1248
Ind. Eng. Chem. Res. 1991,30,1248-1257
Blaisdell, C. T.; Kammermeyer, K. Counter-current and Co-current Gas Separation. Chem. Eng. Sci. 1973,28,1249. Carnahan, B.;Luther, H. A.; Wilkes, J. 0. Applied Numerical Methods; Wiley: New York, 1969; Chapter 6. Gorissen, H. Temperature Changes Involved in Membrane Gas Separations. Chem. Eng. Process. 1987,22,63. Haraya, K.; Hakuta, T.; Yoshitome, H.; Kimura, S. A Study of Concentration Polarization Phenomenon on the Surface of a Gas Separation Membrane. Sep. Sci. Technol. 1987,22,1425. Hwang, S.T.; Thorman, J. M. The Continuous Membrane Column. AIChE J. 1980,26,558. Naylor, R. W.; Backer, P. 0. Enrichment Calculations in Gaseous Diffusion: Large Separation Factor. AIChE J. 1955,1, 95. Pan, C. Y. Gas Separation by High-Flux, Asymmetric Hollow Fiber Membrane. AIChE J . 1986,32,2020. Pan, C. Y.; Habgood, H. W. Gas Separation by Permeation Part I. Calculation Methods and Parametric Analysis. Can. J . Chem. Eng. 1978a,56, 197. Pan, C. Y.; Habgood, H. W. Gas Separation by Permeation Part 11. Effect of Permeate Pressure Drop and Choice of Permeate Pressure. Can. J . Chem. Eng. 1978b,56,210. Quaile, J. P.; Levy, E. K. Laminar Flow in a Porous Tube with Suction. J. Heat Transfer 1975,2,66.
Rautenbach, R.; Dahm, W. Gas Permeation-Module Design and Arrangement. Chem. Eng. Process. 1987,21,141. Shindo, Y.; Hakuta, T.; Yoshitome, H.; Inoue, H. Calculation Methods for Multicomponent Gas Separation by Permeation. Sep. Sci. Technol. 1985,20,445. Spillman, R. W. Economics of Gas Separation Membranes. Chem. Eng. Prog. 1989,85 (l), 41. Terrill, R. M.; Thomas, P. W. On Laminar Flow Through a Uniformly Porous Pipe. Appl. Sci. Res. 1969,21,37. Thorman, J. M.; Hwang, S. Compressible Flow in Permeable Capillaries Under Deformation. Chem. Eng. Sci. 1978,20, 15. Walawender, W. P.; Stern, S. A. Analysis of Membrane Separation Parameters 11: Countercurrent and Cocurrent Flow in a Single Permeation Process. Sep. Sci. 1972, 7,553. Weller, S.; Steiner, W. A. Separation of Gases by Fractional Permeation Through Membranes. J. Appl. Phys. 1950a,21, 279. Weller, S.;Steiner, W. A. Engineering Aspects of Separation of Gases: Fractional Permeation Through Membranes. Chem. Eng. Prog. 1950b,46, 585. Received for review May 29, 1990 Revised manuscript received December 3, 1990 Accepted January 8, 1991
Simple and Accurate Shortcut Procedure To Account for Axial Dispersion in Countercurrent Separation Columns U. von Stockar* and Xiaoping Lu Institute of Chemical Engineering, Swiss Federal Institute of Technology, Lausanne, Switzerland, and Nanjing Institute of Technology, Nanjing, People's Republic of China
A method is proposed for correcting the design of countercurrent separation columns for the impact of axial dispersion effects. After computing the required number of transfer units (NTU) employing conventional procedures based on the plug-flow assumption, the method enables estimation of a correction factor accounting for deviations from plug flow. This factor is used to correct the NTU for such effects. The procedure also works for correcting HTU data that might have been affected by axial dispersion. The method is straightforward and simple to use, and the results agree in simple cases within a few percent with those obtained from a rigorous solution of the model of axial dispersion.
Introduction Textbook design methods for separation processes involving countercurrent contacting of two fluids such as in gas absorption, stripping, rectification, and liquid-liquid extraction usually assume ideal plug flow for both phases in order to evaluate the driving forces throughout the contactor. Flow in real columns is known to exhibit considerable departure from ideal flow, which is usually referred to as "axial dispersion". In partially destroying the countercurrent contacting scheme, it diminishes the mass-transfer driving forces and thus the separation efficiency of the column. Especially the performance of liquid extraction columns are known to be dominated in certain circumstances by axial dispersion rather than by mass transfer (see, e.g., Ricker et al. (1981)). If not properly accounted for, axial dispersion may result in a serious underestimation of the number of required transfer units and may therefore lead to an unsafe design. When the height of a transfer unit is experimentally determined, neglect of axial dispersion and thus overestimating the driving forces will yield high HTU values. Already three decades ago, rigorous analytical solutions for the concentration profiles in two-phase contactors were developed by integrating the basic differential equations
* To whom correspondence should be addressed.
