Solution of bivariate aggregation population balance equation: a comparative study

Abstract

The present work shows a comparative study of two different numerical methods for solving bivariate aggregation population balance equations. In particular, we summarize the cell average technique (Kumar et al. in Comput Chem Eng 32(8):1810–1830, 2008) and the finite volume scheme (Singh et al. in J Comput Appl Math 308:83–97, 2016) for solving pure aggregation population balance equation. The qualitative and quantitative numerical results of various order moments and number density functions are compared with the exact results for analytically tractable kernels. The results reveal that the finite volume scheme approximates the results more accurately and efficiently as compared to the cell average technique. With respect to mixed moments, in particular, the total variance of excess solute (Matsoukas et al. in AIChE J 52(9):3088–3099, 2006), the finite volume scheme is superior to the cell average technique. Additionally, it is also shown that bicomponent moments are more sensitive to the selection of the grid and require finer discretization to reduce errors.

Appendix

Size-independent Kernel

The different exact results for the various initial conditions using size-independent kernel are provided by Gelbard and Seinfeld [10]. For the initial condition, $f(0,u,v)=\frac{16 N_0uv}{{m_{10}^2}{{m_{20}^2}}} \exp \Big [-\frac{2u}{m_{10}} -\frac{2v}{m_{20}}\Big ]$, the exact solution is summarized in the following Table 3.

Table 3 Exact solution for size-independent kernel

Size-dependent Kernel

Fernández-Díaz and Gómez-García [7] has provided different exact solutions corresponding to various initial condition for size-dependent kernel. Here, we have listed for the initial condition, i.e., $f(0,u,v)=\frac{16 N_0uv}{{m_{10}^2}{{m_{20}^2}}} \exp \Big [-\frac{2u}{m_{10}} -\frac{2v}{m_{20}}\Big ]$ in Table 4.

Table 4 Exact solution for size-dependent kernel

Mass conservation

Proposition The finite volume formulation provided in (11) does not conserve the total mass of the system.

Proof Multiply the formulation provided in Eq. (11) by $(u_{p}+y_{q})\Delta u_{p} \Delta v_{q}$ both side and take sum over all p and q. The left hand side gives the first moment at time $t^{n+1}$ and the right hand side can be simplified as

$$\begin{aligned} \sum _{p=1}^{I_1}\sum _{q=1}^{I_2} f_{p,q}^{n+1} (u_{p}+v_{q})\Delta u_{p} \Delta v_{q} = \sum _{p=1}^{I_1}\sum _{q=1}^{I_2} f_{p,q}^{n} (u_{p}+v_{q})\Delta u_{p} \Delta v_{q} + \Delta t^n {T} , \end{aligned}$$

where T is given by the following expression:

$$\begin{aligned} {T} = \sum _{p=1}^{I_1}\sum _{q=1}^{I_2} \Big (&\frac{1}{2} \sum _{(p,r,q, s) \in {\mathcal{L}}^{i,j}}\beta _{p,r,q, s}^{n}f_{p,q}^{n}f_{r,s}^{n}{\Delta u_{p}\Delta v_{q}\Delta u_{r}\Delta v_{s}}(u_{p}+v_{q}) \\&-\sum _{{p}=1}^{I_1}\sum _{{q}=1}^{I_2}\beta _{p,r,q, s}^{n}f_{i,j}^{n}f_{p,q}^{n}\Delta u_{p}\Delta v_{q}\Big ). \end{aligned}$$

(17)

For proving the mass conservation property, it is required to show that $T = 0$ for all times. But, it can be noticed that the two expressions on the right hand side of the above equations are not same. Hence, the mass conservation property does not hold for this formulation.