEMOM with fourth-order Taylor expansion, it is no doubt the reconstruction of size distribution can be also achieved, although the TEMOM with fourth-order Taylor expansion is not recommended.
The advantage of the third-order Taylor expansion model over the fourth-order one attributes to three aspects.Firstly, based on the theory of TEMOM model, the number of governing equations for moments must be the same as the order of Taylor series expansion.In the fourth-order model derived in this work, there is an additional equation as compared to the third-order model, which definitely increases the complex of governing equations; Secondly, in the transfer from the original population balance equation to the governing equations for moments, some fractal moments have to be replaced with approximated models, i.e., Eq. ( 7) for the fourth-order Taylor expansion in this work and Eq. ( 13) for the third-order Taylor expansion in our previous work (Yu et al., 2008a).The fourth-order approximated model inevitably leads to much complicated governing equations for moments due to its more terms.In addition, in the approximated model for any order moment shown in Eq. ( 7), it is obvious the lower moments, such as the zero-th, first and second moments, have to involve the information of higher order moments, such as the third moments, which also reduces the accuracy of the model.In conclusion, the derived model with fourth-order Taylor expansion is much more complicated and prone to errors, thus it is not recommend to be used.

CONCLUSIONS
We first derived a TEMOM model with a fourth-order Taylor expansion, of which the accuracy and application  scope were studied in both the free molecular and continuum regimes.Because the sectional method was selected as a reference, we compared the accuracy of the initial TEMOM with a third-order Taylor expansion and the newly derived TEMOM with a fourth-order Taylor expansion, and found that the initial TEMOM was considerably more accurate.We also demonstrated that the application scope of the newly derived TEMOM with a fourth-order Taylor expansion highly depends on two essential quantities, namely Kn and σ g0 , and is considerably more limited.Finally, we verified that the initial TEMOM with a third-order Taylor expansion is the most reliable model for solving PBEs for agglomerates.

Fig. 1 .
Fig. 1.The comparison of relative errors of M0 , σ g and v g between the TEMOM with fourth-order and third-order Taylor expansion to the referenced SM model in the

Fig. 2 .
Fig. 2.The comparison of relative errors of M 0 , σ g and v g , between the TEMOM with fourth-order and third-order Taylor expansion to the referenced SM model in the free molecular regime.

Table 1 .
The investigated cases with varying geometric standard deviation and Knudsen number.