kinetics of non-isothermal sorption.pdf

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Kinetics of Non-isothermal Sorptionin Molecular Sieve Crystals

A mathematical model is developed to describe non-isothermal sorptionkinetics and used to interpret experimental uptake curves for COB n 4Azeolite and for C02 and nC5H12 n 5 A zeolite. The model assumes that thedominant mass transfer resistance is intracrystalline diffusion while the majorresistance to heat transfer is the external heat transfer between the adsorbentsample and the surroundingj. I t gives a good representation of the kineticbehavior over a wide range of conditions. Both the extreme cases of iso-thermal diffusion and complete heat transfer control are demonstrated ex-perimentally. Intermediate situations, in which the uptake rate is controlledby the combined effects of diffusion and heat transfer, are demonstrated aswell. The parameters derived from the model are consistent, reproducible,and agree well with a prinri estimates. The model provides a useful theoreti-cal basis for the analysis of rapid sorption processes for which the non-iso-thermal approximation is invalid.

DOUGLAS M. RUTHVENLAP-KEUNG LEEand

HAYRETTIN YUCELDepartment of Chemical Engineering

University of New BrunswiekFredericton, N.B., CanadaE3B 5 A 3

The traditional method of measuring diffusivities inporous adsorbents is to measure the transient sorptioncurve generated when a sample of the adsorbent is ex-posed to a change in ambient sorbate concentration. Suchcurves are generally analyzed according to a simple iso-thermal diffusion model. This is a valid approximationwhen diffusion is slow compared with heat transfer, butfor rapidly diffusing systems, the sorption kinetics maybe appreciably influenced by thermal effects,Presented here is a non-isothermal model for an as-semblage of spherical adsorbent particles. The principalmass transfer resistance is intraparticle diffusion whilethe major resistance to heat transfer is the external heattransfer between the adsorbent and the ambient fluid.The model is restricted to small differential changes in

sorbate concentration, since a linear equilibrium relation-ship and constant diffusivity are assumed. These as-sumptions are physically reasonable for many systems.

The model is used to analyze experimental sorptioncurves for COz in 4A and 5A zeolites and for nC5HI2 n5A. The conditions of the experimental study were se-lected to illustrate the main features of the theory. Boththe extreme cases of isothermal diffusion and completeheat transfer control, as well as intermediate cases in-volving combined heat transfer and diffusion control, aredemonstrated. The model provides both a useful criterionfor the validity of the isothermal approximation and atheoretical basis for the analysis of uptake curves forrapidly diffusing systems for which isothermal conditionscannot be achieved.

CONCLUSIONS A N D SIGNIFICANCEThe model shows that the kinetic behavior of the

system is determined by two dimensionless parameters,a and 8. The parameter OL is the ratio of the time constantsfor diffusion and heat transfer, while /3 depends on theheat of sorption, the thermal capacity, and the tempera-ture coefficient of the equilibrium position. By comparingtheoretical uptake curves calculated from the non-isother-ma1 model with the curves for isothermal diffusion, a cri-terion for the validity of the isothermal approximation isestablished.

The model provides a consistent interpretation of ex-perimental kinetic data for a wide range of conditions.In particular, when uptake curves for COz in small (7.3pm) 4A crystals are analyzed according to the non-iso-thermal model, the time constants yield diffusivities con-sistent with the values obtained with larger crystals (in

Current address for H Yucel is Dep-rtment of Chemical Engineering,Middle Eastern Technical University, Ankara, Turkey.0001-1541-80-3135-0016- 01.05. he American Institute of Chemi-cal Engineers, 1980.

which diffusion is practically isothermal). Isothermal anal-ysis leads to apparent diffusivities which are smaller forthe smaller crystals. Apparent diffusivities which vary withcrystal size have been reported in the literature for severalsystems, and the intrusion of thermal effects is a possibleexplanation.The effect of sample configuration on the uptake ratehas been demonstrated experimentally. For certain sys-tems, it is possible to pass from the region of near iso-thermal diffusion to complete heat transfer control, simplyby increasing the depth of the adsorbent sample. Con-stancy of the uptake curves measured with different sam-ple configurations provides a useful experimental test forthe absence of significant thermal effects. But, agreementbetween differentially measured uptake curves determinedover different step sizes and consistency between adsorptionand desorption curves is not a valid criterion of isothermalbehavior. The intrusion of thermal effects is not alwaysevident simply from the form of the uptake curves. Itseems possible that such effects may be responsible forsome of the anomalies in the published literature.

