Equation of Mass Conservation and Its Closure

For the mass transfer at low Reynolds number flow (no turbulence), the conversation equation of a mass species (component substance) is known to be 

In chemical engineering practice, the mass transfer equipments, like distillation, absorption, and many others, are operated under turbulent flow condition. The m, and c in Eq. (3.1) becomes instantaneous value, and their fluctuating character shoidd be considered. 

Similar to the average concept in CFD, the instantaneous concentration c in turbulent mass transfer can be resolved into time-averaged concentration C and fluctuating concentration c  

Substitute foregoing relationship to Eq. (3.1) and take time average of each term, and note that 

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