# Local Fixed Pivot Quadrature Method of Moments for Solution of Population Balance Equation

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

_{min}, V

_{max}]. Using a finite domain to represent an infinite domain will inevitably introduce errors, which are referred to as finite domain errors. As a result, the moments cannot be tracked exactly. This is inherent in almost all DMs. In this sense, SQMOM is a kind of DM more than MOM. (2) If a new born particle with a volume smaller than V

_{min}or larger than V

_{max}is introduced by a certain event (aggregation or breakage), the event will be omitted. Therefore, the new method may influence the physical models in some extreme situations. (3) In SQMOM, the product-different (PD) algorithm [17] must be employed to obtain the abscissas and the weights in each section when a relatively large number of moments (≥6) need to be tracked. However, in PD, the values of abscissas are not restricted to the section they should belong to in mathematics. Hence, with the PD algorithm, the abscissas obtained might be outside the section they originally belong to. Until now, there is no effective solution to such a problem. Thus, it is almost impossible to track many moments with SQMOM.

## 2. Population Balance Equation

**x**, t) is the number density function, describing the particle number distribution in property space referred to as internal coordinates, location, and time space referred as external coordinates, V is the particle volume,

**x**is space location, t is time, and <

**u**

_{i}>

_{V}is the particle velocity in the ith direction conditioned on volume V. S(V; t) describes the micro-behaviors of the particle swarm. In this work, we focus our attention on the solution method for the population balance equation for tracking particle size distribution and its moment due to aggregation and breakage at a given location

**x**. To shorten the symbolic expression, space coordinate

**x**is neglected in the discussion below.

#### 2.1. Sectional Representation for Population Balance Equation

_{i}> is the average velocity. The term on the right-hand side is the moment transformation for the micro-behaviors, aggregation, and breakage, for instance, influencing the evolution of m(

**x**, t) in state space. Before deriving the actual expression of the source term, we introduce a switch function with a Boolean parameter.

#### 2.2. Aggregation

**A**, particles with volume smaller than a;

**B**, particles with volume between a and b; and

**C**, particles with volume larger than b.

**A**+

**A**→

**B:**Particles with volumes lower than a aggregate to form particles with volume in [a, b). The net rate for the variation in m(t) due to this contribution is given by:

**A**+

**B**→

**B:**Particles with volume lower than a aggregate with particles with volume in [a, b) to form particles smaller than b. This contribution is given by:

**A**+

**B**→

**C:**Particles with volume lower than a aggregate with particles with volume in [a, b) to form particles larger than b. This will cause a net loss for m(t) which can be written:

**B**+

**B**→

**B:**Particles with volume in [a, b) aggregate to form particles in the same range. This contribution can be written as:

**B**+

**B**→

**C:**Particles with volume in [a, b) aggregate to form particles larger than b and this contribution is given by:

**B**+

**C**→

**C:**Particles from the domain [a, b) aggregate with particles larger than b to form particles larger than b, which causes a net loss for m(t) in [a, b), and can be written as:

#### 2.3. Breakage

**C**→

**A**+

**B + C:**Particles with volume larger than b break up to form the particles in the domain [a, b). This contribution is given by:

**B**→

**B**+

**C:**Particles in the domain [a, b) break up to form the particles in the same range. This part can be written as:

## 3. Local Fixed Pivot Quadrature Method of Moments

_{i}(t) are the weights; V

_{i}are the quadrature nodes (pivots); n is the number of the quadrature nodes; and δ(V) is the Dirac Delta function.

_{1}), [υ

_{1}, υ

_{2}), …, [υ

_{i}, υ

_{i}

_{+1}), …, [υ

_{N}

_{−1}, ∞), N sections. Note that the section in the present work was equivalent to the cell (or bin) in the discretization methods and resembled the window in STFT. υ

_{i}are the abscissas of the section ends, i.e., nodes. In this work, monomials were adopted as property function, i.e., ψ(V) = V

^{k}(k = 0, 1, 2, …, n − 1). In the ith section, the moment equation (Equation (5)) becomes

_{i}is the number of the pivots in section i.

