Residence Time in Mass Fraction Space

To show that residence time in mass fraction space obeys a linear mixing law, the traditional definition for t written in terms of volumetric flow rate is no longer suitable. Instead, we define, in an analogous fashion, the equivalent mass fraction residence time for a reactor i as follows  

the average mass fraction residence time a is defined as the total reactor volume divided by the total mass flow, 

Expressing all reactor volumes in a similar manner thus gives 

Develop an expression for the mass fraction residence time a in terms of standard volumetric residence time t. 

Converting between t and c is achieved in a straightforward manner when the mixture density is known. Hence, the two residence time expressions are 

---