Isotherm correlation for water vapor on regular-density silica gel

Isotherm correlation for water vapor on regular-density silica gel

Shorter Communications arithmetic average value of 0 over the entire column arithmetic average value of 8’ over the entire column degree of saturation...

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Shorter Communications arithmetic average value of 0 over the entire column arithmetic average value of 8’ over the entire column degree of saturation of surface coverage at the column inlet defined by eq. (24) degree of saturation of column-average surface coverage defined by eq. (23) dimensionless axial distance from the entrance of column (z/L) the value of < at which both the intersection of , with 0 and that of G with K occur as shown in Fig. 3


Aris, R., 1978, Mathematical Modeling Techniques. Pitman, London. Cho, B. K., 1986, Estimation of kinetic parameters for elementary surface processes from integral reactor data: CO oxidation over Pt/AI,O,. Chem. Engng Commun. 47,201217. -_ Cho, B. K., 1987, Application of nonlinear wave propagation theory to adsorption kinetic measurements on supported catalysts. Chem. Engng Sci. 42, 1841-1846. Cho, B. K. and Stock, C. J., 1986, Determination of the coverage-dependent heat of adsorption on supported __ _ catalysts by chromatographic technique: CO adsorption on Pt/Al,O,. Preprints of 10th Canadian Symposium on Catalysis, Kingston, Ontario, 15-18 June, pp. 97-101. Cho, B. K. and West, L. A., 1986, Cyclic operation of Pt/ Al,O, catalysts for CO oxidation. Ind. Engng Chem. Fundam. 25, 158-164.

Chemical Engineering Scimce, Printed in Great Bntain.

Vol. 45, No. 5, pp. 1425-1429,

Isotherm (First

correlation received


ooG%-2509/90 s3.00 + 0.00 0 1990 Pergamon Press plc

for water vapor

1 August

1989; accepted


Regular-density (RD) silica gel is widely used for drying applications of humid air. Industrial drying installations are generally operated as regenerative heat and mass exchangers, involving dynamic adsorption and desorption. The performance of an adsorption unit is determined by the evolution of the properties of the dried gas with time, also known as the breakthrough curve. To model the dynamics of the silica gel bed, an accurate correlation for the sorption isotherm is needed. There are numerous correlations available in the literature; however, they are all based upon a limited set of experimental data. A generalized isotherm correlation for water vapor on RD silica gel, based upon all of the experimental data reported in the literature, is presented. DUBININ-ASTAKHOV


Cvetanovic, R. J. Bnd Amenomiya, Y., 1967, Application of a temperature-programmed desorption technique to catalyst studies. Adu. Catal. 17, 102-149. Falconer, I. L. and Schwartz, J. A., 1983, Temperature programmed desorption and reaction: applications to supported catalysts. Catal. Rev. Sci. Engng 25, 141-227. Golchet, A. and White, J. M., 1978, Rates and coverages in the low pressure Pt-catalyzed oxidation of carbon monoxide. J. Catal. 53, 266-279. Gorte, R. J., 1982, Design parameters for temperature programmed desorption from porous catalysts. J. Catnl. 75, 164-l 74. Gravelle, P. C., 1985, Application of adsorption calorimetry to the study of heterogeneous catalysis reactions. 7%ermochim. Acta 96, 365-376. Herz, R. K. and Marin, S. P., 1980, Surface chemistry models of carbon monoxide oxidation on surmorted Dlatinum catalysts. _I. Catnl. 65, 281-296. __ A Herz. R. K., Kiela. J. B. and Marin. S. P.. 1982. Adsorution effects d&ing iemperature-programmed d&sorption of carbon monoxide from supported platinum. J. Cntol. 73, 66-75. Jones, D. M. and Griffin, G. L., 1983, Saturation effects in temperature-programmed desorption spectra obtained from porous catalysts. J. Catal. 80, 40-46. McCabe, R. W. and Schmidt, L. D., 1977, Binding states of CO on single crystal planes of Pt. Surf. Sci. 66, 101-124. Oh, S. H., Fisher, G. B., Carpenter, J. E. and Goodman, D. W., 1986, Comparative kinetic studies of CO-O, and CO-NO reactions over single crystal and supported rhodium catalysts J. Carnl. 100, 360-376. Tompkins, P. C., 1978, Chemisorption of Gases on Metals. Academic Press. London.


