Calculation of isotherms

Perkins, E. H. and T. H. Brown, 1982, Program path, calculation of isothermal and isobaric mass transfer. University of British Columbia, unpublished manuscript. 

The calculations of isotherms and heats of adsorption at low filling of the cavities of zeolite with adsorbed molecules have been carried out for the systems Ar-NaA and CHj -CaA. Although a simple and rather crude model was used, it was possible to obtain results that agree qualitatively with the experimental data. The equilibrium distribution of molecules among the zeolite cavities also was studied. 

Presently, there are no methods that allow the theoretical calculation of isotherms. However, there are methods available that allow the calculation of mixture isotherms... 

The procedure of Zhdanov and Samulevich enables the calculation of isothermal nucleation rate profiles from determinations of growth rate and crystal size distribution [16,82]. Originally implemented in analyses of zeolite Na-A [83] and Na-X [82] crystallisation, the method has subsequently been applied to other zeolite systems, including silicalite [84,85]. If it is supposed that all the crystals in a batch have the same (known) growth rate behaviour, the total growth time of each crystal can be calculated. Assuming also that the nuclcation point for each crystal can be obtained by linear extrapolation to zero time, the nucleation profile for the whole batch can be determined from their final sizes. 

Figure 3-4 Calculation of isothermal PV work at 350 °C from steam tables. The solid line is a quadratic fit to the points from the tables (see Example... 

Hgure 12 Procedure for calculation of isothermal retention indices. See text for explanation. 

Figure 5.26 Flow diagram for the calculation of isothermal VLE taking into account the nonideal behavior of the vapor phase.

A more elaborate theory of orientation effects was developed by Watts-Tobin and Mott and applied by Bockris, Devanathan, and Muller and Bockris and Habibto the calculation of isotherms for adsorption of organic substances. Solvent molecules were regarded as being present in the double layer in two orientations, up or down (cf. theories of ferromagnetism) at respective surface coverages of N or N[. The sum iVf + is equal to the total coverage of the electrode by solvent molecules. 

The procedure is virtually identical to the calculation of isothermal compressibility. 

The sample cell was loaded with 155.7 mg of Kureha carbon particles. Prior to the adsorption measurements the adsorbent particles were outgassed in situ in vacuum at 623 K for 16 h to remove any adsorbed impurities. The obtained dry sample weight was used in the calculation of isotherm data. Adsorption measurements were subsequently done at different temperatures from 194 to 338 K for ethane and ethene, from 273 to 358 K for propane and propene, and from 298 to 393 K for butane isomers. Five different temperatures for the adsorption of each adsorptive were used to reduce the uncertainty in the derived adsorption parameters. 

Results of model calculation of isotherms of adsorption and desorption of nitrogen on model porous solids parameters listed in Table 1 are shown in Pig. 2. The isotherms in Pig. 2 are numerated as the corresponding model porous solids in the upper line of the table. All isotherms were normalized by limiting values of adsorption for P/P = 0.99. The model porous solid 1 and the isotherm 1 are results of averaging of 40 oalculatlons. 

The calculation of single-stage equilibrium separations in multicomponent systems is implemented by a series of FORTRAN IV subroutines described in Chapter 7. These treat bubble and dewpoint calculations, isothermal and adiabatic equilibrium flash vaporizations, and liquid-liquid equilibrium "flash" separations. The treatment of multistage separation operations, which involves many additional considerations, is not considered in this monograph. 

As a quite different and more fundamental approach, the isotherms of Fig. XI-10 allowed a calculation of X as a function of temperature. The plot of In K versus 1 /T gave an enthalpy quantity that should be just the difference between the heats of immersion of the Graphon in benzene and in n-heptane, or 2.6 x 10 cal/m [141]. The experimental heat of immersion difference is 2.4 x 10 cal/m, or probably indistinguishable. The... 

Fig. XVII-22. Isosteric heats of adsorption for Kr on graphitized carbon black. Solid line calculated from isotherms at 110.14, 114.14, and 117.14 K dashed line calculated from isotherms at 122.02, 125.05, and 129.00 K. Point A reflects the transition from a fluid to an in-registry solid phase points B and C relate to the transition from the in-registry to and out-of-registry solid phase. The normal monolayer point is about 124 mol/g. [Reprinted with permission from T. P. Vo and T. Fort, Jr., J. Phys. Chem., 91, 6638 (1987) (Ref. 131). Copyright 1987, American Chemical Society.]...

