Modeling of silica synthesis in a laminar flame by coupling an extended population balance model with computational fluid dynamics

Figures & data

Table 1. Comparison of reported flame-made silica characteristics found in selected studies in the literature measured with in-situ techniques.

Reference	Experimental technique	Synthesis method	Precursor	$d_p$ [nm]	$R_g$ [nm]	$D_f$
Scheckman, McMurry, and Pratsinis (2009)	Mass-Mobility	DF	HMDSO	$12-93$	n/a	$1.7-2.4$
Camenzind et al. (2008) ‘S-2’	SAXS + TS	LF DF	HMDSO	$5-90$	$52-121$	$1.7-2.9$
Camenzind et al. (2008) ‘S-10’	SAXS + TS	TF DF	HMDSO	$9-12$	$45-93$	$1.75-2.13$
Kammler et al. (2005) ‘Cold’	USAXS	TF PM	HMDSO	$5-11$	$65-135$	$2.1-2.35$
Kammler et al. (2005) ‘Hot’	USAXS	TF PM	HMDSO	$8-27$	$40-170$	$1.95-2.58$
Kim and Choi (2003)	plane laser LS	flat PM	SiCl4	n/a	$50-220$	$1.3-2$
Cho and Choi (2000)	LS + TS	$H_2$ DF	SiCl4	$15-50$	$25-140$	$1.62-1.76$
Choi et al. (1999)	LS + TS	$H_2$ DF	SiCl4	$19-49$	$55-110$	$1.35-1.85$
Chang and Biswas (1992)	LS	DF	SiCl4	n/a	$292-303$	$2.17-2.8$
Chung, Sheu, and Tsai (1992)	laser LS	CF DF	SiCl4	$20-30$	$50-200$	n/a
Zachariah et al. (1989)	dynamic LS	CF DF	SiH4	$20-30$	n/a	n/a
Ulrich and Rieh (1982)	laser LS	PM	SiCl4	$7.6-18$	n/a	n/a

The data presented in the table correspond to: experimental technique (SAXS: small-angle X-ray scattering, USAXS: untra small-angle X-ray scattering, TS: thermophoretic sampling, LS: light scattering), method of synthesis (LF: laminar flow, TF: turbulent flow, CF: counterflow, DF: diffusion flame, PM: premixed flame), material used as a precursor, primary particle $d_p$ and aggregate $R_g$ sizes, fractal dimension $D_f.$

Figure 1. Experimental setup for the synthesis of the SiO₂ nanoparticles with a co-flow diffusion flame burner. Reprinted from ‘Nanostructure evolution: from aggregated to spherical SiO₂ particles made in diffusion flames’, by A. Camenzind, H. Schulz, A. Teleki, G. Beaucage, T. Narayanan and S.E. Pratsinis, European Journal of Inorganic Chemistry, 2008, vol. 6, p. 911–918. Copyright Wiley-VCH GmbH. Reproduced with permission.

Table 2. Global two-step reaction mechanism for HMDSO combustion (Feroughi et al. 2017).

Step	Reaction	A	n	$E_a$
$G_1$	HMDSO + OH → 2SiO + 6CH3 + H	$6\times {10}^{12}$	0.46	15000
$G_2$	SiO + H2O → SiO2(g) + H2	$8.5\times {10}^{10}$	0	5650

The reaction rate constant is given in the form $k=A{T}^n\hspace{0.17em}\exp [-E_a/(RT)],$ where $E_a$ stands for the activation energy. The units of the parameter A are expressed in terms of cm, s, and mol, while $E_a$ is given in cal/mol.

Table 3. Characteristic sintering times for silica nanoparticles tested in the current study.

Reference	Characteristic sintering time, ${\tau }_s$
Tsantilis, Briesen, and Pratsinis (2001)	$6.5·{10}^{-13}{d}_p\hspace{0.17em}\exp \hspace{0.17em}\left[\frac{8.3·{10}^4}{T}\left(1-\frac{d_{p,0}}{d_p}\right)\right]$
Ehrman, Friedlander, and Zachariah (1998)	$6.3·{10}^{-8}{d}_p\hspace{0.17em}\exp \hspace{0.17em}\left[\frac{6.1·{10}^4}{T}\right]$
Kirchhof et al. (2012)	$\frac{(1-1/e)r_s^2}{2^{2/3}(3.5·{10}^{-4})}\exp \hspace{0.17em}\left[\frac{4.4·{10}^4}{T}\right]$
Shekar et al. (2012)	$3.49·{10}^{-14}{d}_p\hspace{0.17em}\exp \hspace{0.17em}\left[\frac{11.44·{10}^4}{T}\left(1-\frac{5.76·{10}^{-9}}{d_p}\right)\right]$

The critical particle diameter in Tsantilis, Briesen, and Pratsinis (2001) expression is $d_{p,0}=1·{10}^{-9}$ m; $r_s$ in Kirchhof et al. (2012) expression denotes the radius of a completely coalesced spherical particle. The units are: m for $d_p,$ $d_{p,0},$ and $r_s,$ K for T and s for ${\tau }_s.$

Figure 2. Contour plots of (a) velocity and (b) temperature fields.

