Linear Isotherm System—Simple Models

One of the simple models used to describe adsorption in a column is the two-phase exchange model. 

Analytical solution of the chromatographic elution curve of this model is given by Eq. (6-68). 

This equation can be approximated in terms of error function for the range S 40 as 

By equating the second moments of the chromatographic elution curves obtained from the general model and the simple model, k in the two-phase exchange model is replaced by the overall mass transfer coefficient, KeOv, which is defined as 

by using the estimated rate parameters of adsorption in a column, a rough estimation of the breakthrough curve is possible by means of the simple model, provided the column is long enough to 

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The analytical solutions presented above are most of all derived on the basis of the very simple Henry isotherm or the more physically sensible Langmuir isotherm. Beside these analytical solutions a direct integration of the initial and boundary value problem of the diffusion-controlled model is possible. To do so differentials are replaced by differences. This approximation leads to linear equation systems for each time step which have to be solved. As... 

We first discuss the various simple models, and start with linear models, favoured for the possibility of analytical solution which allows us to study the system behaviour in a more explicit way. Next we will discuss nonlinear models, and under special conditions such as the case of rectangular isotherm with pore diffusion analytical solution is also possible. Nonisothermal conditions are also dealt with by simply adding an energy balance equation to mass balance equations. We then discuss adsorption behaviour of multicomponent systems. 

Beyond these relatively simple systems and for all other non-linear isotherms, it is necessary to obtain solutions for the breakthrough curves by applying numerical approximation techniques to the model equations. Standard finite difference or collocation methods are commonly used. Table 6.3 provides a brief source list to solutions for plug flow and axially dispersed models with Langmuir, Freundlich or more general isotherms. 

For isothermal, first-order chemical reactions, the mole balances form a system of linear equations. A non-ideal reactor can then be modeled as a collection of Lagrangian fluid elements moving independe n tly through the system. When parameterized by the amount of time it has spent in the system (i.e., its residence time), each fluid element behaves as abatch reactor. The species concentrations for such a system can be completely characterized by the inlet concentrations, the chemical rate constants, and the residence time distribution (RTD) of the reactor. The latter can be found from simple tracer experiments carried out under identical flow conditions. A brief overview of RTD theory is given below. 

The linear equilibrium isotherm adsorption relationship (Eq. 11) requires a constant rate of adsorption, and is most often not physically valid because the ability of clay solid particles to absorb pollutants decreases as the adsorbed amount of pollutant increases, contrary to expectations from the liner model. If the rate of adsorption decreases rapidly as the concentration in the pore fluid increases, the simple Freundlich type model (Eqs. 8 and 9) must be extended to properly portray the adsorption relationship. Few models can faithfully portray the adsorption relationship for multicomponent COM-pollutant systems where some of the components are adsorbed and others are desorbed. It is therefore necessary to perform initial tests with the natural system to choose the adsorption model specific to the problem at hand. From leaching-column experimental data, using field materials (soil solids and COMs solutions), and model calibration, the following general function can be successfully applied [155] ... 

Geochemical models of sorption and desorption must be developed from this work and incorporated into transport models that predict radionuclide migration. A frequently used, simple sorption (or desorption) model is the empirical distribution coefficient, Kj. This quantity is simply the equilibrium concentration of sorbed radionuclide divided by the equilibrium concentration of radionuclide in solution. Values of Kd can be used to calculate a retardation factor, R, which is used in solute transport equations to predict radionuclide migration in groundwater. The calculations assume instantaneous sorption, a linear sorption isotherm, and single-valued adsorption-desorption isotherms. These assumptions have been shown to be erroneous for solute sorption in several groundwater-soil systems (1-2). A more accurate description of radionuclide sorption is an isothermal equation such as the Freundlich equation ... 

If the equilibrium isotherm is linear, analytic expressions for the concentration front and the breakthrough curve may, in principle, be derived, however complex the kinetic model, but except when the boundary conditions are simple, the solutions may not be obtainable in closed form. With the widespread availability of fast digital computers the advantages of an analytic solution are less marked than they once were. Nevertheless, analytic solutions generally provide greater insight into the behavior of the system and have played a key role in the development of our understanding of the dynamics of adsorption columns. 

Freundlich isotherm (Moore, 1963)), the transport equation (Equation 7.6) now becomes non-linear and does not generally have an analytical solution However, the equations are not difficult to solve numerically, and this has been reported by a number of workers for experiments in simple adsorbing systems, e.g. Gupta and Greenkorn (1974). Although it is less convenient to match experimental effluents using a numerical rather than an analytical model, it can still be done relatively straightforwardly. 

Brooks and Richmond - developed a simple surfactant partitioning model that can be applied to isothermal transitional inversions. The linearity and gradient variation of transitional inversion lines in systems containing distributed nonionic surfactant was also explained. The derived model used mixed surfactant theory to predict the slope of the SAD = 0 line with surfactant concentration, for transitional inversion induced by varying the amount of a homogeneous lipophilic and a homogeneous hydrophilic surfactant in an oil-water system. 

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