Linear Equilibrium

Ihe simplest expression for adsorption equilibrium, for an adsorbate A, is given 

Biochemical Engineering A Textbookfor Engineers, Chemists and Biologists, Second Edition. 

Shigeo Katoh, Jun-ichi Horiuchi, and Fumitake Yoshida. 

Biochemical Engineering A Textbook for Engineers, Chemists and Biologists Shigeo Katoh and Fumitake Yoshida 

Copyright 2009 WILEY-VCH Verlag GmbH Co. KGaA, Weinheim ISBN 978-3-527-32536-8 

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The situation in singlet A electronic states of triatomic molecules with linear equilibrium geometry is presented in Figme 2. This vibronic structure can be interpreted in a completely analogous way as above for n species. Note that in A electronic states there is a single unique level for K =, but for each other K 0 series there are two levels with a unique character. 

Simplified Design Procedure for Linear Equilibrium and Operating Lines. A straight operating line occurs when the concentrations are low such that and remain essentially constant. (The material balance is obtained from equation 35.) In cases where the... 

FIG. 16-33 Elution curves under trace linear equilibrium conditions for different feed loading periods audN = 80. Solid lines, Eq. (16-172) dashed line, Eq. (16-174) for Tp= 0.05. 

FIG. 16-34 Elution curves under traee linear equilibrium eouditious with a pulse feed from Eq. (16-174). 

Whenever die rich and the lean phases are not in equilibrium, an interphase concentration gradient and a mass-transfer driving force develop leading to a net transfer of the solute from the rich phase to the lean phase. A common method of describing the rates of interphase mass transfer involves the use of overall mass-transfer coefficients which are based on the difference between the bulk concentration of the solute in one phase and its equilibrium concentration in the other phase. Suppose that the bulk concentradons of a pollutant in the rich and the lean phases are yi and Xj, respectively. For die case of linear equilibrium, the pollutant concnetration in the lean phase which is in equilibrium with y is given by... 

Alternatively, for the case of isothermal, dilute mass exchange with linear equilibrium, NTP can be determined through the Kremser (1930) equation ... 

For a non-linear equilibrium relationship, in which the slope of the equilibrium curve varies with concentration, the magnitudes of the overall mass transfer coefficients will also vary with concentration, even when the film coefficients themselves remain constant. The use of overall mass transfer coefficients in mass transfer rate equations should therefore be limited to the case of linear equilibria or to situations in which the mass transfer coefficient is known to be... 

For a linear equilibrium curve with constant film coefficients, Icl and Icq, the overall coefficient, Kl, will also be constant, but for the case of a non-linear equilibrium relationship, the value of m, which is the local slope of the equilibrium curve, will vary with solute concentration. The result is that the overall coefficient, Kl, will also vary with concentration, and therefore in modelling the case of a non-linear equilibrium extraction, further functional relationships relating the mass transfer coefficient to concentration will be required, such that... 

Franks has suggested that the solution procedure should be based on the concentration value, Yi orX], whichever having the greatest magnitude. Thus for a linear equilibrium in which Yj = m X], the equilibrium equation can be differentiated to give... 

Assuming equilibrium conditions and a linear equilibrium relationship, where Y] = m Xi, and a quasi-steady-state conditions in the gas with dYj/dt = 0 to be achieved, a component balance for the entire two phase system of Fig. 3.56, gives... 

Overall mass transfer coefficients are only constant when both liquid film coefficients are constant and also when the slope of the equilibrium line is constant. Thus, for a non-linear equilibrium relationship, the overall mass transfer coefficient will vary with concentration. How would you implement this effect into the program ... 

We restrict ourselves again to symmetric tetraatomic molecules (ABBA) with linear equilibrium geometry. After integrating over electronic spatial and spin coordinates we obtain for A electronic states in the lowest order (quartic) approximation the effective model Hamiltonian H = Ho + //, which zeroth-order part is given by Eq. (A.4) and the perturbative part of it of the form... 

The linear driving force (LDF) approximation is obtained when the driving force is expressed as a concentration difference. It was originally developed to describe packed-bed dynamics under linear equilibrium conditions [Glueckauf, Trans. Far. Soc., 51, 1540 (1955)]. This form is exact for a nonlinear isotherm only when external mass transfer is controlling. However, it can also be used for nonlinear systems with pore or solid diffusion mechanisms as an approximation, since it provides qualitatively correct results. 

Both species exhibit the expected linear geometry that maximizes the dominant n- - a interaction. However, these isomers are rather perplexing from a dipole-dipole viewpoint. The dipole moment of CO is known to be rather small (calculated Fco = 0.072 D), with relative polarity C- 0+. 40 While the linear equilibrium struc-ture(s) may appear to suggest a dipole-dipole complex, robust H-bonds are formed regardless of which end of the CO dipole moment points toward HF This isomeric indifference to dipole directionality shows clearly that classical dipole-dipole interactions have at most a secondary influence on the formation of a hydrogen bond. 

Figure El2.2a shows the boundary conditions X0 and Yx. Given values for m, Nox, and the length of the column, a solution for Y0 in terms of vx and vY can be obtained Xx is related to Y0 and F via a material balance Xx = 1 - (Yq/F). Hartland and Meck-lenburgh (1975) list the solutions for the plug flow model (and also the axial dispersion model) for a linear equilibrium relationship, in terms of F ...

These equations are applied in the simulation example CHROMDIFF to the case of a two-component separation with linear equilibrium. The situation of a non-linear equilibrium is considered as an exercise in the example. 

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