Ind. Eng. Chem. Res. 1989, 28, 1211-1221
1211
SEPARATIONS Study of Simulated Moving-Bed Separation Processes Using a Staged Model Ulrich P. Ernst and James T.Hsu* Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015
A model for simulated moving-bed separation processes is introduced in which axial mixing caused by the entrainment of liquid in the bed voidage is accounted for via a stagewise backmixing concept. The model has the flexibility t o simulate different modes of operation. The utility of the approach is examined by providing fundamental operating characteristics and guidelines for various applications of the process. The effects of the zone flow ratios, backmixing factor, separation factor, feed rate, and total equilibrium stages on the performance of the process were investigated in this work. The separation factor and the backmixing factor are shown t o be the key parameters for design and operation of the process. These two parameters, along with the type of application, also serve t o define practical operating limits for the internal and external process streams. Adsorptive separations on a commercial scale can generally be classified into batch-contacting and moving-bed operations. Although batch contacting is a relatively simple process and offers operating flexibility, it suffers from the following disadvantages: (i) the whole adsorbent bed is not effectively utilized; (ii) a large amount of desorbate is consumed, resulting in undesirable dilution of the products; (iii) a large difference in the adsorptive selectivity between adsorbates is required; and (iv) the operation is discontinuous, which makes it difficult to integrate with continuous processes. In an effort to overcome the inefficiencies associated with batch operation, continuous countercurrent methods have been developed, in which, theoretically, the mass-transfer driving force is maximized, resulting in more efficient usage of the adsorbent than for an equivalent batch-contacting operation by requiring only a partial separation of components. However, in continuous moving-bed contacting, avoiding nonplug flow of the solid phase has been proven to be difficult, as well as preventing attrition of the adsorbent during mechanical circulation. These problems have been overcome in the simulated countercurrent adsorption process developed by UOP Inc. known generically as the Sorbex process (Broughton, 1968). In this process, rather than cycling the adsorbent, the countercurrent process is effectively mimicked by simultaneously incrementing feed and take-off streams along a fixed column. The apparent countercurrent operation of simulated moving-bed processes is achieved by advancing the feed, eluent, extract, and raffinate points simultaneously at set time intervals in the direction of flow of the mobile phase. The four external process streams effectively divide the system into four zones (defined as zones I-IV in UOP’s Sorbex nomenclature), each of which performs a different function in the overall operation. The desorbent and feed streams enter the system between zones I11 and IV and zones I and 11, respectively. The extract and raffinate streams are removed between zones I1 and I11 and zones I and IV, respectively. Zone I serves to adsorb the more
* To whom all correspondence concerning this paper should be addressed. 0888-5885/89/2628-1211$01.50/0
strongly adsorbed component from solution. In zone 11, the more weakly adsorbed species are displaced from the adsorbent by the more strongly adsorbed species. The desorbent entering a t the front of zone I11 serves to regenerate the adsorbent by desorbing the more strongly adsorbed species. Finally, zone IV can be used either to reduce consumption of the desorbent by displacing the desorbent with the weakly adsorbed species, or it may be operated as a stagnant zone. In order to achieve efficient resolution of feed components, there are flow constraints that must be met within each section. The governing parameters can be defined as solute capacity ratios (rLi) of the two oppositely moving streams, that of the upward moving phase to the hypothetically downward moving solid phase and backmixed portion of the mobile phase. For a binary mixture, the governing parameters are primarily dependent on the distribution coefficients of the two components to be separated. For proper net migration of the components in each zone, the flow rate must fall within prescribed limits, specified by the capacity ratios of the respective components and the function of the particular zone in question. Although processes based on this concept have been developed for a number of commercially important separations (p-xylene from C8 aromatics, fructose from aqueous fructose-glucose mixtures, olefins from paraffins, etc.) (Broughton, 1968, 1977, 1985; Broughton et al., 1970; deRosset et al., 1976), only limited information on the fundamental aspects of the design and performance of such a system has been published. Recently Ching and Ruthven (1985a,b, 1986; Ching et al., 1985) reported the results of an experimental and theoretical study of the separation of fructose and glucose in a simulated countercurrent adsorption unit. They adopted linear and uncoupled equilibrium relationships for both components, thus greatly simplifying the mathematical modeling and making it easier to understand the effects of process variables on performance. Hashimoto et al. (1983) completed an analysis of the simulated countercurrent process in which the adsorbent columns were rotated against fixed external process streams. Two models were introduced. One is an intermittent moving-bed model, which expresses the actual
0 1989 American Chemical Society
1212
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
mode of operation and is useful for the calculation of the transient change of concentration profiles in the bed. The other is a continuous moving-bed model, based on the hypothetical movement of adsorbent, and was found to be convenient in the calculation of steady-state profiles. Barker and co-workers (Barker and Abusah, 1985; Barker and Thawait, 1983; Barker et al., 1984) have also performed separations on synthetic mixtures of fructose and glucose as well as inverted sucrose feedstocks via countercurrent chromatographic techniques. The purpose of this paper is to present a more realistic model for the simulated moving-bed separation process based on stagewise contacting. This approach is flexible and allows for modeling separations that are carried out either with a densely packed or loosely packed adsorbent bed. The case for using stagewise contacting in this process is based on the fact that some packing resins used in chromatographic separations tend to expand and shrink during adsorption and desorption modes. This cycling invariably leads to the appearance of unwanted void spaces, which can be handled by incorporating a stagewise contacting approach. It is interesting to note that, until now, the substantial axial mixing that occurs in this process has not been properly quantified in the open literature. As this plays a crucial role in determining the process performance, it is vital that this phenomenon be accurately defined. By incorporating the entrainment of the mobile phase that occurs upon advance of the external process streams in the backmixing term, it is shown that the axial mixing occurring in this process can more realistically be evaluated.
