ENGINEERING, DESIGN, AND EQUIPMENT
Optimum Reflux Ratio for Fractional liquid Extraction EDWARD G. SCHElBELl Hoffmann-La Roche, /ne., Nufley, N. . I.
I
?r' T H E separation of compounds by fractional liquid extraction the use of reflux will generally decrease the number of theoretical stages required. With a fixed height per theoretical stage, reflux will thus decrease the over-all height of the column. But if the solute concentrations in the column must be maintained below some maximum value, the use of reflux will increase the solvent requirements and therefore require a column of greater cross section. These opposing effects produce an optimum reflux ratio, above which the cost of increasing the diameter to handle the reflux is greater than the saving due to decreasing the column height. A complete study would also include the cost of processing the additional solvent to recover the solute for reflux, and this makes a general solution of the over-all problem very complex. However, the study of a series of stagewise calculations on an ideal system a t varying reflux ratios indicated that it is possible to handle an appreciable amount of reflux with only a slight increase in solvent requirements. Asselin and Comings ( I ) devised a graphical technique for stagewise calculations in a two-solvent fractional liquid extraction with reflux and applied it to the study of ideal systems in which the distribution coefficients are independent of concentration and of nonideal systems in which the distribution coefficients vary with concentration. They observed that some reflux was beneficial in both systems, but for nonideal systems in which the relative distribution decreased with concentration, the effect was much smaller and excessive reflux could even require additional stages. I n ideal systems the stage requirements decrease continuously to a minimum value a t total reflux. I n the case of fractional liquid extraction in an ideal system without reflux the total stages required are given by the empirical equation (6)
Figure 1 shows the variation of total solute quantities in a column designed to separate a mixture of 1.6 parts of component 1 and 1.6 parts of component 2 into 96% pure products. This separation requires an equal number of stages above and below the feed and is called a symmetrical system. Equal reflux ratios are used at the top and bottom of the column. The total number of stages in the column varies and, in order to reduce the ordinate to a common abscissa, the quantities in the stages are located a t the fractional height in the column. Figure 1 shows that the quantities a t the feed stage increase only slightly with increasing reflux ratio. At the end stages the quantities increase linearly with reflux ratio, so that at reflux ratios above 9 the end stages would control the solvent requirements to maintain the concentrations below a maximum value. This maximum value would correspond to the solubility limit of the solute or the concentration a t which the selectivity of the solvents decreases. The reflux ratio giving the same concentrations a t the feed stage and a t the terminal stages makes the most effective use of the necessary amount of solvent, but it is not necessarily the optimum reflux ratio. Figure 2 shows a cross plot of the terminal and feed stage quantities in Figure 1; a t reflux ratios less than 9.3 the quantity in the feed stage controls the amount of solvent and in that way the cross section of the column. For purposes of the subsequent derivations the cost of the extraction column will be considered proportional to the volume. The minimum volume may be determined in any of several ways. Because the volume, V , is equal to the product of the height, h, and the cross-sectional area, a, the minimum volume is obtained when (3)
d(ah) = 0
or adh
+
where n rn is equal to the total stages plus one. I t will be observed that for the usual separations where R1and R; are of the same order of magnitude the last term in the parenthesis is negligible. At total reflux the number of stages is given by the Fenske (3, 4)equation for total reflux in a distillation operation.
+
where n m again represents the total number of stages plus one. Thus, the use of reflux can decrease the stage requirements to somewhat more than half the number without reflux. Distribution data for the 0- and p-chloronitrobenzenes between Skellysolve C and aqueous methanol solutions indicated that this system could be considered ideal with a relative distribution coefficient of 1.62. A similar ideal system was considered by Asselin and Comings ( I ) in their studies on the effect of reflux. 1
Present address, York Process Equipment Gorp., West Orange, N. J.
