## 2 Reaction Rates  
The approach of EDC is to solve partial differential or \"transport\" equations for the mean mass of individual species. Hence, the mean reaction rates have to be expressed. The reaction rate of EDC is the result of a mass balance for the fine-structure reactor, as described by [PERSON] [1; 2]. The inflow has the properties of the \"surroundings\" (denoted by \({}^{0}\)) of the fine-structure reactor and the interior and the outflow also have fine-structure properties (denoted by *), as shown in Figure 1.  
The mass inflow rate divided by the mass of the fine structures is denoted by \(\dot{m}^{*}\). The mass balance of the reactor gives a reaction rate for the reactor [21]  
\[R_{k}^{*}=\rho^{*}\dot{m}^{*}\left(Y_{k}^{*}-Y_{k}^{0}\right) \tag{1}\]  
The ratio of mass in fine structures to the total mass, or the mass fraction of the fine structures, is represented by \(\gamma^{*}\). The intermittent fine structures are assumed to be gathered in certain regions of the total flow, particularly in the interface between the bigger eddies. The ratio of mass in the fine structure regions to the total mass, known as the mass fraction of the fine structure regions, is denoted by \(\gamma_{\lambda}\). Furthermore, not all small eddies are assumed to have conditions that favor proceeding reactions. Therefore, the quantity \(\chi\) is used to denote the fraction of the fine structures that react. Accordingly, \(\gamma^{*}\chi\) is the reactor mass as a fraction of the total mass. In the original version of the EDC [1], \(\gamma^{*}=\gamma_{\lambda}^{3}\). Figure 1: Schematic fine-structure reactor. Reprinted with permission from Ref. [27]. 2022, Elsevier Ltd. The reactions are assumed to proceed in the fine structures. The mean reaction rate, which is the mass source term in the conservation equation for the mean species \(k\), is expressed as [21]  
\[\overline{R}_{k}=\frac{\overline{p}}{\rho^{*}}\gamma^{*}\chi R_{k}^{*} \tag{2}\]  
Substituting Equation (1) to Equation (2) yields  
\[\overline{R}_{k}=\frac{\gamma^{*}\overline{\rho}\chi}{\tau^{*}}\Big{(}\gamma_ {k}^{*}-\gamma_{k}^{0}\Big{)} \tag{3}\]  
where \(\tau^{*}=\frac{1}{n}\) is regarded as the characteristic time scale of the fine structures, or the fine structure residence time. This is known as the fluid dynamics time scale for chemical reactions, and is an important quantity for the treatment of chemical kinetics [3,4]. A mass-weighted average of the fine-structure reactor and its surroundings is  
\[\widetilde{Y}_{k}=\gamma^{*}\chi Y_{k}^{*}+(1-\gamma^{*}\chi)Y_{k}^{0} \tag{4}\]  
then  
\[Y_{k}^{*}-Y_{k}^{0}=\frac{Y_{k}^{*}-\widetilde{Y}_{k}}{1-\gamma^{*}\chi} \tag{5}\]  
Substituting Equation (5) to Equation (3) yields the mean reaction rate  
\[\overline{R}_{k}=\frac{\gamma^{*}\overline{\rho}\chi}{\tau^{*}(1-\gamma^{*} \chi)}\Big{(}Y_{k}^{*}-\widetilde{Y}_{k}\Big{)} \tag{6}\]  
It is convenient to define an \"EDC factor\" \(g_{\text{EDC}}\) as  
\[\overline{R}_{k}=g_{\text{EDC}}\frac{\overline{p}}{\tau^{*}}\Big{(}Y_{k}^{*}- \widetilde{Y}_{k}\Big{)} \tag{7}\]  
For the original version [1], the EDC factor of Equation (7) is  
\[g_{\text{EDC81}}=\frac{\gamma_{\lambda\lambda}^{3}\chi}{1-\gamma_{\lambda}^{3}\chi} \tag{8}\]  
In the 1989/1994 version [2,21], the reciprocal of \(\gamma_{\lambda}\) appeared as a factor in the reaction rate expression. Hence, the EDC factor of Equation (7) is  
\[R_{k}^{*}=\rho^{*}\dot{m}^{*}\left(Y_{k}^{*}-Y_{k}^{0}\right) \tag{1}\]0  
When using a detailed mechanism, [PERSON] and [PERSON] [4] compared the variable \(\chi\) to the simple form \(\chi=1\). The differences in the tested cases were modest. [PERSON] [21] pointed out that, while \(\chi\) was not a unity, it had become a part of the solution.