Calculation of Liquid-Liquid Extraction Processes

as applied to laboratory studies in the fields of simple liquid ex- traction and ... Simple extraction consists of the transfer of a solute from one ...
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ENGINEERING AND PROCESS DEVELOPMENT

Calculation of Liquid-Liquid Extraction Processes EDWARD G. SCHEIBEL Hoffmann-la Roche, Inc., Nutley 10, N. J.

I

NCREASING attention devoted to liquid-liquid extraction in recent years has resulted in simplification of the methods of process calculation, improved equipment design and construction, and improved operating techniques, particularly for laboratory investigations. This paper summarizes all the recent advances as applied to laboratory studies in the fields of simple liquid extraction and in fractional liquid extraction. Multiple Batch Extraction with Fresh Solvent I s Simplest Type of liquid-liquid Extraction

known number of extractions is determined, the value of

$and

subsequently the value of the distribution coefficient, D, may be calculated. If the distribution coefficient is not constant, this method gives an average value over the range of concentrations encountered. More useful data can be obtained if the quantity extracted in the solvent after each step is measured. The fraction remaining in the original solution after each extraction is given by the expression (I

+ LD x)

-1

and the amount extracted in each step is

Simple extraction consists of the transfer of a solute from one liquid solution to another solution. The usual purpose of this transfer is to effect a purification of the desired component. In some cases the desired component is removed from the original solution, which retains unwanted compounds, and in other cases the desired component remains in the original solution, while the impurities are extracted. Another common purpose of this simple extraction is to provide a second solution from which the desired component is more readily recoverable than from the original solution. The second solvent may, in this case, effect a concentration of the desired component or it may be more readily aeparated from the desired component by distillation. One interesting manner in which a concentration may be effected through the use of a chemical reaction is the case of an aqueous eolution of either a weak acid or base. The solvent extract of the aqueous solution can then be treated with a concentrated base or acid solution, respectively, to produce a saturated aqueous solution of the salt, even though the solvent extract is more dilute than the original aqueous solution, Another incidental purpose of the extraction of a desired component from a solution is encountered when the component must be present in some other solvent for the next step in the process, and when a suitable concentration can be obtained, distillation or other more complicated recovery methods are then eliminated. The simplest manner in which a component may be transferred from one solution to another is by successive extractions with fresh solvent. If a Component with a distribution coefficient, D , is brought to equilibrium between L parts of light solvent and H parts of heavy solvent, the ratio of the amount in the light solvent to that in the heavy solvent will be as L D is to H. Thus, the H fraction remaining unextracted in the heavy phase will b e n T L D .

proportional to the amount present in each step. A plot of the logarithm of the amount extracted against the step number should be a straight line with a negative slope such that the ratio, q, between consecutive quantities, is given by the expression,

If the heavy phase is then extracted again with the same amount,

increment of 13 extractions as

L,of light solvent, the fraction remaining will be the square of this

LD

fraction. Thus, Underwood (16) has derived the expression for the fraction unextracted after n extractions with L parts of fresh solvent,

2r=

1

(l+gy

and a chart has been developed to facilitate the application of this equation to the data. If the amount remaining after a

16

&

1 q =

1+,

LD

Thus, it is possible to calculate the distribution coefficient from the slope of the straight line obtained in this type of plot, If the data do not appear as a straight line on the previous plot, either the distribution coefficient varies or the extracted material consists of two or more components with different distribution coefficients. In general, components i n dilute solutions will show constant distribution coefficients, so a continual curvature in the dilute region will indicate additional components. It is possible to determine the initial quantity and distribution coefficient of the different components from the data on successive extractions, and the author of this paper has successfully used the following technique to determine these factors for each of three different components in an unknown mixture. The method is most readily applicable to components which have distribution coefficients differing by a ratio of more than 5. Smaller differences require more extractions, as do greater numbers of components. Figure 1represents a set of typical extraction data on a mixture of equal parts of two components with different distribution coefficients. The circles are the data obtainahle from the composition of the extract solution from each stage. The data for the last five stages lie on a straight line from which the ratio for each stage may be calculated from the terminal conditions for the or 0.907. Solving

Equation 2 for 7,

and in this case the light solvent quantity is one tenth of the heavy solvent quantity, so the value of the distribution coefficient for component B is 1.02. If the quantity of this component is subtracted from the total amounts in the previous extractions, the points represented by the triangles are obtained which also lie on

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ENGINEERING AND PROCESS DEVELOPMENT a strnight line. The ratio from the slope of this line may be cal1.5 1 1 4 LD culated as = 0.497 - = 1.01, and thus for compo-

(2x)

' H

nent -4,D = 10.1. The quantity of each component present in the original solution can be calculated when it is recognized that the sum of the converging geometric series with a ratio of y between terms is equal Ill to -where y,, is the first term. Thus the quantity of compo1-Y 24.8 nent A is equal to -- - or 49.4, and the quxntity of component 0.503 47 B is equal to or 50.6. 0.093

