An Efficient Algorithm Scheme for Implementing the TEMOM for Resolving Aerosol Dynamics

Abstract

This paper presents a new algorithm scheme for implementing the Taylor-series expansion method of moments (TEMOM), which is a general method for solving the population balance equation. Instead of deriving moment ordinary differential equations (ODEs) for particular aerosol dynamics, the new numerical algorithm develops a subroutine that corresponds to types of kernels of aerosol dynamics rather than a particular kernel. Consequently, the TEMOM can be conveniently applied to types of dynamics rather than a particular dynamics, and the derivation from the PBE to moment ODEs is avoided. A key aspect to the new algorithm scheme is that the particle kernels are written in a general form, allowing for a universal expression of aerosol dynamics. The closure of ODEs for moments was accomplished by implementing a new Taylor-series expansion closure function with two varying parameters H and ϕ, which enables the TEMOM applicable to any type of moment sequence. The feasibility of the new algorithm scheme was verified by comparing it with other recognized methods for six classic aerosol dynamics. The new algorithm scheme makes the TEMOM much easier to be used for users as compared to its original version (Aerosol Sci Tech 49(2015):1021–1036). The idea of the algorithm scheme can be applied to other non-quadrature-based methods of moments.

Appendices

Supply Information I

An example for implementing the new numerical algorithm for only coagulation.

We used an example to show how to implement the new numerical algorithm algorithm proposed in this work. If we only consider a coagulation process, whose coagulation kernel is

$$ \beta \left( {v,v{^{\prime} }} \right) = B_{2} \left( {\frac{1}{{v^{{1/D_{\text{f}} }} }} + \frac{1}{{v{^{\prime} }^{{1/D_{\text{f}} }} }}} \right)\left( {v^{{1/D_{\text{f}} }} + v{^{\prime} }^{{1/D_{\text{f}} }} } \right). $$

(36)

It is actually Eq. (5) in the text. For convenience, we write the PBE only involving coagulation process as follows:

$$ \frac{{\partial n\left( {v,t} \right)}}{\partial t} = \frac{1}{2}\mathop \int \limits_{{v^{*} }}^{v} \beta \left( {v - v^{\prime} ,v^{\prime} } \right)n\left( {v - v^{\prime} ,t} \right)n\left( {v^{\prime} ,x_{i} ,t} \right){\text{d}}v^{\prime} - n\left( {v,t} \right)\mathop \int \limits_{{v^{*} }}^{\infty } \beta \left( {v,v^{\prime} } \right)n\left( {v^{\prime} ,x_{i} ,t} \right){\text{d}}v^{\prime} . $$

(37)

To resolve this problem using the improved GTEMOM, at first, we need to convert the PBE in terms of particle size v, as shown in Eq. (37), to a moment equation in terms of the kth moments, m _k:

$$ \frac{{{\text{d}}m_{\text{k}} }}{{{\text{d}}t}} = \frac{1}{2}\mathop \int \limits_{0}^{\infty } \mathop \int \limits_{0}^{\infty } \kappa \left( {v,v^{\prime} ,k} \right)\beta \left( {v,v^{\prime} } \right)n\left( {v,t} \right)n\left( {v^{\prime} ,t} \right){\text{d}}v{\text{d}}v^{\prime} . $$

(38)

The objective of all methods of moments is to find a way to integrate the left term of Eq. (38). To achieve this, in this work, we expanded κ(v, v′, k) using a third-order Taylor-series expansion technique and let it to be a form, as shown in Eq. (10). Then, we write Eq. (36) as a polynomial formed by terms having the same expression, i.e., Eq. (4). At last, the time rate of m _k shown in Eq. (38), i.e., the left term on the right hand of Eq. (3), will be replaced by several separated terms which have a universal expression, as shown in Eq. (12) in the text.

To numerically solve Eq. (8), we only need to write a subprogram according to Eq. (12), and then, any $\frac{{{\text{d}}W_{\text{i}} \left( {\gamma_{\text{i}} ,f_{\text{i}} ,g_{\text{i}} ,k} \right)}}{{{\text{d}}t}}$ term shown in Eq. (8) can be obtained by calling this subprogram, in which $\gamma_{\text{i}} ,f_{\text{i}} ,g_{\text{i}} ,\;{\text{and}}\;k$ are input parameters. When the subprogram is called, some implicit kth moments must be approximated by the basis function, as shown in Eq. (16). Once the subprogram for $\frac{{{\text{d}}W_{\text{i}} \left( {\gamma_{\text{i}} ,f_{\text{i}} ,g_{\text{i}} ,k} \right)}}{{{\text{d}}t}}$ can be successfully called, then the numerical solution for Eq. (8) is achieved.

For simplicity, we provided the subprogram code for calling $\frac{{{\text{d}}W_{\text{i}} \left( {\gamma_{\text{i}} ,f_{\text{i}} ,g_{\text{i}} ,k} \right)}}{{{\text{d}}t}}$ and the subprogram code for calling basis function in attached file. The user only need to specify the kernels they studied, and then write the kernel as a universal form, as shown in Eq. (4). Then, they need to specify $\upgamma$, f, and g. To use the improved GTEMOM, the user should specify both ${\mathcal{H}}$ and ϕ, with them the explicit moments and the number of equations can be determined. For example, if we select ${\mathcal{H}} = 2,$ and ϕ = 1, then the explicit moments required to be solved should be m ₀, m _1, and m ₂. The details how to specify the explicit moments are shown in Table 1.

Supply Information II

The derivation of Eq. (21).

In this supplied information, we only give the derivation about Eqs. (21) and (22) for erosion mechanism, while the other two can be obtained using the same way.

