Rapid Procedures for the Prediction of Fixed-Bed Adsorber Behavior

Ind. Eng. Chem. Process Des. Dev. 1980, 19, 129-137

129

Rapid Procedures for the Prediction of Fixed-Bed Adsorber Behavior. 1. Isothermal Sorption of Single Gases with Arbitrary Isotherms and Transport Modes: Principles and Recommended Methods Dlran Basmadjian Department of Chemical Engineering and Applied Chemistty, University of Toronto, Toronto, Ontario, Canada, M5S 1A4

Rapid and effective methods are presented for predicting fixed-bed adsorption and desorption of single solutes with arbitrary equilibrium curves and transport modes under both isothermal and adiabatic conditions. The methods do not require numerical solutions, or analytical expressions to be fitted to the solute isotherms. A two-step procedure is adopted consisting of a simple graphical construction of the sorption paths and equilibrium curves (both isothermal and adiabatic), followed by a modified application of the Hiester-Vermeulen solutions for a Langmuir isotherm. In part 1 we describe some key features of the method and test its reliability for isothermal systems with complex isotherms and transport mechanisms. The predictions are compared with experimental column data. In part 2 the graphical construction of adiabatic equilibrium curves is developed and predicted results are compared with adiabatic adsorption and desorption data.

Introduction

The literature on gas adsorber analysis and design has over the years accumulated a bewildering mass of material covering a wide range of mathematical sophistication. Representative of one extreme are the numerous computer solutions of PDE models which continue to appear in the literature. Although some of the simpler isothermal results have been cast in convenient graphical or tabular form (Rosen, 1954; Kyte, 1973; Garg and Ruthven, 1973; 1974; Vermeulen et al., 1974), it is clearly difficult to do this for more complex systems (nonisothermal, multicomponent, etc.). In these cases, it is in principle necessary to perform a full numerical solution of each individual problem. Most present-day design methods avoid the numerical work and the attendant need for independent rate measurements by making direct use of laboratory column data, coupled with simple mass and energy balances (Lukchis, 1973). This approach is useful for single-solute adsorption with favorable isotherms but quickly becomes inadequate for systems involving heat effects, multisolute feeds or complex equilibrium isotherms. There is clearly a need for a compromise method which is simple, yet capable of analyzing the more complex features of adsorber operations. The present work represents a first step in this direction. Our original objective was to simplify the prediction of nonisothermal sorption, particularly of the important thermal regeneration step. We had previously analyzed this case in considerable detail using adiabatic equilibrium theory (Pan and Basmadjian, 1971; Basmadjian et al., 1975). The theory utilizes the “method of characteristics” to reduce the model PDE’s to a set of ODE/algebraic equations, the solutions of which represent, in essence, adiabatic equilibrium curves. A mathematical analysis along similar lines was given subsequently by Rhee and Amundson (1972) for the restricted case of a Langmuir isotherm. We started from the simple premise that the conventional solutions developed for single-solute isothermal sorption should be directly applicable to sorption processes along adiabatic equilibrium curves. This proved indeed to be the case, save for modifications made necessary by 0019-7882/80/1119-0129$01.00/0

the more complex shape of the adiabatic curves. These modifications are applicable to complex isothermal systems as well. We next developed a simple graphical method for the construction of adiabatic equilibrium curves based on the equations generated by equilibrium theory. This avoids time-consuming numerical solutions and disposes of the need to fit analytical expressions to the equilibrium isotherms. The resulting two-step procedure (graphical construction plus a simple application of tabulated isothermal solutions) is rapid and effective. With only little practice, one can generate a number of key adiabatic and isothermal sorption curves in a few hours’ work and obtain a quick grasp of column behavior under different operating conditions. In order to test the various components of the method, we start in part 1 with an examination of the isothermal case, paying particular attention to the effect of isotherm shape and diffusional resistances. Some of this is welltrodden territory but there is a need to consolidate the wealth of accumulated material into a few procedures of general validity. In part 2 we provide the framework for adiabatic sorption and develop the graphical construction of the adiabatic equilibrium curves. A detailed comparison of experimental and predicted results is provided in both parts. T h e o r e t i c a l Considerations 1. E f f e c t o f Equilibrium Curve. In isothermal and

adiabatic sorption processes the effective equilibrium curves often show significant departures from the Langmuir or “constant separation factor” form. In many important cases, such as drying of gases with alumina or silica gel, they may inflect as well, yielding sigmoid or “BET”type equilibrium curves of various forms. A further complication in adiabatic systems is the existence of curves with two intersecting branches. One of these is often of the sigmoid type (Pan and Basmadjian, 1971, Figure 5; Basmadjian et al., 1975, Figure 41, and both adsorption and desorption may take place along either branch. For these more complex cases, it is necessary for our subsequent purposes to identify the “favorable” and “unfavorable” segments of the equilibrium curve, i.e., the interval over C 1979 American Chemical Society

130

Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980 ISOTHERMS & SORPTION PATHS c

