Surface Reactions Models

Heterogeneous surface reactions are described using the hydrogen-abstraction–acetylene-addition (HACA) mechanism. Soot growth in the HACA mechanism proceeds through a sequence similar to PAH growth. Hydrogenated armchair sites, \(\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}\), located at the edges of aromatic structures are dehydrogenated through hydrogen abstraction to form radical sites, \(\mathrm{C}_{\mathrm{soot}}^{\circ}\). These radical sites react with \(\mathrm{C_2H_2}\), producing an additional aromatic ring with hydrogenated surface sites.

The radical and hydrogenated surface sites can also react with \(\mathrm{O_2}\) and \(\mathrm{OH}\), respectively, resulting in the removal of carbon from soot particles through oxidation. The elementary reactions used to describe these processes are listed in Table 3.

The soot mass-growth rate through HACA is obtained from the reaction of \(\mathrm{C_2H_2}\) with dehydrogenated surface sites:

(160)\[\omega_{\mathrm{gr}}^i = \alpha^i k_{f,4} [\mathrm{C_2H_2}] [\mathrm{C}_{\mathrm{soot}}^{\circ,i}].\]

Here, \(k_{f,4}\) is the forward rate coefficient of Reaction (172). The concentration of dehydrogenated sites, \([\mathrm{C}_{\mathrm{soot}}^{\circ,i}]\), is obtained by multiplying the surface density of dehydrogenated sites by the total soot surface area per unit mass of gas mixture in section \(i\):

(161)\[[\mathrm{C}_{\mathrm{soot}}^{\circ,i}] = \frac{\rho}{Av} A_{\mathrm{tot}}^i \chi_{\mathrm{soot}}^{\circ}.\]

The surface density of dehydrogenated sites, \(\chi_{\mathrm{soot}}^{\circ}\), is calculated by applying a steady-state approximation to \([\mathrm{C}_{\mathrm{soot}}^{\circ}]\) for the reaction system listed in Table 3:

(162)\[\chi_{\mathrm{soot}}^{\circ} = \frac{ k_{f,1}[\mathrm{H}] + k_{f,2}[\mathrm{OH}] }{ k_{r,1}[\mathrm{H_2}] + k_{r,2}[\mathrm{H_2O}] + k_{f,3}[\mathrm{H}] + k_{f,4}[\mathrm{C_2H_2}] + k_{f,5}[\mathrm{O_2}] } \chi_{\mathrm{soot-H}}.\]

The surface density of hydrogenated sites, \(\chi_{\mathrm{soot-H}}\), is estimated by assuming that the soot surface is composed of outward-facing PAH edges assembled into turbostratic structures [44]. Using an interlayer spacing of 3.15 \(\mathrm{\mathring{A}}\) and two C–H bonds per benzene-ring length gives

\[\chi_{\mathrm{soot-H}} = 0.23\ \mathrm{site\,\mathring{A}^{-2}} = 2.3\times10^{19}\ \mathrm{site\,m^{-2}},\]

which represents the maximum theoretical surface-site density.

In Equation (160), \(\alpha^i\) is the surface-reactivity factor. It ranges from 0 to 1 and represents the reduction in the number of available reaction sites relative to the theoretical maximum because of PAH-layer orientation, particle aging, surface growth, and soot maturity [45, 46]. The surface-reactivity factor has also been observed to depend on the temperature–time history of soot particles [47, 48].

