Extended log-normal method of moments for solving the population balance equation for Brownian coagulation

Figures & data

Figure 1. Various types of size distributions generated from Equation (11)(11) $$n(v,t)=\frac{N\left(t\right)}{6\sqrt{2\pi }v\hspace{0.17em}\text{ln}\hspace{0.17em}{\sigma }^{\prime }\left(t\right)}[\text{exp}\hspace{0.17em}(-\frac{{\hspace{0.17em}\text{ln}\hspace{0.17em}}^2(v/v_{g1}(t\left)\right)}{18{\hspace{0.17em}\text{ln}\hspace{0.17em}}^2{\sigma }^{\prime }\left(t\right)})+\hspace{0.17em}\text{exp}\hspace{0.17em}(-\frac{{\hspace{0.17em}\text{ln}\hspace{0.17em}}^2(v/v_{g2}(t\left)\right)}{18{\hspace{0.17em}\text{ln}\hspace{0.17em}}^2{\sigma }^{\prime }\left(t\right)}\left)\right],$$(11) .

Table 1. Values of $b_0$, $b_2$, and $b_3$ for various $\sigma$.

$\sigma$	$b_0$	$b_2$	$b_3$
1.0	0.7071	0.7071	0.7071
1.5	0.7664	0.7664	0.8029
2.0	0.8617	0.8617	0.9668
2.5	0.9366	0.9366	0.9978
3.0	0.9756	0.9756	0.9999
∞	1	1	1

Table 2. Variations of the zeroth moment, $M_0$, and the second moment, $M_2$, with time predicted by various methods in the continuum regime.

Method	$\tau =1$		$\tau =5$		$\tau =10$
	$M_0$	$M_2$	$M_0$	$M_2$	$M_0$	$M_2$
Laguerre quad.	0.326	5.45	0.0880	22.1	0.0460	42.9
FEM 70 points	0.326	5.48	0.0867	22.2	0.0450	43.1
SM	0.326	5.45	0.0868	22.0	0.0451	42.7
TEMOM	0.327	5.45	0.0879	22.1	0.0459	42.9
LNMOM	0.327	5.45	0.0881	22.0	0.0460	42.8
ELNMOM (present)	0.326	5.45	0.0872	22.0	0.0454	42.8

Figure 2. Relative error of $N$ for the LNMOM, the ELNMOM and the TEMOM for different initial cases in the continuum regime: (a) ${\sigma }_0$=1.1; (b) ${\sigma }_0$=1.4; (c) ${\sigma }_0$=1.7.

Table 3. Maximum relative error of $N$ for the LNMOM, the ELNMOM and the TEMOM for different initial cases in the continuum regime and the free-molecular regime.

Method	RE (continuum)			RE (free-molecular)
	${\sigma }_0$=1.1	${\sigma }_0$=1.4	${\sigma }_0$=1.7	${\sigma }_0$=1.1	${\sigma }_0$=1.4	${\sigma }_0$=1.7
LNMOM	3.01%	3.35%	4.27%	7.55%	7.89%	9.02%
ELNMOM	1.42%	1.60%	3.99%	1.97%	2.06%	6.90%
TEMOM	2.70%	3.05%	--	4.59%	4.89%	6.69%

Figure 3. Relative error of $v_g$ for the LNMOM and the ELNMOM for different initial cases in the continuum regime.

Figure 4. Variations of $\sigma$ with time for different initial ${\sigma }_0$ predicted by the LNMOM, the ELNMOM and the SM in the continuum regime.

Figure 5. Time evolution of the particle size distribution predicted by the LNMOM, the ELNMOM and the SM for ${\sigma }_0$=1.4 in the continuum regime.

Figure 6. Relative error of $N$ for the LNMOM, the ELNMOM and the TEMOM for different initial cases in the free-molecular regime: (a) ${\sigma }_0$=1.1; (b) ${\sigma }_0$=1.4; (c) ${\sigma }_0$=1.7.

Figure 7. Relative error of $v_g$ for the LNMOM and the ELNMOM for different initial cases in the free-molecular regime.

Figure 8. Variations of $\sigma$ with time for different initial ${\sigma }_0$ predicted by the LNMOM, the ELNMOM and the SM in the free-molecular regime.

Figure 9. Time evolution of the particle size distribution predicted by the LNMOM, the ELNMOM and the SM for ${\sigma }_0$=1.4 in the free-molecular regime.

Table 4. Computation time for the LNMOM, the ELNMOM and the TEMOM for ${\sigma }_0$=1.4 in the continuum regime and the free-molecular regime.

Method	Time (continuum)	Time (free-molecular)
LNMOM	0.114s	0.146s
ELNMOM	0.177s	0.244s
TEMOM	0.050s	0.064s