On the use of the adsorbed solution theory for designing adsorption separation units

On the use of the adsorbed solution theory for designing adsorption separation units Renato Rota, Giuseppe Gamba, and Massimo Morbidelli Dipartimento

di Chimica Fisica Applicata,

Politecnico

di Milano, Milano, Italy

The reliable evaluation of multicomponent adsorption equilibria is one of the key problems in the design of adsorbers, which can be pe$ormed through the adsorbed solution theory. This model requires the accurate investigation of pure component equilibria at low-pressure values. Such experiments are not always easily done in practice. Thus, to minimize the experimental effort required to design industrial adsorbers, the sensitivity of the adsorbed solution theory results to the accuracy of the lowpressure experimental data and to the parameter estimation procedure should be reduced. In this article different approaches to the adsorbed solution theory are discussed, which are aimed to reduce its sensitivity and then the experimental effort related to the adsorber design procedure. A comparison between the results of the models and some experimental data are presented, and the main advantages and limitations of such approaches are discussed.

Keywords: multicomponent

adsorption equilibria; adsorbed solution theory; adsorber design

Introduction

l

The importance of the correct representation of multicomponent equilibria in the design of adsorption separation devices is widely recognized. Simple equilibrium relations, such as the Langmuir isotherm, cannot account for important features of multicomponent adsorption equilibria, such as selectivity values depending on the adsorbed phase composition.’ Thus, more accurate models should be used, which arise from a deeper thermodynamic analysis of the involved phenomena.2 One of the most powerful models is the adsorbed solution theory (AST) developed by Myers and Prausnitz.3 Such a model is composed by the following 2NC + 1 equations: l Equality between the fugacities in gas and adsorbed phases Pyi = pp(~)Xiyi(7F, Xi), i = 1)NC l

(1)

Gibbs’ isotherm I+IJ = g

= I,“”

ry(P)d In P, i = 1, NC

(2)

Address reprint requests to Dr. R. Rota at the Dipartimento di Chimica Fisica Applicata, Politecnico di Milano, via Mancinelli 7, 20131 Milano, Italy. Received 10 February 1993; accepted 3 June 1993

230

Sep. Technol.,

1993, vol. 3, October

Stoichiometric

relationship

hlr-

..v

CXi= 1 i=l

From the Xi, py , and r values obtained by solving the preceding equations, it is possible to compute the adsorbed amount of each component. Nonideal behavior of the adsorbed phase can be accounted for by means of suitable activity coefficients tuned on binarymixture experimental data.4 Note that in what follows both r and JI will be called spreading pressure. When the adsorbed mixture behaves like an ideal solution, the activity coefficients approach unity. In this case, the AST allows to predict multicomponent equilibria from single-component equilibrium data. Unfortunately, there is no reliable method to decide a priori if an adsorbed solution can be regarded as ideal. Thus, the designer needs in all cases binary-mixture experimental data, at least to check whether the adsorbed mixture is ideal or not. The main concern about the AST refers to the sensitivity of its predictions to the type and accuracy of the single-component isotherms used to fit the experimental data.&’ This sensitivity arises from the integral on the right-hand side of Equation 2, which strongly depends on the values of the integrand function (i.e., the single-component isotherm) in the range of lowpressure values. Such a low-pressure range, usually 0 1993 Butterworth-Heinemann

Adsorbed solution theory: R. Rota et al. referred to as the Henry region, often lies outside the range investigated experimentally. When this is the case, different single-component isotherms represent equally well the available single-component experimental data, but provide different descriptions of the Henry region, and, consequently, also of the multicomponent mixture behavior. An obvious solution to this problem is to experimentally investigate the Henry region properly and use single-component isotherms able to well represent such experimental data. This is not always easily done in practice, particularly for very strongly adsorbable cornpounds where the Henry region is confined to a range of very low-pressure values. Furthermore, for devices like the fixed-bed adsorbers for bulk separations processes, the designer needs to represent the multicomponent equilibria only at a fixed-pressure value. In this case, the experimental investigation of the Henry region is not convenient because the device is not required to work at such low-pressure values. The aim of this article is to discuss two different approaches to the AST able to remove the problems mentioned previously without involving the knowledge of the Henry region, thus reducing the experimental effort required to design industrial adsorbers.

