A New Method of Moments for the Bimodal Particle System in the Stokes Regime

Abstract

The current paper studied the particle system in the Stokes regime with a bimodal distribution. In such a system, the particles tend to congregate around two major sizes. In order to investigate this system, the conventional method of moments (MOM) should be extended to include the interaction between different particle clusters. The closure problem for MOM arises and can be solved by a multipoint Taylor-expansion technique. The exact expression is deduced to include the size effect between different particle clusters. The collision effects between different modals could also be modeled. The new model was simply tested and proved to be effective to treat the bimodal system. The results showed that, for single-modal particle system, the results from new model were the same as those from TEMOM. However, for the bimodal particle system, there was a distinct difference between the two models, especially for the zero-order moment. The current model generated fewer particles than TEMOM. The maximum deviation reached about 15% for 𝑚₀ and 4% for 𝑚₂. The detailed distribution of each submodal could also be investigated through current model.

1. Introduction

The particulate matter has become one of the most dangerous pollutants to the atmospheric environment and the health of human beings. It will reduce the visibility of the atmosphere and cause the traffic crowding and serious accidents. The fine particles (PM2.5) will also be breathed into the bronchus of human beings, followed by several kinds of respiratory diseases. The lungs will absorb the fine particles and cardiovascular disease will come into being [1]. However, the mechanism of the generation and evolution of the particulate matter still remains to be clarified. Hence, it has both theoretical and realistic senses to study the dynamics of the particulate matter.

Previous study on the aerosol dynamics usually supposes that the particle system is monodispersed (i.e., the system has only one scale) or multidispersed (i.e., the system has multiscales) but is in a log-normal distribution in size [2]. Such kinds of assumptions will greatly simplify the problems, and a series of approximate or precise solutions will be obtained. However, these assumptions are based on the experimental measurement and cannot be applied to all the cases. There is another type of particle size distribution, namely, bimodal or multimodal distribution. For example, the newborn particles together with the background particles compose the bimodal distribution system. Furthermore, the newborn particles may also exhibit a multimodal or bimodal distribution [3]. Pugatshova et al. [4] and Lonati et al. [5] measured the particulate matter in the urban on-road atmosphere in different cities and times. The multimodal distribution was observed. At this time, unacceptable error may appear using mono-dispersed or log-normal assumption.

Take the bimodal system, for example: the particles gather around two independent particle sizes. In order to study such a system, the particle size distribution should be separated into two sub-PSDs [6]. The dynamics of the system may be obtained according to the two subparticle clusters. Under this description, the governing equations of the particle system should be modified to represent the additional coagulation effect [7]; that is, the collision of particles is artificially separated into two kinds: internal coagulation and external coagulation. Because the typical particle diameter of the bimodal system is 5 nm to 2.5 μm, which means that particles lie in different dynamic regimes (free molecular regime, transition regime and continuum regime), the coagulation in such a wide range should also be treated separately. The current study will focus on the continuum (Stokes) regime.

Generally, the particle balance equation (PBE) governs the detailed evolution process of PSD and can be numerically solved. However, because of its huge computation resource to solve the PBE directly, the method of moment (MOM) [2, 8, 9] is often taken into account as an alternation. It takes several moments of PSD in particle volume space and converts PBE into moment equations. Each moment has its physical meaning: zero-order moment represents the number concentration, first-order moment represents the volume concentration, and second-order moment is related to the polydispersity. Although MOM cannot directly give out the evolution of specific PSD, it can obtain the statistical characteristics of particle system and the calculation during this procedure reduces to an acceptable level. As a matter of fact, MOM is widely used in the research of aerosol dynamics for its simplicity and low computational cost.

One limitation of MOM is the closure problem due to the coagulation term in PBE. When PBE is converted into moment equations, the coagulation term will be transformed into fractional moments, which cannot be explicitly expressed and mathematical models should be introduced into MOM to solve this problem, the so-called closure problem. There are typically three kinds of methods: predetermined PSD [2], quadrature method of moment (QMOM) [10], and Taylor-expansion method of moment (TEMOM) [11]. The first class often supposes that the PSD is a log-normal distribution, and the coagulation term can be directly determined. It can only be applied to the log-normal distributed particle system. QMOM utilizes the Gaussian quadrature method to evaluate the coagulation term in the moment equations. The pre-determined PSD is not necessary, but the computation is easy to diverge. TEMOM expands the nonlinear term in the collision kernel using the Taylor expansion. Finally, the coagulation term can be expressed as a linear combination of different moments. TEMOM has its superiority on its easy expression, high precision, and low computational cost. It is widely used in the research of the aerosol dynamics [12–15].

