Ind. Eng. Chem. Process Des. Dev. 1981, 20, 492-495
492
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 (backflow model) (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. A I C M 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. A I C M 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.
‘R. and D. Department, Engineers India Ltd., 4 Parliament St., New Delhi, India. 0196-4305/81/1120-0492$01.25/0
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1981 American Chemical
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Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 3, 1981 493
Miyauchi and Oya (1965) correlated their data for both single and two-phase flow in 3.2 and 5.4 cm diameter columns in terms of E,, the dispersion coefficient for the continuous phase, as follows
=@ i 2fAx (5+l) where 8, the number of “perfectly mixed stages” (i.e., vortices) per compartment is given by @ = 0.57mh(d,2h,)1/3/dh
(2)
The dispersion coefficient is related to the backmixing ratio, . a as follows (Miyauchi and Vermeulen, 1963b). (3)
It was found by photographic studies that @ generally had a value between 1 and 2. However, eq 2 shows that it increases with column diameter as ~i,2/~, and since the plate spacing is usually maintained at 5-10 cm on scale-up, this leads to an unrealistic number of vortices within each compartment for large diameter columns. Further, a combination of eq 1and 2 indicates that backmixing decreases with increasing column diameter, which is not in accord with mass transfer data (Sege and Woodfield, 1954; Logsdail and Thornton, 1957). Rouyer et al. (1974) have given values of E, for columns of 4.5 and 60 cm diameter which indicate that this does not increase with increasing diameter as expected; however, details of flow rates and pulsation intensities were not stated. As this result conflicts with the effect on mass transfer quoted above, it seems probable that it applies to the mixer-settler region in which backmixing is reduced, as compared with the emulsion region, by sealing of the plate openings with dispersed phase droplets at the end of each stroke. It is apparent that there is an urgent need for reliable experimental data on backmixing ratios for larger diameter columns. In the absence of such data, it was considered worthwhile to attempt to derive backmixing ratios indirectly from the available mass transfer data using a method described previously (Pratt and Garg, 1981). Computation of Backmixing Ratios Basis of Method. The mass transfer data of Thornton (1957) and Logsdail and Thornton (1957),covering column diameters of 7.4,15.2, 22.9, and 30.5 cm were used in this study. Although data were given by these workers for two systems and several plate designs, only those for toluene-acetone-water using plates with 3.2-mm holes, 25% free area and 5.08 cm plate spacing were used since these were common to all diameters. It was confirmed that the pulse intensity values, ( f A ) ,were such that operation was always in the emulsion region. The procedure used was to calculate values of NA, the number of transfer units per stage (i.e., compartment), from the data for the 7.4 cm diameter column using the analytical solution to the backflow model together with values of a,calculated from the Miyauchi-Oya correlation, eq 1-3. It was assumed on the basis of careful visual observations of a 7.4 cm diameter pulsed column in operation that backmixing is negligible in the dispersed phase since, although the droplets moved backward relative to the column wall at large pulse amplitudes, their motion was always forward relative to the continuous phase. In this regard, Miyauchi and Oya (1965) presented backmixing data also for the dispersed phase, obtained from residence time distribution measurements; however, it
seems clear that these in fact related to forward dispersion, the effect of which on performance is not influenced by column diameter (Rod, 1966). The resulting derived values of N& were assumed to apply to the larger diameter columns. After smoothing, they were substituted into the same backflow model solution together with the measured exit concentrations of extract phase. Iterative solution then gave values of the backmixing ratio for all column diameters. Derivation of NxValues. The mass transfer data were all obtained using approximately 5% w/v of acetone in toluene as feed (Le., X phase) with acetone-free water as extractant (Y phase); the toluene was dispersed so that interfacial oscillation would have occurred. The exit concentrations were not reported, and these were therefore calculated from the reported values of HWp, the height of an overall plug flow transfer unit based on the continuous phase. The equilibrium data for the toluene-acetone-water system (Thomton and Pratt, 1953) were expressed in linear form as follows over the concentration range used c,* = mcy q (4)
+
where m is a polynomial in temperature over the range 14.5-25.0 “C, given by m=a
+ bt + ct2
(5)
The data were fitted to f3.490 with the following values of the coefficients: a = 0.24948; b = 0.02930; c = -0.000525; q = -0.29218. Using eq 4 the concentrations of the toluene and water phases were expressed in dimensionless units X and Y , respectively (defined under Nomenclature). On this basis the inlet toluene concentration, taken as 5.0 wt % throughout, was X o = 1.0, and the inlet water (acetonefree) was YN+l= 0. The exit toluene concentrations, F, were calculated from the reported H, (i.e. Hd) values using the following modified form ofYthe Kremser-type equation (Miyauchi and Vermeulen, 1963a, Table 1, Case 3) E(exp[N,p(E - 1)1- 11 V= (6) E exp [N,,p(E - 1)1- 1 where Noxp= L/EHoyp. Values of E were taken as the mean of those ab the two ends; to obtain these the exit flows were calculated by iteration from overall mass balances on toluene and water, neglecting mutual solubilities. The dispersed phase holdup, required for use in eq 1, was calculated for each run by means of the usual slip velocity equation (Gayler et al., 1953). Values of the characteristic velocity required in this equation were obtained from the correlation of Thornton (1957) and multiplied by the factor of 2.60 which was found by Logsdail and Thornton (1957) to account for interfacial oscillation in the transfer of acetone from toluene into water. The analytical solution of the backflow model in terms of yl is given in the Appendix (Hartland and Mecklenburg, 1966). Values of N k were obtained by iterative solution of these equations after substitution of the experimental values of V together with a, from eq 1 and 2, E and N. The secant method was used for this purpose, iterating on (Nix)l/zto avoid convergence on negative values (Curtis, 1976) and continuing until Iyl(cdc) - V(exp)l 5 lo4. In all, data for 51 runs were used, of which four gave excessively high values of NA, and were rejected; this was presumably due to the effect of the minor errors in the original data in cases where the backflow operating line and the equilibrium line “pinched”.