AlChE Journal (Vol. 26, No. 1)age 16 January, 1980


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The classical method of studying sorption kinetics in-volves following the transient sorption curve generatedwhen a sample of the adsorbent is subjected to a changein ambient sorbate concentration. The uptake curves havegenerally been interpreted in terms of a diffusion modelon the assumption that the system can be considered asisothermal. See, for example, Barrer (1971) , Loughlin,Derrah and Ruthven (1971) . This is a valid approxima-tion when diffusion is slow but, in faster diffusing SYS-tems, the sorption kinetics may be appreciably influencedby thermal effects, The significance of thermal effects hasbeen demonstrated experimentally by Eagan, Kind1 andAnderson (1971), Kondis and Dranoff (19 71) and byDoelle and Riekert (1 97 7) . Temperature rises as high as20-30C were observed during the sorption of propanein zeolite SA and butane in 13X. However, with the ex-ception of the work of Armstrong and Stannet (1966) ,which appears to have been overlooked in most subse-quent studies, there has been little theoretical analysis.The problem of non-isothermal sorption has recentlyattracted the attention of several different research groups.An approximate analysis based on a linear driving forcerepresentation of the mass transfer rate equation was de-veloped by Chihara et al. (1976) while a solution in termsof the moments of the uptake curves was obtained byKocirik, Zikanova and Karger (197 9) . A more detailedtheoretical analysis was presented by Brunovska et al.( 1978) who solved numerically the simultaneous diffusionand heat conduction equations for a porous sphericalparticle of finite thermal conductivity subjected to a stepchange in sorbate concentration at the external surface.The diffusivity was assumed independent of temperature,external heat transfer resistance was neglected, and theisotherm was considered to be of the Langmuir form.

There are, in principle, two distinct effects which shouldbe considered in the analysis of non-isothermal sorption:1) the temperature dependence of the diffusivity and 2)the temperature dependence of the equilibrium. Forzeolitic sorbents, the temperature dependence of diffusiv-ity generally follows an Arrhenius expression with theactivation energy commonly several kilocalories. The as-sumption of a constant diffusivity is therefore a valid ap-proximation only when temperature changes are small.The effect of the temperature dependence of diffusivity(the dominant effect in the initial stages of the up take) isto increase the rate of adsorption and decrease the rateof desorption, relative to an isothermal process. In laterstages, the effect of the temperature dependence of theequilibrium becomes dominant. This effect retards bothadsorption and desorption rates to similar extents. Non-isothermal adsorption curves, therefore, generally showan initial rapid uptake followed by a slow approach toequilibrium in the long time region.

A theoretical analysis which takes account of the tem-perature dependence of both the diffusivity and theequilibrium has been developed by Lee (1 976) . By re-ducing the size of the step over which the uptake curveis measured, the effect of the temperature dependenceof the diffusivity may be eliminated, but the effect of thetemperature dependence of the equilibrium position re-mains. The problem of theoretical analysis is greatlysimplified by restricting consideration to small differentialchanges in sorbate concentration. Under these conditions,the equilibrium relationship can be considered linear andthe diffusivity constant.

There are, in principle, three distinct resistances toheat transfer which should be considered 1 thermalconduction through the zeolite crystals, 2 ) heat transferAlChE Journal (Vol. 26, No. 1)

or conduction between the individual particles (or zeo-lite crystals) within the adsorbent sample (or pellet)and 3) heat transfer between the sample and the sur-rounding atmosphere. A simple analysis of the time con-stants for these processes shows that for small crystals( - 10 r m ) and for commonly used sample quantities(10-50 mg) the external heat transfer resistance will al-ways be much more important than either intraparticleconduction or inter-particle heat transfer. Under mostpractical conditions it is therefore a good approximationto consider the sample temperature to be uniform andto take account only of the external heat transfer resist-ance.In our work, a mathematical model based on theabove simplifications is developed and then used tointerpret experimental uptake curves for several differ-ent systems under non-isothermal conditions.THEORETICAL ANALYSIS

The assumptions of the mathematical model, whichis similar to the model of Armstrong and Stannett butin spherical geometry, are as follows:1 ) The sample consists of an assemblage of uniformspherical particles (zeolite crystals).

2 ) Thermal conduction through an individual particleand heat transfer between the particles within the sam-ple is rapid, so that the only significant heat transferresistance is at the external surface of the sample. Thisimplies a constant temperature through the sample, al-though, as a result of the heat of adsorption, there willbe a time dependent difference in temperature betweenthe adsorbent and the surroundings. External heat trans-fer is assumed to be governed by Newton's Law.

3) Intraparticle diffusion is the only significant re-sistance to mass transfer. The sorbate concentration atthe surface of each particle in the sample is thereforealways in equilibrium, at the temperature of the sample,with the sorbate concentration in the ambient fluid.4) The diffusivity is assumed constant and the equi-librium relationships are linear, (i.e ., dc /dT constant).

