12, allows the prediction of vapor-liquid equilibria over a wide range of conditions. The proposed equations for the activity coefficients in concentrated solutions are particularly useful for phaseequilibrium calculations. The phase-equilibrium relation for a volatile electrolyte, such as HC1, has the advantage that the electrolyte in aqueous solution is treated as a nonelectrolyte, consistent with its state in the vapor phase. Similarly, in the description of the solid-liquid equilibria, sodium chloride is considered as the same thermodynamic component in both solid and liquid phases. The proposed equations have two advantages over Harned's rule. First, the ternary parameters A23 and A32 are easily extrapolated to new conditions of temperature and composition. Second, eq 11, 12, and 19 can be extended without further assumptions to solutions containing more than two electrolytes. Nevertheless, Harned's rule accurately describes the activity coefficients for a large variety of ternary electrolyte solutions, including those containing electrolytes of complex charge-type. Therefore, although the proposed equations are useful for the HClNaCl-HiO system, they require further study before they can have the general application of Harned's rule. Acknowledgment
L i t e r a t u r e Cited Akerlof, G., Turck, H. E., J. Amer. Chem. SOC.,56, 1875 (1934). Bromley, L. A.,A.l.Ch.E. J., 19, 313 (1973). Hala, E., Proc. Int. Syrnp. Distill., 3, 8 (1969). Harned, H. S.. Owen, B. B., "Physical Chemistry of Electrolytic Solutions," 3rd ad, Chapter 14, Reinhold, New York, N. Y.. 1958. Harned, H. S., Robinson, R. A., "MulticornDonent Electrolyte Solutions," Pergamon. London, 1968. Hirschfelder. J. O., McClure, F. T., Weeks, I. F.. J. Chem. Phys., 10, 201 (19421. "International Critical Tables," Vol. 3, McGraw-Hill, New York, N. Y., 1926. Lewis, G. N., Randall, M., "Thermodynamics," Revised by K. S., Pitzer and L. Brewer, 2nd ed, Chapter 22, McGraw-Hill, New York, N. Y., 1961. Meissner, H. P., Kusik, C. L..A.l.Ch.E. J., 18, 294 (1972). Meissner, H. P., Kusik. C. L., lnd. Eng. Chem. Process Des. Develop., 11, 205 (1973). Meissner, H.P., Kusik, C. L.,Tester, J. W.,A.l.Ch.E. J., 18, 661 (1972). Meissner. H. P., Tester, J. W., lnd. €no. Chem., Process Des. Develop., 11, 128 (1972). O'COnnell, J. P., Prausnitz, J. M . , lnd. €ng. Chem., Fundam., 3, 347 (1964). O'Conneil, J. P., Prausnitz, J. M., lnd. Eng. Chem., Fundam., 9, 579 (1970). Perry, R. H..Chilton, C. H., "Chemical Engineers' Handbook," 5th ed, pp 3-62, McGraw-Hill, New York, N. Y., 1973. Prausnitz, J. M.. "Molecular Thermodynamics of Fluid-Phase Equilibria," Chapter 6, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. Robinson, R. A,, Stokes, R. H., "Electrolyte Solutions." 2nd ed, Chapter 15, Butterworths, London, 1970. Seidell, S..Linke. W. F., "Solubilities," Vol. 2, 4th ed. American Chemical Society, Washington, D. c., 1965.
The author is grateful to "Sociedad Quimica y Minera de Chile" for financial support and to ECOM for the use of their computational facilities. The project was partially sponsored by Latin American Teaching Fellowships, Tufts University.
Received for review January 7, 1974 Accepted M a y 14, 1974
Numerical Investigation of Adiabatic Fixed-Bed Adsorption David 0 . Cooney Department of Chemical Engineering, Clarkson Coilege of Technology, Potsdam, New York, 13676
The energy balance and solute continuity equations which describe behavior in adiabatic adsorption beds were numerically integrated for several cases. A Langmuir equilibrium distribution relation and linear driving force expressions for heat and mass transfer were assumed. Only situations in which "combined wave fronts" exist, as opposed to cases where a "pure thermal wave" precedes the concentration wave, were considered. In particular, the effects on temperature and concentration profiles in both solid and fluid phases due to (a) varying the inlet feed concentration, (b) changing the values of the heat and mass transfer coefficients, (c) changing the heat of adsorption value, and (d) allowing the isotherm to become linear were determined. These effects are discussed in detail. This study illustrates the types of conditions under which definitive "two-zone'' behavior can arise (Le., two distinct mass transfer zones separated by a plateau region), indicates when the assumption of equilibrium behavior is apt to be invalid and demonstrates how rapidly self-sharpening profiles approach an asymptotic (constant pattern) state. The concept of the "effective isotherm" which governs behavior in nonisothermal systems is introduced.
