133
Ind. Eng. Chem. Process Des. Dev. 1988,25, 133-139
Simplification of Multicomponent Fixed-Bed Adsorption Calculations by Species Grouping Seshadri Ramaswaml and Chi Tien' Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, New York 132 10
Simplification of multicomponent fixed-bed adsorption calculations can be achieved by combining adsorbates with similar adsorption affinities into pseudospecies, thus reducing the number of components considered. This study proposes a new criterion which stipulates the formation of a pseudospecies based on (a) the similarity of the reciprocal of the Freundlich exponent of adsorbates and (b) the particle-phase mass-transfer coefficients of adsorbates within the same order of magnitude. The validity of the criterion is substantiated through comparisons with sample calculations and experiments.
In recent years, interest in the study of adsorption has resurged, particularly in the area of multicomponent adsorption. As a result of these activities, algorithms have been developed wich can be used to predict the dynamic behavior of adsoprtion processes involving multiple adsorbates (Wang and Tien, 1982; Larson and Tien, 1984). The availability of these algorithms makes it possible to carry out detailed calculations involving solutions with arbitrarily large numbers of adsorbates. In practical applications of adsorption, the number of adsorbates in an adsorption-treated solution can be exceedingly large. Even with the availability of efficient general algorithms, there exists a practical limit to the number of adsorbates which can be considered in a given calculation. The need for procedures which simplify multicomponent adsorption without seriously comprimking computation accuracy is rather obvious. The use of species grouping to simplify multicomponent adsorption calculations has been examined in recent years (Calligaris and Tien, 1982; Mehrotra and Tien, 1984). The basic idea of species grouping is simple and intuitive. A solution with a large number of adsorbates can be considered to be a solution with a smaller number of pseudospecies, each of which is comprised of a number of adsorbates. Calculations are simplified because the number of species is reduced. In the extreme case, species grouping may reduce a multicomponent adsorption probelm to that of single-species adsorption. To implement the idea of species grouping, the following two questions must be resolved: (a) What conditions determine the formation of pseudospecies from a number of adsorbates? (b) How are the relevent adsorption parameters of the pseudospecies determined? The earlier work of Calligaris and Tien (1982) applied species grouping to calculate equilibrium concentrations in multicomponent adsorption. Their procedure combined adsorbates whose single-speciesadsoprtion isotherm data are within the same order of magnitude into a pseudospecies. The average values of the Freundlich constants of the adsorbates are taken to be those of the pseudospecies. Sample calculation results largely substantiate the validity of this criterion. In a more recent work concerning multicomponent fixed-bed adsorption, Mehrotra and Tien (1984) suggested that adsorbates whose single-species isotherm data and particle-phase mass-transfer coefficients were within a factor of 2 could combine to form a pseudospecies. The validity of their criterion was tested against both sample calculations and experiments.
Similarity in single-species isotherm data does not provide a clear-cut basis of forming pseudospecies. Adsorbates which exhibit similar isotherm data over certain concentration ranges may behave differently beyond that concentration range. The requirement that both the thermodynamic and rate parameters be within a factor of 2 is also rather restrictive. A more precise as well as a more relaxed criterion is obviously needed. The present work develops a more general grouping criterion than that proposed by Mehrotra and Tien for fixed-bed multicomponent adsorption calculations. On the basis that the single-species isotherm data of all the adsorbates involved can be expressed by the Freundlich equation, the condition of grouping is taken to be the similarity of the Freundlich exponent of the adsorbates. The Freundlich constants and the mas-transfer coefficient of the pseudospecies are determined