0

ADSORPTION BED PROFILES

can be accomplished by the following simple rule: For both adsorption and desorption, the slope of the p a t h leading f r o m feed to initial bed concentration must either remain constant or decrease smoothly. Once the sorption paths are established in this manner, equilibrium bed profiles and breakthrough curves are easily constructed using the mass balance relations of equilibrium theory. Thus for the shock portions of the path, one obtains breakthrough time of shock a t position z

w U

DESORPTION BED PROFILES

bed position of shock a t time t Distance in Bed

Figure 1. Representative sorption paths and bed profiles for Type I-Type V isotherms.

which adsorption profiles are of the self-sharpening or broadening type. These segments do not in general coincide with the concave and convex portions of the curve, and a proper “sorption path” must therefore be established. This is conveniently done by means of equilibrium theory. Tudge (1961) has given a partial equilibrium analysis of sigmoid isotherms in a chromatographic context, and brief allusion to their properties is made in the review of Vermeulen et al. (1974, p 16-27). In Figure 1, we have presented a more generalized summary of the results for both adsorption and desorption along isotherms of Type I-V (for a classification of isotherms, see for example, Young and Crowell (1962)). Isotherm types (solid lines) and typical sorption paths (dashed lines) are shown in the top row, followed by plots of the resulting bed profiles under both equilibrium and actual conditions. These diagrams can also be used to identify favorable/unfavorable segments of adiabatic equilibria. The best known of the isotherms, Type I, yields a shock-discontinuity in adsorption which leads to the familiar self-sharpening (“constant-pattern”) profile under nonequilibrium conditions. For desorption, the profile is of the broadening (“proportionate-pattern”) type. Isotherms of Type I11 show the reverse behavior. Equilibrium curves with inflection points (Types 11, IV, and V), give rise to a variety of profiles. In the most general case, they consist of alternate self-sharpening and broadening sections. These may reduce to fronts of a single type depending on the location of feed and initial bed concentrations. For example, the Type I1 isotherm shown in Figure 1 yields, for adsorption, a self-sharpening front followed by a broadening segment. In desorption, the profile consists entirely of a single self-sharpening front, in spite of the fact that the isotherm contains both convex and concave segments. The sorption paths take three distinct forms (Figure 1): chords between points on the equilibrium curve, tangents to it, and segments of the equilibrium curve itself. The chords and tangents provide the self-sharpening sections, the equilibrium curve the broadening portions of the profile. It can be shown from the appropriate theory that the slope of the sorption path is inversely proportional to the propagation velocity of the associated concentration level. This property provides the key to the rational construction of sorption paths: the slopes (Le., velocities) must be continuous and they must allow for an orderly propagation of concentration levels which excludes physically impossible profiles of the “overhanging” type. This

where Aq/AY represents the slope of the sorption path chords or tangents, and Gb and P b are the carrier mass velocity and bed density. In systems with Type I favorable isotherms, z, is identical with the “Length of Equivalent Equilibrium Section” (LES) used in industrial practice, and the quantity t , is referred to as the “stoichiometric breakthrough time” (Lukchis, 1973a). For broadening segments, A q / A Y is replaced by the isotherm slope dq*/dY yielding the following relations breakthrough time of concentration Y a t position z (3) bed position of concentration q a t time t z =

t Gb Pb(dq*/ d Y ) q

(4)

In the absence of column or rate data, eq 1-4 can be used to calculate minimum bed weight, purge gas requirements, or desorption time. They also serve as an important check for the accuracy of experimental column or equilibrium data. In systems with Type I equilibrium curves, for example, t , given by eq 1 must intersect the experimental breakthrough curve at a location which yields equal areas below and above the intersection. It is surprising to find that this important criterion is badly violated in several meticulous and reputable investigations, which nevertheless appear to give quantitative agreement with complex theoretical models. This paradox is likely attributable to a canceling of errors in measured equilibrium data, flow rates, and feed concentrations. In view of these uncertainties, it seems illogical to make a fetish of “quantitative agreement” with precise models which may merely turn out to be fortuitous. The method proposed here is an approximate one and does not lay claim to precise predictions. The results, however, are within the range of errors mentioned and require a vastly reduced computational effort. 2. Effect of Mass Transfer Resistance. Once the equilibrium curves and paths are established, existing isothermal solutions may in principle be used to derive breakthrough and profile curves. Most of the published solutions deal with systems obeying the Langmuir isotherm, Le., those with constant separation factors. The more rigorous solutions based on Fickian diffusion are preferable, but they do not cover a sufficiently wide range of parameters and are limited to a single mode of transport (Garg and Ruthven, 1973,1974). We have preferred to use

Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980

131

1 . 0 7 - -

k

Adsorption 1 Y 1 Y yo50

I

I

A Y A Y -001

Y L

( l - J Y / A Y c ) = 001

r r

Figure 2. Hiester-Vermeulen plots for fractional concentration breakthrough of 0.01. (Reproduced with the permission of T. Vermeulen.)