The topic of adiabatic single-solute adsorption in fixed beds has received considerable attention recently. A variety of analytical studies dealing with behavior in equilibrium systems, constant-pattern systems, and so forth, has been presented in the literature. Likewise, a number of numerical investigations have been reported. It is now well known that the interaction between heat and mass transfer phenomena during saturation or elution of adiabatic fixed beds can lead to situations (a) in which a pure 368
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 4, 1974
temperature wave and then (usually) a temperature plateau region precede a single zone in which both concentration and temperature vary, or situations (b) involving so-called "combined wave front" or "two-zone" behavior, that is, the production of two distinct zones in which concentrations and temperatures both vary, with a plateau region of constant concentrations and temperatures in between. Pan and Basmadjian (1970) have shown that if ( q * / c ) o < c,,/c,,M, then this latter kind of situation
must occur. The present paper deals with systems which satisfy this criterion and hence will deal exclusively with such cases. Leavitt (1962), the first to develop equations describing profile shapes and travel speeds in both zones of “two zone” systems, based his analysis on the assumption of constant-pattern (CP) behavior in both zones. His experimental data on COz adsorption in molecular sieve beds suggested strongly that the trailing zone was a broadening one, however. Leavitt cautioned that such might often be true, in which case his equations (including formulas for determining the interzone conditions) might not be strictly applicable. Later, P a n and Basmadjian (1967), believing Leavitt’s data indicated two constant-pattern zones, presented a similar analysis based on the CP assumption. They were able to prove that favorable isotherms will lead to C P behavior in the leading saturation zone. Difficulties were encountered, however, in deriving existence conditions for CP behavior in the trailing zone. They did point out that in general a leading C P front can exist even when the trailing front is a broadening one. Also, they stated that the equations for uniform CP behavior predict correctly the interzone conditions even when the trailing zone is not a C P one. However, the multicomponent isothermal sorption studies of Klein e t al. (1967), suggest that this may not be the case. In a later paper, Pan and Basmadjian (1970) conclude that the rear wave is generally a broadening one and they mention that no one has yet observed experimentally a system in which both zones are self-sharpening (the studies to be described later in this paper also support this view). However, Pan and Basmadjian (1971) later demonstrated theoretically, using equilibrium theory, that sharp profiles in both zones can occur under certain conditions, e.g., when high temperature feeds are used. A number of investigators have also analyzed adiabatic single-solute sorption using equilibrium theory (Amundson, e t al., 1965; Rhee and Amundson, 1970; Rhee, et al., 1970). Such studies are extremely useful in delineating the basic behavior of such systems, for example, in determining whether sharp or broad fronts exist and in establishing the plateau region conditions. However, in real systems, especially in constant-pattern zones, a close approach to equilibrium often does not pertain, and hence equilibrium analyses are inaccurate. As will be seen from results presented later, profile shapes can change markedly when the departure of a system from equilibrium becomes significant. For obtaining descriptions of systems which are neither in equilibrium or asymptotic states, one must generally resort to the numerical integration of the relevant PDE’s, as was done in the present study. Bullock and Threlkeld (19661, Meyer and Weber (1967), Lee and Weber (1969), Carter (1966, 1968, 1973), and Chi and Wasan (1970) have all carried out numerical analyses of adiabatic adsorption and compared their theoretical results with various data on CH4 adsorption on activated carbon and water sorption by alumina or silica gel. The work by Carter consistently involves “pure thermal wave” type behavior and is thus not directly relevant here (it should be mentioned, however, that very well developed temperature plateaus were obtained both experimentally and theoretically). The work of the other investigators does involve the “combined wave front” systems which are of interest here. However, almost all numerical and experimental results of these investigators indicate temperature profiles with peaks but with no well-formed plateaus and concentration profiles exhibiting no distinct plateaus and no two-zone behavior. This is probably due mainly to the sorbent beds