on the basis that the rate of adsorption of the pseudospecies equals the sum of the rates of adsorption of the individual adsorbates. As stated earlier, application of species grouping simplifies multicomponent adsorption calculations by reducing the number of species to be considered. The resulting problem in most cases is still that of multicomponent adsorption. Consequently, a brief outline of the equations describing multicomponent fixed-bed adsorption and their solutions are first presented followed by the development of the new species-grouping criteria. The validity of the criteria is then tested in sample calculations and against experimental data. Multicomponent Adsorption in Fixed Beds The dynamics of any adsorption process is defined by its stoichiometry, the adsorption equilibrium relationship of the solution-adsorbent system, and the adsorption rate exppessions. For multicomponent fixed-bed adsorption, these equations are ac1
u-
az
+ Pb-ao
aQ1
=0
841 3 4 =( c , - cs) = kS8
a0
UpP,
(1) (QS*
- 4,)
(2)
(3)
where i = 1, 2, ..., N . The appropriate boundary and initial conditions are qi=o, z>o, 010 (4) ci =
(Ci)in,
2
0196-4305l86f1 125-Ol33$Ol.5OfQ 0 1985 American Chemical Society
= 0,
0I0
(5)
134
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
The general algorithm developed by Wang and Tien (1982) applied the ideal adsorbed solution (IAS) theory to estimate the multicomponent adsorption equilibrium which is assumed to exist at the external surface of the adsorbent particles. The interphase concentrations, qs,and c,,, are related to the solution and average adsorbed-phase concentrations, ci and q,, by the relationship
Freundlich exponent, n. Consider the limiting case in which all the adsorbates have the same n value. The Freundlich coefficient of the pseudospecies can be found from the following consideration. Assuming that the single species isotherms of all the adsorbates under consideration can be represented by the Freundlich expression, the concentration of a solution containing the ith adsorbate only, which has the same spreading pressure, a, as the multicomponent solution at equilibrium, is given as
cio =
( & ) l
(7) i=l
2,
=
The corresponding equilibrium concentration in the adsorbed phase, qio, is
qi + 4 i C i
(&)+E where
a qio = -
ni The total adsorbate concentration in the adsorbed phase in equilibrium with a solution of concentrations cl, c2, ..., C N is given as
Substituting eq 15 into 14, one has The parameters a and S are found from the simultaneous solution of the following pair of algebraic equations
E-
i=l
nlzl
(4
(&, + s
n2z2 ++ ... + nNzN]-l a
Since all the ni's are assumed to be the same and equal to n, the following relationship is obtained qT =
For the above set of equations (eq 1-12), the dependent variables are c,, q,, c,,, and q,,. The finite difference approximations developed from eq 1 and 2 permit the estimations of c, and qi at a given point on the z - 0 plane, (1, m),and from the values of c,, qt, c,,, and qs,at the neighboring points (1 - 1, m), I , m - l),and (1 - 1, m - 1). Once qr and c, are known, the corresponding values of c,, and 9% can be found from eq 6-12. The required input parameters for the algorithm are the bed configuration and operating variables as well as the Freundlich constants (A,and n,) and mass-transfer coefficients (k,,and k,J for each of the adsorbates. When species grouping is applied, it is necessary to assign values to these parameters for each of the pseudospecies. The formulas used to estimate these values are discussed in the following sections. Development of Species-Grouping Criterion Both the studies by Calligaris and Tien (1982) and Mehrotra and Tien (1984) have shown the validity of grouping adsorbates on the basis of their similarity of adsorption affinity. If the single-species isotherm data of an adsorbate can be used as an adsorption affinity indicator, similarities in the magnitude of the Freundlich constants A and n may be construed as similarities in adsorption affinity. As a basic premise of the new species-grouping criterion, it is assumed that the conditions under which adsorbates may be combined to form a pseudospecies are the similarity of the reciprocal of the
(16)
a
The mole fraction of the ith adsorbate in the adsorbed phase, zi, in equilibrium with a solution of concentration ci is given as (Calligaris and Tien, 1982) zi
