Modeling of temperature profile in semi-permeable membrane channel at non-isothermal conditions

Abstract

We present a model for thermo- and pressure-driven membrane processes accompanied by transmembrane water and heat fluxes combined in a single unit. High-pressure channel (high-pressure control volume) is characterized by low operating temperature and vice versa. A symmetric plate-and-frame membrane channel was considered in this study. An elementary parallelepiped was selected as a control volume. Balance equations were written over the control volume. A membrane element was represented as a set of conjugate semipermeable volumes. The model is based on the following assumptions: (1) incompressible fluid under steady-state conditions; (2) trans-membrane water flux is opposed to transmembrane heat flux. Transverse velocity profile is approximated by Berman distribution; (3) the mechanisms of transverse transport include: convection resulting from pressure differences and conduction resulting from temperature gradients; (4) the thickness of thermal layer is equal to half height of the channel (δt=H). The temperature polarization (TP) module reveals high sensitivity to thermo-physical properties and hydrodynamics such as to the coefficient of thermal diffusivity a=λ/cρ and to the transverse velocity, V(z). We find that (a) variation of the coefficient of thermal diffusivity in liquid phase from a= 2.0 10−7 m2/s to a= 1.0 10−7 m2/s will reduce TP module from 0.96 to 0.91; (b) variation of transverse velocity at the membrane surface from 5.5 10−5 to 1.5 10−5 m/s can decrease of TP module from 0.95 to 0.79. The developed solution can be used as a sub-model for quantitative estimation of TP phenomena in thermo-driven membrane operations. It can be applied for analysis of pressure-driven processes at non-isothermal conditions and be used as a mathematical algorithm for process analysis and optimization.