A Multiple-Layer Cross-Flow Configuration for Preparative Chromatography of Multicomponent Mixtures Eduardo Wolf
’ and Theodore Vermeulen
Department of Chemical Engineering, University of California, Berkeley, California 94720
Chromatography has proved to be a powerful analytical technique for the separation of multicomponent mixtures. Scale-up of gas chromatography has been described, but low throughput is a severe limitation to commercial operation. The new design proposed involved two modifications of the conventional design. First, instead of eluting all components all the way through the column, cross-flow streams of carrier gas will be introduced at appropriate intermediate points in the column. These streams will flow perpendicular to the bed axis and will elute the products as soon as they have been separated. Secondly, cross flow will allow the use of a multiple layer bed that can retain each component selectively in a separate layer. The performance of the new design is compared with that of a single bed by using a relation between number of plates and feed volume. The calculations show that the throughput of the proposed design is from 2 to 30 times as large as an equivalent single bed unit, depending on the available selectivities for the layered bed and for the reference conventional bed. An example is presented for the separation of fluorohydrocarbons using a Porapak multiple layer bed. The results show that the throughput can be increased about 17 times over the best single bed. The new design can also be applied to other fixed-bed multicomponent separations.
Introduction In preparative chromatography, maximizing the throughput of the column is a primary design objective. Multiple feed cycling and automated fraction collection with recycle of overlapped fractions is one combination used to maximize throughput. In this paper a new design of a multiple layer bed, utilizing cross-flow elution, is presented. The theoretical throughput for a given bed length in the new design will tend to be much larger, in some cases up to 30 times, than for a conventional unit of the same bed length. Three basic characteristics have made GLC a powerful analytical technique and an attractive candidate for use on a preparative scale. First, separation factors attained in GLC are usually larger than those in other separation processes; second, GLC columns exhibit a large number of theoretical plates ( N ) per unit length; third, obvious, but important, multicomponent mixtures can be separated in a single column. However until now, preparative-scale GLC has had limited success, in contrast with analytical applications, because column efficiency (measured by number of plates) decreases as column diameter and feed volume per injection are increased (Baddour, 1966; Ryan, 1968; Timmins, 1964). The consequent low throughput is a severe limitation to commercial operation of large-scale units. Baddour has proposed that the use of special baffles partially solves the problem of loss in efficiency when large-diameter columns are used. In the analysis of column throughput, it is assumed that use of suitable baffles allows column efficiency to be maintained when the diameter is scaled up. Compared with an analytical column, a preparative unit should be equipped for multiple feed cycling, product collection, and carrier gas recycle. The cycling time, t,, must be adjusted to obtain a controlled overlap (depending on specified product purity) between the faster-moving component of the new injection and the slower-moving component of the previous one. For a given flow rate, the column length is determined by the two components that are most difficult to separate (key components). Therefore, nonkey components Address correspondence to this author at the Department of Chemical Engineering, University of Notre Dame, Notre Dame, Ind. 46556.
are fully separated from key components before they reach the column outlet, but this separation increases the cycle time. The objective of the new design is to increase productivity in terms of weight of material separated, both by increasing the volume of each charge and by decreasing the cycle time. A schematic diagram of the proposed design is shown in Figure 1. The design consists of a column, a feed injection system, product recovery units, and a carri-r-gas recycle unit. A four-component mixture is used as an example of a multicomponent feed, having two key components 2 and 3, and two nonkey components 1 and 4. (Here, the components are numbered in the order of increasing affinity for the stationary phase.) After the feed is injected a t the column inlet, the carrier gas elutes the mixture down the column. The first departure from conventional design consist of a multiple-layer bed in which the layers are selective for certain components. Typically, only one component is retained in each layer. Thus, only component four will be adsorbed in bed four, and so on. In practice, all components can adsorb in each bed, but the bed sequence can be arranged so that maximum separation factors are obtained for two components. As the second departure, once all components have been separated, the main carrier gas flow will be suspended temporarily, and side streams of carrier gas flowing perpendicular to the column axis (that is, in cross flow) will be introduced a t particular points in the column. These cross-flow streams elute the components that have already been separated in the column, and there after a new feed cycle begins.