50

I

removal of two unidentified impurities from an aqueous solution of a desired compound by successive extractions. The quantity in the extract solution approached a constant value after about 30 extractions, and this was considered to be due to the distribution of the desired component between the aqueous solution and the solvent. This coefficient was estimated from the limiting constant amount in the solvent and the known concentration in t,he aqueous solution. After subtracting this from the totals for the previous extraction stages, curves similar to that in Figure 1 were obtained. In this particular case, samples of the aqueous solution extracted after different periods of time and analyzed in this manner indicated an appreciable change in the relative amounts of each of the two impurities from which the reaction rate constant could be evaluated. This example demonstrated the application of the batchwise extraction technique with fresh solvent for the analysis and study of mixtures in which the chemical composition of the components may be entirely unknown.

30 Batchwise Countercurrent Extraction Effects More Efficient Utilization of Solvent

The method of extracting with fresh solvent each time does not make the most efficient use of the given quantity of solvent. A better utilization of the solvent can be realized by using the solvent extract from the second stage to extract the solution in the first stage, as shown in the extraction pattern in Figure 2. This figure shows the batchwise technique for establishing steady-state conditions in a continuous countercurrent extraction operation involving five stages.

I

2

3

4

5 6 7 8 91011 EXTRACTION NUMBER

121314

Figure 1. Extraction Data for Equal Parts of Two Components with Different Distribution Coefficients

This represents the accuracy of the graphical technique, because the data were originally calculated for a mixture of 50 parts of each of two components, one with a distribution coefficient of 10.0 and the other with a distribution coefficient of 1.00. When the best straight line through the points is constructed, the limits of accuracy of the experimental measurements must be considered. For example, in Figure 1 after five stages, the amount of component A in the extract is obtained as the difference between two numbers that are nearly equal, and no emphasis should be placed on such points in locating the line for component A . In some cases it is desirable to change the solvent ratio after the major portion of component A is believed to be eliminated, so the curve for component B will have a greater slope for more accurate evaiuation of the distribution coefficient. In this case, the data appear as two discontinuous curves and the amount of component B in the first stage with the altered solvent ratio is calculated as described, and the amount of component B in the previous extracts is calculated from its distribution coefficient and the solvent ratio in the stages. Subtraction gives the amount of component A , which is then correlated to a straight line as previously described. The writer has applied this technique to several similar mixtures of three components in which the third component had a distribution coefficient of about 100. The problem involved the January 1954

rn

484P9513P*4?

Figure 2.

Simple

Batchwise Countercurrent

Extraction

If q represents the fraction remaining in the heavy solution and p represents the fraction removed in the solvent in any stage, it can be shown that p =

E

and q

=

1 E + l

~

(3)

In Figure 2 it can be shown that the quantity of solute remaining in the final heavy solution at any cycle, t, is the sum of the first t terms of the converging series

INDUSTRIAL AND ENGINEERING CHEMISTRY

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ENGINEERING AND PROCESS DEVELOPMENT and The sum of infinite terms of the above series gives the quantity of solute remaining in the heavy solution a t steady-state conditions, based on a unit quantity in the feed. The sum of the terms from t to infinity is a measure of the deviation from steady-state

HEAVY SOLUTION IN (FEED)

-

-

LIGHT SOLUTION OUT (EXTRACT)

LIGHTSOLVENT

F t = F' for t

j1

E' - -1)(E + 1)

E"

(En+'

[x2]'-' (5b) ( E f 1)

>2

where F , is the quantity of solute in the feed solution a t steady state and Ft is the quantity of solute in the solution a t t cycles. The second term inside the brackets in all equations is the fractional increase in the feed concentration. Equation 4 will give a concentration in feed stage less than 10% in excess of the steadystate concentration and will give a more rapid approach to steady conditions than the second pair of equations (Equations 5a and 5b), which will never exceed the steady-state concentrations in any of the cycles. I n general, the approach to steady-state conditions a t the dilute end will be about 90% or better in a number of cycles equal to the number of stages. This technique does not offer any significant advantage in the case of small numbers of stages, but when 10 or more stages are involved the value of the method of operation is obvious. Simple extractions with large numbers of stages are not the usual practice and, since the simple extraction will frequently be carried out in the laboratory to obtain some of the pure material, it may not be possible to increase the feed concentration as required by this operation. Thus, this technique does not have the same practical application as in fractional liquid extraction, but may be considered primarily as an improved method for experimentally verifying steady-state design calculations.

a!

*

Continuous Countercurrent Extractor Allows Multistage Processing of Large Quantities of Materials

Figure 3.

