A theoretical analysis of transport phenomena in membrane concentration of liquorice solutions: a FEM approach

Transport phenomena in membrane processes have been theoretically studied in a lab-scale flat geometry membrane module, fed with liquorice solutions. The proposed model is based on the solution of continuity, momentum and mass transfer equations, accounting for the non-newtonian behavior of liquorice solutions. The system of partial differential equations (PDEs) governing the membrane behavior has been discretized and solved according to the Finite Elements Method, by means of FEMLAB Chemical Engineering Module. The aim of the present study is to determine the range of operating conditions, in the laminar flow regime, where the concentration polarization phenomena, typical of membrane processes, are maintained within acceptable limits. Feed solution flow rate and concentration, membrane hydraulic permeability, operating trans-membrane pressure and fluid rheological properties were shown to play a crucial role for the system performances. These have been analyzed in terms of the permeate flux decay and of the dimensionless concentration of rejected species, measured on the membrane wall. The local mass transfer coefficient at the membrane surface has been also calculated, by definition, as the ratio between the local molar diffusive flux at the membrane surface and the concentration difference (Cw(x) - Cb). On this basis, a theoretical relationship between a local Sherwood number and a combination of characteristic dimensionless parameters has been derived. This relationship ...... shows a remarkable agreement with the theoretical correlation resulting from the solution of wellknown Graetz problem. The proposed approach is general and can be applied, therefore, to characterize the concentration polarization phenomena whenever a solution, whose rheological behavior is modeled by non-newtonian constitutive equation, is to be concentrated in a membrane module.