With the appropriate definition of the particle radius,this model is applicable to either a sample of individualadsorbent crystals, under conditions of intracrystallinediffusion control, or to a macroporous adsorbent pelletunder conditions of intrapellet (macropore) diffusioncontrol. Another practical and important case, that ofan assemblage of crystals in which the mass transfer rateis controlled by diffusion into the sample bed, will bedealt with in a subsequent paper.Subject to these approximations, the system may be

described by the following set of differential equations:

dQ pC, dT ha ( T - To- A H ) -- = -dr c, - o dr (c, - o ) ( D / r Z )(3)

with the initial and boundary conditions:

January, 1980 Page 17


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TABLE .DETAILSF ZEOLITECRYSTALSp 0 2

0 1 2 3 0 1 2 3T= t/r'

Figu re 1. Theore t i ca l up take cu rves ca lcu la ted f rom Eqns . (6) a n d(8) for representat ive va lues of LY a n d p l s o shown i s t h e l i m i t i n gcurve for isothermal d i f fusion (Eqn. 9 ) ---- , and the cu rve fo r

hea t t rans fer con t ro l (Eqn. 10) for a = 0.3 -*-.-..The solution to this set of equations may be obtained

by Laplace transformation. Inversion by the methodof residues yields the expressions for the dimensionlesssorption curve and the temperature history of the sam-ple in terms of the dimensionless parameters Y E hdpC ,and B = ( AH /C , ) . ( a c u / d T ) :

- 3[ (qn cotqn - ) / q n 2 1 exp(- qn2r)- q n cotqn(qn cotqn - ) / q n 2 + 11P 21 3n = l

( 7 )where qn is given by the roots of:

8 )P ( q n ~ ~ t 9 n1 = qn2- YA family of uptake curves, calculated from Equations( 6 ) and ( 8 ) , for sevcral values of 01 and /3 is shown inFigure 1. The limiting case of isothermal behavior is ob-tained whell either Y x or /3 -+ 0. Under these con-ditions the roots of Equation (8) bccome simply qn =na and Equation ( 6 ) reduces to the familiar form:

9)When diffusion is very rapid, the kinetics are con-trolled entirely by heat transfer. The limiting behavior

may be derived by considcring the asymptotic form ofEquation (6 ) for small values of Y.Replacing (qcotqPage 18 January, 1980

MeandiametercrystalZeolite (pm)

4A 3421.5 }7.25A 34

5A 3.6

Composition

Na form, as syn-thesized

As above after4 5 Ca+exchange4 5 Ca+

originSynthesized byYucel usingCharnell'smethodSynthesized byYucel usingChamell'smethodLinde 5ALot 550045

Note: Size fractions were separated using 'Nitek' microsieves.

- 1 in Equation (8) by the series expansion - 92/3+ 9*/4c1+ . . . ) shows that, for small Y, the first rootof Equation (8) is given by q12 = a/ + 8). Theasymptotic form of t he uptake curves is then obtainedfrom the first term of the summation of Equation (6)which becomes:

This is equivalent to the expression derived by King andCassie ( 1940).

For all values of Y and p the uptake curves reduce,in the long time region, to simple exponential curves.For certain values of an d p, the shapes of the curvesare quite similar to the curve for isothermal diffusion(Equation 9) , so it is not always easy to detect the in-trusion of thermal effects simply from the shape of theuptake curves. In practice, thermal effects may be mini-mized by using the smallest possible quantity of ad-sorbcnt spread over as large an area as possible. Further,since Y varies directly with the external area of thesample, varying the size or configuration of the sam-ple provides a convenient experimental test for thermallimitations.

It has been previously assumed (for example Lough-lin, Derrah and Ruthven 1971) that 1 thermal effectsmay be detec ted by varying the size of t he concentra-tion step over which the uptake curves is measured and2 ) that agreement between the uptake curves measuredover different concentration steps and for adsorptionand desorption provides evidence that the system canbe considercd as isothcrmal. This is correct when thediff usivity is temperature dcpendent . However, if theinterval over which the uptake curve is measured be-comes sufficiently small, the effect of the temperaturedcpcndence of diffusivity vanishes. The uptake curvesthen a re independcnt of s tep size and the same for ad-sorption and desorption, even though there may still bean appreciable thermal aff cct arising from the tempera-ture dependence of the equilibrium position. Under theseconditions, identity of the sorption curves for differentstcp si7es docs not provide unequivocal evidence of iso-thermal behavior.