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+ hda
=
0
(3'4)
Any arrangement of this relation can be used to locate the minimum volume, such as
and the minimum thus occurs on Figure 2 when the slope of the controlling quantity curve and the slope of the curve of total stages are in the negative ratio of their ordinates. Also d l n a = -dlnh
(3C)
And, if the logarithms of the quantities and the total stages are plotted against a common function such as reflux ratio, the minimum occurs when the slope of one curve is the negative of the other. In Figure 2 the minimum occurs a t a reflux ratio of about 7 , under which condition the quantities of total solutes in the terminal stages when reflux is used are equal to the quantity in the feed stage a t eero reflux. This latter quantity can be calculated directly and the optimum reflux ratio thus estimated.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 11
ENGINEERING, DESIGN, AND EQUIPMENT The solvent ratio does not vary appreciably from this value if R1 and Ra are of the same order of magnitude, and this is the usual range of desired separations. Substituting these relationships in Equation 5, the optimum reflux ratio is:
Since the rejection ratio for a component is ( 2 ) :
the approximate relationship of Equation 6 gives p n i z ( p m / z - 1) '1 = pn/z - 1 and by a n analogous derivation the retention ratio of component 2 can be derived.
401
/
I
BOTTOM
CENTER (FEED STA6E)
STAGE L O C A T I O N
TOP
- F R A C T I O N A L DISTANCE FROM BOTTOM
Figure 1.
OF COLUMN
Symmetrical system
T E R M l N q L STAGES
The quantity of a solute in the feed stage of a fractional liquid extraction operation can be shown from the relationships of Bartels and Kleiman ( 2 )to be
At a reflux ratio of rp a t the top of the column the quantity of solute 1 in the top stage is given as
0
The same relationships can be derived for solute 2, which is more soluble in the heavy phase, but the quantity in the top stage will be small for a usual degree of separation and can be approximately taken into account by inserting the total overhead product in place of P I . The total overhead product is usually nearly the same as the quantity fi in the feed. The reflux ratio is therefore given as
J
I
I
i
z
i
4
s
e
7
8
~
1I 0
1
1
1
e
REFLUX RATIO
Figure 2. Cross plot of terminal and feed stage quantities of Figure 1
The following relationships can then be developed from Equations 10and11. m+n
It has been shown ( 4 ) that without reflux when R1 = R; the optimum solvent ratio which gives the desired separation in the minimum number of stages, is such that and E 2
November 1955
1 -
dZ
and
mJrn
RIR:+l
= p
INDUSTRIAL AND ENGINEERING CHEMISTRY
-1
(14) 2291
ENGINEERING, DESIGN, AND EQUIPMENT
'1
Table 1. Component 1 2
Feed Parts 2.000 8.000
Material Balance
% 20 80
10.000
STAGES BELOW AND INCLUDING F E E D STAGE m.
W W v)
J 6-
Q
0
/ \,Re4LR=9
3
SOLVENT RATIO
Figure 3.
35
8.404
-
The value of m/n given by this equation is 0.558 and thus m = 10.2 and n = 18.5. The optimum solvent ratio for the separation can be calculated by the relationship (6)
4
45
- EXPRESSED
AS V O L U M E OF W A T E R PER V O L U M E O F ALCOHOL.
Separation of oxalic and succinic acids in n-amyl alcohol and water
Substituting these relationships in Equation 8, noting that fi 4-f2 is the total feed, F , and that as previously mentioned fl is very nearly equal to P, the expression for the optimum reflux ratio a t the top of the column is
I n fractional liquid extraction it is possible to regulate the reflux ratio a t the bottom of the column independent of the reflux ratio a t the top. The corresponding optimum reflux ratio for the bottom section of the column can be derived as
Applying these equations to the previous separation, the reflux ratios at which the quantities of solutes in the terminal stages with reflux are equal to the quantity in the feed stage at zero reflux can be calculated as r p = TB = 7.45; reference to Figure 2 confirms this value. The actual optimum giving the minimum column volume was previously determined a t about 7 and the curve is so flat in this region that the calculated value will be entirely satisfactory. This particular system was a symmetrical case, and for the more nonsymmetrical systems the assumptions in deriving the equations wouId not hold as well. I n order to test a nonsymmetrical case, a feed of 2 parts of component 1 and 8 parts of component 2 was assumed to be separated into a n overhead product 99% pure and a bottoms product 95% pure. The material balance on the column is shown in Table I. Table I gives rejection and retention ratio of R = 3.77 and R,' = 499, and, because the retention ratio of component 2 is over 100 times the rejection ratio of component 1, the separation
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1.596
represents an extremely nonsymmetrical system a t about the limit of the usual separations encountered in practice. The relative distribution was assumed the same as in the previous symmetrical system-namely, 1.62. The total number of stages a t no reflux calculated from Equation 1 is 28.7 and the feed stage location can be calculated by the empirical equation previously developed (6)for a system in which R;>R1
log El = 25
Bot toms Parts % 0.420 5 7.984 95
STAGES A B O V E AND INCLUDING F E E D STAGE n .