osas-5as5/91/263o- 124a$o2.50/o
for a mass-transfer model incorporating axial dispersion in both phases (Miyauchi, 1957; McMullen et al., 1958; Sleicher, 1959; Miyauchi and Vermeulen; 1963). The completely explicit solution for the fractional approach to equilibrium obtained by Miyauchi and Vermeulen (1963) is reproduced in Tables I and 11. As can be seen, such solutions are unfortunately exceedingly complex and are explicit for the performance calculation only. Their application for separation column design therefore requires complicated nested iterations and probably does not reduce drastically expenditure in manpower and computing time as compared to a numerical integration of the basic differential equations. Such calculations are a t best warranted for a final design of a column on a mainframe or a sufficiently powerful minicomputer. For preliminary design work, for feasibility studies, and when optimizing whole plants of which the separation column is just a component, engineers need to be able to assess the importance of such effects as axial dispersion in an approximate, but fast and straightforward way. Hence, several shortcut calculation procedures have been proposed (Epstein, 1958; Sleicher, 1959; Stemerding and Zuiderweg, 1963; Rod, 1969;Watson and Cochran, 1971; Pratt, 1975; Mecklenburgh and Hartland, 1975, Pratt, 1976a,b). Despite the simplifications they afford, these methods remain very involved and difficult to apply although some of them do yield simple procedures in special 0 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30,No. 6,1991 1249 Table I. Rigorous Solution by Miyauchi and Vermeulen (1963): S # 1
II x,
I-- As
Pe,
Pe,
1,
-1
h
Per
Figure 1. Simple countercurrent separation process.
cases such as when the extraction factor m G I L is nearly unity or if the Peclet numbers in both phases are approximately equal, (Stemerding and Zuiderweg, 19631,or for a very low degree of backmixing (Mecklenburgh and Hartland, 1975). Virtually all of the simplified calcuation procedures proposed thus far disregard the fact that to be practically useful any such method must not only be explicit for the design case but it must also provide an equaUy easy and straightforward method for correcting HTU data obtained from a pilot plant experiment for the effect of axial dispersion. Indeed, the overwhelming majority of the HTU data published in the literature implicitly assumes that ideal plug flow in the experimental test column prevailed. The novel computational shortcut procedure proposed in this paper enables correcting the NTU value in the design case and HTU values obtained from pilot plant experiments for the effect of axial dispersion in both phases in a very simple, straightforward fashion. The formulas can be evaluated on a hand-held calculator almost in a matter of minutes. For a case with straight operating and equilibrium line, the proposed design procedure will approximate the rigorous solution to within a few percent.
Development of a Correction Function Consider the extremely simple extraction process depicted in Figure l involving straight operating and equilibrium lines. To design the respective column, one must specify the flow rates of the raffinate and extract phase ( G and L in the figure), the concentrations of the feed streams (yland x 2 ) , and the concentration of the solute remaining in the raffinate (y2). If both phases move through the column in an ideal plug flow fashion, the number of required overall raffinate phase transfer units may be computed according to the following equation, due to Colburn (1939)
Y1- mx2 Y2 - mx2
"= 1 - m G / L
The required column height is then found as the product of the NTU and the HTU characterizing the respective column or packing: hT = HOCNOG
(2)
Equations 1 and 2 are set up for the design calculation. For the so-called performance calculation, the height is specified rather than y2. Equation 2 is then used to calculate the number of transfer units existing in the column
D.43 =
DA4
ai= 1 A' = -0
xz = - -a + 2p'/2 cos -U 3
3
x3 = -a3 + 2p'f2
cos
a x4 = + 2p'/2 cos 3
where
cos u = q / p 9 I 2
p=($+f q =
(;I + f +
a = Pe,
2Y
- Pe,
f l = (NTU)Pe, + Pepex + (NTU)Pe, y
= (1 - F!(NTU)Pe,Pe,
Pe = U i h T / D i ; i = x , y and NTU K m a P h T / G and the performance is computed by solving (1)for y2. In
with
determining HTU from pilot plant experiments, a third type of calculation must be performed, which will be called pilot plant calculation. It involves computing NTU from measured values of y1 and y2 and solving (2) for HTU. In a similar case but with axial dispersion in both phases, the fractional approach to equilibrium attainable in the countercurrent contactor may be computed on the basis of the analytical solution of Miyauchi and Vermeulen (1963)shown in Tables I and 11. The number of transfer units appearing in Tables I and I1 is the true NTU the column would comprise under the influence of axial dispersion as characterized by one Peclet number in each
1250 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
Table 11. Rigorous Solution by Miyauchi and Vermeulen (1963): S = 1
-UPeY 1-
-
DB = DB,
P"Y P3
1-
-
1
1
D B= ~ (eP4- b . h DB4= -(er3 - b3)p3 F3
cL32
NTU
(NTU)Pe,
b3=1+--
P4
P42
NTU
(NTU)Pe,
b4=1+--
;+ (( + q2 =; - ((41+ 8 )
p3 =
112
cL4
a = Pe,
- Per
8 = (NTU)Pe,