Simple Continuous Countercurrent Extraction

conditions and the ratio between suc'cessive terms approaches, but is always greater than 3pp. I t has been shown that for a five-stage fractional liquid extraction the ratio between successive terms is equal to 3pp ( l a )so the aboveseries convergesslightly less rapidly, and it may be concluded that the simple liquid-liquid operation approaches steady-state conditions, less rapidly than the fractional liquid extraction operation. Figure 7 shows the number of cycles required for a 90% approach to steady-state conditions in fractional liquid extraction. This gives an indication of the number of cycles necessary in simple liquid extraction, which will be slightly greater than shown in Figure 7. Since the most economical method for operating a simple liquid extraction process is in the range of extraction factors from 2.0 to 1.25, a fivestage simple extraction operation requires six to eight cycles and a 10-stage operation takes 12 to 30 cycles for a 90% approach to steady state a t the dilute end of the column. A 99% approach requires twice these numbers of cycles. In the case of a large number of stages, it is possible to effect an appreciable decrease in the number of cycles for a given approach to steady-state conditions by adding excess solute to the feed to compensate for the quantity returned in the solvent to this feed stage a t steady conditions. This will be discussed in more detail in the case of fractional liquid extraction, where it has a much greater advantage than in simple extraction. The empirical equations derived for the simple extraction are

The continued operation of a series of batchwise treatments can become very tedious when a large amount of material must be processed in a large number of stages. For this purpose a continuous countercurrent extractor is usually employed. This may consist of a packed column or an internally agitated column (8, I I ) , with or without calming sections for phase separation between the mixing section. *Flow through such a column is shown in Figure 3. The concept of the height equivalent to a theoretical stage has been used in the design of packed columns, although it is not considered as theoretically sound as the concept of the transfer unit (IO). The performance of a packed column varies considerably with different solvent systems and in many cases requires several feet for a theoretical stage even in small diameter columns. On the other hand, the internally agitated column can provide several theoretical stages for a foot of height in the smaller diameter columns. A packed column requires a much longer time to come to steady conditions and leaves a larger amount in process as column holdup than an internally agitated column. Both types of columns present some uncertainty as to the exact number of equilibrium stages in the system as compared to the previous batchwise techniques. However, their simplicity of operation completely offsets this disadvantage when large volumes must be processed. I n addition, the operation of the continuous columns provides data on the performance of units similar to those used in actual full scale production. The relationships for calculating the performance and design of continuous simple liquid extraction have been developed by Underwood ( 1 7 ) xn

Yn+ 1 -7

E-1 E"+' - 1

=xo

or

f o r t = 1 and 2

18

-

gn+ 1 __

D

where x a = initial concentration of solute in the feed, xn = concentration after n stages, and y- + I = concentration of solute in fresh solvent. If the fresh solvent is free of solute, the quantity on the lefb

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ENGINEERING AND PROCESS DEVELOPMENT I 9 6

7 6

5 4

3 2

E .I

.8 .6

.5 .4

-3 -2

.I

hand side of the equation represents the fraction unextracted. The fraction extracted may then be shown to be equal to XO

-=

XO

x,,

E"+' - E

(7)

En+' - 1

Figure 4 shows the graphical representation of Equation 1 for batch extraction with fresh solvent and Equation 6 for continuous countercurrent extraction. This figure is particularly ugeful in observing the relationship between the two methods of operation. Consider, for example, that the batch extraction with five portions a t a solvent ratio of one fifth the volume of feed was found to give 90% extraction of the desired component. According to the *

LD

lower family of curves, the value of - must have been 0.86

H

and the average value of D was therefore 4.3. If this recovery were satisfactory, it could be accomplished in five continuous

L

countercurrent stages a t a value of E = 2.24, whence 2 = 0.52. Thus, the same job can be done with 52% of the total solvent if employed in a countercurrent manner. Also, if the original amount of solvent were employed, E would be equal to 4.3 and the recovery would then be better than 99%. In operating a continuous countercurrent column, as shown in Figure 3, the column is first filled to the top of the packing with the continuous phase. The flow rates of the light and heavy phases are then controlled a t the desired ratio and the interface is maintained by regulating the withdrawal rate from the bottom. If the light phase is desired dispersed, the interface is maintained above the packing a t the top of the column and, conversely, if the heavy phase is to be dispersed, the interface is maintained below the packing a t the bottom. The appearance of a second interface anywhere in the column indicates flooding and requires January 1954

a decrease in the flow rates. In the case of an agitated column, flooding may be also overcome by decreasing the speed of the agitator. The column is then operated until steady conditions are attained, This is determined by constant concentrations in the dilute end. The time may be estimated from the curves of Figure 7 . Thus, a 10-stage column with a total holdup of 1 liter will require about 20 cycles for a 90% approach t o steady state when E = 1.5, and this will involve changing the entire holdup in the column about twice. A total throughput of 2 liters of combined streams will give a 90% approach to steady state and 4 liters will give about 99% approach. This figure is consistent with the data on a 12-inch diameter three-stage column in which steady state was observed to be reached after changing the holdup in the column two or three times (16). This is equivalent to six to nine cycles and, according to Figure 7, a t a value of E = 1, which represents the approximate conditions for this run, three and a half cycles are necessary for a 90% approach to steady state and seven cycles are thus required for a 99% approach. Other data on a 1-inch column ( 1 1 ) have also been consistent with this method of estimation. The over-all material balance on the unit has been taken as the indication of the approach to steady-state conditions. When the products are equal to 90% of the feed, the column has been assumed to approach 90% of steady-state conditions for the run. The number of theoretical stages in the column can be calculated from the performance data by the use of Equation 6 or the curves of Figure 4. The height equivalent to a theoretical stage or the stage efficiency of the agitated column may then be evaluated. The effect of the variables in a packed column has been discussed (IO), and the effect of the variables in the performance of the agitated columns has also been discussed (8, 16).