However, using TEMOM to study the bimodal system has some problems. TEMOM expands the collision kernel function at the average diameter 𝑢₀. For the internal collision, there is no problem, but, for the external collision in bimodal system, this expansion should be extended. For a typical bimodal system, there are two clusters of particles with different diameters, and the total numbers of particles in each cluster are also different. This fact will contribute to the fact that the average diameter of the whole system may lie around the first mode or the second mode or even the mid place of the two modes. If the external collision term also expands at the average diameter of the system, additional error will decrease the accuracy of the simulation. In TEMOM, the convergent region is (0, 2𝑢₀) [16], while, for bimodal system, one mode may lie outside this region if both modes expand at the same point. This possibility may lead to the divergence of the calculation.

Hence, the Taylor-expansion method of moments should be developed to be applied to the bimodal particle system to improve the accuracy and the stability. The current research will focus on the multipoint Taylor-expansion method of moments, and the Stokes regime is preferred for ease.

2. Mathematical Theories

Considering the typical system with Brownian coagulation only, PSD satisfies the PBE as [2] 𝜕𝑁(𝑣)=1𝜕𝑡2^𝑣₀𝛽()𝑁(𝑢)𝑁()()𝑢,𝑣−𝑢𝑣−𝑢𝑑𝑢−𝑁𝑣,𝑡^∞₀𝛽(𝑣,𝑢)𝑁(𝑢)𝑑𝑢, (1) where 𝑁(𝑣) is the size distribution function, which means the number of particles with a volume 𝑣, 𝑢 and 𝑣 are the particle volumes, and 𝛽 is the Brownian coagulation coefficient.

In order to convert PBE into moment equations, the definition of moments is introduced 𝑚_𝑘(𝑡)=^∞₀𝑁(𝑣,𝑡)𝑣^𝑘𝑑𝑣𝑎. (2) Applying (2) to (1), the moment equations are obtained: 𝜕𝑚_𝑘=1𝜕𝑡2^∞₀𝑣₁+𝑣₂^𝑘−𝑣^𝑘₁−𝑣^𝑘₂×𝛽𝑣₁,𝑣₂𝑁𝑣₁𝑁𝑣₂𝑑𝑣₁𝑑𝑣₂. (3)

In current paper, the Stokes regime is studied, and the collision kernel 𝛽 may be rewritten as 𝛽_c𝑣₁,𝑣₂𝑣=𝐵2+²𝑣₁^1/3+𝑣₁𝑣₂^1/3, (4) where 𝐵=2𝑘_𝑏𝑇/𝜇,𝑘_𝑏 is the Boltzmann constant, 𝑇 is the environment temperature, and 𝜇 is the molecular viscosity of gas.

When investigating the bimodal system, PSD can be expressed as 𝑁(𝑣,𝑡)=𝑁_𝑖(𝑣,𝑡)+𝑁_𝑗(𝑣,𝑡). PBE for each sub-PSD can be established. Apply the definition equation (2) to the PBEs. The moment equations can be attained for both cluster 𝑖 and cluster𝑗 listed as follows: 𝜕𝑚^𝑖_𝑘𝜕𝑡=𝐶_𝑘^𝑖𝑖+𝐷_𝑘^𝑖𝑗,𝜕𝑚^𝑗_𝑘𝜕𝑡=𝐶_𝑘^𝑗𝑗+𝐸_𝑘^𝑖𝑗, (5) where 𝐶_𝑘^𝑖𝑖=12^∞₀(𝑢+𝑣)^𝑘−𝑢^𝑘−𝑣^𝑘×𝛽(𝑢,𝑣)𝑁_𝑖(𝑢)𝑁_𝑖(𝑣𝐷)𝑑𝑢𝑑𝑣,_𝑘𝑖𝑗=−^∞₀𝑢^𝑘𝛽(𝑢,𝑣)𝑁_𝑖(𝑢)𝑁_𝑗(𝑣)𝐸𝑑𝑢𝑑𝑣,_𝑘𝑖𝑗=^∞₀(𝑢+𝑣)^𝑘−𝑣^𝑘×𝛽(𝑢,𝑣)𝑁_𝑖(𝑢)𝑁_𝑗(𝑣)𝑑𝑢𝑑𝑣. (6)(7)(8)

