2170
Ind. Eng. Chem. R e s . 1990,29, 2170-2172
monotonically. In much of the enriching section, there is very little difference between y and x , and the incremental increase in enrichment with area is very small in most of the enriching section. A very large area is required to enrich all of the permeate product plus recycle up to the final enrichment. Figure 3 presents the same information for the CCRC. In the flow versus area plot, the vertical lines indicate where recycle and feed are added, while the slopping lines (LH decreases and LL increases) indicate permeation. In the concentration versus area plot, the vertical lines represent the difference between the permeate composition leaving stage n and the composition of the permeate in stage n + 1,at the reject end of that stage. As can be seen, in the CCRC, the more permeable species is stripped from the high-pressure stream by permeation and is “pumped up” to higher concentrations as the permeate is produced and recycled to the next stage in the separation train. The basic reason the CCRC is more efficient than the CMC can be explored by thermodynamic analysis. The CCRC is probably more efficient because the overall process is less irreversible than the CMC and hence produces less entropy increase for the same separation. The entropy increase is a measure of the excess work required to make a separation over the thermodynamic minimum. Although beyond the scope of the present study, a thermodynamic analysis of ideal permeator performance for various designs will be the topic of a future study. Conclusion It is concluded that a countercurrent recycle cascade is theoretically more efficient than a continuous membrane column. Nomenclature F = stage where feed is introduced into the cascade LH = local molar rate on high-pressure side LHIF = molar rate of fresh feed LHI(n = molar rate of feed to the Zth stage LHO(I) = molar rate of reject from the Zth stage LL = local molar rate on low-pressure side LLO(I) = molar rate of permeate product from the Ith stage N = end stage in cascade that produces reject product; number of permeators in a cascade PH = pressure on high-pressure side of membrane PL = pressure on low-pressure side of membrane PA = permeability of most permeable species in membrane
RR = recycle ratio = rate of recycle/permeate product rate X = local high-pressure concentration XIF = composition of fresh feed XI(0 = composition of high pressure stream entering the Zth stage X O ( I ) = composition of reject stream f r o m the Zth stage Y O ( I ) = composition of permeate stream from the Zth stage Y = local low-pressure concentration Greek L e t t e r s a* = ideal separation factor = PA/PB 6 = membrane thickness Op = overall cut = LLO(I)/LHIF O(N) = cut in the Nth stage
Literature Cited Benedict, M.; Pigford, T. H.; Levi, H. W. Nuclear Chemical Engineering, 2nd ed.; McGraw-Hill: New York, 1981;p 685. Blaisdel, C. T.; Kammermeyer, K. Countercurrent and Co-current Gas Separation. Chem. Eng. Sci. 1973,28, 1249-1255. Herbst, R. S. Separation of Boron Isotopes by Gas Phase Permeation of BF,. M.S. Thesis in Chemical Engineering, Montana State University, Bozeman, 1989. Hwang, S.-T.; Thorman, J. M. The Continuous Membrane Column: AZCHE J . 1980, 26 (4),558-566. Kao, Y.-K.; Qui, M.-M; Hwang, S.-T. “Critical Evaluations of Two Membrane Gas Permeator Designs; Continuous Membrane Column and Two Strippers in Series. Znd. Eng. Chem. Res. 1989,28, 1514-1520. Matson, S. L.; Lopez, J.; Quinn, J. A. Separation of Gases with Synthetic Membranes. Chem. Eng. Sci. 1983, 38 (4),503-524. McCandless, F.P. A Comparison of Some Recycle Permeators for Gas Separations. J. Membr. Sci. 1985,24, 15-28. Rautenbach, R.; Dahm, W. Oxygen and Methane Enrichment-A Comparison of Module Arrangements in Gas Permeation. Chem. Eng. Technol. 1987, I O , 256-261. Stern, S.A.; Wang, S.-C. Countercurrent and Co-current Gas Separation in a Permeation Stage; Comparison of Computation Methods. J . Membr. Sci. 1978, 4, 141-148. Ward, W.J., 111; Browall, W. R.; Salemme, R. M. Ultrathin Silicone/Polycarbonate Membrane for Gas Separation Process. J. Membr. Sci. 1976, I , 99-108. Yoshisato, R. A.; Hwang, S.-T. Computer Simulation of a Continuous Membrane Column. J. Membr. Sci. 1984, 18, 241-250.
