1 = of theliquid 0 = at the inlet of the duct t = a t the throat of tlhe venturi T = total Superscript 0 = for the special case when
Vlo
is assumed to be zero
Literature Cited Behie, S. W., Beeckmans, J M., Can. J. Chem. Eng., 51, 430 (1973). Boll, R H., Ind. Eng. Chem., Fundam., 12, 40 (1973). Boll, R . H., Flais, L. R., Maurer, P. W., Thompson, W. L., JAPCA, 24, 934 ( 1974). Calvert, S.,AlChEJ., 16, 392 (1970). Calvert, S.,Lundgren, D., Mehta, D. S.,JAPCA, 22, 529 (1972). Field, R. B., Ph.D. Thesis, Uriiversity of Illinois, Urbana, Ill., 1952.
Gieseke, J. A., Ph.D. Thesis, University of Washington, Seattle, Wash., 1963. Goel, K. C.,Ph.D. Thesis, University of Waterloo, Waterloo, Ont.. Canada, 1975. Hollands, K. G. T., Goel, K. C., Ind. Eng. Chern., Fundam., 14, 16 (1975). Lapple, C.E.,Shepherd, C. B., lnd. Eng. Chem., 32 (5), 605 (1940). Nukiyama, S.,Tanasawa, Y., Trans. SOC.Mech. Eng. (Tokyo), 4, 5, 6 (19381940) (translated by E. Hope, Defence Research Board, Department of National Defence Canada, Ottawa, 1950). Walton, W. H., Woolcock, A., in "Aerodynamic Capture of Particles," E. G. Richardson, Ed., Pergamon Press, New York, N.Y., 1960. Received for review September 18,1975 Resubmitted June 16,1976 Accepted November 29,1976 T h i s w o r k was supported financially by a grant a n d fellowship f r o m t h e N a t i o n a l Research Council o f Canada.
Extraction Parametric Pumping with Reversible Reaction Shlgeo Goto' and Masakazu Matsubara Department of Chemical Engineering, Nagoya University, Chikusa, Nagoya, Japan
Liquid-liquid extraction parametric pumping can be applied to a first-order reversible reaction system in order to obtain1 a conversion higher than that for chemical equilibrium. On the assumption of phase equilibrium, staged parametric pumps are investigated both for a discrete transfer model and for a continuous flow model. The efficiencies of extraction parametric pumping with reversible reaction are compared with chemical equilibrium and the effects of each system parameter on the efficiencies are evaluated. The results indicate that parametric pumpling should be quite effective if the system parameters can be chosen Suitably.
Since Wilhelm et a]. (1966) introduced the idea of parametric pumping, this new technique has been chiefly applied to adsorption processes to separate components of a fluid (Sweed and Wilhelm, 1969; Gregory, 1974; Chen et al., 1974). The adsorption parametric pumping seems to be suitable by two main reasons, that is: (1)the adsorbate equilibrium relationship between liquid and solid phases is greatly changed due to temperature change, and (2) the solid adsorbent can easily be held stationary while a liquid can flow past the solid adsorbent in a packed bed. Recently, Wankat (1'973)extended the parametric pumping to liquid-liquid extraction by devising three techniques for holding one liquid phase stationary while pumping the other liquid back and forth past the stationary phase. The first technique is to coat thle stationary liquid phase on an inert solid support. This can be operated in the same fashion as the adsorption parametric pumping. The second is to put both liquids into a continuous contact column which is inverted 180' every half-cycle. When this method was tried experimentally, the moving phase could always flow due to gravity but no separation was achieved because there was considerable axial mixing. The third is to use a staged system such as a staged column without downcomers inverted every halfcycle, a series of mixer-settlers and a horizontal helix. We can expect that the parametric pumping must be much useful for the combined process of separation and reaction because the conversions of the desired products can be greatly improved by selective sDeparationsof reactants and products. Tam and Miyauchi (1973) investigated the adsorption parametric pumping accompanied by a reversible reaction in a packed bed and the calculated results showed that the parametric pumping effect considerably improved the conversion. Also, Apostolopoulos I( 1975) studied the adsorption para-
metric pumping with reversible reaction and developed a "near-equilibrium'' approach. In the previous papers, the present authors studied the reaction with extraction which was often referred to as the extractive reaction (Goto and Matsubara, 1972a,b, 1974). In this paper the extraction parametric pumping introduced by Wankat (1973) is applied to the system accompanied by a first-order reversible reaction to investigate numerically to what extent the conversion can be improved. Only the staged system which is the third technique of Wankat (1973) for holding one liquid stationary is considered because this seems to be easiest for performing experimental work in the future.