being too short, under the given conditions, to develop flat plateaus and two-zone behavior. An exception is some of the computed profiles of Lee and Weber (1969), which come close to possessing flat interzone regions and which do show two-zone behavior. Also of interest in this connection are the experimental effluent data of Getty and Armstrong (1964) for air drying in alumina beds. Although their temperature us. time curves show peaks but no plateaus, their effluent air dew point us. time curves exhibit very definite two-zone and flat plateau characteristics. Finally, some of the data presented by Pan and Basmadjian (1970) show the leading zone and plateau region very emphatically (data on the trailing combined wave zone are not cited, however). In most analytical or numerical investigations of adiabatic adsorption, parametric studies are absent. Profiles are given for particular systems operated at certain specific conditions, and often little indication is made regarding how system behavior would change under different conditions. Rhee, et al. (1970), have discussed the effect of feed temperature and solid phase heat capacity on equilibrium behavior, and Lee and Weber (1969) have discussed the general effects of nonisothermality on concentration profiles and of gas velocity on temperature peak heights. It might also be noted that the studies of Pan and Basmadjian (1971) constitute a parametric study of the effects of feed concentration, feed temperature, and initial bed loading for the idealized case of equilibrium. Carter (1968, 1973) has, for pure thermal wave systems, shown some of the effects of changing a key parameter ( e . g . , A H by 20%, K , by a factor of 3). However, for “two-zone’’ systems operating under real (i. e., nonequilibrium) conditions, parametric studies are clearly very sparse. In the present paper, we wish to offer further insight into systems in which distinct plateaus and two-zone behavior exist, thereby indicating when such behavior arises, and in particular we also wish to: (1) show how all dependent variables (c, q, q*, T,, T q )vary in the sorbent bed, since most investigators have given information on c and T g only; (2) thereby indicate how the differences T , - TL‘ and q* - q vary in the bed, in order to show when the assumption of equilibrium may be a poor one; this sort of information has not explicitly appeared in the literature; (3) show how rapidly a typical leading-zone profile approaches constant-pattern behavior, so as to indicate when the assumption of such behavior is reasonable; (4) demonstrate the effects of variations in some very important system parameters ( S f , K,, h, feed concentration, and isotherm linearity) on all profiles; and (5) introduce the new concept of the “effective isotherms” which govern profile development in the two transfer zones. We will show that while a solute may possess true isotherms of one shape, the “effective isotherms” governing transport behavior in each zone may be of vastly different shapes. Basic Equations The solute continuity equation for a fixed bed sorption system is
for the case of negligible axial diffusion and using concentration units of mass per unit volume ( e . g . , mol/cc of the phase involved). Since most adiabatic sorption systems of interest are gas-solid systems, in which temperature changes can cause significant gas density variations, it is often preferable to use concentration units of mass/mass ( e . g . , mole fraction) for the gas phase. While in principle any reasonable units may be used, Ind. Eng. Chem., Process
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369
for present purposes we will use c = moles of solute per mole of gas and q = moles of solute per gram of solid. The solute continuity equation then becomes
The energy balance equation for the system is A
In this study, the following mass and energy balances for the solid phase were used