Determining Factors in Throughput of a GC Unit The production rate of a GC unit is given by
Q =
amount of (solutes fed per injection
)
number of
x (injections per unit time
)
(1
The first factor of eq 1 depends on the feed volume and concentration. The second depends on the elution mode. They will be discussed separately. ( i ) Number of Plates Required in Relation to Feed Volume. As the feed volume increases, the resolution ( R )or Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4 , 1976
485
Carrier Gas Recycle System
r
Product Recovery
Cross F l o w Streams
5
Figure 1. Cross-flow multiple layer bed.
Figure 3. Number of plates required for overload elution.
portional to feed volume. (This classification is different than Conder and Purnell’s (1970).) We also assume that peak shapes are Gaussian under elution operation. Consequently the number of plates required for elution is given by a + l 2
N = 16R2(-)a - 1
l+k
(T)
2
=
(F)
(4)
where
W i d t h o f Elution C u r v e Versus N , / G
tR, =
Figure 2. Results from plate theory for large feed volumes.
separation between bands decreases for a given column operating a t a fixed elution rate. The effect of feed-band volume on peak broadening has been analyzed by Van Deemter et al. (1956), Glueckauf (1964),and Purnell and Conder (1970). The relation between peak width and feed volume obtained by Van Deemter is shown in Figure 2. The ordinate is a dimensionless peak width expressed as the band width wi a t the column outlet in cubic centimeters divided by the product of the volume of one theoretical plate and v%. The abscissa is the feed volume in cubic centimeters divided by the plate volume and by v%. The exact solution (solid line) can be approximated by the two asymptotes W,
+
= 4 V ~ ( l ki)TN
N f < 1.5
m
(2)
and
Ind. Eng. Chem., Process Des. Dev., Vol. 1 5 , No. 4, 1976
L + h,) = (1 + k i ) uo
(7)
The term a in eq 4 and 5 is related to the separating ability of the column. cy is the separation factor, t ~the, retention time of component i, L is the column length, G is the carrier gas flow rate, UOis the interstitial velocity, t is the bed porosity, and A is the cross-sectional area. W, is the band width in min
( W , = w,/G).
In the overload mode of operation ( N J m > 1.5) peak shapes are no longer Gaussian. The feed band volume is described by eq 3 and the retention time is given by
Hence, the number of plates required on the overload mode can be obtained by combining eq 3 with eq 5 , 6 , 8 , and 9 to give
N=
The range of applicability of each equation has been moved to the intersection of the two asymptotes (about 1.5). The range of applicability chosen by Van Deemter was N f / m < 0.5 for eq 2 and N f I m > 3 for eq 3. In the above equations VG is the volume of the mobile phase in one theoretical plate, ki = K ~ V L / V is G the capacity coefficient, Ki is the partition coefficient, VL the volume of the liquid phase is one plate, and wi is the peak width measured a t the point where the tangents of the inflection points intersect the base line. Equation 2 represents the case where band width is almost independent of feed volume. Hereafter we call elution mode the operation for which Nf/./IIT < 1.5 and overload elution the operation for which N f / m > 1.5 where band width is pro486
NVG(1 + k i ) L A € = -(1 G G
2T
(10)
+
where X = N f / N and N f = F/VG(l k ) . Equation 10 is similar to the one given by Conder and Purnell. However, eq 10 can be applied to the two modes of operation designated by Purnell as overload and eluto-frontal. I t also includes the resolution R as a parameter, whereas Purnell’s expression was derived for a fixed value of R = 1.5. The ratio of number of plates required for overload elution No and for elution operation N e ,given respectively by eq 10 and 4, is plotted in Figure 3 vs. the quantity XRIa. This quantity combines the information on production per unit bed volume, X the purity of the products R and the separation capacity of the column a. As shown in Figure 3 the asymptotic solution shows no effect on number of plates with feed volume
LNK
~~
LK
HK
HNK
~~
I
I
IC
-L Concn Inside Column
Concn a t Column End
Figure 4. Profile for single bed.