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ENGINEERING AND PROCESS DEVELOPMENT Separation of Two or More Components I s Obtained by Fractional liquid Extraction

If two or more solutes are present in a simple liquid extraction problem, a change in their relative concentrations can be effected. Thus, in the extraction of 90% of a desired solute, any other components with distribution coefficients greater than the desired one can be seen from Figure 4 to be practically completely extracted, so that the residual 10% of the desired component will be substantially free of these other components. Similarly, components with smaller distribution coefficients will be extracted to a lesser degree, so the composition of the solutes ip the residual solution will be much richer in these components. It is possible to obtain a substantially complete separation of the components by methods analogous to those discussed in simple extraction.

Figure

additional stages can be employed to finally eetablish a distinct peak for this impurity (3). In most cases of precise analytical work several hundred extractions will be required, and carrying out these stages in separatory funnels is prohibitive. The method has been made practical by the developments of Craig, who first developed a rotating multitube mechanism for simultaneously making all the extractions necessary for one horizontal line in the triangular pattern ( 2 ) . Thus, in a 100-stage pattern only 100 transfers are required to obtain the necessary 5000 extractions. Another development of Craig consists of a series of glass extraction tubes mounted adjacently on a rocking frame (3). The tubes are so designed that a partial movement of the frame causes the two liquid phases to mix with each other and come to equilibrium. An inversion of the frame causes the light phase to flow to the adjacent tube to effect a transfer equivalent to one horizontal line in the pattern. Any number of tubes may be set up in this manner by increasing the length of the frame. The most recent development has been the construction of a robot mechanism to carry out the entire operation of all the transfers automatically (4). This technique allows an analysis of a multicomponent mixture utilizing only one property of the components which may be common to all. Thus, the weight of residue in each tube may be measured or the light absorption or any other property of the solutions can be taken as the meamre of the quantities in each tube.

5. Multiple Batch Extraction with Fresh Solvent

Multiule Batch Extraction with Fresh Solvent. Bv introducing fresh solvent along the sides of the triangular extraction pattern shown in Figure 5 and by introducing the feed mixture to be separated into the stage a t the apex it is possible to effect a segregation of the components in the different stages along the base of the triangular pattern. Craig (8, 3) has shown that the fraction of each feed component in each of the stages along the base of this triangle is given by the expansion of the binomial (p q)"--l, where p and q represent the previously defined fractions for the particular component and n represents the number of stages along the base of the triangle (the same as the number along a side). Thus, for the five-stage operation the expansion gives

+

pr

+ 4 p P + 6paqa+ 4pqa + 9'

as indicated in Figure 5. When E = 1, so that p = q = 0.5, the distribution of the component will be symmetrical with a maximum a t the center stage. With E greater than unity the maximum will be displaced toward the left and with E less than unity the maximum will be displaced toward the right. By carrying this pattern to a sufficient number of stages it is possible to separate the peaks in the distribution curve sufficiently to isolate the pure components by combining the solutions in the stages in the vicinity of the maxima. When the distribution coefficients do not differ appreciably it is necessary to carry out the extraction pattern to many more stages in order to separate the components into two distinct peaks so the purified products may be isolated. This method of purification is relatively inefficient because 50 stages are necessary to obtain a separation of components having distribution coefficients differing by a factor of 3. The technique can be used to best advantage for analytical purposes. It is possible to calculate the theoretical distribution curve along the base of the triangle from the concentration and location of the maximum, Any significant deviation from the theoretical curve on either side of the peak indicates the presence of an impurity, and 20

Figure

6.

Batchwise Countercurrent Fractional Liquid Extraction

The calculations have been fully described and summarized

(a). The subsequent methods for studying fractional liquid extraction require a specific method of analysis of the mixture so the respective amounts of each component may be determined. Hence the foregoing technique may also serve as the method of analysis in the following studies.