Note that 𝐶_𝑘^𝑖𝑖 and 𝐶_𝑘^𝑗𝑗 are only related to 𝑁_𝑖 or 𝑁_𝑗. These two terms represent the internal coagulation effect in the cluster 𝑖 or 𝑗. As a result, the single point binary Taylor expansion is used to deal with these two terms (at 𝑢₁ or 𝑢₂). The results from the typical TEMOM can be directly used. In (9) 𝑚_𝑘 represents 𝑘th moment of PSD 𝑁_𝑖 or 𝑁_𝑗. Consider 𝐶₀^𝑖𝑖=𝐵𝑚₀^𝑖281𝑚₁^𝑖2−151𝑚₁^𝑖4−2𝑚₂^𝑖2𝑚₀^𝑖2−13𝑚^𝑖₂𝑚₁^𝑖2𝑚^𝑖₀,𝐶₁^𝑖𝑖𝐶=0,_2𝑖𝑖=−2𝐵₁81𝑚₁^𝑖22𝑚₂^𝑖2−13𝑚^𝑖₂𝑚₁^𝑖2𝑚^𝑖₀−151𝑚₁^𝑖4. (9)

The approximation of 𝐷_𝑘^𝑖𝑗 and 𝐸_𝑘^𝑖𝑗 will be deduced in the following part. Substitute (4) into (7) and (8). A lot of fractional moments will appear in the expression of 𝐷_𝑘^𝑖𝑗 and 𝐸_𝑘^𝑖𝑗, which can be approximated through the Taylor expansion of 𝑣^𝑝 (𝑝 is fraction) at 𝑢₁ or 𝑢₂. Consider 𝑚_𝑝≈𝑢^𝑝−2𝑝²−𝑝2𝑚₂−𝑢^𝑝−1𝑝²−2𝑝𝑚₁+𝑢^𝑝2𝑝²−3𝑝+2𝑚₀. (10) Making use of (10), 𝐷_𝑘^𝑖𝑗 and 𝐸_𝑘^𝑖𝑗 can be expressed as a linear combination of 𝑚^𝑖_𝑘 and 𝑚^𝑗_𝑘. Moreover 𝐷₀=−∑𝑎_𝑚𝑛𝑚^𝑖_𝑚𝑚^𝑗_𝑛81,𝐷₁=∑𝑏_𝑚𝑛𝑚^𝑖_𝑚𝑚^𝑗_𝑛,𝐷81₂=∑𝑐_𝑚𝑛𝑚^𝑖_𝑚𝑚^𝑗_𝑛81,𝐸₀𝐸=0,₁=∑𝑑_𝑚𝑛𝑚^𝑖_𝑚𝑚^𝑗_𝑛81,𝐸₂=−∑𝑒_𝑚𝑛𝑚^𝑖_𝑚𝑚^𝑗_𝑛.81 (11)

The exact expressions of the coefficients in 𝐷₀, 𝐷₁, 𝐷₂, 𝐸₁, and 𝐸₂ are listed in the appendix.

3. Tests and Discussion

In order to verify the deduction, both theoretical and numerical validations are performed, respectively.

Note that, if 𝑁_𝑖=𝑁_𝑗=𝑁/2, (5) turns into two sets of moment equations with monomodal distribution. If (5) and set 𝑚_𝑘=𝑚^𝑖_𝑘+𝑚^𝑗_𝑘, the theoretical systematic moment equations are attained: 𝜕𝑚_𝑘𝜕𝑡=4𝐶_𝑘^𝑖𝑖. (12)

Substitute 𝐷₀,𝐷₁,𝐷₂,𝐸₁, and 𝐸₂ into (5), and set 𝑢₁=𝑢₂=𝑚₁/𝑚₀, 𝑟=1. The right side of new equation just equals 4 times of (9), which is consistent with the theoretical equation (12).

