Solid-Liquid Extraction in Countercurrent Cascade with Retention of Liquid by the Solid Spherical Geometry Constant Diffusivity Florencio P. Plachco’ and Maria E. Lago Escueb de lngenierja Q u h c a , Facultad de Ingenieria, Universidad de Los Andes, Merida, Venezuela
The analytical solution and the method of calculation are presented for a cascade of N perfectly mixed batch extractors operating countercurrently. The solid is assumed to have spherical geometry and to retain solvent on its surface when passing from one extractor to the next. The results obtained, assuming constant diffusivity and a linear adsorption isotherm, are presented in dimensionless plots for direct use in design; N varies from 2 through 10.
Introduction The countercurrent solid-liquid extraction in a cascade of batch extractors, perfectly mixed and with finite volume, is widely used for its mechanical simplicity, good efficiency, and flexibility. However, a description of this type of process in terms of the basic Transport phenomena cannot be found. A thorough bibliographical review of solid-liquid extraction and current design methods appears in works by Rickles (1965)and Schneider (1968). This arrangement is used in very different extractive industries such as oil extraction from oil-bearing seeds, beetsugar, essential oils, tannin from “quebracho”, several soluble substances in fish, toxins from certain products of biological origin, etc. In this present treatment, during the ith extraction, the following fundamental assumptions are considered to prevail: (1)the diffusivity is constant; ( 2 )the adsorption isotherm is linear; (3) the resistance to mass transfer in the film is negligible; ( 4 )the mixing is perfect; ( 5 ) the solid has a given size and geometry which do not alter during the extraction; (6) the quantity of solvent present in the solution retained by the solid when passing from one extractor to the following one is constant; (7) all particles are retained in a stage for the same length of time; (8) retention time is the same for all stages; ( 9 ) the kinetics of adsorption-desorption is much more rapid than the diffusion inside the porous medium. It is well known that the products of biological origin have variable diffusivities and therefore give rise to nonlinear problems, the solution of which, often numerical, is highly specific. According to Plachco and Krasuk (1970),there are cases when it is possible, considering adequate diffusivity, to treat these problems as linear; the results obtained compare favorably with the experimental values. This must be considered as a first approximation to the solution of the problem. Assumptions 2 and 3, have been confirmed experimentally by Oplatka (1954),Yang and Brier (1958), and Krasuk et al. (1967). The fourth and fifth assumptions are inherent to practical extraction. The sixth assumption is approximate because when the concentration of the solution varies, its viscosity also varies and the quantity of liquid retained by the solid varies accordingly. The solvent in the solution, expressed by units of volume, varies inversely with the concentration and generally tends to compensate partially for the first variation.
The most adequate procedure is to take an average value of the amount of solvent retained along the whole cascade in order to make use of the results presented here. In this present treatment the solid has been considered without consideration of liquid retained when entering into the first extractor. The results are given in the form of graphs which permit the (with redirect calculation of the nonextracted fraction, spect to the maximum extraction possible with the initial solution in infinite batch) in terms of the total dimensionless time, Fa,the quotient of volumes, J , the fraction of solvent retained by the solid, P , and the number of active extractor in the cascade, N .
c*
Model The extraction from solids of spherical geometry taking place in the ith extractor of a cascade can be represented by the following set of dimensionless equations.