The value of \(\alpha\) has been represented using constant, application-specific values and empirical expressions based on particle size and flame temperature. A detailed review is provided in Chapter 4 of Veshkini [49]. Omnisoot can calculate \(\alpha^i\) using the empirical expression proposed by Appel et al. [1]:

(163)\[\alpha^i = \tanh \left[ \frac{ 12.56-0.00563T }{ \log_{10} \left( \frac{\rho_{\mathrm{soot}}Av}{W_{\mathrm{carbon}}} \frac{\pi}{6} \left(d_p^i\right)^3 \right) } - 1.38 + 0.00068T \right].\]

Alternatively, \(\alpha^i\) can be related to the H/C ratio of soot particles by assuming that all hydrogen atoms reside on the particle surface [31]:

(164)\[\alpha^i = \frac{ H_{\mathrm{tot}}^i }{ C_{\mathrm{tot}}^i }.\]

The HACA contributions to the carbon and hydrogen source terms are calculated from the HACA growth rate by accounting for the two carbon atoms in \(\mathrm{C_2H_2}\) and the relative numbers of armchair and zigzag hydrogenated sites on the soot surface [10]:

(165)\[I_{C_{\mathrm{tot}},\mathrm{haca}}^i = \frac{ 2\omega_{\mathrm{gr}}^i }{ \rho }.\]

(166)\[I_{H_{\mathrm{tot}},\mathrm{haca}}^i = \frac{ 0.25\omega_{\mathrm{gr}}^i }{ \rho }.\]

The rates of change of the concentrations of \(\mathrm{C_2H_2}\) and H radicals due to HACA growth are

(167)\[\left( \frac{ \mathrm{d}[\mathrm{C_2H_2}] }{ \mathrm{d}t } \right)_{\mathrm{gr}} = -\sum_{i=1}^{n_{\mathrm{sec}}} \omega_{\mathrm{gr}}^i.\]

(168)\[\left( \frac{ \mathrm{d}[\mathrm{H}] }{ \mathrm{d}t } \right)_{\mathrm{gr}} = 1.75 \sum_{i=1}^{n_{\mathrm{sec}}} \omega_{\mathrm{gr}}^i.\]

The HACA surface reactions are

(169)\[\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H} + \mathrm{H} \overset{k_{f,1}}{ \underset{k_{r,1}}{\rightleftharpoons} } \mathrm{C}_{\mathrm{soot}}^{\circ} + \mathrm{H_2}.\]

(170)\[\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H} + \mathrm{OH} \overset{k_{f,2}}{ \underset{k_{r,2}}{\rightleftharpoons} } \mathrm{C}_{\mathrm{soot}}^{\circ} + \mathrm{H_2O}.\]

(171)\[\mathrm{C}_{\mathrm{soot}}^{\circ} + \mathrm{H} \overset{k_{f,3}}{\longrightarrow} \mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}.\]

(172)\[\mathrm{C}_{\mathrm{soot}}^{\circ} + \mathrm{C_2H_2} \overset{k_{f,4}}{\longrightarrow} \mathrm{C}_{\mathrm{soot}}{-}\mathrm{H} + \mathrm{H}.\]

(173)\[\mathrm{C}_{\mathrm{soot}}^{\circ} + \mathrm{O_2} \overset{k_{f,5}}{\longrightarrow} 2\mathrm{CO} + \mathrm{product}.\]

(174)\[\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H} + \mathrm{OH} \overset{k_{f,6}}{\longrightarrow} \mathrm{CO} + \mathrm{product}.\]

Table 3 Arrhenius rate coefficients for the HACA surface reactions, \(k=AT^n\exp[-E/(RT)]\).

Reaction	Pathway	Direction	\(A\) [\(\mathrm{m^3\,mol^{-1}\,s^{-1}}\)]	\(n\)	\(E/R\) [K]
(169)	\(\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}+\mathrm{H}\rightleftharpoons\mathrm{C}_{\mathrm{soot}}^{\circ}+\mathrm{H_2}\)	Forward	\(4.17\times10^7\)	0	6542.52
(169)		Reverse	\(3.9\times10^6\)	0	5535.98
(170)	\(\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}+\mathrm{OH}\rightleftharpoons\mathrm{C}_{\mathrm{soot}}^{\circ}+\mathrm{H_2O}\)	Forward	\(1.0\times10^4\)	0.734	719.68
(170)		Reverse	\(3.68\times10^2\)	1.139	8605.94
(171)	\(\mathrm{C}_{\mathrm{soot}}^{\circ}+\mathrm{H}\rightarrow\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}\)	Forward	\(1.0\times10^4\)	0.734	719.68
(172)	\(\mathrm{C}_{\mathrm{soot}}^{\circ}+\mathrm{C_2H_2}\rightarrow\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}+\mathrm{H}\)	Forward	80	1.56	1912.43
(173)	\(\mathrm{C}_{\mathrm{soot}}^{\circ}+\mathrm{O_2}\rightarrow2\mathrm{CO}+\mathrm{product}\)	Forward	\(2.2\times10^6\)	0	3774.53
(174)	\(\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}+\mathrm{OH}\rightarrow\mathrm{CO}+\mathrm{product}\)	Forward	\(\gamma_{\mathrm{OH}}=0.13\)	—	—