Staged Parametric Pump Consider a staged parametric pump in Figure 1operating in the direct mode where the temperature of the system can be changed immediately when the flow direction of the moving phase is changed. The stages are numbered from 1 to N starting at the hot reservoir. The hot reservoir is numbered zero and the cold reservoir N 1. For simplicity, we assume that the dead volumes of the hot and cold reservoirs are negligible. Also, the parametric pump is assumed to start from the hot half of the cycle where the moving phase flows from the cold reservoir to the hot reservoir through N stages while the system is hot. Then, the temperature is changed from hot to cold and the direction of the flow is reversed. Both solvents in the moving and the stationary phases are assumed to be immiscible with each other. The volumes of both phases in each stage, VM and V s are assumed to be constant. The first-order reversible reaction is taking place only in the stationary phase of each stage.
+
Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
193
- MC
M~~1,l= . p MRJN
-
(13)
where MAJNand MA,^,],^ indicate the moles of A before and after the reaction step, respectively, in the stage i after the transfer step p in the cycle j. AMH and AMc are the mole changes due to the reversible reaction during the reaction step for the hot and cold halves of the cycle. AM can be calculated as
Flow while system is hot Flow while system is cold
Figure 1. Schematic of staged parametric pump.
AM
= rll
- exp(-dJ)l
(14)
where The reactant, A, and the product, R, are partitioned between the stationary and moving phases. The concentrations of A and R in both phases are assumed to be instantaneously equilibrated with each other through the operation; that is,
- ~ R ) M R , I N- (1 - f A ) M A , I N K ( 1 - f R ) + (1- fA) dJ = f i ( K ( 1 - f R ) + (1 - f A ) I / P T
y=
K(1
(17)
6=kT
(18)
Here, y is the mole change which will occur as I$ tends to infinity. Then, 4, K , and 6 are the normalized reaction time for one transfer step, the reciprocal of chemical equilibrium constant, and the forward reaction rate constant multiplied by the period of half-cycle, respectively. The values for the hot and cold halves of the cycle are indicated by the subscripts H and C, respectively, in AM, y,+ , 6 , and K . The details of derivation of eq 14 are presented in the Appendix. The boundary conditions on recursion relations 6-13 are written fAH *MA,N+l,j,p-I fRH
Discrete Transfer Model In the case of the discrete transfer model, all of the moving phase in a stage is transferred at one time and then brought to equilibrium with the stationary phase in the next stage. p~ transfer steps are repeated in each half-cycle until the feeding reservoir becomes empty. Each transfer step is followed by each reaction step where the reversible reaction occurs batchwise in the stationary phase for the time of T l p ~T. is the time required for each half-cycle. Define f A and f R as the fraction of A and R in the moving phase after the reaction step. By the assumption of phase equilibrium, eq 2 and 3, we have
= CAM,N+1,1-1
*
M ~ , ~ + i , ] , p -= i CRM,N+l,]-l
=
fAC ’ M A , o , ] , ~ - I fRC
*
CAM,O,]
M~,o,l,p-i= CRM,O,J *
VM
(19)
VM
(20)
VM
(21)
VM
(22)
After the last reaction step of each half-cycle the temperature is changed instantaneously and phase equilibrium is reestablished immediately. Then, the first transfer step in the other direction is made before any further reaction can occur. Therefore at the end of each half-cycle the values of fA and f R change due to temperature change while the total moles of A and that of R in each stage do not change. These give the initial conditions for the next half-cycle. The feed concentrations of A and R from the hot and cold reservoirs for the next half-cycle are CAM,O,J
and f R are constant for all stages in each half-cycle but are different for the hot and cold halves of the cycle because m A and m R shift with temperature. Mass balances of A and R can be written for the stage i in the cycle j. For the transfer step p in the hot half of the cycle j,we obtain
(16)
= k’lk
K
where m A and m R are distribution coefficients of A and R, respectively. As the temperature of the system is increased, m A and m R are increased or decreased dependently on the system. If mass transfer rates between two phases are quite high relative to reaction rates and the flow rate of moving phase, the assumption of instantaneous phase equilibrium may be acceptable. Two different models which simulate flow patterns of moving phase in discrete stages will be considered below. They are the discrete transfer model and the continuous flow model. A series of mixer-settlers may be simulated by the former and a horizontal helix system by the latter (Wankat, 1973).
