733
I n d . Eng. Chem. Res. 1991,30, 733-739
case in which all the dimensionless terms are of the same order. In general, the terms of larger relative size will tend to show greater accuracy at the expense of the smaller terms. Literature Cited Carbonell, R. G.; McCoy, B. J. Moment Theory of Chromatographic Separations: Resolution and Optimization. Chem. Eng. J. 1975, 9,115-124. Felder, R. M. Estimation of Gas Transport Coefficients from Differential Permeation, Integral Permeation, and Sorption Rate Data. J. Membr. Sci. 1978,3, 15-17. Felder, R. M.; Ma, C.-C.; Ferrell, J. K. A Method of Momenta for Measuring Diffusivitiea of Gases in Polymers. AIChE J. 1976,22, 724-730. Goodknight, R. C.; Fatt, I. The Diffusion Time-Lag in Porous Media with Dead-End Pore Volume. J. Phys. Chem. 1961, 65, 1709-1712.
Kang, Yong-Soo. Diffusion of Organic Vapors in Glaeey and Heterophase Polymer Systems. Doctoral Thesis, Chemical Engineering, Tufts University, Medford, MA, May 1986. Meldon, J. H.;Kang, Y.-S.; Sung,N.-H. Analysis of Transient Permeation through a Membrane with Immobilizing Chemical Reaction. Znd. Eng. Chem. Fundam. 1986,24,61-63. Michaels, A. S.; Bixler, H. J.; Fein, H.L. Gas 'I'ransport in Thermally Conditioned Linear Polyethylene. J. Appl. Phys. 1964, 35, 3165-3 178. Noble, R. D. Analysis of Facilitated Transport with Fixed Site Carrier Membranes. J. Membr. Sci. 1990,50,207-214. Tsuchida, E.; Nishide, H.; Ohyanagi, M.; Kawakami, H. Facilitated Transport of Molecular Oxygen in the Membranes of PolymerCoordinated Cobalt Schiff Base Complexes. Macromolecules 1987,20,1907-1912.
Received for review June 7, 1990 Revised manuscript received October 10,1990 Accepted October 31, 1990
Generalized Design Equations for Backmixed Liquid Extraction Columns with Nonlinear Equilibria H.R.Clive P r a t t * a n d Geoffrey W . Stevens Department of Chemical Engineering, University of Melbourne, Parkville, Victoria 3052, Australia
Two methods are described, in terms of the backflow model, for calculating explicitly the number of nonideal stages required in a backmixed multicompartment extraction column with nonlinear equilibria to achieve a given performance. These are analogous to methods described previously for differential extractors in terms of the diffusion model. In both cases the equilibrium curve is represented by two or three straight line segments, and the resulting equations are expressed in generalized form to cover both models. Worked examples indicate that the methods are of adequate accuracy for both design and scale-up of extraction equipment. One of the methods is also of particular value for the computation of backmixed columns for dual-solvent extraction processes. Introduction It is now generally accepted that axial dispersion plays an important part in obtaining the performance of most types of countercurrent liquid extraction column. This takes the form both of backmixing, especially of the continuous phase, and of forward dispersion ("forward mixing") of the dispersed phase arising from the range of droplet sizes present, with consequent spectrum of droplet velocities. Sleicher (1959, 1960) proposed two models to account for backmixing, termed respectively the "diffusion model", applicable to differential type contactors such as packed columns, and the "backflow model", applicable to staged (i.e., compartmental type) columns. Analytic solutions of these models for systems exhibiting linear equilibria are complex; however, Pratt (1975,1976b)obtained simplified solutions of adequate accuracy for all but impracticably short columns, which were later combined into a generalized form applicable to both models (Pratt, 1983). Unfortunately, few systems exhibit linear equilibria, and exact analytic solutions are not obtainable for the nonlinear cases. However, Pratt (1976a) proposed two alternative methods for the calculation of column height in terms of the diffusion model, involving the approximation of the exact equilibrium curve by two or three linear segments, and the solutions for the simpler of the two methods were later combined into a generalized form (Pratt, 1983a). This method is not always applicable, however, as in many cases one or more of the linear equilibrium line segments may intersect the operating line on extrapolation. This is of no consequence with the alternative method, which in-