494
Ind. Eng. Chem. Process Des. Dev., Voi, 20, No. 3, 1981
HOLD UP( x 1 Figure 1. Correlation of N L values for 7.4-cm diameter column.
A study of the results showed that N L was dependent on both the holdup, x , and the pulse intensity, (fA). After several trials the results were found by minimization of the average deviation to be satisfactorily correlated by the following expression, with a mean deviation of 22.8%
Nix = al
+ az(fA) + a3x + a,x(fA) + U ~ ( ~ +Aa6x2 )~ + ~ , ( f A ) ~ x+: asx2(fA) + ~ g ~ ( f A(7)) ~
EOUATION T O LINE
aC:d: * o Z ( f A ) o ' o ' ( O
170+0 302Ud/UJ
A comparison of this expression with the computed NAx values is shown in Figure 1. Backmixing Ratios. Values of a, were next computed from the data for all four columns (including the 7.4 cm diameter unit) by iterative solution of the equations in the Appendix after substitution of the NAz values from eq 7 together with F,E and N . The same iterative procedure was used, with convergence to IF(calc) - F(exp)I 5 lo4. In all, 124 data points were obtained, of which 8 were rejected. Plots of the resulting a, values against U,/ ud for each column diameter and (fA) value appeared to be of hyperbolic form. Several models were tested using this as a basis, leading finally to the following dimensional equation for cy, a, = ~ ! ~ ~ ~ ( f A ) ~ ~ ~+~0.3017Ud/ ~ ( 0 . 1 7U,) 03
(8)
where d, is expressed in cm and (fA) in cm/s. Values of the exponents and constants in this expression were obtained by minimizing the squares of the residuals using the Levenberg-Marquardt algorithm (Marquardt, 1963). This gave a multiple correlation coefficient, R2,of 74% and a standard deviation of 2.49. A comparison of eq 8 with the computed a, values is shown in Figure 2. Discussion The considerable scatter shown by the final correlation of CY, values is not unexpected in view of the sensitivity of mass transfer data in general to experimental inaccuracy. Despite this, the significance of the relatively large exponent of 0.80 on column diameter is demonstrated by the closeness of the 95% confidence limits on the exponent, which were 0.62 to 0.99.
Figure 2. Final correlation of backmixing coefficients, a,.
Of the two remaining variables, the effect of pulse intensity, (fA) is weak, as shown by the exponent of 0.101 with 95% confidence limits of -0.123 to +0.325. On the other hand the flow ratio, UdlU,, had a large effect, especially at values above unity where a, increases rapidly. These last two results are explained by the work of Anderson and Pratt (1978), who showed that continuous phase backmixing results mainly from a circulatory flow induced by the dispersed phase droplets in dissipating their potential energy; the backmixing would therefore be expected to be influenced to a major extent by the relative flow rate of dispersed phase, but not by its degree of dispersion. Equation 8 is at present the only available source of cy, values for pulsed plate columns of up to 30.5 cm diameter
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 3, 1981 405
operating in the emulsion region, and it may well be suitable for limited extrapolation to larger diameters. It is, however, subject to two limitations, the first of which concerns the reliability of the Miyauchi-Oya correlation. Thus, eq 1 and 2 indicate that a, decreases with increase in column diameter, whereas eq 8 shows the opposite. Also, the number of data points obtained by these workers for two-phase flow was relatively small, and they were obtained for very low values of the flow ratio, U,/U,. It appears, therefore, that eq 8 is better suited to providing relative a,values, e.g., for scale-up purposes, rather than absolute values. The second limitation concerns the effect of physical properties, which are not included in eq 8. In particular, it would seem that density difference is an important variable, since it determines the amount of potential energy dissipated by the dispersed phase (Anderson and Pratt, 1978). It is evident on both counts, therefore, that there is an urgent need for reliable experimental data on continuous phase backmixing for two-phase flow in pulsed plate columns of relatively large diameter. Appendix The analytical solution for Y,, for the present case, i.e., a, = 0; ay,N& and N finite; E # 1.0 is Y , = A1 A3~3/.~3" A 4 ~ 4 / . ~ 4 "
