William N. Gill

REVIEW OF REVERSE OSMOSIS MEMBRANES AND TRANSPORT MODELS

Abstract After a brief introduction to membrane processes in general, and the reverse osmosis process in particular, the structure and properties of membranes and membrane transport theory are described. The mechanism of salt rejection and transport properties of membranes are discussed in detail. Solubility, diffusivity, and permeability of membranes to solutes and solvents are reviewed critically and compared with each other. Special attention is given to two particular types of membranes, cellulose acetate (CA) and aromatic polyamide (AP) membranes, which are often used for water desalination. The major portion of this article is devoted to the review and discussion of membrane transport theory with application to the reverse osmosis and ultrafiltralion processes. It is shown that the solvent flux can be represented reasonably well by linear models such as the solution-diffusion model (Lonsdale, et al., 1965). The contribution of pore flow to the solvent flux is small. The solute flux, however, is not ...

Hollow fiber reverse osmosis: experiments and analysis of radial flow systems

The performance of a hollow fiber reverse osmosis system was determined experimentally by measuring the fraction of feed recovered as product, φ, and the concentration reduction ratio, θp. A predictive model for these two quantities was developed which yields results that agree with the experimental data very well for applied pressures of 300–450 psig., feed rates from 75 to 525 cc/sec and feed concentrations up to 15000 ppm of sodium chloride. n nThe concept of distributed sinks in a continuum is used for the shell side, and a diffusion model is used to describe solute transport across the fiber wall. The principal assumptions made are negligible concentration polarization, small radial pressure drop in the shell and small solute transport, compared with solvent transport, through the fiber. Simple closed form expressions for φ and θp, given by eqns (43a) and (51), were obtained which are accurate over the range of experimental conditions employed.

Foced convection heat and momentum transfer to dendritic structures (parabolic cylinders and paraboloids of revolution)

Abstract Three fundamental theoretical flow models including Oseen type rectilinear flow, potential flow, and Oseens viscous flow approximation, are developed for forced convection heat transfer to parabolic cylinders and paraboloids of revolution which are relevant to dendritic growth. Closed-form results are obtained and presented in terms of Nusselt numbers at the stagnation point which are related to downstream values by analytical expressions. The heat flux from paraboloids of revolution is much larger than that for parabolic cylinders, especially at smaller values of the Peclet number. Local expressions for the heat flux are consistent with the shape preserving growth of these configurations as isothermal dendrites in forced convection fields. Diffusion of momentum and energy extends to significantly greater distances into the flow for parabolic cylinders than paraboloids of revolution. Boundary-layer assumptions would lead to large errors for the conditions studied here, especially for low Pr fluids. The results are useful for describing the diffusion of mass also.

HYDRODYNAMIC CHROMATOGRAPHY: THREE DIMENSIONAL LAMINAR DISPERSION IN RECTANGULAR CONDUITS WITH TRANSVERSE FLOW

Experimental observations1,9 indicate much poorer separations than are predicted by two dimensional theory. The purpose of this work is to explain these differences and suggest ways in which system performance can be improved. The large effect of span-wise variation in axial velocity caused by side walls on hydrodynamic separations carried out in rectangular conduits with transverse flow is studied theoretically. As the aspect ratio increases, the steady stale retentivity (convection coefficient) approaches an asymptotic value obtained by neglecting side wall effects. However, the dispersion coefficient does not reduce to that for a flow with no side walls. Indeed, the asymptotic steady state dispersion coefficient is at least six times larger than that obtained by two dimensional theory which neglects side wall effects. As the transverse Peclet number increases, the effect of side walls on the dispersion coefficient becomes much larger. The present three dimensional theoretical predictions, in contrast t...

RADIAL FLOW HOLLOW FIBERREVERSE OSMOSIS: EXPERIMENTS AND THEORY

Abstract The performance of a hollow fiber reverse osmosis system is studied both theoretically and experimentally. Experiments were carried out for applied pressure ranging from 200 to 400 psig, feed rates varying from 75 to 380 cc/sec and for feed concentrations up to 34,000xa0ppm of sodium chloride. A mathematical model is proposed to predict productivity, ϕ, and product concentration, θp. The model involves solving membrane transport equations simultaneously with the hydrodynamic equations. The solubility-diffusion-imperfection, or pore diffusion model, is used to describe solute and solvent transport across the membrane. The axial gradients of shell side concentration, neglected in previous investigations, are taken into account. The differential equations are solved numerically by the 4th Order Runge-Kutta method. Predicted values of ϕ and θp agree within 8% and 17% respectively, with experimental data over the entire range of operating conditions. However, membrane transport coefficients were found t...