F.P . McCandless Department of Chemical Engineering Montana State University Bozeman, Montana 5971 7 Received for review April 3, 1990
Accepted July 9,1990
ADDITIONS AND CORRECTIONS Comments on “Model for Hold-Up Measurements in Liquid Dispersions Using an Ultrasonic Technique” Sir: In a paper published by our group (Yi and Tavlarides, 1990), a model for hold-up measurements in liquid dispersions is discussed. Continuing our research efforts in the same area, we realized that even though the model can predict the dispersed-phase holdup of liquid dispersions within 7 % relative error (see Tables IV and V of the cited article), still there are some aspects that need further 0888-5885/90/2629-2170$02.50/0
clarification. In the cited paper, sound-wave refraction and reflection phenomena were considered in order to estimate the path length of sound in the dispersed phase as
L&,i = L&?gd,i
i = 1, 2
(1)
(see &O eq 12 of the cited article), where L is the distance
0 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 2171 Sound-wave path length changes in the droplets create path length changes in the continuous phase as shown in Figure 1. As the path length inside the droplet changes from q(r) to p ( r ) ,due to sound-wave refraction, the length s(r) has to be subtracted from the sound-wave path length in the continuous phase. This length is given by the following relation:
I
' I n . .I
s(r) = p ( r ) cos (a- P) - q(r) = &)(cos a cos (3 + sin a sin P) - q ( r )
-u
(6)
where
p ( r ) = 2a cos P
Transmitted wave
Incident wave
Dispersed phase droplet
The subscripts 1 and 2 refer to the cases of y < 1 and y 3 1, respectively. Here y is the ratio of the sound velocities in the dispersed and continuous phases given by
The notation used here differs from the one used in the cited work, to specify the selection of eq 2 or eq 3 based on the sound-velocity ratio in the phases and not whether the dispersed phase is organic or continuous. The latter notation does not account for the situation when the sound velocity in the organic phase is greater than in the aqueous phase. The sound-wave path length in the continuous phase was assumed by the authors of the cited article to be the remainder of the distance the ultrasound wave travels between the ultrasonic transducers and is given by
(see also eq 13 of the cited article). This assumption does not give the exact path length change of the sound wave in the continuous phase for single droplet encounters and also neglects the elongation of the path length due to directional changes after the sound wave emanates from the drop as depicted in Figure 1. It is recognized that when the wave undergoes multiple droplet encounters before reaching the receiving transducer, the sound-wave path length experiences further elongation in both the dispersed and the continuous phase. These types of path length elongations are second-order effects and are not considered here. Therefore, as a first-order correction of the timeaverage model, only sound-wave path length changes in the droplets are considered. The path length correction in the dispersed phase is given by eq 12 in the cited paper, whereas the path length correction in the continuous phase is derived here in place of eq 13.
(7)
From the sound-wave refraction law, given by eq 7 of the cited article,
Continuous phase
and
cos a = (1 - P/a2)'12
sin a = r / a
sin- a - -Ucont - - -1 sin P ud& y
Figure 1. Plane wave propagation through the dispersion.
between the transducers (see Figure 1, cited paper), 4 the dispersed phase fraction, and gd,i a correction factor given by the following relations:
q(r) = 2a cos a
(8)
sin P and cos P can be estimated as follows:
Substitution of eqs 7 and 9 into eq 6 and integration over the droplets' projected surface provides an estimate for an average correction for the continuous-phase path length per each droplet. This result is
= "lr"r a2 o
[(4a2 - 4r2y2)'12[
74
(l-I?))2 + -
((
1-
5)
X
1
- (4a2 - 4r2)'12 dr (11)
and yields 4 hi(a) = -ag',,i 3
i = 1, 2
(12)
where
g',,2 = 2 57
.['
1 - (1 -
$)'I
[ (
gyz 1 -
1 - -: 2 ) 2 ]
+ for y 2 1 (13b)
If the path length through the entire drop population is normalized by the ratio of the projected area of the dispersion droplets to that of the transducers in the direction of the transmitted wave, then similarly to the case of the dispersed phase (eq 11 of the cited paper), the total length that has to be subtracted (for y 6 1) from or added (for y 3 1) to the continuous-phase path length is given by