_{i}− 1 are adopted

_{i}, υ

_{i}

_{+1}) as long as each location has a unique value. It is worth pointing out that the main concern here is the numerical accuracy of the quadrature over [υ

_{i}, υ

_{i}

_{+1}) or [υ

_{n}

_{−1}, ∞). It is well known that numerical integration of arbitrary smooth functions over a bounded interval is most accurate when carried out with quadrature points chosen as zeros of orthogonal polynomials. Several classical orthogonal polynomials are available [19] such as Gauss-Legendre over (−1, 1) with weighted function W(x) = 1, Gauss-Chebyshev over (−1, 1) with weighted function W(x) = (1 − x

^{2})

^{−1/2}, Gauss-Laguerre over (0, ∞) with weighted function W(x) = x

^{α}e

^{−x}(α > −1), and Gauss-Jacobi over (−1, 1) with weighted function (1 − x)

^{α}(1 + x)

^{β}. The parameters, α in Gauss-Laguerre and α and β in Gauss-Jacobi, controlling the actual form of the polynomials can be used to optimize the distribution of the particle ensembles according to the actual distribution. Note that the orthogonal domains of Gauss-Legendre, Gauss-Chebyshev, and Gauss-Jocobi are not [υ

_{i}, υ

_{i}

_{+1}). Thus, a transformation is required before these polynomials can be used in the simulation. The following relation can serve this purpose if υ

_{i+}

_{1}≠ ∞

_{i}, υ

_{i}

_{+1}). The zeros of the orthogonal polynomial of Gauss-Laguerre with α = 0, are taken as the locations of pivots after transforming t in [0, ∞) to s in [υ

_{N}

_{−1}, ∞) using a simple linear relation s = t + υ

_{N}

_{−1}.

_{0}= 0 and υ

_{N}= ∞. s is the scale factor.

#### 3.1. Aggregation

_{k}in each section. In this work, monomials were adopted as the property functions, i.e., ψ(V) = V

^{k}(k = 0, 1, 2, …, n − 1). In the lth (l = 0, 1, …, N − 1) section, after introducing the approximation of Equation (16) and some manipulations, Equation (8) yields

_{i}is the count of the pivots in section i; u

_{ij}are the abscissas of jth pivot in section i; and ω

_{ij}is the weight of jth pivot in section i.

#### 3.2. Breakage

_{k}(t) in the pure breakage process. In the lth (l = 0, 1, …, N − 1) section, after introducing the approximation of Equation (16) and some manipulations, Equation (14) yields

_{k}(t) due to the breakage of particles is the summation of Equations (29) and (30).

#### 3.3. Reconstruction of the Moments

#### 3.4. Reconstruction of the Particle Size Distribution

## 4. Test Cases

_{1}= 0.01 and s = 1.2. In each section the six pivots were specified (i.e., six moments were tracked). A time step of 0.01 s was adopted. All simulations in this work were carried out on a desktop computer with Intel Core i5 CPU (3.1 GHz) and 8 G memory.

#### 4.1. Pure Aggregation

_{0}= 1, together with exponential initial particle size distribution

_{0}= 1, V

_{0}= 1. In this test case, an analytical solution was available [20]

_{0}N

_{0}t/(2 + C

_{0}N

_{0}t). The moments could be evaluated analytically. Figure 3 depicts the evolution of the percentage errors for the first six moments with: (A) LFPQMOM; (B) QMOM; and (C) FPQMOM. In the figure, the evolution for the relative errors for all the moments were similar in both methods, but a little larger than FPQMOM. Errors increased rapidly initially and leveled off after 2 s. The maximum percentage error with QMOM was 1.12 × 10

^{−9}%, whereas with the new method, it was 1.10 × 10

^{−9}%, a small decrease. FPQMOM had the highest accuracy. Higher rank moments had larger percentage errors with both methods. It is worth pointing out that, with LFPQMOM, the relative errors for m

_{1}were always zero within the simulation time, but with QMOM, they were lower than 10

^{−12}%, but larger than zero. With the new method, m

_{1}could be predicted without any error. In the figure, percentage errors for m

_{1}were not included due to their low values.

_{0}= 1, but with Gaussian initial particle size distribution, having the form

_{0}= 1 and V

_{0}= 1. In this test case, analytical particle size distribution is given by [20]

_{0}N

_{0}t/(2 + C

_{0}N

_{0}t). The moments can be evaluated analytically.