+ PeyPe, + (NTUIPe,
phase. The two tables are thus analogous to (1)but solved for the fractional approach to equilibrium. Thus these equations are straightforward, albeit cumbersome to use for performance calculations. Application of the solution of Miyauchi and Vermeulen to the more important design and pilot plant calculations requires extensive iterations. If one wanted to determine the true HTU for a given packing and system, y1 and yz would have to be measured on a pilot plant separator and hence the fractional approach to equilibrium would be specified for the calculation. The extent of axial dispersion in the test packing would have to be either measured directly on the pilot plant by characterizing the residence time distributions or grossly estimated from the literature correlations. The true number of transfer units NTU contained in the pilot column could then be found iteratively from Table I or 11. Dividing the known height of the test column by this number yields of the true height of one transfer unit. Design calculations become even worse because the Peclet numbers in the analytical solution contain hT.as a characteristic length, which is the very variable one wshes to compute in this case. Use of Miyauchi and Vermeulen's solutions thus requires that both NTU and hT be determined by iterations. In order to simplify these calculations,we propose to use an efficiency fador measuring the effect of axial dispersion
on mass transfer as follows: efficiency factor = NTU,,/NTU
(3)
NTU,, is the "exterior apparent" number of overall raffinate phase mass-transfer units. By "exterior apparent" the number is meant that would be required without axial dispersion. If both the operating and equilibrium lines are straight, NTU,, can be calculated from (1). NTU is the actually required number appearing in Table I. The same ratio has already been introduced by Sleicher (19591, who called it "column efficiency". Similar efficiency measures have been proposed in terms of hT and HTU by Epstein (1958), Watson and Cochran (1971), and Sherwood et al. (1975). The definition according to (3) uses strictly only NTU values. Axial dispersion affects the driving force and thus should be accounted for by correcting NTU rather than HTU. The efficiency thus defined can be used for correcting the apparent number of transfer unitsfor the effect of axial dispersion if the efficiency is known as a function of those variables tht are specified in a given case. When one wishes to determine HTU from pilot plant experiments, these variables are the measured fractional approach to equilibrium, the extraction factor, and the two Peclet numbers. In order to devise a true correction procedure, the fractional approach should be replaced by NTU,, assuming that a user will determine NTU, from the measured performance of his separator employing a conventional design procedure of his choice. It is important to stress that the efficiency factor must be given in terms of NTU,, rather than NTU. Equations correlating the efficiency factor to NTU such as the one proposed by Sleicher (1959) cannot be used for either pilot plant or design calculation without time-consuming iterations, because NTU is unknown in such cases. Hence the efficiency must be known as a function of the following variables: NTUap/NTU = fpp(NTU,,,SSe,~ey) (4) where S denotes the extraction or the stripping factor mG/L. For the present study, function (4) was computed numerically by calculating the fractional approach to equilibrium from NTU, by means of (1)and by then solving the equations of Table I numerically for the true NTU. This calculation was repeated for several thousand seta of values for the independent variables. Typical results of such calculations are plotted in Figure 2. For the design calculation different types of Peclet numbers were defined in order to eliminate the unknown column height from them: Pemu = UHOG/D (5) In order to be of direct use for the design calculations, values of the efficiency factor must be available as a function of the following variables: NTU,,/NTU = fD(NTU,,,SSeHTvy,PeHTv~) (6) This function was numerically evaluated in the same way as explained for the pilot plant calculation but by replacing all Peclet numbers in Tables I and I1 by the following function UhT Pe = D (7)
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1251
-
At (NTU,,) m the efficiency factor defined by (4) becomes zero (see Figure 2). A linearized plot of the efficiency factor vs (NTU), would thus be described by NTUap NTUap --1NTU (NTUap)NTu-=I-
Y."
0
5
15
10
NTU
b
NTU,lp
25
20
,
NTUap 1nS S-1
(9)
Pe,Pe,
+
Pe,
+ S(Pe,)
1 .o
It can be shown that (9) is indeed correct for S = 1. In other cases, the curves are concave upward or concave downward depending on the exact values of S, Pe,, and Pe,. An empirical expression was fitted to represent these curves as follows:
0.8
NTU 0.6
0.4
I
0.2
+B
(10)
A
0
5
15
10
NTU C
NTU,
20
where A and B are defined as
,
1 .o
s < 1:
0.8
NTU
X
0.8
0.4
(NTU, - 0.9352)0.a32 \2)0.4364
0.2
I . . . . I . 0
5
10
15
, , ] 20
NTU,
Figure 2. Correction factor for evaluating the true number of transfer unita in pilot columns. Pey = 20 in all cases. Solid lines, rigorous computation; broken lines, approximative formulas. (a) mGIL = 0.4;(b) mGIL = 0.8; (c) mGIL = 1.5.
