where N is the number of sampling points on the cross-section; Xi and Xm represent the mass fraction of sampling point i and the optimal mass fraction, respectively.σ denotes the standard deviation of concentration and σmax is the maximum standard deviation of concentration in the specific cross-section perpendicular to the flow direction. M (0 ≤ M ≤ 1) is the mixing index in the specific cross-section. M = 0 corresponds to no mixing (for which σ = σmax) and M = 1 a complete mixing (for which σ = 0). As the mixing index increases, the species concentration in the cross-section becomes more uniform and the mixing efficiency is improved.