Dispersion coefficient, apparent

Apparent dispersion coefficient, Z) (assnmed to be equal to molecular dilTusivity) 0.00033 cm ... 

The apparent dispersion coefficient in Equation 10.8 describes the zone spreading observed in linear chromatography. This phenomenon is mainly governed by axial dispersion in the mobile phase and by nonequilibrium effects (i.e., the consequence of a finite rate of mass transfer kinetics). The band spreading observed in preparative chromatography is far more extensive than it is in linear chromatography. It is predominantly caused by the consequences of the nonlinear thermodynamics, i.e., the concentration dependence of the velocity associated to each concentration. When the mass transfer kinetics is fast, the influence of the apparent axial dispersion is small or moderate and results in a mere correction to the band profile predicted by thermodynamics alone. 

The band profile obtained as a numerical solution of Equation 10.10 gives the concentration distribution as a function of the reduced time at the column end, i.e., at location x= 1, regardless of the column length. The band profile depends only on the column efficiency, the boundary conditions, the phase ratio, and the sample size (which is part of the boundary conditions). The mobile phase velocity has been eliminated from the mass balance equation and the apparent axial dispersion coefficient has been replaced by the plate number. 

In Reprint C in Chapter 7, the behavior of a tracer pulse in a stream flowing through a packed bed and exchanging heat or matter with the particles is studied. It is shown that the diffusion in the particles makes a contribution to the apparent dispersion coefficient that is proportional to v2 fi/D. The constant of proportionality has one part that is a function of the kinematic wave speed fi, but otherwise only a factor that depends on the shape of the particle (see p. 145 and in equation (42) ignore all except the last term and even the suffixes of this e, being unsuitable as special notation, will be replaced by A. e is defined in the middle of p. 143 of Chapter 7). In this equation, we should not be surprised to find a term of the same form as the Taylor dispersion coefficient, for it is diffusion across streams of different speeds that causes the dispersion in that case just as it is the diffusion into stationary particles that causes the dispersion in this.7 What is surprising is that the isothermal diffusion and reaction equation should come up, for A is defined by... 

Because of their structural and conformational complexity, polypeptides, proteins, and their feedstock contaminants thus represent an especially challenging case for the development of reliable adsorption models. Iterative simulation approaches, involving the application of several different isothermal representations8,367 369 enable an efficient strategy to be developed in terms of computational time and cost. Utilizing these iterative strategies, more reliable values of the relevant adsorption parameters, such as q, Kd, or the mass transfer coefficients (the latter often lumped into an apparent axial dispersion coefficient), can be derived, enabling the model simulations to more closely approximate the physical reality of the actual adsorption process. 

Comparing this with the slow reaction case, we note that the effective velocity has increased (by a factor 1.83), the dispersion coefficient is reduced by a factor 3 while the apparent reactor scale Damkohler number changed from Das to 8jp. 

In this equation, c is the concentration in the fluid phase and q is the quantity in the solid phase. The column porosity e (expressed as phase ratio f = (1 -e)/e) defines the fraction of the fluid phase in the column. Furthermore, u stands for the linear velocity and t and x are the time and space coordinates, respectively. All contributions leading to band-broadening are lumped in a simplifying manner into an apparent dispersion coefficient, D p. In equation (21-2), it is assumed that the two phases are constantly in equilibrium expressed by the adsorption isotherms. Due to the nonlinear character of the isotherm equations, the solution of equation (21-2) requires the use of numeri-... 

On the other hand, Settari et al. (50) used a finite-element analysis in examining the consec[uences of both velocity-dependent and constant dispersion coefficients during a two-dimensional displacement. They found that fingers in the concentration distribution developed when the permeability was homogeneous, so long as the dispersion coefficients were sufficiently small. This was apparently the first successful use of truncation and round-off errors to play the roles of physical perturbations in initiating instabilities. Russell (51) later had a similar experience. 

Ee Apparent longitudinal dispersion coefficient of gas in emulsion phase based on empty vessel... 