(15)
=
fAH
a
M ~ , i , , p l p ~V M
p=l
(23)
fA
+ (1 - f A H )
MA,i,j,p-l
(6)
+ (1 - f R H ) ’ MR,i,j,p-l = MAJN+ M H
(7)
MAJN= fAH
’MA,i+l,j,p-l
MRJN= f R H
*
*
MR,i+l,j,p-l
MA,i,j,p
MR , .i j., p - MRJN- MH
(8)
(9)
and for the transfer step p in the cold half of the same cycle
MAJN= fAC
MA,i-l,j,p-l
MRJN= fRC
’ MR,i-l,j,p-l
MA,i,j,p
194
+ (1- f A C )
MA,i,j,p-l
(1- f R C ) M R , i , j , p - l
= MAJN+ M
C
Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
(10) (11) (12)
CAM,N+l,j
CRM,N+lj
9 = 9 =
fAC
M A , N ~ , ~ / P *TV M
(25)
fRC
M R , N ~ , ~ ~ PVTM*
(26)
p=l
p=l
The initial conditions at the beginning of parametric pump are that the concentrations of A and R are C A M , N + ~ , Oand C R M J , T + ~ ,for O the cold reservoir and zero for all stages and that the volumes of the cold and hot reservoirs are p~ VM and zero, respectively. If the system parameters, N , PT, KH, KC, 6H, 6C, fAH, fAC, f R H , fRC, CAM,N+I,O, and CRM,N+1,0 are given, CAM,OJ, C R M , O ~ , ~ easily be obtained by solving CAM,N+l,j, and C R M , N + ~ ,can sequentially eq 6-26 as follows. Use eq 6-9 with boundary and CRM,OJ by conditions, eq 19 and 20, to determine CAM,OJ eq 23 and 24. Then, use eq 10-13 with boundary conditions, eq 21 and 22 to find C A M , N + ~ and , ~ C R M , N + l , j by eq 25 and 26.
The numerical results will be shown later. The dimensionless concentration, x,will be calculated from dividing C by the total initial concentration, (CAM,N+I,O CRM,N+~,O).
+
Continuous Flow Model Next, we consider tlhe continuous flow staged parametric pumping in which the flow rate of moving phase, F, is constant for the hot and cold halves of the cycle. Mass balances of A and R around stage i become
I I
, ,
I
to-’
I IIIIII
I
I IIIIII
for the hot half of the cycle and
1
+ VS(kcCAS,i,j- ~ c ’ ~ R s , L , J )(30) for the cold half of the cycle. The feed concentration of A from the hot reservoir for the next half-cycle becomes the average value of the concentration received from stage 1 during the time of half-cycle, T.
S,
10.’
T
cAM,O,r
=
cAM,iJ
d t l ~
(31)
Similarly, the feed concentration of A from the cold reservoir becomes
The feed concentration of R is obtained by replacing the subscript A in eq 31 and 32 by R. At the end of each half-cycle the temperature is changed instantaneously and the phase equilibrium between moving and stationary phases is immediately reestablished. The direction of the flow is reversed just after the temperature change. Since the total moles of A and R in each stage remain unchanged, the initial concentrations in the moving phase for the next half-cycle can be obtained from the mass balances for each stage. After substituting eq 2 and 3 on the assumption of phase equilibrium, the ordinary differential equations 27-30 can be numerically integratedl If the volumes of reservoirs for the continuous flow system are chosen equal to those for the discrete transfer system, that is, FT = PTVM, the same system parameters can be used for these two systems. Results a n d Physical Interpretations The performance of extraction parametric pumping with the reversible reaction is investigated by the simulation study for two models describled above. The system parameters involved in the models can be classified into three categories: (1) those governing chemical reactions, (2) those governing phase equilibrium, and (3) operational parameters. The chemical reaction parameters are the forward and backward reaction rate constants of first order for the hot and cold haves of the cycle, k ~kc, , kH‘, and kc’. In the models, however, di, KH, KC are used in place of mensionless parameters, 6 ~ 6c, them. Here, 6~ and 6c are the forward reaction rate constants
x 10.‘
10
j
10‘
10’
Figure 3. Effect of cycles for discrete transfer model: N = 20, PT = 11, K H = 1,6H = 0.01, bC = 0, f A H = 0.5, f A C = 0.6,f“ = 0.6, f K C = 0.5; solid lines: X A , N + I , O = 1, XR,N+I,O = 0; dotted lines: XA,N+I,O = 0, XR,N+I,O
= 1.