Alternative approaches to the AST The classic approach the following steps: l l

to AST can be summarized

in

Single-component and binary-mixture data are collected. The parameters of each single-component isotherm (e.g., Ki and IT (i = l,NC), when using the Langmuir isotherm in Table I) are evaluated independently by fitting each single-component set of experimental data.

Table 1 Single-component

isotherms

Langmuir

P(P)

= SP

Freundlich

P(P)

= AP

Fritz

P(P)

= iy$

Toth

r-O(P) =

IB +*h

The single-component isotherms are used in the ideal AST model (IAST), i.e., Equations l-3 with yi = 1 (i = 1,NC), for estimating binary-mixture equilibria. If the IAST model predictions do not agree with the binary-mixture data, then the parameters accounting for the dependence of the activity coefficients on composition and spreading pressure are estimated by fitting these data. The multicomponent equilibria are predicted through the AST model (Equations l-3). An approach to the AST, aimed to remove the sensitivity mentioned earlier, was discussed by Gamba et al4 This procedure allows to disregard the Henry region fully, which is the source of such a sensitivity, by replacing the lower extreme of integration in Equation 2 with a finite pressure value p* > 0, lying outside the Henry region. In this case, Equation 2 reduces to AJ, - A$: = /y

rp(P)d In P

where A$ = I/I - I/I?, and +I$ is the spreading pressure value at which the first component is in equilibrium with a pure gas at pressure equal to p*; A+; = I/$ I/J: is the spreading pressure difference between the pure compounds i and 1 at p*. Note that in this approach the spreading pressure does not appear as the

Notation A surface area of the adsorbent K Langmuir equilibrium constant number of components NC p”(r), p”(t,!~) single-component adsorption equilibrium pressure P pressure Pm;) pi PEW PE R ideal gas constant T temperature x adsorbed-phase mole fraction fluid-phase mole fraction Y Greek letters Y ri

activity coefficient adsorbed amount of the ith component in the mixture

Tp

& 44 lr t: JIP

adsorbed amount of the ith single component saturation value of the adsorbed amount $ - Ji: 41 - +: spreading pressure spreading pressure (?rA/RT) spreading pressure of the ith single component at P = p* spreading pressure of the ith single component at P = P,

Subscripts talc exp i, j m

calculated experimental ith or jth compound mixture

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231

Adsorbed

solution theory: R. Rota et al.

I l

l Pressure,P

-

Figure 1 Spreading pressure as a function of the absolute pressure for two pure compounds.

absolute value, JI, but only as the difference A$ with respect to a reference value selected as JIT. It appears that if the low-pressure region is not involved in Equation 4, the sensitivity of the AST results on the choice of the single-component isotherm is removed. This is shown in Figure 1 where the spreading pressure is reported as a function of pressure for two pure compounds. Let us consider a mixture of these compounds at pressure P, . If they do not form azeotropes, the spreading pressure equilibrium value of the mixture, $, should lie; between $7 and $20, depending on the mixture composition. Thus, the upper extreme of integration in Equation 4, &‘($I), lies, between P; and P, for the first, compound, and between P, and Pf, for the second one, as shown in Figure 1 (dashed lines). If the arbitrary reference value is P; < p* < Pfj (for instance, p* = P,), the lower pressure value at which the single-component equilibrium has to be investigated is then given by P; for the first component and by p* ( = P,,,) for the second one. For many practical systems this means that the Henry region can be totally disregarded by truncating the pure-component experiments at a pressure value outside this region. In what follows such an approach will be referred to as the truncated AST (TAST). The price to be paid by TAST is the introduction of the NC - 1 new parameters A+; = $T - @. These are adjustable parameters referring to the single component equilibrium behavior, which can also be estimated by comparison with binary data. To summarize, the use of the TAST model involves the following steps: Single-component and binary-mixture data are collected. The parameters of each single-component isotherm are evaluated independently by fitting each singlecomponent set of experimental data. A reference pressure value, p* (usually equal to the operating pressure), and a reference compound (usually the first one), are selected. The NC - 1 parameters A$; are estimated by com232

Sep. Technol.,

parison with the available binary-mixture data, without involving the activity coefficients. If the agreement between the experimental data and the model results is not satisfactory, the activity coefficients are introduced. Thus, both the NC - 1 parameters AJli”;and all the parameters involved in the adsorbed activity coefficient model are estimated by comparison with the binary-mixture data. Note that, as previously mentioned, to account for the dependence of the activity coefficient on the spreading pressure, several adjustable parameters are required, thus strongly complicating this step.4 The multicomponent equilibria are predicted by the TAST model (Equations 1, 3, and 4).