The adsorption potential theory was originally proposed by Polanyi (1932). Using the gas-kinetic theories of

on regular-density in revised


3 October

silica gel 1989)

Boltzmann, Polanyi arrived at a theoretical relationship between the adsorbed amount and the average potential energy of the adsorbed molecules, expressed in the form W = f(A),

where A = R Tin


where A is the adsorption potential. The plot of the adsorbed amount as a function of A is called the characteristic curue if this curve is independent of temperature. In this case, using sorption thermodynamics, it can be shown that the adsorption potential is the difference in free energy between the adsorbed phase and the saturated liquid phase of the adsorptive at the same temperature, and thus is a true thermodynamic property, characteristic for the physical state of the adsorbate layer. Dubinin (1972, 1975) examined experimental sorption equilibrium data for many systems, and showed that the characteristic curve holds for many microporous adsorbents and can often be approximated by a Weibull expression:


Shorter Communications


woexp[ -($y


where W, is the total micropore volume, and E, is the characteristic energy of adsorption. For most microporous systems, the parameter n = 2 is recommended, in which case eq. (2) is referred to as the Dubinin-Radushkevich expression. Stoeckli (1977) commented on the application of this latter form and showed experimentally that eq. (2) with n = 2 applies only to uniform pores, i.e. pores with equal dimensions. A non-uniform pore system can be approximately visualized as the superposition of independent uniform microporous substructures. Often a two-site DubininRadushkevich expression is used to represent the alternate sorption in the smaller ultramicropores and the larger supermicropores:


WO.lexp[ - (&y] +


-($-y]. (3)

This expression is sometimes referred to as the Dubinin-Astakhov equation and is used to correlate the experimental equilibrium sorption data of water vapor on RD silica gel presented in the following discussion.



A literature survey on the sorption properties of RD silica gel (Van den Buick, 1987) yielded nine papers which presented original experimental sorption data. Representative test parameters for each of the publications are listed in Table 1. Alhlberg’s (1939) data were among the first experimental sorption equilibrium data for water vapor on commercial silica gel. The accuracy of the reported data may be questionable, probably due to the dynamic character of the measurement techniques. Ahlberg’s data are not included in the analysis presented here. Taylor (1945) determined the static sorption equilibrium isotherm for water vapor on an unspecified type RD silica gel. Kester and Lunde (1973) commented on Taylor’s experiments that the reported data may be in error due to experimental inaccuracies for T > 60°C. For the present analysis Taylor’s data are used for T < 60°C. Hubard (1954) presented an update of Taylor’s data and added original experimental sorption data at intermediate temperatures. The data are presented in graphical form, and for this analysis 84 data points read from the graphs in Hubard’s publication are retrieved. Jury and Edwards (1971) published a detailed set of experimental data for grade 01 Davison silica gel. These authors showed experimentally the existence of a character-


1. Test parameters

Source Ahlberg Taylor Hubard Mikhail er al. Jury et a!. Kester el al. Rosas Pendleton et al. Fraioli

istic curve for this sorption pair. The experimental data were curve-fitted with a two-site Langmuir expression with good accuracy. Kester and Lunde (1973) and Lunde and Kester (1975) analyzed Taylor’s data and also presented a limited set of original data at intermediate temperatures. Their equilibrium model assumed chemisorption for the first layer and physisorption for the subsequent layers. All sorption stages were characterized by an equilibrium constant and an enthalpy of reactionTemperature effects were included with the van’t Hoff relationship. The parameters were fitted to the experimental data, and a good overall curve fit was obtained. Rosas (1980) presented experimental equilibrium sorption data on Davison PA40 silica gel. The data were correlated with the BET isotherm correlation for multilayer adsorption, which is strictly valid only for type II isotherms, and a close agreement was nevertheless reported. Pendleton and Zettlemoyer (1984) studied the adsorption of water on a microporous silica. The experimental data were curve fitted with the two-sited Dubinin-Astakhov equation with close agreement. Unfortunately, the silica gel in their study had a maximum water uptake of only 0.20 kg/kg, which indicates that the properties of that gel are considerably different from the RD silica gel that is commonly used as desiccant. Seven of the reported data points with P/P, < 0.1, representing the filling of the ultramicropores, are retrieved for this study. Fraioli (1984) reported a set of carefully equilibrated, experimental sorption data. No correlation or analysis was presented.