Model Presented is a discrete model for simulated moving-bed adsorption processes that incorporates the stagewise backmixing concept as developed for continuous rotating disk contactors by Mecklinburgh and Hartland (1969). Figure 1 is a schematic diagram of a cell backmixed adsorbent bed, consisting of N cells divided into four zones. The total number of cells in the system is equivalent to the number of contacting stages. For fixed-bed operation, each cell constitutes the height equivalent to a theoretical stage (HETP). The extract, feed, raffinate, and desorbent streams establish the bounds of the four process zones. Cells 1 to p comprise zone 111. The point of extract withdrawal lies between cells p and p + 1. Cells p + 1 to n - 1 comprise zone 11, and feed is added between cells n - 1 and n. Cells n to m comprise zone I, with the withdrawal of raffinate occurring between cells m and m + 1. Cells m + 1 to N represent zone IV. Desorbent is added between cells N and 1. The conceptual definition for flow rates and masstransfer terms used here are adopted from the modeling approach for countercurrent contacting devices proposed by Ricker et al. (1981). F D J and F A L are the constant net phase flow rates in zone L; f D , ] , and f A , j are variable backmix flow rates leaving cell J. In this process, the desorbent flow out of the Nth stage is recycled to the first stage. Correspondingly, the hypothetical adsorbent flow is cycled from the first stage to the Nth stnge. In analogous fashion, the backmixing terms coming out of the first and Nth stages are cycled to their respective stages. The effect of axial mixing is incorporated into the model via the backmixing parameter. The net desorbent flow rates in each zone are dependent on the values chosen for the external process streams and the magnitude of the backmixing term. The adsorbent flow rate can be calculated from the adsorbent inventory and the system cycle time. By definition, the ouerall flow rates for each phase are
I Extract
............
j
Desorbeni FD
IO
cell N
Figure 1. Schematic diagram of simulated countercurrent adsorption operation.
specified as the sum of the net phase flow and the amount backmixed upon advance of the external process streams (i.e., F D L + f D , j and F A & + f A ,j ) . The model is predicated on the assumption that mass-transfer limitations between phases are negligible. In addition, nonporous adsorbent particles, constant volumetric flow rates, isothermal conditions, and linear distribution coefficients of all components are assumed to hold throughout the entire bed. A steady-state mass balance of the mobile phase for component i around stage j in zone L is given by (FD,L
+ f D , j ) c D , i , j-1 - ( F D L + f D , j + f D ,j + l ) c D , i , j
+ f D , j + l c D , i , j + l - qi, j = 0 (l)
Similarly, a mass balance for the adsorbent phase is fA, j-lcA,i, j-1 -
+ fA, j-1 + f A , j
)CA.i, j
+
+ f A ,j ) C A , i , j + l + q i , j = 0 (2) where qi,, is the net interphase flux of species i in the j t h (FAL
stage. The equilibrium distribution between phases is represented by (3) C A , ij = K i , jcD,i, j where K,, is the distribution coefficient of the ith species and may, in general, be a function of bulk concentration (i.e., Ki,j = f l C D , i , j , C A , i , j ) , i = 1, 2, ...,I). In order to get at the salient features of the model, we limit the discussion from here on to the hypothetical case where the distribution coefficients of all constituents involved are constant ( K i , j = Ki). In this case, the concentrations of the adsorbed phase can be eliminated by adding eq 1 and 2 and combining with eq 3 to give the total mass balance for the ith com-
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1213 Chart I
B1
c1
A2
B2
C2
An
Bn
Cn AN-1
Z N
ponent around the j t h stage solely in terms of the liquidphase concentrations:
+ f D , j + K i f A , j-l)cD,i, j-1 (FDJ. + f D , j + f D , j+l + KiFAJ. + KifA, j-1 + K i f ~ j, ) c D , i , j + ( f D , j+l + K i F A J + KifA, j ) c D , i , j+l =
(FDJ.
(4)
For computational purposes, eq 1-4 are normalized with respect to the adsorbent flow rate, F A & , and the concentration of the component highest in the feed composition, CFi,,, to yield
+ P D , j ) x D , i , j-1 - (@DJ. + OD, + P D , j+l)XD,i,
(@DJ.