b 0
Overhead Parts % 1.580 99 0.016 1
log' 1+ 41s
and the value of E1 at the optimum solvent ratio is 1.165 and Ez is thus 0.719. The optimum reflux ratio at the top of the column calculated from Equation 15is 22 and the optimum reflux ratio at the bottom is 3.4. Stagewise calculations a t these reflux ratios and a t the same solvent ratio previously determined as optimum for the case of no reflux give values of m = 9.2 and n = 12.8. The total quantity of solutes a t the feed stage under these conditions was 77.7 as compared to 66.9 a t no reflux, as calculated from Equation 4. The column volume using these reflux ratios was thus decreased by 12%, which was the identical decrease obtained by the use of reflux in the symmetrical case. I n the nonsymmetrical case a t the calculated reffux ratios the ratio of the total quantity of solutes in the bottom stage to the total quantity in the feed stage was 0.827 and the ratio of the total quantity of solutes in the top stage to the total quantity in the feed stage was 0.888. The values compare favorably with the ratio of 0.875 for both ends in the symmetrical case a t the calculated reflux ratio. Thus it appears that the equations derived for the symmetrical system yield substantially the same relative conditions of operation in the nonsymmetrical system within the usual range of separations encountered in practice. Also the curve of column volume against reflux ratio is so flat in the region of the minimum that the same saving will be obtained a t the calculated reflux ratios and the theoretically exact optimum reflux ratios. This criterion for selecting the optimum solvent ratio is based solely on the cost of the column. If the cost of separating the solute from the solvent is a significant fraction of the total cost, the most economic reflux ratio would be less than this calculated value; and this latter cost might be large enough to preclude the use of reflux entirely. Figure 3 shows the results of a study on the separation of oxalic and succinic acids in n-amyl alcohol and water, previously reported by Asselin and Comings ( 1 ) . This system is nonideal and it is first necessary to determine the optimum solvent ratio a t all the different reflux ratios. The optimum solvent ratio was taken a t the minimum of the curve of total stages. The stagewise calculations without reflux indicated a relative distribution of 3.0 a t the feed stage. Using this value in Equations
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Vol. 47, No. 11
I
ENGINEERING, DESIGN, AND EQUIPMENT 15 and 16 for the given separation gives r p = r B = 3.4. Cross plotting the theoretical stages at the optimum solvent ratio and also the total solvents required a t this point against the reflux ratio indicated that the optimum reflux ratio giving the minimum column volume was in agreement with this calculated value. The curve of total volume was again so flat that very little difference wm observed a t reflux ratios between 3 and 5, so it would appear that the equation can be used as the starting point for a study on a nonideal system. It requires a preliminary set of stagewise calculations to evaluate the relative distribution a t the feed stage unless this value can be estimated satisfactorily by a superficial inspection of the distribution data. The total decrease in the column volume between no reflux and the optimum reflux ratio in this case was only 5%. The sharper minima in the curves of Figure 3 indicate that the soivent ratio is much more critical at t h e higher reflux ratios and that improper control may give a poorer separation than is attainable without reflux. The over-all advantages of reflux in this system may be questionable. On the other hand, in the ideal systems previously studied the column volume was decreased by 12% through the use of reflux. The proposed equation for the calculation of optimum reflux ratio can be applied to effect a saving in column size in ideal systems and also in nonideal systems, if the proper relative distribution is used. Nomenclature
a
= cross-sectional area of extraction column
B
=
total solute product in heavy solvent leaving column
concentration in light phase distribution coefficient, concentration in heavy phase E = extraction factor = L D / H = quantity of component in feed = total quantity of feed h = height of extraction column H = flow rate of heavy solvent L = flow rate of light solvent m = stages below and including feed stage n = stages above and including feed stage P = solute product in light solvent leaving column Q = quantity of solute in feed stage T = reflux ratio R = rejection ratio = ratio of quantity of solute in light solvent product stream to quantity in heavy solvent product stream R‘ = retention ratio = ratio of quantity of solute in heavy solvent product stream to quantity in light solvent product stream = 1 / R p = relative distribution = 0 1 / 0 2 Subscripts 1 and 2 refer to components more and less soluble in the light solvent, respectively. Subscripts P and B refer to the top and bottom of the column, respectively.