Two simple bimodal systems are simulated to validate the current model. The single point and multipoint expansion methods are both taken into account and the results are compared with each other to show the validity and accuracy. The initial size distributions both satisfy the log-normal distribution as follows: 𝑁(𝑣,𝑡)=𝑁₀−expln²𝑣/𝑣_𝑔/2𝑤²_𝑔√2𝜋𝑣𝑤_𝑔. (13)

For Case I, 𝑁^𝑖₀=𝑁^𝑗₀=1.0, 𝑣^𝑖_𝑔=𝑣^𝑗_𝑔=√3/2, and𝑤^𝑖_𝑔=𝑤^𝑗_𝑔=ln(4/3) [8], which represents a monomodal system and the PSD is separated into two equal sub-PSDs. For Case II, 𝑁^𝑖₀=1.0, 𝑣^𝑖_𝑔=√3/2 and 𝑤^𝑖_𝑔=ln(4/3) and 𝑁^𝑗₀=0.1𝑁^𝑖₀, 𝑣^𝑗_𝑔=1000𝑣^𝑖_𝑔, and 𝑤^𝑗_𝑔=0.1𝑤^𝑖_𝑔, which represents a bimodal system consisting of two log-normal sub-PSDs.

Figure 1 shows the results of Case I for both single point TEMOM and multipoint TEMOM. From the figure, a good agreement is obtained. This is because the particle system is, in the final analysis, a mono-modal system. The consistency between two methods is just as the same as the theoretical analysis at the beginning of this paragraph.

Figure 1 

The evolution of moments for Case I using different expansion schemes.

Figure 2 shows the results of Case II for both single point TEMOM and multi point TEMOM. From the figure, an obvious deviation is found. It shows that, for a typical bimodal system, the particle size difference between different models can not be neglected. The value for multipoint TEMOM is always smaller than that for TEMOM especially for 𝑚₀. This means that the original TEMOM model will underestimate the coagulation effect for the particle number concentration (𝑚₀). Another interesting phenomenon is that 𝑚₁ is the same for both of the two models. The reason is that 𝑚₁ physically represents the volume fraction of particles. The particle collision (coagulation) will not change the total volume or the mass of particles. Hence, 𝑚₁ is a constant from the beginning to the end.

Figure 2 

The evolution of moments for Case II using different expansion schemes.

Define the error function as 𝐸_𝑘=𝑚^m_𝑘−𝑚^s_𝑘𝑚^s_𝑘, (14) Where 𝑚^m_𝑘 represents the moments in multi-point TEMOM and 𝑚^s_𝑘 represents the moments in original TEMOM. The exact tendency of 𝐸_𝑘 is shown in Figure 3. According to the figure, the maximum deviation for 𝑚₀ will be about 15% and 4% for 𝑚₂. For 𝑚₀, the error function 𝐸₀ will increase in a very short time, reach the maximum, and then decrease slowly. This phenomenon indicates that the difference in particle size will lead to a relatively large deviation at the very beginning of coagulation for bimodal particle system when the TEMOM is selected for the bimodal particle system.

Figure 3 

The variation of error function 𝐸𝑘 versus dimensionless time 𝜏.

Figure 4 shows the different moments in modes 𝑖 and 𝑗 using the technique proposed in current paper. According to the figure, an obvious reduction is found for each moment 𝑚_𝑘 in mode 𝑖, which means that the coagulation will lead to the decrease of 𝑚₀ (particle number concentration) and 𝑚₁ (particle volume fraction). Particularly the volume fraction of particles, 𝑚₁, no longer keeps a constant because of the external collision with particles in mode 𝑗 and the new birth of bigger particles. For particles in mode 𝑗, the internal coagulation in mode 𝑗 will lead to the decrease of 𝑚₀, while the external coagulation between mode 𝑖 and mode 𝑗 will take no effect on 𝑚₀. As a result, the slope of curve is flatter than that in Figure 2. However, 𝑚₁ and 𝑚₂ are comparable with those in Figure 2, because these two parameters are related to the particle volume tightly. The average volume of particle in mode 𝑗 is much bigger than that in mode 𝑖, according to the initial condition. In general, such a result indicates the importance of current technique, giving more accurate result and more detail for the complex bimodal particle system.

Figure 4 

The evolution of moments for Case II with different modes.

4. Conclusions

The current research showed a multipoint Taylor-expansion method of moments for the bimodal particle system in the Stokes regime. A theoretical deduction was performed and brief results are given. Both theoretical validation and numerical tests are implemented. The results show that, for a single-modal system, there is no difference between the two methods. However, for a bimodal system, although the evolution of moments has the same tendency, there is obvious deviation between the two methods. For the case investigated in current paper, the maximum deviation for 𝑚₀ is about 15% and 4% for 𝑚₂. Each moment 𝑚_𝑘 in mode 𝑖 will decrease. The technique proposed in this paper will bring in the accuracy and details of particles. This method can be further extended to the multi-modal system