volved the use of arbitrary boundary conditions between the terminations of each of the equilibrium line segments. In order to complete the range of available methods, therefore, the corresponding relations are derived below for the backflow model and expressed in generalized form covering both models. Review of Methods Linear Equilibrium. The derivations of the two models have been presented by several workers, e.g., Sleicher (1959,1960);Hartland and Mecklenburgh (1966); Pratt and Baird (1983). The solution for the feed- (i.e., X-) phase profile, in dimensionless units, is as follows for the diffusion model: X = A I A2eA2'HZ A4eAs'HZ A4euHz (1) and for the backflow model: X = Ai + A ~ J L+~ "A3p3" + A,," (2) The values of A1-A4 differ for the two models. The corresponding solutions for the extractant- (i.e., Y-)phase profile are obtained by multiplying A2-A4 in (1)and (2) by ~ 2 - a 4 ,respectively; the latter are defined in Table IV. Values of the A' and p in (1)and (2) are given by the roots of a characteristic equation, which, when backmixing occurs in both phases, is cubic, as follows: K3 - f f K 2 - @K - y 0 (3) where K corresponds to either A' or p. This reduces to a quadratic equation when backmixing occurs in only one phase. Values of the coefficients a,8, and y for the two models are summarized in Table I.
+
+
Q8SS-5SS5/91/2630-0733$02.50/0 @ 1991 American Chemical Society
+
734 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 Table 1. Coefficients of Characteristic Equations backmixed phase(s) eauation model" a both eq 3
D
X
D
K~
coefficients 8 P, + EP, PIPy
Y
+-dch2
Hoxdch
E + -PI
- f l -~ y = 0
dch
B
B
OD = diffusion model; B = backflow model. Table 11. Roots of Characteristic Equation value of root" backmixed phase(s) both X
K2
a / 3 + 2p'/2 cos ( 4 3 ) P / 2 + [(6/2)*+ Y11'2
K3
a13 --m
-1
(Dmodel) (Bmodel)
Y
-m
neither
-@/a - [ ( B / 2 ) 2- Yl''z
-m
4-m
-1
K4
a / 3 + 2p1l2COS ( u / 3 + 240') 612 - [ ( 8 / W + Y11'2
+ 2p'/2 cos ( u / 3 + 1200)
-8l2 + [(8/2)2- Y1'/2 (E - l)/& (Dmodel) Mo,(E'(B model)
(Dmodel)
5
(Bmodel)
1+ P
OD = diffusion; B = backflow. p = ( ( ~ / 3+)613; ~ q = ( ~ t / 3+) a@/6 ~ + 712; u = cos-' ( q / ~ ' . ~ ) . The constants A in (1)and (2), obtained by substitution of the four boundary conditions, i.e., for each phase at each end of the column, into (1)and (2), lead to lengthy expressions for X that cannot be solved explicitly for column height or number of stages. However, by taking into account the properties of the roots, K~ of (3), Pratt (1975, 1976b) showed that (1)and (2) could be reduced to simplified forms that can be solved explicitly for column height, H,or number of real (i.e., nonideal) stages, N. He further showed (1983a) that the two solutions can be combined into a unified form, as follows:
O X
I
CY
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
CX
where C = K 4 (diffusion model) or In ( K + ~ 1) (backflow model), and Yo is the exit extractant phase composition in dimensionless units. Values of the roots K Z - K ~ in (4), obtained by solving (3) with the appropriate values of the coefficients CY, @, and y given in Table I, are listed in Table 11. If backmixing occurs in only one phase, (3) reduces to a quadratic also given, together with the roots, in Tables I and 11. The value of iVoxappearing in the backflow model equations (Table I), i.e., the number of transfer units per stage, is related as follows to the Murphree stage efficiency, EMI, when the individual stages are fully backmixed (Pratt, 1983b). (5) E M x = wox/(l + wax) Although approximations are involved in deriving (4), it was found that the errors are negligible for column
Figure 1. Operating diagram for example 1 and for the corresponding diffusion model example from Pratt (1976a). Operating line for backflow model plotted from matrix solution and points ( 0 ) from simplified solutions.