Comparing the time constants from Equations (6 )and (9 ) shows that thc system may be considered iso-

AlChE Journal (Vol. 26, No. 1)


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06

@ = a 6 5D/rrsO.036 S

Pressure Step :95 92 ori0.4

-= I1 \0.080.061

0.04) I I I I I I I I f I0.0 40 120 200 280 360 440T i r n e , s e cFigure 2. Uptake curves for sorpt ion of COz i n 4A z e o l i t e ( 3 4 ~ m )crysta ls at 273K, measu red over the same pressur e steps. 0 , 36m9sample; + 13mg sample. Theoret ical curve is calculated for iso-thermal d i f fus ion w i th D/rz = 1.3 x sec-'.thermal if a/ 1 + 8 ) >> . This condition is unneces-sarily severe, since if the value of 6 is sufficiently small,thermal effects are importan t only in the very last stagesof the uptake curve. By directly comparing the uptakecurves calculated from Equation (6 ) with the isothermalcurves (Equ ation 1 9 ), it may be shown that the errorin the time constant arising from the use of the isother-mal approximation will not exceed 15 over the range0435 uptake (provided that d / 3 > 60). This condi-tion is equivalent to and somewhat less severe than thecriterion developed by Chihara, Suzuki and Kawazoe(1976) from the linear driving force approximation. Forsmall values of p, the criterion may be relaxed somewhatfurther.EXPERIMENTAL

Sorption curves were measured gravimetrically using a Cahnvacuum microbalance system. We followed the change inweight of a small sample of adsorbent when subjected to asmall differential step change in sorbate pressure. Details ofthe zeolite crystals are given in Table 1. In the systems studied,heats of sorption are relatively large and thermal effects are

1o

0.3

\ 0.1

I=2- 0.03

t1 o I I I I I I I

323K. Pr25-20Tori

important. The experimental data illustrate the main featuresof the non-isothermal model.C02-4A

Uptake curves for COz in 34.pm 4A crystals, measured oversimilar pressure steps, with two different depths of adsorbentsample are shown in Figure 2. There is good agreement be-tween the uptake curves showing that, under these conditions,the system is essentially isothermal and the macro-diffusionalresistance of the sample is negligible. F o r a more accuratecomparison with the experimental data, we calculated thetheoretical curve for isothermal diffusion for an assemblage of

TABLE.PARAMETERSBTAINEDROM NON-ISOTHERMALNALYSISOF SORPTIONURVESFOR C02 IN 4A ZEOLITE (FIGURE)

Non-isothermal analysisCrystalradius D T ~ D h d f P(elm) (see-1) (cm2 * sec-1) sec-1) aK )

371-80 Torr)2.9 x 10-3 8.4 x 10-9 288.5 x 10-3 9.8x 10-9 } 0.08 0.5 9.4E.5

{ E;1.2.067 8.9 x 10-9

5.2 x 10-95.18x 10-90.036 4.8 x 10-9AlChE Journal (Vol. 26, No. 1)

7.3410.073 0.65 16.22.0

January, 1980

1.8 x 10-34.5x 10-3-22323orr )

IsothermalanalysisD / r z(see-1)

2.7x 10-3

1.7 x 10-34.4x 10-3

Page 19

8 x 10-30.015

0.012


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TABLE. ANALYSISF UPTAKECUHVESOR C02-5A UNDER CONDITIONSF HEATTRANSFERONTROLEQN.10)

PI - Z(Torr) Cmol/cavity )

263-232150-140102-9574-6944-39

213-205147-13188-8350-4530-27

234-204119-10843-3920-174.3-3.6

43-3721-1812-84.6-4.2

114-9863-5527-238.5-8.24.4-3.7

4.053.262.772.401.91

8.07.436.745.724.87

10.19.578.577.565.22

9.08.57.526.66

8.27.86.95.54.2

From equilib.PCP From uptake curves( c a l g-1 W p C p* deg-1) ( s - 1 )

(34pm) T hic kbe d 34mg 371K~~~

0.38 0.0520.35 0.0520.30 0.0490.25 0.0460.18 0.018

Av. c).U49(34pm) T hic kbe d 37mg 323K

0.24 0.0340.29 0.0370.33 0.0360.36 0.0340.4 0.035Av. 0.036

(34pm) T hic kbe d 34mg 273K0.240.260.330.510.76

0.0310.0310.0310.0320.03Av. 0 032

( 3 4 ~ m ) T h i n b e d 12.5mg 273K0.34 0.0530.48 0.0620.65 0.0610.058.75

v. 0.06Linde 5A (3.6fim) Thick bed 38mg 273K

0.260.290.420.620.76

cubic particles with the size distribution of this sample (seeRuthven and Loughlin 1971), rather than from Equation ( 9 ) .This curve fits the experimental curve well.Further evidence that, for this system, sorption rates arecontrolled by intra-crystalline diffurion, comes from a compari-son of the uptake curves for different crystal size fractions(Figure 3) . The sorption curves for the 34pm and 21.5~mcrystals can be well represented by the theoretical curves forisothermal diffusion, The time constants ( D / r z ) are in theratio of the square of the crystal radii, so that essentially thesame diffusivities are obtained for both crystal sizes. Sorptionrates in the 7.3pin sample are higher than in the large crystals,but not as high as would be expected from the difference incrystal size. This apparent anomaly can be explained by theintrusion of heat transfer resistance, The uptake curves werematched to the non-isothermal model (Equations 6 and 8) ,using values of p calculated from the equilibrium isothermsand values of ( h a / & , ) determined from measurements forsimilar systems under conditions of complete heat transfercontrol (Equation 10). This yields values of D/r2 , and hencevalues of D , which are consistent with the near isothermaldata for the larger crystals (s ee Tab le 2 ) .