_{2}after 6 s and m

_{4}and m

_{5}after 9 s began to rise, but for m

_{0}and m

_{3}they began to fluctuate after 7 s. With QMOM, the relative errors for all the moments rose sharply initially, and leveled off after 2 s, the evolution of which was very similar to that in the previous case. With the new method, the prediction for the moments of m

_{2}, m

_{4}, and m

_{5}was poor relative to QMOM, but was lower 10

^{−6}% within the simulation time, which could still satisfy industrial needs. With FPQMOM, the numerical prediction was the best. In this test case, m

_{3}could be predicted exactly without any error within the simulation time with the new method. The percentage errors for m

_{3}were not included in the figure due to their low values with both methods. Figure 6 gives the particle size distribution predicted by LFPQMOM at 0, 2, 5, and 10 s, along with analytical solutions for comparison. In the figure, the numerical PSDs with our new method agreed well with the analytical PSDs. Again, the new method made an excellent prediction for both moments and PSD. The computational time was about 71,766, 5024, and 1709 for LFPQMOM, QMOM, and FPQMOM, respectively, in this test.

_{3}, we calculated the m

_{3}with m

_{3i}of the whole PSD using Equation (32) first, and then compared it with its analytical counterpart where m

_{3i}is the tracked variable not m

_{3}. In each section, a fixed pivot quadrature method of moment (FPQMOM) was used. FPQMOM was more accurate than QMOM, as can be seen in the figure. As a result, the relative errors of the moments with FPQMOM do not deviate from QMOM or LFPQMOM too much. If the relative error for a tracked variable is too small, the values of the relative errors can be random. The compound relative errors can also be random. The fluctuations in the errors of m

_{0}in Figure 5A may be related to this.

_{0}= 0.01, together with the exponential initial distribution, i.e., Equation (36) with N

_{0}= 1 and V

_{0}= 1. Analytical solution was given by [20]

_{0}and τ = 1 − exp(−C

_{0}N

_{0}V

_{0}t). I

_{1}(x) is the modified Bessel function of the first kind. Moments can be evaluated using the numerical integration method.

^{−5}%. It should be noted that the relative errors for m

_{1}were not lower than 10

^{−12}%, even though the conservation of the mass was considered in the new method. This is because no analytical solutions for the moments were available. The numerical errors for m

_{1}were not caused by the new method or QMOM, but came from the evaluation of the moments through numerical integration. The sharp drops in the relative errors were related with the numerical errors of the evaluation of the moments. Figure 8 illustrates the particle size distribution at 0, 2, 10, and 30 s. It can be observed that the numerical predictions with the new method were in excellent agreement with the analytical counterparts. Number density with dimensionless volume larger than 10 approached zero. The computational time was about 86,314, 5570, and 1948 ms for LFPQMOM, QMOM, and FPQMOM, respectively.

_{0}= 0.01, together with the exponential initial PSD, i.e., Equation (36) with N

_{0}= 1 and V

_{0}= 1. Analytical solution for this test case was given by [20]

_{0}and Γ (x) is the gamma function.

_{0}, m

_{1}, and m

_{2}are given in Table 1. Based on the solution for m

_{2}, the actual value for t

_{gel}can be calculated

_{gel}is 50 s in this test case. To predict the moments as exactly as possible, especially near the gelling point, two nodes were adopted for QMOM and three pivots in each section for LFPQMOM. The percentage errors of the first three moments are shown in Figure 9A–C for LFPQMOM, QMOM, and FPQMOM, respectively. An interesting feature with the new method is that the simulation can be carried out for as long as possible before the gelling point. The maximum time that can be used as the simulation time is t

_{gel}− Δt. Δt is the time step. In this test case, a time step of 0.01 s was adopted, thus any time of t ≤ 49.99 s could be adopted as the simulation time. With QMOM, only an even number of moments could be tracked, four moments in this test case. We could only carry out the simulation before the time of 49.6 s when m

_{3}diverged. After that time, QMOM was not appropriate, and the new method could also make a good prediction, with a relative error smaller than 10

^{−5}%, as can be seen in Figure 9A. Relative errors with FPQMOM were similar to those with LFPQMOM. The relative errors for m

_{1}with all methods here were lower than 10

^{−13}%, hence were not included in the figures. Figure 10 shows a good agreement of numerical PSD predicted by the new method at 0, 10, 20, and 50 s with the analytical solution. Number density approached zero with a dimensionless volume larger than 10. The computational time was about 143,962, 8957, and 3199 ms for LFPQMOM, QMOM, and FPQMOM, respectively, in this test.