=
(--) Aini
" i ci
Summing up all the mole fractions, one has
Again, when the condition that all the nl's are the same and equal to n is invoked, the following expression is obtained
(9)'
=1
(20)
Upon rearrangement, one has
The equilibrium relationship of the pseudospecies is assumed to have the form
Accordingly, the Freundlich coefficient of the pseudospecies, A, can be found by equating eq 21 and 22, or
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
135
N
C ni
n = -i=l N To obtain the expressions to calculate kl and k , for the pseudospecies, the equations analogous to eq 1 and 2 with the total adsorbate concentrations ct and qT as the dependent variables are
On the other hand, the summation of eq 2 for all the adsorbates gives
The requirement that eq 25 be consistent with (26) implies N
Ckib(ci- c,) = ki(CT - CsT)
(27)
i=l
N
Cks,(qs&- si) = ks(qs, - qT)
i=l
(28)
The condition of eq 26 is approximately satisfied because the kz's are essentially constant for most of the adsorbates. The satisfaction of eq 27 is unlikely, if not impossible. As an approximation, it is required that the equality condition be met at least initially (i.e., qi = 0), on the grounds that this requirement is important in determining the initial part of the effluent concentration history (breakthrough curve) where the bed was relatively unsaturated. Furthermore, if the particle-phase mass transfer is the ratecontrolling step, eq 28 becomes
where the superscript * denotes the value in equilibrium with the bulk liquid phase. The particle-phase mass-transfer coefficient of the pseudospecies k , therefore can be defined to be N
Cks,qi* k , = -i=l- Ck,,zi N
C qi*
(29b)
i=l
i=l
Combining eq 17, 18, and 29b, one has N
N
i=l
The expression of A and k , (eq 23 and 30) were obtained on the basis that the n values for all the adsorbates are equal in magnitude. In reality, two adsorbates would rarely have identical Freundlich exponents. For practical considerations, the condition of combining adsorbates to form pseudospecies can be relaxed to be that the adsorbates should have approximately the same Freundlich exponent. The n value of the pseudospecies is assumed to be
In addition, the kl of the pseudospecies can be considered
There are two unresolved questions concerning the criteria of species grouping developed above. First, both eq 23 and 30 give average values with c i / C z l c i as the weighting factor (or part of a weighting factor). Except at the inlet of the bed, this information is not available. One is therefore forced to evaluate k, and A of the pseudospecies based at the inlet condition. Second, the grouping criterion stipulates that adsorbates whose n values are sufficiently close may combine to form a pseudospecies. A quantitative definition of the closeness of the n values, which permits grouping, is left unanswered. The resolution of these two questions was made empirically as shown below. Sample Calculations A number of sample calculations were made to implement the species-grouping criterion developed above. In these calculations, total adsorbate concentration histories of effluents were calculated exactly and by applying species grouping. The comparisons between the results according to these two different kinds of computations provide a measure of validation of the criterion. The adsorbates considered in the sample calculation are listed in Table I. Also included in the tables are the Freundlich constants for these adsorbates as well as their mass-transfer coefficients (both liquid and particle phase). These Freundlich constants correspond to the singlespecies adsorption of the adsorbate onto Calgon F-300 granular activated carbon and were reported by Dobbs and Cohen (1980). For the mass-transfer coefficients, kl,'s were estimated by using the expression suggested by Vermeulen et al. (1973) with the bulk-phase diffusivity of the adsorbates estimated by the Wilke-Chang equation. Once the bulk-phase diffusivity of an adsorbate is known, one can estimate the pore diffusivity of the same adsorbate, from which k , can be calculated in the manner suggested by Hsieh et al. (1977). Details of these calculations can be found in Ramaswami's thesis (1984). The 19 compounds listed in Table I can be classified into 4 groups if the closeness of the n values