heights corresponding to two or more transfer units or actual stages (Pratt, 1975,1976b). Values of the roots are also included in Table I1 for the case of zero backmixing in either phase; the resulting solutions are then identical with the exact expressions for plug flow. The backflow model solution also reduces to that for theoretical stages when IPoI w. Nonlinear Equilibrium (Method 1). As stated earlier, analytical solutions to the models are not obtainable when the equilibrium relationship is nonlinear. However, results of sufficient accuracy are obtainable by representing the equilibrium curve by means of two or three straight segments, as shown for two segments in Figure 1. The simpler method for this purpose is that described as the "second method" by Pratt (1976a). In this procedure the equilibrium curve is represented initially by the two
-
Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 735
-.-
n R
straight lines ABC and GBD, intersecting at B. The height or number of stages required over the full concentration range AC is first calculated by using (4); the corresponding values for the range AB are then calculated from the following expression, obtained by rearrangement of the simplified profile equation
I
=Y
00
The same procedure is applied using equilibrium line GBE, calculating the height or number of stages for the full concentration range, and the proportion for section GB; the value for section BD is then obtained by difference and addition of the values for AI3 and BD gives the total height or total stages. Finally, the inlet "jump" concentrations are calculated from the following expressions:
'1
(8) In Figure 1,the jump concentration cy^ is used to relocate the equilibrium line segment GBD to G'BD' noting that only the portion of the equilibrium curve above the jump concentration is effective for mass transfer. The example shown in Figure 1has been described fully for the diffusion model by Pratt (1976a) and Pratt and Baird (1983), who obtained a total height of 9.82 ft, revised to 9.96 f t using equilibrium line G'BD'; the exact value, obtained numerically, was given as 10.00 f t by Mecklenburgh and Hartland (1967b). Alternative Procedure For Nonlinear Equilibrium (Method 2): Development of Equations. The extension of the foregoing method to the case where the equilibrium curve is represented by three linear segments is obvious. However, this method has limitations when applied to systems of considerable nonlinearity, such as that shown in Figure 1. Thus, the performance indicated by that example is poor, with a solute recovery of only 48.8%, and to obtain a higher recovery it would be necessary to employ a third equilibrium line segment such as EF in Figure 2. However, as shown, both the upper and lower segments intersect the operating line so that a solution is not obtainable by this means. Although a somewhat extreme case, such as situation can often arise when a close approach is required with systems where the nonlinearity is quite small. In an alternative approach for the diffusion model, Pratt (1976a) treated each column section as an independent entity, each with dimensionless feed and extractant inlet concentrations normalized to 1.0 and 0, respectively. As previously stated, it is not possible to specify exact boundary conditions at the junctions of the operating lines corresponding to the individual column sections. However, as empirical approximations, it was suggested that the
01
02
03
04
05
06
07
08
09
10
CX
Figure 2. Operating diagram for example 2. Operating line plotted from matrix solution and points ( 0 )from simplified solution.