multiplied by the period of half-cycle and KH and KC are the ratios of the backward to the forward reaction rate constant, that is, the reciprocals of chemical equilibrium constants. The phase equilibrium parameters are the distribution coefficients, mAH, mAC7 mRH, and mRC but the factions,fAH,fAC,fRH, and fRC defined by eq 4 and 5 are used instead of them. The value off is monotonously increased from zero to unity as the value of m is increased from zero to infinity. Since we assume instantaneous phase equilibrium, it is unnecessary to consider the parameters concerning interphase mass transfer rates. The operational parameters are the number of stages, N and the total number of transfer steps per half-cycle, p ~The . value of pT can be regarded to be equal to FTIVMfor the continuous flow model. At the start of parametric pumping, raw materials involving A and R are filled in the cold reservoir whose volume is PTVM while the hot reservoir is empty. Two insoluble solvents containing no raw materials are provided for the stationary and moving phases of each stage. The initial concentrations of A and R in the cold reservoir are expressed by XA,N+I,O and XR,N+l,O. From the definition shown in the NoInd. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
195
- r
m, xA,O.~
0.3
1 0.1
0.5
f,
0.7
0.9
menclature section, the sum of them is unity. Since the concentrations at the final state are independent of the initial concentrations (see Figures 2 and 3), we can assign arbitrary values to the latter for the calculations of the former. Define efficiencies of extraction parametric pumping with the chemical reaction for evaluating its performance. The qualities of the product a t the final state in the hot and cold reservoirs can be expressed by the ratios of the concentration of R to that of A as follows. For the hot reservoir Q
- CR,o,m
0--=CA,O,-
XR,O,m
(33)
10
XA,O,m
and for the cold reservoir QN+I
=
CR,N+l,m
- XR,N+l,m
CA,N+l,m
XA,N+ 1,m
(34)
One of QO and Q N + is ~ expected to be much greater than the other due to separation effects of parametric pumping and the product is withdrawn from the hot reservoir if QO > Q N + ~and from the cold reservoir if Qo < Q N + ~ .In the operation of extractive reaction where the extractive phase (the moving phase for the parametric pumping) and the reactive phase (the stationary phase) are in cocurrent or countercurrent continuous flow a t the constant temperature corresponding to the hot half-cycle (Goto and Matsubara, 1972a), the quality of the product which is withdrawn from the extractive phase a t the exit can be expressed by QER
=
(cRM/cAM)exit
(35)
-3
l10 1 010-3
= mAHcAS;
CRM
=
14'
2
I I l l 1I l l SH
i I
1/KH
I I I I I I Il
1
1 0 ' '
SH
7=
Qo XROrnax(70, VN+~);7 0 = -= K H -; QPR
XA,O,-
QN+I = XR,N+l,m 7 N + l = - KH QPR XA,N + 1, m ~
QER = RH TER = QPR
(38) (39)
mAH
The efficiency of parametric pumping relative to extractive reaction can be defined in a similar manner and is equal to ~ V E R .
Then, we have QER
= mRH/mAHKH
(36)
In another operation of pure reaction where only the reactive phase is in a continuous flow at the constant temperature corresponding to the hot half-cycle, the quality of the product can be represented as 196
i
I
QPR = (CRS/CAS)exit = 1/KH (37) on the assumption of chemical equilibrium at the exit. Now define the efficiencies of parametric pumping and extractive reaction relative to pure reaction as
= mRHCRS
Further, assume that chemical equilibrium of reversible reaction is reached a t the exit, that is (CRS/CAS)exit
I Illllll
Figure 6. Effect of 6~ for discrete transfer model: N = 20, p~ = 11, KH = 1, 6C = 0 , f A H = 0 . 5 , f ~ = c 0 . 6 , f R H = 0.6,fRc = 0.5.
On the assumption of instantaneous phase equilibrium, we have CAM
I
Ind. Eng. Chem., Fundarn., Vol. 16, No. 2, 1977
Since there are ten system parameters in each of two models for the extraction parametric pumping with chemical reaction, it is difficult to investigate the effects of them completely. So, based on the results of some trial calculations, the standard values of parameters are set up as follows: reaction parameters: BH = 0.01, 6c = 0, K H = 1 , KC = arbitrary finite value; phase equilibrium parameters: fAH = 0.5, f ~ =c 0.6, RH = 0.6, ~ R C
t 0’
1
1 0’
10’
16’
1o2
r
H
l0’k
Figure 7. Effect of 6c for discrete transfer model: N = 20, PT = 11, K H = KC
= 1, 6~ = 0.01, /AH = 0.5, ~
A C =
0.6, /RH = 0.6, /RC = 0.5.
Figure 9. Effect of p T or FTIVMfor discrete transfer model and for continuous flow model: N = 20, K H = 1, b~ = 0 . 0 1 , 6 ~= 0, f A H = 0.5, /AC
Figure 8. Effect of K H for discrete transfer model: N = 20, PT = 11, 6~ = 0.01, 6 ~ =; 0, /AH = 0.5,
/AC
= 0.6, /RH = 0.6, /RC = 0.5.
’“1 1-
5
9
13
N 17
21
25
29
= 0.6, /RH = 0.6, /RC = 0.5.
