MATHEMATICAL ANALYSIS O F FRACTIONAL CRYSTALLIZATION
573
When t = 24 hr.,
Let q.h = 1 cm.*: Q
o.12
{
-5 + Q
e-6
subsequent terms having been neglected, i. e., terms containing p = 1 , 2 , 3 , etc. ' 1 .
= 0.1198 mole of potassium chloridc per c n 8
T h e mean concentration in the slits a t the end of 24 hr. is 0.9998 X 10F moles of potassium chloride per cm.3 T h e equilibrium concentration is 1 X IO-o moles of potassium chloride per cm.8 The true s t a t e of affairs as regards a diffusion process in such a system is somewhere between those of our two limiting assumptions, i.e., that, on the one hand, diffusion into slits takes place solely from the extensions of the slits and that, on the other hand, the concentration of the solution above the slits remains constant. T h e calculations indicate t h a t at 24 h r . the change of concentration in the slits would have been such as to have been easily detected, i.e., a change of concentration of somewhere between 3 and 11 per cent. Since no change was observed, we conclude t h a t at zero time there was no bulk deplct,iou in the slits m c h as was postulated by McBain.
FRACTIOSAL CRYSTALLIZATION A MATHEMATICAL ANALYSISOF FRACTIONAL CRYSTALLIZATION PROBLEMS I N 'n-COMPONENT S Y S T E M S
H J. GARBER Department of Chemzcal Engineering, [Jniuerszty of Cincznnatz, Cmcznnatz, Ohzo ASD
A W GOODM.15
Department of Mathenatzcs, Cniuerszty of Czncznnatz, Czncannatz, Ohao Receaved February 21, 1940
Problems in fractional crystallization which deal with multicomponent systems are customarily solved by the Ese of special methods developed for the particular case being considered. Blasdale (1) and Hildebrand (5) use combinations of two- and three-dimensional graphical and algebraic methods. Morey (7) advocates the use of solid geometry to solve problems in phase equilibria. In the chemical industry, graphical solutions are used whenever possible. Since there seems to be a predilection for variety
574
H. J. GARBER AND A. W. GOODMAN
in this branch of application of the phase rule, a complete systematic plan for the study of fractional crystallization processes is rarely given. As a result, this important topic has suffered unwarranted neglect. In the following discussion, the quantitative effect of evaporating water from a solution a t constant temperature is considered (see table 1 for the meanings of the symbols used). I t is assumed that the coexistent phases are in equilibrium a t all times; with these assumptions the phase rule is
P + F = C + 2 The law of conservation of matter is also used, since it is implied whenever a material balance is made. Later an assumption of linearity for certain unknown functions involved is introduced. This assumption simplifies numerical estimations and leads to errors, the magnitudes of which can be determined only when complete experimental data for the system are available. In any discussion the choice of the axioms used always determines the forms of the mathematical propositions which are deduced. Thus, in this problem the phase rule and the law of conservation of matter automatically dictate the final results. As a consequence, this discussion does not presume to discover anything inherently new; instead, it proposes to develop a general analytical method for making calculations involving fractional crystallization. For four-component systems, a suitable analytical method arises from the use of vector algebra (3, 4).' For a general system of n-components, ordinary algebra is employed. The results thus obtained assume a form analogous to those for the four-component system; hence vector algebra in this case performs a double duty, simplifying the problem of fractional crystallization for four-component systems, and pointing to the generalization for the n-component system. For a system containing n ions in a water solution in equilibrium with s solid phases and a vapor phase, the independently variable components are
C = n - l + (ions)
1 = n (water)
Only n - 1 of the ion concentrations can be independently varied, since the amount of the nthion is determined from the relationship: 2 Equivalent weights of positive ions = 2 Equivalent weights of negative ions The coexistent phases are P = 1 (liquid)
+
s
(solid)
+
1 = s + 2 (vapor)
1 Where vector algebra is employed, the notations and terminology of J. W. Gibbs are used.
.