Typical results are plotted in Figure 3.
Correlation of Results in Terms of Simple Mathematical Functions Since the correction factors expressed by (4) and (6) are functions of four independent variables, they cannot be published exhaustively in the form of graphs or tables. Mathematical functions were thus developed and fitted to the numerical results in order to convey the correction functions in a form easy to use. In trying to find a suitable algebraic form for the pilot plant calculation, it is useful to recall that in a contactor with a specified degree of axial dispersion, defined in terms of Pe, and Pe,, there exists an upper limit for NTU, at which its performance will be totally governed by the axial dispersion rather than by mass transfer. At that limit the driving force will approach zero and the true NTU will tend to infinity while NTU,, stays at a finite value. It can be shown (Miyauchi and Vermeulen, 1964) that this upper limit follows from Table I as
s L 1:
B=O (lob) This equation fitted 2367 sets of data with both Pe values ranging from 2 to 100 and with S values form 0.3 to 2 with a standard deviation of 0.024. The equation is plotted as broken lines in Figure 2 and is seen to agree reasonably well with the exact calculation (solid lines). At very low stripping factors expression 10 appeared to deviate somewhat more from the correct values. Hence a more accurate expression for S = 0 was developed: -NTUap =1-[1+
NTU
]
-1 1.438Pe,0.9444
(11)
NTUa,'*048
Since backmixing in the extract phase is irrelevant if the solute has a near-infinite solubility in it (S = 0), the correction factor does not depend on Pe, in this case. When (11) was used at S = 0 and (10) for stripping factors ranging from 0.1 to 2, a set of 2722 data points with Pe values ranging from 2 to 100 was reproduced with a standard deviation of 3.5 absolute %. In the case of the design calculation the efficiency factor does not go to zero because the Peclet numbers Pem are defined differently. As can be seen from (7), the absolute degree of backmixing, expressed by Pe, is constantly reduced as one increases NTU. The efficiency factor
1252 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 a N
,
1.0
I
3
NTU 0.8
NTV, NTU 0.6
0.0 I
, ........................................
0.4
D
0.2'
"
'
"
0
"
"
'
0.4
r-l
................................
0.5
30
I .o
1
I
parun&=: P b H N X
0.8
8
NTU,
l 4 j
NTU 0.6
2
,...............
......
a ............................. ..__
. . . . . .............
....................................... 0.4
....... .......................
0.2 20
30
T--+ 20
10
0.2 ID
-
NTUq
d
i,
0
20
t 2
NTU-
I ....................
10
"
'
30
.......................................................
0.8
"
I
0.8
NTU
"
20
10
1.0
NTU-
.....
...'...,.........
0.4
0.5 30
NTU,
NTUq
Figure 3. Correction factor for evaluating the true number of required transfer units in design calculation. Solid and broken lines, see Figure 2. (a) mG/L = 0.4, Pemy = 2; (b) mG/L = 0.8, Pew, = 8; (c) mG/L = 0.8, PemY = 1; (d) mG/L = 1.5, P e m , = 2.
therefore tends toward a constant limit (cf. Figure 3). Ita dependency on NTU, may be computed analytically by substituting (7) into (97 and by solving for NTU. Dividing NTU, by this result yields NTUan -= NTU
For extractors containing more than just a few transfer units, the efficiency factor may also be estimated grossly from the limit value (12) tends to when NTU, approaches infinity: P~HTU,@HTU, NTUap -(14) NTU peHTU,.$eHTU,x + PeHTUy + S(PeHTU,x) This simple equation will approximate the correction factor to a few percent in many cases provided that NTU, > c8. 5.