computer. The Appendix outlines the particular solution scheme which was employed. In the cases we investigated, stability considerations required that a time increment of about 0.5 sec or less be used. The size of the distance increment had a lesser effect on numerical stability, as long as it was not too small. Convergence considerations required that the distance increment be no larger than about 0.5 cm in order to give results which would change by less than 1%upon any further reduction in the size of the distance increment. For our “base case,” we considered feeding a 0.1 mole fraction, 20°C gas stream at a flow rate of 100 cc/min to a cylindrical bed of adsorbent 3.5 cm in diameter. The bed, initially solute-free and also at 20°C throughout, was considered to be packed with 0.1-cm diameter particles ( a = 60 cm2/cc of solid) at a void fraction of 0.36. The system pressure was assumed to be 500 psig throughout (negligible pressure drop). Add_itionally, the following parameter values were selected: C,, = 9.0 cal/mol”C; = 0.25 cal/g”C; i s = 1.3 g/cc; h = 0.0001 cal/cm2 sec”C; K , = 0.0001 g/cm2 sec; AA = 15,000 cal/mol. The values of a, and jsare typical of common gases and adsorbent materials. K, and h were chosen somewhat arbitrarily for this base case, and their values were varied later for other cases. Even so, the cited values are realistic and typical. Available data on heats of adsorption suggest that AA values of 5000-25,000 cal/mol are most common. For the base case, a value of 15,000 cal/mol was specified. In this study, If?was assumed to be independent of the extent of adsorption. As Hassler (1963) points out, this is often a good assumption, especially for compounds having fairly high boiling points. For certain solute/adsorbent pairs, however, this assumption may be invalid. A pressure of 500 psig was chosen because many, if not most, current “pressure swing” adsorption operations are carried out at pressures of this order (see Stewart and Heck, 1969). Also, the coefficient multiplying dq/at in equations 2 and 3 and multiplying dT,/at in eq 3 contain p g in the denominator. For pressures near atmospheric, these coefficients are very large. The coefficient of aq/at in eq 3 is especially large since A H appears in the numerator. Its value at one atmosphere pressure is about 90,000,000 for the base case parameter values. This causes any errors in the finite-difference approximation for aq/at to be enormously magnified and results in severe instabilities. At a pressure level of 500 psig, the coefficients mentioned are a factor of 35 smaller, enabling stability to be achieved. Pressures somewhat lower than this could certainly have been used. The isotherm expression employed for the base case had the classical Langmuir form
eps
e,,, eps,
In these equations, the rates of heat and mass transfer are characterized as linear functions of overall driving forces. Such linearized expressions are generally accepted to be reasonably accurate. As boundary and initial conditions, we assume uniform temperature and zero adsorbate concentration everywhere in the bed for t < 0, and uniform feed conditions for t > 0. The neglect of axial diffusion of mass and energy is often unjustified for gas phases. Such diffusion can be incorporated into the analysis, however (exactly for linear systems, and approximately for nonlinear systems) by decreasing the values of KP and h according to the formulas developed by Klinkenberg and Sjenitzer (1956), for example. It should also be pointed out that because ija and u will vary with axial distance in adiabatic adsorbers as a result of temperature variations, these quantities should really be included inside the differentials in eq 2 and 3 (the more minor variations of K , and h can be neglected). Alternatively, one can leave them outside the differentials if one uses the correct pointwise average values (as determined by the prevailing T,)during each step of the numerical integration. By defining dimensionless variables t+ = t u / L , z+ = z / L , c+ = C / C O , q+ = q/qo* and T+ = ( T - To)/To, one can rewrite eq 2 through 5 in dimensionless form. The parameters which appear as multipliers of the derivatives can be shown to consist of the following groups or products of these groups
q* =
0.15Kc 40Kc
1
+
where the temperature dependence is incorporated in the coefficient K , which is given by Therefore, these quantities (or any derived set of groups obtained by combination) determine the basic behavior of adiabatic adsorption systems. Since a better feel for the systems in question is achieved if values are cited for the individual items (such as Kp, h, u, A g , etc) rather than values for the arbitrary dimensionless groupings, we will confine ourselves to dealing with the dimensional parameters only in all that follows. Numerical Studies
Equations 2 through 5 were written in terms of finite differences and numerically integrated on an IBM 360/44 370
Ind. Eng. Chern., Process
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1974
h,-K K298°K
-
-
AR 1 R ( E
1
-m)
(7)
K 2 9 p ~was taken to be equal to unity (since only the ratio of the K‘s is important). This isotherm is shown in Figure 1. It should be noted here that in cases when AH was changed, the new value was used in the equilibrium relation above, as well as in eq 3 and 5.