Most calculations of f(Q) for a heterogeneous surface, using an adsorption isotherm assume a patchwise distribution of sites. Explain for what kind of local isotherm functions,/((2,P, T) this assumption is not necessary, and for which it is necessary. Give examples. 

A Type II isotherm indicates that the solid is non-porous, whilst the Type IV isotherm is characteristic of a mesoporous solid. From both types of isotherm it is possible, provided certain complications are absent, to calculate the specific surface of the solid, as is explained in Chapter 2. Indeed, the method most widely used at the present time for the determination of the surface area of finely divided solids is based on the adsorption of nitrogen at its boiling point. From the Type IV isotherm the pore size distribution may also be evaluated, using procedures outlined in Chapter 3. 

When it is desired to evaluate the specific surfaces of a set of closely related samples of solid, however, only one of the samples needs to be calibrated against nitrogen (or argon), provided that all the isotherms of the alternative adsorptive can be shown to have indentical shape. A simple device for testing this identity, by use of the a,-plot, is described in Section 2.13 by means of the a,-plot it is also possible to proceed directly to calculation of the specific surface without having to assign a value to or to evaluate the BET monolayer capacity, of the alternative adsorptive. 

The BET method for calculation of specific surface A involves two steps evaluation of the monolayer capacity n from the isotherm, and conversion of n into A by means of the molecular area a . 

In calculations of pore size from the Type IV isotherm by use of the Kelvin equation, the region of the isotherm involved is the hysteresis loop, since it is here that capillary condensation is occurring. Consequently there are two values of relative pressure for a given uptake, and the question presents itself as to what is the significance of each of the two values of r which would result from insertion of the two different values of relative pressure into Equation (3.20). Any answer to this question calls for a discussion of the origin of hysteresis, and this must be based on actual models of pore shape, since a purely thermodynamic approach cannot account for two positions of apparent equilibrium. 

Calculation of pore size distribution (Roberts Method"). Worked example from desorption branch of nitrogen isotherm on... 

Fig. 3.28 The Kiselev method for calculation of specific surface from the Type IV isotherm of a compact of alumina powder prepared at 64 ton in". (a) Plot of log, (p7p) against n (showing the upper (n,) and lower (n,) limits of the hysteresis loop) for (i) the desorption branch, and (ii) the adsorption branch of the loop. Values of. 4(des) and /4(ads) are obtained from the area under curves (i) or (ii) respectively, between the limits II, and n,. (6) The relevant part of the isotherm.

The table convincingly demonstrates how the unsuspected presence of micropores can lead to an erroneous value of the specific surface calculated from a Type II isotherm by application of the standard BET procedure. According to the foregoing analysis, the external specific surface of the solid is 114m g" the micropore volume (from the vertical separation of isotherms A and E) is 105 mm g but since the average pore width is not precisely known, the area of the micropore walls cannot be calculated. Thus the BET figure of 360m g calculated from isotherm E represents merely an apparent and not a true surface area. 

The a,-plots were based on the standard isotherms of Nj and CCI respectively, on Fransil. For calculation of. 4,(BET), the value u fCCU) = 37 was assumed. 

Another view is given in Figure 3.1.2 (Berty 1979), to understand the inner workings of recycle reactors. Here the recycle reactor is represented as an ideal, isothermal, plug-flow, tubular reactor with external recycle. This view justifies the frequently used name loop reactor. As is customary for the calculation of performance for tubular reactors, the rate equations are integrated from initial to final conditions within the inner balance limit. This calculation represents an implicit problem since the initial conditions depend on the result because of the recycle stream. Therefore, repeated trial and error calculations are needed for recycle... 

Knowing the experimental retention times, the previous equation allows the calculation of experimental concentration on the solid phase. Parameters of adsorption isotherms, can then be determined by fitting experimental and calculated concentrations. 

Calculation of TMB flowrates To calculate TMB flowrates, linear behavior of the adsorption isotherms for a feed concentration of 1 g is assessed. To check this point, we will use the criterion given in Equation (10). 

Calculation of the SMB flowrates The criterion C + K - Cg<0. (Equation 10) equaling now approximately 0.22, the system is operating under nonlinear adsorption isotherm conditions. 

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