Figure 5. Profile for cross-flow single bed.
for ARIa < 0.37 (corresponding to N J m < 1.5). Beyond this point the number of plates required to obtain a given purity increases rapidly as X increases keeping all other variables constant. As ARIa approaches 0.6, the overload mode requires twice the column length as at the limit of elution (AR/a = 0.37) while production increases only by a ratio of about 1.6. I t is clear that a t this condition the production can be doubled by using two columns of length L instead one column of length 2L. Consequently there is an optimum for ARIa which is determined by economic factors (Conder, 1975). This optimum is approximated in subsequent computations by choosing ARIa = 0.4 as an operating point. The exact solution obtained from numerical computation of Van Deemter’s solution is also shown in Figure 3. The asymptotic solutions become coincident with the exact solution as XRIa diverges from 0.37. The previous discussion indicates the limitations in feed volume existing in chromatographic separations. A simple way to increase X without increasing N is to keep XRIa constant. This can be done by increasing e. Selective bed allows use of the best value of cy available for the separation of two adjacent components. Frequently, the best bed for separating one pair does not give the best separation factor for separating other components. On the other hand, use of selective beds implies larger retention times and therefore an increase in t,. Combining cross-flow elution with a multiple bed allows independent control on these two factors. (ii) Effect of Cross-Flow on Throughput. Once a relation between N and F has been established, it is possible to relate F with the amount of mixed solutes fed in each injection. Under ideal gas conditions
maximum. The cycling time is given by
weight soluteshjection =
4
4
1
1
C mi = F C p,”, F P T RT
(11)
For the case of a multicomponent mixture, the column length is determined by the pair of components that have a separation factor closest to unity (key components). Therefore, X will be determined for the key components
F
=
+
N f v ~ ( 1h ~ )
(12)
Substituting the plate volume by eq 7 gives
F
+k
= h~(AcL)(l
~ )
Considering from equation 2 that tain
w,= 4 t ~ , / V % , we ob-
or for simplicity tc = f ( t R 4 - tR1)
(17)
where f will be an adjustable parameter ranging from 1 to 1.6 for the case of a binary mixture and base-line separation. For most cases where N is large (100-lOOO), f will be smaller than 1.2. Then the throughput (weightitime) is (18)
Case 2: Single Bed with Intermittent Cross Flow (SBCF). Concentration profiles inside the column at the moment when cross-flowelution will start are shown in Figure 5. The fastest moving component (LNK) of the latter charge has overtaken the slowest moving component (HNK) of the preceding charge, at a point x inside the column at the time where separation between the key components has been achieved at the column end. The time elapsed between injections will be given by tc =f(tx4
- t x i ) + tCF
(19)
where t,, is the time required for component i to travel the distance x; t c is~the time required to carry out the cross-flow elution and f has the same definition as in eq 17. It will be assumed that tCF will equal the time necessary for component 4 to travel a column diameter D at the flow rate selected for this step. Furthermore, it will be assumed that the cross-flow design does not decrease column efficiency. Then throughput will be given by
(13)
and total throughput (weightitime) then is given by
where t, is the elapsed time between successive injections.
Throughput Calculations Four cases will be analyzed by comparing throughput of beds having same length: first, throughput of a single bed operating in a conventional mode; second, a single bed in cross flow; third, a multiple layer bed without cross flow, and fourth, a multiple layer bed in cross-flow operation. Throughput equations will be derived for the case of a four-component mixture with a light nonkey (LNK), a light key (LK), a heavy key (HK), and a heavy nonkey (HNK). Case 1: Conventional Single Bed (SB). The effluent of a single-bed column is shown in Figure 4. For this case, it is assumed that the operation is carried out at XRIa = 0.4, i.e., in the region where the ratio feed volumelbed volume is a
where D is the column diameter and Uo’ is the interstitial velocity used in the cross-flow elution. For the case that t , ~ > 20 then QSBCF= a 4 3 Q ~Therefore, ~. for this case, cross-flow operation increases throughput by a factor approximately equal to ~ 4 3 . Case 3 Multiple Layer Bed without Cross Flow (MLB). A separation column consisting of selective beds can be used as a means to obtain larger separation factors. However, in general, the heavy components will be more strongly held when passing through a bed which is more selective for the light components. This effect will result in longer cycling times, which will decrease the throughput. Concentration profiles in an MLB are shown in Figure 6. Bed 4 will separate selectively component 4 from 3,2, and 1. Bed 3 will separate component 3 from 2 and 1.Finally, component 2 will be separated from 1in bed 2. It is assumed that no significant separation occurs between the nonkey compoInd. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
487
Table I"
+X+
Retention times, min '
14
L3 ' Concn lnstde Column
Lz
Column type
Figure 6. Profile for multiple layer bed (cross flow and through flow). Most difficult separation occurs in bed 3. Significant separation occurs only between selectively adsorbed components.