Figure 10. Effect of stages at optimal value of p r for discrete transfer C 0.6, /RH = 0.6, /RC = 0.5. model: KH = 1 , b =~ O.OI,/AH = 0.5, ~ A = = 0.5; operational parameters: N = 20, PT = 11. The effect of each system parameter on the efficiency 7 was investigated varying one of the parameters while fixing the others a t the standard values. The calculated results are shown in Figures 2-10. It will generally be true that the values of k c and kc’ are quite small relative to those of kH and kH’. In addition, the efficiency 7 is insensitive to 6c(= k c T ) especially in the range of 6c/6H (= kc/kH) smaller than about 0.1 (see Figure 7 ) .So, the standard value of 6c was set equal to zero. Since 6c = kcT, T # 0, KC = kc’lkc, the values of k c and kc’ must be equal to zero for 6c = 0 and KC being finite. This means that the reaction rates a t the cold half-cycle are assumed to be negligible in comparison with those at the hot half-cycle. The standard value of total number of stages, N = 20 is so chosen as to obtain a nearly maximal value of the efficiency 7 by use of a nearly minimal value of N while the maximal value of 7 is attainable a t an infinitely large value of N (see Figure 10).The standard value of total number of transfer steps per half-cycle, PT = 11 is chosen equal to the value which maximizes the value of 7 at N = 20 (see Figure 9). Remaining standard values are chosen suitably so as to make the parametric pumping effective. Since our sirnulation study indicates that the continuous flow system is generally less effective than the discrete
transfer system and that the tendencies of performances for both systems are not so different from each other, the performance of the former is shown in comparison with that of the latter only in Figure 9. Transient Characteristics.Figures 2 and 3 show examples of transient characteristics of parametric pumping. Every parameter in these figures was set equal to the standard values except the value of 6~ in Figure 2 which was set equal to 0.1. Two extremes of initial conditions are considered in these figures, that is, XA,N+1,0 = 1, XR,N+1,0 = 0 (solid lines) and X A , N + i , o = 0, XR,N+1,0 = 1 (dotted lines). As the cycles are increased, each solid line tends to the corresponding dotted line and becomes constant. Therefore, the value of x at the final state, x- which is independent of the initial conditions can be taken as a measure of performance of parametric pumping with the reversible reaction. The number of cycles required for the value of x to be close to its final value is about 100 cycles for 8~ = 0.1 (Figure 2) and about 1000 cycles for 6~ = 0.01 (Figure 3). So, roughly speaking, the required number of cycles seems inversely proportional to the value of 8 ~ This . may be explained as follows. If 6~ is inInd. Eng. Chem., Fundarn., Vol. 16, No. 2, 1977
197
creased, mole changes will increase because of increase in the reaction rates during the hot half-cycle. On the other hand, during the cold half-cycle where no reaction take place due to 6c = 0, only instantaneous mass transfers due to the change in the phase equilibrium from the hot to the cold half-cycle occur. Therefore, the concentration changes with time are mainly governed by the reaction rates during the hot half-cycle and in the consequence the required number of cycles decreases with the increase in b ~ . T h e Effects of Phase Equilibrium Parameters. Figures 4 and 5 show the effects of phase equilibrium parameters, fAH, fAC, fRH, and fRc on the efficiencies and concentrations in the hot and cold reservoirs. The separation of R from A in each stage is dependent on the difference between fR and fA. We define the degree of separation as
the other hand, the efficiency of extractive reaction for the same values of parameters is ~ E = R 1.5. Thus, if we can choose a solvent having a large value of SHand a small value of Sc, the operation of extraction parametric pumping can yield the product with strikingly high quality relative to both operations of extractive reaction and pure reaction. T h e Effects of Reaction Parameters. Figures 6,7, and 8 show the effects of reaction parameters, 6 ~ bc, , K H , and KC. The effect of 6~ which is proportional to the forward reaction , in Figure rate constant during the hot half-cycle, k ~is shown 6. As b~ is increased, the efficiency 70 is quickly decreased because X A , O , ~is rapidly increased while the yield XR,O,= remains nearly unchanged. The backward reaction rate constant during the hot half-cycle, k ~ is’ always proportional to the forward constant k~ because the value of KH = k H ’ / k H is held constant. As both forward and backward reaction rates are s = fR - f A (40) , reversible reaction beincreased with the increase in b ~the tween A and R approaches chemical equilibrium in any stage. Since 0 < fR < 1and 0 < fA < 1, we have -1 < S < 1. In too Therefore, separation effects due to parametric pumping extreme cases = 1 ( f =~1and fA = 0 ) and s = -1 ( f =~0 action are reduced and the efficiency 70 is accordingly deand fR = l),the product R and the reactant A are perfectly creased. In the extreme case where b~ tends to infinity, separated between two phases. In the former case, R is in the chemical equilibrium will be attained in any stage and the moving phase and A in the stationary phase and vice versa in efficiency 70 will become unity. the latter case. Such an ideal situation will, however, never be The effect of d c / 6 ~is shown in Figure 7 for the standard encountered in practice. value of SH = 0.01. When 6c # 0, it is necessary to specify the To improve the efficiency of parametric pumping with revalue of KC and in this figure KC is set equal to K H . As 6c/bH is action, it is desirable to accumulate A in one reservoir and R increased, the yield XR,O,- is decreased slowly while XA,O,- is in the other reservoir as efficiently as possible. For this end, increased and in the consequence the efficiency 70 is deone of SHand S c which are the degrees of separation in the creased. As both forward and backward reaction rate conhot and