Figure 4. Hiester-Vermeulen plots for fractional concentration breakthrough of 0.5, (Reproduced with the permission of T.vermeulen~)

Adsorption i Y I Y - 09

Desorpt on (1-iY i Y )

09

01 I

1-2

i

0 01

11-1Y 1 Y ; ) = 0 7

~

r

r



the approximate solutions of Hiester and Vermeulen (1952, 1963) which are based on the reaction-kinetic model and are available in extensive tabular and graphical form (Hiester and Vermeulen, 1952b; Vermeulen, 1974). The kinetic parameters have been expressed in terms of diffusivities by matching the solutions with those of the Glueckauf rate equation (1955). Several investigators have confirmed the validity of these solutions for systems with constant separation factors (Hiester and Vermeulen, 1952a; Lightfoot et al., 1962, p 177). Part of our purpose in this work is to extend their use to arbitrary equilibrium curves and transport modes and to test their validity particularly for nonisothermal sorption of gases.

(a) Systems with Constant Separation Factors and Transport Coefficients. Graphical solutions for this case are given by Vermeulen e t al. (1974, p 16-34, 16-35) and in more detailed form in the supplement of Hiester and Vermeulen (1952b). Further extensions of these solutions in tabular form have recently become available (Vermeulen, 1979). Several key graphs from the compilation of Hiester and Vermeulen (195213) are reproduced in Figures 2-6. These were chosen for their particular ease of interpreting breakthrough data and can be used, with some modification, for the prediction of bed profiles as well. In essence, they allow a direct reading of breakthrough times for five

132


total desorption time for nonlinear isotherms. f(r) in eq 6 is the principal correction factor for the kinetic treatment and is given to a good approximation by the following equations (Vermeulen et al., 1974, p 16-21). 2 f(r) = -for 0.2 < r 5 1 (8) r + l and

10

f(r) = 2-1

01

i Y i Y -099 t l - i Y i Y 1 099

0 01

1

0

1

1

l

1

1

1

,

20

10 r


key dimensionless concentration ratios: 0.01,0.10, 0.5,0.90, and 0.99. The graphical representations are in terms of three dimensionless groups: the throughput parameter 2 (or 7‘) representing the stoichiometric mass balance (eq l ) ,the number of reaction units NR which is a measure of the diffusional resistance of “sharpness” of the front, and the separation factor r. The latter is used to express the equilibrium isotherm, and is similar in definition to the familiar relative volatility of vapor-liquid systems. We have simplified the more elaborate original expressions (Vermeulen et al., 1974, pp 16-16, 16-25, 16-27) by neglecting secondary correction factors and made appropriate changes to render them suitable for sorption systems. With these modifications, the equations become

154z NR = - - f(r) for pore diffusion R2 v 150s Z NR=--

R2 r = r =

.-P b . - . f(r) for solid diffusion &o

pP

Ay,

(AY/AY,)(l (Aq/Aqo)(l

-

&/&J

AY/AY,)

for adsorption

(6a) (6b) (7a)

( A q / M o ) ( l - sY/AY,) for desorption (7b) (AY/AYo)(l - &/AYo)

where 0 < r < m , the two extremes corresponding to a rectangular or “irreversible” equilibrium curve; r = 1 applies only a t the points of transition between favorable and unfavorable segments (see Figure 1) or for equilibrium curves which are linear over the entire operational concentration range. It does not, in general, apply near the origin, even though the slope there is known to approach a constant value (“Henry’s constant”). It is therefore incorrect to use linear solutions (Rosen, 1954), as is sometimes done, to predict either initial breakthrough or

1 -

4

for r 2 1

For particle or film resistance controlling, the equations result in a maximum error of 10% in computed NRvalues. Larger deviations occur in systems with combined transport resistances, and then only for the highly irreversible case (r < 0.2). In these cases, however, the sorption front becomes fairly steep and the attendant error in estimating breakthrough time or bed length is usually of little practical significance. The quantities AY, and Aq, describe the concentration range over which the sorption paths are favorable or unfavorable and are determined from the constructions shown in Figure 1. For adsorption on a clean bed along a Type I isotherm, for example, LY, is equal to the feed concentration and Aq, is the corresponding equilibrium loading. AY/AY, represents the fractional breakthrough concentration ( = X in Vermeulen et al., 1974) for a particular segment of the sorption path. For use of the Hiester-Vermeulen plots, Figures 2-6, LY/lY, is set in turn equal to 0.01, 0.1, 0.5, 0.9, and 0.99 to obtain the full breakthrough curves. The important transport parameter is the particle diffusivity which can be of the solid-phase or pore diffusion mode (Vermeulen et al., p 16-18). The Hiester-Vermeulen plots yield identical results for either mode, provided the proper diffusivity-rate law combination is used in obtaining transport coefficients from rate measurements. One has, in other words, the choice of using a solid phase Fick’s law to extract solid diffusivities or obtain pore diffusivities using the gas phase version. Care must be exercised, however, in extracting the proper diffusivity value from measured “time constants”. These points are illustrated further in part 2. For systems with several internal diffusion modes, adequate results can be obtained by using a single equivalent pore diffusivity (see subsequent treatment of data Bowen et al. (1963,1973) and Pan (part 2 ) ) . (b) Systems with Variable Separation Factors. We now turn to the problem of developing an approach valid for systems with arbitrary isotherms. Average constant values of r are often used in cases where the separation factor does not change by more than -0.4 unit between feed concentration and origin. Attempts to accommodate a wider range of variations have also appeared in the literature, notably the equilibrium analysis of Tudge (1961), and indications in the works of Hiester and Vermeulen (1954a, 1963, p 16-33) that the Langmuir solution may be applied in piecewise fashion to systems with varying r and N R values. This latter suggestion, adopted here in modified form, has not, to our knowledge, been tested in any meaningful way. To provide a theoretical basis for our approach, it is convenient to revert to the asymptotic equilibrium theory. We had shown in the preceding section that a properly constructed equilibrium sorption path will, for a general isotherm, result in alternating favorable and unfavorable segments. These will make their appearance sequentially as shocks or broadening fronts in bed profiles or break-


