Ind. Eng. Chem. Process Des. Dev. 1981, 20, 489-492
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Scale-up of Backmixed Liquid Extraction Columns H. R. C. Pratt” and M. 0. Garg‘ Department of Chemical Engineering, University of Melbourne, Parkvilb, Victoria 3052, Australia
A method is described for the design of liquid extraction columns by scaling up from the results of small scale pilot plant tests. This is based on the assumption that H,, or is independent of column diameter, and that
&
increased backmixing accounts entirely for differences in performance on small and large scales. The method is Illustrated by means of a worked example, for a pulsed plate column.
Introduction The design of liquid extraction columns involves the prediction of two independent factors, viz. (i) the diameter required to accommodate the desired throughput, and (ii) the height necessary to give the desired degree of separation of the feed components. Ideally, these should be determined by theoretical procedures, without recourse to experimental measurement except of phase equilibria. This is in fact often possible in the case of the diameter, since satisfactory correlations of limiting throughput (i.e,, floodpoint) are available for most types of columns in the absence of interfacial turbulence arising from interfacial tension gradients (Marangoni effect). The prediction of column height is more difficult, involving as it does the interfacial area of contact of the phases, which in turn is determined by the droplet size and holdup, the individual phase mass transfer coefficients,and in most cases the degree of axial dispersion in either or both phases. Predictive methods have in fact been proposed for the perforated plate column, in which axial dispersion is absent (Skelland and Conger, 1973), and for the packed column (Pratt and Anderson, 1977),provided that interfacial oscillation does not occur. Even in such cases, however, the cumulative effect of the inevitable errors which arise in the correlation of the relevant variables is likely to lead to considerable inaccuracy. It is therefore preferable to base the design on previous experience, carrying out tests with a small-scale pilot unit in the case of new processes. In the past, scale-up has usually been based on the conventional plug-flow model, in terms of either theoretical stages or transfer units. Such methods, although satisfactory for true mixer-settlers and unpulsed perforated plate columns, are unlikely to give reliable results for most other contactor types due to the adverse effect of axial dispersion, i.e., of both true backmixing and forward dispersion. Of these the former generally predominates in the continuous phase, taking the form mainly of a circulatory backflow (Wijffelsand Rietema, 1972; Anderson and Pratt, 1978), although forward dispersion is normally included by “force fitting” of residence time distribution data to a backmixing model. Backmixing can also occur in the dispersed phase, e.g., due to droplet circulation in rotary agitated columns. However, a more general effect is that due to polydispersivity, i.e., to the wide range of droplet sizes present, each with a different specific area and velocity, and therefore a different mass transfer rate (Olney, 1964). This effect, loosely termed “forward mixing”, results in a reduced performance, although this loss is offset by R. and D. Department, Engineers India Ltd., 4 Parliament St., New Delhi, India 0196-4305/8111120-0489$01.25/0
droplet coalescence and redispersion (Chartres and Korchinsky, 1975); however, it is of a totally different nature from backmixing and, being a property of the dispersion itself rather than of column diameter (Rod, 19661, does not require to be taken into account in scaling up. Two types of backmixing model have been proposed by Sleicher (1959,1960) and have been further developed by others (Miyauchi and Vermeulen, 1963a; Hartland and Mecklenburgh, 1966);these are known as the diffusion and the backflow models, applicable respectively to differential and to stagewise (Le., compartmental) columns. These models form the basis of the proposed scale-up method. Scale-up Procedure Basis of Method. The mass transfer rate in differential contactors is determined primarily by the mass transfer coefficient, the interfacial area, and the concentration driving force. The first two can conveniently be combined and expressed in terms of H,, (or H0$, the height of a “true” overall transfer unit which in view of the polydisperse nature of the process is given as follows, assuming the X (Le., feed) phase to be dispersed Hox = W ( k o , a ) , (1)
where CP(Ac,) I1.0 corrects for the effect of polydispersivity on driving force. For any given system the values of kox,iwill be effectively constant for each droplet size. Consequently Hoxwill also be constant for a steady-state size distribution, and the concentration gradient, as given by de, Hex= (c, - c,*) (3) dz will be a maximum when the driving force, (c, - cx*),is a maximum. Column operation is most efficient when the phases are in countercurrent plug flow, and the height required for a given duty, as obtained by integration of eq 3, is then a minimum. Any deviation from this ideal flow pattern due to backmixing of either phase (or both) results in an operating line which is convex toward the equilibrium curve, with consequent reduction in driving force (Miyauchi and Vermeulen, 1963a). The proposed scale-up method for differential columns, using the diffusion model, is therefore based on the assumption that H, has the sqne value in pilot and full scale columns, and that differences in performance result entirely from differing degrees of backmixing. This requires in the case of mechanically agitated columns that the intensity of agitation be the same in both cases. 0 1981 American Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 3, 1981