J
P D , j + l x D , i , j+l P A , j - l x A , i , j-1 - (@A&
+ P A , j-1 + P A , jlxA,i, XA,i, j
= KixD,i,
+ OD, j + K i P A , j - l ) X D , i ,
(@DJ
P D , j+l
j-1
+ Ki@AJ + KiPA, (OD, j+l
j
- Qi, j
+ = 0 (5)
+
PA, j ) x A , i , j+l
(@A&
j
+ Qi, j = 0 (6) (7)
j
+ PD,j + + KiPA, j )xD,i, j +
- (@D&
j-1
+ K i @ A J + K i P A , j ) x D , i , j+l = 0
(8)
The mass balance for certain stages require the addition of extra terms to account for the external process streams or have special terms due to their unique position relative to the overall operation. desorbent stage, j = 1:
+ PD,1 + K i P A , N ) X D , i , N - (@D,III + PD,1 + 8D,2 + Ki@A,III + K i P A , N + K i P A , l ) X D , i , l + @D,2
BN-1 AN
matrix in Chart I resulting from eq 8. Because there is no transfer of adsorbent from one cell to an adjacent cell, PA,, = for j = 1,2, (14)
...a
Assuming each cell or stage holds an equal amount of adsorbent, the hypothetical adsorbent rate is constant, FA,I= FAJI= FA,III = FA,IV = FA (15) which yields @A,I
+ KiPA,l)XD,i,2
=
@A,II
+ P D , p + KiPA,p-l)XDj,p-l - (@D,II + P D , p + P D , p + 1 + @E + Ki@A,III + &PA,p-l + K i P A , p ) X D , i , p + ( P D , p + 1 + K i @ A , I I + KiPA,p ) X D , i , p + l = 0 (lo)
(@D,III
+ PD,n + KibA,n-l)XD,i,n-l
- (@DJ + 6D.n + PD.n+1 +
+ KiPA,n-1 + KiPA,n lXD,i,n + + K i @ A , I + K i P A , n ) X D , i , n + l = -@FXFi
Ki@A,II (PD,n+l
(11)
raffinate stage, j = m:
+ @D,m + KiPA,m-l)XD,i,m-l - (@D.IV + PD,m + P D , m + l + @R + Ki@A,I + K i P A , m - l + K i P A , m )XD,i,m + (12) (PD,m+l + Ki@A,IV + KiPA,m )XD,i.m+l =
@D,III - @E
@DJ
=
@DJI
+D,IV
(@D,IV
+ P D , N + KiPA,N-l)XD,i,N-l PD,1
- (@D,IV + P D , N +
+
[email protected],IV + K i P A , N-1 + K i P A , N )xD,i, @D,1 + Ki@A,III + & P A , N ) X D , i , l
N
+ = 0 (13)
Since all physical parameters are assumed known and the mass balances for separate components are uncoupled, the liquid-phase concentration profiles may be directly obtained for the individual components by solving the
=
@A
=1
(16)
=
+ @F (18)
@D,I - @R
Thus, A,, B, , and C, can be reduced to the following terms: A, = @DJ + PD,]
BJ = -(@D,L + PD,, + PD,]+l + Kt)
c, = PD,,+1
(19)
+ Kt
The AI and C N elements are a result of the dependency of the first and last stages on each other, and these are obtained as for A, and C, coefficients, respectively. For coefficients with special terms, j=1
Ai = @DJV + P D , ~
(@DJ
Nth stage, j = N :
@A,IV
=
feed stage, j = n: (@D,II
=
@,,I1
(9)
extract stage, j = p :
@A,III
For steady-state operation, the process streams may be calculated from the following balances: @D,III = @D,IV + @D
+
-@DxD,i
=
The net mobile-phase flow rates are dependent on the amount of backmixing that occurs within each cell. In order to compare the effect of backmixing on the performance of a simulated moving-bed operation, we define the dimensionless backmixing factor, a, which may be calculated from the ratio of the backmixed mobile-phase flow to the apparent adsorbent flow rate. The amount of backmixing of the mobile phase corresponds to the residual liquid held up in each stage. Typical values determined experimentally for the dimensionless term range from 0.55 to 5.0. If constant backmixing values are assumed for all stages, PD,, = a (17)
(@D,IV
Ki@A,III
=
for j = p =
Bp = -(@D,II
@D,III
+ PD,p
+ 0D.p + P D , p + l
+ @E + K c )
for j = n An
@D,II
+ PD,n
Am
= @D,I
+ PD,m
for j = m Bm
= -(@D,IV + P D , n + P D , n + l + @R + Kt)
1214 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
for j = N
terms. Here, it is defined as the net capacity ratio, yL,i,
With the exception of the first and last rows, the coefficient matrix has tridiagonal form. By providing the necessary storage terms for the Al and C N terms, efficient solution of the matrix can be accomplished by employing standard solution techniques available for sparse matrix systems with nonzero, off-diagonal components (Hofeling and Seader, 1978).
Equations 21 and 22 can be combined to provide the following bounds nominally governing the flow ratios in all four zones:
Discussion Broughton et al. (1970) have considered the degrees of freedom available in undertaking the design of a simulated countercurrent adsorption process. At steady-state conditions, with the knowledge of the proportion of backmixing, if three of the four internal (or external) process streams are fixed, the remaining flow rates may be obtained. In addition, cycle time and the adsorbent inventory in each of the four zones must be specified. In general, cycle time is dependent upon the mass-transfer efficiency and needs to be determined experimentally. In the limiting case considered here, in which mass-transfer limitations are assumed negligible, cycle time with not affect the results obtained. The imposition of bounds on the flow ratios (+DJ) within each zone is necessary to achieve the proper separation of the components. In order to attain the optimal process efficiency of a conventional separation, one where downstream processing is the prime consideration and in which both maximal purity and recovery of one or both products is desired, suitable choices for flow ratios of each zone are nominally confined to limits specified by the distribution coefficients of the species in question. Given the simplifying assumptions made earlier, the actual governing parameter, which defines the net movement of a particular component within a zone L, is specified by the overall capacity ratio, rL,i. This parameter is based on the solute capacity ratio of the sum of upward moving streams to that of the s u m of downward moving streams moving past an arbitrary point within each zone:
Since it is necessary that the net direction of transfer of species i be toward either the extract or the raffinate removal port, then rL,imust be specified to be either greater than or less than unity, depending on the function of the zone. If separation of a binary mixture is taken into consideration, nominal constraints on the flow ratios in each zone are derived from the intersection set of the following pairs of inequalities obtained from the capacity ratios for each component: zone I11 2 11n irII1,, 2 11
wIII,l
By appropriate rearrangement of terms, the overall capacity ratio can be reduced such that it is based solely on the net internal flow rates of the two major streams, that of the net upward moving mobile phase to that of the downward moving solid phase. It is therefore possible to determine this parameter directly without requiring a priori knowledge of the magnitude of the backmixing