1993, vol. 3, October

It is worth mentioning that in the case of strongly adsorbable compounds it can be assumed that the single-component isotherms are at saturation for any pressure value between p* and pp, that is, T;(p) = r:. Furthermore, for many practical applications, the spreading pressure of the mixture ranges in a narrow field. Thus, the spreading pressure influence on the activity coefficients can be neglected. This is the case in which the TAST model gives the greatest advantages with respect to the classic AST, as Equations 1 and 4 can be rearranged leading to4**

(5) These relations can be further simplified if the saturation loads for the various involved compounds are similar, i.e., I; = r-. The use of Equation 5 coupled with Equation 3 requires only binary-mixture experimental data, and the saturation load of each single compound. Thus, the experimental characterization of single-component equilibria as a function of pressure can be avoided. For strongly adsorbable compounds, this is a significant advantage. Despite its simplicity, this is a reliable approach that was successfully used in adsorber design.’ The NC - 1 parameters A$:, which have been introduced previously as adjustable parameters, can be alternatively estimated through the isobaric integral from suitable binary-mixture experimental data at p* (= Pm).

(6) This approach has the advantage to avoid the fitting procedure for the AJii?;, but it requires experimental data of the binary-mixture at pressure equal top * properly distributed over the entire composition of the fluid phase. When the effect of spreading pressure on the activity coefficients cannot be neglected, one should know the absolute value of the spreading pressure, and not only its difference from a reference value as given by the TAST model. In these cases, disregarding the influence of spreading pressure on the activity coefficients can introduce errors of the order of 30%.9 An alternative

Adsorbed

approach, which is able both to remove the AST sensitivity to the Henry region data and to give the absolute value of the spreading pressure, is presented in the following. The basic idea is to constrain the isotherms employed for representing the single-component behavior to be thermodynamically consistent with the binarymixture experimental data. This is done through the value of the spreading pressure difference between the pure compounds at a given temperature and pressure, A$$, which can be evaluated either from the experimental binary-mixture equilibrium data at the same temperature and pressure using Equation 6, or from the single-component isotherms as follows: A$; = 1:’

(r:(p) - rp))d

In P

(7)

The isobaric integral (Equation 6) can be computed numerically from the binary experimental data through suitable interpolation and quadrature formulas, whereas the integral on the right-hand side of Equation 7 can often be evaluated analytically as a function of the single-component isotherm parameters. Because the integral on the right-hand side of Equation 7 is very sensitive to the values of the integrand function in the low-pressure region, small deviations of the isotherm values from the true ones induce large errors in the value of A$$ computed by this equation, which in turn does not agree with the value computed from Equation 6 using the binary experimental data. In other words, small errors in characterizing the Henry region of one of the pure compounds induce a thermodynamic inconsistency between the experimental binary-mixture data and the isotherms used for representing the single-component behavior. This is an undesirable situation, because single-component isotherms that do not satisfy the previous thermodynamic constraint cannot provide an accurate prediction of the multicomponent equilibria. To avoid this problem, the parameters of all the NC single-component isotherms can be estimated by simultaneously fitting all the experimental single-component data with the constraint of satisfying the NC!/ 2(NC - 2)! equalities between the right-hand sides of Equations 6 and 7 for all the binary-mixtures that can be formed with the NC compounds. For example, if NC = 2 one has to fulfill only one such constraint (2!/ 2(2 - 2)! = l), i.e., dy2,exp

=

I

B’

(rp) -

r:(P))d In P

(8)

where the left-hand side is the experimental value of A@, (i.e., the value estimated from the binary-mixture experimental data), whereas in the right-hand side the single-component isotherm parameters are involved. For instance, when using the Langmuir isotherm, Equation 8 reduces to

w.exp

solution theory: R. Rota et al.