For compartment-type (i.e., stagewise) contactors it is more appropriate to use the backflow model, which assumes that each stage is well mixed and that backflow occurs by mutual entrainment of the phases on leaving the stage. In this case the mass transfer is expressed in terms of the parameter Nt,, the number of “true” transfer units per compartment, defined as follows Nk = (~,a)*&c/Uc (4) This is related to the Murphree efficiency by the relation (Treybal, 1963) N x / ( l+ N&) (5) The value of N L may be expected to remain unchanged under conditions of similar agitation, as does H,,, and its use for scale-up is therefore proposed. There is, however, a proviso in this case for columns, e.g., of rotary agitated type, which require the compartment height to be increased on scale-up, as will be shown later. Pilot Scale Testing. The present treatment is limited to simple ternary systems, and use of a pilot scale column will be assumed; however, it is evident that the method can equally well be used to scale up (or down) from an existing full scale installation. It will also be assumed that the aim of the pilot scale experiments is to obtain design data for a commercial unit, rather than to simulate the latter on a reduced scale. This requires that the experimental program be designed to yield data of a fundamental nature including, if appropriate, the effect of concentration on the controlling parameters. On this basis the following requirements must be kept in mind. (1)The pilot unit should have a throughput of preferably not less than one-tenth that of full scale, subject to a lower limit on column diameter of about 8 cm (3 in.), and a length equivalent to not less than about 3-6 true transfer units (about 2-3 plug flow transfer units) or, with compartment-type contactors, a minimum of 8-10 stages. (2) The design of the column internals should be such as to maintain the same dispersion characteristics in pilot and full scale units. Rules for three common contactor types are as follows: (a) Packed Columns. The packing size can be reduced in the test column subject to it not being smaller than the “critical size” (Gayler et al., 1953). (b) Reciprocating Columns. For oscillating and pulsed plate columns the plate design and spacing, and the product of pulse amplitude and frequency, (fA), are maintained constant. (c) Rotary Agitated Columns. For rotary disk and multi-impeller extractors the column internals are made geometrically similar except for the compartment height, which varied as d,2/3(Pratt, 1955). The agitator speed is selected so that the characteristic droplet velocity (and hence the flooding rate) is the same in both columns; for the rotary disk column, this is obtainable from the correlation of Logsdail et al. (1957). (3) If the equilibrium relationship is nonlinear, the test should if possible be conducted over one or more concentration ranges for which it can be approximated by straight lines, to permit the use of analytical methods for scale-up. (4) If interfacial oscillation is considered likely, tests should be conducted over the smallest practicable concentration ranges to enable the effect of concentration on performance to be assessed. It should be noted in this regard that the effects of interfacial oscillation on both mass transfer and flooding rate diminish as equilibrium is approached, i.e., toward the low concentration (extractant inlet) end of the contactor. Occasions will arise when some of the above requirements cannot be met simultaneously, necessitating compromise. This is particularly true if interfacial oscillation
occurs, since there is then conflict between the requirements 1 (for length) and 4. It must be emphasized in this regard that interfacial turbulence is the most intractable problem encountered in the extractor design, and it is difficult to foresee the development of a satisfactory theoretical method of predicting its effect on performance and throughput. The test procedure, using the actual system of interest, is then as follows. (1) If considered necessary (e.g., due to the likelihood of interfacial oscillation), measurements are made of the flooding rate as a function of throughput and, if relevant, the agitation rate. These are then interpreted in terms of the characteristic velocity (Logsdail et al., 1957); alternatively, the latter can be obtained from measurements of holdup of dispersed phase (Gayler et al., 1953; Logsdail et al., 1957). (2) Mass transfer runs are then carried out at the same flow ratio and column loading as for the full scale unit, varying the agitation rate (if relevant) to determine the optimum combination of throughput and performance. (3) Having selected the best conditions, repeated runs are carried out under steady-state conditions, taking samples of inlet and exit streams for analysis. The results are used to calculate overall material balances on solute; only those which close to within 2-3% can be considered acceptable. Modeling of Results. The recommended procedure consists in substituting