zone 111 zone I1 zone I
@D,III
K2 5
'
@D,H
K1
5 Ki
K2 I @D,I IK1 zone IV @D,IV IK2 (23) where K1 and K2 are distribution coefficients for the more preferentially and less preferentially adsorbed species, respectively. For certain applications (e.g., separation of xylenes and monosaccharide isomers), the operating bounds established above do not necessarily represent an optimal processing strategy, especially when only one product is desired and a recycle of the other products to upstream isomerization units can be implemented. Depending on which product is valued, appropriate choices for flow ratios of certain zones may fall outside the above constraints without adversely affecting the purity of the desired product. In this case, a significant portion of the preferred species is withdrawn with the less desired product, but this is recycled upstream where it is reprocessed. Hence, an increase in overall productivity can be realized, albeit accompanied by a decrease in apparent recovery for the separation process. In addition to three of the flow rates, the adsorbent inventories for zones I-IV (i.e., the number of cells in each zone) and the apparent adsorbent flow rate may also be independently set. Each cell may be treated as a theoretical equilibrium stage, conceptually equivalent to that employed in other staged fractionating operations. The apparent adsorbent flow rate is a function of the cycle time required. Whereas the relative performance optimization of the system with respect to mobile-phase flow rates can be detailed in a straightforward manner, the adsorbent inventories for zones I-IV and the apparent adsorbent flow rate are not easily melded to nonspecific optimization since there are no defined bounds to which these terms must conform. Strategic choices for these variables are dictated primarily by absolute performance considerations. Limits for these parameters may evolve from a compromise on such disparate factors as product value, purity requirements, process economics, and the characteristic chemical and physical properties of the adsorbent but need to be specified a priori before practical analysis can be initiated. In keeping within the framework of the above analysis, we shall therefore focus on obtaining a more complete understanding of the effects individual process streams and key process parameters (namely, backmixing factor and distribution coefficients) have on system performance. Flow Rates. A hypothetical case has been created which models the separation of an equimolar, binary mixture. Species 1, the more preferentially adsorbed component removed as extract product, has been arbitrarily designated as the desired product. Thus, only optimization of the extract product with respect to product purity, recovery, productivity, and concentration (downstream processing) is considered. In an effort to facilitate exposition of the process performance, the adsorbent inventories for zones I-IV have been set to reflect one possible design. The total number of equilibrium stages has been fixed a t 20. Zones 1-111 each consist of six stages,
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1215
-
d
Z
,
-8
..
Q
= 0.67
0,
= 0.2
-3
-
Y
a
4
e
.6-.
; V
a
-
region of operation nominally allowed under constraints imposed by Eq. 23.
.6
region of operation nominally allowed under constraints imposed by Eq. 23.
Y
; i
.41
-
T
e
.g
- 8
.2
*)
E
k
.o 0
P H
80.
% -
h;i +%
?
-
zQ 6 0 .
\
"
2'
'
40.
40.
f
20. *)
0.
c
.o
0.4
: .2
.6
.4
.a
.o
1.0
Figure 2. Effect of the fraction of mobile phase removed as extract product on extract purity and concentrationfor a fixed feed rate, aF = 0.2.
while zone IV consists of the remaining two. The distribution coefficients have been set at K1 = 2.0 and K2= 1.0, respectively. The backmixing value for each stage has been specified to be a = 0.67, which represents a typical value for a packed column. Combining the flow relationships outlined by eq 18 and the constraints governing internal process streams in eq 23 yields the following bounds on the external process streams: zone I11 @D,W + @D 2 2 zone I1
15
@D,III
- @E 5 2
zone I
15
@D,II
+ @F
@DJV
51
.2
-4
.6
.8
1.0
Q,IP,+@,)
*E/PD+OF)
zone IV
:
5 2 (24)
The three pertinent external process streams, desorbent, extract, and feed flows, which directly influence system performance, have been studied over an extended range of operation. Figures 2-4 directly relate the purity and the fractional extract concentration with the fraction of mobile phase removed as extract product, @E/(@D + @F), for a set of feed rates aF = 0.2, 0.6, and 1.0 covering the
Figure 3. Effect of the fraction of mobile phase removed as extract product on extract purity and concentrationfor a fixed feed rate, = 0.6.
extent of permissible feed input as specified by eq 24. The shaded portions represent the flow regime that falls within the constraints provided by eq 24. Figure 5 gives the corresponding purity to recovery relationships for @F = 1. Constant lines of the extract to desorbent flow ratio (aE/aD)have been added to facilitate use in conjunction with Figure 4. The flexibility for operating internal process streams shrinks substantially as the feed ratio is increased. By eq 24, for the proper separation of components, the flow ratios in zones I and I1 are bounded by the same values. Consequently, the maximum feed input represented by = K1 - K 2 can only be employed if the flow ratio in zone I1 is held to the minimum allowable value (i.e., @D,II = K2).In Figure 4, for a feed ratio aF = 1, which represents the maximum allowable feed input in this case, selection of the desorbent input rate effectively fixes the value of extract removal. This results out of the necessity to adhere to the minimum allowable flow in zone 11, such that this feed rate may be accommodated. For a fixed value of extract recovery, it is evident from Figure 5 that employing a greater desorbent input yields higher extract purities. However, beyond a desorbent flow ratio greater than aD= 6, no appreciable gain in purity is observed, and as is apparent in Figures 2-4, increasing
1216 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1.0 0
0,
H
a 80.
r; k +P
2 . N
I
60.
2
40.
.-b
K , / X , = 211 Q = 0.67
u
z PI
.2 --
20. -.
u
p
@F
= 1.0
b *
8 :
0. T
Extract Recovery
-
XD,l$JXp,IQF.
2
100
Figure 5. Extract purity-recovery relationshipfor a fixed feed rate, a p = 1.0.