Results for the Base Case Figure 2 presents computed c, q * , q, T,, and T,- profiles for the base case conditions cited above. The interaction
I 160
1+40KC
REDUCED FEED t = 2400
CONC’N SEC
CASE
1 450
% 8
‘ i 140 n +-3 Lz
“30 5
1 :
30 DISTANCE ALONG
50
40 COLUMN
120 60
CM
Figure 3. Profiles for 2% feed. 0 0.02 0.04 0.06 008 0.10 GAS-PHASE SOLUTE CONTENT, (MOLEIMOLE)
Figure 1 . Langmuir isotherm for base case. 010-
I
I
1
I
30
40
50
60
,?O
A20 0
IC
20
70
DISTANCE ALONG COLUMN, CM
Figure 2. Column profiles for the base case.
between concentration and temperature causes generation of a very decided plateau region under these conditions. As one proceeds downstream from the top of the bed into the upper mass transfer zone (MTZ), one finds that c, q , and q* decrease with distance as usual. Under isothermal conditions one would find q* to be consistently greater than q, due to the finite nature of mass transfer rates. For sorption accompanied by significant heat release, however, any adsorption of solute by the solid phase causes a rise in T,. This in turn tends to lower q * , decrease the driving force q* - q , and therefore impede further mass transfer. Hence, adsorption becomes self-limiting. Finally, when T, is sufficiently high, q* is lowered to the point where c and T , both must decrease due to limits imposed by material and energy balance considerations. We will show later that behavior in the trailing MTZ is governed by an “effective isotherm” which is nearly linear (hence this zone roughly broadens in proportion to distance travelled down the bed), and that the leading MTZ is governed by a strongly favorable “effective isotherm.” This zone therefore rapidly develops constant-pattern behavior ( e . g , by the time (-450 sec) that the middle concentration c = 0.05 mol/mol reaches roughly 20 cm down the column, asymptotic frontal shapes are virtually 100% attained). From Figure 2 one may also note that ( a ) since K,, and h are reasonably high, the differences T , - T , and q* - q are relatively small in the upstream MTZ, and therefore an assumption of equilibrium behavior would be approximately correct for this zone; (b) although T g - T , is small in the downstream MTZ, q s - q is quite large, and therefore the system is (from a mass transfer standpoint) very far from equilibrium in this zone; and (c) the T,, T,, and
c profiles in the downstream MTZ are remarkably similar. Note also the marked q* peak which occurs in the downstream zone. This peak shows quite well how complex the interaction between concentration and temperature can be. Upstream of the peak, the influence of decreasing T , outweighs the effect of decreasing c, and q* rises with distance. Finally, as c and T , get low enough, the relative influences of each reverse, and q* thereafter declines with distance. In the upstream MTZ, the effects of c and T s on q* are in the same direction of course, so the possibility of a peak in q* is excluded. It is because the equilibrium value of q* depends on both c and T,, whereas the equilibrium value of Ts depends only on T,, that it is possible for large departures from mass transfer equilibrium to occur in the downstream MTZ a t the same time that heat transfer equilibrium is closely approached. Finally, it might be indicated that the values of lOOq* and lOOq at zero axial distance are 0.291 and 0.282, respectively, for this case, a t t = 1200 sec.
Effect ofLowering Inlet Feed Concentration from 10% to 2% Figure 3 shows that, for a solute which follows a Langmuir isotherm, the rates of travel of concentration and temperature waves are markedly reduced a t lower feed concentrations. Note that Figure 3 shows results for t = 2400 sec, rather than 1200 sec as in the previous plot. The lower rates of travel are, of course, due to ( a ) the greater relatiue degree of sorption a t lower concentrations which one has with Langmuir solutes, and (b) the fact that less absolute sorption gives a lower total heat release and lower temperature rises (thus, less decline in q” due to temperature effects). The slower movement of the profiles clearly causes the development of the plateau region to be concomitantly slower. Because, for Langmuir solutes, ( q * / c ) o increases for lower feed concentrations, it is sometimes observed that a transition to pure thermal wave behavior can occur a t sufficiently low feed concentrations. However, for the isotherm and system properties employed here, ( q * / c ) o is less than M ) for any possible feed concentration, and therefore only ‘(twozone” behavior can arise.
(e,,,/e,,
Effect of Reducing K , a n d h Figure 4 shows the effects of decreasing KP and h to 2570 of the base case levels (we do not wish to imply, however, that under actual conditions K , and h vary in such harmony). This causes the profiles to spread out considerably, as one would expect. A severe “erosion” of the plateaus takes place on both ends. The highest values of T , and T , attained are lower than in the base case, because Ind. Eng. C h e m . , P r o c e s s D e s . Develop., Vol. 13, No. 4 , 1974
371
REDUCED K F 8 h CASE t = 1200 SEC
0.02-
z W U 8
0
j
I
I
I
I
j
in
20
30
40
50
60
.20 70
I
0
10
20
Figure 4. Profiles for Kp = h = 2.5 X
lob
40
30
DISTANCE ALONG
DISTANCE ALONG COLUMN, CM
50
60
70
COLUMN, CM
Figure 6. Profiles for linear isotherm.
