A reaction rate model (Equation 8) was obtained which fits the experimental data with a 3.8% standard deviation and is consistent with the proposed reaction mechanism. The energy of activation for the reaction was 24.5 kcal per mole, well. within the range of values typical of surface reactions. Although thermodynamics and the reaction of ethylene oxide to form dichloroethane support the proposed reaction mechanism, operation over a wider range of process conditions would be desirable to test the proposed rate model more severely. literature Cited Arganbright, R. P., Yates, W. F., J . Org. Chem. 27, 12058 (1962). Carrubba, R. V., Sc.D. thesis, Columbia University, New York, 1968. Draper, N., Smith, H., “Applied Regression Analysis,” pp. 26-32, Wiley, New York, 1966. Ernst, O., Wahl, H. (to I. G. Farbenindustrie A.-G.), Ger. Patent 486,952 (Nov. 9, 1922). Fontana, C. M., Gorin, E., Kidder, G. A., Meredith, C. S., Id. Erg. Chem. 44, 363-80 (1952). Gorin, E., Fontana, C. M. Kidder, G. A., I d .Eng. Chem. 40, 2128-38 (1948).
Laine, F., Kuziz, C., Wetroff, G. (to Pechiney Compagnie de Products Chimiques et Electrometallurgies), Fr. Patent 1,301,990 (Aug. 24, 1962). Meissner, H. P., Thode, E. F., I d . Eng. Chem. 43, 129 (1951). Parthasarathy, R., Ph.D. thesis, University of Florida, Gainesville, Fla., 1960. Prahl, W., Mathes, W. (to F. Raschig G.m.b.H.), U.S. Patent 2,035,917 (March 31, 1936). Price, J. L. (to Monsanto Chemical Co.), U.S. Patent 3,010,913, Applied (Sept. 23, 1957). Pye, D. J. (to Dow Chemical Co.), U. S. Patent 2,752,402 (June 26, 1956). Reitlinger, O., U. S. Patent 2,674,633 (April 6, 1954). Tizard, H. T., Chapman, D. L., Taylor, R., Brit. Patent 214,293 (Dec. 14, 1922). Vdovichenko, V. J., Galenko, W. P., Gaz. Prom. J . 5 (4), 37 (1960). Yoshida, F., Ramiswami, D., Hougen, O., A.1.Ch.E. J . 8, 5 (1962).
RECEIVED for review May 16, 1969 ACCEPTED February 24, 1970
Solid-liquid Countercurrent Extractors Florencio P. Plachco and Julio H. Krasukl Laboratorio de Operaciones Unitarias y Procesos, Departamento de Qutmica, Universidad de Chile, Casilla 2777, Santiago, Chile The mathematical problem involved in the countercurrent extraction of a solid of plane sheet geometry with an initial concentration distribution (constant diffusivity) is solved. The resulting integral equation was solved using a numerical iterative method which coincides, within the range of possible error, with the analytical solution for the case of uniform initial concentration. Graphical solutions relating the variables of practical interest are presented. Concentration profiles in the liquid and solid phases along the extractor are also included. The numerical method was applied to several situations with previous extraction where variable initial concentration distributions arose. Previous theory is applied to the extraction of natural products, specifically sugar from beet. The theory foresees accurately the total mass extracted, facilitating the design of countercurrent extractors for products of natural origin.