conditions dnX/dZn = 0 and dnY/dZn = 0 could be applied, where n I 2. It was found in fact that satisfactory agreement with the exact results was obtainable for n = 3, and this procedure was therefore used to obtain appropriate design equations. It is apparent that there is a need for a comparable method for the backflow model. This has now been accomplished by replacing the above boundary conditions in terms of third-order differentials by the corresponding third-order differences;an outline of the derivation is given in the Appendix. It has also been found that the resulting expressions can be combined with those for the diffusion model (Pratt, 1976a), giving the generalized equations summarized in Table 111. These are applicable to all combinations of backmixed and plug-flow phases, using the appropriate values of the roots ~i listed in Table 11. Description of Procedure. It is assumed that the relevant concentrations and parameter values have been specified in the form shown in the worked examples described later. The operating diagram is first drawn on (cy, c,) coordinates, showing the equilibrium curve and plug flow operating (i.e., balance) line. The equilibrium curve is then approximated by two or three straight line segments intersecting at Y-phase compositions cy& and, if appropriate, cy,r;the slopes and intercepts of each are measured precisely and used to calculate the coordinates of the intersections. The successive steps of the procedure are then carried out as follows: 1. The exit extractant phase composition cy,o is calculated from the overall mass balance and expressed in dimensionless form, yf,o,for the feed inlet section. The extraction factor for this section, Et, is then calculated, followed by the characteristicequation roots and a, values (Tables I, 11,and IV). Finally, the number of stages, Nf, the dimensionless X-phase exit composition, XNf, and, if desired, the X-phase jump concentration, X,, are calculated using (9)-(11) in Table 111. 2. The value of cIoa ( = c a s ) is calculated from XNf and used to convert the Y-phase exit concentration, c for the middle section into dimensionless form, Yo,. &e value of E, is then calculated, followed by the K, and a, values, and these are used in (12) and (13)to obtain N , and X N p 3. The value of cXoJ(=cfia) is obtained from XNa and used to calculate X , ,from c,A, the specified feed phase composition leaving the raffinate outlet section. Values of E,,K,, and a, are then calculated and used in (14) and (15) to determine N , and YJ,r. 4. The total number of stages is then given by N = Nf + N , + N,. 5. Finally, the Y-phase exit jump concentration, cy8+1, is calculated from Y,s, and used to determine whether the
736 Ind. Eng. Chem. Res., Vol. 30, No. 4,1991 Table 111. Generalized DesiPn Eauations' Feed Inlet Section height or number of stages
Table IV. Definitions of Parameters parameter diffusion model
backflow model
exit feed phase composition xl,f or xN.f
=
- (cyN+l- q ) l m cy - cyN+l mc,o - (cy' - q )
c,o
feed phase "jump" concentration
Table V. Results for Example 1' section of equilibrium line no. of stages column m q E method 1 method 2 feed inlet 1.2623 -0.3938 0.7917 6.371 5.369 raffinate outlet 0.909 -0.200 1.0994 3.731 4.627 10.102
total stages
Middle Section
Values of the Y-phase jump concentrations were almost identical at cyj = 0.100 by the two methods.
height or number of stages
H, or ( N-, - 2) = iCl n (
[
-
a4( a3
($ -
- a3a4[
Computer Program. The foregoing methods are particularly amenable to solution by computer, as they can form the basis of an iterative procedure for the scale-up of columns from pilot plant data. Thus, programs have been written in both BASIC and FORTRAN which permit solution in any of three modes, viz., (i) design, for the computation of column height or number of stages; (ii) performance,for the calculation of H,,or No, from pilot plant data, using an iterative Newton-Raphson procedure; and (iii) operation, for calculating exit concentrations for a particular column, given the values of the mass-transfer parameters, HOxor NOx, the backmixing parameters, and the inlet compositions. Either the diffusion or the backflow model can be selected, with linear or nonlinear equilibria; in the latter case, either of the present methods can be used. The extraction factor, E, can be calculated in all three modes, or specified in modes i and iii. When calculated, allowance is made for changes in solvent ratio due to solute transfer, to give a conservative design (Pratt, 1983a).
11
[
ads.? - a2S$( 9 ] ( 1 - Yo,,)
exit feed phase composition XI,
or
XNm
=
Raffinate Outlet (Extractant Inlet) Section height or number of stages
(14)
extractant phase "jump" concentration
Worked Examples As a first example, a case similar to one described previously for the diffusion model (Pratt, 1976a) will be considered and will be solved by both methods. Example 1. The specification for this example is as follows: equilibrium relation: cy = c,2 cx0 = 1.00
c X a= 0.512 a
9.996
See Table IV for definitions of parameters,
placement of the equilibrium line segment for the raffinate outlet section is satisfactory (Pratt, 1976a). If the column is divided into only two sections, step 2 is omitted from the above procedure. Although the above account relates to the backflow model, its application to the diffusion model is self-evident.
cy0 = 0.0
F,/Fy = 1.0 LY, = aY = 2.0
No, = 0.50
The equilibrium relationship is approximated by the lines AB and BE in Figure 1. The corresponding values of the slopes and intercepts are given in Table V, together with the number of stages calculated by the two methods. For comparison, the performance was calculated by the exact matrix method described by Ricker et al. (1981),
Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 737
mated by finite steps, or numerical integration can be used.