Appendix

The coefficients in (11) are listed with the definition 𝑟=(𝑢₁/𝑢₂)^1/3. Consider𝑎₀₀=70𝑟+70𝑟⁻¹+162,𝑎₀₁=35𝑢₁⁻¹2𝑟²−𝑟⁴,𝑎₀₂=𝑢₁⁻²10𝑟⁷−14𝑟⁵,𝑎₁₀=35𝑢₁⁻¹2𝑟−𝑟⁻¹,𝑎₁₁=−35𝑢₁⁻²𝑟⁴+𝑟²,𝑎₁₂=𝑢₁⁻³10𝑟⁷+7𝑟⁵,𝑎₂₀=𝑢₁⁻²10𝑟⁻¹−14𝑟,𝑎₂₁=𝑢₁⁻³7𝑟⁴+10𝑟²,𝑎₂₂=−2𝑢₁⁻⁴𝑟⁷+𝑟⁵;𝑏₀₀=𝑢₁14𝑟−10𝑟⁻¹,𝑏₀₁=−7𝑟⁴−10𝑟²,𝑏₀₂=2𝑢₁⁻¹𝑟⁷+𝑟⁵,𝑏₁₀=−112𝑟−40𝑟⁻¹𝑏−162,₁₁=𝑢₁⁻¹56𝑟⁴−40𝑟²,𝑏₁₂=8𝑢₁⁻²𝑟⁵−2𝑟⁷,𝑏₂₀=𝑢₁⁻¹5𝑟⁻¹−28𝑟,𝑏₂₁=𝑢₁⁻²14𝑟⁴+5𝑟²,𝑏₂₂=−𝑢₁⁻³4𝑟⁷+𝑟⁵;𝑐₀₀=𝑢²₁5𝑟⁻¹−28𝑟,𝑐₀₁=𝑢₁14𝑟⁴+5𝑟²,𝑐₀₂=−4𝑟⁷−𝑟⁵,𝑐₁₀=𝑢₁98𝑟−25𝑟⁻¹,𝑐₁₁=−49𝑟⁴−25𝑟²,𝑐₁₂=𝑢₁⁻¹14𝑟⁷+5𝑟⁵,𝑐₂₀=−196𝑟−25𝑟⁻¹−162,𝑐₂₁=𝑢₁⁻¹98𝑟⁴−25𝑟²,𝑐₂₂=𝑢₁⁻²5𝑟⁵−28𝑟⁷;𝑑₀₀=𝑢₁10𝑟⁻¹−14𝑟,𝑑₀₁=7𝑟⁴+10𝑟²,𝑑₀₂=−2𝑢₁⁻¹𝑟⁷+𝑟⁵,𝑑₁₀=112𝑟+40𝑟⁻¹𝑑+162,₁₁=𝑢₁⁻¹40𝑟²−56𝑟⁴,𝑑₁₂=8𝑢₁⁻²2𝑟⁷−𝑟⁵,𝑑₂₀=𝑢₁⁻¹28𝑟−5𝑟⁻¹,𝑑₂₁=−𝑢₁⁻²14𝑟⁴+5𝑟²,𝑑₂₂=𝑢₁⁻³4𝑟⁷+𝑟⁵;𝑒₀₀=𝑢²₁−28𝑟+5𝑟⁻¹+4𝑟⁻²+4𝑟⁻⁴,𝑒₀₁=𝑢₁14𝑟⁴+5𝑟²+16𝑟−32𝑟⁻¹,𝑒₀₂=−4𝑟⁷−𝑟⁵−2𝑟⁴−8𝑟²,𝑒₁₀=𝑢₁98𝑟−25𝑟⁻¹−32𝑟⁻²+16𝑟⁻⁴,𝑒₁₁=−49𝑟⁴−25𝑟²−128𝑟−128𝑟⁻¹𝑒−324,₁₂=𝑢₁⁻¹14𝑟⁷+5𝑟⁵+16𝑟⁴−32𝑟²,𝑒₂₀=−196𝑟−25𝑟⁻¹−8𝑟⁻²−2𝑟⁻⁴𝑒−162,₂₁=𝑢₁⁻¹98𝑟⁴−25𝑟²−32𝑟+16𝑟⁻¹,𝑒₂₂=𝑢₁⁻²−28𝑟⁷+5𝑟⁵+4𝑟⁴+4𝑟². (A.1)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors gratefully acknowledges the financial support from the National Natural Science Foundation of China under Grant no. 11302070, the National Basic Research Program of China (973 Program) under Grant no. 2010CB227102.