._
2.0-
I
d
---
-
K , / K , = 211 a
= 0.67
region of operation nominally allowed under constraints imposed by Eq. 23.
Pi
*
: 20. 0.4
.o
.2
.6
.4
.e
1.0
a$(@D + @ A .o
Figure 4. Effect of the fraction of mobile phase removed as extract product on extract purity and concentration for a fixed feed rate, @F = 1.0.
the desorbent input results in lowered product concentrations. The value of aD = 2.67 represents the minimum allowable flow rate of desorbent as given by the flow constraint on zone 111. This apparent lower limit for desorbent input necessarily accounts for the portion of the entering desorbent that is backmixed upon advancement of the external process streams. Below this value, the attainable purity drops off sharply. From the flow constraints on zone III provided by eq 24, it follows that the apparent flow rate in zone IV for all cases considered here is fixed a t *D,IV
=
(25)
as is obtained from the lower bound placed upon the flow rate in zone I11 by eq 23 (i.e., @D,~IIh = 2). With this operating approach, the flow rates in zone IV, in actuality, are simulated as the apparent cocurrent downward movement of both phases. Because the apparent flow rate for zone IV cancels with a backmixing term,this leaves the net stagewise transfer of liquid properly described by the remaining backmixing terms. A judicious choice for external process streams requires that consideration be given to the coupling action observed
.2
.4
.6
Mobile Phase Concentration
.8
-
1.0
XD,,,,
Figure 6. McCabe-Thiele diagram for species 1 for a fixed feed rate, @p
= 1.0.
among all three performance variables. A desorbent flow rate should be chosen which does not result in substantially lowered product purity while also conforming to minimized dilution of the extract product. The feed and extract flow rates should be within the constraints imposed but should be set with the stipulation to maximize productivity. These guidelines serve to restrict the practical range of operation to a narrow regime. As an example, we have chosen the following representative conditions to simulate system performance. From the performance curves in Figure 4,it appears that an acceptable trade-off between product concentration and purity may be obtained a t a desorbent flow ratio of aD = 4. A feed flow rate of a F = 1 has been chosen which represents the maximum allowable feed input. Having set these two flow rates, as has been argued previously, the extract flow ratio must be set = 2.33, for the backmixing value given. at McCabe-Thiele Analysis. The results from the simulation of the above operating conditions are shown on a representative McCabe-Thiele diagram for component 1 in Figure 6. It should be noted that, while it would appear
K , / K , = 211 = 0.67 = 4.0 = 2.53
Q
1.21
0, 0, 0, 0,
0, 0,
= = = =
2.0 1.0
0.6 0.2
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1217
--
100.
0 , = 0.
e
-
0,
1.2
= 1.6
M
c+
4
c
*c
BO.
\
"2 6
'.c
,
0.
4.
12.
E.
Stage
20.
16.
No.
Figure 7. Theoretical concentration profiles for varying feed rates.
that the operating lines have a simple linear relationship with the desorbent flow ratio of the particular zone, this is not necessarily the case since the mass balances around each zone must include the respective backmixing terms which include additional independent variables: zone I11
+ (@D,IV + a ) X D , i , N + axD,i,p+ll l(@D,III + a ) x D , i , p + a x D , i , l ) = XA,i,l - XA,i,p+l
{@DxD,i
zone I1 {(@D,II
: b
d"
40.
Y
u I.
u
wX
20.
0.
.o
.4
.B
Feed Rate
1.2
1.6
2.0
- aF
Figure 8. Effect of the feed rate on extract purity and recovery.
+ a)xD,i,p + aXD,i,n) ((@D,II
+ a ) x D , i , n - l + (yxD,i,p+l)
=
XA,i,p+l
- XA,i,n
zone I {@FXF,i
+ (@DJ + a)XD,i,n-l + aXD,i,m+l) {(@D,I + a)XD,i,m
+ axD,i,n)
= XA.i,n - XA,i,m+l
zone IV {(@D,IV
60.
+ a ) X D , i , m + axD,i,ll
l(@D,IV
-
+ a ) x D , i , N + a X D , i , m + l l = XA,i,m+l - X A , i . l
(26)
Depending on the function of the zone, these terms may vary sizably across adjacent stages. It should also be pointed out that the operating line of zone IV, in which there is cocurrent flow of phases, should be expected to have a slope with a negative value. However, since no net flux of phases past one another occurs in zone IV (i.e., apparent velocity of the mobile phase is equivalent to that of the adsorbent phase), the operating line for zone IV appears only as a point on the equilibrium curve. Feed Rate. There are two sets of operating constraints nominally governing the specification of the feed rate, either of which may apply to a given separation. The case can occur in which, due to the inadequate capacity of the adsorbent in relation to the feed concentrations CF,i(Le., a small Ki),the maximal feed rate as specified from eq 24
cannot be maintained without resulting in decreased resolution of the feed components. This is defined here as a capacity constraint, requiring that the upper bound on the feed input be changed accordingly. The other case involves operation where the adsorbent capacity is high enough to be able to process a maximal feed rate. As given by eq 24, this is designated as a volume constraint. Comparison of the effect of increasing or decreasing the feed ratio from the value selected above is illustrated in Figure 7, which includes the theoretical mobile-phase concentration profiles obtained for the system using the feed ratios examined earlier. As the feed is increased, the product concentration of species 1in the extract is raised and the adsorbent is more efficiently engaged in executing the requisite functions of an adsorptive separation, that of adsorption, purification, and desorption. Although it is desirable to operate a t the higher feed rate when considering productivity, there must be a concomitant increase in the number of stages if both product purity and recovery are to remain unchanged. The notion that operating with a lower feed input may yield simultaneous improvement in both purity and recovery does not hold. Figure 8 examines the extract purity and recovery over a range of feed rates for the design and desorbent input specified above. In general, with in-
1218 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 100.