Table VI. Results for Examde 2 equilibrium line
section of column m Q E 1.2623 -0.3938 0.5282 feed inlet 0.909 -0.200 0.7334 middle raffinate outlet 0.511 -0.0588 1.3047 total stages
9.37 8.71 12.53
X-phase concn at inlet 0.8975O 0.6084 0.4126
30.61
0.0273b
no. of stages
X-phase inlet jump. bY-phase inlet jump.
which gives the concentration profile for a specified number of stages. Assuming N = 10 stages, this gave a value of 0.504 for the exit feed phase concentration, in good agreement with the specified value of 0.512. Example 2. In this example the same equilibrium relationship will be assumed, but with a much closer approach to equilibrium at the raffinate outlet. As a consequence it will be necessary to represent the equilibrium curve by three straight segments AB, BC and CD, as shown in Figure 2. It is observed that those for the feed inlet and raffinate outlet sections both intersect the balance line within the operating range when extended, so that only method 2 can be used. The specification for this example is as follows: equilibrium relation: cy = c,2 c,O = 1.00 ~ f=i0.250
cy” = 0.0
FJF, = 0.667 (Y,
= (Y, = 2.0
Nox = 0.5
The resulta of the computations are summarized in Table VI. Assuming a total of N = 31 stages, the exit feed phase composition obtained using Ricker et al.’s matrix method (1981) was 0.254, in close agreement with the specified value. The operating line shown in Figure 2 was plotted from the resulting concentration profile.
Discussion Reference to Table V shows that both methods give effectively the same result for the total number of stages in example 1, although there is a difference of about one stage between the individual values for each section. This probably arises from the effects of the third-order differences used as boundary conditions between sections, as these extend over a finite number of stages. This is in contrast to the diffusion model, for which the corresponding boundary conditions are point values, and negligible differences were in fact observed between the heights of the individual sections calculated by the two methods for the comparable example studied by Pratt (1976a). It can be concluded that both methods provide reliable solutions for design purposes provided that care is taken in the fitting of the equilibrium line segments, especially in the vicinity of the intersections; however, the first method is to be preferred when applicable. As alternatives to the present methods, both the “boundary iteration” and the matrix method might be considered. The former, described originally in graphical form by Rod (1964), involves guessing the inlet “jump” concentration at one or the other end of the extractor and calculating the profile by stage-to-stage computation until the other end is reached; if the required boundary conditions are not met, the calculation is repeated using a fresh initial value of the jump concentration. For differential extractors the continuous profiles can be approxi-
Unfortunately, this method suffers from the extreme sensitivity of the profiles to the initial guess of the inlet concentration; thus, Mecklenburgh and Hartland (1967a) showed that under some circumstances, using a 39-bit computer, convergence was not obtainable with a 1-bit error in the starting value, i.e., with an accurancy of 1in 10l2. This problem can be overcome, if backmixing occurs in only one phase, by starting the computation at the entry of the phase that is not backmixed; however, circumstances can arise, as in dual-solvent extraction (see below) where this is not possible. The matrix inversion method referred to earlier, for which a computer program is available (Ricker et al., 1981), is restricted to stagewise extractors, although differential extractors can be approximated using finite steps. However, the method yields the concentration profile for a given number of stages, so that it is necessary to assume the latter and to iterate on this until the desired exit concentrations are obtained. Additional iteration, using a Newton-Raphson procedure, is also required for nonlinear equilibria, and it is then necessary to provide gradients of the distribution coefficient throughout the extractor. The method is therefore tedious and may require considerable computer time; however, the present methods could be used profitably to provide an initial estimate of the stage requirements. An important application of the second of the present methods is for the computation of the total stages, or height, required in dual-solvent extraction (Dongaonkar et al., 1988). This involves successive iterations of stage numbers and exit compositions for two solutes in the two sections