Numerical Solution of a Modified Moment Model for Particle Population Balance Equation in the Continuum-Slip Regime T.T. Kong, Q. He, and M.L. Xie
Abstract
In the present paper, numerical solution of a modified moment model for particle population balance equation in the continuum-slip regime is proposed based on the Taylor expansion method of moments (TEMOM). No simplification of the slip correction factor in the collision kernel makes this solution have a higher accuracy than the previous work. Results illustrate that the numerical solution is also applicable when the Knudsen number is extended up to 10. Keywords
Numerical solution regime
1
Moment model
Introduction
The air pollution increasingly influences the health of human beings, and aerosol particles are recognized as the unhealthiest components. It is significant and urgent to pay more attention on the particle size and concentration. Muller [5] first established the particle population balances equation (PBE) describing the particle evolution. The general form of population balance equation with mono-variant is proposed to be [2]: @nðx; tÞ 1 ¼ @t 2
Zv bðv1 ; v v1 Þnðv1 ; tÞnðv v1 ; tÞdv1 0
ð1Þ
Z1
bðv1 ; vÞnðv; tÞnðv1 ; tÞdv1 0
where n(v,t) is the number of particles per unit spatial volume with particle volume from v to v + dv at time t, and β is the collision kernel of coagulation which depends on the T.T. Kong Q. He M.L. Xie (&) State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, China e-mail:
[email protected]
Population balance equation
Continuum-slip
particle sizes, concentrations, and transport mechanisms in the system. Unfortunately, PBE cannot be achieved to obtain the exact analytical solution for the strong nonlinear terms with the same mathematical structure as Boltzmann’s transport equation. The numerical simulation is another way proposed to solve the PBE, e.g., direct bin methods [3, 7], moment-based method [4], and Monte Carlo method [6, 8]. Recently, Yu et al. [12] proposed a new moment method called the Taylor series expansion method of moments (TEMOM) for the PBE undergoing Brownian coagulation. The main idea of this method is using the Taylor series expansion technique to close the moment evolution equations. This method has no prior requirement for particle size spectrum with low computational cost, which meets the demand of modern, complicated particulate industries perfectly. In this study, a modified moment model for particle population balance equation due to Brownian coagulation in the continuum-slip regime is proposed. The nonlinear forms in the slip correction factors in the collision kernel are not neglected, and present moment model can be reduced to the linearized moment model proposed by Yu et al. [13]. The results show that this solution remains good accuracy when the Knudsen number is extended to 10.
© Springer Science+Business Media Singapore and Tsinghua University Press 2016 G. Yue and S. Li (eds.), Clean Coal Technology and Sustainable Development, DOI 10.1007/978-981-10-2023-0_57
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Modified Moment Model
with B is a constant as:
As mentioned above, the TEMOM has been regarded as a promising method for modeling the aerosol particle balance equation. It is proposed to solve the moments of the size distribution in the present study. The kth-order moment mk of the particle distribution is defined by: Z1 mk ¼ vk nðvÞdv ð2Þ 0
If we multiply both sides of the PBE with vk and integrate over all particle sizes, a system of transport equations for mk are obtained as a result. Finally, the particle moments in a spatially homogeneous system can be described by: dmk 1 ¼ 2 dt
Z1 Z1 h 0
i ðv þ v1 Þk vk vk1 bðv; v1 Þnðv; tÞnðv1 ; tÞdvdv1
ð3Þ
0
In the continuum-slip regime, the collision kernel is: ! CðvÞ Cðv1 Þ 1=3 1=3 b ¼ B2 1=3 þ 1=3 v þ v1 ð4Þ v v 1
