Please find the papers (doi) and links from where the first formula is taken:
Also, researchers who are using population balance to study mass transfer. For example,
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