of a column, above and below a central feed point. The results given by the present method for a specific system were confirmed by use of the matrix method; however, the use of the latter alone, without prior knowledge of the stage requirements, would require much iteration and would be very wasteful of computer time. In practice, backmixing of continuous phase has been found to occur in the majority of extraction column types, and extensive data on backmixing parameters are available in the literature; these have mostly been obtained by tracer injection methods in the absence of mass transfer. Many workers have obtained axial dispersion data for the dispersed phase by the transient tracer injection (i.e., residence time distribution) method and have reported the results in terms of a backmixing parameter. However, evidence is accumulating to the effect that in fact such dispersion is not due to backmixing, but to “forward mixing” of the droplets due to the range of their sizes, and hence of their corresponding velocities in the droplet swarm; such is the case for the packed column (Gayler and Pratt, 1957), pulsed plate column (Aufderheide and Vogelpohl, 1986; Prvcic et al., 1989), and Kuhni column (Dongaonkar, 1990) and probably applies in whole or part to most column types. As shown by Levenspiel and Fitzgerald 11983), it is unsound to interpret tracer responses in such cases in terms of a backflow parameter. The effect of such forward dispersion, due to droplet polydispersivity,is to reduce column performance in comparison to that for a monodispersion. Further, the effect is a property of the dispersion itself rather than of the extractor diameter. For scale-up purposes, therefore, the value of the mass-transfer parameter H, or Nosobtained by the present method from pilot-scale data with Ped or ffd set equal to zero will be based on a value of the overall mass-transfer coefficient averaged over all droplet sizes (an consequent slip velocities) and reduced in accordance with
738 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991
the degree of forward mixing. The resulting value should therefore provide a satisfactory basis for scale-up purposes subject to the superficial phase velocities being the same and the appropriate value of the continuous-phase backmixing parameter being used, if different from that in the pilot column.
Nomenclature Note. See Table IV for definitions of parameters. Ai = constants in (1)and (2) ai = constants (Table IV) c . = concentration in phase j, kg m-3 d, = characteristic dimension, e.g., packing size, compartment height, m Ej = effective longitudinal diffusion coefficient in phase j , m2 23-1
E = extraction factor, UJmU or FJmF, F. = flow rate of phase j, m3 syi If = height of column, m m = dcy*/dcx,slope of equilibrium line N = total number of actual stages (i.e., compartments) n = number of stage counted from feed inlet Nox= number of overall X-phase transfer units per stage Pi = d,Yi/E,, Peclet number q = -mq intercept of equilibrium line on cy axis q’ = intercept of equilibrium line on c, axis U . = superficial velocity of phase j, m s-l x‘ = dimensionless concentration in feed phase (Table IV) Y = dimensionless concentration in extractant phase (Table IV) 2 = z / H , fractional length within extractor z = length within extractor measured from X-phase inlet, m Greek Symbols a, 0, y = coefficients of characteristic equations (Table I) aj= backmixing ratio for phase j , i.e., ratio of backflow velocity to uj K~ = ith root of generalized characteristic equation X i = ith root of characteristic equation (diffusion model) ~. l1~ = ith root of characteristic equation (backflow model) Subscripts d = dispersed phase f = feed inlet section I = feed phase exit end (diffusion model) i = number of root of characteristic equation j = X or Y phase “jump” concentrations (Table IV, Figures 1 and 2) k , 1 = intersections of equilibrium line segments m = middle section n = number of stage counted from feed inlet end N = feed phase exit stage (backflow model); N + 1refers to fictitious end stage o = feed phase inlet end within column (diffusion model), or stage 0 (fictitious end stage), backflow model r = raffinate outlet section x = feed phase y = extractant phase
Superscripts * = value at equilibrium I = extractant phase entering column (diffusion model) N + 1 = extractant phase entering fictitious end stage N + 1 (backflow model) O = feed phase entering column or fictitious end stage (either model)
Appendix. Derivation of Backflow Model Equations Following Sleicher (1960), fictitious end stages 0 and N + 1,in which no mass transfer occurs, are assumed at the ends of each column section.