= r; ln(1 + K,p*) - I: ln(1 + Klp*)

(9)

which represents a constraint for the values of K, , K, , I’:, and I;. Note that the simultaneous fitting of all the single-component adjustable parameters requires a nonlinear, multiresponse regression procedure, which should be able to account for nonlinear constraints among the involved parameters. It is worth noting that along these same lines Myers” proposed to use Equation 8 for computing the Henry constant of the more strongly adsorbable component in a binary-mixture as a function of the experimental value of 4%. expand of the value of the Henry constant of the less strongly adsorbable component. This procedure is recommended when reliable values of the equilibrium parameters for the less adsorbable component can be obtained from single-component equilibrium data. Thus summarizing, to compute multicomponent equilibria involving NC compounds the constrained approach to the adsorbed solution theory (CAST) involves the following steps: l l

l

l

Single-component and binary-mixture data are collected. The A$$, eXPvalues are computed from binary equilibrium data through Equation 6 for the NC!/2(NC 2)! binary mixtures, which can be formed from the NC compounds. The parameters of each single-component isotherm are simultaneously estimated by fitting all the singlecomponent data together and enforcing the NC!/ 2(NC - 2)! constraints in the form of Equation 8. The obtained parameter values are introduced in the AST model (Equations l-3), and the same steps previously discussed for the classical approach are followed.

To summarize, the main differences between TAST and CAST are related to the parameters At,!~il;,as well as to the model structure. In the CAST, the values of the A$$ parameters are always computed through the isobaric integral and then used for constraining the single-component isotherms to be thermodynamically consistent with the binary data. This affects only the estimation of the single-component from thereafter the AST equilibrium parameters; model is employed for computing multicomponent equilibria according to the classic procedure. Extension to real adsorbed solutions, even in the case of activity coefficients depending on the spreading Following the pressure, follows straightforwardly. classic approach, binary-mixture data are required to estimate the activity coefficients values when the adsorbed solution behavior is nonideal. In this regard, the proposed approach and the classic one can be regarded as fully equivalent. Conversely, the TAST is based on a rearrangement of the model aimed to remove the absolute value of the spreading pressure and to introduce the parameters AI,/I~ . In many cases this leads to a simpler analytical model, but it cannot Sep. Technol.,

1993, vol. 3, October

233

Adsorbed

solution theory: R. Rota et al.

Table 2

Parameter average percentage

values of single-component isotherms employed for the computations errors in reproducing single-component equilibrium data9

Part

Component

Isotherm

1

YS

Langmuir Freundlich Fritz Toth Langmuir Freundlich Fritz Toth Langmuir Freundlich Fritz Toth Langmuir Freundlich Fritz Toth Langmuir Freundlich Fritz Toth Langmuir Freundlich Fritz Toth Langmuir Freundlich Fritz Toth Langmuir Freundlich Fritz Toth

CO2

2

H2S

(32

3

H2S

co2

4

H2S

co2

reported

in Table 3 and correspondent

Parameters K A K A K A K A K A K A K A K A K A K A K A K A K A K A K A K A

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

103.60 l-” = 2.14. 1O-3 3.09.10-3 t = 0.22 2.16 F” = 2.96.10-3 t = 5.71. lO-3 8 = 0.15 C = 7.56 l-” = 2.41. 1O-3 2.22. 1O-3 t = 0.35 0.91 I= = 4.63.10-3 t = l.05.10-2 B = 0.36 C = 96.91 r” = 2.15.10-3 3.46.10-3 t = 0.27 6.90 I-” = 2.67. 1O-3 t = 3.51. 1O-3 B = 0.10 C = 7.69 r” = 2.41.10-3 2.19.10-3 t = 0.32 0.71 5.29.10-a t = 1.44.1O-2 B = 0.40 C = 49.85 r” = 2.36. 1O-3 2.97. 1O-3 t = 0.20 0.92 r- = 5.91.10-3 t = 1.40. lO-2 B = 0.19 C = 3.90 I-” = 2.77.10-3 2.19.10-3 t = 0.30 0.87 I-- = 4.72. 1O-3 t = 6.55.10-3 B = 0.37 C = 57.67 r” = 2.31. 1O-3 3.16.10-3 t = 0.23 2.66 I= = 3.72.10-3 t = 5.04.10-3 B = 0.14 C = 3.89 I= = 2.81. 1O-3 2.18.10-3 t = 0.26 0.34 r= = 8.57.10-3 t = 2.81. 1O-2 B = 0.39 C =

r- =

E%

0.37 0.19

0.50 0.21

0.53 0.33

0.47 0.18

0.29 0.11

0.50 0.29

0.40 0.22

7.03 2.21 1.04 0.95 16.67 6.47 0.97 1.23 7.21 6.83 2.57 1.94 18.00 10.68 2.00 1.56 2.91 0.54 0.42 0.43 1.64 0.69 0.14 0.16 3.27 2.12 0.55 0.51 1.64 1.67

0.38 0.13

Part 1: without constraint; all the experimental data were considered. Part 2: with constraint (Equation 8); all the experimental data were considered. Part 3: without constraint; only the experimental data in the range [r5/2, r”] were considered. Part 4: with constraint (Equation 8); only the experimental data in the range [Y/2, Yl were considered.

handle in a simple way adsorbed solutions involving activity coefficients depending on spreading pressure.