where the constant B2 = 2kBT/3μ, kB is the Boltzmann constant, T is the gas temperature, and μ is the gas viscosity. The slip correction factor C is defined as follows: A3 C ¼ 1 þ Kn A1 þ A2 exp ð5Þ Kn with A1, A2, and A3 as constants. For air [1], the constants can be taken as A1 = 1.257, A2 = 0.400, A3 = 0.55, respectively. The slip correction factor is used to accommodate the gas slip effects on small particles with Knudsen number between the continuum regime and free molecular flow. Kn is the Knudsen number, which is the ratio of mean free path to particle radius, namely Kn = λ/a. The nonlinear terms in the slip correction factor are recorded as E(v) = exp(−A3/Kn). It can be expanded with Taylor series at the point (the particle mean volume u = m1/m0) and truncated by the first three teams as: EðvÞ ¼ f0 ðuÞ þ f1 ðuÞ ðv uÞ þ f2 ðuÞ ðv uÞ2 Eðv1 Þ ¼ f0 ðuÞ þ f1 ðuÞ ðv1 uÞ þ f2 ðuÞ ðv1 uÞ2
ð6Þ
and f0, f1, and f2 are defined as: A3 u1=3 B f0 ðuÞ ¼ e
A3 u1=3 B A3 1e f1 ðuÞ ¼ 3 Bu2=3 A3 u1=3 B A3 ðA3 u1=3 þ 2BÞ 1 e f2 ðuÞ ¼ 18 B2 u5=3
ð7Þ
B ¼ Kn0 ð
m1 1=3 Þ m00
ð8Þ
in which the initial Knudsen number Kn0 is: Kn0 ¼
k 4pm00 1=3 ¼ kð Þ u0 3m1
ð9Þ
In the TEMOM, the first three moments, m0, m1, and m2, are required to close the particle balance equation. In this study, the ordinary differential equations for moments obtained by the Taylor expansion technique has the form: 9 8 2 2 5 9m2 m0 180m2 m21 m0 þ 171m41 > 3 > > > BA f u 4 2 2 > > m1 > > > > > > 5m22 m20 64m2 m21 m0 103m41 > > > > þ BA f > > 2 0 1=3 m4 u > > 1 = < 2 2 2 4 dm0 B2 2 5m2 m0 64m2 m1 m0 103m1 ¼ m0 þ BA 4 1 u1=3 m1 > dt 81 > > > > > 2m22 m20 13m2 m21 m0 151m41 > > > > þ 4 > > m1 > > > > 2 2 2 4 > > 6m2 m0 93m2 m1 m0 þ 87m1 2 > > : 3 ; BA f u 4 2 1
ð10aÞ
m1
dm1 ¼0 dt
9 8 2 2 6m2 m0 þ 69m2 m21 m0 75m41 > > > > BA f ðuÞ 2 1 > > 2u2=3 m41 > > > > 2 2 2 4 > > m2 m0 2m2 m1 m0 80m1 > > > > BA f > > 2 0 1=3 m4 u > > 1 = < dm2 4B2 2 m22 m20 2m2 m21 m0 80m41 ¼ m1 BA 4 1 1=3 u m1 > > dt 81 > > > > > > m22 m20 13m2 m21 m0 151m41 > > 4 > > m > > 1 > > 2 2 2 4 > > 9m2 m0 þ 63m2 m1 m0 72m1 > > ; : BA f 2 2 5=3 4 u
ð10bÞ
ð10cÞ
m1
Introducing mc ¼ m0 m2 =m21 (where m0, m1, and m2 are the first three moments, respectively), Eq. (10) has the following expression: 9 8 ð2m2c 13mc 151Þ þ > > > > > > > > 2 1=3 > > ð5m 64m 103ÞBA u þ > > c 1 c = < dm0 B2 2 2 1=3 ¼ m0 ð5mc 64mc 103ÞBA2 f0 u > dt 81 > > > > > > ð6m2c 93mc þ 87ÞBA2 f1 u2=3 þ > > > > > ; : 2 5=3 ð9mc 180mc þ 171ÞBA2 f2 u dm1 ¼0 dt 9 8 ð2m2c 13mc 151Þ þ > > > > > > > > 2 1=3 > > 2ðm 2m 80ÞBA u þ > > c 1 c = < dm2 2B2 2 ¼ m1 2ðm2c 2mc 80ÞBA2 f0 u1=3 > > dt 81 > > > > > ð6m2c þ 69mc 75ÞBA2 f1 u2=3 þ > > > > > : 2 5=3 ; 2ð9mc 63mc þ 72ÞBA2 f2 u ð11Þ
Numerical Solution of a Modified Moment Model for Particle …
Neglecting the nonlinear term in the slip correction factor, the functions f0, f1, and f2 are simplified as f0 = f1 = f2 = 0. Then, the corresponding moment model is reduced as dm0 B2 2 ¼ m0 dt 81
(
ð2m2c 13mc 151Þ þ
425
The ordinary differential Eq. (11) for particle moment evolution is reduced as: dm0 B2 2 ¼ m0 ð2m2c 13mc 151Þ dt 81 dm1 ¼0 dt dm2 2B2 2 ¼ m ð2m2c 13mc 151Þ dt 81 1
)
ð5m2c 64mc 103ÞBA1 u1=3
dm1 ¼0 dt ( ) dm2 2B2 2 ð2m2c 13mc 151Þ þ ¼ m dt 81 1 2ðm2c 2mc 80ÞBA1 u1=3
ð12Þ
The simplified moment model is same as Yu’s model [13]. In some cases, it is assumed that C(v) = C(v1) = C(u). The collision kernel can be written as: ! 1 1 1=3 1=3 b ¼ B2 CðuÞ 1=3 þ 1=3 v þ v1 ð13Þ v v 1
Then, the ordinary differential equations for particle moment evolution can be simplified as: dm0 B2 2 1=3 ð2m20 m22 13m0 m21 m2 151m41 Þ ¼ m0 u u dt 81 m41 dm1 ¼0 dt dm2 2B2 2 1=3 ð2m20 m22 13m0 m21 m2 151m41 Þ ¼ m u u dt 81 1 m41 ð14Þ
which is the moment model in the continuum regime, and the simplified particle moment evolution Eqs. (14) and (16) can be reduced to the same form. This result is the same as the asymptotic analysis by Xie and He [10], and Xie and Wang [11] have given out the analytical and asymptotic solutions of the moment model in the continuum regime in their papers.