Feed Enters Center Stage of All Cycles in Batchwise Countercurrent Fractional Extraction

Multiple batch extraction with fresh solvent is a fractionation method based on a single feed introduction. In order to establish the peaks most accurately, the solvent ratio is chosen so the value of E will be close to unity and the desired component will accumulate near the center tubes. In the countercurrent operation the feed is introduced into the center stage a t all the cycles and

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ENGINEERING AND PROCESS DEVELOPMENT the study of the operation is based on the quality of the products from the end stages. No consideration is given to the intermediate concentration pattern except to observe that the maximum allowable concentration of solutes is not exceeded. In the previous method the maximum concentration is attained in the initial feed stage. In this method of operation the final concentration in the feed stage may be many times that of the initial feed stages, because i t is necessary for the concentration to increase to such proportions that the unit feed quantity will finally be expelled at the ends of the pattern. It is possible to calculate the feed-stage concentration at these steady-state conditions and the feed quantity must be chosen so the maximum allowable concentration will not be exceeded. It has been observed that the approach to steady-state conditions using the given feed quantity is very slow and the following equation has been proposed (12) for estimating the fractional deviation of the product streams from steady-state conditions after t product cycles:

procedure, however, gives the same approach to steady state in only a fraction of the previous number of cycles (IS). The method is based on the fact that the feed quantity a t each cycle can be varied to maintain the steady-state concentration at the feed stage at all times. RiForous calculation of this variation is difficult for a large number of stages, but two simple approximate methods have been developed. If the feed quantity of a component at cycle t is determined by the equation Ft=F*

1

1 + 2 (E

2E + 1) [m]'-'/

En - E l)(E"

-

where F , is the feed quantity a t steady state, the feed stage concentration will not vary by more than 10% from that a t steady state and a 90% or better approach to steady state can be obtained in a number of product cycles equal to the number of stages. The equation gives the same value for E and for its reciprocal, so that at the optimum solvent ratio the quantity required of each component is the same ratio as in the feed. Thus, the total feed may be used in the equation. At solvent ratios

-

when the total number of stages is equal to 2, 1. This equation has been found by Peppard and Peppard (9) to be appreciably in error a t large numbers of stages and cycles, and they proposed two other empirical equations, with increasing accuracy and complexity. Their more accurate equation is

-

log S = [(0.100n2 0.55n

+ 0.80)E-0.9 - t ]

They demonstrated its applicability up to 31 stages and 250 cycles. These equations were developed for the symmetrical system with center feed. The rigorous solutions of the symmetrical and nonsymmetrical systems have also been developed (14); however, for most practical applications, the curves of Figure 7 , which are based on the rigorous calculations of Peppard and Peppard ( 4 , will be adequate. The curves represent the cycles required for a 10% deviation from steady-state conditions. When E is unity the largest number of cycles is required. The curves are also the same for values of E and the reciprocal of E. This is the condition for optimum solvent ratio when two components are to be separated such that the fractional amount of each component removed in each of the two product streams will be the same. For this condition it can be shown that E1 = p and E1 = -1 Also, the ratio of the quantity of a component in the light

B'

Figure 7.

product stream to the quantity in the heavy product stream will be

(F)"

= E"

at all cycles independent of the approach to steady-

state conditions. Thus, at the optimum solvent ratio, the ratio of the two components in each product stream will be the same at all cycles and only the total quantities will vary. These factors are important in evaluating a batchwise countercurrent operation because they allow a calculation of the separation factor, p , from the first cycles, and it is only essential that sufficient product be obtained for the analysis. This may be done by combining a series of consecutive product streams. The smaller the separation factor the greater the number of stages required for a given separation. Thus, for a mixture of equal parts of two components with a value of p = 1.10, E, = 1 1.05, and EZ = 1.05,about 30 stages would give only about SO% pure products at each end and, from Figure 7 , about 200 cycles would be required for a 90% approach to steady conditions. The laboratory investigation of steady-state conditions in the previous case is practically prohibitive. A recently developed January 1954

Cycles Required for 90% Approach to SteadyState Conditions in Fractional Extraction

other than the optimum the use of the value of E furthest from unity in the equation will give a conservative value of total feed and i t will not approach steady state as quickly. The pattern of Figure 6 shows that the feed cycle number is not the same as the product cycle number because of the delay at the top of the pattern before the first product is obtained. This equation gives a maximum feed stage concentration in the second cycle which will be less than 10% in excess of the final concentration. If this excess cannot be tolerated, the following equation may be used for the first two cycles:

F'

=

E"

[l +

(E"

- E'

+ l ) ( E - 1)(E + l ) t - l

The subsequent cycles may be calculated by the equation

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ENGINEERING AND PROCESS DEVELOPMENT This method does not exceed the final feed stage concentration for any of the usual cases investigated but requires a few more cycles for the same approach to steady-state conditions. The application of this batchwise technique of fractional liquid extraction in a known number of theoretifa1 stages allows a more accurate evaluation of small values of p than possible with a single stage because of the limits of accuracy of the analysis. Thus, the

-

LIGHT SOLVENT RICH IN COMPONENT 1

HEAVYSOLVENT

EXTRACTION COLUMN

FEED MIXTURE, COMPONENTS I AND 2

G H T ,IL SOLVENT

__HEAVY SOLVENT RICH I N COMPONENT 2

Figure 8.