through curves (Figure 1). It is clear from this analysis that each segment can be treated as an independent entity, its shape and position entirely determined by the local slopes of the sorption paths. In the presence of a transport resistance one would expect a similar approach to be valid, each segment being subject to eq 5-9 over the appropriate range. It is then usually sufficient to divide the path into favorable/unfavorable regions, and within each segment to compute point values of r and NR at the dimensionless concentration levels of' 0.01-0.99. Hiester and Vermeulen (1954) suggest a somewhat different and lengthier approach. They evaluate "average" values of r and NR a t AY/AYo = 0.03, 0.3, 0.7, and 0.93 within each region and assign these values in turn to the end points of the ratio span AY/AYo = 0.01 - 0.1,O.l - 0.5, etc. Throughput ratio differences A2 for each interval are read from Figures 2 4 and their partial sums EA2 plotted against AY/AYo. The true 2 values are established by setting 2 = 1 a t the stoichiometric midpoint of the curve. That point is obtained by intersecting a rectangle of height AY/AYo = 1 and width = J,'Z d(AY/AYo)with the drawn curve. The net effect of this more elaborate and precise procedure is to weight r and NR values obtained from the entire theoretical breakthrough curve of a particular segment. In practice we have found the results of this approach to be nearly identical with those obtained from the more rapid point-by-point calculations of r and NR proposed here. We show this in part 2 by considering the rather extreme example of a system with r values ranging from 0.3 to 1, NR from 4 to 6.5. This is the most drastic change in parameter values we encountered in either part 1or 2. The more moderate changes found in the remaining systems could therefore be accommodated without loss of accuracy by using our point-wise calculation of r and NR. ( c ) Systems with Several Transport Resistances. While most industrial processes operate under conditions of predominant particle resistance, it is occasionally necessary, because of pressure drop and other limitations, to operate a t lower than average velocities. Both film resistance and axial dispersion become significant under these conditions. A rigorous treatment for combined external/internal isothermal diffusion has so far been provided only for the linear case (Rosen, 1954; Masamune and Smith, 1964, 1965). For nonlinear isotherms, a frequent practice is to resort to the classical linear combination of phase resistances, appropriately corrected for departures from linear equilibrium. A treatment of this type has also been suggested for the reaction-kinetic model. The various correlations for fluid phase transport coefficients given in the review of Hiester et al. (1963) and Vermeulen et al. (1974) have not been significantly changed over the years and it was felt useful to reassess them in the light of more recent data. Film Resistance. Since the publication of the recommended correlation for kp by Wilke and Hougen (1945) several careful reevaluations have appeared in the literature. Of these, the work of Petrovic and Thodos (1968) is perhaps most appropriate since it covers the range of particle Reynolds numbers (3-230) encountered both in laboratory and industrial practice. The work has been further refined in a recent publication of Hsiung and Thodos (1977). In contrast to earlier work, both these studies carefully separate axial diffusion effects from film resistance. Several key correlations have been rearranged for direct use and are reproduced for convenience below. Spherical particle shape is assumed throughout.

133

Wilke and Hougen (1945)

Lightfoot et al. (1962, p 128)

Petrovic and Thodos (1968)

These equations were tested in the interpretation of the breakthrough data of Wicke (1939), which show significant contributions due to both film resistance and axial dispersion. Equations 10 and 11give similar results, but differ from the values predicted by eq 12 by about 35% a t the lowest Reynolds numbers tested (1-3). The combination of the Petrovic-Thodos correlation for kp and axial Peclet number given in the later work of Hsiung and Thodos (1977) appeared to give the best agreement with experiment. Axial Dispersion. The effect of axial dispersion on mass transport can be approximated by an appropriate resistance term l/kda which is added to the film and particle resistance components. Vermeulen et al. (1974, p 16-20) have suggested an expression containing both eddy and molecular diffusion contributions

where Pe = limiting axial Peclet number (2 for gases) and g is a correction factor to account for departure from linear equilibrium. Various recent measurements involving gas flow through packed beds indicate that kda is considerably lower than had previously been thought. Limiting Peclet numbers calculated from column adsorption data have been reported to be as low as 0.1-0.3 (Chao and Hoelscher, 1966; Ozil and Bonnetain, 1977). Hsiung and Thodos (1977), in their reevaluation of transport in inert beds, extracted the following correlation for axial Peclet number containing both the eddy and molecular diffusion contributions Pe = 0.34.Re0." or by analogy to eq 13

0.1 > Re

> 100

(14)