_{0}= 0.01, but with a Gaussian initial PSD, i.e., Equation (38) with N

_{0}= 1 and V

_{0}= 1. Analytical solution for PSD is given by [21]

_{0}and Γ (x) is the gamma function. The analytical moments are listed in Table 1. The gelling time t

_{gel}was 16.67 s. Similar to the previous case, two nodes were adopted with QMOM and three pivots in each section with LFPQMOM to track the moments exactly. Figure 11 depicts the evolution of the relative errors for the first six moments predicted by: (A) LFPQMOM; (B) QMOM; and (C) FPQMOM. The gelling time was 16.67 and a simulation time of 16.67 − Δt (=16.66) could be carried out with the new method, when m

_{3}diverged. A maximum simulation time of 16.65 s could only be adopted with QMOM. Within a simulation of 16.65 s, the percentage errors with both methods were comparable, which were lower than 10

^{−6}%, but, after that time, QMOM was not appropriate for this test case. Similar to the previous case, the relative errors for m

_{2}near the gelling point rose sharply. The relative errors for m

_{1}were not included in the figure for their low values. Again, numerical errors with FPQMOM were similar to LFPQMOM. Figure 12 gives the particle size distributions at 0, 2, 10, and 16.66 s predicted by LFPQMOM along with their analytical counterparts for comparison. The numerical PSDs were in excellent agreement with the analytical PSDs on all the time points sampled. The computational time was about 46,042, 2818, and 996 ms for LFPQMOM, QMOM, and FPQMOM, respectively, in this test.

#### 4.2. Pure Breakage

_{0}= 0.1. Uniform distribution was used as the daughter distribution

_{0}= 1 and V

_{0}= 1 as the initial distribution, the analytical solution for the moments was given by [22]

^{−9}%. The percentage errors for m

_{3}were not included in the figure due to their low values. Similar to the previous cases, m

_{3}could be predicted without any error with the new method. Numerical predictions with FPQMOM were the best in this test. Figure 14 depicts the particle size distribution at 0, 2, 10, and 50 s. Due to lack of analytical PSD, a comparison of the numerical PSDs with their analytical counterparts was impossible, and thus, in this test case, only numerical PSDs are given at 2, 10, and 50 s. The computational time was about 14,850, 1037, and 345 ms for LFPQMOM, QMOM, and FPQMOM, respectively, in this test.

_{0}= 0.1, together with a uniform daughter distribution, i.e., Equation (47), exponential initial distribution, i.e., Equation (36) with N

_{0}= 1 and V

_{0}= 1. Analytical solution was given by [22]

_{4}and m

_{5}for instance. However, with the new method, the relative errors rose relatively slowly at incipient time, as can be seen in Figure 15A. With the simulation time, the maximum error for all the moments with FPQMOM was 1.88% whereas with QMOM, the maximum was 3.66%. With the new method, the numerical accuracy was improved relative to QMOM in this test case. The relative errors for m

_{0}were comparable. Again, with the new method, m

_{1}was predicted without any error, but not with QMOM. Relative errors for m

_{1}were not included in the figure. FPQMOM was again the best in numerical prediction, and the relative errors for all the moments were less than 0.1% within the simulation time. Figure 16 gives the particle size distribution at 0, 2, 5, and 10 s, from which we observed that the numerical predictions with the new method agreed very well with the analytical PSDs at the time sampled. The computational time was about 3020, 190, and 66 ms for LFPQMOM, QMOM, and FPQMOM, respectively, in this test.

_{0}= 0.01. Uniform distribution was used for the daughter distribution, i.e., Equation (47), together with the exponential initial distribution, i.e., Equation (36) with N

_{0}= 1 and V

_{0}= 1. Analytical solution was given by [22]

_{4}and m

_{5}, and rose for m

_{0}–m

_{3}. Within the simulation time, all the relative errors were lower than 2%. However, with QMOM, all the relative errors for the moments except m

_{0}and m

_{1}rose up to 0.1 within 5 s initially. Within the simulation time, the maximum percentage error for all the moments was 13.8%. With the new method, the numerical accuracy was improved in this test case. Again, LFPQMOM made an exact prediction for m

_{1}, similar to the previous cases. The relative errors for m

_{1}were not included in the figure. Again, LFPQMOM had the smallest errors. Similar to the third case for the aggregation process, the sharp drops in the relative errors were related to the numerical evaluation of the moments. Figure 18 depicts the particle size distribution predicted by LFPQMOM at 0, 10, 20, and 50 s along with analytical counterparts for comparison. The numerical predictions exactly agreed with the analytical PSDs at different time points. The computational time was about 15,050, 876, and 338 ms for LFPQMOM, QMOM, and FPQMOM, respectively, in this test.