TABLE 1
Table of symbols All concentrations are expressed in terms of the equivalent weights of the ion-former per 1000 moles of water SYMBOL
A, B, C,
..
... c. . . . . . . . . . . . a i , b ; , c;
F................
f;. . . . . . . . . . . . . . . i J
.................
n
P R S,
tl
,t2,
ts
.........
v. . . . . . . . . . . . . . . . xi... . . . . . . . . . . . . x i , ..
.............
x i o . .. . . . . . . . . . . . . a , @ , y , S.
........
x................. ,El. . . . . . . . . . . . . . . p.................
.............. ...............
“i..
wj.
YEANINQ
The invariable points for the system under consideration, and the invariable points for the simpler systems which arise as special cases when the concentration of ane or more of the ions is zero The concentration of ion X;a t the invariable point A, B, C, ... The number of components in the system whose concentration can be independently varied The number of degrees of freedom for the system An unknown function of thevariables t l t t ts . . . . which gives the concentration of ion i in the solution whenever the values of t 1 , t 2 , t 3 . . are given An index which has integral values i = 1, 2, 3, ... n, any particular value of which designates a specific ion An index which has integral values j = 1, 2, ... s, any particular value of which indicates a specific solid The number Jf simple ions for t,he system, considering the solvent as 100 per cent un-ionized The number of phases in equilibrium The point representing the composition of the initial mixture The various solid phases which are stable in the region of temperature and concentration with which we are concerned; also some point which indicates the ratios of the number of equivalent weights of the ions in a molecular weight of the solid The independent variables which, when specified, determine the system completely The point representing the composition of the solution a t any time The ions present, such as CY, Na+, . . . . where the subscript i indirates a particular ion. Also, for a four-component system any three can be considered as mutually perpendicular unit vectors The concentration of ion X; in the solution at any time; the coijrdinates of the point V The initial concentration of ion Xi in the solution; the coordinates of the point R Xumerical multipliers which play the r6le of t , t 2 . . . . when the assumption of linearity is made The ratio of the number of moles of solvent a t any time t o the initial number of moles of solvent The number of equivalent weights of ion Xi in a molecular weight of the solid S i The number of moles of water which have been removed from the initial solution by evaporation The number of moles of solid S, which have crystallized The number of moles of water of crystallization in solid S i per mole of the solid S,
..
.
For a four-component system, the phase diagram is a space model, and for this special case an arrow placed over a capital letter represents the vector from the origin t o the point given by t h a t capital; thus
A = the position vector of the point A The standard notation of Gibbs is used in all discussions employing vector algebra. 575
-
576
H. J. GARBER AND A. W. GOODMAN
Thus the degrees of freedom are F =n
+ 2 - + 2) = n - .s (S
This discussion will be restricted to operations a t constant temperature, thus fixing one variable and reducing the degrees of freedom by 1; hence
F = n - s - l (1) It should not be supposed that fixing the temperature limits the generality of this discussion or impairs its usefulness. I n a later section this restriction will be reconsidered and will be shown to be only apparent. In this discussion an invariable point is defined as that solution composition which causes F in equation 1 to vanish, or s = n - 1 (2) The first crystallization end point is defined as that composition of solution obtained on isothermal evaporation of water a t which a second solid phase just begins to appear. The rth crystallization end point is defined as that composition of solution obtained on isothermal evaporation a t l)thsolid phase just begins to appear. which an (T It is also assumed, throughout the entire paper, that the salts which are formed by the evaporation process are of constant composition, Le., the pure salts are insoluble in each other in the solid state. This rules out systems involving isomorphous salts. (In such a system, evaporation of water produces crystals containing two salts in proportions dependent upon the concentrations of the salts in the mother liquor.)