where
A = - -In - S s-1
B=- S +- 1 (1%) P ~ H T UP~~ H T U ~ Although based on the linear form of (lo), (12) correlates the correction factors sufficiently well also for S values deviating from unity provided that (NTU,/NTU) > 0.1. For a set of 6866 data points covering S values from 0.25 to 2 and Pe values from 0.2 to 8, the standard deviation was 0.045 (see also Figure 3). Equation 12 does not correlate the data for lower S values well; at an extraction factor of S = 0.1 the standard deviation often exceeded 10%. At S = 0, the following equation (13) is recommended. It correlated 806 sets of data with a standard deviation of 4.2%: NTU,, 1-. -1NTU e ~ p ( P e ~+ ~ ~ ~ . ~- 0.166 ~ ) NTUa,1.46 (13)
Application When trying to assess the importance of axial dispersion, HTU data from pilot plant experiments or from the literature will first have to be corrected for this effect. The procedure proposed to this effect is shown graphically in Figure 4, for an example in which the raffinate phase is a gas and the extract phase is liquid. HTU data from the literature or from pilot plant experiments come in terms of apparent values of the gas-phase and liquid-phase heights of a mass-transfer unit or as the apparent values of the (overall) gas- and liquid-phase numbers of masstransfer units prevailing in the pilot plant column (see top of Figure 4). The degree of backmixing occurring in both phases in the test column must be estimated in terms of Peclet numbers from the literature or determined by residence time distribution (RTD) experiments. Correlations for predicting Peclet numbers in commercial-scale gas/liquid contactors have been prwnted by Dunn et al. (1962,1977), Woodburn (1974), Richter (1978),and others. Some of the available data has been described by Sherwood et al. (1975), and von Stockar and Cevey (1984) reviewed published correlations for PeL. Hanson (1971) summarized data on axial dispersion in liquid extractors. More recent data has been published by many authors, including Pratt
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1253 Pilot plant experiments or literature
NOG,ap
'""\ t
t
1
t
1 ih.l hi'
Figure 4. Schematic procedure for finding the true HTU values from pilot plant experiments or from literature values. The graph was set up assuming that the raffinate phase is a gas.
and Anderson (1977), Boyadzhiev and Boyadjev (1973), Venkantarama et al. (1980),Haug (1971), and Hody (1975). An extensive detailed review of axial dispersion in liquid/liquid contactors has been written by Steiner (1988). As literature correlations often use the dimension of the packing elements as a characteristic length (Pe ) these values have to be converted to column Peclet numgrs (Pe) based on the known height h of the pilot plant column: Pe = Pe,(hT/dp)
(15)
Equation 10 (or (11)for S = 0) is then used to estimate the correction factor. In cases where a gross assessment suffices, the correction factor can also be estimated from the much simpler (9). Subsequently the true number of transfer units contained in the test column is determined by dividing the measured NTU, by the correction factor. Alternatively, the true HTU v&es can directly be computed as follows: HTU = HTU,,(NTU,,/NTU) (16) This procedure, carried once for each of the phases, yields the "true* individual phase heights of a transfer unit (HG and HL) that are to be used for design calculations. When designing a new separator, conventional design procedures are employed to compute the apparent number of required transfer units, NTU, (see Figure 5). Packing Peclet numbers for both phases are then estimated from the literature and converted into their HTU equivalents
PeHW = Pe,(HTU/d,)
(17)
On the basis of these estimates, the efficiency factor is evaluated from (12) (or (13) for S = 0). For a gross assessment the much simpler (14) may often suffice. Dividing NTU,, by the efficiency converts it to NTU, which upon multiphcation with the true HTU yields the required column height. Since the proposed procedure only corrects NTU values, it is believed that a reasonable assessment of the effect of axial dispersion will be obtained irrespective of the method employed to compute NTU,,. The accuracies cited in the former section will however only be possible for straight operating and equilibrium lines.
Hook
Figure 5. Design procedure based on the true required NTU. The graph was set up assuming that the raffinate phase is a gas. Table 111. Diffusivities of Various Gases solute solvent D,m2 s-l reference SO2 air 1.31 X Billet and Schultes (1988) SO2 water 1.87 X 10" Billet and Schultes (1988) 0, air 1.78 X 10" Sherwood and Pigford (1952) 0, water 2.83 X lo4 Weast (1977) NH3 air 2.17 X lob Sherwood and Pigford (1952)