is taken to mean that the maximum difference between the n value of a given compound cannot be more than 10% from the average value of the group. For the sample calculations, four kinds of solutions each of which contained all the adsorbate of a given group were considered. Since all the adsorbates considered for each calculation (ranging from 2 for the first group to 7 for the fourth group) belong to one group, application of species grouping reduced the calculation to that of single-species adsorption. The A , n, kl, and k , values of the pseudospecies formed were calculated according to eq 23,30,31 and 32 and listed in Table I. The adsorption bed was assumed to have a height of 20 cm, packed with Calgon F-300 carbon (average diameter 0.16 cm) and with a liquid superficial velocity of 0.0612 cm/s. The calculated effluent concentration (total adsorbate) histories for the four cases were compared with those from exact calculations as shown in Figures 1-4. The results of the comparison are mixed. For the first case with aqueous solutions of carbon tetrachloride and
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
136
Table I. Adsorbates Used i n Species-Grouping Computationsn
group I I1
I11
compounds carbon tetrachloride cyclohexanone values for pseudospecies composed of group 1 substances bis(2-ch1oroethoxy)methane uracil 5-chlorouracil bis(2-chloroisopropy1) ether values for pseudospecies comprised of above four substances acrolein 1,1,2-trichloroethane values for pseudospecies comprised of above two substances values of pseudospecies comprised of group I1 substances tetrachloroethylene phenol bromoform values for pseudospecies comprised of above three substances 1,l-dichloroethane values of pseudospecies comprised of group I11 substances
DDT
IV
n 1.2048 1.3330 1.3213 1.5385 1.5873 1.7241 1.7544 1.6511 1.5385 1.6667 1.5502 1.6293 1.7857 1.8519 1.9231 1.8523 1.8868 1.8549 2.0000 2.3810 2.2220 2.3000 2.4390 2.2727 2.4390 2.3256 2.3690 2.3465
Ab
4.715 1.970 2.199 1.811 1.920 3.078 2.630 2.492 0.293 0.820 0.3755 2.164 5.360 2.590 1.377 2.787 0.206 2.667 17.101 7.910 7.582 7.856 4.332 2.075 2.913 4.382 3.641 5.413
naphthalene 1,3-dichlorobenzene values for pseudospecies comprised of above three substances dimethyl phthlate toluene 2-chlorophenol nitrobenzene values for pseudospecies comprised of above four substances values of pseudospecies comprised of group IV substances
k,, l / s 1.7 X 10“ 3.3 x io” 3.0 X 10“ 5.0 X 10” 5.1 X lo4 1.8 X 10” 1.9 x 104 2.8 X 10” 1.2 x 10-4 2.1 x 10-5 7.8 x 10-5 3.8 X 10” 5.7 x 10-7 2.1 x 10-6 5.1 X 10” 1.8 x 10” 2.2 x 10-4 1.6 x 10-4 1.3 x 10-7 5.3 x 10-7 7.6 x 10-7 6.2 x 10-7 15.3 x 10-7 14.9 X lo4 6.3 X 10” 2.3 X 10” 3.4 x 106 1.4 X 10“
kl,cm/s 1.6 X 1.5 x 10-3 1.5 X 1.3 X loT3 1.5 X 1.4 X 1.3 x 10-3 1.4 X 1.6 x 10-3 1.6 x 10-3 1.6 x 10-3 1.4 X 1.5 x 10-3 1.7 x 10-3 1.5 X 1.6 x 10-3 1.6 x 10-3 1.6 x 10-3 1.1 x 10-3 1.4 x 10-3 1.4 x 10-3 1.4 x 10-3 1.2 x 10-3 1.5 X 1.5 X 1.4 X lov3 1.4 x 10-3 1.4 X
influent concn, mmol/L 0.01 0.10
0.10 0.10 0.10 0.10
0.10 0.01 0.01 0.10 0.01
0.01 0.001 0.10 0.10 0.10 0.10 0.10 0.10
aFor computations involving Table I1 carbon, F300; bed height, 20 cm; superficial velocity, 0.0612 cm/s. bValues of A based on in units of mmol/L and mol/(g “C), respectively.
2 to \
t
e
w” 0 g os-
06-
5
04r
and q
10-
w“
F
u
’5
c
h
08-
q
cokulattons ____ Exact Specles Gwm (me pseudo-sceesi
i too
OO
xx)
xy)
400
CORRECTED TIME. h f S
Figure 1. Total effluent concentration history of group I.
\wu-
3. Total effluent concentration history of group 111.
L
”-
1.0
10-
? CI
2
08-
h2
06-
u 4 - _ _ Species __ prou6u.q (one preudoqrwpl
e
e5 0 2 P ” 01 0
I
I
I
I
100
200 CORRECTED TME, hrr
300
400
04-
/’ I
100
//’
I
200
I
I
I
573 X?€I 403 CORRECTED TIME, h s
I
I
6tT)
700
I E(C0
Figure 2. Total effluent concentration history of group 11.
Figure 4. Total effluent concentration history of group IV.
cyclohexanone (group I compounds), the agreement displayed in Figure 1 is excellent. The breakthrough curve for the case group I11 compounds calculated from the use
of species grouping agrees reasonably well with the exact calculations (see Figure 3). However, the initial part of the curve (up to 100 h) certainly warrants improvement.
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
137
Table 11. Experimentally Determined Values of A , II and
k
.'