Page 20 January, 1980

0.960.920.820.640.49

0.921.041.221.31.38

0.670.791.131.562.33

0.470.670.791.08

0.022 0.740.022 0.890.020 1.20.020 2.00.023 2.3-Av. 0.022

CP( c a l - g-1* deg-1)

0.400.380.370.390.37-AV. 0.3130.260.280.270.280.29

Av. 0.28

0.360.330.300.330.33Av. 0.32

0.720.720.830.71Av. 0 8

0.350.330.340.310.33Av. 0.34-

If non-isothermal effects are ignored, and the uptake curvesfor the 7.3.um crystals are matched to the isothermal model,erroneously low values of D/r2 are obtained. This results indiffusivities which seem to decrease with decreasing crystalsize, Diffusivities which vary in this way with crystals sizehave been reported by Karger and Car0 (1976), Gupta, Maand Sand (1971), Satterfield and Frabetti (1967), and Brandtand Rudloff (1964), and the intrusion of thermal effects offersa possible explanation.C02-5ADiffusion in the 5A zeolite is much faster than in 4A be-cause of the larger effective window aperture. Heat transferlimitations become more significant, even in the larger crystals.Sorption curves were measured over a wide range of condi-tions in the Linde 5A crystals (3.6pm) and in the Ca+ ex-changed ( 5 A ) form of the 3 4 ~ m rystals ( - 65 C a + ).Two different sample bed configurations were used: a thinbed of about 10-12 mg spread as thinly as possible over theentire area of the sample pan, and a thick bed sample ofabout 35 mg on a smaller pan.For the thick bed, sorption rates were essentially controlledby heat transfer under all conditions. The sorption curves,

AlChE Journal (Vol. 26, No. 1)


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TABLE. COMPARISONF PARAMETERSALCULATEDROM THERMALLYONTROLLEDUPTAKE URVESDATA F TABLE )Heat capacities

EstimatedWt of sample Wt of pan CP Experimental

Sample T K ) (mg) (mg) (cal - g-1 deg-1) CPThick bed 34 20 0.320.38Thin bed 273 12 35 0.8 0.8

Note: Heat capacity of sieve cal.gm-l.deg-1) is 0.19 at 273K and 0.22 at 371K. Heat capacity of pan is 0.21 at 273K and 0.225 at 371K.

Heat transfer coefficientsTotalexternal

Theoretical estimates Experimental area c m 2 1= d P )a/P x wtofh a / p C ~ (cm2 g-1 samplew x 104 la x 104 h x 104Sample T ( K ) (cal cm-1 sec-1 deg-1) s-1)Thick bed 273 1.54 0.9Thin bed 273 3.22 0.9

371 2.54 2.25

0.031 t I I0 20 40 60 80 1 0 0 120 14t (sec)

Figure 4. Experimental sorpt ion curves fo r C 02 i n 5A zeol i te (34pmcrystals) at 273K, under condi t ions of complete heat t ransfer control :(a) 34 mg sample (thick bed); up take curve for Linde 5A crystalsover the pressure step 4.4-3.7 T or r i s also indicated for comparison;(b) comparison of experimzntal curve for pressure step 20-17 T or rf rom (a) wi th theoret ical curve for isothermal di f fus ion; (c) 12.5 m gsample (thin bed) (Numbers on curves denote pressure steps in Torr).

1.4.44 0.0324.79 0.0493.12 0.06 153 1.9

measured over similar pressure steps with similar adsorbentsample quantities (34-4Omg),for the small Linde 5A crystalsand the large Charnel1 crystals, showed very little difference,despite the order of magnitude difference in crystal size. Thisis convincing evidence of the absence of any significant intra-crystalline diffusional resistance, under the experimental con-ditions. Representative sorption curves are shown in Figure4 and comparative data are given in Tables 3 and 4.The uptake curves (both adsorption and desorption) showa rapid initial uptake followed by a slow exponential approachto equilibrium, in accordance with Equation 10. Because ofthe larger external area/volume ratio for the thin bed sample,the heat transfer rate is faster, And, the uptake is controlledby the combined effects of both heat transfer and diffusion,rather than by heat transfer alone (as for the thick bed). De-pending on the precise conditions, one or other of these effectsmay be dominant. This is illustrated in Figure 5.