Feed Inlet Section. The two boundary conditions at the feed inlet end (n = 0) are the same as those for the linear equilibrium case (Pratt, 1976a; Pratt and Baird, 1983). These are listed below, in dimensionless concentration units, together with the appropriate conditions at the feed-phase exit end (n = N + 1). 1. X phase, n = 0: A balance around stage 0, where there is an inlet “jump”, gives x , - a,(Xt - X,) = 1 Substitution of the X values from ( 2 ) then gives 4
+ i=2C A i ( 1 - U x ~ i =) I
A1
(All
2. Y phase, n = 0: A similar balance around the Y-phase exit, noting that there is no mass transfer in stage 0, gives Yo = Yo = Y1 Hence on substitution of ( 2 ) A
C
A
~
i=2
3. Y phase, n = N
= ~ o~
K
~
+ 1: As there is no inlet “jump”
yN+1 =
yN+1 = 0
4
+ i=2C A i u i ( ~+i l)N+l= 0 (A3) 4. X phase, n = N + 1: Taking third-order differences and equating to zero ( x N + 1 - 2 x N + XN-1) - (x, - 2xN-1+ XN-2) = 0 :. x N + 1 - 3xN + 3XN-l - XN-2 =o :.
A,
Substitution of ( 2 ) and collecting terms gives finally 4
X A ~ K ? (+K 1)N-2= ~ 0
i=2
(A4)
Equations Al-A4 are solved for the values of A1-A4,which are substituted into ( 2 ) . The resulting expression for X N is then simplified, taking into account the properties of roots K 2 and K~ (Pratt, 1976b). The corresponding expression for YN,Le., incorporating ~ 2 - 0 4 into the last three terms of (21, is solved for N . A comparison of the resulting expressions for N and X N with those for the diffusion model (Pratt, 1975) then gives the generalized expressions listed in Table 111. The same procedure is used for the middle and raffinate outlet sections, using the boundary conditions given below. Middle Section. For this section there are no “jumps” at either phase inlet, but zero third-order differences are assumed at both phase outlets. The relevant equations are therefore as follows: 1. X phase, n = 0 x,o = x,, = 1.0 4
CAI = 1.0
2.
i-1
2. Y phase, n = 0: Y3 - 3Y2
.:
+ 3Y1 - Yo = 0 C4, A ~ U= 0~ K ~ i=2
3. X phase, n = N
+ 1:
Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 739 4
:.
+
C A ~ K F ( K1)N-2 ~ =0
i=2
4. Y phase, n = N
(A7)
+ 1:
yN+l
yN+1 = 0
4
:.
Al
+ i=2C A ~ U ~+(l)N+l K~ =0
(A81
Raffinate Outlet Section. 1. X phase, n = 0 x,o = x , ,= 1.0 4
:.
CAI = 1.0
2. Y phase, n = 0: Y3 - 3Yz
C A ~ U ~=K0? is2
+ 1:
xN+l 4
:.
+ 3Y1 - Yo = 0 4
:. 3. X phase, n = N
(A9)
ill
= x N + I = XN
+
C A ~ K ~ (l)N K ~= 0
i=2
(All)
+
4. Y phase, n = N 1: In this case there is an inlet "jump*, and hence, from a balance around the stage: YN+' = 0 = YN+1 - ay(YN - YN+1)
:.
4
Al
+ i=2C A ~ U ~+ (l K) N~[ +l ~ i (+l cyy)]
(A12)
Inlet "Jump" Concentrations. The X jump is obtained by substituting n = 0 in the simplified expression for the X profile in the feed inlet section. Similarly, the Y jump is obtained by substituting n = N 1 in the expression for the Y phase in the raffinate outlet section. A comparison of the reulta with those obtained in a similar manner for the diffusion model then gives the generalized expressions in Table 111.
+
Literature Cited Aufderheide, E.; and Vogelpohl, A. A Convective Model to Interpret Dispersed-Phase Residence Time Measurements in Pulsed Liquid/Liquid Extractors. Chem. Eng. Sci. 1986, 41, 1747-1757. Dongaonkar, K. R. Mass Transfer and Axial Dispersion in a Kuhni Liquid Extraction Column. Ph.D. Thesis, University of Melbourne, 1990.
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Received for review May 8,1990 Revised manuscript received September 26, 1990 Accepted October 9, 1990