(cgs units).
REDUCED AH
,’
I 1.0-
t-
-30
P 50 DISTANCE
Figure 5 . Profiles for AH =
ALONG
COLUMN,
60
120 70
CM
- 7500 cal/mol.
of the spreading effect, and therefore q* tends to be generally higher than previously. The differences T , - T , and q* - q are uniformly much larger, as also anticipated. Effect of Reducing AH When AH is cut in half (to 7500 cal/mol), the profiles take on the shapes shown in Figure 5 . The reduced AH causes less heat release per mole of solute adsorbed, which keeps the adsorption capacity ( q * ) from declining so drastically. In general, more adsorption per unit bed volume takes place and therefore the rate of movement of solute (and hence thermal energy) down the bed is considerably slower than in the base case (note that Figure 5 is for t = 2400 sec). Surprisingly, however, the maximum T , and T z rises attained are greater (by more than 40%) than those for the base case. Since the profiles in both Figures 2 and 5 are reasonably well developed, this unexpected behavior does not seem to be just a temporary phenomenon which occurs only at early times. Obviously there must be a point which reducing A H further would cause the maximum Tq and T , values to decrease. We have not determined this point, however. Because of the greater degree of adsorption, it is clear that more time (or distance traveled) must elapse to permit significant plateaus to develop in this situation. Finally, it might be noted that for this case the interactions are such that only a very faint maximum in q* develops. Effect of Linear Isotherm Using a linear isotherm of the form q* = 0.03Kc (when plotted in Figure 1, this isotherm, for 20°C, connects the origin and the upper right hand corner) gives rise to the profiles shown in Figure 6. Note that t = 780 sec for this figure. In general, the profiles are more spread out than in the base case, as one might have predicted. However, part 372
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 4, 1974
B o
I
0
I
I
I
I
002
004
006
008
GAS-PHASE
I
010
SOLUTE CONTENT, MOLEIMOLE
Figure 7. Effective isotherms for the various cases. of this is due to the faster rate of movement of mass and energy down the column, since the amount of solute adsorbed with this isotherm is considerably less than with the convex Langmuir type of isotherm used in the base case. Qualitatively the profiles here are very similar to the base case, however. Effective Isotherms Since we are considering highly nonisothermal systems, it is clear that the “effective isotherms” (Le., q* us. c relationships) which govern mass transfer zone behavior are not the lines of constant temperature (true isotherms) shown in Figure 1. Rather, the relations presented in Figure 7 are those which pertain. This figure shows that: (a) even for the “linear isotherm” cases, the effective isotherm is highly nonlinear; ( b ) the highly convex parts of these curves extending from the origin to the inflection points govern behavior in the leading MTZ; (c) the nearly linear portions of these curves extending from the inflection points sharply upward determine behavior in the trailing MTZ; (d) the inflection points repreBent the plateau concentrations q* and c; (e) the values of the plateau q* and c concentrations are not affected by variations in K , and h, which is as expected, but are affected by alterations to AH (and also by changing the mathematical form of the isotherm-but this is obvious); and (f) the approach of the effective isotherm to an asymptotic condition is fairly rapid, especially at low q* and c and a t high q* and c; that is, a t either ends of the T , and T , “humps” wherethe T rises are lowest. Variation of Other Parameters Other parameters ( e . g . , u, a ) could have been varied in this study. In most cases, however, the effects can be directly inferred from the results presented already, and
therefore specific consideration of the effects of these other parameters need not be undertaken. For example, changes in a would have the same consequences as changes in K , and h. Alterations in gas velocity would be expected to affect the degree of spread of the profiles and influence rates of frontal travel in the usual fashions. Varying the system pressure at constant volumetric feed rate would change the solute input rate and also affect the volumetric heat capacity of the gas phase. Even in this case, one can pretty well judge the overall_ effects. Wide changes in physical properties (e.g., cup.,Cps, ps, t ) cannot be justified if one restricts oneself to common materials. Therefore, for purposes of the present study, it is felt that the cases treated represent a reasonable number and choice of conditions. Conclusions Besides giving general insight into the basic nature of adiabatic adsorption, the results presented here lend themselves to the formulation of a few specific conclusions. 1. Unless an adsorbent bed is reasonably long, K , and h must be respectably high in order to generate a significant plateau region. Nonlinearities in the governing isotherm (in particular, favorable or convex ones) also assist in establishing plateau regions. 2. The maximum temperature rises which occur can actually be larger for solutes having smaller heats of adsorption, under certain conditions. 3. The shape of the q* us. c relationship which determines profile development can be quite different from the true isotherm shapes. General conclusions (e.g., decreasing K , and h causes profile spreading, decreasing A H leads to slower frontal movement rates, etc.) are much more obvious from the foregoing discussion and need not be reiterated here. Appendix. Outline of Numerical Solution Procedure Replacing each time derivative by a simple first-order forward finite difference, e . g .