cold halves of the cycle, respectively, should be large stants during the cold half-cycle are increased due to the inand the other small. The greater the difference ISH- Scl is, the more 7 will be improved. The product R will be concen- * crease in bc while holding KC constant, the backward reaction becomes predominant in stages near the hot reservoir because trated in the hot reservoir if SH> Sc and in the cold reservoir R is richer than A. This is a reason why the efficiency 70 at the if SH< Sc. Therefore, generally, 7 = 70 if SH> S c and 7 = hot reservoir is decreased as bc is increased. However, if S c / b ~ q ~ if +SH~< Sc. In addition to the improvement of efficiency, < 0.01,70 is independent of 6c and is almost equal to 70 a t 6c we must strive for the improvement of the yield of R which in = 0. Even if bc/bH = 1,that is, the reaction rate constant k is our case corresponds to one of dimensionless concentrations independent of temperature, the efficiency 70 is much greater in the hot and cold reservoirs, XR,O,- and XR&+l,m. The higher than unity. Therefore, the effect of 6c on the performance of concentration of A in the stationary phase during the hot parametric pumping is not important. half-cycle where the reactions mainly proceed is required to The effect of KH is shown in Figure 8. Since KH is the ratio obtain the higher yield. If SHis large, this requirement will of the backward to the forward reaction rate constant, the be satisfied. Thus, it is desirable to choose a solvent having a backward reaction becomes higher than the forward reaction large value of SHand a small value of S c for simultaneous as KH is increased. Thus, the yield XR,O,- is decreased rapidly improvement of the efficiency and the yield. Since SH= 0.1 and the efficiency 70is decreased, too. However if KH < 1 0 , ~ o and s c = -0.1 for standard values o f f , f A H = fRC = 0.5 and is almost constant. It is noted that XA,O,- and XR,N+l,m have fAC = fRH = 0.6, we can expect that the parametric pumping peaks. When K H 0, the backward reaction becomes negliwill be effective for these values. gible and A can convert to R completely. The product R is The effect of fAC and fRH which are always held equal to concentrated in the hot reservoir due to the parametric each other is shown in Figure 4 for the standard values of fAH pumping action and XR,O,- tends to unity while XA,O,=, XAfl+l,m = fRC = 0.5. When fAC = fRH = 0.5, no separations of A and and XR,N+l,- approach zero. On the other hand, when KH R occur because SH= SC = 0 and SH- S c = 0. Therefore, we a, the backward reaction becomes so large that A cannot have 7 = 70 = 7 ~ + =1 1where the parametric pumping cannot convert to R. The reactant A is concentrated in the cold resbe effective. ervoir and X A , N + l , m tends to unity while XA,O,-, XR,O,-, and When fAC = fRH > 0.5, both the efficiency ~0 and the yield XR,N+1,= approach zero. XR,O,- are high because SH> 0 and S c < 0. On the other hand, T h e Effects of Operational Parameters. Figures 9 and when fAC = fRH < 0.5, the yield XR,N+l,m is low although the ~ high because SH< 0 and Sc > 10 show the effects of operational parameters, p~ and N . The efficiency q ~ is+relatively effect of PT is shown in Figure 9 for both discrete transfer 0. model and continuous flow model. The value of p~ for the The effect of change in fAC is shown in Figure 5 for the former model corresponds to FT/VMfor the latter. For the standard values of f A H = fRC = 0.5 and fRH = 0.6 where the former model, there exists an optimal value of p~ = 11 that c 0.4, there is value of SHis always equal to 0.1. When f ~ = almost no separation of A and R because SH= S c = 0.1 and maximizes 70 although the yield XR,O,- is increased monotoSH- SC = 0. So, the efficiency 7 is nearly equal to unity. When nously as p~ is increased. Since the period of half-cycle, T is fAC > 0.4, both the efficiency 70 and the yield XR,O,- are high ~ fixed, the reaction time during each reaction step, T l p is because SH> SC.When fAC < 0.4, the yield XR&+I,- becomes decreased as p~ is increased. Therefore, mole changes during extremely low although the efficiency 7 ~ + is1 high because SH each reaction step are decreased with the increase in p~ while < sc. separation effects are increased and in consequence the effiThe efficiency of parametric pumping for standard values ciency 70 is increased. However, when p~ becomes too large, of parameters is 7 = 70 = 92.6 as seen in Figures 4 and 5. On the reaction time is so small that mole changes during each
s
-
-
198
Ind. Eng. Chem., Fundam.. Vol. 16, No. 2, 1977
reaction step are not sufficient and the efficiency 70 is decreased through a maximal value. For the continuous ilow model, there is an optimal value of FT/VM FZ 7 that maximizes 70.The value of 70 for the continuous flow model is about )5 y4 of that for the discrete transfer model. The yield XR,O,= for the former is also inferior to that for the latter. Since the moving phase flows continuously in the case of the continuous flow model, separation effects are reduced by the continuous mixing of fresh and old fluids in each stage. 011 the other hand, in the case of the discrete transfer model, since the moving phase can be transferred at a time followed by the batchwise reaction during each reaction step, the separation of R from A is effective. The effects of other system parameters for the continuous flow model are not indicated in this paper because they are similar to those for the discrete transfer model. The effect of N is shown in Figure 10 for the optimal value of p ~ As. N is increased, 70 is increased and converges to a constant value for N larger than 20. The yield XR,O,- is decreased until N = 5 and then increased for N larger than 5. The optimal value of p~ is roughly equal to half of N . Although the product R is rich in stages near the hot reservoir and the reactant A rich in stages near the cold reservoir, the difference of concentration between adjacent stages cannot be so large that the sufficient number of stages is required to improve the efficiency.