aCi - - - _aWi _ _ aR
3JaFo;
Ci = Wi
( R = l ; F o i> 0 )
( R = 1;Foi
> 0)
(4)
(6)
where 1 Ei - El1 K w~ ci = K ciw l-f -E lE11 ;Cl = ; wi = K w l f - El1 KWIf - C l l
SK Dti R = r/rOi; Ji = -; Foi = Li roi
(7) (8)
It should be noted that w l f appears in the demensionless terms defined by (7).This value is unknown a priori, but as shown by Munro and Amundson (1950), Kitaev et al. (1967), Plachco and Krasuk (19701,and Plachco and Lago (1972),in countercurrent arrangements, it is possible to solve for it with an overall mass balance. The solution to set (1) through (61,according to Plachco and Lago (1972),is written
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1975 361
Equation 17 can be writen in dimensionless form
where
PW(,-l,,f - WlI + (1 - P )W(l+l),f= 0
(18)
Form equations ( 5 ) and ( 6 )it may be written
W(l-i),f = C(,-i),j(R = 1,Fof(l-1)) W1,I = Or
then it is found that
(19)
Wl+l,f = C(l+l),f(R= 1,FOf(L+l))
-6
c
+ + +
(p,) * p n
[9Ji2(l/pn 1) 6Ji [9Ji2 9Ji p n 2 ] (3Ji p n 2 )
E"nZFoi COS
+
n=l
+
+
X J 1 ~ - g ( i ) s i n ( p n 7 ) d 7 + - (13J'J ; )
s1 o
T2-g(7)
+ pn2]
By eq 19 the dimensionless balance (18) can be expressed p[8(l-i)a(l-i)(Foj(I-i), R = 1)
+ P(i-l)(F~f(L-l), R = I)] + (1 - P ) [B(,+l)a(i+l)(Fof(i+l), R = 1) + P ( r + l ) ( F o f ( i + l )R, = 111 = 0 81
dr
(12)
Calculation Procedure If a quantity ( L - SA) of solvent enters the N t h extractor as fresh solution and if the solid retains SA of solvent on its surface when passing from extractor N - 1 to N, then the quantity of solvent present in extractor N is L . Since the solid being transferred from extractor N carries with it SA of solvent, only L - SA of solvent is transferred from extractor N to N - 1. Together with the solvent SA, which the solid retains on its surface on transfer from extractor N - 2 to N - 1,again we find that the quantity of solvent present in extractor N 1 is L . Following this argument for the rest of the extractors it can be shown that extractors N , N - 1, N - 2 , . . . , 3,2 contain the quantity L of solvent. Because of the assumption that the solid entering the cascade does not have solvent retained on its surface, the content of solvent present in the first extractor is L - SA. This leads to J1 = J / ( 1 - P),Jz = J3 = . . . Jn-l = J , = J . In order to simplify the expression it is convenient to define
(20)
It is obvious from (20)that it is sufficient to know 8(1-1), 8,, g ( l - l ) ( R ) and , g(,+l)(R)in order to calculate 8(1+1), since for , (3(1+1) can be computed. this case C X ( ~ - ~ )f,l ( l - l ) ,a ( l + l )and In the calculation sequence 81,82, gz(R), andg3(R) are obtained; substituting i = 2 in eq 20, the value of 83 follows immediately. C3(R,Fof3)can be calculated for arbitrary values of R and similarly, once g3(R) and 83 are known, then g l ( R ) may be obtained. The calculation proceeds sequentially through i = 3,4, etc. until the N t h extractor is reached, when g N ( R ) and 8~ are thus known. With these later values the concentration profile in the solid coming out from the last extractor can be calculated. I t might be more useful to know the mean dimensionless concentration given by expression ( 1 2 ) . It is convenient to express the mean dimensionless concentration in the solid a t its exit from the cascade once the calculation of C N ( F O N is ~) finished, by the following dimensionless value
1
c ~ f e*
In order to establish the relation between and it is necessary to consider their definitions and those of W,j, 8 ~ ,
Based on definitions (7) and ( 5 ) and the equality (6), it may be shown that
Biai(Fof1,R = 1) + Pi(Fofi,R = 1) = 1
(15)