At first, we write the PBE involving breakage process:

$$ \underbrace {{\frac{{\partial n\left( {v,t} \right)}}{\partial t} = \mathop \int \limits_{v}^{\infty } a\left( {v{^{\prime} }} \right)b\left( {\left. v \right|v{^{\prime} }} \right)n\left( {v{^{\prime} },t} \right){\text{d}}v{^{\prime} } - a\left( v \right)n\left( {v,t} \right)}}_{\text{breakage}}. $$

(39)

To use the improved GTEMOM, we multiply v ^k on both sides of Eq. (39) and then integrate it from 0 to ∞. After introducing the definition for the kth moments in terms of particle size v, as shown in Eq. (3) in the text, we obtain

$$ \underbrace {{\frac{{{\text{d}}m_{\text{k}} }}{{{\text{d}}t}} = \mathop \int \limits_{0}^{\infty } \bar{b}_{\text{i}}^{\text{k}} a\left( v \right)n\left( {v,t} \right){\text{d}}v - \mathop \int \limits_{0}^{\infty } a\left( v \right)n\left( {v,t} \right){\text{d}}v}}_{\text{breakage}}. $$

(40)

The expressions of b(v|v ^′), $\bar{b}_{\text{i}}^{\text{k}}$ and a(v) have been clarified in the text. For the erosion mechanism, $\bar{b}_{\text{i}}^{\text{k}}$ is a power function, as shown in Table 2. Thus, we cannot integrate directly without further disposition because of power expression. Similar to our study on proposing the improved GTEMOM, we expanded the power function shown in $\bar{b}_{\text{i}}^{\text{k}}$, i.e., (v − v _p)^k, using a third-order Taylor-series expansion technique, and we obtained:

$$ \begin{aligned} \bar{b}_{\text{i}}^{\text{k}} & = \frac{1}{2}\left( {u_{0} - v_{p} } \right)^{k - 2} \left\{ {\left( {k^{2} - k} \right)v^{2} + \left( { - 2k^{2} u_{0} - 2kv_{p} + 4ku_{0} } \right)v} \right. \\ & \quad \left. { + \left( {k^{2} u_{0}^{2} + 2ku_{0} v_{p} - 3ku_{0}^{2} + 2v_{p}^{2} - 4v_{p} u_{0} + 2u_{0}^{2} } \right)} \right\} + v_{\text{p}}^{\text{k}} , \\ \end{aligned} $$

(41)

where u ₀ is the Taylor-series expansion point, which is identical to that in the text, i.e., (m ₁/m ₀)^1/ϕ, or others having the same physical dimension as (m ₁/m ₀)^1/ϕ. As Eq. (41) is introduced into Eq. (2), we can integrate out Eq. (40) as follows:

$$ \begin{aligned} \frac{{{\text{d}}m_{k} }}{{{\text{d}}t}} & = \frac{\zeta }{2}\left( {u_{0} - v_{p} } \right)^{k - 2} \left\{ {\left( {k^{2} - k} \right)m_{{2 + \frac{3}{{D_{f}^{2} }}}} + \left( { - 2k^{2} u_{0} - 2kv_{p} + 4ku_{0} } \right)m_{{1 + 3/D_{f}^{2} }} } \right. \\ & \quad \left. { + \left( {k^{2} u_{0}^{2} + 2ku_{0} v_{p} - 3ku_{0}^{2} + 2v_{p}^{2} - 4v_{p} u_{0} + 2u_{0}^{2} } \right)m_{{3/D_{f}^{2} }} } \right\} \\ & \quad + \zeta v_{p}^{k} m_{{3/D_{f}^{2} }} - \zeta m_{{k + \frac{3}{{D_{f}^{2} }}}} . \\ \end{aligned} $$

It is also Eqs. (21) and (22) in the text. In this converted ordinary differential equation, it is clear that there are several implicit moments, such as $m_{{2 + 3/D_{f}^{2} }}$ and $m_{{1 + 3/D_{f}^{2} }}$. To achieve the final solution for this equation, the basis function for approximating arbitrary kth moments shown in Eq. (16) must be used.

Supply Information III

The derivation of Eq. (21).

$$ \begin{aligned} \left. {\frac{{{\text{d}}m_{k} }}{{{\text{d}}t}}} \right|_{\text{BR}} & = \mathop \int \limits_{0}^{\infty } \bar{b}_{\text{i}}^{\text{k}} a\left( v \right)n\left( {v,t} \right){\text{d}}v - \mathop \int \limits_{0}^{\infty } v^{k} a\left( v \right)n\left( {v,t} \right){\text{d}}v \\ & = \mathop \int \limits_{0}^{\infty } \left( {\varepsilon v^{k + f} + \varepsilon_{0} } \right)\zeta v^{{\frac{3}{{D_{f}^{2} }}}} n\left( {v,t} \right){\text{d}}v - \mathop \int \limits_{0}^{\infty } v^{k} \zeta v^{{\frac{3}{{D_{f}^{2} }}}} n\left( {v,t} \right){\text{d}}v \\ & = \mathop \int \limits_{0}^{\infty } \left( {\varepsilon v^{k + f} + \varepsilon_{0} - v^{k} } \right)\zeta v^{{\frac{3}{{D_{f}^{2} }}}} n\left( {v,t} \right){\text{d}}v \\ & = \varepsilon \zeta m_{{k + f + \frac{3}{{D_{f}^{2} }}}} + \varepsilon_{0} \zeta m_{{\frac{3}{{D_{f}^{2} }}}} - \zeta m_{{k + \frac{3}{{D_{f}^{2} }}}} . \\ \end{aligned} $$

(41)