-
Conclusions Extraction parametric pumping was applied to the system with the first-order reversible reaction. If the parametric pumping is used for suitable combinations of a reactant, a product and two immiscible solvents, the quality of the product can be remarkably improved in comparison with both the extractive reaction and the pure reaction without loss of the yield. However, since an alternate repetition of heating and cooling is required for the parametric pumping, a consumption of heat energy will be much higher than that for other two operations. After all, if the parametric pumping is adopted, the quality of the product can be improved at the cost of an increase in the consumption of heat energy. Therefore, the parametric pumping will be adopted if heat energy is not only abundant but also available at a low price. I t is most desirable for the parametric pumping to choose such exceptional systems where mA and mR are decreased and increased, respectively with increase in the temperature. For example, as the temperature is increased from 30 to 50 “C, the distribution coefficient of trilinolein is decreased while that of triolein is increased for nearly immiscible two solvents of furfural and n-heptane (Perry and Chilton, 1973). However, the present authors are not sure if the reversible reaction between trilinolein and triolein exists since the relating kinetics are not available. On the other hand, for most systems, mA and mR will both increase or both decrease with increase in the temperature. In these systems, the parametric pumping is still effective if dependencles of mA and mR on temperature are considerably different with each other. The comparison of two models indicates that the discrete transfer system is more effective than the continuous flow system. However, for the continuous flow model, we assumed the complete mixing of fresh and old fluids in each stage and then the performance inay be lowered. If an axial mixing for both moving and stationary phases is negligible, it is considered that the performance may be identical with that for the discrete transfer model if the number of stages tends to infinity. Appendix Mole Change Due to the Reversible Reaction at a Re-
action Step. The reaction is taking place batchwise in each stage. Thus, we have
Since the total moles of A and R are not changed during the reaction step, we have
+
+
MA MR = MAJN MRJN= constant
(A21
and from eq 4 and 5 , we have
VSCAS= (1- ~ A ) ( V M C A + MVSCAS)= ( 1 - ~ A ) M A(A31
VSCRS= (1- ~ R ) ( V M C R + MVSCRS) = ( 1 - ~ R W R(A41 Eliminating CASand CRS by eq A3 and A4 from eq A1 and using eq A2 yield
+ k’(1 - ~ R ) ( M A ,+I NMR,IN)
(A51
By integrating eq A5 from zero to T/PTand determining the integral constant in terms of the initial condition, MAJN,we obtain
AM = MA - MAJN= y l l - exp(-4)l
(14)
where
Y = ( 4 1- ~ R ) M R , I N - (1 - ~ A ) M A , I N ~-/ f( K~(+~) (1- /A)) 4 = 6141 - f ~+ )(1- ~ A ) J / P T
(15) (16)
= k’/k
(17)
6 = kT
(18)
K
Nomenclature C = concentration, mol/cm3 F = flow rate of moving phase, cm3/s f = fraction solute in moving phase, dimensionless k = forward reaction rate constant of first order, s-l k’ = backward reaction rate constant of first order, s-l M = mole solute in stage, mol A M = mole change due to reaction, mol m = distribution coefficient, dimensionless N = total number of stages p~ = total number of transfer steps per half-cycle Q = quality of product, dimensionless r = reaction rate, mol/cm3 s S = degree of separation, dimensionless T = time required for half-cycle, s t = time, s V = volume, cm3 Greek Letters y = ( 4 1 - ~ R ) M R , I-N( 1 - ~ A ) M A , I N~ f/ M~+ ~ )(1- f ~ ) 1 ; y, K , f~ and f~ have the same subscript of C or H 6 = kT; 6 and k have the same subscript of C or H 7 = efficiency of parametric pumping defined by eq 38 70 = KH(XR,O,m/XA,O,m) 7N+1 = KH(XR,N+l,m/XA,N+l,-) ~ E R= efficiency of extractive reaction defined by eq 39 K = k’/k; K,k and k’ have the same subscript of C or H - f ~ () 1 ~A))/PT; 4,6, K , f ~and, f~ have the same 4 =
+ -
subscript of C or H
XA,i,j XR,i,j
= =
+ CRM,N+1,0) + CRM,N+1,0)
CAM,i,j/(CAM,N+l,O CRM,i,j/(cAM,N+l,O
Subscripts A = reactant C = cold halfofcycle ER = extractive reaction H = hot half of cycle Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
199
Goto, S.,Matsubara, M., J. Chern. Eng. Jpn., 5, 87 (1872a). Goto, S.,Matsubara, M., J. Chem. Eng. Jpn., 5, 90 (1972b). Goto, S.,Matsubara, M., J. Chem. Eng. Jpn., 7, 213 (1974). Gregory, R . A., A.LCh.E. J., 20, 294 (1974). Perry, R. H., Chilton, C.H., "Chemical Engineers' Handbook," 5th ed, p 15-7, Mc(;raw-Hlll, New York, N.Y., 1973. Sweed, N. H., Wilhelm, R. H.,Ind. Eng. Chem., Fundam., 8, 221 (1969). Tam, L. V., Miyauchi, T., J. Chem. Eng. Jpn., 6, 271 (1973). Wankat, P. C., Ind. Eng. Chem., Fundam., 12, 372 (1973). Wilhelm, R. H., Rice, A. W., Bendelius, A. R., Ind. Eng. Chem., fundam., 5, 141 (1966).
