Zone spreading mass transfer

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. 

In the case of an unfavorable isotherm (or equally for desorption with a favorable isotherm) a different type of behavior is observed. The concentration front or mass transfer zone, as it is sometimes called, broadens continuously as it progresses through the column, and in a sufficiently long column the spread of the profile becomes directly proportional to column length (proportionate pattern behavior). The difference between these two limiting types of behavior can be understood in terms of the relative positions of the gas, solid, and equilibrium profiles for favorable and unfavorable isotherms (Fig. 7). 

Figure 2.6. Illustration of zone spreading due to mass transfer (K = 2.0).

Also, the theory is much more complex than just presented. Bly36 has classified the process into three mechanisms steric exclusion, restricted diffusion, and thermodynamic considerations, and the process has been thoroughly studied. The rate equation is also different for SEC. For polymers, the longer retained peaks have smaller peak dispersivities, H, than early peaks, in direct contrast to normal LC expectations. In part this is due to the fact that the smaller molecules that elute last have higher diffusion coefficients and therefore less mass transfer zone spreading. 

Band broadening The tendency of zones to spread as they pass through a chromatographic column caused by various diffusion and mass transfer processes. 

Resistance to mass transfer, is by far the major contributor to sample zone spreading within the column. This term is minimized by a column packing that attains equilibrium of the analyte between the mobile and stationary phases as quickly as possible. A moderate linear flow rate, u, should be employed because cr mt increases with flow rate. 

Axial dispersion (sometimes referred to as backmixing) is a spreading of the concentration profile in the axial direction due to flow variations within the adsorbent bed (see the pulse analysis section in Appendix C). This effect can also contribute to the spreading of the mass transfer zone. 

Since SEC is an essentially linear system, the theories developed in Sections 14.1-1 and 14.1-2 are applicable. A chromatographic separation thus looks like Fig. 14.1-2 with zone spreading added on. Since the mass transfer terms in Eq. (14.1-12) are inversely proportional to the diffusivity, large solutes thee penetrate the porea have high H values. Desalting is an easy separation and has a relatively low M since the large molecules do not penetrate the potes. 

Usually the desorbent must be removed from the A and B product streams. Increasing the amount of desorbent will increase the cost for this removal and will also increase the diameters of the columns requiring more adsorbent. Thus, the ratio of desorbent to feed, D/F, often controls the cost of SMB systems. For an ideal system with no zone spreading (no axial dispersion and very fast mass transfer rates) the solute movement theory can be used to calculate D/F by solving Eqs. tl8-29al to tl8-29dl simultaneously with Eq. (18-15) and the mixing mass balances with constant density. 

For an isotherm with a Langmuir shape, if the column is initially loaded at some low concentration, c q, (ciow = 0 if the column is clean) and is fed with a fluid of a higher concentration, C] (see Figure 18-IZA), the result will be a shock wave. The feed step in adsorption processes usually results in shock waves. Experiments show that when a shock wave is predicted the zone spreading is constant regardless of the column length (a constant pattern wave). With the assunptions of the solute movement theory (infinitely fast rates of mass transfer and no axial dispersion), the wave becomes infinitely sharp (a shock) and the derivative dq/dc does not exist. Thus, the Aq/Ac term in the denominator of Eq. f 18-141... 

One characteristic of the solutions for Eqs. [18-541 and [18-551 for linear isotherms is mass transfer resistances and axial dispersion both cause zone spreading that looks identical if the mass transfer parameters or axial dispersion parameters are adjusted. Thus, from an experimental result it is impossible to determine if the spreading was caused solely by mass transfer resistances, solely by axial dispersion, or by a combination of both. This property of linear systems allows us to use sinple models to predict the behavior of more complex systems. 

In linear systems the variances (o ) from different sources add. This is equivalent to stating that the amount of zone spreading from different sources is additive. Mathematically, this ability to add variances is the reason we can use an effective diffusion coefficient to model a system where mass transfer resistances are inportant. 

In most adsorption systems the isotherm is favorable for adsorption and therefore unfavorable for desorption. In desorption the mass transfer zone is therefore dispersive, leading to a continuously spreading concentration prOfife (proportionate-pattern behavior) while in adsorptioii the mass transfer zone is compressive, leading to constant-pattern behavior. For example, for a system which obeys the Langmuir isotherm (Eq. (8.6)] I... 

The major sources of zone spreading within the column are often lumped together in a term called resistance to mass transfer . This term includes several factors that contribute to peak broadening. 

The classic Van Deemter equation and its modifications have been used to describe zone spreading in GC and HPLC in terms of eddy diffusion, molecular diffusion, and mass transfer. The efficiency of a zone in HPTLC is given by the equation (34)... 

Fig 7 4), which corresponds to the Langmuir equilibrium with separation factor rl in the Freundlich isotherm system In these cases, a constant profile of mass transfer zone is established while adsorption proceeds in a column (Fig 7 5) This is in contrast to linear isotherm systems where mass transfer zone continues to spread with increase of traveling time The reason for the formation of a constant pattern is explained as follows The speed of movement of the point on the mass transfer zone whose concentration is C, F(C), can be related to the equilibrium through the basic equation... 

It is fortunate that for many fixed bed adsorption processes of commercial interest the shape of the mass transfer zone remains unaltered as it progresses through the majority of the bed because this leads to substantial simplifications in design. For a favourable isotherm, particularly one of Type I, the mass transfer wave spreads from a shock front as it progresses through the initial region of the bed. As explained earlier in this chapter the... 

Frohlich (1947) based his calculations on the hypothesis of the energy-level scheme shown in Fig. 6.1, where conduction electrons are derived from impurity levels lying deep (V = 1 eV or more) in the forbidden zone. There is also a set of shallow traps spread below the conduction-band edge (F> AF> kT). In outline, the theory of breakdown is then as follows. In an applied electric field E, energy is transferred directly to the conduction electrons (charge e, mass m) at a rate A = jE, where j is the current density. If we suppose that each electron is accelerated in the field direction for an average time 2r between collisions at which its energy is completely randomised, then the mean drift velocity of the conduction electrons in the field direction... 

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