In applying this expression to the sorption experiments of Wicke (1939) one obtains kda values approximately three times lower than those given by eq 13. This improves the agreement with the experimental breakthrough curves, but it is clear that further careful evaluation of this aspect of transport resistance remains to be done. Overall Transport Resistance. In the reviews of Hiester et al. (1963) and Vermeulen et al. (1974), individual resistances are added linearly, corrected in various ways for nonlinearities of the isotherm. We have omitted some of these corrections because of the greater uncertainties in the transport coefficients and rearranged the result for direct use in eq 6. Thus 1 1 = - 1+ - + 1 (16) 15D,,/R2 kda k p 15D,/R2

134


Table I. Details and Parameters of Isothermal Sorption Experiments work of Wicke ( 1 9 3 9 ) adsorbent solute carrier gas

carbon

length, m diameter, m density, kg/m3 temperature, K pressure, Pa

0.72 7 x 10-3 360 273

work of Bowen et aLa

silica gel N2

alumina water vapor air

- 10‘

0.72 7 x 10-3 850 273 -105

u p t o 0.60 1.1 x 625 303 -105

radius, m density, k g / m 3 D p / R 2 s, - ’

Particle 1 . 5 x 10-3 550 4

7 x 1100 0.265

10-3

concentration, mol % velocity, m / s equilibrium loading, kg/kg

Feed adsorption 1 4 desorption 0 0.0105-0.25 0.0107-0.25 0.040 0.0179

particle, s fluid film, s axial dispersion, s

Mass Transfer Resistances 0.017 0.25 0.915-0.0121 0.0178-0.002 0.715-0.015 0.385-0.008

co2

co

N,

2

Bed

a

1.2-3.0 0.43-3.9 0.093-0.175 0.038-0.32 see D p / R 2

Bowen and Donald ( 1 9 6 3 ) ; Rimmer and Bowen ( 1 9 7 2 ) ; Bowen and Rimmer ( 1 9 7 3 ) .

where the sum of resistances is expressed in terms of an “overall” pore diffusivity D,, for use in eq 6a. A similar expression may be used to derive an equivalent overall solid phase diffusivity Dso.

Analysis of Experimental Data In choosing column data from the considerable mass of published material, we limited ourselves to studies which included independent rate and equilibrium measurements and involved systems with complex isotherms and transport resistances. This narrowed the choice to the work of Bowen and Donald (1963), Rimmer and Bowen (1972), and the classical study of Wicke (1939). Experimental parameters and other details for these systems are summarized in Table I. 1. Data of Bowen and Donald (1963) and Rimmer and Bowen (1972). The work of this group, extending over the period 1963-1973, appears to be the only comprehensive sorption study involving a complex isotherm. Measurements were made on the system alumina-water vapor-air for which the equilibrium curve is a Type IV isotherm with two inflection points (Figure 7). Both bed profiles and breakthrough curves are reported. Bed profiles given in the 1972 work were successfully simulated but are not reported here because of apparent discrepancies between bed saturation values and the isotherm reported there (Rimmer and Bowen, 1972, Figure 1). We chose instead the less complete profiles reported in the earlier paper. Breakthrough curves were taken from the 1972 work, and the isotherm used for both sets of data came from Figure 5 of the 1963 study. The sorption path includes both favorable and unfavorable segments, with r values ranging from -0.45 to -1.9. Diffusivity values were taken from separate measurements of Bowen and Rimmer (1973) made on single pellets. Unfortunately, the pellet size was somewhat larger than the bed granules (6/7 vs. 8/14) and some heat effects were unavoidable a t the higher concentration levels. A significant contribution due to surface diffusion was an added complication. The authors nevertheless managed to extract both pore diffusivity (0.012 cm2 s-l) and total combined diffusivities from their data which they used successfully in computer simulations of the column results.

Water VaDOr-Alumina at 30°C

0 3-

0 1-

161

L

0

20

&’

Y (KgH,O/Kg AIR)x10-” ~- , 40 60 80 R H Oh

1

100

Figure 7. Data for sorption of water vapor on activated alumina: isotherm of Bowen and Donald (1963); diffusivities of Bowen and Rimmer (1973).

We used the same values (converted to pore diffusivities), and adjusted particle radius to 1 mm (average for 8/14 mesh), obtaining combined D, f R2 values of 1.2-3.0 s-l for the range of humidities in question (Figure 7). These values are in good agreement with the measurements of Carter (19681, who obtained D,/R2 = 0.815 s-l for somewhat larger granules ( R = 1.5 mm) of the same origin. (a) Breakthrough Curves (Rimmer and Bowen, 1972). The reported breakthrough curves had been obtained on beds of different length fed a t a constant rate with air of 97.5% humidity. This concentration is at the extreme upper end of the isotherm (Figure 7) and gives rise to a central broadening segment flanked by two self-sharpening fronts. Although the breakthrough runs were not carried to saturation, they are sufficiently long to identify two of the three anticipated segments (Figure 8b). The initial constant-pattern front is clearly distinguishable from the broadening portion of the curve which breaks through a t R.H. -50% and coincides nicely with the lower tangent point of the sorption path. The long

Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980 Fig 8a Bed Profiles

I

n 04'

Y

! 94 4

a

Relat

be

*urnlaity

These values have been entered in Figure 7 and are used, along with the appropriate D,/R2, for the calculation of N R from eq 6a. Thus, at the midpoint of the leading zone