OL a) ( b ) k (C)T=17JK, P.97-W T - j t J K , P-234-107 , T-l93K, P.63-31

\ d= in4

\?\ 8 ; 015\

\ \.\ \ \

\\ \

20 40 60 0 20 40 60 0 20 40 60t ( sec )

Figure 5. Sorption curves for COz i n the thin bed sample (-12 mg)o f 5A zeol i te (34rm crystals): ( a) and (b) show curves in which theint rus ion of heat t ransfer res is tance causes s igni f icant deviat ion f romthe isothermal di f fus ion curve; a t 193K di f fur ion i s slower and theuptake curve shown in (c) i s practically isothermal; theoretical non-isothermal curves calculated from Eqns. 6 and 8 with the indicatedparameters, - , sothermal curves for same values of P/ r2-._. , asymptotic curves for heat transfer control, -----. A t193K the isothermal and non-isothermal curves are nearly coinc ident .Paints are experimental.AlChE Journal Vol. 26, No. 1) January, 1980 Page 21


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HI2- 5 A 523K P:27- 21lor

0.03 -I I I0 10 20 30 40 50 60

t ( s e c )Figure 6. Uptake curves for n-pentane-Linde 5A a t 250C, measuredover pressure step -27-21 Torr, with differ ent sample bed depths.Theoret ical non-isothermal curves are calculated f romEqns. 6 ond 8 with the parameters given in Table 5. The isothermalcurve -----I i s from Eqn. 9 with the same value of D / rz = 0.018

sec-1).

In the long-time region, all curves represented by Equation( 6 ) or ( 9 ) approach an exponential decay. 'I'heretore, for cer-tain values of the parameters ha/pC, and p the curves for com-plete heat transfer control (Equation 10) may appear similarto the curves for isothermal ditfusion, except in the initial re-gion, as illustrated in Figure 4b. For rapidly diffusing systems,the initial uptake rate is difficult to measure with accuracy, soit may be difficult in practice to distinguish unequivocally be-tween heat transfer and diffusion control simply from the formof an individual uptak e curve.However, such a distinction is relatively easy when a seriesof curves measured over dif fere nt pressure steps is available.The curves are plotted in semilogarithmic coordinates andvalues of th e appa ren t parame ters p and ha/pC, are calculatedaccording to Equation 10 from the slopes and intercepts. Valuesof pC, A H a c / d T ) are calculated from the equilibriumdata and values of the effective heat capacity C, may the n befound from the kinetic data for each run. If the uptake curvesare indeed thermally controlled, the values of both heat ca-pacity and heat transfer coefficient should be independent ofpressure. Confirmation may be obtained by comparing thevalues of heat transfer coefficients and heat capacities derivedin this way with the values estimated from physical data. Theresults of such calculations are summ arized in Ta ble 3. Onlyrepresentative data are given; the mean values are averages ofabout 20 runs at each temperature.Th e parameters (C, a nd ha / pCp ) derived from the analysisof the uptake curves are compared in Table 4 with values es-timated a priori. The total effec tive heat capacity is estimatedfrom C, = heat capacity of sieve + ( W t of pan/W t of sieve)x heat capacity of pan. The values estimated in this way areclose to t he experimental values. The overall h eat transfer coeffi-cient h is estimated as the sum of the contributions from radia-tion (Stefan's Law with emissivity 0.8: h = 4.4 x 10-12 T3)and from conduction in a stagnant fluid (h u = 2 or h =k / R ) , where k is the thermal conductivity of the gas ( inde-pendent of pressure) and R is the equivalent radius of theadsorbent sample, estimated from sample wt/density = (4/3)d 3 . Using the estimated heat transfe r coefficient, together

TABLE . PARAMETERSOR NON-ISOTHERMALPTAKE-27-21 Tom SEE FIGURE)

Sample ha/&, D / P

CURVES FOR n-PKNTANE-5A AT 523K, PRESSURE STEP

w t ( m g) U ( s - 1 ) ( s - 1 )_-0.018

13 0.23 7.6 0.13621 0.25 5.1 0.09243 0.29 3.4 0.061

with the experimental values of C , and ha/&,, gives a valuefor the total external surface area of the sample. This value iscomparable with the geometric area.It is evident that t he mod el of com plete heat transfer con-trol fits the kinetic data for the thick bed sample over theentire range of conditions and the parameters derived fromthis model are similar for the two dlfferent crystal sizes andin good agreement with estimated values. When data for thethin bed sample are analyzed in this way, the values obtainedfor C a nd ha/ , show regular trends with pressure. Detailedanalysis shows that except for a limited range of pressure at0 C ( 5 - 5 0 Torr) the diffusional resistance was significant. So,the uptake rate in the thin bed is controlled by both diffusionand heat transfer. The behavior of the thin bed sample is il-lustrated in Figure 5. At 273K and 323K, the initial portionsof the uptake curves lay close to the isothermal diffusioncurve, but the deviation from isothermal behavior becomespronounced in the long-time region. These non-isothermal dif-fusion curves are well represented by the theoretical model(Equations 6 and 8 ) . At the lowest temperature (193K), dif-fusion is slower and, since the sorbate concentration is ap-proaching the saturation limit, the value of p is small. Underthese conditions the uptake curve is practically isothermal.n-P en t ane -L i nde 5A 3.6p.m crystals) at 2 5 0 C