the experimental value of the exit extract phase composition, together with the estimated backmixing parameters, the column length or number of compartments and the extraction factor E into the appropriate model solution and solving for H, or NL. This value is then assumed to apply to the full scale unit and is substituted together with the corresponding backmising parameters and desired exit composition into the same model to obtain the height or number of stages required. Analytical solutions to the two models are available when the equilibria can be represented by the linear relation c,* = mcy + q (6) For this purpose the exit compositions of feed and extract phases are expressed in terms of the dimensionless compositions X and Y, respectively (defined under Nomenclature); the inlet compositions in this form are XO = 1.0, Y‘ = 0.0. The solution for the X-phase profile for the diffusion model is then as follows (Sleicher, 1959; Miyauchi and Vermeulen, 1963a)
X = AI + A 2 e X s+ A3eXg+ A 4 e u
(7)
where 2 is the fractional height, the A, are the roots of a cubic characteristic equation and the Ai are constants; for the Y-phase profile the A, are multiplied by constants ai. In this model the backmixing is expressed in terms of the Peclet numbers P = U,d/E,, and these together with N , (= LIH,,), B, and E are contained in the coefficients of the characteristic equation. The corresponding solution for the backflow model is as follows (Sleicher, 1960) X,
= Ai’
+ Aip; + A3’pf + A4‘pt
(8)
where n is the stage number and the p, are the roots of a cubic characteristic equation in (p - 1);the backmixing is expressed in terms of backmixing ratios a),the ratio of backflow to net forward flow. Somewhat simpler solutions to both models, with quadratic characteristic equations, apply when backmixing
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 3, 1981 491
occurs in only one phase, or when E = 1.0. The various solutions to the diffusion model are given by Miyauchi and Vermeulen (1963a) and Pratt (1975), and to the backflow model by Pratt (1976b). A full summary including many special cases is l?(= 1 - X)and #(= Y/E). Expressions for the exit extract phase composition YO or Y‘ are obtained by substitution of 2 = 1.0 or n = 1, respectively, in the corresponding Y-profile equation. The scale-up procedure is then as follows. (1)Obtain values of the backmixing parameters Pj or a* for pilot and full scale columns from appropriate correlations. (2) Substitute the exit value of Y obtained for the pilot scale column, together with the backmixing parameters and E, into the appropriate expression for YO or Y‘ and its associated characteristic equation and solve for H , or NL; this can best be done directly by computer using the secant method, iterating on (H,)’12 or (NL)l12to avoid convergence on negative values (Curtis, 1976). (3) Substitute the resulting value of H , or N L together with the desired exit value of Y and the backmixing parameters for the full scale column into the same equation for YO or Y‘ and solve for length or number of stages; alternatively to avoid iteration, the simplified direct solutions for length or number of stages given by Pratt (1975, 1976b) can be used. When using this method in practice it is necessary to determine which model is the more appropriate. Although Miyauchi and Vermeulen (1963b) have shown that the two models become identical when the number of stages is large, Le., when each stage is effectively differential, this condition seldom applies in practice. It is therefore recommended that the diffusion model be used for differential extractors, e.g., packed and baffle plate columns, and the backflow model for compartment-type extractors such as rotary disk, pulsed, and oscillating plate columns. The above procedure is not directly applicable when the equilibria are nonlinear. However, in such cases the equilibrium curve can generally be approximated by two or three straight line segments. If this is so, the length of the pilot scale column should be adjusted so that runs can be carried out in turn with each segment, giving values of H , or Nix for each. The length of the full scale contactor can then be obtained for the diffusion model by means of either of two procedures described by Pratt (1976a) in which the column is divided into sections. The second of these procedures is also applicable, with simple modification, to the backflow model. Worked Example. It is desired to treat 1.30 m3/h of toluene containing 50 kg/m3 of acetone with 1.54 m3/h of water in a 0.305 m diameter pulsed plate column at 21 “C to reduce the acetone content to 1.0 kg/m3. Determine the number of plates required, assuming these to have 3.2-mm holes and 24.6% free area, with a spacing of 5.0 cm. Experiments were reported by Thornton (1957) for a 7.4 cm diameter column provided with 18 plates of the same type, operating with the toluene phase dispersed. The following results were obtained at 21.5 “C using a pulse amplitude of 16.0 mm and frequency of 3.0 Hz with the same relative phase flow rates as the present full scale unit, i.e., 81.6 and 96.5 L/h of toluene (i.e., X phase) and water ( Y phase), respectively, with a mass balance error of -1.7% c$ = 50.0; c y 1 = 0.0;
c i = 36.90 kg m-3
The equilibrium relationship for this system at 21.5 O C , in concentration units of kg m-3, was expressed as c,* = 0 . 6 3 7 0 ~-~0.2922 Hence the value of c i in dimensionless units is F = 0.4674.