100. C
0
0
-
2
bl
4 80.
80.
4
60. k
t
*x
X , / X , = 211 u
iDD
aF
= 0.67 = 4.0 = 1.0
Adsorbent ratio: Zone 1II:II:I:IV
QD
= 0.67 = 12.
aF
= 1.0
(I
-
3:3:3:1
.-*x 2
P, * u
40.
4
‘ =6/l
I
20. --
i. *
wX 0. 4 0.
1 20.
40.
Extract Recovery
60.
-
100.
80.
XD,,,p@E/d’F9,@F
I
100
0.
I 0.
I
I
20.
60.
40.
Extract Recovery
-
80.
100,
XD,l,pE/X~,I*~
Figure 9. Extract purity-recovery relationship for designs involving geometric increases in the total equilibrium stages.
Figure 10. Effect of the separation factor (K,/K,) on extract purity and recovery.
creasing feed input, purity improves but recovery declines. As can be seen, there is a practical ceiling on the product purity that can be attained for this particular system irrespective of the feed rate. From Figure 8 as well as Figures 2-4, the maximum attainable extract purity for this design lies at approximately 95%. In Figure 7 , as the feed ratio is decreased, there is a corresponding drop in the concentration profile along the adsorbent bed. However, the residual concentration of species 2 at the extract stage does not fall off proportionately. Consequently, operating with a decreased feed rate does not result in an increase in purity. Equilibrium Stages. The above analysis suggests that a critical number of stages are necessary in order to carry out a properly functioning separation given a strict set of process requirements. Should the initial design not meet product specifications in terms of purity and recovery, the analysis of a design with less stages may be used to augment the design of a version with a greater number of stages. The obvious benefit of performing an initial parametric study with less stages is that it reduces the arbitrariness of determining an optimal ratio of adsorbent inventory in zones I-IV for a given separation. Figure 9 compares purity-recovery relationships for three geometrically similar versions of the example adopted for discussion. As expected, better performance may be realized as the number of stages increases. For set operating conditions, proportional changes are observed in both parameters as the number of equilibrium stages is increased, but the percentage increase becomes smaller for successive multiples of the initial number of stages. The dotted line (@E/@,, = 0.58) represents the characteristic performance for the operating conditions evaluated in Figure 6. Doubling the number of stages from 20 to 40 results in roughly a 10% increase in purity and a 15% increase in recovery. Increasing the number of stages again from 40 to 60 results in lesser gains for both parameters. Purity is improved by approximately 5% and recovery by 10%. This trend is observable for additional increases in the system size. Because of residual concentration levels inherent to the operation a t these operating conditions, there is an operating “pinch observed when higher values of both purity and recovery are desired. Unless the design
incorporates an inordinately large number of stages, specifying the operation to improve one variable will mean a decrease in the other. Consequently, for a fixed number of stages, to significantly improve the performance of one variable without resorting to a decrease in the other requires evaluation of alternate design configurations involving the stage ratios per zone or improvement in the separation chemistry. Distribution Coefficients. It is evident from eq 24 that distribution coefficients play a predominant role in determining the system operating limits. The effect of changing the separation chemistry is shown in Figure 10 for a selected set of operating parameters and a fixed value of KZ. As is apparent, rapid improvement of the product purity and recovery is observed as the separation factor ( K l / K , ) increases. For a separation factor of K 1 / K , I 6 for these operating conditions, there is an operating pinch observed when both high purity and recovery are desired. No real improvement is obtained for higher separation factors. Again, this may be attributed to the presence of residual concentrations. While a larger separation factor results in gains for both product purity and recovery, of equal importance is the magnitude of the individual distribution coefficients since they serve to control the allowable feed rate and hence, productivity. As K , - K 2 increases, the limits of feasible operation change, enabling a higher allowable feed throughput and thereby improving process efficiency. Backmixing. The other parameter that directly affects performance is the backmixing factor. A greater amount of axial mixing occurs when operating with larger backmixing terms, consequently resulting in less satisfactory separation. Predictably, both product purity and recovery are adversely affected as backmixing increases. This is reflected in Figure 11, where, for specified operating conditions, the effect of varying the backmixing factor over the range outlined earlier is illustrated. Productivity. Where recycle of the undesired product to an upstream isomerization unit may be incorporated, extract productivity replaces extract recovery as a primary dependent variable. In this case, the extract recovery obtained for the separation step alone does not represent a meaningful figure since it does not reflect the actual
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1219 100.
E H I
EO.
E x + c
. d
k
60.
K , / K , = 211
c
I
GD
= 12.
eF
= 1.0
40.
x Y
.c
/'
/
5
k *
20.
U
n 4
A 0. 0.
20.
60.
40.
Extract Recovery
-
XD,l,pE/XF,,GP
BO. I
100. 100
Figure 11. Effect of backmixing factor (a)on extract purity and recovery.