Since the initial concentration profile is assumed ( g l ( R )= 01, al(Foj1, R = 1)and P1(Fofl,R = 1)are values which can be calculated therefrom. From eq 15 81 can then be solved. When calculating a1 and P1 it is necessary to remember that J1 = J / ( 1 - P ) . With this value C1(R,F o l f ) can be calculated, once 81 is found, using eq 9 for any value of R; in this case nine equally spaced points were chosen and used to determine the coefficients in an (8th order) polynomial expression for g2(R). The coefficients in this expansion are found directly by matrix inversion. Knowing gZ(R), remembering that w11 = w2f and using equality ( 6 ) ,then 82~2@'0f2,R
= 1) + Pz(Fo,fz, R = 1) = 01
(16)
Thus, as before, once 0 2 and /32 are found 02 can be calculated. Knowing 8 2 and g2(R), C2(R, F 0 f 2 )for arbitrary values of R can be calculated thus determining g d R ) . If a mass-balance in the liquid phase is made over extractor i (i > 1)for Foi = 0, it is found that
SAW,-~,J + ( L - SX)w,+l,f= Lull 362
Ind. Eng. Chem., Process Des. Dev.. Vol. 15,No. 3, 1976
(17)
J , and P as well as the following overall balance
s
C'lI
+ ( L - SA)wr = ( L - 2SA)Wlf
+ ShUNf + ScNf
(22)
After some tedious algebra it is found that
-
c*=
J l+(--(1 - 2P) l+-
I)
J
eNf
+-(1- 2P)
(1- 2P) C N f
A mass-balance in the liquid phase a t FON = 0 in the n extractor gives SAWN-l,f
+ ( L - SA) W I = LWNI
- ith (24)
The following generalization, by means of dimensionless values, can now be written WI = ON - P*CN-l,f(R= 1 , Fof,N-1) (1 - P)
(25)
F i g u r e 1. Schematic diagram of the average concentrations in the different extractors as a function of time.
P = 0125
F i g u r e 3. Dimensionless average concentration in the solid vs. F O ~ ' ~ , for P = 0.250 and cascades with N = 2(1)10, with J as parameter.
--D
00
i
Figure 2. Dimensionless average concentration in the solid vs. FoliZ, for P = 0.125 and cascades with N = 2(1)10, with J as parameter.
c*
405
Between eq 25 and 23 the relation between and c N f is established since ON, CN-l,f(R = l,FOf,N-l), and CN,f(R = 1, F o f N ) are found during the calculation procedure.
Results and Conclusions The results presented in the figures refer to solid-liquid extraction in a countercurrent cascade of N batch, finite active, perfectly mixed extractors. The solid is assumed to have spherical geometry and to retain some solution when passing from one extractor to the other. The consideration that the Foi and Pi are the same in all of the extractors and the fact that the solid enters dry to the first extractor leads to
Figure 1 shows in a schematic way part of the nomenclature and the evolution, wih time and with the different extractions, of the mean concentration in the solid, E, and the concentration in the liquid, w. Figures 2 , 3 , and 4 represent vs. F01/2with N = (2(1)10}, J = (O.O(O.2)l.O)for P = (0.125,0.250,0.375),respectively. From
c*
IO4
I
N \ \
'\
--
-- _
_
Figure 4. Dimensionless average concentration in the solid vs. F O ~ ' ~ , for P = 0.375 and cascades with N = 2(1)10, with J as parameter. Ind. Eng. Chem., Process Des. Dev.. Vol. 15, No. 3, 1976
363
J
I
10
i
P= 0250
J = 0.6
I
Figure 5. Dimensionless average concentration in the solid vs. the proportion of solvent retained by the solid for different parameters of N , J , and F O " ~ .
I
\ \ \
t Figure 7. Dimensionless average concentration in the solid vs. N with = 0.250, and J = 0.6.
""I
Fo as parameter for P - - - p = 0.250 -P * 0 3 7 5
Table Ia Fo = 0.36
N I -I
P
J
2
3
4
5
6
7
8
9
10
0.125 0.2 0.465 0.430 0.420 0.415 0.410 0.410 0.410 0.405 0.405 0.250 0.4 * * 0.765 0.655 0.615 0.590 0.575 0.56 0.55 * * * * * * * * 0.375 0.6 *
-
The asterisks indicate that under these conditions the desired extraction cannot be obtained even at Fo *.