Continuous Countercurrent Fractional Extraction

observation of a value of p of 1.07 in a single stage contacting with a possible analytical error of 2% of the actual value would only establish the value of p between 0.99 and 1.15 while an 11-stage operation giving p6 = 1.50 & 0.12 would establish p = 1.07 & 0.015. This latter value is satisfactory for process calculations. Continuous Countercurrent Production Unit Carries Out Full Scale Extraction

The batchwise method of operation is applicable to the processing of limited quantities of material primarily for the purpose of obtaining basic equilibria data and investigating the feasibility of the fractional liquid extraction process. The full scale operations would be carried out in a continuous unit, and in processing larger quantities on a laboratory or pilot plant scale a continuous countercurrent unit would be employed. The performance data on such a unit also serve as a basis for the design of the full scale equipment. The flow through such a unit is shown in Figure 8. Numerous types of continuous extractors have been described for multistage operation. The simplest device for obtaining a large number of stages in a small unit employs mechanical mixing by means of a series of paddle-type agitators on a centrally located shaft. Data have been presented on such a unit with wire mesh packing (11,16) to provide phase separation between the agitated chambers. Another unit is equipped with baffles to provide the necessary mixing and disengaging sections in the column (8). Imparting a pulsating motion to the liquid in a packed column has also been reported to increase appreciably the efficiency of the packed column, and such a design may also be used for fractional liquid extraction. In general, however, an ordinary packed column without mechanical agitation does not provide sufficient theoretical stages in a reasonable height to be used for this process. 22

On a continuous production unit the rate of approach to steadystate conditions is of little consequence, but the studies on laboratory or pilot plant unit are made for a limited time and it is necessary to have some idea of this approach a t the time the measurements and analyses are taken. I t can be deduced from the relationships in Figure 6 and those discussed in the foregoing section that the fraction of the feed obtained in the two product streams may be taken as the approach to steady-state conditions. It is, of course, equally essential in fractional liquid extraction, as in simple extraction, that the flow rates of the two liquid phases be maintained constant. On the other hand, the rate of feed of the mixture to be separated is frequently so small on the laboratory size columns that it cannot be readily regulated. In this case, an intermittent feed every few minutes to give the desired average rate will be satisfactory. Because the amount of solute in the feed stage is so many times larger than the feed quantity, thi$ method will not produce any major disturbances in the concentration pattern in the column. With a constant average feed rate, the approach to equilibrium may be estimated in advance, by a method similar to that suggested for the simple liquid extraction column. The number of cycles may be calculated as the product of the number of theoretical stages in the column and the number of times the total column holdup has been changed in the course of the run. The approach may be calculated from Equation Sa or the time for a 90% approach to steady state may be determined from Figure 7. In the case of the same solvent flow per unit of cross-sectional area, the time required for a given approach to steady state will be proportional to the height of a theoretical stage. Thus, in addition to the physical advantage of keeping the height of a theoretical stage to a minimum, it is also desirable from an operational standpoint, It was previously observed that there was only a slight difference in the approach to steady-state conditions in a five-stage operation in the simple liquid extraction and in the fractional liquid extraction with center feed. Simple liquid extraction may be considered as a fractional liquid extraction with the feed introduced a t the end stage. Thus, it may be deduced that the fractional liquid extraction with feed stages intermediate between the center and the ends will approach steady state a t the same rate, and thus Figure 7 can be applied with reasonable reliability to any fractional liquid extraction column. The feed quantity may also be varied in a manner similar to the batchwise technique to establish steady-state conditions in the minimum time. The initial feed rate may be calculated from Equation 9 fort = 1.

E" - E

Fr =

F8

[1 +

(E"

+ l ) ( E - 111

(94

The feed rate may then be decreased stepwise according to Equation 9, if the time for a cycle is calculated as the holdup per theoretical stage divided by the total solvent flow rate. This gives the time required to change the entire holdup in the stage. A simple approximate method of decreasing the excess feed quantity by a factor of 0.5 over successive periods of time has also been used, but in this case the time cycle must be shorter, particularly for the larger values of E. This method approaches the more theoretically desirable technique of continuously varying the feed quantity, which does not lend itself readily to manual operation. After the column has been thus operated to attain at least 90% of steady-state conditions, the product streams are collected to obtain sufficient product sample for analyses. The determination of the number of theoretical stages in the unit may be made by the stage-to-stage calculations described elsewhere (11). In the case of ideal systems where the distribution coefficients of both components are constant in the column and the quantity of solvent, if any, introduced along with the feed mixture is small compared with the amount of this solvent introduced a t the end of

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ENGINEERING AND PROCESS DEVELOPMENT the column, the relations derived by Klinkenberg and coworkers (6, 7) may be applied. While a range of solvent ratios will give the desired separation, there is one solvent ratio which requires less theoretical stages than any other. This is defined as the optimum solvent ratio ( I I ) , and Klinkenberg (7) has empirically correlated this optimum solvent ratio with the ratio of the quantity in the light solvent to the quantity in the heavy solvent for both components being separated. This ratio is analogous to the rejection ratio used in distillation calculations. The parameter log E1 of the curves is the ratio of -

