The mass transfer problems in the hollow-fiber membrane extractor module with concurrent- and countercurrent-flow were investigated theoretically and experimentally in this study. A two-dimensional mathematical model of the hollow-fiber membrane extractor module was developed theoretically and the shell side flow described by Happel's free surface model was taken into account. The analytical solution is obtained using an eigenfunction expansion in terms of the power series and an orthogonal expansion technique. The theoretical predictions were represented graphically with the mass-transfer Graetz number (Gz), flow pattern and packing density (phi) as parameters and the theoretical results were also compared with those obtained by experimental runs. The highest extraction rate, extraction efficiency and mass transfer efficiency can be achieved by arranging the packing density phi = 0.3. The results show that the device performance of the countercurrent-flow device is better than that of the concurrent-flow device. The experiments of the extraction of Cu2+ by using D2EHPA with PVDF hollow fibers is also set up to confirm the accuracy of the theoretical predictions. The accuracy of the theoretical predictions for concurrent- and countercurrent-flow are 5.87 x 10(-2) <= E-1 <= 6.69 x 10(-2) and 2.46 x 10(-2) <= E-1 <= 3.48 x 10(-2), respectively, for Gz(a) = 40.8. (C) 2009 Elsevier Ltd. All rights reserved.