IN = initial condition
i = stagenumber j = cyclenumber M = moving phase N 1 = cold reservoir 0 = hot reservoir PR = pure reaction p = transfer step number R = product S = stationaryphase m = infinity
+
Literature Cited Apostolopoulos, G. P., Ind. Eng. Chem., fundam., 14, 11 (1975). Chen, H. T., Lin. W. W., Stokes, J. D., Fabisiak, W. R., A.I.Ch.E. J., 20, 306 (1974).
Received for review October 17, 1975 Accepted September 17,1976
An Empirical Correlation among Areotropic Data Keith M. Seymour," Ralph H. Carmichaeh Jerry Carter,2James E I Y ,Eric ~ Isaacs,2Jerry King, Robert NaylorI2 and Tina Northern2 Department of Chemistry, Butler University, Indianapolis, Indiana 46208
+
An equation log [ 10 ( N A / N ~ = ) ] -0.234At 0.874 serves to calculate NAfor the 251 reported alcohol-alkane, azeotropes with an average ANAof 0.050. The calculated value of NAfor 1108 azeotropes involving 15 different types of azeotropes has an average ANA of 0.056. A modification of the equation gives an average ANA of 0.028 for the alcohol-alkane azeotropes and an average ANA of 0.046 for the 1108 azeotropes.
Introduction It was early recognized that there is a relationship between the difference between the boiling points of the components of an azeotrope and its composition. Lecat (1918) used a power series to relate the composition of the azeotropes formed between members of an organic family of compounds with a single fixed component, e.g., n-alkanes and ethanol. In eq 1, X A is the weight fraction of component A and At is the difference between the boiling points of the two components. x A = A0
+ A l ( A t ( + A 2 E 2 + As(E'31+ . . .
(1)
The necessity of using essentially all of the s m d l number of members of a given family forming azeotropes with a fixed component in order to determine the constants in eq 1made this of little predictive value. In the 1940's, Mair et al. (1941) and Skolnik (1948) described graphical methods of correlating the composition of the azeotropes within an organic series, which were of limited utility. Meissner and Greenfield (1948) reported another graphical correlation which led to eq 2. These workers
~OONA = 55 - 0 . 9 1 5 [ T ~ ~ ( TT~A ) ]
(2)
found the graph useful for hydrocarbons (except terpenes) and halogenated hydrocarbons with alcohols, phenols, and cresols. Using 50 in place of 55 as the constant in this equation, azeotropes between halogenated hydrocarbons and ketones and aldehydes could be correlated. A decade later, Johnson and Madonis (1959) used variations of eq 2 to include a number
of other series of compounds. Because of the many variations in the form of their equations and in the constants required, this treatment suffered from the limitations of Lecat's method. A graphical correlation for azeotropes involving several series of compounds was proposed by Seymour (1946)in which At was plotted against N A . These curves were fitted quite closely to eq 3. 1 A t /3 N A = 0.5 . -arcsinh(3)
+
a
(
3
log 10correspondence to this author a t 810 Gonzalez Drive, Apt. 2L, San Francisco, Calif. 94132. Petroleum Research Fund Fellows. 1 Address
*
200
Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
Y
This type of equation gave useful correlations but because of the variety of constants required, it was limited in the same way as the treatments of Lecat (1918) and Johnson and Madonis (1959). In an effort to find a more useful correlation, the data for the azeotropes between 1-alkanols and n-alkanes from Horsley's (1947) collection were combined with some data from this laboratory and the values of At were plotted against a variety of functions of different properties of the constituents of the azeotropes. The most useful results were obtained by plotting At against log NAINB.The curves obtained were largely linear with slight sigmoid curvature a t each end. The data for the azeotropes formed from the first three 1-alkanols and n-alkanes available a t that time are plotted in Figure 1. The lines drawn through the points were calculated from the data for each alcohol by the method of least squares and fit eq 4. It is observed in Figure 1that there is a rather systematic
=mat+ b
(4)
shift in the slope of the lines as the chain length of the alcohol is increased. This suggested that a variable parameter related