Numerical Design Example Problem Statement. Sulfur dioxide is to be removed from an aqueous solution by countercurrent stripping with air a t atmospheric pressure and 298 K. To this effect a packed column containing 2-in. Raschig rings has to be designed such that a liquid loading of 4000 lb ft+ h-I (5.42 kg m-2 s-l) and a gas flow rate of 270 lb ft" h-' (0.366 kg m-2 s-l) can be handled. The liquid inlet mole fraction is 1.4 X lo4 and the gas stream has to be loaded with SOz to 98% of the equilibrium value. Is it conceivable that axial dispersion will have an appreciable influence on the design? Properties of System and Packing. The equilibrium line will be taken as straight with a slope of m = 15.9 (Liley et al., 1984). The diffusivities of the involved substances are given in Table III. HTU values for 2-in. Rashig Rings have been published by Sherwood and Holloway (1940) and Fellinger (1941) for conditions very similar to those envisaged in this design problem. Data on axial dispersion in commercial columns are scarce. The Peclet numbers have been estimated from the data due to Dunn et al. (1962,1977), which has been reproduced in graphical form by Sherwood et al. (1975, Figure 11.10). Design Based on Plug-Flow Assumption. In this problem, the raffinate phase is the liquid. The extraction factor is therefore given by L/mC and (1)has to be rewritten as follows:
1254 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 Table IV. Mass-Transfer Characteristics of 2-in.Raschig Rings reference HL 0.42 m Sherwood and Holloway (1940),run 152 HG 0.27 m Fellinger (1941) Pep,L 0.21 Dunn et al., (1962),as reproduced in Figure 11.10 by Sherwood et al. (1975) Dunn et al. (1962), as reproduced in Figure 11.10 Pep,c 1.8 by Sherwood et al. (1975)
A balance gives x2 = 4.85 X y2 = 0, and the extraction factor is evaluated as being 1.5. The apparent number of transfer units according to (18) follows as 1 -0.5
(NOL)ap = -In [-0.5(2.88) + 1.51 = 5.74 Estimating the HTU values (Table IV) for SO2 by taking them inversely proportional to the square root of the respective molecular diffusion coefficients (Table 111)yields the following values: HL
= 0.42( =1.87 )I2
HG = 0.27(
Hence
s) 112
= 0.517 m
(19)
= 0.245 m
(20)
= HL + (L/mG)HG ( H o L )=~0.517 ~ + 1.5(0.245) = 0.885 m HOL
(21)
h~ = ( N d a p ( H O d a p hT 5.74(0.885) = 5.077 m (22) Pilot Plant Calculations. The HL experiment of Sherwood and Holloway (1940) from which the value of HLwas taken (run 152) was carried out in a packed bed 20.3 in. high. The apparent NTU was therefore
-
(20.3 in.)(0.0254 m in-l) 0.42 m = 1.228
(23)
This corresponds to NOC,ap at the top of Figure 4, which was set up for a gaseous raffinate phase. The column Peclet number can be computed based on (15) (see also upper middle of Figure 4): hT
PeL = Pep,LdP 20 3 PeL = 0.21=2 2 This corresponds to a large degree of backmixing in the liquid phase that might have affected HTU. Since oxygen is very poorly soluble in water (m ;c: 4.4 X lo4), the extraction factor was essentially zero and the gas-phase backmixing was irrelevant. The column efficiency factor may be evaluated approximately from (11). However, one has to bear in mind that the liquid was the raffinate phase and that the liquid-phase Peclet number has to be substituted for Pe, in (ll),(lower middle of Figure 4).
[
NTUap -1NTU NTUap = 1NTU
1-t
[+ 1
I
1.438PeLo.9444 (NOL)ap1.048
1.438
20.9444
1
-1
1.2281.048
= 0.7
(24)
This efficiency figure can now be used to correct the apparent HLvalue obtained from (19) and employed in the design when assuming plug flow. According to (16) the true HTU for SO2 in water is (HL)true = (HL)ap(NTUap/NTU) ( H L ) ~ ~=,0.517(0.7) ,~ = 0.362 m (25) or 30% lower than the apparent value. (The same correction calculation could also have been performed as shown in the lower part of Figure 4.) In order to correct the apparent HTU values for the extract phase in a similar way (right-hand side of Figure 4), one has to assess the influence of backmixing on Fellinger's data. This is more difficult because the height of the packed bed for his experiments is not readily available. The stripping factor prevailing during the measurement may however be estimated. For the system ammonia/ airlwater at low concentrations,and at 20 "C,m = 0.8 (von Stockar, 1973), whence -mG- - 0.8(270)(18) = 0.03 L 4000(29) At such a low stripping factor the ammonia is absorbed so easily that the bed must have been quite shallow in order to avoid total absorption. Already with 3 apparent gas-phase transfer units 95% of the entering ammonia would have been removed from the gas, thus making the determination of the driving force at the top of the packing a demanding analytical task. A reasonable value would thus have been = 2. Hence hT
=
(NOG)ap(HOG)ap
(26)
(NOG)ap(HG)ap
where is the apparent value of the gas-phase height of a transfer unit published by Fellinger (see Table IV): hT = 2(027 m) = 0.54 m The column Peclet numbers are again estimated according to (15), based on the Pep measurements by Dunn et al. (see Table IV and top middle of Figure 4): 0.54 m PeG = 1.8 = 38 2 in.(0.0254 m in.-l) 0.54 m PeL = 0.21 =5 2 in.(0.0254 m in.-l) Using again (ll),but keeping in mind that this time the gas is the raffinate phase (lower middle of Figure 4):
[
I
NTUap peG0.9444 -= 1 - 1 + 1.4338NTU Nap1.0&
[
NTUap -= 1 - 1 + NTU
380.9444 1.433821.048
= 0.96
1
(27)
The correct HTU value for design is thus (HG)true = (HG)apN"ap/NTU) (28) where (HG)ap is Fellinger's apparent value, but corrected for SOz (see 20)). (HG)true = 0.245(0.96) = 0.235 m Thus, Fellinger's HTU value cannot have been affected much by deviations from plug flow. Indeed, one might intuitively suspect that the influence of axial dispersion
