Ind. Eng. Chem. Res. 1997, 36, 419-423
419
A Simple Way To Describe Nonisothermal Adsorption Equilibrium Data Using Polynomials Orthogonal to Summation Syed M. Taqvi† and M. Douglas LeVan* Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22903-2442
Nonisothermal adsorption equilibrium data are described using the virial equation written as a polynomial function in adsorbed-phase loading and reciprocal temperature. This equation is written in terms of polynomials that are orthogonal to summation. As a result, application of the least-squares method to correlate experimental data becomes straightforward and simple, not even requiring any matrix algebra. Ill-conditioning of the normal equations for high-order polynomials is avoided. The model is successfully applied to describe nonisothermal data for different systems over a wide range of temperature and pressure. Introduction Data for adsorption equilibrium of pure components are often obtained at different temperatures. An equation with temperature-dependent parameters that can accurately describe such data over a wide range of temperature and pressure and is very easy to use would be useful for engineering applications. Several models are available in the adsorption literature to correlate temperature-dependent equilibrium measurements. For example, relations based on the Polanyi pore-filling concept have an explicit temperature dependence. These can be used to correlate experimental data at different temperatures to obtain a temperature-independent characteristic curve. Thus, determination of isotherms at different temperatures becomes possible. An alternative description of temperature-dependent vapor-adsorbate equilibrium is possible using the virial formalism. The virial description has been used by Haydel and Kobayashi (1967), DeGance (1992), DeGance et al. (1993), and others to model isothermal multicomponent adsorption equilibria. The parameters in the virial equation are functions of both temperature and molecular interactions. Barrer and Davies (1970) assumed a linear dependence of the virial coefficients and an exponential dependence of the Henry’s law constant on temperature to correlate temperaturedependent pure component adsorption equilibrium data. Recently, Zhang et al. (1991) also used the virial description to correlate temperature-dependent data. They expanded all of the parameters in the equation in a series in reciprocal temperature. In single-variable regression problems, polynomials orthogonal to summation are often used to eliminate matrix calculations and ease mathematical computations (Golub and Ortega, 1992). The method for generating the orthogonal polynomials in one variable was first described by Forsythe (1957) and extended to multivariables by Weisfeld (1959). In this work, we apply the concept of orthogonal polynomials to the virial formalism to develop a simple method for correlating adsorption equilibrium data at different temperatures. The form of the virial equation that we use is similar to that of Zhang et al. (1991). * Author to whom correspondence should be addressed. † Current address: UOP, P.O. Box 5016, Des Plaines, IL 60017. S0888-5885(96)00366-1 CCC: $14.00
Thus, our equation is in the form of a power series in loading and reciprocal temperature. Such a description is mathematically very versatile as the number of terms in the series can be adjusted as required for an accurate description of the experimental data. By rewriting the virial equation in terms of polynomials that are orthogonal to summation, we eliminate the need for solving a possibly ill-conditioned system of equations to obtain the parameter values. Evaluation of the parameters is therefore direct and trivial and makes this a powerful and robust technique for correlating temperature-dependent data. Mathematical Method Virial Equation. The virial equation for a pure adsorbed component is
ln(p/n) ) A +
3 2 Bn + Cn2 + ... A 2A2
(1)
The coefficients in the above equation are functions of temperature. Expressing the temperature dependence in terms of 1/T, we can write
A ) A(0) +
A(1) A(2) + 2 + ... T T
(2)
B ) B(0) +
B(1) B(2) + 2 + ... T T
(3)
C ) C(0) +
C(1) C(2) + 2 + ... T T
(4)
and so forth. Thus, the right-hand side of eq 1 is a polynomial function in n and 1/T. Least-Squares Method. In a typical least-squares approach, the parameters in eqs 1-4 would be evaluated from experimental measurements by minimizing the objective function N