Comparison with the experimental data The experimental data selected to investigate the performance of the aforementioned models refer to the system &S - CO2 on H-mordenite at 16 kPa and 303 K.g As shown earlier@ the results of the IAST when applied to such an ideal system exhibit a strong sensitivity to the type of single-component isotherm used to represent the experimental data. In particular, the four single-component isotherms summarized in Table I were considered. It should be stressed that the approach presented here is not aimed to well represent the Henry region of each single component but to find a thermodynamically consistent set of parameters for the single-component isotherms. In fact, two of the isotherms reported in Table I (Freundlich and Fritz) cannot represent properly the Henry region.2 Despite this limitation, when their parameters fulfill the constraint (Equation 8), these isotherms also can give good results in terms of multicomponent equilibria predictions, as shown later. 224

Sep. Technol.,

1993, vol. 3, October

Following the classic approach, the adjustable parameters of the single-component isotherms are estimated through a nonlinear optimization procedure using as objective function the sum of the squared errors of the adsorbed amounts. The values of such parameters, together with the average percentage errors, are reported in the first part of Table 2. It appears that, with the exception of the Langmuir isotherm, which usually shows a larger average error, all the other isotherms are able to represent the experimental data with a comparable accuracy. However, such isotherms, when employed to predict the binary-mixture behavior according to the classic approach to AST (i.e., Equations l-3), provide rather scattered results. This is shown in the first part of Table 3, where such results are summarized in terms of average percentage errors in reproducing both the adsorbed mole fraction and the total adsorbed amount. The first step of the approach proposed in this work (i.e., CAST) is the estimation, through a cubic spline quadrature procedure, of the value of AI/J&from the binary equilibrium data. This value is equal to -4.05 - 10e3 mol/g. Next, the parameters involved in the single-component isotherms are estimated by com-

Adsorbed Table 3 Average percentage equilibrium data9

Part

errors in reproducing

Singlecomponent isotherm Langmuir Freundlich Fritz Toth Langmuir Freundlich Fritz Toth Langmuir Freundlich Fritz Toth Langmuir Freundlich Fritz Toth

E%

(X)

15.31 41.90 8.71 20.25 15.43 13.08 15.00 14.92 14.29 14.63 14.48 14.47 14.87 13.83 14.57 14.57

H$

-

E%

CO2

r,,1

5.50 4.46 19.98 2.12 5.21 4.79 2.34 2.18 5.11 1.66 1.77 1.76 4.61 2.16 2.16 2.07

Part 1: classical AST, with the single-component equilibrium parameters reported in part 1 of Table 2. Part 2: CAST, with the single-component equilibrium parameters reported in part 2 of Table 2. Part 3: TAST, with the single-component equilibrium parameters reported in part 3 of Table 2. Part 4: CAST, with the single-component equilibrium parameters reported in part 4 of Table 2.

Table 4 Values of A$; calculated through Equation 7 using the single-component equilibrium parameters reported in the first and second part of Table 2

Isotherm Langmuir Freundlich Fritz Toth

Parameters as in part 1 of Table 2 -4.10*10-3 -6.03.10-3 -3.17.10-3 -4.78. lO-3

Parameters as in part 2 of Table 2 -4.05.10-3 -4.05.10-3 -4.05.10-3 -4.05.10-3

parison with the single component experimental data and by forcing them to fulfill condition 8, with the A@, value reported earlier. The second part of Table 2 reports the obtained values of the adjustable parameters, together with the correspondent average errors in reproducing the experimental equilibrium data. Note that both the parameters and the average error values reported in parts 1 and 2 of Table 2 are not very different from each other. The main difference is that the parameters in part 1 minimize the error in representing single-component data but fail in reproducing the correct value of A+$. Conversely, those in part 2 exhibit a larger error in reproducing single-component data because they also have to satisfy the constraint imposed by the A$!, value. This is confkmed by the values of A$& calculated using both sets of singlecomponent parameter values as reported in Table 4. As expected, when the parameters reported in the first

solution theory: R. Rota et al.

