Abstract
An investigation of the growth and the distribution of the fouling layer at the interface of pressure-driven membranes is presented. Also, its effect on the flow variables is studied. A model of the flow in the channel, the fouling layer and the membrane skin is developed. Then, the equations governing the flow and the rate of change for the fouling layer thickness are non-dimensionalized. Consequently, two parameters characterizing the effects of the inertia terms and the fouling layer are obtained. The undisturbed solution (Re(,w) = 0, S = 0) is first found, then the first-order perturbation problems for both Re(,w) << 1 and S << 1, are solved.
It is found that for relatively large values of E (which correspond to the ultrafiltration regime with moderate values of (DELTA)P across the porous media); the pressure, the suction velocity, and the fouling layer thickness decay exponentially along the channel and the present solution can not be approximated by similarity solutions. Whereas for small values of E (which correspond to some range of the hyperfiltration regime) the pressure, suction velocity and the fouling layer thickness decrease linearly along the X-axis and the solution is in good agreement with the similarity solutions. Also, it is noticed that the build-up of the fouling layer tends to decrease the driving pressure force and the presence of the inertia terms leads to larger pressure drop P(,i)-P(,e). To validate the previous solutions, another method is used to solve the problem and the restrictions made on Re(,w) and S are removed. The resulting equations are solved numerically. The results obtained using the integral method are in good agreement with those of the perturbation method (difference averages from 0.2% to 0.8%). Furthermore, the computional results show that the rate of growth of the fouling layer thickness declines in time to reach some steady state value. Finally, the onset and the location of the flow separation are predicted by the integral method and they are strongly affected by the profile of the fouling layer as well as the specified boundary conditions for both the flow and the fouling layer.