1 2 3 4 5 6 7 8 9 10
A
adsorbate 2,4-dichlorophenol p-nitrophenol p-chlorophenol o-cresol phenol benzoic acid cyclohexanone adipic acid uracil acrolein
147.0 160.3 142.3 109.8 46.6 28.7 9.4 13.3 11.1 0.63
n 4.9432 6.7934 5.5741 4.5956 3.3389 2.3849 2.1110 2.2326 1.9410 1.4417
4.0 x 6.9 x 2.9 x 1.3 x 3.3 x 2.7 x 4.6 x 1.3 x 7.1 x 5.0 x
10-5 10-5 10-5 10-5 10-5 10-5 10-5 10-4 10-5 10-4
'Note: values for A have been calculated based on c and q in units of mg of TOC/L and mg of TOC/g, respectively.
Figure 5. Schematic diagram of experimental apparatus.
For the other two cases as shown in Figures 2 and 4, substantial differences were found between the calculated results based on species grouping and exact calculations. To improve the agreement between the calculated results based on specific grouping and exact calculations, it is necessary to place additional restrictions on parameters other than n. After a number of trial attempts, the following additional conditions were imposed on species grouping. Adsorbates which form a pseudospecies must also have their respective particle-phase mass-transfer coefficient within the same order of magnitude. With these conditions, compounds listed under group I1 should from two pseudospecies; one contains acrolein and 1,1,2,trichloroethane, while the second pseudospecies has the other four adsorbates. Similar groupings were also made for compounds of groups I11 and IV. The calculated effluent concentration histories according to those new grouping procedures gives much better improvement as shown in Figures 1-4. The validity of the new conditions of grouping is further tested in experimental work described below. Experimental Confirmation The experimental work conducted consisted mainly of passing aqueous solutions of multiple adsorbates (up to eight adsorbates) through columns packed with granular activated carbon. The total adsorbate concentration histories of the effluent determined (in terms of mass of organic carbon per unit volume of solution) were then compared with calculations based on species grouping. Furthermore, in order to provide a perspective in assessing the degree of agreement, exact calculations of the concentration histories were also made and compared with experimental data. The experimental apparatus and procedure used in this work were similar to those employed earlier by Mehrotra and Tien (1984) with some modifications. Details of these can be found in h a s w a m i ' s thesis. A schematic diagram of the apparatus is shown in Figure 5. The organic compounds used in the experimental work are given in Table 11. For convenience, these compounds were arranged in Table I1 in the ascending order of their n values. The Freundlich constants of these compounds were determined from single-species batch-containing measurments. The particle-phase mass-transfer coefficients were determined by matching the experimentally determined single-bed breakthrough curve against calculated results. Details of these determinations can be found elsewhere (Ramaswami, 1984). An examination of Table I1 reveals the need to relax the condition of the n values for species grouping used in the sample calculation. The adherence to the requirement that
Carbon Bed Iwght Inlet mnc (totd)* Superficial vebctfy
Computed (exact1 Computed lpseudo-group1 A Experimental data O2
0
/IA
IO
30
20
4/
40
50
60
70
ELAPSED TIME,hrS
Figure 6. Effluent concentration profile of adsorbates in run 1: (1) 2.4-dichlorophenol, (2) p-chlorophenol, (3) p-nitrophenol, and (4)
u
y
10-
L
a
l.?
0 08-
k 06-
g
*Cln
11
(Cln 12
U
ec
Carbon Bed hetght Inlet c a m (tot011. Superficial velocity : 76 :
76
F403
10 cm
152 ~ ~ T C C N OO5cm/s
mpTOC/f "
04-
02-
A
Compued (pseudo-grwpl ExpertmeM data
J lL
01 0
5
IO
I
I5 ELAPSED
20
t
25
I
I
30
35
TIME, hrs
Figure 7. Effluent concentration profile of adsorbates in run 2: (1) phenol and (2) benzoic acid.