part of Table 2 are used for representing the binary mixture behavior in the context of the AST, different isotherms give uneven results. The average errors in representing the binary equilibrium data through the proposed approach are summarized in the second part of Table 3. It is worth stressing that the developed approach allows to remove the influence of the single-component isotherm on the AST performance, but it renounces to provide a true prediction of the binary-mixture behavior from single-component experimental information, because binary data have to be used for evaluating A$$, . However, note that the isotherm parameters were not tuned on binary data. The only information used involving such data is the value of At/&. This approach provides true predictive results when applied to multicomponent equilibria involving more than two components. As mentioned earlier, for nonideal adsorbed solutions also the classic approach renounces to predict the two-component equilibria from single-component equilibrium data only. Similar results are obtained using the TAST, where no single-component experimental data in the Henry region are needed. In the third part of Table 3 the results obtained through this approach are reported. Note that in fitting the single-component isotherm parameters only those single-component equilibrium data where the adsorbed amount is larger than one half of the saturation value were used. The obtained parameter values and the correspondent percentage errors are reported in the third part of Table 2. It appears that the obtained errors are significantly lower than those shown in the previous parts of Table 2. This was expected, because a lower number of equilibrium data, falling in a narrower pressure range, was considered in the fitting procedure. This accounts also for the larger differences in the estimated values of the model parameters. Nevertheless, the results for the binary equilibria reported in Table 3, part 3, are very similar to those obtained through the CAST and reported in part 2 of the same table. Thus, the two approaches are both equally efficient in removing the sensitivity of the multicomponent equilibria predicted by the AST from the choice of the model used for reproducing the singlecomponent equilibrium data. As a final test for the CAST, we consider the case in which experimental data in the Henry region are not available. This is done by considering, in the fitting procedure of the single-component isotherms, only the single-component data in the high-pressure region (i.e., adsorbed amount larger than one half of the saturation value) and simultaneously enforcing the constraint represented by Equation 8. The obtained parameter values are summarized in the fourth part of Table 2, whereas the correspondent results in reproducing binary equilibrium data are reported in Table 3, part 4. It appears that, even in the case in which the Henry region is not experimentally investigated, the CAST provides reasonable predictions of the multicomponent equilibria, which are not influenced by the particular choice of single-component isotherm. In other words, the repSep. Technol.,

1993, vol. 3, October

235

Adsorbed t

solution theory: R. Rota et al.

also the sensitivity of the AST results to the choice of the objective function in the regression method. Similarly to Richter et al. ,’ the following three objective functions were considered:

5.10-3

A

0.2

0.4

Vapor

0.6

Phase Mole of Ethane -

0.8

1.0

Fraction

4 5.10-31

I

Vapor

The sum of the squared differences between experimental and predicted values of the ratio P/IO(P) (linearized form) The sum of the squared differences between experimental and predicted values of the adsorbed amount, IO(P) (original form) The sum of the squared differences between experimental and predicted values of the ratio I”(P)IP (spreading pressure form) The selection of one or the other of the preceding objective functions can be based on the final use of the developed single-component equilibrium model, such as the evaluation of the spreading pressure value through Equation 2. However, in general, when I’(P) and P are the measured variables, the second objective function should be used to get an unbiased estimate of the equilibrium parameters. Figures 2a and 3a compare the experimental data of the binary system at 5 and 20 bar with the AST results obtained using the classic approach and the three objective functions earlier (same parameter values as in reference 7). A large scatter among the predic-

Phase Mole Fraction of Ethane -

Figure 2 Amount adsorbed for the system methanelethane on activated carbon at T = 303 K and P = 5 bar. Experimental data from Richter et al.‘: 0 = methane; 0 = ethane. (A) Classical IAST; (6) TAST. - = First objective function (linearized form); __ = second objective function (original form); -a- = third objective function (spreading pressure form).

resentation of the Henry region seems to be not essential as long as the single-component isotherms are consistent with the binary mixture data.

