General Column Mass Balance

The following assumptions on the behavior of the chromatographic systems are made  

Molar volumes of the analyte and mobile-phase components are constant, and compressibility of the hquid phase is negligible. 

Adsorbent is characterized by its specihc surface area and pore volume, which are evenly distributed axially and radially in the column. (This assumption is equivalent to the assumption of column homogeneity.) 

The column void volume, Vo, is dehned as the total volume of the liquid phase in the column and could be measured independently [18]. Total adsorbent surface area in the column, S, is determined as the product of the adsorbent mass and specific surface area. 

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The phenomenological description of the retention mechanism discussed above is only applicable for the system with single partitioning process and well-defined stationary and mobile phases. A more general method for the derivation of retention function is based on the solution of column mass balance [17]. 

In order to better understand chromatography it is important to study the underlying mass transfer operations which are occurring. These mass transfer phenomena are well studied (20-27), and analytical solutions exist to most limiting cases. In general, a mass balance for one component in a packed column is given by ... 

To determine the value of time f (= t) when the feed solution breaks through a column of length L, a solute mass balance is carried out over the column from time f = 0 to the time t of breakthrough. For the sake of generality, the mass balance is carried out in a column which may have some solute present in the column at f = 0, i.e. 

The products are analysed by gas chromatography usually on five different columns in order to detect CO, H2, C02, H20 and CrCn hydrocarbons (alkanes, alkenes) and alcohols. The mass balance for carbon, based on CO consumption, generally lies within 85-95 ... 

Generally for modelling chromatograph systems, component mass balances are required for each component in each phase. The differential liquid phase component balances for a chromatographic column with non-porous packing take the partial differential equation form... 

Novel general expressions were developed for the description of the behaviour of the height equivalent of a theoretical plate in various chromatographic columns such as unpacked (open capillary), packed with spherical nonporous particles and packed with spherical porous adsorbent particles. Particles may have unimodal or bimodal pore size distribution. The expression describing the mass balance in open capillaries is... 

Although simple, the conceptual base of the method should be discussed. Consider a stage P somewhere in a column of stages. In general, for a vapor-liquid process, a vapor stream would enter the stage from stage P — 1 and a liquid would enter from stage P -t- 1. The differential equation for the mass balance on component i would be... 

In most adsorption processes the adsorbent is contacted with fluid in a packed bed. An understanding of the dynamic behavior of such systems is therefore needed for rational process design and optimization. What is required is a mathematical model which allows the effluent concentration to be predicted for any defined change in the feed concentration or flow rate to the bed. The flow pattern can generally be represented adequately by the axial dispersed plug-flow model, according to which a mass balance for an element of the column yields, for the basic differential equation governing llie dynamic behavior,... 

Table A.2 is model output for seawater freezing at 253.15 K. Beneath the title, the output includes temperature, ionic strength, density of the solution (p), osmotic coefficient amount of unfrozen water, amount of ice, and pressure on the system. Beneath this line are the solution and gaseous species in the system. The seven columns include species identification, initial concentration, final (equilibrium) concentration, activity coefficient, activity, moles in the solution phase, and mass balance. The mass balance column only contains those components for which a mass balance is maintained. The number of these components minus 1 is generally the number of independent components in the system (in this case, 8 — 1 = 7). The mass balances (col. 7) should equal the initial concentrations (col. 2). This mass balance comparison is a good check on the computational accuracy.

The pattern of flow through a packed adsorbent bed can generally be described by the axial dispersed plug flow model. To predict the dynamic response of the column therefore requires the simultaneous solution, subject to the appropriate initial and boundary conditions, of the differential mass balance equations for an element of the column,... 

It should be understood that this rate expression may in fact represent a set of diffusion and mass transfer equations with their associated boundary conditions, rather than a simple explicit expression. In addition one may write a differential heat balance for a column element, which has the same general form as Eq. (17), and a heat balance for heat transfer between particle and fluid. In a nonisothermal system the heat and mass balance equations are therefore coupled through the temperature dependence of the rate of adsorption and the adsorption equilibrium, as expressed in Eq. (18). 

In general, the overall balance for the mass transport streams (Eqs. 6.23 and 6.24) at the column inlet and outlet has to be fulfilled. In Eq. 6.92 the closed boundary condition is obtained by setting the dispersion coefficient outside the column equal to zero. In open systems, the column stretches to infinity and in these limits concentration changes are zero. 

The detailed study of the mass transfer kinetics is necessary in certain problems of chromatography in which the column efficiency is low or moderate. Complex models are then useful. The most important ones are the General Rate Model [52,62] and the FOR model (see next Section) [63]. To study the mass transfer kinetics, these models need to consider separately the mass balance of the feed components in the two different fractions of the mobile phase the one that percolates through the bed of the solid phase (column packed with fine particles or monolithic column) and the one that is stagnant inside the pores of the packing material. 

The transport approach has been used very early, and most extensively, to calculate the chromatographic response to a given input function (injection condition). This approach is based on the use of an equation of motion. In this method, we search for the mathematical solution of the set of partial differential equations describing the chromatographic process, or rather the differential mass balance of the solute in a slice of column and its kinetics of mass transfer in the column. Various mathematical models have been developed to describe the chromatographic process. The most important of these models are the equilibrium-dispersive (ED) model, the lumped kinetic model, and the general rate model (GRM) of chromatography. We discuss these three models successively. 

In the equilibrium-dispersive model of chromatography, however, we assume that Eq. 10.4 remains valid. Thus, we use Eq. 10.10 as the mass balance equation of the component, and we assume that the apparent dispersion coefficient Da in Eq. 10.10 is given by Eq. 10.11. We further assume that the HETP is independent of the solute concentration and that it remains the same in overloaded elution as the one meastued in linear chromatography. As shown by the previous discussion this assxunption is an approximation. However, as we have shown recently [6], Eq. 10.4 is an excellent approximation as long as the column efficiency is greater than a few hundred theoretical plates. Thus, the equilibriiun-dispersive model should and does account well for band profiles under almost all the experimental conditions used in preparative chromatography. In the cases in which the model breaks down because the mass transfer kinetics is too slow, and the column efficiency is too low, a kinetic model or, better, the general rate model (Chapter 14) should be used. 

This chapter deals essentially with the apphcations of the theory of chromatography to the calculation of solutions of the SMB model in different cases of general interest. The theoretical tools required are a general model of the SMB process and a model for its columns. The former is an integral mass balance that is easy to write. The possible column models were described in the previous chapters. Finally, an accurate model of the competitive isotherms of the feed components is necessary. 

If we count the equations listed, we will find that there are 2n + 4 equations per stage. However, only 2 n + 3 of these equations are independent. These independent equations are generally taken to be the n component mass balance equations, the n equilibrium relations, the enthalpy balance, and two more equations. These two equations can be the two summation equations or the total mass balance and one of the summation equations (or an equivalent form). The 2n + 3 unknown variables determined by the equations are the n vapor mole fractions the n liquid mole fractions, the stage temperature 7 and the vapor and liquid flow rates LJ and Ly. Thus, for a column of 5 stages, we must solve s 2n + 3) equations. 

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