+
+
where l" = Yl is obtained by setting n = 1. The roots, pr of the characteristic equation and the coefficients are as follows (P3 - 11, (P4 - 1) = -@/2 f [(@/2)2- Y ] ' / ~(positive sign for w4) where
B=
1
+ ",(2
-E
+ aY)
(1+ ay)(l+ " A
of an overall plug flow transfer unit based on phase j , m h, = compartment height, m L = column length, m m = reciprocal slope of equilibrium line, dc,*/dcy m, = fractional free hole area of plates N = number of stages (i.e., compartments) Nolp = number of overall plug flow transfer units based on Phase j Noj = 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 Pi= Peclet number for phase j , Ujh,/Ej q = intercept of equilibrium line t = temperature, O C U . = superficial velocity of phase j m s-l x'= [c, - (mcYN+'+ q ) ] / [ c ;- (mc$l + q ) ] , dimensionless X-phase concentration x = fractional holdup of dis ersed phase Y = m(cy- cYN+l)/[c(1- (mcY8l + q ) ] , dimensionless Y-phase concentration Greek Letters a, = backmixing ratio for phase j, i.e., ratio of backflow to Uj fl = number of vortices per compartment Subscripts c = continuous phase d = dispersed phase j = X or Y phase; c or d phase n = stage number counted from feed phase inlet x = X (i.e., feed) phase y = Y (i.e., extractant) phase Superscripts N = exit X phase (external to column) N + 1 = inlet Y phase (external to column) 0 = inlet X phase (external to column) 1 = exit Y phase (external to column) * = equilibrium value Literature Cited
NL(1 - E) Y =
Holy = height
Anderson, W. J.; Pratt, H. R. C. Chem. Eng. Sci. 1978, 33, 995. Baird. M. H. I. Can. J. Chem. Eng. 1974, 52. 750. Cdwn, R. M.; Beyer, 0. H. Chem. Eng. Frog. 1953, 49, 279. Curtis, A. R. In "Optlmization in Action"; Dixon, L. C. W., Ed.; Academic Press: London, i976; p 48. Gayler. R.; Roberts, N. W.; Pratt, H. R. C. Trans. Inst. Chem. Eng. 1953,
(1 + aJ(1 + Nix)
Also
__.
31. 57.
1)1
A = pulse amplitude (peak to peak), m or cm B = L/hc c . = concentration of solute in phase j , kg m-3 = column diameter, m or cm = longitudinal diffusion coefficient for phase j , m2 s-l = extraction factor, mU,/U, f = pulse frequency, s-l Hoi = height of a "true" overall transfer unit based on phase 1, m
4
Hartland. S.; Mecklenbwgh, J. C. Chem. Eng. Sci. 1966, 21, 1209. Ingham, J. "Recent Advances In LiqukKlqukl Extraction"; Hanson, C., Ed.; Pergamon Press: Oxford, 1971: Chapter 6. Karr, A. E.; Lo, T. C. Roc, Int. Solvent Extr. Conf. 1971, 1 , 299. Logsdall, D. G.: Thornton, J. D. Trans. Inst. Chem. Eng. 1957, 35, 331. Marquardt. D. W. SlAM J. Appl. Math 1963, 1 1 , 431. Miyauchi, T.; Vermeulen, T. Ind. Eng. Chem. Fundem. 1963a. 2 , 113. Miyauchl, T.; Vermeulen. T. Ind. Eng. Chem. Fundem. I963b, 2 , 304. Miyauchi, T.; Oya, H. A I W J. 1965, 11, 395. Novotny, P.; Prochazka, J.; Landau, J. Can. J. Chem. Eng. 1970, 48, 405. Watt, H. R. C.; Garg, M. 0. Ind. Eng. Chem. Process Des. Dev. 1981, accompanying article in this Issue. Rao, K. V. K.; Jeelani, S. A. K.; Balasubramanian,G. R. Can. J . Chem. Eng. 197%.56, 120. Rod, V. Brit. Chem. Eng. 1966, 1 1 , 483. Rower, H.; Lebohlloc, J.; Henry, E.; Michei, P. Roc. Int. Solvent Exh. Conf. 1974, 3, 2339. Sege, 0.; Woodfield. F. W. Chem. Eng. Frog. Symp. Ser. No. 73, 1954, 50. - - la\ ,-, 14: Ibl 179. Thornton, J. D.'-Trans. Inst. Chem. Eng. 1957. 35, 316. Thornton, J. D.; Pratt, H. R. C. Trans. Inst. Chem. Eng. 1953, 37, 289. I
Received for review July 14,1980 Accepted March 5, 1981