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1255 Table V. Summary of Design Example with axial dispersion pilot-plant calcn HL,m HG,m
plug flow apparent values 0.517 (eq 19) 0.245 (eq 20)
design calcn HOLl m NoL (=NTU) hm m
plug flow apparent values 0.885 (sq 21) 5.74 (eq 18) 5.077 (eq 22)
Pe Pe, 2 38
pe,
NTU,/NTU 0.7 (eq 11) 0.96 (ea 11) with axial dispersion
5
Pe
Pemy
Pemr
3
25
in the liquid is not important for systems with gas-film control. However the effect could be surprisingly large. Measurements could, e.g., readily be done a t mG/L = 0.5 and with 6 apparent transfer units. This would make the height of the packed bed 6(0.27) = 1.62 m, and the column Pe numbers would be recalculated according to (15) as: Pec = 57 PeL = 6.7
A simple estimation of the column efficiency could be carried out by using (9) _.. -
-- -
I\ITU
-1-
Pe,
. . -+ 6.7(57) -0.5 57 + OA(6.7)
= 0.222
(9)
0.66 (eq 12)
-NTU,, NTU
P~HTU,L~~HTU,G PeHTU,fleHTU,G
NTU,
-NTU
-
+ PeHTU,L + s(peH"u,C)
3(25) = 0.645 3(25) + 3 + 1.5(25)
(29)
Equation 12 is more accurate:
--
2
0.715 m = 25 2 inJ0.0254 m in.-l) 0.715 m P e H T U , L = 0.21 =3 2 in.(0.0254 m in,-l)
where: A = - In S = -in 1.5
For a more accurate estimation, one would evaluate A and B in (10): A = 1.09 B = 0.03 whence, according to (10): NTU,, --1+ 0.03= 0.318 NTU 7.714(1.09) Backmixing in the liquid phase can thus almost completely destroy the performance even of a gas-film-controlledabsorber. Design Calculation. The true overall height of a raffinate phase unit is calculated according to (21) as follows (see also middle left of Figure 5): ( H O L )=~0.362 ~ + 1.5(0.235) = 0.715 m The apparent number of raffinate phase transfer units obtained by assuming plug flow (-5.74, see (18)) must now be corrected for the influence of axial dispersion in the new column. (This number corresponds to NmSpin Figure 5 because the gas is the raffinate phase in that figure.) The HTU Peclet numbers are computed by employing (17) (see middle part of Figure 5): P'2Hq-u = Pep(HTU/dp) PeHTU,c
corrected valuea 0.715 (ea 21) 8.7 (eq $0) 6.22 (eq 31)
+ S(Pe,)
6 In --- n.5 -.-
NTU,,/NTU
approximate value for the correction factor. It has to be kept in mind that the liquid is the raffinate phase:
NTU
-+ S-1
corrected values 0.362 (eq 25) 0.235 (ea 28)
= 1.8
(17)
Equation 14 may be used in order to come up with an
S-1 0.5 = 0.811
and 1 B=- S +P~HTU,L P~HTU,C
- -1.5 + -1
3 25 = 0.54
1
AB 0.811(0.54) +B - = 1 + 0.54 NTUap 5.74 = 1.464
NW, -NTU
2
+
(1.464)2 4
5.74
= 0.66 Thus (lower right in Figure 5):
(Nodtrue = (NoL)ap/(NTUap/NTU) = -5.74 0.66 = 8.7
(30)
h~ = (NoL)true(HoL)true = 8.7(0.715) = 6.22 m (31) An overview of these calculations is provided in Table V. The required packed height is therefore some 23% higher
1256 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
than when working with the plug-flow assumption. Although this might not seem an overwhelming difference, it must be borne in mind that the result could have been entirely different due to several reasons. If backmixing in the liquid had for instance dominated the HC measurement, the true HoLmight have been as low as HoL= 0.362 + 1.5(0.245)(0.3)= 0.472 m Because of lower HTU Peclet values, the column would have been even less efficient, requiring as many as 10 actual overall raffinate phase transfer units. The high NTU requirement would however have been overcompensated by the low HTU value, thus yielding a lower packed height (4.7m) than the simple plug-flow assumption. Axial dispersion effects are thus seen to have sometimes the opposite effect of what is normally expected. Conclusions Neglecting deviations from plug flow in countercurrent contactors results in overestimated HTU values when evaluating pilot plant experiments and in an underestimation of the required number of transfer units when designing a new separator. In many situations these two effects tend to compensate each other, but there are obviously cases in which they lead to a serious underdesign. Such dangers are especially present when HTU data obtained in a mass-transfer-controlled regime is used to design a contactor being largely dominated by axial dispersion effects as sometimes occurs in liquid-liquid extraction columns. It is also possible that axial dispersion effects give rise to a lower final column length than the plug-flow assumption, which is in contrast to normal expectations. Thus even if it is true that compensation effects tend to attenuate the impact of axial dispersion, neglecting them will introduce a very considerable design uncertainty. It is difficult to decide in a given case whether a design based on the plug-flow assumption is more or less correct or whether it could be completely invalidated by axial dispersion effects. It is in deciding this question that the proposed shortcut design procedure should prove especially useful. It enables a reasonably accurate estimation of the impact of these effects on the design in a simple and straightforward way. In case deviations from plug flow have a small or moderate effect, the proposed procedure can be used to improve a simple design without much effort in manpower and computing time. If it turns out that axial dispersion has an overwhelming effect, one might decide to repeat the calculations using a rigorous computer model.