ac _ -
~ ( ti ,+ A t )
at
-
~ ( ti) ,
At
initial conditions in the sorbent bed, one has values for c ( i , t ) , Ts(i,t),etc., for every axial position (i.e., each i value) for t = 0. One then calculates c(i,t + A t ) , T,(i,t A t ) , etc, for the first time level ( t = A t ) for every axial position, starting a t i = 1 and ending at i = N, where N is the number of axial increments (i = 0 is the bed inlet). The sequence of computations at each axial position is as follows: (1) solve eq 4’ for q(i,t + A t ) ; (2) solve eq 5’ for T,(i,t A t ) and eq 2‘ for c(i,t A t ) ; ( 3 ) solve eq 3’ for T,(i,t A t ) . When done in this order, all quantities which one needs to know are always in hand. After calculating values for every i at the current time level, one repeats the whole procedure for additional time levels as many times as desired.
+
+ +
+
Nomenclature a = particle specific surface area (area per unit volume of particle) c = fluid phase solute concentration C, = heat capacity h = overall heat transfer coefficient AH = heat of adsorption K = coefficient in Langmuir isotherm, eq 6 K , = overall mass transfer coefficient L = length of adsorbent bed M = molecular weight of the adsorbate MTZ = abbreviation for “mass transfer zone” q = solid phase solute concentration q* = value of q corresponding to equilibrium with fluid phase solute content R = gasconstant t = time T = absolute temperature z = axial distance in adsorbent bed
Greek Symbols = void fraction p = density Subscripts 0 = value corresponding to bed inlet g = fluid phase (gas) value s = solidphasevalue Superscripts + = dimensionless variable ‘ = concentration in mass per volume units per unit mass - =-_ value value per mole A
and each distance derivative by a simple backward finite difference, e.g. ac _ -
c ( i , t ) - c(i
az
-
Literature Cited
1, t )
A2
in eq 2-5 leads to the following, after rearrangement
~ ( it , + A t ) = ~ ( it ),
+
( e A t / h z ) [ ~ (-i 1 , t )
~ ( it)I, + A[q(i,t ) Tg(i9t
+
A t ) = T g ( i ,t )
At)]
+
+
q(i,t
+
+
- T g ( i ,t ) ]
D [ q ( i ,t ) - q ( i , t
A t ) = EAt[q”(i,t )
(2’)
At)]
B [ T , ( i , t) - T s ( i ,t
( r A t / A z ) [ T , ( i - 1, t )
+ q(i, t
+
-
-
+
At)]
(3’)
- q ( i , t ) ] + q ( i , t ) (4’)
T s ( i ,t + A t ) = F A t [ T , ( i , t ) - T s ( i ,t ) ]
T s ( i ,t ) + G [ q ( i ,t ) - q ( i , t
+
+
At)]
(5’)
where A = b s ( l - t ) / & t ; B = C,,A/c,,; D = ARA/c,,; E = Kt,a/bs; F = h a / j s C U s and ; G = AR/C,,. From the
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Received f o r reuiew .January 21, 1974 Accepted J u n e 10, 1974
Ind. Eng. Chem., Process Des. Develop., Voi. 13, No. 4 , 1 9 7 4
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