Feasibility of colloidal particles separation by potential barrier chromatography

Abstract Computations are carried out to show that potential barrier chromatography based upon the existence of a surmountable potential barrier between particles and deposition surface may be useful in separating particles smaller than about 2500 A. The method appears to have such a high sensitivity that it is able to perform separation of particles which differ by only 5% in size or by about 1% in surface potential, or 1% in Hamakers constant.

A note on fluoride removal by reverse osmosis

Abstract It is shown that reverse osmosis membrane transport data for sodium fluoride solutions are correlated by a linear plot of 1/R vs. 1/Jv as predicted by Pusch [9]. The slope of the plot is independent of feed concentration over a 5-fold range from about 360 to 1800 ppm. The reject coefficient at infinite volume flux, R∞, is slightly less than unity.

An experimental study of the complete-mixing model for radial flow hollow fiber reverse osmosis systems

Abstract The complete mixing model of hollow fiber systems, suggested by Soltanieh and Gill [1–3] predicts that 1/ R is linear in Sm/Fp, regardless of the membrane transport model which is included in it, and this is confirmed experimentally for the units studied here. The plug flow model, which has been used by all previous investigators, fails to predict this observed fact. It is shown experimentally, by using two radial flow hollow fiber modules of different lengths, that the linear relationship between 1/ R and Sm/Fp exists at all levels of concentration (from 1000 to 35,000 ppm), pressure (from 150 to 400 psi), and feed flow rate (from 75 cm3/s to 400 cm3/s). Furthermore, at higher concentrations, plots of 1/ R vs Sm/Fp become independent of concentration and are represented by a single line. The pure water permeability coefficient (A) at 25°C ranges from 7.393 × 10−12 to 9.277 × 10−12 cm2/s/g for the large module, and from 4.764 × 10−12 × 10−12 to 4.950 × 10−12 cm2/s/g for the small module when pressure decreases from 400 to 200 psi. Furthermore, A is not a strong function of pressure; it decreases by about 20% for the large module (the more permeable membrane) and about 4% for the small module, when pressure increases by a factor of two. The solute permeability coefficient (K2) determined by two methods is relatively constant with pressure and is a function of concentration only up to a certain level, after which it remains constant. Based on the complete-mixing model, K2 for the large module remains constant at about 1.48 × 10−5 cm/s above a concentration of about 10,000 ppm and for the smaller module (with less permeable membrane) K2 becomes constant at about 0.216 × 10−5 cm/s above a concentration of about 5000 ppm. The solute permeabilities calculated from the complete-mixing model are 10 to 25% lower, but show the same trend with concentration as those obtained by using the plug flow approach of Dandavati et al [4].

Unsteady reverse osmosis or ultrafiltration in a tube

Abstract The unsteady behavior of a tubular hyperfiltration system, with both a pulse and a continuous feed, is analyzed using a significant extension of miscible dispersion theory. The extension permits one to treat unsteady convective diffusion in systems with convection at interfacial boundaries in a relatively straightforward manner. It is shown that such convection can effect an enormous reduction in axial dispersion in such systems. A superposition integral is used to generalize the solution for the pulse input to reverse osmosis systems with continuous feeds which may be subject to arbitrary fluctuations in feed concentrations.

Coverage dependent rate of cell deposition.

A quantitative criterion is developed for the applicability of a previous diffusional model for cell deposition on surfaces, using as a starting point a modified Fokker-Planck equation. The previous model for cell deposition through a stagnant liquid is extended to include the coverage-dependent effect of the deposited cells on subsequent deposition. This effect is shown to be responsible for an experimentally measured increase in the rate of cellular deposition with time, in contrast with the ever-decreasing rate of deposition predicted by the previous model. The present model also predicts that the decay in the fraction of cells in solution depends on the initial number of cells, again in contrast with the prediction of the previous model.