heat transfer processes in solids, in a countercurrent arrangement, occur in such different situations as in the metallurgic industry (heating of solids with and without chemical reaction), in the extraction of solids of biological origin (sugar from sugar beet, oil from oilseeds), and in food dehydration plants (tunnel dryers with truck and tray). The first approach to this type of problem was reported in relation to heat transfer in moving beds by Munro and Amundson (1950). They referred to the heating of solid spheres with uniform initial temperature with and Present address, Chemical Engineering Department, University of California, Davis, Calif. 95616
without energy sources. However, their solutions were not presented in a practical graphical or numerical form. Graphical and tabulated solutions to this problem, considering a finite heat transfer coefficient, have been given by Kitaev et al. recently (1967). The theory related to solid-liquid countercurrent extraction is still more restricted. The extraction of cosettesplane sheet geometry-in the diffusion tower of the beet sugar industry has not been worked out completely: Oplatka (1954) quoted an approximate solution; Yang and Brier (1958) developed a semiempirical method; Zabel and McKibbins (1961) gave a solution obtained by means of an analog computer, but the variables as well as the assumptions are not clearly set up. In any case these Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 3, 1970 419
references are concerned with the less complex case of constant initial concentration throughout the solid. However, in many instances in industrial extraction a nonuniform concentration distribution exists in the solid as it enters the extractor countercurrently-for example, in the beet sugar industry, where the solid enters the diffusion tower after being heated and extracted (Figure 1). This pre-extraction creates a nonuniform concentration distribution throughout the solid. The proper evaluation of the influence of this initial distribution may improve design of industrial countercurrent extractors. Important Basic Hypotheses. 1. Constant effective diffusivity in the solid. 2 . Linear adsorption isotherm between the concentrations in the extracting solution and on the external surface of the solid. 3. Diffusional resistance negligible in the external boundary layers of the cosettes. 4. No backmixing in the liquid and solid phases; longitudinal diffusion negligible in the liquid phase. 5. Q and pi. constant along the extractor. The first assumption is far from the real case where the variation with concentration of the effective diffusivity of the solid along the extractor is large. However, this is a first approach to the problem. On the other hand, experimental data can be fairly well represented, if an average effective diffusivity along the extractor is used. The second and third assumptions have been ascertained in practical situations (Krasuk et al., 1967; Oplatka, 1954; Yang and Brier, 1958). Conditions 4 and 5 are necessary, so that the model may be treated mathematically. Assumption 5 is valid for dilute, solutions in the solid being extracted. The average of these values (Q, p r ) along the equipment may be taken as representative of the process for concentrated solutions. The case of countercurrent extraction of cosettes of plane sheet geometry, considering initial concentration distributions in the solid, is here discussed. The numerical results are compared with the analytical solution for the case of uniform concentration distributions. Graphs cover the range of practical interest for the case of uniform initial concentration. F'igures are shown which permit the evaluation of the average concentration in the extracted solid when a nonuniform initial concentration distribution exists at the extractor solid inlet. Finally, the theory developed for a flat concentration profile is applied to a definite situation such as the extraction of sugar beet, and the results are compared with existing experimental data.
Extraction with Variable External Conditions
The extraction of solids in a co- or countercurrent arrangement can be treated by solving the case for a solid which during a batch extraction would be submitted to boundary conditions ident,ical with those encountered when the solid is displaced along the extractor. Thus, the problem of countercurrent extraction for solids of plane sheet geometry and thickness 21 is described by the following equations and conditions.
The average concentration, E", as based on Carslaw and Jaeger (1959) can be written
The evaluation of the first element in Expression 2, when the particle moves with constant velocity along the extractor ( t = Yt,), requires the knowledge of w,X = [$(A) - w,]/(EO - zq),which is likewise an integrand in a series. The problem reduces to a Volterra (1959) integral equation, because $(A) is unknown beforehand during coor countercurrent extraction, and it may be solved by successive iterations (Kantorovich and Krylov, 1958). If w: and [g(Z) - w,]/ (E" - w,) may be expressed as polynomials of the eighth degree in Y , the following expressions are written.
2_
n = O
Y;
0
Y.1
Figure 1. Diagram of continuous countercurrent extractor
420
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 3, 1970
2
d,
H, (-l)'-'
d:'-'
-t
If the extractor is now divided into eight equal stages and a concentration profile, ws",is postulated, lo: can be calculated for nine points of the extractor according to Equation 3. Dimensionless concentrations in the liquid phase, cz can be evaluated and, hence, new values of W: to start another iterative cycle if a mass balance is set up; this balance depends on the experimental arrangement of extraction being considered. Countercurrent Extractor