From the overall material balance the exit toluene and aqueous phase flows are 76.8 and 101.3 L/h, respectively, giving E = 0.5130 and 0.5072 at the toluene and water inlet ends, respectively, with a mean of 0.5101. The estimated value of the backmixing ratio, ay,for the continuous phase is 2.39, and that for the dispersed phase is zero (Garg and Pratt, 1981). Substitution in the backflow model equations (Harland and Mecklenburgh, 1966, eq XXXIII-XXV) and solving by iteration gave N L = 0.3812. It will be assumed that the 0.305 m diameter column will be operated with the same pulse frequency and amplitude as the small column. The value of NL will therefore be taken as the same for both units, Le., 0.3812. The estimated value of ayfor the large unit is 7.46 (Garg and Pratt, 1981). From the overall material balance c: = 39.33 kg/m3, Le. Y’ = 0.4982, and E = 0.50777; substitution of these values into the simplified equation for the backflow model (Pratt, 1976b, eq 37) gives N = 58.11 stages. Rounding off upwards, 59 stages (i.e., plates) are therefore required. Discussion Scope for use of the above scale-up method is restricted at present due to the lack of axial dispersion data for large diameter columns. However, it has a much sounder theoretical basis than methods based on the plug flow model, and should be used by preference once such data become available. This limitation is less restrictive in the case of packed columns, since backmixing in these appears to be virtually independent of column diameter, as indicated by the mass transfer data of Gayler and Pratt (1957) for columns of 7.6-30.5 cm diameter with Raschig ring packings. Packing size does, however, appear to have some effect and there is a need for further axial dispersion data, particularly for the larger sizes. In the case of stagewise columns the present method assumes that, while the column diameter is increased on the full scale to maintain the same superficial phase velocities, the compartment height remains unchanged. This is in fact generally the case for pulsed and reciprocating perforated plate columns, but for rotary disk and multiimpeller types it is necessary to increase the height, e.g., as d?l3 (Pratt, 1955) in order to retain a satisfactory flow pattern. The present scale-up method should therefore give a “safe” design, due to the increase in residence time. On the other hand, eq 4 indicates that NAz increases in proportion to the compartment height; while this may well be the case, it would appear unwise to allow for this until the effect of compartment geometry on axial dispersion is better understood. Nomenclature A = pulse amplitude (peak to peak), L a = superficial area of contact of the phases, L-’ B = L/d = concentration of solute in phase j , ML-3 or (mol) L-s = characteristic dimension (e.g., packing size or compartment height), L di = diameter of droplet of size i, L d , = column diameter, L E . = longitudinal diffusion coefficient for phase j , L 2 T 1 d& = Murphree efficiency based on phase X E = extraction factor mU,/U, f = pulse frequency, T’ f i = volume fraction of droplets of diameter di in holdup Hoi = height of a “true” overall transfer unit based on phase 1, L h, = compartment height, L k = overall mass transfer coefficient baaed on phase j , LT’ Lo’= column length, L
2
Ind. Eng. Chem. Process Des. Dev. 1981, 20, 492-495
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m = reciprocal slope of equilibrium line dc,*/dcy N = number of stages (i.e., compartments) N:j = number of overall transfer units per stage (i.e., compartment) based on phase j n = number of stage (i.e., compartment) counting from feed phase inlet P, = Peclet number for phase j , U,d/Ej q = intercept of equilibrium line U . = superficial velocity of phase j , L T ’ x‘ = dimensionless X-phase composition = [c, - (mc: + Q ) ~ / [ c :- (mc: + 41 for diffusion model or (mcYN+l+ q ) ] / [c, - (mcYN+’+ q ) ] / [ c : - (mcf + q ) ] for backflow model x = fractional holdup of dispersed phase Y = dimensionless Y-phase composition = m(c c ; ) / [ c : - (mc: + q ) ] for diffusion model or m(c, - cyNY+’)/ [c: - (mcYN+’+ q ) ] for backflow model 2 = fractional length along column from X-phase inlet (= z / L ) z = distance along column, L Greek Letters cyj = backmixing ratio for phase j , Le., ratio of backflow to U, i i = roots of characteristic equation for diffusion model p, = roots of characteristic equation for backflow model Subscripts av = average i = number of root of characteristic equation i = droplet size fraction j = X or Y phase n = number of stage Superscripts Z = inlet Y or exit X phase (diffusion model) (external to column)