\

I

20.4 Figure 4 shows that for this value of N R and a separation factor of 0.48, 1 - 2 = 0,001, so that 2 N 1 to a very close approximation. The breakthrough time t can now be directly computed from eq 5

40

z 0

Fig 8b Breakthrough Curves 8 7 -

(0.083)(0.1005)(625) 1 = 34.6 min t = AY0Gb60Z = (0.013)(0.194)(60) Aqo2pb

g BED LENGTH

9

135

c

""

Similar calculations can be carried out for the other segmental values of AY/AYo, which are then converted to overall breakthrough ratios based on total feed humidity. Thus, for the midpoint of the leading zone, the overall ratio AY/AYo is (0.5)(0.013)/(0.013 + 0.007 0.0066) = 0.245. The results have been plotted in Figure 8b and agree reasonably well with the experimental values. The results of the Hiester-Vermeulen method are nearly identical but involve lengthier calculations. (b) Bed Profiles (Bowen and Donald, 1963). For the profile measurements, both bed length and flow rate were kept constant while inlet humidity was progressively reduced from 94.4 to 10% R.H. This procedure brings out very nicely the effect of feed concentration on sorption path and bed profiles. Thus, at the highest inlet humidity, the path again consists of a central unfavorable segment flanked by two favorable intervals (Figure 7). The corresponding profile should therefore exhibit two relatively steep and self-sharpening segments on either side of a shallower and broadening central portion. This is confirmed by the results shown in Figure 8a. Discrepancies between experiment and predictions near the inlet are probably due to heat effects which are particularly noticeable at the highest concentrations. Similar differences arise in predicting breakthrough from a short 3-cm bed which was omitted in Figure 8b. When feed humidity is dropped to 75.170, the upper favorable region of the sorption path is lost and the predicted profile now consists of a single steep portion near the bed outlet followed by the flat (unfavorable) segment which now occupies some 80% of the bed. Predicted profiles beyond a bed weight of 13 g are identical for both the higher humidities. Unfortunately, the measured data do not extend past this point, but Figure 3 in the more detailed 1972 work shows the trend quite clearly. Simulations of this case (not shown) give good semiquantitative agreement but because of the discrepancies in the isotherm are to some extent displaced from the data. Finally, with the feed humidity at 26.3% (or lower), only a single favorable portion of the path is retained and the profile shows most of the bed at saturation, with a short mass transfer zone near the bed outlet. Calculations of theoretical profiles utilize the same equations as before but in this instance a short trial and error procedure is needed since the desired quantity, bed weight or length z , appears in both eq 5 and 6. This is accomplished without any difficulty. 2. Data of Wicke (1939). This classical study, executed with great care, includes measurements of both adsorption and desorption breakthrough curves of carbon dioxide on columns of silica gel and activated carbon. In a previous analysis of two of these curves, Vermeulen et al. (1973, p 16-37) extracted an average separation factor and transport coefficient from the midpoint behavior of the adsorption

+

TIME MIN

Figure 8. Comparison of predicted (-1 and experimental ( - - - ) profiles and breakthrough curves of water vapor on beds of alumina; data of Bowen and Donald (1963), and Rimmer and Bowen (1972).

drawn-out form of these curves attests to the strong influence of the unfavorable domain. To illustrate the calculations involved in obtaining theoretical breakthrough curves for these more complex equilibria, sample computations are presented for various concentration levels over the entire isotherm range. We start by establishing the sorption path and the associated favorable/unfavorable segments. This is quickly done by invoking the rule that slope must diminish or remain constant in passing from feed to the initial bed condition. At the high feed humidity used in the breakthrough experiments, this leads to a central unstable zone flanked by two tangents drawn from feed and initial loading points to the isotherm (Figure 7). For the lowest feed humidity shown by the bed profile data (Figure 8a, 26.3%), the sorption path reduces to a single self-sharpening segment defined by a chord connecting origin and feed point. The span, Aqo, AYo,of each segment is identified next, and corresponding r values calculated at its midpoint (AY/AYo = 0.5) using eq 7a. One obtains for the leading zone Aqo = 0.083; (Aq)o,5 = 0.056; AYo = 0.013

AY/AYo(l - Aq/Aqo) - (0.5)(1 - 0.056/0.083) (0.5)(0.056/0.083) (1- AY/AY,) (A4 /Aqo) 0.48 for the central broadening zone

r=

Aqo = 0.089; (A4)o.S = 0.035; AYo = 0.007

(0.5)(1 - 0.035/0.089) r = = 1.53 (0.5)(0.035/0.089) for the rear zone Aqo = 0.195; (A4)o.j = 0.118; AYo = 0.0066 r=

(0.5)(1 - 0.118/0.195) = 0.66 (0.5)(0.118/0.195)

136

Ind. Eng. Chem.

Process Des. Dev., Vol.

