Nonisothermal Behavior of Fixed-Bed Adsorbers A Mechanistic Explanation Friedrich Helfferich Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802
Physical chemistry and chemical engineering are full of phenomena whose mathematics have long been worked out, but whose physical mechanisms are neither immediately apparent nor explained in a simple manner in standard texts. T o find such explanations is a challenge to the educator, for a convincing mechanistic picture is retained more easily by the student than are mathematical formulas or statements of facts. One such challenge is being picked up here: to explain in simple, nonmathematical terms the behavior of fixed-bed adsorbers under nonisothermal conditions. Actually, more than that is being attempted, namely, to introduce the student in a more general manner to certain typical facets of propagation phenomena in systems with more than one dependent variable. Fixed-bed adsorption columns are used for many purposes in the laboratory and on a commercial scale, especially for removal of components from fluids and chromatographic separations. In small laboratory columns, in adsorption from liauids (with higher heat cavacitv), . . and if the heat of adsorption is low, temperature vnrialions usually remain unimr~onant.This is often not so in large heds, adior~rionfrom gases, and with high heats of adsorpiion. under s&h conditions, temperature rise can even become a limiting factor in design. Commercial-scale drying of gases on alumina or molecular sieves is a typical example. The effects of such temperature excursions will be examined here. Under isothermal conditions the behavior of the adsorption column is simplicity itself, a t least as long as we are not concerned with finer details. Imagine a fluid containing a sorbable solute at constant composition being passed through an adsorbent bed initially free of sorbate (see Fig. 1)and assume isothermal behavior. The solution saturates the first layers of the bed and, in doing so, is s t r i ~ o e dof solute. Additional solution Dasses through the saturated (equilibrated) layers without ihange in COGDosition and saturates further lavers farther downstream, and Ho on. At any time the bed can be viewed as consisting of two distinct zones, one of saturated and one of still unsaturated adsorbent, with a more or less sharp transition in between. This transition traveling through the bed may be called an "adsorption wave." The velocity of the adsorption wave must be less than that of the fluid because adsorption keeps removing solute from
646
Journal of Chemical Education
the latter, converting solution to solute-free fluid and thus eroding the advancing front of solution (see Fig. 2). A simple material balance consideration for the solute in a layer Az through which the wave passes. uSCAt - 0 = (C + C)S& (1) inflow outflow accumulation yields, on rearrangement, the velocity of the adsorption wave: uada=
AdAt
= uC/(C
+ C)
(2)
(see list of symbols at end of paper). The retardation, relative to the fluid velocity u, by the factor C / ( C + C ) reflects the stripping of solute from the solution to saturate the adsorbent. Essentially equivalent derivations are given in almost any elementary chromatography texts ( I ) . As an aside, in eqns. (1)and (2) we have used concentrations per unit volume of packed bed in order to obtain the mathematical relations in their simplest form. Our concentrations
Solute-Free out
+,I",
t
-
Solute Concentmtim
Solution in Figure 1. isothermal operation of fixed-bed adsorption column. Left physical system: right: solute concenhation profile.
Temperature
Solute Concentmtim ~n F l u ~ d
-
Solute Concentration on Msorbent
Wove Faster Than Adsorption Wove
+
Figure 2.Retardation of adsorption wave relative to fiuid flow through removal of solute from fluid by adsorption. (Material balance requires shaded areas in both diagrams to be equal.)
Amount of Fluid Entered
-
Solute Concentmtim-
Figure 3. Nonis&rmal operations with fan temperature wave. Left: b'ajajectaies of waves and heat in diagram of distance versus amount of fluid entered; right: soiute concentration and temperature profiles
Symbols used in Equations
c
C C
9
S t
v z 6
p
from an enthalpy balance strictly analogous to the material balance (1):
concentration of solute in fiuid phase, per unit voiume of fiuid phase concentration ol soiute in fiuid phase, per unit volumie of bed concentration of salute on adsorbent, per unit volume of bed heat capacity of fluid, per unit voiume of bed heat capacity of adsarbent (plus adsorbate, if any) per unit voiume of bed heat capacity of column wail, per unit volume of bed concentration of soiute on adsorbent, per unit weight of adsorbent cross-sectional area 01 bed time velocity of fiuid in bed distance fmm inlet end of bed fractional void voiume of bed density of adsorbent
(4) + iS, + F,) The retardation by the factor C , / ( C , + rp + (T,) reflects the Ute,,
ronsumntion of heat from the fluid for heatina of adsorhent and w d . As in the case of the adsorption wave, tbe mechanism of retardation is the erosion of the advancing front, here by removal of heat from the fluid; Figure 2 applies if the concentration scales are replaced by temperature scales and "heat exchange" is suhstituted for "adsorption!' [Note that the heat capacities in eqn.'(4) refer to unit volume of bed and are related t o the respective specific heat capacities by equations corresponding to eqns. (3).]