adsorbates of the same pseudospecies should have n values not different by more than 10% of that of the pseudospecies would not lead to any significant simplification. Consequently, it was decided to relax the 10% restriction to 20%. With this relaxation, the nine compounds listed into Table I11 can be grouped into three pseudospecies containing compounds 1-4, compounds 5 and 6, and compounds 7-9, respectively. Of the seven experimental runs conducted, the first six were made when using aqueous solutions with adsorbates only from a given group. In run 7, eight adsorbates for all these groups were considered. The experimental conditions of the seven runs as well as the parameter values of the pseudospecies are given in Table 111. The results of the experimental work and the calculated breakthrough curves are shown in Figures 6-11. In all
138
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
Table 111. Experimental Conditions Granular Activated Carbon F 400 bed height, run no. cm u,cm/s adsorbates 10 0.05 2,4-chlorophenol (1) p-chlorophenol (2) p-nitrophenol (3) o-cresol (4) 10 0.05 phenol (1) benzoic acid (2) 10 0.04 uracil (1) cyclohexanone (2) 10, 15 0.05 adipic acid (1) cyclohexanone (2) 10 0.05 uracil (1) cyclohexanone (2) adipic acid (3) 10 0.05 2,4-dichlorophenol (1) p-nitrophenol (21, p-chlorophenol (31, o-cresol (4), phenol (5), benzoic acid ( 6 ) , cyclohexanone (71, adipic acid (8)
Carbon Bed helght Inktcurc (total)* Superficial velcaty 0 8-
*IC, I,,,
Lc2 J
,,,
:4 :
< lot
F400 10 cm 111 rrgTOC/I 0 0 4 cm/s
w" 0
3 mgTOC/t
68
inlet concn, mg of TOC/L
F400 10 cm
"
z 0
Inlet c o w (totall Sucefficol velocity
0120mgTOC/l 05cm/s
06-
04-
0 3-
/// A
A
OL
1
0
5
-Computed (exact)
- - _ -Computed /pse&-groupI A Experimental data
I'
/'
I
I
1
I
I
I5
20
25
30
35
I
10
ELAPSED T I M E , h m
ELAPSED TIME, hrs
Figure 8. Effluent concentration profile of adsorbates in run 3: (1) uracil and (2) cyclohexanone.
-___
2
10cm
,r
-Complted (exacti A Experlmental Computed(pseuja-gmupi d'o/;t
Figure 10. Effluent concentration profile of adsorbates in run 6 (1) uracil, (2) cyclohexanone, and (3) adipic acid.
IrlL
Carbon
FIryl
,I
ti 5
/
W
co ban
8ed height
F400 IO. lScm
Inlet cum (totall' Superficial velwity
005cm/sec
I
0
5
IO
15
-
0
Computed (exact)
150mgTCC/t
Comphd (three pseudo-groum) A Experimental data
IW
I
20
25
3 0
35
E L A P S E D T M E . hs
Figure 9. Effluent concentration profile of adsorbates in rum 4 and 5: (1)adipic acid and (2) cyclohexanone.
these figures, the agreement between the calculated results based on species grouping and experimental results are very good. In fact, the degree of agreement achieved by predictions based on species grouping is as good as, if not better than, that of the exact calculations. It is obvious that simplifications in multicomponent adsorption calculations by species grouping does not necessarily compromise its accuracy. Both the sample calculations and experiments demonstrate that species grouping offers a simple straightforward way of simplifying multicomponent adsorption calculations. It is true that the sample calculations and the experiments performed utilized adsorbates whose single-
0 0
IO
I
I
20
30
I
40
I
1
I
50
60
70
ELAPSED T I M E , hrs
Figure 11. Effluent concentration profile of adsorbates in run 7 (eight adsorbates).
