## 1 Introduction  
Natural and environmental engineering systems are considered to be homogeneous systems in which the distribution of matter in the system is neglected at microscopic molecular scale movements and is mainly influenced by macroscopic physical migration phenomena, which affect the reaction energy and rate of reaction. In the homogeneous system, the conversion rate of the chemical is controlled either chemically or kinetically. Generally, it is very complicated to calculate the nonlinear chemical reaction rate in the heterogeneous system, so it tends to be ignored or simplified when assuming the homogeneous system [1]. However, the interaction between two molecules where an actual reaction takes place in a real system is a heterogeneous system in which small microscopic molecular scale turbulent migration phenomena are considered to be important [1]. The asymmetric mixing process strongly controls the chemical reaction rate when the timescale of the chemical reaction is of the same order or faster than the turbulent timescale [1; 2; 3; 4; 5; 6; 7; 8]. The chemical reaction rate requires intimate mixing of the species at the molecular-scale (microscale), while dispersion is effected primarily by large-scale turbulent eddies with a nonlinear cascade process resulting in micro-mixing [8]. The largest mixing scale in the turbulent reacting flow is represented by the turbulent Damkohler (\(D_{t}\)) (or non-dimensional reaction rate), which is a ratio of the turbulent time scale (\(\tau_{t}\)) to chemical time scale (\(\tau_{ch}\)). The \(D_{t}\) affects the segregation between the species in the asymmetric mixing process related to the chemical reaction [9]. Especially, the segregation of the species represents the influence of incomplete mixing on the chemical reaction rate when the chemical reaction time scale is similar to the turbulent time scale (\(|D_{t}|\geq 1\)). If the chemical reaction is extremely slow relative to the mixing processes (\(|D_{t}|\ll 1\)), the reactants become effectively well mixed. Also, the reacting species remain segregated in the case of very fast chemical reactions relative to mixing (\(|D_{t}|\gg 1\)). In a heterogeneous system, it is limited to the overall reaction rate due to the microscopic mass transfer phenomenon and can be expressed as the separation intensity, \(I_{s}\)[3; 4; 5; 7]. The intensity of segregation \(I_{s}\) clearly depends on the turbulent Damkohler number and it can be related to the concentration fluctuation \(\overline{c^{\prime}_{A}c^{\prime}_{B}}\) and the mean concentrations \(\overline{C}_{A}\) and \(\overline{C}_{B}\) of the reactive species \(A\) and \(B\); \(I_{s}=\overline{c^{\prime}_{A}c^{\prime}_{B}}/\left(\overline{C}_{A}\overline {C}_{B}\right)\). The concentration fluctuation needs to be well specified because the turbulent mixing effect on the chemical reaction rate can be measured by the concentration fluctuation. The effective (nonlinear) reaction rate is varied with the concentration covariance (fluctuation) that is a measure of segregation between two reactants, is directly related to the micromixing phenomena, and then is lowered or enhanced depending on whether the covariance is negative (non-premixed reactants) or positive (premixed reactants). Thus, the effective reaction rates, \(R_{e}\), in terms of the covariance can be rewritten in terms of the intensity of segregation \(I_{s}\). \[R_{e}=-k_{f}\left(\overline{C}_{A}\overline{C}_{B}+\overline{c^{\prime}_{A}c ^{\prime}_{B}}\right)\]  
\[R_{e}=-k_{f}(1+I_{s})\overline{C}_{A}\overline{C}_{B}\]  
The turbulent mixing effect on the chemical reaction rate can expressed as a function of the intensity of segregation \(I_{s}\) (\(I_{s}=-1\) for the complete segregation between the species, \(I_{s}=0\) for the perfectly mixed species by the turbulence, and \(I_{s}>0\) or \(I_{s}<0\) for the enhanced or lowered reaction rate compared to the perfectly mixed case).