19, No. 1, 1980

Fig 9a Carbon - Adsorption

Z

0 F a a:

II

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t-

z

I

U

0

I

I

50

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BI

I

, i

0

500

1200

3000

3600

TIME, SEC

600

1200 3000 TIME. SEC

3600

4200

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0

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0

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TIME, SEC

Figure 9. Predicted (-1 and experimental (- - -) adsorption and desorption breakthrough curves of carbon dioxide on activated carbon; data of Wicke (1939b).

front to predict the complete adsorption and desorption curves under identical flow conditions. Here we consider an entire series of runs with both adsorbents in which bed length and feed concentration were kept constant, but velocity varied from 1 cm/s to 25 cm/s (Table I). Under these conditions, fluid phase resistance was a significant factor for charcoal even at the highest flow rates and became predominant for both adsorbents a t l cm/s. The broadening effect this has on the adsorption breakthrough curve is clearly evident in the results shown in Figure 9 and 10. Use of a single transport coefficient would give erroneous results in this case. We combined instead the independent pore diffusivity measurements of Wicke (1939a) with appropriate fluid phase coefficients to arrive a t an overall diffusivity for use in eq 6a. Equilibrium isotherms for the two adsorbents were of Type I, with r values ranging from approximately 0.4 to 0.8 (or the inverse for desorption). Fluid film mass transfer coefficients and an equivalent axial dispersion coefficient were calculated from the correlations of Thodos et al., eq 12 and 14. The correction factor g was omitted in these calculations because of the larger uncertainties in the correlation coefficients. The resulting ka values are given in Table I and show a surprisingly large dispersion contribution. The combined resistances, however, reflect reasonably well the observed broadening of the adsorption curves, with perhaps an overestimation of the effect at the lowest velocity. Some discrepancies are found between predicted and experimental stoichiometric times, particularly for the silica gel bed. Wicke (1939b, p 304) has attributed this to fluctuations in flow rate and feed compositions, which are often unavoidable in even the most meticulous work. The desorption curves show the expected proportionate-pattern behavior, and the agreement between theory and experiment is good for the charcoal bed and less satisfactory for silica gel a t the intermediate velocity. It is difficult to state whether this is due to shortcomings in the model or the notorious difficulty of predicting the

TIME. SEC

Figure 10. Predicted (-) and experimental (- - -) breakthrough curves: carbon dioxide on silica gel; data of Wicke (193913).

trailing end of desorption fronts. We used equilibrium and diffusivity values measured by Wicke at 1torr (Le., a t the 1%level) but a discrepancy of several hundred seconds still remained for the silica gel case. However, other charcoal desorption curves taken from the same work (not shown here) gave good agreement over the entire concentration range. This seems to confirm the basic reliability of the method. Conclusions A simple two-step method, consisting of a graphical construction of sorption paths and the use of modified solutions for the Langmuir isotherm, can be applied with acceptable accuracy to predict various isothermal column sorption data. Systems with equilibrium isotherms of arbitrary shape and complex transport resistances can be accommodated. The method is attractive as a rapid means of assessing isothermal bed behavior. Acknowledgment The author gratefully acknowledges the help derived from several discussions with Theodore Vermeulen. Nomenclature D = particle diameter, m D , = gas-phase diffusivity, m s-* D , = pore diffusivity, m s-* D,, = equivalent overall particle diffusivity, m s-* (eq 16) D , = solid phase diffusivity, m g = correction for nonlinear equilibrium, eq 15 Gb = inert carrier mass velocity, kg m-2 s-l kda = volumetric mass transfer coefficient equivalent to axial dispersion, s-l k p = fluid film volumetric mass transfer coefficient, s-l NR = dimensionless number of reaction units, eq 6 Pe = dimensionless Peclet number, D u / D t b q = adsorbate concentration, kg of solutejkg of solid q* = equilibrium loading, kg of solute/kg of solid r = dimensionless separation factor, eq 7 R = particle radius, m Re = Reynolds number, Dup /lg Sc = Schmidt number, c(,/pPbg t = time, s

Ind. Eng. Chem. Process Des. Dev. 1980, 19, 137-144

superficial gas velocity, m s-l Y = gas phase solute concentration, kg of solute/kg of carrier z = distance from column inlet 2 = dimensionless throughput parameter, eq 5 -~ u =

~

Greek Symbols q, = bed void fraction pg = gas viscosity, N s m-2 Pb = bed density, kg/m3 p g = gas density, kg/m"

L i t e r a t u r e Cited Basmadjian, D., Ha, K. D., Pan, C. Y., Ind. Eng. Chem. Process Des. Dev., 14, 328 (1975). Bowen, J. H., Donald, M. B., Chem. Eng. Sci., 18, 599 (1963). Bowen, J. H., Rimmer. P. G.. Chem. Eng. J . , 6, 145 (1973). Carter, J. W., Trans. Inst. Chem. Eng., 46, T213 (1968). Chao, R., Hoelscher, H. E., AIChE J . , 12, 271 (1966). Garg, D. R., Ruthven, D. M., Chem. Eng. Sci., 28, 799 (1973). Garg, D. R., Ruthven, D. M., Chem. Eng. Sci., 29, 1961 (1974). Giueckauf, E., Trans. Faraday Soc., 51, 1540 (1955). Hiester, N. K., Vermeulen, T., Chem. Eng. Prog., 48, 505 (1952a). Hiester, N. K., Vermeulen, T.. Document No. 3665, American Documentation Institute, May 14, 1952b. Hiester, N. K., Vermeulen, T., Klein, G., in "Chemical Engineers' Handbook", J. H. Perry, Ed., 4th ed, Chapter 16, McGraw-Hill, New York, N.Y., 1963.

