Equilibrium isotherms

Fig. 11. (a) Equilibrium isotherm and (b) dimensionless equilibrium diagram showiag favorable, linear, and unfavorable isotherms. 

Two variations of the technique exists isocratic elution, when the mobile phase composition is kept constant, and gradient elution, when the mobile phase composition is varied during the separation. Isocratic elution is often the method of choice for analysis and in process apphcations when the retention characteristics of the solutes to be separated are similar and not dramaticallv sensitive to vei y small changes in operating conditions. Isocratic elution is also generally practical for systems where the equilibrium isotherm is linear or nearly hnear. In all cases, isocratic elution results in a dilution of the separated produces. 

The chromatographic resolution of bi-naphthol enantiomers was considered for simulation purposes [18]. The chiral stationary phase is 3,5-dinitrobenzoyl phenyl-glycine bonded to silica gel and a mixture of 72 28 (v/v) heptane/isopropanol was used as eluent. The adsorption equilibrium isotherms, measured at 25 °C, are of bi-Langmuir type and were proposed by the Separex group ... 

For nonlinear systems, however, the evaluation of the flow rates is not straightforward. Morbidelli and co-workers developed a complete design of the binary separation by SMB chromatography in the frame of Equilibrium Theory for various adsorption equilibrium isotherms the constant selectivity stoichiometric model [21, 22], the constant selectivity Langmuir adsorption isotherm [23], the variable selectivity modified Langmuir isotherm [24], and the bi-Langmuir isotherm [25]. The region for complete separation was defined in terms of the flow rate ratios in the four sections of the equivalent TMB unit ... 

The separation of bi-naphthol enantiomers can be performed using a Pirkle-type stationary phase, the 3,5-dinitrobenzoyl phenylglycine covalently bonded to silica gel. Eight columns (105 mm length) were packed with particle diameter of 25 0 fiva. The solvent is a 72 28 (v/v) heptane isopropanol mixture. The feed concentration is 2.9 g for each enantiomer. The adsorption equilibrium isotherms were determined by the Separex group and already reported in Equation (28) [33]. 

Figure 3. Carbon formation Equilibrium isotherms (°F) at 30 psia...

Partial equilibrium isotherms (°F) at 270 psia partial pressure of hydrogen, carbon monoxide, carbon dioxide, and steam... 

Figure 6.17. Effective double layer adsorption equilibrium isotherms for kj=l and various values of the parameter Ajll.

Plasma Type Non-Isothermal (non-equilibrium) Isothermal (equilibrium)... 

The equilibrium isotherms were favorable type and the Langmuir equation represents our experimental data very well. 

It should be stressed that in the case of linear isotherm, the peak broadening effect results from eddy diffusion and from resistance of the mass transfer only, and it does not depend on Henry s constant. In practice, such concentration profiles are observed for these analyte concentrations, which are low enough for the equilibrium isotherm to be regarded as linear. 

Axial dispersion coefficient Constant in equilibrium isotherm... 

Example 14 Calculation of Band Profiles in Displacement Chromatography An equimolar mixture of two components (concentrations cf = cf = 1 arbitrary unit) is separated with a displacer with concentration Ci = 2. The equilibrium isotherm is ... 

Woodburn silt loam soil, 1.9% organic matter, equilibrium isotherm-GC, Chiou et al. 1983)... 

In dilute solutions of surfactants adsorption processes are controlled by transport of the surfactant from the bulk solution towards the surface as a result of the concentration gradient formed in the diffusion layer the inherent rate of adsorption usually is rapid. For non-equilibrium adsorption the apparent (non-equilibrium) isotherm can be constructed for different time periods that are shifted with respect to the true adsorption isotherm in the direction of higher concentration (Cosovic, 1990) (see Fig. 4.10). 

When a specific feed composition is given, the constraints on m1 and m4 as well as the complete separation region in the (m2, m3) plane can be determined,since these depend only on the parameters of the adsorption equilibrium isotherms and the feed composition itself. Based on these values an operating point can be selected, i. e. a set of four values of = 1,..., 4 fulfilling the complete separation requirements. Since the flow rate ratios are dimensionless groups combining column volumes, flow rates and switching intervals, the constraints on the flow rate ratios are independent of the size and productivity of the SMB unit. 

Many solutions are available for equation 17.70 and its refinements. Three cases are considered to illustrate the range of solutions. Firstly, it is assumed that the bed operates isothermally and that equilibrium is maintained between adsorbate concentrations in the fluid and on the solid. Secondly, the non-equilibrium isothermal case is considered and, finally, the non-equilibrium non-isothermal case. 

Equilibrium, isothermal adsorption in a fixed bed, single adsorbate... 

Figure 18.3. Equilibrium isotherm for the ion exchange A + 2B(S) = A(S) + 2B. The parameters are values...

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