Where t is time, z are the axial position in the column, qt is the concentration of solute i in the stationary phase in equilibrium with Cu the mobile phase concentration of solute /, u is the mobile phase velocity, Da is the apparent dispersion coefficient, and F is the phase ratio (Vs/Vm). The equation describes that the difference between the amounts of component / that enters a slice of the column and the amount of the same component that leaves it is equal to the amount accumulated in the slice. The fist two terms on the left-hand side of Eq. 10 are the accumulation terms in the mobile and stationary phase, respectively [109], The third term is the convective term and the term on the right-hand side of Eq. 10 is the diffusion term. For a multi component system there are as many mass balance equation, as there are active components in the system [13],... 

As already mentioned, the effects of several parameters are often lumped into one (see also Section 6.5.3.1). In this case, all band broadening effects are included in a dispersion coefficient. The so-called apparent dispersion coefficient Dapp is used here to distinguish from the axial dispersion coefficient, Dm, which is assumed to be independent of concentration and only influenced by the quality of the packing. The lumped parameter Dapp includes peak broadening effects caused by the fluid dynamics of the packing (axial dispersion), as well as by all other mass transfer effects that might occur, and was first introduced by van Deemter et al. (1956). 

For nonlinear isotherms, the apparent dispersion coefficient will generally also depend on the concentration. Like most lumped parameters, it is generally dependent on the interstitial velocity (Section 6.5.3.1). 

Notably, Eq. 6.58 can be defined in similar fashion to Eq. 6.44 with the velocity um of Eq. 6.43, while all constant parameters such as volume fractions are included in the apparent dispersion coefficient, which then differs from the one defined in Eq. 6.58 ... 

The dispersion coefficient D (Section 6.5.6.2) is assumed to depend only on the packing properties and flow conditions (Eq. 6.24) and, therefore, generally differs from the apparent dispersion coefficient Dapp defined in Section 6.2.4.I. 

Equation 6.139 illustrates the meaning of the apparent dispersion coefficient as a lumped parameter. 

The axial plate height is related to the apparent axial dispersion coefficient by 2 Da... 

Chapters 10 to 13 review the solutions of the equilibrium-dispersive model for a single component (Chapter 10), and multicomponent mixtures in elution (Chapter 11) and in displacement (Chapter 12) chromatography and discuss the problems of system peaks (Chapter 13). These solutions are of great practical importance because they provide realistic models of band profiles in practically all the applications of preparative chromatography. Mass transfer across the packing materials currently available (which are made of very fine particles) is fast. The contribution of mass transfer resistance to band broadening and smoothing is small compared to the effect of thermodynamics and can be properly accounted for by the use of an apparent dispersion coefficient independent of concentration (Chapter 10). 

Haarhoff and Van der Linde [68] have given a more direct mathematical demonstration of this result in the case of a moderately overloaded column, with a parabohc isotherm. It leads to the fundamental equation of the equilibriiun-dispersive model, in which the diffusion coefficient in the diffusive term of the mass balance equation (Eq. 2.2) is replaced by the apparent dispersion coefficient (Eq. 2.38). 

In the equilibrium-dispersive model, we assume that the mobile and the stationary phases are constantly in equilibrium. We recognize, however, that band dispersion takes place in the column through axial dispersion and nonequilibrium effects e.g., mass transfer resistances, finite kinetics of adsorption-desorption). We assume that their contributions can be lumped together in an apparent dispersion coefficient. This coefficient is related to the experimental parameters by... 

In contrast to the equilibrium-dispersive model, which is based upon the assumptions that constant thermod3mamic equilibrium is achieved between stationary and mobile phases and that the influence of axial dispersion and of the various contributions to band broadening of kinetic origin can be accounted for by using an apparent dispersion coefficient of appropriate magnitude, the lumped kinetic model of chromatography is based upon the use of a kinetic equation, so the diffusion coefficient in Eq. 6.22 accounts merely for axial dispersion (i.e., axial and eddy diffusions). The mass balance equation is then written... 

---