#### 4.3. Aggregation and Breakage

_{0}= 1, a linear kernel for breakage, i.e., Equation (49) with a

_{0}= 0.1, and uniform daughter distribution, i.e., Equation (47), along with the following initial particle size distribution were adopted

_{0}= 1, V

_{0}= 1. In this test case, analytical PSD was available [23]

_{0}V

_{0}/C

_{0})

^{1/2}/N

_{0}, and τ = C

_{0}N

_{0}t. Moments could be evaluated analytically. Evolutions of percentage errors for the moments are shown in Figure 19 with: (A) LFPQMOM; (B) QMOM; and (C) FPQMOM. In the figure, the evolutions for the relative errors were very similar with the three methods. Initially, all the relative errors rose sharply and leveled off after the time of 20 s, except the errors for m

_{3}, which had a continuous drop as the time advanced. With the new method and FPQMOM, there were two fluctuations in the drop of relative errors for m

_{3}, but not with QMOM. Within the simulation time, the maximum relative error with both methods were comparable. In the figure, the relative errors for m

_{0}and m

_{1}were not included due to their low values. Figure 20 gives the particle size distributions at 0, 2, 10, and 50 s. It was observed that the numerical PSDs agreed very well with the analytical PSDs. Note that the PSD at 10 s coincided with the PSD at 50 s, proving that a steady state had been reached by the time of 10 s. Note that, with different physical parameters, the steady state can be different. The computational time was about 497,950, 32,275, and 10,374 ms for LFPQMOM, QMOM, and FPQMOM, respectively, in this test.

_{0}= 1, a linear kernel for breakage, i.e., Equation (49) with a

_{0}= 0.25, and uniform daughter distribution, i.e., Equation (47), along with the following initial particle size distribution were adopted

_{0}= 1 and V

_{0}= 2. When the following relation was satisfied, the analytical solution was available [23]

_{1}(τ) = 7 + τ + exp(−τ), K

_{2}(τ) = 2 − 2exp(−τ), L

_{2}(τ) = 9 + τ − exp(−τ), p

_{1,2}= 0.25(e

^{−t−τ}− 9 ± (d(τ))

^{0.5}), d(τ) = τ

^{2}+ (10τ − 2τe

^{−τ}) + 25 − 26e

^{−τ}+ e

^{−2τ}, τ = C

_{0}N

_{0}t. The moments can be evaluated analytically.

_{2}–m

_{5}with: (A) LFPQMOM; (B) QMOM; and (C) FPQMOM. The figure is very similar to the previous case. Initially, the errors rose sharply and leveled off for m

_{2}, m

_{4}, and m

_{5}, but decreased for m

_{3}continuously. There was one difference from the previous case: the maximum percentage error was 0.42% with the new method, and 1.00% with QMOM. The numerical accuracy was improved with the new method relative to QMOM, which was similar to FPQMOM. Relative errors for m