DISCUSSION The experimental data retrieved from the literature are analyzed with the two-sited Dubinin-Astakhov eq_ (3). First, the parameters W,.,, Eoal, Wov, and W,,, are determined for each individual data set using a least-squares curve fit of the experimental data within that set. Because the functional form of the isotherm equation has a high degree of flexibility. the curve fit procedure indicates that, for equal mean square value of the error, there exists an interdependence among the four curve fit parameters. One of the parameters may vary over a wide range, while the other three parameters are then well determined for a minimum root mean square error. The results indicate that there exists two groups of data, each characterized by a different interdependency between the parameters. The experimental data from Taylor (194% Jury and Edwards (1971), Kester and Lunde (1973) and Fraoli (1984) belong to a group which has a consistently higher E,., than a second group, formed by the data from Hubard (1954) and Rosas (1980). The selected data from Pendleton and Zettlemoyer (1984) give values only for We,, and Ea.,, and belong to both groups. In a second step, the experimental data of the alternate

for published experimental sorption equilibrium silica gel and water vapor

Silica gel Davison Davison Davison Davison Davison Davison Davison Unilever Davison

GR-03 GR-01 CR-05 PA-40 G200 GR-40





O-1.00 o-o.73 O-l.00 O-1.00 O-O.67 o-o. 54 o-0.90 O-O.83 0-1.00

8-40 24-232 4-93 35 28-93 26-121 25-40 25 20

data for RD

Number of points 35 87 84 48 74 10 19 24 18

Shorter Communications sets within the same group are combined and curve fitted with eq. (3). Figures 1 and 2 show the characterisitc curves for the adsorption of water vapor on RD silica gel in comparison with the experimental data points for both groups. The parameters which give the best curve fit are listed in Table 2, where also the root mean square values of the fit are given. The scatter of the experimental data about the curve fits is about 0.0071 kg/kg for the group with high E 0.2 and about 0.011 kg/kg for the group with low E,., . The maximum water uptake is, respectively, 0.348 and 0.354 kg/kg for both groups. Table 2 also lists the F-statistics for each of the data sets, curve-fitted with the parameters of the first (1) and second (2) combined sets. For all data sets except those of Jury and Edwards (1971) and Kester and Lunde (1973), the F values indicate that the parameters of the combined set belong to


the 95% or better confidence region. The parameters of the combined set not containing the data set fall outside the confidence region. For the data sets of Jury et al. and Kester et al., the F, values are still significantly higher than the F, values. The sorption isotherm with low E,,,, illustrated in Fig. 3, shows the curvature of the isotherm at the low and high vapor pressures. These curvatures determine the speed with which the transfer zones propagate through a desiccant matrix, and are therefore important for predicting the performance of regenerative driers. The isotherm expression with low E,., was used in a model for analyzing experimental, dynamic sorption data of water vapor on RD silica gel (Van den Buick, 1987). It is shown that this isotherm gives a close agreement between selected experimental and predicted results. As an example, an experimental break-

Fig. 1. Characteristic

curve of water on RD silica gel in comparison E 0,2’

with selected experimental

data, low

Fig. 2. Characteristic

curve of water on RD silica gel in comparison E 0.2.

with selected experimental

data, high


Fig. 3. Sorption



of water on RD silica gel in comparison E 0.2’

through curve is shown in Fig. 4. The mixed constantpattern-xpansion character of this wave can only be explained by the peculiar curvature of the isotherm.

with selected


Acknowledgements-The author wishes to thank the Solar Energy Laboratory of the University of Wisconsin-Madison for the financial support for this work. The author also acknowledges Dr A. Fraioli of Argonne .National Laboratory for the release of experimental data.

CONCLUSION The isotherm expression given by eq. (3) with the values listed in Table 2 is an accurate correlation of various independent sets of experimental data for the sorption of water vapor on RD silica gel. The expression is straightforward to use for engineering modelling. Also the heat of adsorption can be readily derived from eq. (3).

data, high




Mechanical Engineering Department Katholieke Universiteit te Leuven Celestijnenlaan 300A B-3030 Heverlee, Belgium


P P0 R T W *o

adsorption potential characteristic energy of adsorption defined for (p, n - p) degrees of freedom as S--S^n-p where S is the error sum of squares of -T-T’ the c&e fit, and s^ is the minimum values of S (Draper and Smith, 1966). water vapor pressure saturation pressure of water vapor universal gas constant absolute temperature water uptake by silica gel, kg water/kg dry desiccant microscopic volume REFERENCES



Fig. 4. Experimental breakthrough curve of desorptiod of water from RD silica gel. Initial conditions (31”C, W = 0.333 kg/kg). Inlet conditions (63”C, W = 0.076 kg/kg).

Ahlberg, J. E., 1939, Rates of water vapor adsorption from air by silica gel. Ind. Engng Chem. 31, 988-992. Draper, N. R. and Smith H., 1966. Applied Regression Analysis, pp. 272-274. John Wiley, New York. Dubinin, M. M., 1972, Fundamentals of the theory of physical adsorption of gases and vapours in micropores, in Adsorption-Desorption Phenomena (Edited by F. Ricca), pp. 3-17. Academic Press, New York. Dubinin, M. M., 1975, Physical adsorption of gases and vapors in micropores. Prog. SurJ Membrane Sci. 9, t-70. Fraioli, A. V., 1984, Rates of Adsorption of Water on Manganese Oxides and Silica Gel. Report ANL-84-98, Argonne National Laboratory, Argonne, IL. Hubard, S. S, 1954, Equilibrium data for silica gel and water vapor. Ind. Engng Chem. 46, 356-358.