The sorption curves for this system provide a clear illustra-tion of th e effect of varyin g the dep th of the adsorbent sam-ple. Desorption curves, measured over essentially the samepressure step with three different depths of sample are shownin Figure 6. Up to about 75 completion, the curves for thethree samples are similar and quite close to the isothermalcurve, calculated from Equation 9. However, there are con-siderable deviations in th e later portions of th e curves. As maybe expected, the smallest sample, which has the highest ex-ternal area/volume ratio, lies closest to the isothermal curve.The greatest deviation is shown by the largest sample. Allthree curves are well approximated by the theoretical non-isothermal curves, calculated according to Equations 6 a nd8, with the parameters given in Table 5.The value of D / r 2 derived from the non-isothermal model(0.018 sec-1) agrees well with the value obtained by iso-thermal analysis of the initial region of the curves. The valuesof p are consistent with the values estimated from equilibriumdata, using experimental values of the effective heat capacity.Values of ha/&, agree well with those obtained at the sametemperature for faster diffusing systems, under conditions ofcomplete hea t transfer control.C O N C L U S I O N

The non- i so the r m a l m ode l p r ov ide s a consis ten t in-te rpre ta t ion of t he up t a ke be ha v io r ove r a wide r a n g e ofconditions. The effects of va r y ing t he a dso r be n t c r y s t a lsize , the configuration of t h e a d s o r b e n t s a m p l e a n d theso r ba t e c onc e n t r a t i on ( w h ic h a f f e c t s t he pa r a m e te rth r ough the t e r m & /aT) a r e c o r r e ct l y p r e d i c t e d . Thepa r a m e te r s de r ive d f r om the m ode l agree well witha priori estimates.Most of the sorp t ion k ine t ic data r e po r t e d in thel i t e r a tu r e ha ve be e n i n t e r p r e t e d a s sum ing i so the r m a lbchnvior. W he n d i f fus ion i s r ap id or heat t r ansfe r s low,s igni f icant depar tures f rom iso the rmal behavior may oc-

AlChE Journal (Vol. 26, No. 1)age 22 Jonuary, 1980


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cur. The intrusion of heat transfer resistance is not al-ways obvious simply from the shape of the uptake curve,since for certain values of th e parame ters Y and p, theform of the non-isothermal curve resembles closely theform of the curve for isothermal diffusion. Analysis ofsuch an uptake curve according to the isothermal as-sumption will lead to an erroneous value for the d 8 u -sional time constants.For systems such as hydrocarbons in 13X zeolite orin mordenite, diffusion is rapid and heats of sorption arerelatively large. Under these conditions, the uptake be-havior is likely to be significantly modified by heat ef-fects. The uptake curves for such systems show the formtypical of thermal limitation, a rapid initial uptake fol-lowed by a slow final approach to equilibrium. See,for example, Satterfield and Frabetti (1967) and Satter-field and Katzer (19 71 ). I t is probable th at some anoma-lies in the published diffusivity data may arise fromfailure to take adequate account of heat transfer limita-tions. A similar conclusion was reached by Zikanovaand Kocirik (1979), in their recent study of cyclohexanesorption in 13X zeolite compacts.The non-isothermal model provides a convenient basisfor the analysis of non-isothermal uptake curves. The

mode1 makes it possible to d eterm ine a reliable criterionfor validity of the isothermal approxima tion. Th erm aleffects may be minimized in practice by using the small-est possible quantity of adsorbent sample spread overas wide an area as possible, and by maximizing the ef-fective thermal capacity, possibly by adding inert mate-rial to the sample. Varying the configuration and parti-cle size of the adsorbent provides a convenient directexperimental test for the significance of heat transferresistance. H owe ver, even if hea t trans fer resistancecannot be entirely eliminated, it is possible to derive re-liable diffusional time constants from the u ptak e curve susing the non-isothermal model, provided that diffusionis not so fas t that t he up take is entirely controlled b yheat transfer (Eq uatio n 10 .ACKNOWLEDGMENT

The uptake curves for n-pentane reported in figure 6 weremeasured by Andreas Vavlitis in our laboratory.NOTATIONuc 7, ) = adsorbed phase concentrationc , ( T ) = adsorbed phase concentration at crystal surface