4
Error% ¼ ð15Þ
Using the definition of dimensionless particle moment mc as above, Eq. (15) has the following expression: dm0 B2 2 1=3 ¼ m0 u uð2m2c 13mc 151Þ dt 81 dm1 ¼0 dt dm2 2B2 2 1=3 ¼ m u uð2m2c 13mc 151Þ dt 81 1
3
ð16Þ
It can be found that: lim u ¼ 1;
lim f1 u2=3 ¼ 0;
t!1
lim f0 u1=3 ¼ 0
t!1
lim f2 u5=3 ¼ 0
t!1
ð17Þ
/ /0 100% /0
ð19Þ
Here, φ and φ0 are the variables in present paper and in other paper, respectively.
5
Asymptotic Analysis
t!1
Computation
The programs are written using the MATLAB programming language and performed with the MATLAB compiler. The fourth-order Runge–Kutta method with a fixed time step of 0.001 is applied to solve the set of ordinary differential equations numerically. In all simulations, without loss of generality, the multi-disperse system is selected as the same initial case, i.e., m00 = 1; m10 = 1; m20 = 4/3, and the constant B2 is assumed to be unit for convenience. The relative error of any variable in the present paper to that in the other paper is calculated as following form:
With ϕ is represented as: A3 u1=3 B þ u1=3 u ¼ A1 B þ A2 Be
ð18Þ
Results and Discussions
In order to make the study much more representative, four initial Knudsen numbers, i.e., 0.0001, 0.1000, 5.0000, and 10.0000, are selected and investigated. The variations of the first three moments m0, m2, and mc with four initial Knudsen numbers are shown in Fig. 1. According to the paper by Xie [9], the asymptotic solution for the particle moments in the −1 continuum regime is given as: m0|CR → (81/169)B−1 2 t , 2 m2|CR → (338/81)B2m1t, and the asymptotic value for dimensionless particle moment is mc∞ → 2. The asymptotic solution in the continuum regime is also compared with the numerical solution in the continuum-slip regime in the figure. It can be noted from the figure that, with a decrease of the initial Knudsen number, the two moments m0 and mc increase, while the moment m2 decreases. It is also obvious
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Fig. 1 Comparisons of present results in the continuum-slip regime and the asymptotic solution in the continuum regime: a zeroth particle moment; b relative errors of zeroth particle moment compared with Yu; c second particle moment; and d dimensionless particle moment
that, in the log–log coordination, the variations of the first three moments m0, m2, and mc in the continuum-slip regime in this paper all gradually tend to the asymptotic solution in the continuum regime at long time, which is consistent with the asymptotic analysis and proves the present numerical solution is accurate. However, it will cost a much long time to reach asymptotic solution in the continuum regime due to a very small growth rate. Yu et al. [13] simplified the slip correction factor in collision kernel as C = 1 + AKn and proposed a new analytical solution for the population balance equation in the continuum-slip regime. In this paper, the nonlinear terms in the slip correction factor E(v) are not neglected in order to make the collision kernel more accurate, and as a result, the moment model has a much more complicated form. The obtained moment model can be reduced to the simplified moment model by Yu et al. [13] if we neglect the nonlinear
terms, but with a little difference in the coefficient. As is shown in the figure, the evolution of m0, m2, and mc remains right even when the Knudsen number is up to 10, while the solution proposed by Yu et al. is not suitable when the Knudsen number is more than 5. Therefore, the present numerical solution is more applicable and reasonable than previous works when solving the particle balance equation in the continuum-slip regime. What is more, from the comparison with Yu’s solution in Fig. 1b, it is obvious that the relative error is very small. In this case, the solution by Yu et al. is good enough for practice, and our solution is not convenient in terms of its much more complicated model. In a word, this modified solution for PBE in the continuum-slip regime is an ideal one, which yields a higher accuracy and has a wider scope of application. However, the analytical solution is difficult to be obtained due to the introduction of the nonlinear terms in the slip correction
Numerical Solution of a Modified Moment Model for Particle …
factor. We will give out the analytical solution of this modified moment model for the PBE in the continuum-slip regime in near future.
6
Conclusion
In the present study, numerical solution of a modified moment model has been proposed for the PBE in the continuum-slip regime. The solution is achieved based on the performance of Taylor expansion method of moments with no simplification of the slip correction factors, which makes it yield higher accuracy than the results obtained before. Results show that this solution remains right even when the Knudsen number is up to 10, while it is 5 in the past papers. Acknowledgments This work is supported by the National Natural Science Foundation of China with Grant No. 11572138, the Fundamental Research Funds for the Central Universities (Project No. 2013TS078), and the Foundation of State Key Laboratory of Coal Combustion (Project No.FSKLCCB1401).
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