)
exp 2 [ln(pcalc ∑ i /ni) - ln(pi /ni)] i)1
(5)
where the summation is over all of the data and N is the number of data points. For to be minimum it is necessary for its partial derivative with respect to each parameter to be zero. Therefore, for an M parameter equation, this is a set of M simultaneous algebraic © 1997 American Chemical Society
420 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997
equations in M unknowns. The coefficient matrix obtained from these equations usually tends to become ill-conditioned for large M. For single-variable problems ill-conditioning usually occurs at M ) 6 (Golub and Ortega, 1992). Even for smaller values of M the coefficients calculated at one M have no relationship to the coefficients calculated at another M, and the whole calculation has to be repeated as M is increased gradually to obtain a better fit. Method of Polynomials Orthogonal to Summation. The computational difficulties mentioned above can be overcome by the use of orthogonal polynomials. We do not use eqs 1-4 directly because the terms are not orthogonal to summation. Instead, we construct a series of orthogonal polynomials and evaluate the coefficients in the new series by the least-squares method. Following this, we can determine the coefficients in eqs 1-4 if desired. First, we rewrite eq 1 in a more general form as M2
ln(p/n) )
M1
∑ ∑a j )0 j )0
(x˜ )
( 1,j2)
(11)
u(0,0) ) 1
(12)
For j2 ) 0, the recurrence formula is
u(j1,0)(x˜ ) ) x1u(j1-1,0)(x˜ ) -
Rk˜j˜ uk˜ (x˜ ) ∑ k˜ 0, we have
u(j1,j2)(x˜ ) ) x2u(j1,j2-1)(x˜ ) -
Rk˜j˜ uk˜ (x˜ ) ∑ k˜ M2, and proceed as described below. These conditions on M1 and M2 and the order of summation in eq 6 ensure that the u(j1,j2)(x˜ ) polynomials or, equivalently, the (j1,j2) pairs appear in a proper sequence. Considering the indices (j1,j2), we order terms first by second index and then by first index. Thus, if j1 were to run from 0 to 2 and j2 from 0 to 1, terms would be added to the series in the order (0,0), (1,0), (2,0), (0,1), (1,1), and (2,1). This ordering is useful in the orthogonalization of u(j1,j2)(x˜ ) described later in this section. Equation 6 can be written in a more compact form as
∂
N
a˜j
where
uj
(j1,j2)
2
written for each ˜j as
(10)
for ˜j * k˜ . This condition reduces all but one term on the right-hand side of eq 9 to zero. Thus, eq 9 can be
The summations in eqs 13 and 15 are over all previously obtained terms in the series. The R˜kj˜ ’s give orthogonality so that eq 10 is satisfied; in determining a new polynomial u˜j, new R˜kj˜ ’s are determined, one for each k˜ , to make the new polynomial orthogonal to the previously determined ones. The procedure described above for the generation of orthogonal polynomials in two variables is an adaptation of the method proposed by Weisfeld (1959). However, our method of arranging ˜j’s is different from that suggested by Weisfeld. We order ˜j’s to avoid getting extraneous terms in the model equation. In Weisfeld’s ordering procedure, the summation in eqs 13 and 15 would be over all k˜ < ˜j for which [(j1 + j2) - (k1 + k2)] e 2. For example, if we wanted the virial equation to be cubic in n with a linear dependence of the virial coefficients on 1/T, we would get a 1/T2 term as a result of orthogonalization using the ordering method of Weisfeld, whereas the ordering scheme outlined above will not give such undesired terms in the series. Calculation Procedure. We first obtain all ˜j’s in the proper order knowing the highest power of loading and reciprocal temperature in the virial equation. For example, suppose we want ln(p/n) to be quadratic in
Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 421 Table 1. Variances for the Virial Equation Truncated at Different Points N (∆ ln p)2 var ) (1/N)∑i)1
(n, 1/T) (n, 1/T2) (n2, 1/T) (n2, 1/T2) (n3, 1/T) (n3, 1/T2) H-mordenite CO2 H2S C3H8 BPL carbon CH4 C2H6 C2H4 CO2
0.0322 0.0100 0.0636
0.0318 0.0099 0.0611
0.0090 0.0088 0.0188
0.0085 0.0082 0.0146
0.0037 0.0042 0.0044
0.0028 0.0026 0.0030
0.0103 0.0382 0.0247 0.0072
0.0103 0.0356 0.0240 0.0070
0.0030 0.0065 0.0132 0.0020
0.0029 0.0058 0.0128 0.0017
0.0020 0.0033 0.0129 0.0012
0.0013 0.0011 0.0121 0.0003
Figure 2. Adsorption isotherms for H2S on H-mordenite (Talu and Zwiebel, 1986): (O) 10.0, (0) 30.0, (b) 65.0, and (4) 95.0 °C.
Figure 1. Adsorption isotherms for CO2 on H-mordenite (Talu and Zwiebel, 1986): (O): 10.0, (0) 30.0, and (b) 50.0 °C.