C and C ner unit bed volume are related to the fluid-nhase concentration c per unit volume of fluid phase, and adsorbed concentration q per unit weight of adsorbent, by C = re, C = (1 - r ) p q
= UC,/(C,
(3)
The Problem: Nonlsothermal Adsorption
If we drop the premise that, despite release of heat of adsorption, temperature variations will remain negligible, we have t o answer the questions: (1) What happens to the released heat? (2) Which part of the column is heing heated?, and (3)How does this affect adsorption? Mathematical derivations of the effect haw been &en by various authors inchtding Short, Smith, and l'wigg (21: Hhee. Heerdt, and Amundsm 131. Ruthven. Care. and Crawforrl t4).nnd Vermeulen. Klein, and Hiester '(5). Here, a simple kiplanation of the mechanism will he attemnted. For simolicitv. the areument will be presentedfor the abiahatic case,and &at tranifer will be assumed to occur onlv hv convection of fluid and heat exchange between fluid, adsorbent, and column wall. A Simpler Case: Pure Temperature Wave
First, let us examine a simpler problem, that of a temperature wave traveling through a bed in which no adsorption or desorption occurs. Assume that a fluid of higher temperature enters a bed initially a t amhient temperature. The fluid has to heat the adsorhent and column wall to the higher temperature and, in doing so, is itself cooled to the lower temperature. The process is exactly analogous to that of the solution saturating an adsorhent and heing stripped of the sorhahle solute. At any time the bed consists of two zones, of higher and ambient temperatures, separated by a traveling "temperature wave," that is, a temperature variation advancing in the direction of flow. The velocity of that wave can be calculated
Combinations of Adsorption and Temperature Waves We can now attemnt to niece toeether the behavior of an adsorption column uider nbnisoth&mal conditions. We shall do s1, for a bed. initiallv free of sorbate and at ambient temperature, recei;ing a sb~utionof constant composition, ambient temperature, and containing a solute which is adsorbed with positive heat of adsorption. (The behavior in other cases is easilv deduced once the mechanism is understood.) In esienre, there will again be an adsorption wave, but this wave will now act a s a travelingsource 1,t heat, releasing heat of adsorption in each layer of bed as that layer is saturated. The heat released will, in general, travel at a velocity different from that of the adsorption wave-there is no physical reason that the velocities of temperature and adsorption waves should he the same. Therefore, two cases can be distinguished: the temperature wave may he faster or slower than the adsorpt~nonwave. For lar e scale gas-phase operations for which C Cp/iTp
(5)
If the temperature uraue is faster, the heat released by the adsorption wave travels faster than the latter, leaving it behind. All the released heat, therefore, is contained in a zone ahead of the adsorption wave (see Fig. 3). The front of this zone, entailing no adsorption, is a pure temperature wave with velocity according to eqn. (4) (there can he no solute ahead of the adsorption wave). The rear of the hot zone coincides with the adsorption wave. This wave thus entails both adsorption and temperature variation. However, adsorption remains unaffected because the only place where the temperature is different is ahead of the wave, where no solute is present anvwav. The retardation of the wave relative to fluid flow. caisedby the requirement to saturate the adsorhent, remains unaffected by the temperature variation. (In the material Volume 59
Number 8
August 1982
647
T m p m t m Wave S i a c r Than Adsorption Wove Tmperot~~
-
the temperature of the "hot zone" must be established. Procedures applicable for such purposes can he found in earlier publications (3-5). Here, only one striking facet shall he emphasized. The temperature in the "hot zone," whether ahead or behind the main adsorption wave, depends on the size of that zone. Under otherwise comparable conditions (i.e., same heat release, same heat capacities), the narrower the zone in which the heat is stored, the greater is the temperature rise caused hv that heat. Accordinelv. .. .. the hiehest temoeratures may he expected if the velwitiesuf the a&orptiun &r leqn. C ~ J and I temperature wavr .Iron. . (4j1 are verv similar. In fact. if these tu,o ;.t.lucitiw are equal. an infinite iemperature risi would be calculated unless he:11conduction is t3ken inta~arcount as an additional heat transfer mechanism [as was pointed out as early as 1954 by Short et al. (2)]. To he sure, the arguments presented here are not a conclusive "derivation" and have glossed over details. They are merely a rationalization, intended to help us understand the mechanism of a behavior which mathematics predicts and observation confirms. ~
