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The predicting equations for the mass-transfer rate and mass-transfer efficiency in a concentric circular membrane extractor module under concurrent- and countercurrent-flow with various barrier locations were derived theoretically by calculating the mass balance on each subchannel. The analytical solution was obtained by using the separation variable and eigen-function expansion in power series. The mass-transfer efficiency enhancement in this study is represented graphically with the volumetric flow rate and permeable-barrier location as parameters. The theoretical predictions show that the improvements in the extraction rate, extraction efficiency, and mass-transfer efficiency can be achieved by setting the barrier location moving away from the  ¼ 0.5. The influences of the barrier location, aqueous phase flow rate, and the concentrations of Cu2þ on the mass-transfer efficiency enhancement are also discussed.

The mathematical formulation of mass transfer through a laminar concentric circular membrane extractor with inserting permeable barrier has been investigated analytically. The analytical solution was solved by using the separation variable and eigen-function expansion in power series. The dimensionless concentration profiles in both subchannels, extraction rate, extraction efficiency and mass-transfer efficiency were Gz b Fig. 6. Extraction efficiency of the Cu2þ concentration and flow pattern as parameters.