recovery of the desired species for the overall operation. Performance curves as in Figures 2-4 are presented for a feed ratio of eF = 2 in Figure 12. From comparison with Figure 4, it is apparent that, for corresponding desorbent input ratios, there is a notable gain in extract concentration over that attainable from operating within the governing constraints of eq 24. The representative purity versus productivity plot is given in Figure 13. Increased productivity requires increased feed input. Since only one product is targeted and the objective is to maximize extract productivity, the bounds on internal flow ratios given in eq 24 do not necessarily apply. As can be seen in Figure 7 for aF = 2, when the feed input is increased beyond the limits specified by eq 24, the steep concentration profile of species 1 in zone I disappears. This occurs because the available stripping capacity of the adsorbent for species 1 in this zone is insufficient. The result is decreased separation of components within this zone, and the advantage gained in throughput is offset by the necessity for recycle of the raffinate, which contains a substantial concentration of the desired species. For proper purification of species 1, it is crucial that the bounds set by eq 24 for zone I1 still be applied, likewise for process streams in zones I11 and IV. Although productivity can be enhanced by setting a higher than allowed extract rate, operating below the specified constraints on the flow ratio in zone I1 yields the converse phenomenon of the above by reducing the rectifying capacity of the mobile phase in this zone and resulting in an unwanted decrease of extract purity. This may be ascertained from Figure 13 by examining the effect of changing the extract rate for a fixed feed input. Alternately, operating at higher flow ratios within zone I1 results in better rectification and, consequently, improved purity but also results in lowered productivity since less volume is removed as extract. For a fixed removal rate, there is a finite productivity rate that can be achieved. From Figure 13, above a feed rate of @F = 3, there is no appreciable gain observed for this system. This phenomenon is readily explained from the concentration profiles for species 1 in Figure 14 covering a range of feed rates. While there may be an attendant growth of the overall mobile-phase concentration
.o
.2
.4
.6
.E
1.0
@E/(@D+%)
Figure 12. Effect of the fraction of mobile phase removed as extract product on extract purity and concentrationfor a fixed feed rate, bF = 2.0.
profiles in purification (zone 11) and adsorption (zone I) sections for higher feed rates, desorption curves for species 1 in zone I11 remain essentially constant beyond a feed rate of @F = 3, yielding only marginal increases in the concentration at the extract stage. Coupled with a fixed extract removal rate, this serves to effectively limit productivity. Again, the optimal flow ratios are set by determining a suitable trade-off between performance factors, in this case, extract purity, productivity, and concentration. A McCabe-Thiele diagram has been constructed in Figure 15 for species 1, which illustrates the performance of the system with a feed ratio of aF = 2, employing the same desorbent and extract flow ratios used in the previous case. Comparison with the McCabe-Thiele plot in Figure 6 highlights the contrasting objectives. Higher concentrations at all corresponding stages are evident for operation with increased feed rate. Points E and R represent the conditions at the extract and raffinate stages, respectively. While there is roughly a 30% increase in extract concentration observed corresponding to the increased feed input from @F = 1.0 to 2.0, there is approximately a 160% increase in raffinate concentration of species 1. As can be
1220 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
I
100.
1
40.
-.
1
.-x 5
K , / K , = 211
Y
PI u
v m
= 0.67 = 4.0
(1
QD
20. -.
.o
.2
.4
.a
.6
1.0
Y
4 0.
+
J
:
Productivity
-
X,,,,,OE
Figure 13. Effect of varying the feed rate on the extract purityproductivity relationship.
K,/K, = (I = @D = PE =
0
4.
211 0.67 4.0
2.33
8
12
16.
20
Stage No. Figure 14. Theoreticalconcentration profiles for species 1 obtained for varying feed rates.
determined from Figure 13, increasing the feed rate beyond aF = 2 for these operating conditions does not yield any real improvement in performance. Instead, a higher feed rate will only necessitate an increased recycle of the raffinate stream to the upstream isomerization unit.
Conclusions The modeling of simulated countercurrent adsorption processes seems appropriately suited to the conceptual approach of employing backmixing to account for the axial
Mobile Phase Concentration
~
XD,l,J
Figure 15. McCabe-Thiele diagram for species 1 for a fixed feed rate, OF = 2.0.
mixing resulting from entrainment of the mobile phase. The consistency of the model is due to the strong physical analogy backmixing has with the actual mode of operation when considered over a time average. In addition, the cyclical nature of the operation is satisfactorily described by the inclusion of the proper terms in the computational method. The utility of the model has been explored by offering a fundamental analysis of the principle design considerations for this process. Numerical simulation has been carried out by using a hypothetical operation with a relatively small number (20) of equilibrium stages. In this way, the dependence of relevant process variables on specific operating parameters is enhanced and may be better elucidated. The design configuration (Le., the number of stages required in each zone) of the process is governed primarily by which product(s) are desired and any accompanying product specifications. Separation chemistry, the physical mode of operation, and also the type of separation being performed, as illustrated in the discussion, are required to establish the bounds for practical operating limits. Selected performance characteristics for the possible applications have been presented to provide insight as to the process limitations and the necessary compromises on the performance factors that must be weighed when determining feasible operating conditions. For the application in which both product streams continue to downstream processing, the three critical performance factors for each product that must be evaluated are component purity, recovery, and concentration. For the application involving recycle of one product stream to an upstream isomerization unit, the relevant performance factors include component purity, productivity, and concentration. It is tacitly understood by the authors that the model gives only a first-order approximation of the system behavior. Rigorous analysis of the process requires extension of the model and computational method to accommodate for added complexity, such as nonlinearities in process variables and intraparticle and interphase mass-transfer resistances. Although not presented here, a more sophisticated general model has been developed (Ernst and Hsu, 1989).
