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Hello Prashanth,

Please find the papers (doi) and links from where the first formula is taken:

https://doi.org/10.1016/j.ces.2007.01.065

https://doi.org/10.1016/j.ces.2018.01.009

https://ris.utwente.nl/ws/portalfiles/portal/5529968/numerical_modeling.pdf

https://www.hzdr.de/publications/PublDoc-10321.pdf

Also, researchers who are using population balance to study mass transfer. For example,

https://doi.org/10.1016/j.ijmultiphaseflow.2012.09.005

In this papers the (phi_b_dot) represent mass transfer rate, which is written in the form of molar concentration (not in the form of species fraction), that i similar to first formula.   

I can share more papers also where first formula is applied without taking volume fraction into consideration. So, I think everyone cannot be wrong. But I am not able to figure out why volume fraction is not a part of mass transfer rate.

Any lead will helpful

Thanks in advance

Sincerely

Krishna