Components
P
QS
R
3.85# 3.96 3.33 5.6 6.99 5.71# 1.41 14.03 11.71 2.02 23.35 19.14 Dichlorotetrafluoroethane 10.43 1mt X 2.3 mm i.d.; T = 30 O C ; flow rate = 25 cm3/min. CRC Handbook, 1972. Difluoroethane Chlorodifluoromethane Dichlorodifluoroethane Chlorodifluoroethane
1.15
A B C D E
,
1.21#
Table 11. Separation Factors for Adjacent Components
Keys E-D D-C C-B B-A
P
QS
R
1.05
1.15 1.66 2.014 2.099 0.88 (1.15)
1.019 1.63 2.05 1.019
5.16
1.43 1.16 1.05
1.41
Optimum bed sequence P-R/QS-R Bed number 4-3-2
("324
Figure 7. Throughput ratio vs. separation factor cated by a dagger k,j
j/i 4 3
4 60 60 80
2
3 12 40
2 10 40/a 24
28
keys
R = 1.5
4-3 3-2
tCF - 0.1 G -
1
5 33/a 8
t).Hence (24)
Then the feed band volume can be calculated by eq 11, which describes the overload operation
2-1
-
nents in each bed. The bed sequence was chosen so that no reversal in separation factor (i.e., no change beyond unity) occurs when passing from one bed to the other. Each bed will have its own pair of key components and the most difficult separation (the one with lowest CY)will occur in bed 3. Other bed sequences can be chosen depending on the particular case being analyzed. In some cases it might be convenient to change the bed order; however, the sequence presented here ensures that no mixing will occur of components already separated. Throughput will be described by equations similar to case 1, except that the total retention time of i will be the sum of the retention times for each bed j and will be characterized by a capacity coefficient k;j. The throughput is given by QMLB
= x3L3(1 + k33)AtPT/tc
tc =
f ( t R 4 - tR1)
(21)
+ tCF
(22)
or QMLB
N4 (k44
- k14)
(G(1 + k33)
+
hAtP~Uo/f (k43 - k i d 3 (k42 - h4i) I (1+ k33) N3 (1 + k33) +
~ C F
~ R J
(23) The MLB will require fewer plates than the best single bed. Hence, to compare throughput on the basis of beds having the same number of plates ( N ~ =BNMLB)the MLB can operate in the overload mode. T o simplify let us call M the ratio of total plates to those required to operate a t XR/a = 0.4 (indi488
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
Then, combining eq 23 with eq 26, we obtain
1 6 4
(k44 - h14) (1+ hxd
+
(k43 - hi31 (1 + h33)
+
( h 4 2 - k4i) Ngt (1 + k33)
~
~ C F
~ R J
(27) Results are expressed in terms of the ratio QMLB/QSB. Calculations were made by using the values of k,, given in Figure 7 . N S Bwas calculated for the two values of ( ~ ~ 3 2in) s dicated. ( a 3 2 ) 3 was taken as a variable parameter corresponding to beds having different separation factors. N, was calculated from eq 4 for one of the two key components (redefined for each layer) for which that layer is more selective. a3 was calculated from eq 5 for the appropriate value of (a&. The results, shown in Figure 7, indicate that a MLB is advantageous whenever ( ~ ~ 2 3can ) s be improved significantly over CYSB.However, for small improvements in CY,in difficult separations, the MLB gives equivalent or lower throughput than the single bed due to the increase in cycle time. Case 4: Cross Flow in a Multiple-Layer Bed (MLCF). This case is the superposition of cases 2 and 3, on which full advantage of both systems is attained. Concentration profiles inside a MLCF are shown in Figure 6. The same bed sequence described for the MLB is used. Cross flow will start after separation of all components have been achieved (as shown
~
Cross Flow Streams
Table 111. Throughput of a Porapak Layer Bed Equivalent to a Porapak QS Monolayer Beda Single bed
Layered bed
QS
Bed 1.15
Olk
N,+ 2.89 x 0.031
P
R
5.16
2.05
2.099
135 0.354 0.054AcU o p
56
0.0031AcUop = 17.46.