Figure 3. Contour plots of (a) silica volume fraction and (b) primary particle diameter fields.

Table 4. Inlet boundary conditions, where $u_{\mathit{inlet}}$ denotes the stream bulk inlet velocity in the streamwise direction.

Stream	Profile	$u_{\mathit{inlet}}$ [m/s]	T [${}^{°}$C]	Mole fractions [%]
Inner	Laminar	$U=6.715$	75	$X_{C{H}_4}=61.338;$ $X_{N_2}=36.803;$ $X_{\mathit{HMDSO}}=1.859$
$1^{st}$ annulus	Laminar	$U_1=2.057$	75	$X_{N_2}=100$
$2^{nd}$ annulus	Laminar	$U_2=8.57$	75	$X_{O_2}=100$
Co-flow air	Uniform	$U_{\mathit{air}}=0.4$	20	$X_{N_2}=76.83;$ $X_{O_2}=23.17$

The inlet velocity profile at the inflow is presented in Figure S2 of the supplementary information.

Figure 4. Comparison of numerical predictions (two-PBE model) with the experimental FTIR data of Camenzind et al. (2008) for flame temperature along the centerline and effect of radial averaging. An average experimental error of $\pm$60 K given by Kammler et al. (2002) was used in the figure for the FTIR data.

Figure 5. Comparison of numerical predictions (two-PBE model) with experimental data for flame temperature and effect of emission of thermal radiation from gas species and from particles. Results of the radial maximum temperature for each case are compared with the experimental FTIR data of Camenzind et al. (2008). An average experimental error of $\pm$60 K given by Kammler et al. (2002) was used in the figure for the FTIR data.

Figure 6. Comparison of the experimental SAXS data of Camenzind et al. (2008) with numerical predictions for (a) primary particle diameter, (b) radius of gyration of aggregates, (c) primary particle number concentration and (d) average number of primary particles per aggregate along the pathline of maximum silica volume fraction. Results were obtained with the two-PBE, the extended one-PBE, and the monodisperse model (described in Section 2).

Figure 7. Comparison of the experimental SAXS data of Camenzind et al. (2008) with numerical predictions for the geometric standard deviation of aggregate (agg) and primary particle (pp) size distributions along the pathline of maximum silica volume fraction.

Table 5. Definition of timescales for processes encountered in silica flame synthesis.

Process	Nucleation	Aggregation	Sintering	Flow
Timescale, $\tau$	$\frac{{\int }_0^{\infty }n(v)dv}{J}$	$\frac{{\int }_0^{\infty }n(v)dv}{\left|-\frac{1}{2}{\int }_0^{\infty }{\int }_0^{\infty }n(w)n(v)\mathit{dwdv}\right|}$	$\frac{{\int }_0^{\infty }{n}_p(v)dv-{\int }_0^{\infty }n(v)dv}{\left|-\frac{3}{{\tau }_s}\left(\frac{V}{v_{p,av}}-\frac{M_{23}}{v_{p,av}{}^{2/3}}\right)\right|}$	${\int }_0^x\frac{1}{u_x(x^{\prime })}d{x}^{\prime }$

Figure 8. Rates (absolute values) of kinetic processes (a) and timescales (b) along the pathline of maximum silica volume fraction. Results were obtained with the two-PBE model. Note that, for the sintering process in (a), the rate refers to the number concentration of primary particles, $N_p.$

Figure 9. Effect of sintering time on (a) primary particle diameter, (b) aggregate radius of gyration. Results were obtained with the two-PBE model. Comparison is also made with the experimental data of Camenzind et al. (2008).

Figure 10. Effect of nucleation model on (a) primary particle diameter and (b) aggregate radius of gyration. Three models are compared: instantaneous nucleation (InstNuc), classical nucleation theory (CNT) and the dimers-based approach that was used for the rest of the results. Results were obtained with the two-PBE model. Comparison is also made with the experimental data of Camenzind et al. (2008).

Table 6. Comparison of PBE models for describing aerosol dynamics and particle morphology.

Model	two-PBE	Extended one-PBE	Mono
No. of equations	116	61	3
PSD of aggregates	polydisperse	polydisperse	monodisperse
PSD of primary particles	polydisperse	monodisperse	monodisperse
CPU time (s) / timestep	1.14	0.32	0.187

The total CPU time per timestep includes all processes (flow field, scalar convection/diffusion, gas phase reaction, and PBE solution). Tabulation of the aggregation kernel was employed for the one-PBE model.

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