Acknowledgment
We acknowledge the impulse for this work provided by the late Professor T. K. Sherwood. The incentive to tackle this question came from a discussion between him and one of the authors many years ago at Berkeley.
Nomenclature a = specific interfacial area, m D = axial dispersion coefficient of the phase considered, m2 S-1
Di = axial dispersion coefficient, (i = x , y, L, G),m2 s-l d = nominal dimension of packing, m
d=
gas flow rate, mol m-2 s-1 HTU = true raffinate phase height of a transfer unit, m HTU, = raffiiate phase height of a transfer unit uncorrected for axial dispersion Hi = height of a transfer unit, true value if nothing else is specified (i = L, G, OL, OG), m hT = packed tower height, m
Km = overall gas-phase mass-transfer coefficient, mol s-l m-2 Pa-'
L = liquid flow rate, mol m-2 s-l m = slope of equilibrium line NTU = true number of overall raffinate phase transfer units NTU, = exterior apparent number of overall raffinate phase transfer unit, uncorrected for axial dispersion Ni = number of transfer units, true value if nothing is specified (i = L, G,OL, OG) Pei = UihT/Di,column Peclet number (i = x , y, L, G) Pep,i = Uid /Di, packing Peclet number (i = x , y, L, G) PeHmi= d(HTU)/Di, HTU Peclet number (i = x , y , L, G) S = mG/L, stripping factor U = superficial velocity of the considered phase, m s-l Vi = superficial velocity (i = x , y, L, G), m s-l
Subscripts y = referring to raffinate phase x = referring to extract phase G = referring to gas phase L = referring to liquid phase OG = overall gas phase OL = overall liquid phase ap = apparent value true = true value
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Stemerding, S.; Zuiderweg, F. J. Axial Mixing and ita Influence on ,Extraction Efficiency: A Simplified Method for Design Calculations. Chem. Eng. 1963, May, CE-156-CE-160. von Stockar, U. Gasabsorption mit Wheeffekten, Dissertation No. 4917, ETH-Ziirich, 1973. von Stockar, U.; Cevey, P. F. Influence of the Physical Properties of the Liquid on Axial Dispersion in Packed Columns. Znd. Eng. Chem. Process Des. Dev. 1984,23, 717-724. Venkantamara, J.; Degaleeshan, T. E.; Laddha, G. S. Continuous Phase Axial Mixing in Rotary Disk Contactors. Can. J. Chem. Eng. 1980,58, 206-211. Watson, J. S.; Cochran, H. D. A Simple Method for Estimating the Effect of Axial Backmixing on Countercurrent Column Performance. Ind. Eng. Chem. Process Des. Dev. 1971,10, 83-85. Weast, R. C., Ed. CRC Handbook of Chemistry and Physics, 58th ed.; CRC Press: Cleveland, OH, 1977. Woodbum, E. T. Gas Phase Axial Mixing at Extreme High Irrigation Rates in a Large Packed Absorption Tower. AZChE J. 1974,20,
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Receiued for reuiew February 8, 1990 Revised manuscript received November 1, 1990 Accepted November 23, 1990
Extraction Kinetics of Palladium(I1) with Bis(2-ethylhexyl) Sulfoxide from Hydrochloric Acid Media Hanzhang Wang,* Jianhua Pan, and Jiansheng Gu Department of Chemistry, Suzhou University, Suzhou 215006, People’s Republic of China
Investigation was made on the extraction kinetics of palladium with bis(2-ethylhexyl) sulfoxide (DEHSO) from hydrochloric acid solution by the use of a constant interfacial area cell, which permits continuous and automatic measurements. Effects of concentrations of Pd(II), C1-, and DEHSO, and the effect of stirring speed and specific interfacial area A (interfacial area divided by the bulk volume) on the extraction rate were studied, along with the interfacial adsorption, biphase distribution, and polymerization characteristics of DEHSO. The observed extraction rate expression -d[Pd] /dt = K’Aa,[Pd][DEHSO] supports a mechanism in which the rate-determining step is (PdC13L)[ L (PdC12L2)i+ C1-, with an activation energy of 44.7 kJ/mol, where a3 = [PdC1