loading and linear in reciprocal temperature, i.e.,
(
ln(p/n) ) A(0) +
) (
)
A(1) 2 (0) B(1) + B + n+ T A T C(1) 2 3 (0) C + n (17) T 2A2
(
)
From above we have ˜j ) (j1,j2), u(j1,j2)(x1,x2) with x1 ) n and x2 ) 1/T, and j1 and j2 represent highest powers on n and 1/T, respectively. Thus, from eq 6, we have M1 ) 2 and M2 ) 1, and the ordering of the ˜j’s as given previously is (0,0), (1,0), (2,0), (0,1), (1,1), and (2,1). Next we write ln(p/n) in terms of orthogonal polynomials as in eq 6 and evaluate the parameters using eqs 11-16. Then, if desired, we obtain the virial parameters in eq 17 by collecting terms of like powers in eq 6. Results The virial model was used to correlate the data of Talu and Zwiebel (1986) for the adsorption of carbon dioxide, hydrogen sulfide, and propane on H-mordenite and the data of Reich et al. (1980) for the adsorption of methane, ethane, ethylene, and carbon dioxide on BPL activated carbon. The following variance was defined
var )
1
) N
N
∑1 (∆ ln p)2
N
(18)
where ∆ ln p is the difference between a calculated and
Figure 3. Adsorption isotherms for C3H8 on H-mordenite (Talu and Zwiebel, 1986): (O): 9.9, (0) 30.0, and (b) 51.1 °C.
an experimental value of ln p. The virial equation was truncated at different points with different values of highest powers of n and 1/T. The variances calculated for all examples considered are shown in Table 1. As is evident from the table, increasing the temperature dependence beyond linear does not improve the quality of the fit significantly. However, on increasing the highest power of loading in the equation, there is a significant decrease in the variance and a better fit is obtained. Figures 1-3 show the fits obtained for adsorption on H-mordenite when ln (p/n) is quadratic and cubic in n with a linear dependence on 1/T. In all figures, pressure is plotted on a logarithmic scale to clearly show the quality of the fit at low pressures. The correlation describes the data well over several decades of pressure. Tables of virial coefficients for all figures are given in the Appendix. The correlations for adsorption on BPL activated carbon are shown in Figures 4-7. Both the quadratic and the cubic equation accurately describe the data for
422 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997
Figure 4. Adsorption isotherms for CH4 on activated carbon (Reich et al., 1980): (O) : 212.7, (0) 260.2, and (b) 301.4 K.
Figure 6. Adsorption isotherms for C2H4 on activated carbon (Reich et al., 1980): (O) 212.7, (0) 260.2, and (b) 301.4 K.
Figure 5. Adsorption isotherms for C2H6 on activated carbon (Reich et al., 1980): (O) : 212.7, (0) 260.2, and (b) 301.4 K.
Figure 7. Adsorption isotherms for CO2 on activated carbon (Reich et al., 1980): (O) 212.7, (0) 260.2, and (b) 301.4 K.
methane on activated carbon as shown in Figure 4. For the adsorption of ethane on activated carbon, both equations work well for the isotherms obtained at 301.4 and 260.2 K (Figure 5). However, the cubic equation does not seem to work well in extrapolation to low loadings at 212.7 K. The fits obtained from the cubic equation after removing the circled points from the data set are also shown. The cubic equation appears to work much better without these points. Thus, the correlation is very sensitive to all of the data, particularly points at extremes. This is especially true for higher-order fits. For ethylene and carbon dioxide, both the quadratic and the cubic equations accurately describe the experimental data at the three different temperatures (Figures 6 and 7). For carbon dioxide, a decrease of 75% in variance is observed on increasing the temperature dependence of the cubic equation from linear to quadratic in 1/T. Therefore, for this case we have also shown the fits obtained when the series is truncated at M1 ) 3 and M2 ) 2.
Table 2. Coefficients for Virial Equation Truncated at (n2, 1/T) H-mordenite
(0,0) (1,0) (2,0) (0,1) (1,1) (2,1)
BPL carbon
CO2
H 2S
C 3H 8
CH4
C2H6
C 2H 4
CO2
15.74 1.021 -0.5768 -4748 472.7 55.08
15.07 -1.338 0.6115 -5354 1091 -192.0
18.73 -10.83 9.129 -5820 3895 -2088
9.622 0.4766 0.0012 -2001 -57.00 8.809
9.681 0.3581 0.0949 -2803 -58.07 -0.8052
11.36 -0.2092 0.1073 -3210 151.3 -15.37
10.92 0.1489 0.0311 -2660 -0.1654 -3908
Conclusions We have presented a simple approach for correlating adsorption equilibrium data at different temperatures. The virial equation is in the form of a power series and is mathematically very versatile. We have been able to accurately correlate experimental data on different adsorbents and over wide ranges of temperatures and pressures using this model. For most systems, the virial equation truncated after the third coefficient (quadratic
Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 423 Table 3. Coefficients for Virial Equation Truncated at (n3, 1/T) H-mordenite (0,0) (1,0) (2,0) (3,0) (0,1) (1,1) (2,1) (3,1) a
BPL carbon
CO2
H2S
C3H8
CH4
C2H6
C2H6a
C2H4
CO2
14.47 7.219 -6.477 1.500 -4488 -970.2 1457 -357.4
19.52 -14.02 11.09 -2.618 -6628 4714 -3182 747.6
24.57 -45.46 67.42 -29.56 -7354 12820 -17060 7625
10.59 -0.2867 0.1340 -0.0028 -2314 208.5 -46.66 2.700
4.497 4.884 -1.016 0.0815 -1322 -1341 312.4 -22.91
8.169 2.876 -0.6780 0.0644 -2433 -713.5 201.8 -16.86
10.48 0.7569 -0.1832 0.0260 -3027 -52.56 46.02 -5.500
11.62 -0.3135 0.1112 -0.0038 -2900 165.5 -34.65 1.614
Without circled points.