= external surface area per unit volume of ad-sorbent sampleinitial (stLady sta te) a dsor bed phask concentra-tionfinal steady state adsorbed phase concentrationequilibrium adsorb ed phase concentrationeflective heat capacity of adsorbent sample (in-cluding containing pan )iniracrystalline diffusivityexternaI he at transfer coefficientroots of Equation (8)= dimensionless adsorbed phase concentration

c - co) / c , - o)-Q ( T ) = average value of Q def ined by Equat ion (2)Qs 7 = dimensionless adsorbed phase concentration atQ = equilibrium value of Q (for specilied ambientrAlChE Journal (Vol. 26, No. 1)

3 ractional uptake m t / m l )crystal surface = (C, - o)/ (c, - g)concentration)= equivalent radius of zeolite crystal

2 = temperatureToGreek Lettersq = dimensionless radial coordinater = timensionless time D t /p-AH = heat of adsorptiona = dimensionless parameter ha / pC ,f iLITERATURE CITEDArmstrong, A. A. and V. Stannett, Tem perature Effects dur-ing Sorption of Water Vapour in High Polymers, Diehlakromolekulare Chemie, 90, 145-160 (1966) and 95, 78-91 ( 1966 ) .Barrer, R. M., Intercrystalbne Diffusion, Ado. Chem., 102,1 (1971) .Brandt, W. W . a nd W. J . Rudloff, Diffusion of Ethane inModified Synthetic Zeolites, J. Phys. Chem. Solids, 25, 167( 1964) .Brunovska, A., V. Hlavacek, J. Ilavsky and J. Valtyni, Analy-sis of Non-Isothermal One-Component Sorption in a SingleAdsorbent Particle, Chem. Eng. Sci., 33, 1395 (1978) .Chihara, K., M. Suzuki and K . Kawazoe, Effect of Heat Cen-eration on Measurement o f Adsorption Rate by GravimetricMethod, Chem . Eng. Sci., 31, 503 1976) .Doelle, H. J . and L. Riekert, Kinetics of Soiption and Diffu-sion of n-Butane in Zeolite NaX, Am. Chem. Sac. Symp.Ser., 40, 401 ( 1977) .Eagan, J. D., B. Kind1 and R. B. Anderson, Kinetics of Ad-sorption on A Zeolites: Temperature Effects, Ado. Chem.,102,164 ( 1971) .Gupta, J. C., Y. H. Ma and L. B. Sand, Diffusion of SO2 inSynthetic Mordenjte and Natural Erionite, AlChE Symp.Ser., 67 (117) , 51 ( 1971 ) .Karger, J . and J. Caro, Influence of a Surface Barrier onSorption in 5 A Zeolite, 2. Chemie, 16, 331 ( 1976 ) .Karger, j., . Caro and 91.Bulow, Relation of Kinetics of AI-

kane Sorption by NaCaA Zeolite to Size of Crystals, IZOAkad. Nauk S S S R Ser. Khim. (12), 2666 (1977) .King, G. and A. B. D. Cassie, Propagation of TemperatureChanges through Textiles in Humid Atmospheres: Part IRate of Adsorption of Water Vapour by Textile Fibres,Trans.Faraday Sac., 36 , 445 ( 1940 ) .Kocirik, M., A. Zikanova and J. Karger, Influence of Ad-sorption Heat Generation on Uptake Behavior in BidisperseAdsorbents, I Chem. Tech. Biotech., in press,Kondis, E. F. and J. S. Dranoff, Non-Isothermal Sorption ofEthane by 44 Molecular Sieves, AlChE Symp. Ser., 67(117),25 (1971) .Lee, Lap-Keung, Ph.D. thesis, Univ. of New Brunswick, Fred-ricton, Canada 19 76 ).Loughlin, K. F., R. I. Derrah and D. 51. Ruthven, On theMeasurement of Zeolitic Diffusion Coefficients, Can. JChem. Eng., 49, 66 19711 .Ruthven, D. hl. and K. F. Loughlin, Effects of CrystalShape and Size Distribution on Diffusion Veasurements inMolecular Sieves, Chem. Eng. Sci., 26, 577 1971 ).Satterfield, C. N. and A. J. Frahetti, Sorption and Diffusionof Gaseous Hydrocarbons in Syn thetic Mordenite, AIChE J,,13, 731 196 7) .Satterfield, C. N. a nd J. R. Katzer, Counter-Diffusion of LiquidHydrocarbons in Type Y Zeolites, Ads. Chem., 102, 193( 1 9 7 1 ) .Zikanova, A. and 11. Kocirik, Diffusion of Cs Hydrocarbonsin NaX and Mg.4 Zeolites, Conference on Properties andApplications of Zeolites, London, England (April 18-20,1979) .

= initial (a nd final) s teady state temperature

= effective density of adsorbe nt samp le= dimensionless para meter A H / C , ) ( a c* / aT ) ,

Mnnrrscript receiced arch 19, 1979; retiision receiced JuZy 10 andaccepted July 27, 1 9 7 9 .Jonuary, 1980 Page 23

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