Carbon atoms on the soot surface are oxidized through reactions with \(\mathrm{O_2}\) and \(\mathrm{OH}\), represented by Reactions (173) and (174), respectively. These pathways decrease the total carbon content of soot and release gaseous products.

The \(\mathrm{O_2}\)- and \(\mathrm{OH}\)-oxidation rates are

(175)\[\omega_{\mathrm{ox,O_2}}^i = \alpha^i k_{f,5} [\mathrm{O_2}] [\mathrm{C}_{\mathrm{soot}}^{\circ,i}].\]

(176)\[\omega_{\mathrm{ox,OH}}^i = \gamma_{\mathrm{OH}} \beta_{\mathrm{OH}}^i Av [\mathrm{OH}] [\mathrm{soot}^i].\]

Here, \(\gamma_{\mathrm{OH}}=0.13\) is the reaction probability for collisions between OH radicals and soot particles [1]. The collision frequency between OH and soot particles, \(\beta_{\mathrm{OH}}^i\), is calculated from kinetic theory:

(177)\[\beta_{\mathrm{OH}}^i = \sqrt{ \frac{\pi k_B T}{2} \left( \frac{1}{m_{\mathrm{agg}}^i} + \frac{1}{m_{\mathrm{OH}}} \right) } \left( d_c^i+d_{\mathrm{OH}} \right)^2.\]

The mass and equivalent diameter of an OH radical are \(m_{\mathrm{OH}}=2.824\times10^{-26}\) kg and \(d_{\mathrm{OH}}=0.3\) nm, respectively [50].

The oxidation contribution to the total-carbon source term is calculated by accounting for the number of carbon atoms removed through each pathway:

(178)\[I_{C_{\mathrm{tot}},\mathrm{ox}}^i = \frac{ 2\omega_{\mathrm{ox,O_2}}^i + \omega_{\mathrm{ox,OH}}^i }{ \rho }.\]

The rates of change of the concentrations of \(\mathrm{CO}\), \(\mathrm{O_2}\), \(\mathrm{OH}\), and H due to oxidation are

(179)\[\left( \frac{ \mathrm{d}[\mathrm{CO}] }{ \mathrm{d}t } \right)_{\mathrm{ox}} = 2 \sum_{i=1}^{n_{\mathrm{sec}}} \omega_{\mathrm{ox,O_2}}^i.\]

(180)\[\left( \frac{ \mathrm{d}[\mathrm{O_2}] }{ \mathrm{d}t } \right)_{\mathrm{ox}} = -\sum_{i=1}^{n_{\mathrm{sec}}} \omega_{\mathrm{ox,O_2}}^i.\]

(181)\[\left( \frac{ \mathrm{d}[\mathrm{OH}] }{ \mathrm{d}t } \right)_{\mathrm{ox}} = -\sum_{i=1}^{n_{\mathrm{sec}}} \omega_{\mathrm{ox,OH}}^i.\]

(182)\[\left( \frac{ \mathrm{d}[\mathrm{H}] }{ \mathrm{d}t } \right)_{\mathrm{ox}} = \sum_{i=1}^{n_{\mathrm{sec}}} \omega_{\mathrm{ox,OH}}^i.\]