110
i
2
3
4
5
6
7
8
9
10
x i-i
Figure 6. Dimensionless average concentration in the solid vs. N with J as parameter for P = 0.250, P = 0.375, and Fo = 0.36.
these figures and those corresponding to P = 0 (Plachco and Lago, 1972), the negative influence of the liquid retained by the solid backmixing on the extraction becomes evident. In order to emphasize this effect, Figure 5 showing vs. P with Fol/*,N, and J as parameters, reveals the greater influence of a higher P at higher values of J . Another significant fact is that a t high values of P and J the vs. Fo curves intersect. This means that for this condition a lower extraction is obtained, in spite of the increase in the number of extractors. This may be interpreted from a conceptual point of view by the inefficiency resulting from a greater liquid retention being greater than the efficiency gained by adding another extractor; in other words, there is a strong backmixing. In order to show more clearly the behavior of the present extractive arrangement, Figures 6 and 7 are presented. From Figure 6 it can be seen that for P = 0.250, Fo = 0.36, and J = 1, the "curve" becomes horizontal when N = 3; however, for J = 0.6 this is not obtained even with 10 extractors, and for J = 0.2 approximately 9 extractors suffice. For the case of P = 0.375, it is notorious that for J = [1.0,0.8,0.61 the vs. N curves present a minimum, indicating that for N > Nminan increase in N for a given cascade produces an inverse effect. The other fact which arises is the great difference in extraction existing between both P values.
c*
c*
c*
364
Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 3, 1976
From Figure 7 it can be inferred that for values of P = 0.250 and J = 0.6 the effectiveness attained through the addition of another extractor increases with higher values of Fo. I t is worthwhile to analyze the effect on the total time of extraction, Fo, of N for the case when pure solvent is being used (w1= 0), and it is desired to extract, for instance, 85% of = 0.15) and the the solute initially present in the solid solid-liquid system is characterized by J = 0.2, P = 0.125. For such a situation with X and K remaining constant the following table results.
(c*
J 0.2 0.4 0.6 P 0.125 0.250 0.357 in Table I can be obtained for = 0.15 The values of and for the parameters J and N as exemplified in Figures 2, 3, and 4.
c*
Nomenclature c = concentration of solute in the solid, g of solute/g of inert
solid c = refers to the average concentration in the solid, g of solute/g of inert solid C = dimensionless concentration in the solid defined by eq 7 = average dimensionless concentration in the solid, defined - by eq 7 C* = average dimensionless concentration defined by eq 21 D = effective diffusivity, cm2/s Fo = E ,: Pol/ = overall Fourier number of the cascade Foi = Fourier number defined by eq 8 g( ) = dimensionless initial concentration profile in the solid J = effective fraction of the solvent volumetric flows, inside the pores of the solid and in the liquid phase defined by eq 8
k = equilibrium adsorption constant, g of pure solvent/g of inert solid
K = ( c p ~ / p+ ~ )k, concentration relation expressed on a liquid
and solid base, g of pure solvent/g of inert solid L = amount of solvent in the liquid phase, g of pure solvent P = SX/L fraction of solvent retained by the solid when passing from one extractor to another r = radial coordinate, cm ro = external radius, cm R = r/ro = dimensionless radial coordinate S = amount of inert solid in the extractor, g of inert solid t = time, s w = concentration of solute in the liquid phase, g of solute/g of pure solvent W = dimensionless concentration in the liquid phase defined by eq 7 Greek Letters a( ) = function defined by eq 13 /3( ) = function defined by eq 14 t = porosity of the solid 8 = dimensionless initial concentration of the solution X = mass of pure solvent retained per gram of inert solid, g of pure solvent/g of inert solid p~ = apparent density of the solvent, g of pure solvent/cm3 of solution p n = roots of transcendental eq 10 ps = apparent density of the solid, g of inert solid/cm3 of total solid
$(
) = initial concentration profile in the solid, expressed as a function of the dimensionless coordinate R , g of solute/g inert solid
Subscripts f = refers to t equal to the time of residence in the extractor being considered i = refers to the ith extractor I = refers to t = 0 in a given extractor L = refers to the liquid phase n = dummy index N = refers to the last active extractor of the cascade s = refers to the solid Literature Cited Kitaev, B. I.,Yaroshenko, Yu. G., Suchkov, U. D.. "Heat Transfer in Shaft Furnaces, p 31. Pergamon, London, 1967. Krasuk, J. H., Lombardi, J. L., Ostrovsky, C. D., lnd. Eng. Chem., Process Des. Dev., 6, 187 (1967). Munro, W. D.. Amundson, N. R., lnd. Eng. Chem., 42, 1481 (1950). Oplatka, G., 2.Zuckerind., 79, 471 (1954). Plachco, F. P., Krasuk, J. H., lnd. Eng. Chem., ProcessDes. Dev., 9, 419 (1970). Plachco, F. P., Lago, M.E.. Can. J. Chem. Eng., 50, 611 (1972). Rickles. R. N., Chem. Eng., 15, 157 (March 1965). Schneider. F., "Technolcgie des Zuckers", p 173, M. und H. Schaper, Hannover, 1968. Yang, H. H.. Brier, J. C., AlChEJ.. 4, 453(1958).
