Friedrich G. Helfferich

Non-linear waves in chromatography. I: Waves, shocks, and shapes

Abstract Cause and effect in non-linear chromatography are examined from the point of view of wave theory. This first of four instalments is restricted to single-component systems and examines monotonic concentration variations and chromatographic peaks and bands. It uses the wave equation, which states the velocity at which a given concentration advances, to establish the properties of “waves,” that is, monotonic concentration variations. Depending on the sense of curvature of the isotherm, a wave may be self-sharpening or nonsharpening. A self-sharpening wave remains, or sharpens to become, a shock layer; a nonsharpening wave spreads indefinitely, eventually in proportion to traveled distance. The concentration profile of a shock layer depends on the shape of the isotherm and on the dispersive effect of non-idealities, of which resistance to mass transfer usually is the most important. Mass-transfer resistance in the moving phase causes “fronting:” mass-transfer resistance within the stationary sorbent causes “tailing.” It is therefore in general not possible to model shock layers with only a single, lumped mass-transfer parameter. The concentration profile of a nonsharpening wave depends almost exclusively on the shape of the isotherm. The knowledge of wave behavior is used to examine peak shapes in elution under overload conditions and bands in displacement. The peak shape in elution is almost entirely determined by the degree of overload and the shape of the isotherm. Wave theory confirms a rule previously stated by Knox that, in columns exceeding a certain length, samples containing the same amount of solute give peaks of essentially the same shape under conditions of predominant concentration overload, predominant volume overload, or any combination of the two. In displacement development, the final pattern can be established by determination of the lengths of the bands of the individual components according to Tiselius and separate calculation of the shock-layer profiles.

Non-linear waves in chromatography II. Wave interference and coherence in multicomponent systems

This second instalment on non-linear waves examines in detail the effects caused by interactions between different solutes in multicomponent systems. Emphasis is on concepts and qualitative insight, and mathematics is kept at a minimum. To avoid unnecessary complications, the assumption of ideal theory are taken for granted. In the discussion, no restrictions are imposed on the nature of equilibria between the moving and sorbent phases and on the number of components. However, all examples are of two- and three-component systems with competitive sorption equilibria.

Mass transfer and kinetics of ion exchange

These papers were written by leading scientists and technologists working in the field of ion exchange, and they discuss the synthesis, equilibria, transport phenomena, kinetics and design of ion exchange resins and membranes, both systematically and progressively. This volume also reflects recent developments in the field, with sections dealing with novel areas of application for ion exchange, such as, pollution control, energy conservation and the recovery and recycling of non-renewable materials.

Migration and fate of pollutants in soils and subsoils

Experts in soil and environmental sciences as well as in the theory of wave propagation and numerical modelling methods provide a comprehensive account of different aspects of pollutant migration in soils, aquifers and other geological formations. Emphasis is laid on the analysis of contributing phenomena and their interactions, modelling and the practical use of such knowledge and models for guidance in disposal operations, preventive measures to minimize ecological damage, the prediction of consequences of seepage, and the design of remedial actions. Topics covered include the chemical behaviour of soils, sorption and retardation, biochemistry of pollutants, ion exchange and kinetics of reactions in soils, measurement of adsorption and desorption, multiphase hydrodynamics, multicomponent wave theory and the coherence concept, nonlinear wave propagation in geological formations, multiphase convective transport, diffusion and fast reaction, modelling pollutant transport, numerical methods, dispersion of contaminants from landfills, risk analysis, water reuse and the radioactive soil contamination at Chernobyl.

Ion Exchange Kinetics--Evolution of a Theory

The theory of ion exchange kinetics has come a long way from its origins oven, a century ago but still has muck distance to cover. In the earliest speculations about “base exchange” in soils, forces other, than those of inanimate nature were suspected to have their part. This view was superseded by an almost equally speculative one regarding ton exchange as a chemical reaction of a kinetic order corresponding to its stoichiometric coefficients. Only the concentrated, extensive, theoretical and practical studies in connection with the Manhattan project in World War II revealed ion exchange as essentially a statistical redistribution of ions by diffusion, with a rate limited by mass transfer resistances in either the particle or the external fluid. The next step ahead ms taken in the 1950’s with the realization that ions, as carriers of electric charges, are subject to the electric field their own diffusion generates. In other words, they obey the Nernst-Planck equations more closely than Fick’s simpler laws. Actually, this was a rediscovery since the Nernst-Planck equations had been proposed as early as 1913 for ionic diffusion in glasses and had become stock in trade of membrane science ever since Teorell and K. H. Meyer published their pioneering studies in 1935.

Non-linear waves in chromatography III. Multicomponent Langmuir and Langmuir-like systems

This third instalment on non-linear waves examines in detail the chromatographic behavior of multicomponent systems whose sorption equilibria are uniformly competitive and without selectivity reversals. This very common class includes systems with multicomponent Langmuir isotherms, but is not restricted to them. An examination of particle velocities and wave velocities leads to a set of rules for what may and may not happen in a column if sorption equilibrium is of this type. As an example, all qualitative features of frontal analysis are deduced without calculation. The arguments also illustrate how, more generally, wave theory can be used to deduce chromatographic behavior from given equilibrium properties. In addition, an easy-to-use mathematical procedure is presented with which column responses in systems with Langmuir sorption isotherms can be calculated. The procedure is based on a transformation of the concentration variables. Compositions of plateau zones between waves, wave velocities, and sharpening criteria are obtained as simple algebraic expressions or conditions in terms of the values of the new variables in only the initial and entering fluids. The new variables are easily transformed back into concentrations where this is desired. Three fully calculated examples of five-component frontal analysis, separation of a three-component mixture by displacement development, and two-component elution from an overloaded column show how analytical solutions can be obtained for many cases of practical interest. As in Part II, the assumptions of ideal chromatography are taken for granted.

Models and physical reality in ion-exchange kinetics

Abstract Four cases of ion-exchange kinetics are pointed out in which premises of a mathematical model are incompatible with facets of physical reality. The cases are (1) calculation of nonlinear gradients in an unstirred Nernst “film”, (2) shell-core models in the absence of a mechanism that can produce shell-core behavior, (3) reaction-controlled shell-core models, and (4) models for macroporous beads in which the wrong diffusion step becomes rate-controlling.

Ion-exchange equilibria of amino acids on strong-acid resins: Theory

Abstract Equations describing equilibrium uptake of neutral, acidic, and basic amino acids by strong-acid cation exchangers as a function of pH and concentrations of amino acid and added electrolyte or buffer are derived.

Theory of multicomponent chromatography a state-of-the-art report

Abstract Chromatography at high concentration and with large sample size, as desired for preparative separations, involves non-linear isotherms and interference of sorbable components with one another. Theoretical developments of the last fifteen years and their implications for separation methods are reviewed in the light of the concepts of wave propagation and coherence.

Generalized model for multispecies ion-exchange kinetics including fast reversible reactions☆

Abstract The Nernst-Planck model for intraparticle-diffusion controlled ion exchange, describing ion exchange as interdiffusion of ions subject to electric coupling in a quasi-homogeneous phase, has been extended to multispecies systems with fast, reversible reactions at local equilibrium. A computer model has been set up, based on a reformulation of the Nernst-Planck equations and a “reaction matrix” composed of stoichiometric coefficients and reaction-coupling factors. The computer model is applicable to spherical beads and planar membranes.