Nomenclature Ai = matrix coefficient for cell j with respect to the variable for cell j - 1
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1221
B, = matrix coefficient for cell j with respect to the variable for cell j C, = matrix coefficient for cell j with respect to the variable for cell j + 1 CA,,, = concentration of the ith component for cell j in the adsorbent phase, kg/m3 of adsorbent CD,,,, =. concentration of the ith component for cell j in the mobile phase, kg/m3 CD,,= concentration of the ith component in the desorbent, kg/m3 CF,,= concentration of the ith component in the feed, kg/m3 apparent adsorbent flow rate, m3/s FDL = net mobile-phase flow rate for zone L , m3/s FD = desorbent flow rate, m3/s FE = extract flow rate, m3/s FF = feed flow rate, m3/s FR = raffinate flow rate, m3/s f A , , = volumetric backmixing flow rate for cell j in the adsorbent phase, m3 of adsorbent/s f D , , = volumetric backmixing flow rate for cell j in the mobile phase, m3/s K , = distribution coefficient of the ith component, (kg/m3 of adsorbent) / (kg/m3) q,,, = net interphase flux of the ith component for cell j , kg/s Q,-,*, = dimensidess net interphase flux of the ith component for cell j (=qij/FACF,i,,) Xa-,.,, = dimensionless concentration of the ith component for cell j in the adsorbent phase (=CA,,~/CF,, ) XD,,,= dimensionless concentration of the rtrcomponent for cell j in the mobile phase (=CD,,~/CF,,-! XD,, = dimensionless concentration of the cth component in the desorbent (=CD,,/CF,,,) XF,,= dimensionless concentration of the ith component in the feed (=CF,,/~F,,,) Greek Symbols a = dimensionless backmixing constant BAJ = dimensionless volumetric backmixing flow rate in the adsorbent phase (=fAJ/FA= 0) pD = dimensionless volumetric backmixing flow rate in the desorbent phase (=fDJ/FA= a) rLf= overall capacity ratio of the oppositely moving streams in zone L for the ith component (=(@DL + p D , , ) / ( @ A h K , + ~
BD,,))
yL, = capacity ratio of the primary, oppositely moving phases
in zone L for the ith component (=@DL/@AK,) = dimensionless hypothetical adsorbent flow rate (= FA,L,/FA= 1) = dimensionless mobile-phase flow rate for zone L (= FD,L/FA) @D = dimensionless desorbent flow rate (=FD/FA) @ E = dimensionless extract flow rate (=FE/FA) aF= dimensionless feed flow rate (=FF/FA) aR = dimensionless raffinate flow rate (=FR/FA) Subscripts A = refers to the adsorbent phase D = refers to the mobile phase E = refers to the extract stream F = refers to the feed stream Z = number of solute species L = zone index N = refers to the last cell
R = refers to the raffinate stream i = index of the solute species j = cell index m = refers to the position of the raffinate stream max = denotes the highest concentration, species i, in the feed
n = refers to the position of the feed stream p = refers to the position of the extract stream I = refers to zone I
I1 = refers to zone I1 I11 = refers to zone I11 IV = refers to zone IV
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Barker, P. E.; Thawait, S. Separation of Fructae from Carbohydrate Mixtures by SemicontinuousChromatography. Chem. Ind. 1983, 817.
Barker, P. E.; Irlam, G. A.; Abusah, E. K. E. Continuous Chromatographic Separation of Glucose-Fructose Mixture Using Anion Exchange Resins. Chromatographia 1984,18(10), 567. Broughton, D. B. Molex: Case History of a Process. Chem. Eng. Prog. 1968, 64(8), 60. Broughton, D. B. Bulk Separations via Adsorption. Chem. Eng. Prog. 1977, 73(10), 49. Broughton, D. B. Production-ScaleAdsorptive Separations of Liquid Mixtures by Moving-Bed Technology. Sep. Sci. Technol. 1985, 19, 723.
Broughton, D. B.; Neuzil, R. W.; Pharis, J. M.; Brearley, C. S. The Parex Process for Recovering Paraxylene. Chem. Eng. Prog. 1970, 66(9), 70.
Ching, C. B.; Ruthven, D. M. An Experimental Study of a Simulated Countercurrent Adsorption System-I Isothermal Steady State Operation. Chem. Eng. Sci. 1985a, 40(6), 877. Ching, C. B.; Ruthven, D. M. An Experimental Study of a Simulated Countercurrent Adsorption System-11 Transient Response. Chem. Eng. Sei. 198513, 40(6), 887. Ching, C. B.; Ruthven, D. M. An Experimental Study of a Simulated Countercurrent Adsorption System-IV: Non-Isothermal Operation. Chem. Eng. Sci. 1986, 40(12), 3063. Ching, C. B.; Ruthven, D. M.; Hidajat, K. An Experimental Study of a Simulated Countercurrent Adsorption System-I11 Sorbex Operation. Chem. Eng. Sci. 1985, 40(8), 1411. deRosset, A.; Neuzil, R. W.; Korous, D. J. Liquid Chromatography as a Predictive Tool for Continuous Countercurrent Adsorptive Systems. Ind. Eng. Chem. Process Des. Dev. 1976, 15(2), 261. Ernst, U. P.; Hsu, J. T. A Rigorous Model for Analysis of Nonlinear Adsorption in Simulated Moving-Bed Separation Processes. In preparation. Hashimoto, K. S.; Adachi, S.; Noujima, H.; Maruyama, H. Models for the Separation of Glucose/Fructose Mixtures Using a Simulated Moving-bed Adsorber. J. Chem. Eng. Jpn. 1983,16(5), 400. Hofeling, B. S.; Seader, J. D. A Modified Naphthali-Sandholm Method for General Systems of Interlinked,Multistage Separators. AIChE J. 1978, 24(6), 1131. Mecklinburgh, J. C.; Hartland, S. Design of Differential Countercurrent Extractors with Backmixing Using Stagewise Models. Can. J. Chem. Eng. 1969,47, 453. Ricker, N. L.; Nakashio, F.; King, C. J. An Efficient, General Method for Computation of Countercurrent Separation Processes with Axial Dispersion. AZChE J. 1981, 27(2), 277. Received for reuietu December 28, 1988 Revised manuscript receiued April 20, 1989 Accepted May 11, 1989