1% P

*

I t is possible to derive the equations of his curves theoretically from the equation derived by Bartels and Kleiman (1) to give the rejertion ratio for a component

function regardless of actual sign. This makes the equation symmetrical about the 45" line in Klinkenberg's correlation of total stages (6) when R1 = Rz and gives agreement within the accuracy of reading and interpolating the empirical curves. The location of the feed stage in the same correlation may be evaluated from the empirical equation

if Rz < R1, and

if RI < Rz By defining the retention ratio, R', of a component as the reciprocal of the rejection ratio, this equation becomes

R;

=

1 - E', E'; E",'"

-

If component 1 is the one which is more soluble in the light solvent and component 2 is more soluble in the heavy solvent, both R1 and Rh must be greater than unity for a reasonable separation. In this case, E2 will be less than unity and the signs of both numerator and denominator in Equation 10a are changed to give positive numbers. If n is large, so E: is substantially the same as E: - I, Equation 10 becomes R1 = E:

-1

(11)

Similarly, if m is large ET becomes negligible with respect to unity and Equation 10a reduces to

If both n and m are large, the values of unity in the above equations will be negligible with respect to the first terms. Solving for n and m, differentiating with respect to El, and noting that a t dn dm the minimum number of stages - and also that E, = dE1 = - dE1 @E2, the optimum solvent ratio can be expressed as log El

=

log

l+d%

This equation is in perfect agreement with Klinkenberg's empirical curves (6) except for his two extreme curves where the values of his parameter are highest and lowest. In this region, the equation gives values closer to the 45" line in his chart. Since the equation is exact for large values of n and m, this indicates that the empirical curves in these extreme regions are not straight lines, as shown, but must curve toward the 45" line at higher rejections and retentions. The discrepancy between the above equation and the empirical curves appears when the ratio between rejection and retention ratio is less than 0.1 and greater than 10, and since these conditions are outside the usual operating range, the equation is applicable over the practical range of separations. Similarly, over this range the total number of stages a t the optimum solvent ratio has been correlated to the empirical equation

1

where the expression, log January 1954

R

r ,indicates the positive value of this

The ranges of rejections and retentions specified in these equations make them symmetrical about the R1 = R2 line, which is a requirement of empirical curves of Klinkenberg (6). Equations 13 and 14 are in agreement with Klinkenberg's empirical curves over the practical range of operation. Outside of this range the calculated feed stage location is in better agreement with the optimum solvent ratio calculated from Equation 12 than from Klinkenberg's curves, since the former relationship was used in developing the equations. Nevertheless, the number of total stages, even in this region, agrees with Klinkenberg's curves practically within the accuracy of reading or interpolating them. iill these empirical equations are of a form that can be theoretically derived for the case of R1 = R,, which is the condition of the previous symmetrical case. Optimum Solvent Ratio Is Reliable Basis for Design Calculations

In interpreting laboratory or pilot plant performance data, if the feed stage location calculated from the previous equations is not in reasonably good agreement with the actual feed stage location, the optimum solvent ratio has not been employed and a larger number of theoretical stages would be obtained by s t e p wise calculations. Thus, the design based on the calculated theoretical stages will be conservative. In order to prevent excessive errors in this respect, additional data should be obtained closer to the optimum solvent ratio. It is preferable to analyze all data in this manner to prevent the fictitiously large stage efficiencies which result from operating far from the optimum solvent ratio. Such operation produces a "pinch" between the operating curve and the equilibrium curve and is too sensitive to the accuracy of the actual operating or equilibrium data to be reliable. The writer of this paper prefers t o base all design calculations on the optimum solvent ratio, and the best efficiency can be calculated on this basis only when the column has actually been operated close to this ratio. Operation of the column a t any other solvent ratio and subsequent calculation from these equations will give a lower apparent efficiency than actually obtained in the column, and the discrepancy will be indicated by an appreciable deviation of the calculated feed stage location from the actual location. If the two locations agree within 20% of the total column height, the calculated efficiency will be sufficiently accurate for all practical purposes, because the total theoretical stages will not vary appreciably with solvent ratio or feed stage location over this range. A more complete 'quantitative analysis of this subject can be deduced from the works of Klinkenberg et al. (6-7). The choice of the optimum solvent ratio for the full scale design is obvious because i t is the ratio which requires the minimum number of theoretical stages to obtain the desired separation. In order to obtain a reliable full scale design, it is thus desirable