137

Hsiung, T. H., Thodos, G., I n t . J . Heat Mass Transfer, 20, 331 (1977). w . S,'Chem. Eng. Sei., 28, 1853 (1973). Lightfoot, E. N., Sanchez-Palma, R. J., Edwards, D. O., in "New Chemical Engineering Separation Techniques", H. M. Schoen, Ed., Interscience, New York. N.Y.. 1962. Lukchis, G. M., Chem. Eng., 79(14), 111 (1973a). Lukchis, G. M.. Chem. Eng., 79(16), 83 (1973b). Lukchis, G. M., Chem. Eng., 79(18), 83 (1973~). Masamune, S.,Smith, J. M., Ind. Eng. Chem. Fundam., 3, 179 (1964). Masamune, S.,Smith, J. M., AIChE J . , 11, 41 (1965). Ozil, P., Bonnetain, L., Chem. Eng. Sci., 32, 303 (1977). Pan, C. Y., Basmadjian, D., Chem. Eng. Sci., 26, 45 (1971). Petrovic, L. J., Thodos, G., Ind. Eng. Chem. Fundam., 7, 274 (1968). Rhee, H. K., Amundson, N. R., Chem. Eng. J., 3, 121 (1972). Rimmer, P. G., Bowen, J. H., Trans. Inst. Chem. Eng., 50, 168 (1972). Rosen, J B., Ind. Eng. Chem., 46, 1590 (1954). Tudge, A. P., Can. J . Phys., 39, 1611 (1961). Vermeulen, T., Klein, G., Hiester, N. K., in "Chemical Engineers' Handbook", J. H. Perry, Ed., 5th ed, Chapter 16, McGraw-Hill, New York, N.Y., 1974. Vermeulen, T., private communication, 1979. Wicke, E., Koiloid Z.,86, 167 (1939a). Wicke, E., Kolloid Z.,86, 295 (1939b). Wilke, C. R., Hougen, 0. A., Trans. Am. Inst. Chem. Eng., 41, 445 (1945). Young, D. M , Crowell, A. D., "Physical Adsorption of Gases", p 4, Butterworths, 1962.

Receiued for revieu, March 13, 1979 Accepted October 15, 1979

Rapid Procedures for the Prediction of Fixed-Bed Adsorber Behavior. 2. Adiabatic Sorption of Single Gases with Arbitrary Isotherms and Transport Modes Diran Basmadjian Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Ontario, Canada, M5S lA4

A simple prediction method outlined previously (part 1) is extended to simulate concentration and temperature fronts in t h e adiabatic adsorption and desorption of single solutes in a fixed bed. The two-step procedure involves the

graphical construction of adiabatic equilibrium curves from given isotherm data and a subsequent modified application of the Hiester-Vermeulen solutions for isothermal sorption along a Langmuir isotherm. The method is capable of accommodating complex equilibria and transport resistances. It does not require any numerical solutions or advance knowledge of profile shape. Predicted results show satisfactory agreement with the data of Carter and measurements from this laboratory.

Introduction We had previously shown in part 1 that isothermal fixed-bed sorption of single solutes with arbitrary isotherms could be described, with reasonable accuracy, by a piece-wise application of the Hiester-Vermeulen solution for a Langmuir isotherm. Complex transport resistances and variable diffusivities could also be accommodated. Construction of the sorption paths for complex isotherms was accomplished in simple fashion by using the principles of isothermal equilibrium theory. A similar approach may be used for nonisothermal sorption by applying the Hiester-Vermeulen solutions along appropriate adiabatic equilibrium curves. These curves can in principle be derived by combining the equilibrium isotherms with the conservation equations of adiabatic equilibrium theory (Pan and Basmadjian, 1971; Basmadjian et al., 1975a). The only simplification in this approach is the assumption of local thermal equilibrium, i.e., negligible particle to fluid heat transfer resistance. In order to retain the simplicity of our prediction method, we devised a graphical/algebraic solution of these equations which avoids any computer work and does not require analytical expressions for the equilibrium iso0019-7882/80/1119-0137$01 .OO/O

therms. It is merely necessary to draw a set of experimental isotherms at regular or arbitrary intervals in a q-Y diagram (solid-gas concentration) and use these for a stepwise construction of the adiabatic equilibrium curve. The mass and energy conservation equations are recast into an appropriate form for this purpose. Adiabatic equilibrium curves are generally more complex than their isothermal counterparts. As shown in our previous analyses, equilibrium profiles can consist of single solute/temperature fronts preceded by a thermal wave, or take the form of two combined fronts flanking a plateau. The former situation is most common in adsorption, the latter in thermal regeneration processes. The associated equilibrium curves will consequently be either single curves or will have two branches intersecting at the plateau concentration. Each of these curves can in turn be of the favorable or unfavorable type or exhibit a more complex form with inflection points. The self-sharpening fronts must in principle be generated by the algebraic conservation equations of equilibrium theory, while the differential equations are used to construct the broadening segments. In practice there is little difference in the curves obtained by either procedure so 0 1979

American Chemical Society

Rapid Procedures for the Prediction of Fixed-Bed Adsorber Behavior

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