H i = L+gg,i
i = 1, 2
(14)
2172
Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990
Table I. True Holdup versus Corrected Holdup for the Water/Toluene System' re1 corrected re1 measd 40rg, $qj, &%A, error, @DTA, eq error, Sauter mean '7n rpm 90 % % 17, % % diameter, mm 9.2 0 9.2 9.2 300 11.1 9.7 5.4 9.1 -1.1 0.783 400 10.8 2.1 -4.3 9.4 0.553 8.8 -1.1 5.4 ,500 11.1 9.7 0.374 9.1 0.286 600 11.1 9.7 -1.1 9.1 5.4 12.6 0 12.6 12.6 300 14.0 12.2 11.7 -7.1 -3.2 400 14.6 12.8 12.0 -4.8 0.592 1.6 12.4 -1.6 ,500 15.2 13.3 0.415 5.6 12.0 -4.8 600 14.6 12.8 1.6 22.U 0 21.6 21.6 1.8 21.9 6.4 300 26.8 23.4 -0.5 1.023 21.9 400 26.8 23.4 -0.5 0.560 6.4 21.9 500 26.8 23.4 -0.5 6.4 21.9 6.4 600 26.8 23.4 -0.5 27.8 0 37.8 27.8 27.5 -1.1 1.252 300 33.6 29.4 5.8 -1.1 27.5 0.856 5.8 400 33.6 29.4 -1.1 27.5 0.524 5.8 500 33.6 29.4 27.5 -1.1 5.8 600 33.6 29.4 30.0 0 30.0 30.0 1.214 6.7 0 0.686 300 36.6 32.0 30.0 400 36.6 32.0 0 6.7 0.297 30.0 6.7 0 500 36.6 32.0 30.0 0 6.7 600 36.6 32.0 30.0 39.3 0 39.3 39.3 37.6 -4.3 300 45.9 40.2 2.3 -4.3 400 45.9 40.2 2.3 37.6 -4.3 500 45.9 40.2 2.3 37.6 -4.3 GOO 45.9 40.2 2.3 37.6 a
Table 11. True Holdup versus Corrected Holdup for the 0.2 M HN0,/30 vol 70 TBP in n -Dodecane System' re1 corrected re1 measd &'&A, &%A, error, @DTA, eq error, Sauter mean % rpm % 90 90 17, % % diameter, mm 0 5.2 5.2 5.2 300 6.0 5.1 -1.9 4.7 -9.7 400 6.4 5.5 5.7 5.1 -2.5 500 6.6 5.5 7.7 5.1 -2.5 600 6.6 5.6 7.7 5.1 0 13.0 0 12.7 12.7 300 15.9 13.6 4.6 12.5 -3.6 0.455 400 15.9 13.6 4.6 12.5 -3.6 4.6 500 15.9 13.6 12.5 -3.6 600 15.9 13.6 4.6 12.5 -3.6 26.0 0 26.7 26.7 300 31.7 27.1 4.2 25.0 -4.0 0.595 400 32.3 27.6 6.2 25.4 -2.2 0.421 500 32.3 27.6 6.2 25.4 -2.2 600 32.3 27.6 6.2 25.4 -2.2
v,
Experimental information is provided in the cited article.
is recommended over eq 14 of the cited paper. Furthermore, it is our opinion that for some liquid systems with larger differences in sound velocities, the advantage of employing eq 17 will prove greater. Also, we wanted to make it clear that the model described in the cited article cannot predict the hold-up value at phase inversion, since there is no relation between the sound velocity and the phase inversion of a system. What it can do, however, is to tell when phase inversion occurred, by inspection of the travel time of sound through the dispersion. The travel times prior (t* ) and after (t*,) the phase inversion are given below for tLe case when y < 1:
Experimental information is provided in the cited article.
L& ld!,
Then the actual path length of the sound wave in the continuous phase becomes
t*p = -
+
udis,p
and L&d,2
gc,, 1 + g 'c,, The time-average model given by eq 1 in the cited paper becomes
(18)
Kont,p
t*, = -
where
L(1 - &?c,l)
Udis,a
(16)
+
L(1 - &c,2)
(19)
Ucont,a
where udis,p = Ucont,,and Ucont,p = Udi?,a. t*p and t*, do not have to be equal. Therefore, any discontinuity in the measured travel time of sound through the dispersion indicates that phase inversion occurred. Acknowledgment
This equation is more accurate than eq 14 of the cited article. Here d denotes the dispersed phase; c denotes the continuous phase; t*, t,, and t d denote the travel times through the dispersion, the continuous phase, and the dispersed phase, respectively; and gd,i represents the path length correction in the dispersed phase and is given by eqs 2 and 3. Applying the above equation to the data listed in Tables IV and V of the cited paper, one can obtain the results shown in Tables I and 11. . It can be seen that, for the systems under investigation, the assumption that led to eq 13 of the cited article is reasonable. The exact estimation of the sound-wave path length in the continuous phase improved slightly the prediction of the dispersed phase holdup, and therefore, eq 17 of this correspondence
The suggestions by Mr. J. Yi and financial support for one of us (C.T.) by Phaedron Technologies, Inc., are gratefully acknowledged. Literature Cited Yi, J.; Tavlarides, L. L. Model for Hold-Up Measurements in Liquid Dispersions Using an Ultrasonic Technique. Znd. Eng. Chem. Res. 1990, 29, 475-482.
* T o whom correspondence should be addressed. C. Tsouris, Lawrence L. Tavlarides* Department of Chemical Engineering and Materials Science Syracuse University Syracuse, New York 13244