INDUSTRIAL AND ENGINEERING CHEMISTRY

23

ENGINEERING AND PROCESS DEVELOPMENT that the interpretation of the laboratory and pilot plant data be made on the same basis as the final design. All of the equations in the present paper have been based on a constant distribution coefficient, and it has been assumed that when the coefficient varies, an average value may be estimated. In selecting average values, care must be taken that the average coefficients selected have the average re!ative distribution, since it is possible in an arbitrary selection of absolute values to obtain ratios outside of the range of their actual relative distributions. The relative distribution controls the number of theoretical stages required for a given separation. In some nonideal systems, the distribution coefficients vary almost logarithmically with concentration, and it is impossible to estimate an average distribution coefficient. In these cases, the trial-and-error procedure of detailed stagewise calculations ( 11 ) must be employed. Probably the best approach would be to evaluate the proper average value of D in nonideal systems to be introduced in the equations given in this paper. This average value of D may possibly be calculated for a linear and a logarithmic variation of the coefficient with concentration. These relationships have not yet been developed and an accurate solution of the nonideal cases requires the tedious stagewise calculation. Nomenclature I) = distribution coefficient

LD E = extraction factor = H Ii‘ = feed quantity per cycle in batch extraction feed rate in

continuous extraction H = heavy solvent quantity per cycle in batch extraction = heavy solvent rate in continuous extraction L = light solvent quantityper cycle in batch extraction = light solvent rate in continuous extraction m = stages below and including the feed stage in fractional liquid extraction = n with center feed n = stages in column in simple extraction = stages above and including feed stage in fractional liquid extraction E p = fraction of component in light solvent = __ E f l 1 p = fraction of component in heavy solvent = E+1

R

=

E’

=

t

= = =

x 21

rejection ratio of component = quantity leaving in light solvent quantity leaving in heavy solvent 1 retention ratio = R number of cycles concentration or quantity in heavy solvent concentration or quantit in light eolvent

-

Jl

relative distribution = D2 6 = fractional deviation from steady state

p

=

Subscripts 1 refers to component more soluble in light phase 2 refers to component more soluble in heavy phase o refers to solute-rich end of extraction system n refers to solute-lean end of extraction system s refers to steady-state conditions t refers to cycle number t 4

Literature Cited

(1)Bartels, C. R., and Kleiman, G., Chem. Eng. Progr., 45, 589 (1949). (2) Craig, L. C., J. Bid. Chem., 155, 619 (1944). (3)Craig, L. C., and Craig, D., in “Technique of Organic Chemistry,” Vol. 111, New York, Interscience Publishers, Inc., 1950. (4) Craig, L. C., Hausmann, W., Ahrens, E. H., Jr., and Harfenist, E.J., Anal. Chem., 23, 1236 (1951). (5) Klinkenberg, A., Chem. Ew.Sci., 1, 86 (1951). (6) Klinkenberg, A., IND. ENG.CHEM.,45,653 (1953). (7)Klinkenberg, A., Lauwerier, H. A., and Reman, C. H., Chem. Eng. Sci., 1,93 (1951). (8) Oldshue, J. Y., and Rushton, J. H., Chem. Eng. P r o p . , 48, 297 (1952). (9) Peppard, D. F., and Peppard, M. A., IND.ENG.CHEM.,46, 34 (1954). (10) Perry, J. H.,ed., “Chemical Engineer’sHandbook,” 3rd ed., pp. 713-53, New York, McGraw-Hill Book Co., 1950. (11) Scheibel, E. G., Chem. Eng. Progr., 44, 681-90, 771-82 (1948). (12) Scheibel, E. G.,IND.ENQ.CHEM.,43, 242 (1951). (13)Zbid., 44, 2942 (1952). 46,43 (1954). (14)IW., (15) Scheibel, E. G.,and Karr, A. E., Ibid., 42, 1048 (1950). (16) Underwood, A. J. V., I d . Chemist, 10, 128 (1934). (17)Ibid., 10, 129 (1934). RECEIVED for review March 6, 1953. AOO~PTED September 2,1958.

(BATCHWISE FRACTIONAL LIQUID EXTRACTION)

Double Withdrawal and the EDGAR L. COMPERE‘ AND ADA L. RYLAND2 College of Chemistry and Physics, Louisiana Sfafe University, Baton Rouge, La.

I

NCREASING interest in the use of fractional solvent extraction and countercurrent distribution for the separation of closely related compounds has led to attempts to establish theoretical calculations for use in predicting the results of these experiments. Stene (16) and Craig (4, 17) have made significant contributions to the field by developing theories applicable to 1

Present address, Chemistry Division, Oak Ridge National Laboratory,

Oak Ridge, Tenn. Present address, Polychemimla Department, Du Pont Experimental Station, Wilmington, Del.

24

the basic and single withdrawal operations. Scheibel (1.8) considered the relationship between batchwise and continuous countercurrent extraction processes, and Johnson and Talbot (9, 10) derived equations for predicting the distribution of a fixed quantity of feed in a series of mixers and settlers. The accompanying papers by Auer and Gardner (I), Peppard and Peppard (11), and Scheibel ( 1 4 ) also are concerned with various aspects of the problem. The discrete stage operations considered in this paper represent a n approach to continuous countercurrent extraction processes,

INDUSTRIAL A N D ENGINEERING CHEMISTRY

Vol. 46, No. 1