If for this case a balance is set up between the entrance and outlet of the extractor, for Fof m --f
Q(CL- CO) = M(wo - wi)
(CL - co) = J ( C f - co) (19) Setting up a balance now, at a finite Fof, between planes i and ( i + 1) (Figure 1) and using Expressions 18 and 19
-
Relation between w:and c*. For a point over the surface of the solid, the expression relating 4 ( h ) and c, assuming a porosity c for the solid and a lineal adsorption isotherm, is -
+ k] c = KC
+(A) = [E
(7)
for this arrangement
Cf
=
may be written. For the case J 2 1 and Fo, 3 the ends of the extractor gives
(8)
using the definitions of w?and c", c - co w,*=- c*
(91
- co
is obtained. Analysis of Limiting Concentrations. It is convenient to analyze the limiting concentrations which result for different flow relations. The bulk mass balance is given by MAwL = QAcL -
(10)
-
aw,= wo- w,;i = L
OF
f
ac, = c, - co; i = L or f
(11) (12)
From Equations 8, 11, and 12 it is inferred that A c ~= (AW,/K) (13) Since always during extraction C L 5 cf and W L 2 wf, inequalities ACL 5 Acf and awf 2 AWL are satisfied; with these expressions plus 10 and 13
J 5 (Awf/AwL) may If with with
(14)
J 2 (ACL/&) (15) be written. for F o-+~ m the solid a t its outlet is in equilibrium the incoming liquid, necessarily J 5 1, in accordance Equation 14. WL=
wfonly if For
-
the balance between
Q(c/ - CO) = M(Go - W L )
(21)
Substituting Expression 8 in 21
= cf only if Fof+
m,
03
(22)
When setting up a balance, at finite Fof,between planes i and ( i + 1) (Figure 1) and considering Expressions 21 and 22, Equation 20 is ag+ obtained. It is therefore concluded that for every J >= 1 the dimensionless mass balance between any two planes is given by Equation 20. The simplicity of Equations 9 and 20 justifies the choice of the dimensionless concentrations, w" and c". Calculation Procedure. The calculation starts with an arbitrary set of W$ (i = 1,.. . , 9) values, within the range 0 to 1, which by means of the well-known technique of matrix inversion produce coefficients b,. These coefficients are then used in Equation 3, permitting calculation of the corresponding wP values for given Fof and Y,. The new c,* values are obtained by Equation 20. Based on Equation 9, this concentration distribution in the liquid enables the starting of a new iterative cycle. The process is stopped arbitrarily when two successive profiles differ by less than a predetermined value. Analytical Solution for Uniform Initial Distribution in the Solid ( Y = 0)
Formulation of Problem. The equations and conditions which represent the problem (Figure 1) are
J 5 1
(16) Inversely, for the liquid at its outlet to be in equilibrium with the incoming solid, J 2 1 follows, in accordance with Equation 15. CL
m,
(Go - w / ) = J(Go - w L )
Lo/K; wf= Kco
Cf
-
w: - w:+1 = 1/J c: - c:, 1
PS
Since the definitions of cf and are
(18)
is obtained. Replacing Equation 8 in 18
J 2 1
x=l,y>=O
w ( x ) = Go = wo
0 5 x 5 1
c = Co
t=O y=L
(17)
Expressions 16 and 17 indicate the necessary conditions prevailing when equilibrium exists between the solid and liquid phases in a countercurrent extractor. These relations are independent of the geometry or other conditions, such as type of diffusivity, of the solid being extracted. Mass Balance
First, the balance for J S 1 is considered, because definite limiting concentrations exist, depending on the value of J .
x=o
w = K-c
x=l
Y 2 0
It is convenient to effect the transformation y .A . E . p s / M and to choose the following dimensionless variables (Kitaev et al., 1967).
w - wo W= KcL - wo
-
w - WQ U'I, - wij ;w= ; w,= (29) KC^. - L L ' ~ KcL - wo
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 3, 1970 421
I n these dimensionless concentrations an unknown concentration, CL, is included; however, according to Munro and Amundson (1950), this still permits the solution allowing for a bulk mass balance. By substitution of the dimensionless parameters defined by Equations 29 and 30 into 23 through 28, these latter become
a2w aw dZ2 - aFO
0 5 Z 5 1,
Fo
0
(31)
0 5 Z 5 1,
Fo = 0 Fo = 0
(33) (34)
2 = 0,
Fo 2 0
(35)
where the second m mber represents the summation of the residues of [h.e(’O*P)]. T o evaluate the residues, the techniques detailed by Churchill (1958) and Arpaci (1966) were employed. CASE I. J < 1. There is one simple pole in p -+ 0; for p real and positive, other poles cannot exist; if p is real and negative, simple poles could exist as long as the following holds. pn
= J-tan
(pn);
t~ =
1, 2, 3. . . . ; pl” =
ipn
(45)
The residues, for this case, were evaluated by means of Equation 7-225 (Arpaci, 1966). Summing up all the poles, Expressions 66 and 67 are obtained.
aw - dC J--a2 dFo w=o c=1
-aw dZ
-0
Laplace Transform of Solution. The Laplace transforms of W and C are defined by Expressions 37 and 38 as
L [ W ( Z ,F O ) = ]
s
m
e-PFo W ( Z , Fo)dFo=
0
h(Z,P ) = h
(37)
m
L[C(Fo)]= J e-PFoC(Fo)dFo= H ( p ) = H (38) 0
Transforming Equation 31 with respect to Fo:
(39)
where p. is given by Equation 45. For J -+ 0 (constant concentration of the extracting solution), Equation 46 is coincident with the solution for a batch extraction in an infinite volume (Crank, 1957). Likewise, it is possible to conclude that Solution 46, for Fa -+ m ( J < 1) leads to the same result as in the concentration of the solid phase a t the solids outlet: In this case the concentration of the solid a t its exit is in equilibrium with the concentration of the entering liquid phase according to Expression 16. Equation 46 satisfies Equations 31, 32, and 35. CASE 11. J = 1. For this situation there is a pole of order 2 for p -+ 0; by applying formula 7-220 (Arpaci, 1966) the corresponding residues are obtained; for p real and negative, new poles exist if Equation 45, with J = 1, is satisfied. Summing up the corresponding residues gives solutions 48 and 49.