Q
QS
R
1.41 435
a QMLCF/QSB
t in Figure 6) and component 1reaches component 4 without overlap a t some point x in bed 3. Throughput will be given by QMLCF
tc =f(tx4
- t x l ) + tCF
(29)
or rearranging
:i
Figure 8. Schematic diagram of a cross-flow section.
= x3L3(1 + ~ ~ s ) A ~ P T / ~ c (28)
where
t c = f - (1
To Product Recovery
L3 + k24) + (1 + u 0
k33)
Following the same reasoning as in case 3 (MLB),the MLCF can be operated in the overload mode; then eq 26 will be valid and combining with eq 28 and 30 we obtain
A plot of the ratio of QMLCFIQSB given by the ratio of eq 31 and 15 is presented in Figure 7 . The calculations were made by using the same numerical values as in case 3. The results show that throughput can be increased as much as 30 times depending on the selectivities that can be attained in the layered bed. Moreover, a significant increase in throughput can be achieved with a small increase in selectivity. Also, the increase in throughput is more important = 1.2). It is aswhen the separation is more difficult ( ( a 3 & j ~ sumed that ( C Y ~ Z ) M L>B (a32)SB corresponds to a case where the key bed in the MLB cannot be used as a single bed because of its inability to separate one or both of the nonkey components. The ratio of QMLCF/QCFis about one-fourth as large as QMLCF/QSB, since cross flow itself increases throughput by that factor. Illustrative Example. The advantages of the MCF can be illustrated by the separation of fluorohydrocarbons in Porapak layers of different polarities. A mixture of these solutes was used for the calculations mainly because appropriate data were available. Table I shows the retention times of five fluorohydrocarbons in three different Porapak beds. When each is used as a single bed, the key component (indicated by a # ) will be A-B for beds P and QS, and B-C for bed R. Retention times were converted to separation factors (by eq 7 and 8)
which are shown in Table 11. For two adjacent components, one can select the bed that has the largest separation factor. Accordingly, the bed sequence P-R-QS-R, having the separation factors italicized in Table I1 will be optimal. The smallest separation factor of the MLCF corresponds to the pair A-B that will be taken as keys. The separation of B, C, and D will be performed in a bed containing half R and half QS to counteract the effect of reversal in separation factor for the pair A-B in bed QS. The best single bed is taken to be Q-S. The results, summarized in Table 111, indicate that a = 17.4 can be obtained. ratio of QMLCF/QSB
Conclusions A new process arrangement for production-scale gas chromatography has been presented. To evaluate the performance for this arrangement, an improved relation between feed volume and number of plates has been derived from the plate theory. The relations obtained indicate that i t is not advantageous to operate a GC column beyond XRIa = 0.4. Throughput equations were obtained for three cases, assuming that cross flow does not decrease the efficiency. The calculations show that throughput can be increased from 2 to 30 times with respect to a single bed having the same number of plates. An illustrative example shows that in a typical case where selective beds are available-the separation of fluorohydrocarbons in Porapak beds-throughput can be increased about 17 times by using the MLCF. Calculations show a significant potential for equipment development, as well as the need for chemical synthesis of new selective stationary phases for the separation of homologous components. A column having a square section and internal baffles, as shown in Figure 8, seems to be the most appropriate one to carry out the cross-flow configuration. The aim of the internal design should be to maintain column efficiency when cross flow is used. Nomenclature a = separation parameter as defined by eq 5 A = column cross-sectional area, cm* D = column diameter, cm F = feed volume, cm3 f = width correction in elapsed time G = carrier gas flow rate, cm"min K , = partition coefficient; solute liquid phase/solute in gas phase h, = capacity coefficient of component i k,, = capacity coefficient of component i in bed j L = column length, cm m, = mass of solute ihnjection, g N = number of theoretical plates Nf= number of theoretical plates Q = throughput, g/min Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
489
R = resolution, as defined by eq 6 tR, = retention time of component i, min t , = time elapsed between injections, feed cycling time, min
VL = VG = UO = W, =
liquid phase volume in one plate, cm3 mobile phase volume in one plate, cm3 interstitial velocity, cm/min peakwidth,min
Greek Letters separation factor X = Nf/N PT = total solute concentration in feed volume, g/cm3 e = bed porosity q j =
Subindexes i = componentnumber j = bednumber K = key component
Literature Cited Baddour, R. F.,U S . Patent 3 250 058 (1966). "CRC Handbook of Chromatography", CRC Press, Cleveland, Ohio, 1972. Conder, J. R., Chromatographia, 8 , 60 (1975). Glueckauf, E., Trans. Faraday Soc., 51, 729 (1964). Purnell, J. H., Conder, J. R., Chem. Eng. Sci., 25, 353 (1970). Ryan, J. M., Timmins, R. S..O'Donnell, J. F., Chem. Eng. Prog., 54,8 (1968). Timmins, R. S..Mir, L., Ryan, J. M.. Chem. Eng., 170 (May 19, 1969). Van Deemter, J. J., Zuiderweg, F. J., Klinkenberg, A,. Chem. Eng. Sci., 5 , 271 (1956).