_{0}and m

_{1}were not included in the figure given their low values. The numerical and the analytical particle size distributions at 0, 0.5, 2, 10, and 20 s are shown in Figure 22. The numerical PSDs agreed well with the analytical PSDs. The PSDs at 10 and 20 s coincided with each other, demonstrating that, by the time of 10 s, a steady state had been reached. Initially, the PSD was a Gaussian distribution, but, as the time advanced, the distributions became exponential, as can be seen in the figure. With the other parameters, the particle size distributions and the steady state could be different. The computational time was about 198,260, 12,048, and 4198 ms for LFPQMOM, QMOM, and FPQMOM, respectively, in this test.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Hounslow, M.J.; Ryall, R.L.; Marshall, V.R. A discretized population balance for nucleation, Growth, and Aggregation. AIChE J.
**1988**, 34, 1821–1832. [Google Scholar] [CrossRef] - Kumar, S.; Ramkrishna, D. On the solution of population balance equations by discretization—I. A fixed pivot technique. Chem. Eng. Sci.
**1996**, 51, 1311–1332. [Google Scholar] [CrossRef] - Kumar, S.; Ramkrishna, D. On the solution of population balance equations by discretization—II. A moving pivot technique. Chem. Eng. Sci.
**1996**, 51, 1333–1342. [Google Scholar] [CrossRef] - Nicmanis, M.; Hounslow, M.J. Finite-element methods for steady-state population balance equations. AIChE J.
**1998**, 44, 2258–2272. [Google Scholar] [CrossRef] - Rigopoulos, S.; Jones, A.G. Finite-element scheme for solution of dynamic population balance equation. AIChE J.
**2003**, 49, 1127–1139. [Google Scholar] [CrossRef] - Qamar, S.; Warnecke, G. Solving population balance equations for two-component aggregation by a finite volume scheme. Chem. Eng. Sci.
**2007**, 62, 679–693. [Google Scholar] [CrossRef] - McGraw, R. Description of aerosol dynamics by the quadrature method of moments. Aerosol Sci. Technol.
**1997**, 27, 255–265. [Google Scholar] [CrossRef] - Rong, F.; Marchisio, D.; Fox, R.O. Application of the direct quadrature method of moments to polydisperse gas solid fluidized beds. Powder Technol.
**2004**, 139, 7–20. [Google Scholar] - Su, J.W.; Gu, Z.L.; Li, Y.; Feng, S.Y.; Xu, X.Y. Solution of population balance equation using quadrature method of moments with an adjustable factor. Chem. Eng. Sci.
**2007**, 62, 5897–5911. [Google Scholar] [CrossRef] - Su, J.W.; Gu, Z.L.; Li, Y.; Feng, S.Y.; Xu, X.Y. An adaptive direct quadrature metod of moment for Population Balance Equation. AIChE J.
**2008**, 54, 2872–2887. [Google Scholar] [CrossRef] - Alopaeus, V.; Laakkone, N.M.; Aittamaa, J. Numerical solution of moment-transformed population balance equation with fixed quadrature points. Chem. Eng. Sci.
**2006**, 61, 4919–4929. [Google Scholar] [CrossRef] - Gu, Z.L.; Su, J.W.; Jiao, J.Y.; Xu, X.Y. Simulation of Micro-behaviors Including Nucleation, Growth, Aggregation in Particle System. Sci. China Ser. B Chem.
**2009**, 52, 241–248. [Google Scholar] [CrossRef] - Smith, M.; Matsoukas, T. Constant-number Monte Carlo simulation of population balances. Chem. Eng. Sci.
**1998**, 53, 1777–1786. [Google Scholar] [CrossRef] - Tandon, P.; Rosner, D.E. Monte Carlo simulation of particle aggregation and simultaneous restructuring. J. Colloid Interface Sci.
**1999**, 213, 273–286. [Google Scholar] [CrossRef] [PubMed] - Irizarry, R. Fast Monte Carlo methodology for multivariate particulate systems—I. Point ensemble Monte Carlo. Chem. Eng. Sci.
**2008**, 63, 95–110. [Google Scholar] [CrossRef] - Attarakih, M.M.; Drumm, C.; Bart, H. Solution of the population balance equation using the sectional quadrature method of moments (SQMOM). Chem. Eng. Sci.
**2009**, 64, 742–752. [Google Scholar] [CrossRef] - Gordon, R.G. Error bounds in equilibrium statistical mechanics. J. Math. Phys.
**1968**, 9, 655–663. [Google Scholar] [CrossRef] - Golub, G.H.; Vanloan, C.F. Matrix Computations, 3rd ed.; Johns Hopkins University Press: Baltimore, MD, USA, 1996; pp. 183–188. [Google Scholar]
- Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. Numerical Recipes in C++, 2nd ed.; Cambridge University Press: Cambridge, UK, 1992. [Google Scholar]
- Scott, W.T. Analytic studies of cloud droplet coalescence I. J. Atmos. Sci.
**1968**, 25, 54–65. [Google Scholar] [CrossRef] - Smit, D.J.; Hounslow, M.J.; Paterson, W.R. Aggregation and gelation—I. Analytical solutions for CST and batch operation. Chem. Eng. Sci.
**1994**, 49, 1025–1035. [Google Scholar] [CrossRef] - Ziff, R.M.; McGrady, E.D. The kinetics of cluster fragmentation and depolymerisation. J. Phys. A Math. Gen.
**1985**, 18, 3027–3037. [Google Scholar] [CrossRef] - McCoy, B.J.; Madras, G. Analytical solution for a population balance equation with aggregation and fragmentation. Chem. Eng. Sci.
**2003**, 58, 3049–3051. [Google Scholar] [CrossRef]

**Figure 3.**Evolution of the relative errors for the first six moments with: (

**A**) LFPQMOM; (

**B**) QMOM; and (

**C**) FPQMOM (square: m

_{0}; up triangle: m

_{2}; down triangle: m

_{3}; left triangle: m

_{4}; right triangle: m

_{5}).

