Isotherms mathematical description

Many models have been proposed for adsorption and ion exchange equilibria. The most important factor in selecting a model from an engineering standpoint is to have an accurate mathematical description over the entire range of process conditions. It is usually fairly easy to obtain correcl capacities at selected points, but isotherm shape over the entire range is often a critical concern for a regenerable process. 

A systematic, rational analysis of both isothermal and nonisothermal tubular systems in which two fluids are flowing must be carried out, if optimal design and economic operation of these pipeline devices is to be achieved. The design of all two-phase contactors must be based on a firm knowledge of two-phase hydrodynamics. In addition, a mathematical description is needed of the heat and mass transfer and of the chemical reaction occurring within a particular system. 

Predominantly, Freundlich s fitted adsorption isotherms computed by means of simple linear regression were proposed for the mathematical description of the process studied. Unlike the Langmuir equation, the Freundlich model did not reduce to a linear adsorption expression at very low nor very high solute concentrations, as above resulted. 

Empirical Models vs. Mechanistic Models. Experimental data on interactions at the oxide-electrolyte interface can be represented mathematically through two different approaches (i) empirical models and (ii) mechanistic models. An empirical model is defined simply as a mathematical description of the experimental data, without any particular theoretical basis. For example, the general Freundlich isotherm is considered an empirical model by this definition. Mechanistic models refer to models based on thermodynamic concepts such as reactions described by mass action laws and material balance equations. The various surface complexation models discussed in this paper are considered mechanistic models. 

The two-phase kinetic model developed by Karickhoff (65) is capable of fitting either the sorption or desorption of a sorbing solute. For linear isotherms, the mathematical description given by Karickhoff (1) and others (67, 70, 71) is virtually identical to that of a mass transfer process (72). 

A mathematical description of a pressure swing system has been presented by Shendalman and Mitchell153 who assumed that isothermal equilibrium adsorption takes place and that the isotherm is linear, with the feed consisting of a single adsorbate at low concentration in a non-adsorbed carrier gas. 

When measured adsorption data are plotted against the concentration value of the adsorbate at equilibrium, the resulting graph is called an adsorption isotherm. The mathematical description of isotherms invariably involves adsorption models described by Langmuir, Freundlich, or Brauner, Emmet and Teller (known as the BET-model). Discussion of these models is given in Part 111, as conditions relevant to chemical-subsurface interactions are examined. 

In addition to these characterizations of adsorption curves, mathematical descriptions of adsorption isotherms, based on physical models, often are used to study solid interactions with contaminants. The main adsorption isotherms include those of Langmuir, Freundhch, and Brunauer-Emmet-TeUer (BET) they are depicted in Fig. 5.2. 

Beside the theoretically derived Gibbs adsorption isotherm, a large number of models have been developed that empirically describe a relationship between the interfacial coverage, the surface tension, and the surfactant concentration in the bulk phase. These adsorption isotherms are known under the names of the authors that first described them—i.e., the Fangmuir, Frumkin, or Volmer isotherms. A complete mathematical description of these isotherms is beyond the scope of this unit and the reader is encouraged to consult the appropriate literature instead (e.g., Dukhin et al., 1995). 

A high value of C(ss 100) is associated with a sharp knee in the isotherm, thus making it possible to obtain by visual inspection the uptake at Point B, which usually agrees with na derived from the above equation to within a few per cent. On the other hand, if C is low (<20) Point B cannot be identified as a single point on the isotherm. Since Point B is not itself amenable to any precise mathematical description, the theoretical significance of the amount adsorbed at Point B is uncertain. 

For the mathematical description of isotherms on heterogeneous surfaces the following equation may serve as a starting point ... 

Equation 4.37 is the mathematical description of the adsorption isotherm of an anion, A" , at a controlled pH. It has the form of the well-known Langmuir equation used to describe chemisorption. 

The physical and mathematical description of the ribbon extrusion process was first given by Pearson [24], who simplified the conservation equations by using a onedimensional, isothermal, Newtonian fluid approach, and neglected the effects of polymer solidification. As in the case of blown film processes, several modifications and models have been proposed for the ribbon extrusion process (Table 24.2). 

The L-curve isotherm is by far the one most commonly encountered in the literature of soil chemistry. The mathematical description of this isotherm almost invariably involves either the Langmuir equation or the van Bemmelen-Freundlich equation. Ths Langmuir equation has the form... 