Amount at Fluid Entand
-
SOiYle Cancsntmtim-
Figure 4. Nonisothermal operation wRh slow temperature wave. Diagrams correspond to those in Figure 3. halance and velocity eqns. (1)and (3, the values of C and C remain unchanged, and the temperature is ambient aeain behind the wave.) If the temperature waue is slower than the adsomtion wave. the hehavio; is more complex (see Fig. 4). Here, the released heat of adsorption is left behind by the adsorption wave. The "saturated" adsorbent behind the wave is now a t higher temperature and thus contains less solute than a t ambient temberature. (If the heat of adsorption is positive, as presumed, an increase in temperature decreases adsorption.) Accordingly. the wave will travel faster, namely, at the velucity of an isuthrrmal adsorption wavr at the higher temperature. Turning" to the rear of the "hut zone" we mieht " at first exoect that wave to travel at the velocity of a pure temperature wave; it would do so if no adsorotion or desor~tionwould occur a t that wave. However, sinEe the tempeiature drops hack to ambient at that wave. additional adsorotion occurs. and the adsorbent now hecomes saturated at ahhient trmperature. This additional adsorotion is arrom~aniedbv release of heat. This heat, generatedit the rear of the "hot"zone" travels a t the temperature velocity, that is, a t the same velocity as all the heat released earlier; therefore, it is neither carried forward nor left behind hut, rather, keeps the layer temporarily from cooling and thereby retards the wave. The "hot zone" can in fact he viewed as consisting of two portions: one portion. hetween the first adsorptionkave and a plane traveling a t the pure temperature velocity, contains the heat released hv the first qdsoiption wave; the other portion, between that plane and the second wave, contains the heat released hy the latter wave (see Fig. 4). A comparison of the two cases brings out a significant difference. In the first, with faster temperature wave, the temperature variation has little effect on the adsorption hehavior of the bed (the onlv effect heine that the adsorotion wave will be less shaip and breakthrou& of small quadtities of solute ahead of the main front will occur a little earlier). In the second case, with slower temperature wave, the adsorption hehavior is profoundly affected; the main adsorption wave is faster than under isothermal conditions and the operating capacity of the bed therefore is significantly decreased. For a calculation of adsorption hehavior in the second case
648
Journal of Chemical Education
~
A Look Bevond
The hehavior of fixed-bed adsorhers under nonisothermal conditions is a typical manifestation of a feature encountered more generally in propagation phenomena in systems with more than one dependent variable. That is. a sinele variation a t a starting point-here, the column i n l e t i s propagated in form not of a single wave, hut of a set of waves traveling a t different velocities. The number of waves in such a set is related to the number of dependent variables (mathematically, to the degrees of freedom of the system). In our example there are two, solute concentration and temperature, and two waves aregenerated. Iftwuiolutes were pre~sent,fvra tc,ral of three dependent variahles, three waves u , o ~ ~he l d ohserved, etc. If the variables interact with one another, as do solute concentrations and temperature, all variables in general vary across each wave (except any variable may be zero on bothsides of a wave). Such hehavior, for example, is an essential feature of multicomponent chromatography (6) and multiphase flow in porous media (7). Nonisothermal adsorption in fixed beds is a manifestation in which the underlying mechanisms is relatively easy to perceive, and is therefore a good example to introduce students to problems of this kind. Literature Cited
.".
(61 Helffetieh, F.,and Klein. G..'"MultieomponcntChromatograpy? Marcel Dekker. New York. 1970. (7) Heifferich. F.'Theory of Multimmponenf MultiplhaacDiiplecementin P o r n Medi*" SPEJ. 21,61 (1981).