**Figure 4.**Particle size distribution predicted by LFPQMOM at 0, 2, 5, and 10 s (square: numerical PSD at 0 s; circle: numerical PSD at 2 s; up triangle: numerical PSD at 5 s; down triangle: numerical PSD at 10 s; line: analytical PSD).

**Figure 5.**Evolution of relative errors for the first six moments with: (

**A**) LFPQMOM; (

**B**) QMOM; and (

**C**) FPQMOM (square: m

_{0}; up triangle: m

_{2}; down triangle: m

_{3}; left triangle: m

_{4}; right triangle: m

_{5}).

**Figure 6.**Particle size distributions at 0, 2, 5, and 10 s (square: numerical PSD at 0 s; circle: numerical PSD at 2 s; down triangle: numerical PSD at 5 s; up triangle: numerical PSD at 10 s; line: analytical PSD).

**Figure 7.**Evolution of the percentage errors for the first six moments with: (

**A**) LFPQMOM; (

**B**) QMOM; and (

**C**) FPQMOM for the pure aggregation with sum kernel and exponential initial PSD (square: m

_{0}; circle: m

_{1}; up triangle: m

_{2}; down triangle: m

_{3}; left triangle: m

_{4}; right triangle: m

_{5}).

**Figure 8.**Particle size distribution at 0, 2, 10, and 30 s for the pure aggregation with sum kernel and exponential initial PSD (square: numerical PSD at 0 s; circle: numerical PSD at 2 s; up triangle: numerical PSD at 10 s; down triangle: numerical PSD at 30 s; line: analytical PSD).

**Figure 9.**Evolution of percentage errors of m

_{0}and m

_{2}for the pure aggregation process with product kernel and exponential initial PSD: (

**A**) LFPQMOM; (

**B**) QMOM; and (

**C**) FPQMOM (square: m

_{0}; up triangle: m

_{2}).

**Figure 10.**Particle size distribution at 0, 10, 20, and 50 s for the pure aggregation process with product kernel and exponential initial PSD (Square: numerical PSD at 0 s; circle: numerical PSD at 10 s; up triangle: numerical PSD at 20 s; down triangle: numerical PSD at 50 s; line: analytical PSD).

**Figure 11.**Relative errors for m

_{0}and m

_{2}for the pure aggregation process with product kernel and Gaussian initial PSD with: (

**A**) LFPQMOM; (

**B**) QMOM; and (

**C**) FPQMOM (square: m

_{0}; up triangle: m

_{2}).

**Figure 12.**Particle size distributions at 0, 2, 10, and 16.66 s for the pure aggregation process with product kernel and Gaussian initial PSD (square: numerical PSD at 0 s; circle: numerical PSD at 2 s; up triangle: numerical PSD at 10 s; down triangle: numerical PSD at 16.66 s; line: analytical PSD).

**Figure 13.**Evolution of relative errors for the pure breakage process with constant kernel, exponential initial PSD, and uniform daughter distribution with: (

**A**) LFPQMOM; (

**B**) QMOM; and (

**C**) FPQMOM (square: m

_{0}; up triangle: m

_{2}; down triangle: m

_{3}; left triangle: m

_{4}; right triangle: m

_{5}).

**Figure 14.**Particle size distributions at 0, 2, 10, and 50 s for the pure breakage process with constant kernel, exponential initial PSD, and uniform daughter distribution (square: numerical PSD at 0 s; line: analytical PSD at 0 s; circle line: numerical PSD at 2 s; up triangle line: numerical PSD at 10 s; down triangle line: numerical PSD at 50 s).

**Figure 15.**Evolution of the first six moments for the pure breakage process with linear kernel, exponential initial PSD, and uniform daughter distribution: (

**A**) LFPQMOM; (

**B**) QMOM; and (

**C**) FPQMOM (square: m

_{0}; up triangle: m

_{2}; down triangle: m

_{3}; left triangle: m

_{4}right triangle: m

_{5}).