Shorter Communications Table

2. Summary

of the parameters


of the Dubinin-Astakhov

sorption RMS




Taylor Jury et al. Kester et al. Pendelton et al. Fraoli Total Hubard Rosas Pendelton



19 74 10 7 18

7310 8550 9880 9750 8670

2980 3120 3570



84 19 7 110

et al.





(4, 15) (4, 70) (44,6) (2, 5) (4, 14)

3.7 38 58 0.9 0.9

21 333 159






7.3 3.5 1.0 2.5 6.9






(4, 124)


7840 7050 9750

2600 2140

0.109 0.152 0.085

0.246 0.202

0.355 0.354

11 12 2.5

(4, 80) (4, 15) (2, 5)

40 16

0.7 1.2 5.3







(4, 106)




45. No.

Effects of particle


0.335 0.371 0.355

Symp. Ser. 69(134),



0.228 0.264 0.288

Lunde, I’. J. and Kester, F. L., 1975, Chemical and physical gas adsorption in finite multimolecular layers. Chem. Engng Sci. 30, 1497-1505. Pendleton, P. and Zettlemoyer, A. C., 1984, A study of the mechanism of microporous filling. II Pore filling of a microporous silica. J. Colloid Interface Sci. 98, 439-446. Polanyi, M., 1932, Section III. Theories of the adsorption of

Chemical Engineering Science, Printed in Great Britam.



0.117 0.107 0.067 0.085 0.117

Jury, S. H. and Edwards, H. R., 1971, The silica gelwater vapor sorption therm. Can. J. them. Engng 49, 663666. Kester, F. L. and Lunde, P. J., 1973, Chemisorption-B.E.T. model for adsorption of water vapor on silica gel. A.1.Ch.E.


5, pp.



gases. A general Trans. Faraday


and some


Sot. 28, 316333.

Rosas, F., 1980, Pure vapor adsorption of water on silica gels of different porosity. MS thesis, Colorado School of Mines, Boulder, CO. Stoeckli, H. F., 1977, A generalization of the DubininRadushkevich equation for the filling of heterogeneous micropore systems. 1. Colloid Inte@ace Sci. 59, 184-185. Taylor, R. K., 1945, Water adsorption measurements on silica gel. Jnd. Engng Chem. 37, 649-652. Van den Buick, E, 1987, Convective heat and mass transfer in compact regenerative dehumidifiers. PhD thesis, University of Wisconsin-Madison, Madison, WI

ooos2509/90 13.00 + 0.00 C2 1990 Pergamon Press plc


on bubble rise and wake in a two-dimensional fluidized bed

(First received 30 June 1989;




INTRODUCTION Various flow characteristics of a gas-liquid-solid fluidized bed, such as bed expansion/contraction, solids entrainment and solids mixing, largely depend on the local behavior of the solid particles (Fan, 1989). The most crucial mechanism responsible for such solids behavior is probably the interaction of the particles with gas bubbles as well as their wakes. Bubble wake properties have been recently studied by Tsuchiya and Fan (1986) and Kitano and Fan (1988) for relatively large single bubbles rising through a two-dimensional liquid-solid fluidized bed. Tsuchiya and Fan (1986) found through visualization that the bubble wake flow is cyclic and governed by a vortex formation shedding process. They identified the primary wake as an “‘effective” wake region responsible for the solids transport. Kitano and Fan (1988) directly measured the local solids holdup distribution in the near wake using an optical fiber probe, and estimated the mean solids holdup in the primary wake. Both studies.

in revised form

10 October



however, were conducted over rather limited ranges of particle size and density and thus were not conclusive about the effect of particle properties on vortex shedding frequency, primary-wake size and solids holdup in the wake. Although these wake properties are inherently, probably most, sensitive to the liquid flow condition prevailing in the vicinity of the flow separation points, the extent of disturbance in the liquid flow caused by the presence of the particles can be significant if the particle inertia is either very large or very small. When the particle inertia is large, the trajectories of solid particles deviate from those of liquid elements, resulting in extensive impaction of the particles upon the bubble roof (Levich, 1962) and also establishing a solids concentration gradient and a region of high turbulence level in the wake (Tsuchiya and Fan, 1986). For particles of small inertia, the liquid-solid suspension tends to behave like a pseudo-homogeneous medium and the behavior of the bubble and the