Table 4. Coefficients for Virial Equation Truncated at (n3, 1/T2) BPL carbon CO2 (0,0) (1,0) (2,0) (3,0) (0,1) (1,1)
10.64 0.7378 -0.3198 0.0488 -2616 -149.3
BPL carbon CO2 (2,1) (3,1) (0,2) (1,2) (2,2) (3,2)
124.0 -20.27 -13950 17720 -13860 2252
Appendix: Coefficients for Virial Equation Virial coefficients for all figures in the paper are given in Tables 2-4. The first column in a table gives the powers of n and 1/T, respectively, that the coefficient multiplies. Thus, a coefficient in the row (2,1) would be multiplied by simply n2/T and summed with other terms to give ln(p/n). Literature Cited
in loading) with a linear dependence of the coefficients on reciprocal temperature works well. For more complex systems, higher virial coefficients or a stronger temperature dependence are easily included in the model as shown. The concept of orthogonal polynomials in several variables provides an efficient and easy way to correlate experimental data using the virial description. Evaluation of parameters is direct and does not involve any matrix calculations or other extensive computations. Acknowledgment We are grateful to the U.S. Army ERDEC for financial support. Nomenclature a: parameter in least-squares polynomial A, B, C, ...: virial coefficients A: specific surface area of adsorbent, m2/kg ˜j, k˜ : 2-tuples of nonnegative integers M: number of parameters [(M1 + 1) × (M2 + 1)] M1: highest power of n in virial equation M2: highest power of 1/T in virial equation n: adsorbed-phase concentration, mol/kg N: number of data points p: pressure, Pa T: temperature, K x˜ : vector of two independent variables Greek Letter
Barrer, R. M.; Davies, J. A. Sorption in Decationated Zeolites I. Gases in Hydrogen Chabazite. Proc. R. Soc. London A 1970, 320, 289-308. DeGance, A. E. Multicomponent High-Pressure Adsorption Equilibria on Carbon Substrates: Theory and Data. Fluid Phase Equilib. 1992, 78, 99-137. DeGance, A. E.; Morgan, W. D.; Yee, D. High Pressure Adsorption of Methane, Nitrogen, and Carbon Dioxide on Coal Substrates. Fluid Phase Equilib. 1993, 82, 215-224. Forsythe, G. E. Generation and Use of Orthogonal Polynomials for Data Fitting with a Digital Computer. J. Soc. Ind. Appl. Math. 1957, 5, 74-88. Golub, G. H.; Ortega, J. M. Scientific Computing and Differential Equations; Academic Press Inc.: San Diego, CA, 1992. Haydel, J. J.; Kobayashi, R. Adsorption Equilibria in the MethanePropane-Silica Gel System at High Pressures. Ind. Eng. Chem. Fundam. 1967, 6, 546-554. Reich, R.; Ziegler, W. T.; Rogers, K. A. Adsorption of Methane, Ethane, and Ethylene Gases and Their Binary and Ternary Mixtures and Carbon Dioxide on Activated Carbon at 212-301 K and Pressures to 35 Atmospheres. Ind. Eng. Chem. Process Des. Dev. 1980, 19, 336-344. Talu, O.; Zwiebel I. Multicomponent Adsorption Equilibria of nonideal mixtures. AIChE J. 1986, 32, 1263-1276. Weisfeld, M. Orthogonal Polynomials in Several Variables. Numer. Math. 1959, 1, 38-40. Zhang, S.-Y.; Talu, O.; Hayhurst, D. T. High Pressure Adsorption of Methane in NaX, MgX, CaX, SrX, and BaX. J. Phys. Chem. 1991, 95, 1722-1726.
Received for review June 28, 1996 Revised manuscript received November 4, 1996 Accepted November 6, 1996X IE960366D
: least squares error Subscript i: data point index
X Abstract published in Advance ACS Abstracts, January 1, 1997.