**Figure 16.**Particle size distribution at 0, 2, 5, and 10 s for the pure breakage process with linear kernel, exponential initial PSD, and uniform daughter distribution (square: numerical PSD at 0 s; circle: numerical PSD at 2 s; up triangle: numerical PSD at 5 s; down triangle: numerical PSD at 10 s; line: analytical PSD).

**Figure 17.**Evolution of the relative errors for the first six moments for the pure breakage process with square kernel, exponential initial PSD, and uniform daughter distribution: (

**A**) LFPQMOM; (

**B**) QMOM; and (

**C**) FPQMOM (square: m

_{0}; up triangle: m

_{2}; down triangle: m

_{3}; left triangle: m

_{4}; right triangle: m

_{5}).

**Figure 18.**Particle size distribution at 0, 2, 5, and 10 s for the pure breakage process with square kernel, exponential initial PSD, and uniform daughter distribution (square: numerical PSD at 0 s; circle: numerical PSD at 10 s; up triangle: numerical PSD at 20 s; down triangle: numerical PSD at 50 s; line: analytical PSD).

**Figure 19.**Relative errors for m

_{2}–m

_{5}for the aggregation and breakage combined process with constant aggregation kernel, linear breakage kernel, uniform daughter distribution, and exponential initial PSD with: (

**A**) LFPQMOM; (

**B**) QMOM; and (

**C**) FPQMOM (up triangle: m

_{2}; down triangle: m

_{3}; left triangle: m

_{4}; right triangle: m

_{5}).

**Figure 20.**Particle size distributions at 0, 2, 10, and 50 s for the aggregation and breakage combined process with constant aggregation kernel, linear breakage kernel, uniform daughter distribution, and exponential initial PSD (square: numerical PSD at 0 s, Circle: numerical PSD at 2 s; up triangle: numerical PSD at 10 s; down triangle: numerical PSD at 50 s; line: analytical PSD).

**Figure 21.**Evolutions for the relative errors of m

_{2}−m

_{5}for the aggregation and breakage combined process with constant aggregation kernel, linear breakage kernel, uniform daughter distribution, and Gaussian initial particle size distribution: (

**A**) LFPQMOM; (

**B**) QMOM; and (

**C**) FPQMOM (up triangle: m

_{2}; down triangle: m

_{3}; left triangle: m

_{4}; right triangle: m

_{5}).

**Figure 22.**Particle size distributions at 0, 0.5, 2, 10, and 20 s for the aggregation and breakage combined process with constant aggregation kernel, linear breakage kernel, uniform daughter distribution, and Gaussian initial particle size distribution (square: numerical PSD at 0 s; circle: numerical PSD at 0.5 s; up triangle: numerical PSD at 2 s; down triangle: numerical PSD at 10 s; left triangle: numerical PSD at 20 s; line: analytical PSDs).

**Table 1.**Analytical solution for the first three moments for the aggregation process with product kernel.

Moments | Analytical Solution |
---|---|

m_{0}(t) | ${m}_{0}(t)={m}_{0}(0)-\frac{1}{2}{C}_{0}{m}_{1}^{2}(0)t,\hspace{0.17em}\hspace{0.17em}0<t<{t}_{gel}$ |

m_{1}(t) | ${m}_{1}(t)={m}_{1}(0),\hspace{0.17em}\hspace{0.17em}0<t<{t}_{gel}$ |

m_{2}(t) | ${m}_{2}(t)=\frac{{m}_{2}(0)}{1-{C}_{0}{m}_{2}(0)t},\hspace{0.17em}\hspace{0.17em}0<t<{t}_{gel}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Su, J.; Le, W.; Gu, Z.; Chen, C.
Local Fixed Pivot Quadrature Method of Moments for Solution of Population Balance Equation. *Processes* **2018**, *6*, 209.
https://doi.org/10.3390/pr6110209

**AMA Style**

Su J, Le W, Gu Z, Chen C.
Local Fixed Pivot Quadrature Method of Moments for Solution of Population Balance Equation. *Processes*. 2018; 6(11):209.
https://doi.org/10.3390/pr6110209

**Chicago/Turabian Style**

Su, Junwei, Wang Le, Zhaolin Gu, and Chungang Chen.
2018. "Local Fixed Pivot Quadrature Method of Moments for Solution of Population Balance Equation" *Processes* 6, no. 11: 209.
https://doi.org/10.3390/pr6110209