species carbon adsorption isotherm can be approximated by the Freundlich expression and consequently their adsorption affiiity can be characterized by the values of their Freundlich constants. The same idea can be readily extended to systems obeying other theories. For example, for a mixture whose adsorption equilibrium is represented by the Langmuir expression, species grouping can be made on the basis of the Langmuir constants. It should be pointed out that the use of the ideal adsorbed solutions theory does not necessarily require that the single-species isotherm of the individual adsorbate obey the Freundlich expression extending to sufficiently low concentrations as
i39
Ind. Eng. Chem. Process Des. Dev. 1986, 25, 139-142
pointed out in the earlier work of Wang and Tien (1982). The use of a single Freundlich expression for the single species isotherm data of the individual adsorbate in the present work was merely for the purpose of convenience and clarity in presenting the basic idea of species grouping. Acknowledgment
This study was performed under Grant CPE 83 09508, National Science Foundation. Nomenclature
A aP C
2 k, n 9 qT S
U
z ti
Freundlich coefficient radius of adsorbent particles adsorbate concentration in the solution phase total adsorbate concentration in the solution phase liquid-phase mass-transfer coefficient particle-phase mass-transfer coefficient reciprocal of Freundlich exponent adsorbate concentration in the adsorbed phase total adsorbate concentration in the adsorbed phase quantity defined by eq 9 superficial velocity axial distance mole fraction of the ith adsorbate in the adsorbed phase
Greek Letters defined by eq 10 Pb bulk density of adsorbent PP density of adsorbent particles a spreading pressure % corrected time
4i
Subscripts i ith adsorbate in inlet condition S adsorbent-solution interface Superscript single-speciesstate
0
L i t e r a t u r e Cited Calllgarls, Mary Beth: Tien, C. Can. J. Chem. Eng. 1982, 6 0 , 772. Dobbs, R. A.; Cohen, J. M. EPA-600/8-80-023, 1980. Hsieh, J. S. C.; Turian, R. M.; Tien, C. AIChE J. 1977, 23, 263. Larson, A. C.; Tien, C. Chem. Eng. Commun. 1984, 2 7 , 339. Mehrotra, A. K.; Tien, C. Can. J. Chem. Eng. 1984, 6 4 , 632. Ramaswaml, S. M. S. Thesis, Syracuse University, Syracuse, NY, 1984. Vermuelen, T.; Klein, G.; Helster, N. K. I n “Chemical Engineers Handbook”, 5th ed.;McGraw-HIII: New York, 1973; p 1. Wang, S . G . : Tien, C. AIChE J. 1982, 2 8 , 565.
Received for review October 15,1984 Accepted May 31, 1985
Group Contribution Method To Predict Critical Temperature and Pressure of Hydrocarbons Joseph W. Jalowka and Thomas E. Daubed’ Department of Chemical Englneering, The Pennsylvanle State University, University Park, Pennsylvania 16802
A group contribution model was developed to predict critical pressures and critical temperatures of hydrocarbons using second-order, Benson-type groups. The critical temperature model utilizes the normal boiling point and the groups present In the molecule as parameters. When compared to established methods published by Lydersen and Ambrose, the model produces more accurate results for all families of hydrocarbons except alkanes. The critical pressure model uses the normal boiling point, the critical temperature, and the groups present in the molecule as parameters. The results again are more accurate than the Ambrose or Lydersen models using either an experimental critical temperature or a critical temperature estimated by using the model developed in this study.
Critical properties are important parameters in many calculations involving phase equilibria and thermal properties. These properties are difficult to measure experimentally since equipment capable of rapidly and accurately producing and maintaining high temperatures and pressure is required. Also, for those compounds which contain high numbers of single bonds, the problem of thermal decomposition is a major obstacle to measurement of critical properties. Thus, prediction methods are important and in many cases are the only means by which these properties can be determined. Some of the most successful prediction methods employ group contribution models requiring the structure of the molecule to estimate critical properties. Lydersen’s (1955) model to predict critical temperature is given by Tb
- = 0.567
+ (CAT)- (CAT)2
(1)
TC where Tb = normal boiling point in kelvins, T,= critical temperature in kelvins, and CAT= summation of group
0196-4305/86/1125-0139$01.50/0
increments using T,and Tb in the specified units. Lydersen also developed a similar model to estimate the critical pressure by developing an equation which has the form
(y)”’ + = 0.34
CAP
where MW = molecular weight, P, = critical pressure in atmospheres, and CAP= summation of group increments. Both models work extremely well for a wide variety of organic compounds. Another method which can be generalized to a wide range of organic compounds and produces accurate results was developed by Ambrose (Ambrose et al., l974,1978a,b; Ambrose, 1978). The major difference between the Ambrose and Lydersen model is the inclusion of the Platt number. The Platt number is the number of carbon atoms three bonds apart and is an indicator of the degree of branching in the molecule (e.g., the Platt number of an n-alkane is equal to the number of carbon atoms minus three). Ambrose developed equations of the following form 0 1985 American Chemical Society