Received for review July 17, 1972 Resubmitted January 19,1976 Accepted March 6,1976
Ther maI Decomposition of Manganese Sulfate John P. McWilliams and A. Norman Hixson" Department of Chemical and Biochemical Engineering. University of Pennsylvania, Philadelphia, Pennsylvania 19 174
When carbon is added to MnS04 in a mole ratio between 0.5 and 1.O, the thermal decomposition of MnS04 is substantially promoted, yielding good rates (-60% conversion in 1 h) in the range 1250-1300 O F . The products of decomposition consist of almost pure manganous oxide, MnO, and a gas phase containing average values of 67 YO SO2 and 33 YOCOP.Increasing the carbon ratio above 0.5 (up to 1.O) gives an increased decomposition reaction rate but does not alter the ratio of SO2 to COP in the product gases. The overall decomposition kinetics may be described by a multiple step model which includes a sulfate pre-reduction step.
Objective The higher oxides of manganese (MnO:! and MnzO,) react readily with sulfur dioxide in aqueous solution or in a gaseous state, forming MnS04. Such diverse processes as manganese ore beneficiation (Allen, 1954; Ravitz et al., 1946; Vedensky, 1946; Wilhite and Hollis, 1968) and removing SO:! from stack gases (Bienstock et al., 1961; Ludwig, 1968) take advantage of this property. Economical recovery of the SO:! from the MnS04 would be advantageous either for recycle purposes or for further reduction to produce sulfur. MnS04 can be decomposed thermally, but on a commercial scale, the problems associated with the necessary high temperatures 2000-2300 O F are manifold. An extensive investigation of manganese ore beneficiation was carried out during World War I1 at the Three Kids Mine in Nevada (Vedensky, 1946).Ring formation and agglomeration in the rotary kilns used for the MnS04 decomposition forced frequent shutdowns and poor process efficiency.
Pechkovsky (1955,1956,1957,1959)and others (Suchkov et al., 1959) have reported work on an MnS04 decomposition. Apparently, manganese ore is beneficiated in Russia using an SO:! process that includes a thermal decomposition of MnS04 in horizontal kilns at 2000-2300 O F . Pechkovsky investigated the use of carbon to reduce the temperature and found beneficial effects at 1475 O F using up to a maximum C/Mn molal ratio of 1/1. Inasmuch as the published data were incomplete and shed no light on the reduction mechanism, this work was undertaken to study in detail the carbon-aided reduction of MnS04. It was a further purpose to determine the maximum SO:! content obtainable in the exit gases which could be furnished to another process to produce sulfur as an integral part of an SO2 recovery system for stack gases.
Chemical Equilibrium for the System The theoretical chemical equilibria for the manganese sulfate-carbon system were developed in order to guide the Ind. Eng. Chem., Process Des. Dev.. Vol. 15, No. 3, 1976
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