The solution of 39 that satisfies Condition 35 is
h = K1 cosh (p”’2)
(40)
Now transforming Equation 32 and taking 34 into account
dh J-- - p . H - 1 , dZ
z=1
(41)
Also from Equation 36
n = l
h=H
Z=1
Replacing Equation 42 in 41, taking 40 into account, Transform 43 is finally obtained.
h=
p 1 ’ 2 * ~(p”2Z) ~~h [p”2scosh (p’“) - J - s i n h ( p 1 ’ 2 ) ] s p
(43)
Inversion of Transform, Equation 43. Three situations may be distinguished when inverting Equation 43: the most general case in industry where J < 1, and cases where J = 1 and J > 1. Since Expression 43 is analytical except for a number of poles to the left of a line with a constant real part, W ( 2 ,F o ) is given by
422
n = 1
I t can be proved that Equation 48 satisfies 31 and 32 as well as Condition 35. Likewise, it can be concluded from 49 for Fo -+ m that
KCr. = W Q= Kcj .*. C L = C j
(50)
The liquid phase, when issuing, presents a concentration in equilibrium with the concentration of the entering solid. From a specific mass balance throughout the extractor and Equation 50, the following can be obtained, for J = 1: W L = KC^ = Wf (51) Expressions 50 and 51 agree with previous conclusions summarized by Relations 17 and 16, respectively. CASE 111. J > 1. There is one simple pole p 0; for p real and positive, there is now another simple pole if Equation 52 holds: -+
m
W ( z ,Po) =
Pn
(42)
Res(bn)
(44)
Ind. Eng. Chern. Process Des. Develop., Vol. 9, No. 3, 1970
y =
J . t a n h ( y ) ; y2 = p
(52)
This residue is obtained by applying expression 7-225 (Arpaci, 1966). For p real and negative infinite simple poles as for the previous cases are found. Therefore, 2Je(Fo.yz)
W ( Z ,Fo)
1
= ___
(1 - J )
(y2
+
2J.
COS
+J
cash (rz) cosh (7) -
J2)
(PnZ)
y is the value satisfying Equation 52 @"'
inmediately from the analytical solution, the numerical method was verified by comparison with the values of the average concentration of the solid a t its outlet. It is therefore necessary to find the relation between W L and EL From the balance given by Expression 10 and Definition 29, considering Equation 8 and the definition of parameter J , the following is obtained after rearranging. -
w; = y; p real
and positive); p n corresponds to the infinite values which satisfy Equation 45. From 53 IT' becomes
For J + m Expression 53 shows that there is no extraction ( w = wo); a t the same time it satisfies Equations 31 m : . KCL= wo, in other and 32; for Fo + a , W words CI, = e/, or the concentration of the liquid phase, a t the outlet of the extractor, is in equilibrium with the initial concentration on the solid. This result also agrees completely with the discussion already presented and summarized by Equation 17.
