In this study

an optimization method is proposed to enhance the gas–liquid mass transfer in bubble column reactor based on the entropy generation extremum principle. The mass transfer–induced entropy generation can be maximized with the increase of mass transfer rate

based on which the velocity field can be optimized. The oxygen gas–liquid mass transfer is the major rate–limiting step of the toluene emissions biodegradation process in bubble column reactor

so the entropy generation due to oxygen mass transfer is used as the objective function

and the conservation equations of the gas–liquid flow and species concentration are taken as constraints. This optimization problem is solved by the calculus of variations

the optimal liquid flow pattern is obtained and the relationship of the maximum mass transfer enhancement on viscous dissipation is revealed

which can be used to improve the design of internal structure of the bubble column reactor.