N = exit X phase (backflow model) (external to column) N + 1 = inlet Y phase (backflowmodel) (external to column) 0 = inlet X phase; exit Y phase (diffusion model only) (external to column) 1 = exit Y phase (backflow model only) (external to column) * = equilibrium value Literature Cited Anderson. W. J.; Pratt, H. R. C. Chem. Eng. Sci. 1978, 33, 995. Chartres, R. H.; Korchinsky, W. J. Trans. Inst. Chem. Eng. 1975, 53, 247. Curtis, A. R. In “Optimization in Actlon”; Dixon, L. C. W., Ed.; Academic Press: London, 1976; p 48. Garg, M. 0.; Pratt, H. R. C. Ind. Eng. Chem. Process Des. Dev. 1981. accompanying article In thls issue. Gayler, R.; Pratt, H.R. C. Trans. Inst. Chem. Eng. 1957, 35. 273. Gayler, R.; Roberts, N. W.; Pratt, H. R. C. Trans. Inst. Chem. Eng. 1953, 31, 57. Hartland, S.; Mecklenburgh, J. C. Chem. Eng. Sci. 1068, 21, 1209. Logsdeil, D. H.;Thornton, J. D.; Pratt, H. R. C. Trans. Inst. Chem. Eng. 1957, 35, 301. Miyauchi. T.; Vermeulen, T. Ind. Eng. Chem. Fundam. 1083a, 2 , 113. Miyawhi, T.; Vermeuien, T. Ind. Eng. Chem. Funck?m. 1963b. 2 , 304. Olney, R. 0. AICM J. 1984, 10, 827. Ran, H. R. C. Ind. Chem. 1955, 31. 505, 552. Pratt. H. R. C. Ind. Eng. Chem. Process Des. Dev. 1075, 14, 74. Pratt, H. R. C. Ind. Eng. Chem. ProcessDes. Dev. 19768, 15, 34. Pratt, H. R. C. Ind. Eng. Chem. Process Des. Dev. 1976b, 15, 544. Pran, H. R. C.; Anderson, W. J. Proc. rnt. Solvent Extr. Conf. 1977, 1 , 242. Rod. V. &it. Chem. E m . 1088. 1 1 . 483. Skeknd, A. H. P.; Con&, W. L. Ind: Eng. Chem. Process Des. Dev. 1973, 12, 445. Slelcher, C. A. AICM J . 1959. 5 , 145. Sleicher, C. A. AICM J. 1980, 6 , 529. Thornton, J. D. Trans. Inst. Chem. Eng. 1957, 35, 316. Treybai, R. E. “Liquid Extraction”, 2nd ed.; McGraw-HIII: New York, 1963; Table 10.1. Wijffels, J.-B.; Rietema, K. Trans. Inst. Chem. Eng. 1972, 50, 224.
Received for review July 14,1980 Accepted March 5, 1981
Effect of Column Diameter on Backmixing in Pulsed Plate Columns M. 0. Garg‘ and H. R. C. Pratt * Department of Chemical Engineering, University of Melbourne, Parkville, Victoria 3052, Australia
Values of the continuous phase backmixing ratio, a,, for the toluene-acetone-water system in pulsed plate cdumns of 7.4 to 30.5 cm diameter operating in the emulsion region have been obtained numerically from the mass transfer data of Thornton (1957) and Logsdail and Thornton (1957). For this purpose values from Miyauchi and Oya’s correlation (1965) were used for the 7.4 cm diameter column, and it was assumed on the basis of visual observation that dispersed phase backmixing was negligible. The results were correlated as follows: a, = d~80(fA)0~10(0.170 0.302Ud/Uc),where d, is in cm and (fA) in cm/s. This expression applies to plates with 3.2 mm holes, 25% free area, and a spacing of 5.08 cm; it is best used to obtain relative values of a,, i.e. for scale-up purposes, rather than absolute values.
+
Introduction It is well established that the performance of both pulsed and reciprocating plate columns, as measured by the H. T.U., passes through a minimum with increasing pulse intensity, ( f A ) ,due to the competing effects of simultaneous increases in interfacial area and continuous phase backmixing (Cohen and Beyer, 1953; Sege and Woodfield, 1954a). The value of this minimum also increases with increasing column diameter, as shown in the case of the pulsed column by Sege and Woodfield (1954), and for the
reciprocating plate column by Karr and Lo (1971). The adverse effect of column diameter is undoubtedly the result of increasing backmixing of the continuous phase (Sege and Woodfield, 1954b; Logsdail and Thornton, 1957). Values of backmixing ratios have been reported by a number of workers, who obtained these from both steady-state and transient tracer techniques and from measured concentration profiles along the column during mass transfer. A review by Ingham (1971) indicated that all work up to that time had been done with columns up to only 5.8 cm diameter, in some cases using a single phase only; this is true also of later reported work (Novotny et al., 1970; Rao et al., 1978),apart from that of Baird (1974), who used a 15 cm diameter column.
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1981 American Chemical
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