D
=
fF
literature cited (1) Asselin, G. F., and Comings, E. W., IND. ENG.CHEM.,42, 1198
(1950).
(2) Bartels, C. R., and Kleiman, G., Chem. Eng. Progr., 45, 689
(1949). (3) Fenske, M. R., IND.ENG.CHEM.,24, 482 (1932). (4) Klinkenberg, A., Ibid.,45, 653 (1953) (5) Scheibel, E, G., Ibid., 46, 16 (1954). RECEIVED for review
January 29, 1966.
A C C E P T ~July D 23, 1965.
Heat Transfer in Packed Beds Analytical Solution of Temperature Profiles in Fixed- and Moving-Bed Reactors and Heat Exchangers ANDREW PUSHENG TING’ Cafalyfic Construction Co., Philadelphiu, Pa.
I
S THE design of nonadiabatic packed reactors, temperature
profile must be known as a function of radial position and height, in order t o account for the effect of temperature upon the reaction rate, and to calculate the heat-transfer surface required and the maximum skin temperature of the reactor tubes. In the case of fixed- or moving-bed heat exchangers, it is also more logical t o size the equipment based upon the bed-edge temperature instead of the average bed temperature. Thus, the knowledge of temperature profile in such heat exchangers is also important if a rational and accurate design is required. Investigations of radial heat-transfer rates in fixed beds have followed two different methods of approach. In the first method, only the uniform inlet temperature and the bulk mean outlet temperature have been measured. The results have been correiated as heat-transfer coefficients, h, or over-all effective thermal conductivities, kea. Coefficients h are based upon the log mean of the difference between a “constant” wall temperature and the uniform gas temperature at the inlet, and the difference between the “constant” wall temperature and the bulk mean gas temperature a t the exit. The works of Colburn (6),Leva and others ( f I - I 3 ) , Brinn and others ( 2 ) , Hougen and Piret (9),Singer and Wilhelm (15), and Vershoor and Schuit (16) are of this type. The work of Colburn and of Leva is based purely on dimensional 1
Piesent address, Chemical Construction Corp., New York, N.
November 1955
Y.
analysis. With modifications Hougen, Pigford, Schuit, and Wilhelm use the analytical solution of Graetz (8),which is a study of heat transfer with liquid flows in empty tubes. Wilhelm gives solutions of point temperature as well as bulk temperature. The Graetz equations require the use of a uniform jacket temperature and assumptions of an infinite thermal conductivity for the tube wall and infinite heat transfer coefficients for film resistances on both sides of the tube wall. The second method is based upon the measurement of temperature profiles within the beds. Coberly and Marshall (a), Felix and Neill (7), Bunnell, Irvin, Olson, and Smith (9), Irvin, Olson, and Smith (IO), Schuler, Stallings, and Smith ( l a ) , and Argo and Smith (I) made studies in this way. Results were reported in terms of effective thermal conductivities, k,, of the gas-solids bed. The knowledge of temperature profiles permitted evaluation of k , as a function of radial position. It was found that the resistance to heat transfer increased greatly near the wall. Coberly and Marshall, and Felix and Neill accounted for this by postulating that an additional resistance to heat transfer existed a t the wall over that in the bed proper. These investigators could reproduce their measured temperature profiles by utilizing a wall heat-transfer coefficient and a constant effective thermal conductivity throughout the bed. Smith handled the problem by allowing the effective thermal conductivity to vary in a con-
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