## 4 EDC Model Constants and the Effects on Reaction Rate  
The constants used in the EDC model include the primary constants, \(C_{\mathrm{D1}}\) and \(C_{\mathrm{D2}}\), and the secondary constants, \(C_{\gamma}\), \(C_{\tau}\), and \(C_{\beta}\). By comparing the expressions of \(w^{*}\) and \(q^{*}\) in Equations (15) and (16) to those in [1,2], the relations between \(C_{\mathrm{D1}}\) and \(C_{\mathrm{D2}}\) can be expressed as  
\[\left\{ \begin{matrix}\frac{3}{2}C_{\mathrm{D1}}=6\zeta^{2}\\ C_{\mathrm{D2}}=15\zeta^{2}\end{matrix} \right. \tag{29}\]  
where \(\zeta\) is a numerical constant. Introducing \(\zeta=0.18\)[1,2] yields the accurate original values of the primary constants \(C_{\mathrm{D1}}=0.1296\) and \(C_{\mathrm{D2}}=0.486\). The accurate secondary constants can also be obtained, i.e., \(C_{\gamma}=2.1584\), \(C_{\tau}=0.4025\), and \(C_{\mathrm{R}}=11.57\). By comparing the expressions of \(u^{*}\) and \(L^{*}\) in Equations (17) and (18) to those in [1,2], the relations between \(C_{\mathrm{D1}}\) and \(C_{\mathrm{D2}}\) can be expressed as  
\[\left\{\begin{matrix}\left(\frac{C_{\mathrm{D2}}}{3C_{\mathrm{D1}}^{2}}\right) ^{1/4}=1.74\\ \frac{2}{3}\left(\frac{3C_  
where \(\frac{2}{3}C_{\rm D1}(=C_{\beta})\) is equal to the constant \(C_{\mu}=0.09\) in the widely used \(k\)-\(\varepsilon\) model [32]. This is also the reason for the approximation of \(C_{\rm D1}\) from 0.1296 or 0.1357 to 0.135.  
Based on the work of [PERSON] and [PERSON] [33], [PERSON] [21] concluded that the range of the constant ratio \(\frac{C_{\rm D2}}{C_{\rm D1}}(=C_{\alpha})\) is from 6 to 50, followed by \(C_{\rm D1}=0.135\), and \(C_{\rm D2}\) is from 0.11 to 0.91, where small values imply low Reynolds numbers. This \(C_{\alpha}\) value range might be larger, e.g., from 1 to 100. For example, an MILD combustion furnace had been successfully modelled with smaller \(C_{\alpha}\) values of 1 and 2 by [PERSON] et al. [27]. Since combustion commonly increases small-scale turbulence and dissipation at high wavenumbers (smaller scales) [5], \(C_{\rm D2}\) should be decreased.  
The mean reaction rate, Equation (28), can also be expressed as  
\[\overline{R}_{k}=\frac{\overline{p}Re_{\rm T}^{\frac{3}{2}}}{C_{\mu}\left(Re_ {\rm T}^{\frac{3}{2}}-C_{\gamma}^{3}\right)}\frac{\varepsilon}{k}\left(Y_{k} ^{*}-\widetilde{Y}_{k}\right) \tag{32}\]  
The effect of \(C_{\gamma}\) on \(\overline{R}_{k}\), as shown by Equation (28) or Equation (32), is not straightforward due to the strong coupling of \(\overline{R}_{k}\), the EDC constants, and the flow variables. Therefore, the effect of \(C_{\gamma}\) on \(\overline{R}_{k}\) in Equation (32) can be found from the partial derivative of \(\overline{R}_{k}\) with respect to \(C_{\gamma}\), i.e.,  
\[\frac{\partial\overline{R}_{k}}{\partial C_{\gamma}}=\frac{3C_{\gamma}^{2}}{ Re_{\rm T}^{\frac{3}{2}}-C_{\gamma}^{3}}\overline{R}_{k} \tag{33}\]  
It is reasonable to require that the mass fraction of the fine structure regions' \(\gamma_{\lambda}\) value be less than unity. \(C_{\gamma}^{4}\leq Re_{\rm T}\) is obtained from Equation (21) or Equation (23). This means the proposed EDC constants are constrained by the turbulent Reynolds number. In this case, the denominator of Equation (33) is positive. Therefore, Equation (33) shows the comparatively simple coupling between \(C_{\gamma}\) and the change rate of \(\overline{R}_{k}\) and indicates that increasing \(\overline{R}_{k}\) would require increasing \(C_{\gamma}\), and vice versa. Meanwhile, the largest proposed \(C_{\gamma}\) is limited to the lowest \(Re_{\rm T}\) value in the space. By reviewing the proposed EDC modifications and giving an analysis of the proposed EDC constants, [PERSON] [21] presented three criteria to evaluate the proposed constants. The first is the fine-structure region fraction \(\gamma_{\lambda}\). If \(\gamma_{\lambda}\) is closed to unity for a high turbulent Reynolds number \(Re_{\rm T}\), the set of constants can be ignored.