US stat ent can be validated both through model experiments and with rigoroi mathematical analyses of the properties of adsorption isotherm equations. However, the point that these equations are not unique mathematical descriptions of surface reactions in soils can be made effectively by a simple counterexample. Consider a typical batch sorption experiment in which o-phosphate is reacted with soil material and suppose that in this reaction an amorphous aluminum phosphate phase is formed. 

Additional bibliographies should also be mentioned. The Handbook of Food Isotherms contains more than 1000 isotherms, with a mathematical description of over 800 [162]. About 460 isotherms were obtained from the monograph of Ref. [163]. Data on sorption properties of selected pharmaceutical materials are presented in Ref. [98]. 

The equations of Freundlich, Langmuir as well as Brunauer, Emmet, and Teller, which were presented in Chap. 2, are suitable for the mathematical description of sorption isotherms. 

Next, a mathematical description of T is given for a quasi-isothermal run. This type of run does not only simplify the mathematics, it also is a valuable mode of measuring Cp as described in Sect. 4.4.5. In addition, standard TMDSC with y O is linked to the same analysis by a pseudo-isothermal data treatment as described in Sect. 4.4.3. 

The primary information about chemical transformations is obtained by the measmement of the chemical composition, that is, concentration measurements. The chemical processes occurring are complex and do not consist only of chemical reactions but also of physical phenomena, such as mass and heat transport. The major goal of chemical kinetic studies is to extract intrinsic kinetic information related to complex chemical reactions. Therefore, the transport regime in the reactor has to be well defined and its mathematical description has to be reliable. A typical strategy in kinetic experiments is the minimization of the effects of mass and heat transport on the rate of change of the chemical composition. In accordance with this, a kinetic experiment ideally has to fulfill two main requirements isothermicity of the active zone and uniformity of the chemical composition, which can be accomplished by, for example, perfect mixing within the reaction zone. 

In order to make the process useful in industrial practice, it is necessary to propose a mathematical description. Equilibrium data, commonly known as adsorption isotherms, are basic requirements for the design of biosorption systems (Aksu, 2005). A variety of models have been used in the literature to describe the equilibrium between adsorbed metal ions on the biomass ( eq) and metal ions in solution (Ceq) at constant temperatures (Table 7.1). 

Abstract. This article describes a hydrodynamic model of collaborative flnids (oil, water) flow in porons media for enhanced oil recovery, which takes into account the influence of temperature, polymer and surfactant concentration changes on water and oil viscosity. For the mathematical description of oil displacement process by polymer and surfactant injection in a porous medium, we used the balance equations for the oil and water phase, the transport equation of the polymer/surfactant/salt and heat transfer equation. Also, consider the change of permeabihty for an aqueous phase, depending on the polymer adsorption and residual resistance factor. Results of the numerical investigation on three-dimensional domain are presented in this article and distributions of pressure, saturation, concentrations of poly mer/surfactant/salt and temperature are determined. The results of polymer/surfactant flooding are verified by comparing with the results obtained from ECLIPSE 100 (Black Oil). The aim of this work is to study the mathematical model of non-isothermal oil displacement by polymer/surfactant flooding, and to show the efficiency of the combined method for oil-recovery. 

In view of the eomplexity of adsorption phenomena and the different mechanisms contributing to physisorption, none of the current theories of adsorption is capable of providing a mathematical description of an experimental isotherm over its entire range of relative pressure. In practice, two procedures have been used to overcome this problem. The first approach involves the apphcation of various semiempirical isotherm equations with different analytical forms and different ranges of validity. The second procedure makes use of standard adsorption isotherms obtained with selected nonporous reference materials and attempts to explain differences in the isotherm shape in terms of the different mechanisms of physisorption [2]. In particular, physisorption of gases by nonporous sohds, in the vast majority of cases, gives rise to a type II isotherm [2,5,13]. From the type II isotherm of a given gas on a particular sohd, it is possible, in principle, to derive a value of the monolayer capacity of the solid, which, in turn, can be used to calculate the specific surface of the solid. Here, we describe briefly both procedures and their application to rare-gas adsorption. 

The interaction between polymers (P) and amphiphiles (A) can be concretely represented by a binding isotherm, and analysis of the isotherm yields an estimate of the change of the thermodynamic variables upon binding. Many mathematical descriptions of binding have appeared but only a few of the more common models are introduced in this section. 

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