-
=
1 + WL ( J - 1) 1+ VL.J
(55)
T o obtain the analytical values of WL, it was necessary to prepare detailed tables covering the roots of Equations 45 and 52. The analytical and numerical results were compared, the last showing acceptably good accuracy (relative error < 0.170). The results obtained numerically, for uniform initial concentration distribution in the solid, are shown in Figures 2 through 7. Figure 2 summarizes the information of the utmost practical interest, indicating &i as a function of J and Foi. One conclusion is that the efficiency of the extraction depends only on these dimensionless parameters; for a solid of a given geometry being extracted, a constant Fordetermines a constant ti = H , / M ; i,?i remains constant if
Verification of Numerical Method and Presentation of Results for Uniform Initial Concentration Distribution
Since the evaluation of the average concentration profiles in the solid, along the extractor, does not follow
Therefore, the yield of a diffusion column can be in-
Figure 2. Average concentration in solid a t outlet of extractor
Ind. Eng. Chern. Process Des. Develop., Vol. 9 , No. 3, 1970 423
IC 09
Qa 0.7
06 n
I Q5
m i
1 3OA a3 02 01
Of
c
01
0.2
0.3
0.6
0.4
0.7
Y, (-> 0.5 Figure 3. Average concentration in solid along extractor J = 0.2
0.:
\
0.1E
n
I
*-
U
0
0:
0.0:
01 I
Figure 4. Concentration in liquid phase along extractor J = 0.2
424
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 3, 1970
OB
0.9
1.o
creased by ( M 2 / M 1 ) for , the same efficiency (GI,,,= & ), if for a given value of M 2 / M I = Q L / Q l , a holdup ratio Hs,/Hs, = M2/Ml is employed, provided that the total area of the particles is exposed. For a J value of 0.2 ( J < l ) , the solid is extracted more rapidly a t the entrance to the extractor, as expected (Figures 3 and 4). For J > 1--i.e., J = 1.4- -the extraction depends on the value of Foi: For small values, the situation is analogous with the one for J < 1; the opposite is true if Fo, increases, which causes a higher concentration in the liquid phase (Figures 6 and 7 ) . For J = 1 and high values of Foj, the rate of extraction along the extractor is constant; otherwise, it coincides with that represented for J < 1 (Figure 5 ) . Countercurrent Extraction with Nonuniform Initial Concentration Distributions in Solid (Null Concentration of Entering liquid Phase to Extractor)
The practical interest of the extraction of a solid with initial concentration distribution [g(z) # Go] a t the entrance to the extractor has been considered. Two situations were considered: F = 1 and F < 1. The concentration distributions for F = 1, extraction in an infinite volume with null concentration in the extracting liquid, were calculated using solution 4-17 given by Crank (1957). The distribution thus obtained, g(Z)/ w,,was expressed in polynomial form, in Z , of degree 8, using nine equally spaced points. The evaluation of Go/w,and consequently of g ( z ) / W O ,made by means of this polynomial, enabled the initiation of the calculation procedure. This procedure assumes uy = 0, in other words, null concentration of the liquid entering the extractor countercurrently. T o extend the range covering possible initial
concentration profiles, the situations for F = 0.5 and F = 0.7 were also considered; these latter cases are examples of previous extraction occurring batchwise in a finite volume. The concentration distributions for a given F l lwere ~ obtained from expression 4-45 (Crank, 1957). properly modified for the present case of extraction. From this solution, for a given F , [w,- g ( % )I/[u; 1, Equation 52 solid porosity, dimensionless mute variable, Expression 2 values defined by Expression 45 mass concentration of pure solvent in solid pores, g. of pure solventicu. cm. of solution apparent density of inert solid, g. of inert solidicu. cm. total solid density of inert solid, g./cu. cm. 3.1415.. . concentration function in solid in 2 = 1 for t 2 0 (boundary condition in 1)
SUPERSCRIPT ._
= average value SUBSCRIPTS
i = plane 1 L = outlet of continuous extractor of phase literature Cited
Arpaci, V. S., “Conduction Heat Transfer,” p. 372, Addison-Wesley, Reading, Mass., 1966. Carslaw, H. S., Jaeger, J. C., “Conduction of Heat in Solids,” p. 104, Oxford, 1959. Chorny, R. C., Krasuk, J. H., IND.ENG. CHEM.PROCESS DESIGNDEVELOP. 5 , 206 (1966). Churchill, R. V., “Operational Mathematics,” p. 149, McGraw-Hill, New York, 1958. Crank, J., “Mathematics of Diffusion,” p. 45, Oxford, 1957. Hodgman, Charles, Ed., “Handbook of Chemistry and Physics,” Chemical Rubber Publishing Co., Cleveland, Ohio, 1955. Kantorovich, L. V., Krylov, V. l., “Aproximate Methods of Higher Analysis,” p. 150, Interscience, New York, 1958. Kitaev, R. I., Yaroshenko, Yu. G., Suchkov, V. D., “Heat Transfer in Shaft Furnaces,” p. 31, Pergamon, London, 1967. Krasuk, J. H., Lombardi, J. L., Ostrovsky, C. D., IND. ENG. CHEM.PROCESS DESIGNDEVELOP.6, 187 (1967). Munro, W. I)., Amundson, N. R., Znd. Eng Chem. 42, 1481 (1950). Oplatka, G., Z Zucherind. 79, 471 (1954). Volterra, V., “Theory of Functionals and of Integral and Integro-Differential Equations,” Dover, New York, 1959. Yang, H. H., Brier, J. C., A.1.Ch.E. J 4, 453 (1958). Zabel, L. W., McKibbins, S. W., Chem Eng. 68, 121 (1961). RECEIVED for review May 19, 1969 ACCEPTED April 3, 1970
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 3, 1970 433