Received for review June 3,'1975 Accepted M a y 21,1976
Mechanisms of Coal Particle Dissolution J. Guin, A. Tarrer;
L. Taylor, Jr., J. Prather, and S. Green, Jr.
Department of Chemical Engineering, Auburn University, Auburn, Alabama 36830
The rate of dissolution of coal in vehicle solvent is of primary importance in coal conversion processes. In this study, dissolution of individual coal particles is followed using photomicroscopy in a sequence of batch experiments. At 350 'C, coal particles are observed to disintegrate into smaller units. Effects of temperature, gas phase composition, and solvent hydrogen donor activity on the extent of particle breakup are studied. Providing sufficient hydrogen is available, the process of disintegration is observed to occur very rapidly, with the qualitative appearance of a fluid-solid surface reaction having a high activation energy. Implications of this evidence are discussed and interpreted in light of previous investigationsof coal dissolution.
Introduction
A study of proposed processes for the production of liquids from coal reveals that a large number begin with the dispersion of solid coal particles in a carrier oil vehicle, usually derived from the coal itself. Subsequently, these processes follow various routes depending upon the presence or absence of an external catalyst and whether the liquefaction occurs in the presence or absence of a hydrogen gas phase. A review of progress in several coal conversion processes has been recently presented (Chem.Eng. B o g . , 1975). In the catalytic processes, the coal particles must presumably dissolve before any interaction with the catalyst surface would be possible. The initial slurry-forming step with subsequent dissolution of coal particles is common, moreover, to most noncatalytic processes, such as the promising solvent refined coal (SRC) process. Thus the rate of solid to liquid transformation is indeed important in coal conversion processes. Because a primary objective of most coal liquefaction processes revolves around sulfur and mineral matter removal, considerable attention has been focused upon the hydrodesulfurization and hydrogenation of coal-vehicle slurries. These investigations have employed primarily techniques of homogeneous reaction kinetics (Liebenberg and Potgieter, 1973; Wiser, 1968) although since the initial reaction mixture is a slurry, it would seem that the initial conversion kinetics might well be governed by heterogeneous mechanisms and, thus, fluid-particle interactions should a t least be examined before being considered negligible. Previous Studies. At the present time, it appears that very 490
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
few detailed examinations have been made of the initial stages of coal particle dissolution in vehicle solvents with emphasis on elucidating the particle-fluid interactions, for most studies have been directed toward ascertaining the final amount of coal converted under some given set of conditions. The highly porous structure of coal suggests that intraparticle mass transfer and diffusion could play a vital role in its dissolution. Likewise, external fluid-particle transport phenomena cannot be ruled negligible a priori. If these effects are appreciable, the size of the coal particles should be an important operational variable; for the larger the size of the particles, the longer the diffusional path and the smaller the ratio of external to internal surface area. Typically, however, as reported by Curran et al. (1967) and Kloepper et al. (1965), the rate of dissolution has been observed to be independent of particle size. On the other hand, investigations by Ashbury (1934) and Jenney (1949) showed particle size to be a variable of some significance. Thus, some clarification in this area would seem desirable. Hill et al. (1966) have applied Eyring absolute reaction rate theory to the kinetics of the dissolution of bituminous coal in tetralin. Average heats of activation and apparent entropies of activation are presented, and marked changes are shown to occur in the entropy of activation as dissolution proceeds. This is interpreted to imply that the fraction of reaction sites available for reaction increases drastically during dissolution. The exact manner and rate of particle breakup to form these additional reaction sites, however, was not clearly established. Along somewhat more theoretical lines, several plausible
