Osmosis equation

For the near wall solution, from the slipping plane to 3 Debye lengths from the wall, the solution is restricted to cases where there are no inertial effects within this region. In other words the left hand side of (5) goes to zero and we are left with an equation of the same form as the DC electro-osmosis equation but now with an AC electric field. The solution to this equation is... 

In this expression the particle is considered to be exposed to a homogeneous electric field of strength . The shape of the particle is of no importance just as the electro-osmosis equation (11) is valid for an arbitrary shape of the porous plug. Restrictions on the applicability of expression (26) are similar to the case of electro-osmosis y. the double-layer must be thin compared with the dimensions of the particle, p. the particle must be insulating and the surface conductance at the interface must be so small that the distribution of the external field is practically uninfluenced by it. 

The 2eta potential (Fig. 8) is essentially the potential that can be measured at the surface of shear that forms if the sohd was to be moved relative to the surrounding ionic medium. Techniques for the measurement of the 2eta potentials of particles of various si2es are collectively known as electrokinetic potential measurement methods and include microelectrophoresis, streaming potential, sedimentation potential, and electro osmosis (19). A numerical value for 2eta potential from microelectrophoresis can be obtained to a first approximation from equation 2, where Tf = viscosity of the liquid, e = dielectric constant of the medium within the electrical double layer, = electrophoretic velocity, and E = electric field. 

The performance of reverse osmosis membranes is generaUy described by the water and salt fluxes (74,75). The water flux,/ is linked to the pressure and concentration gradients across the membrane by equation 4 ... 

The salt flux, across a reverse osmosis membrane can be described by equation 5 where is a constant and and < 2 the salt concentration differences across the membrane. 

Equations (22-86) and (22-89) are the turbulent- and laminar-flow flux equations for the pressure-independent portion of the ultrafiltra-tion operating curve. They assume complete retention of solute. Appropriate values of diffusivity and kinematic viscosity are rarely known, so an a priori solution of the equations isn t usually possible. Interpolation, extrapolation, even precuction of an operating cui ve may be done from limited data. For turbulent flow over an unfouled membrane of a solution containing no particulates, the exponent on Q is usually 0.8. Fouhng reduces the exponent and particulates can increase the exponent to a value as high as 2. These equations also apply to some cases of reverse osmosis and microfiltration. In the former, the constancy of may not be assumed, and in the latter, D is usually enhanced very significantly by the action of materials not in true solution. 

Transport equations, for the surface force-pore flow model, 21 640—641 Transport gasifier, 6 798 Transport models, reverse osmosis, 21 638-639... 

The thermodynamic approach does not make explicit the effects of concentration at the membrane. A good deal of the analysis of concentration polarisation given for ultrafiltration also applies to reverse osmosis. The control of the boundary layer is just as important. The main effects of concentration polarisation in this case are, however, a reduced value of solvent permeation rate as a result of an increased osmotic pressure at the membrane surface given in equation 8.37, and a decrease in solute rejection given in equation 8.38. In many applications it is usual to pretreat feeds in order to remove colloidal material before reverse osmosis. The components which must then be retained by reverse osmosis have higher diffusion coefficients than those encountered in ultrafiltration. Hence, the polarisation modulus given in equation 8.14 is lower, and the concentration of solutes at the membrane seldom results in the formation of a gel. For the case of turbulent flow the Dittus-Boelter correlation may be used, as was the case for ultrafiltration giving a polarisation modulus of ... 

The main emphasis in this chapter is on the use of membranes for separations in liquid systems. As discussed by Koros and Chern(30) and Kesting and Fritzsche(31), gas mixtures may also be separated by membranes and both porous and non-porous membranes may be used. In the former case, Knudsen flow can result in separation, though the effect is relatively small. Much better separation is achieved with non-porous polymer membranes where the transport mechanism is based on sorption and diffusion. As for reverse osmosis and pervaporation, the transport equations for gas permeation through dense polymer membranes are based on Fick s Law, material transport being a function of the partial pressure difference across the membrane. 

Taft equation (eq 16 in reference (36)) and reverse osmosis data on solute transport parameter Dam/K6 (defined by eq 12 later in this discussion) for different solutes and membranes (44,45,46), and (iv) the functional similarity of the thermodynamic quantity AAF+ representing the transition state free energy change (36) and the quantity AAG defined as... 

All symbols are defined at the end of the paper. Equation 10 defines the pure water permeability constant A for the membrane which is a measure of its overall porosity eq 12 defines the solute transport parameter D /K6 for the membrane, which is also a measure of the average pore size on the membrane surface on a relative scale. The Important feature of the above set of equations is that neither any one equation in the set of equations 10 to 13, nor any part of this set of equations is adequate representation of reverse osmosis transport the latter is governed simultaneously by the entire set of eq 10 to 13. Further, under steady state operating conditions, a single set of experimental data on (PWP), (PR), and f enables one to calculate the quantities A, Xy 2> point... 

The object of the foregoing discussion is two-fold eq 19 to 24, together with Figure 12, show how one can obtain the values of Daji/k6 of solutes for a very large number of membrane-solution systems from Dam/k6 data for a single reference solute such as sodium chloride they also show how the physicochemical parameters characterizing solutes, membrane-materials and membrane-porosities are integrated into the transport equations in the overall development of the science of reverse osmosis. 

The above equations show that for a reverse osmosis system specified In terms of y, 9, and X, any one of the six quantities (performance parameters) C], C2, C3, C3, X or X or T, and A uniquely fixes all the other five quantities (112). Further, since the relationships represented by the set of eq 34 to 41 Involve 8 equations with 12 unknowns, namely, y, 9, X, Z, A, C-j, C2, C2, C3, C3, C3 and X or X or T, by fixing any four Independent quantities Included In the above unknowns, eq 34 to 41 can be solved simultaneously to obtain the remaining 8 quantities. The utility of this approach to system analysis for reverse osmosis process design and predicting the performance of reverse osmosis modules Is Illustrated In detail In the literature (6d,105,107,108,111,112,113). 

Gibbs adsorption equation is an expression of the natural phenomenon that surface forces can give rise to concentration gradients at Interfaces. Such concentration gradient at a membrane-solution Interface constitutes preferential sorption of one of the constituents of the solution at the interface. By letting the preferentially sorbed Interfacial fluid under the Influence of surface forces, flow out under pressure through suitably created pores in an appropriate membrane material, a new and versatile physicochemical separation process unfolds itself. That was how "reverse osmosis" was conceived in 1956. 

In the literature, there are many transport theories describing both salt and water movement across a reverse osmosis membrane. Many theories require specific models but only a few deal with phenomenological equations. Here a brief summary of various theories will be presented showing the relationships between the salt rejection and the volume flux. 

Plate and frame systems offer a great deal of flexibility in obtaining smaller channel dimensions. Equations 4 and 5 show that the Increased hydrodynamic shear associated with relatively thin channels Improves the mass-transfer coefficient. Membrane replacement costs are low but the labor involved is high. For the most-part, plate and frame systems have been troublesome in high-pressure reverse osmosis applications due to the propensity to leak. The most successful plate and frame unit from a commercial standpoint is that manufactured by The Danish Sugar Corporation Ltd. (DDS) (Figure 15). 

One can transcribe the phenomenon in the form of an equation following the same thinking as for electro-osmosis. One says A current density j results not only from an electric field but also from a pressure difference AP, and, for small X and AP,... 

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