About the regression procedure of singlecomponent equilibrium data Richter et a1.7 pointed out that not only the choice of the single-component isotherm but also the choice of the regression method used to evaluate its adjustable parameters can strongly affect the prediction of multicomponent equilibria provided by the AST. In particular, they showed that when the Langmuir isotherm is employed to represent the single-component experimental data of both methane and ethane on activated carbon at 303 K, the binary-mixture equilibria predicted by the AST strongly depend on the objective function used in the optimization procedure. The origin of such a strong sensitivity is in the different weight attributed in particular to the data in the low-pressure region, which, as discussed earlier, strongly affect the results of the multicomponent equilibria calculations. Because both TAST and CAST are able to disregard the low-pressure behavior of the single-component isotherms, one should expect them to be able to reduce 236

Sep. Technol.,

1993, vol. 3, October

Vapor

0.2

Phase Mole of Ethane -

0.4

0.6

Fraction

0.6

Vapor Phase Mole Fraction of Ethane Figure 3 Amount adsorbed for the system methanelethane on activated carbon et T = 303 K and P = 20 bar. Experimental data from Richter et al.’ Definitions as in Figure 2.

Adsorbed Table 5 Values of the Langmuir isotherm parameters used in the computations shown in Figures 26 and 36. Only single-component experimental data in the range [P-/2, Pm] were considered, using three different objective functions: &(okkcalc- qkW)* Objective function ok = P/I’#p) ok =

r!(p)

or = P$p)/P

Component Ethane Methane Ethane Methane Ethane Methane

K

l-”

1.41 0.19 1.77 0.19 2.64 0.22

4.67.10-3 5.00.10-3 4.56.10-3 4.96. 1O-3 4.08. 1O-3 4.75.10-3

at p*

WI = 10 bar

-9.06.

1O-3

-9.77.10-J -9.57.10-3

tions obtained using different objective functions is evident. The results shown in Figures 2b and 3b were obtained using the TAST procedure, where AI,@,is regarded as an adjustable parameter, and the data in the Henry region are disregarded. It appears that the scatter owing to a different choice of the objective function is significantly reduced. Note that in this case the binary data are fitted and not predicted. As discussed elsewhere,4 when the TAST is able to reduce the scatter for the fitted binary data, it reduces also the scatter for the predicted multicomponent equilibria. Thus, the preceding conclusion can be regarded as fully general. The parameters used in Figures 26 and 36 are summarized in Table 5. Note that the same value of A$,*, estimated from the binary data at p* = 10 bar was used for computing both the isobaric binary-mixture equilibria at 5 and 20 bar. This was made possible by selecting in both cases the lower limit of the Gibbs isotherm integral (Equation 4) at p* = 10 bar. Consequently, both the sets of binary-mixture experimental data were simultaneously used for estimating the At@i value. A small scatter among the values estimated was found; this is probably due to the lack of experimental data of binary-mixtures properly distributed in the fluid-phase composition range.

of the spreading pressure on the activity coefficients is negligible. In both these procedures the single-component data in the Henry region can be disregarded. Of course, if experimental data in the Henry region are available, and thermodynamically consistent isotherms able to fit such experiments in all the pressure range are used, the discussed approaches are not necessary because the classic procedure gives good results. However, if the opposite is true, either TAST or CAST can be useful to avoid wrong predictions of the multicomponent equilibria. The price that must be paid is the loss of predictivity for binary-mixture data because such data are necessary to estimate the value of the isobaric integrals, AI/J:. However, if the adsorbed solution is nonideal, the same information is needed, also by the classic approach, to estimate the activity coefficients .

In general, one can decide which approach is most suited for a particular situation by computing multicomponent equilibria following the classic approach with different single-component isotherm models fulfilling the thermodynamic requirements discussed by Talu and Myers.’ If the predictions of the IAST are quite different when different single-component isotherms, able to reproduce the experimental data with the same accuracy, are used, then the use of TAST or CAST is recommended. This allows the designer to avoid further experiments to investigate the Henry region. In this case the sensitivity of the AST results to the choice of the objective function used in the regression method of the single component data is also reduced.

Acknowledgments Financial support of CNR-Progetto Finalizzato ica Fine is gratefully acknowledged.

In designing industrial adsorbers one needs to describe multicomponent adsorption equilibria. The more accurate this description, the more reliable the final design. A correct representation of such equilibria through the AST requires an adequate experimental investigation of the pure components behavior inside the Henry region. This experimental work can be difficult if some compound is strongly adsorbable, so that its Henry region is confined to very low-pressure values. To reduce the experimental effort required to design an industrial adsorber, an alternative approach to the AST, which is able to reduce its sensitivity to the model used to represent the single-component experimental data, was developed. It leads to results similar to those obtained through a previously presented modified approach, the TAST model, which is particularly suited for strongly adsorbable compounds when the influence

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