+
THE FOUR-COMPONENT SYSTEM
Consider a four-component system whose phase diagram for some constant temperature is given by figure 1 ; while the diagram is for a specific case, the discussion and deductions which follow are general. Upon evaporating vater isothermally from a mixture whose initial composition is indicated by the point R 1 , a quantity u4 of the solid S4 crystallizes. Let the composition of this solid be represented by Sa = XI
XZ%t4XSs t , X4 ,I,(HzOL,
This generalized solid includes all possible cases. since any one or any combination of the subscripts may be set equal to zero to give any special case. The only refitriction is that ,& must not be used as a divisor in the subsequent work. A material balance gives
ZIJL
+
+
~ 2 ~ x 2SSJS
=
+
+ lOOOHzO
~4~x4
+
Xz Xs X4 ,E,(HZO)~, pHzO -t X(ZIXL 23x2 ZSXS+ 54x4 + 1000HzO)
0 4 x 1 It4
+
+
MATHEMATICAL ANALYSIS OF FRACTIONAL CRYSTALLIZATION
577
It is proper to reiterate that, with any system cont)aining n ions, the concentrations of only n - 1 of the ions can be varied independently; thus in setting up any material balance, only n - 1 of the ions need be
I
GIG.1.
Typical equilibrium diagrnm for a four-cwmponent system at constant temperature
considered in detail. Although the remaining irm serws to deemninethe nature of the system, it yields only redundaiit inforn t'ion. Since the total amount, of any ion or ion-former i5 cmstant, the coefi-
57 8
H. J. GARBER AND A. W. GOODMAN
cients for each of the n equations
- 1 ions may be equated, obtaining the set of
p =
lOOo(1 - A) - 6 4 0 4
The above expression gives the composition of the solution a t any time V = ( 5 1 , x2 , xg), after salt S 4 has crystallized as a result of evaporation of water. The position vector from the origin to this variable point V is
T h e interpretation o j the above identity i s that in a n y case where a singAe solid crystallizes, the position vector of the solution composition at a n y time, V , .+ the position vector of the original solution, R, and the position vector of the salt crystallizing, S,, are always coplanar. Since the composition of the solution a t equilibrium in contact with the solid crystallizing must lie on the saturation surface of the solid crystallizing, the above theorem also identifies the path taken by the residual solution during evaporation as the intersection of the plane passing through points representing the origin, the original solution, and the salt crystallizing, with the saturation surface. This fact indicates the direct manner in which to compute the composition of the residual solution a t any time, V , as well as defining the isothermal crystallization path. The problem of finding the first crystallization end point reduces to finding the point a t which the characteristic plane of the solute crystallizing cuts the curve which represents the two-solid equilibrium involved. A way of visualizing this property, as well as the mechanism by whihh it arises, is afforded if equation 4 is rewritten in vector form as 4
MATHEMATICAL ANALYSIS OF FRACTIONAL CRYSTALLIZATION
579
This equation can be generalized. Thus, for more than one salt crystallizing
Inasmuch as X is always positive, less than unity, and monotonically decreasing with the evaporation of solvent, while ui is always positive and monotonically increasing with the evaporation of solvent, the above vector equation aids visualization. The progressive steps for the one-solid case can be diagrammed as shown in figure 2.
e84
c-
FIG.2. Plane vector diagram for crystallization of one solid
In the diagram 0 represents a parameter which increases with the amount of water removed. The vector equation for the one-solid case, together with the diagrammatic representation, shows that the composition of the residual solution tends away from the line representing the solid crystallizing. The actual curve taken depends on the solubility characteristics of the system being considered. The corresponding vector diagram for the case of crystallization of s solids is difficultto construct but can be visualized. This geometric concept aids in selecting the proper two-solid equilibrium curve. Returning to figure 1, if the initial composition Ro as diagrammed is given, then the first crystallization end point is the inter-
580
H. J. GARBER AND A . W. GOODMAN +
section of the plane of S 3 and GORith the curve GD, since the solid Sa is crystallizing. If another initial composition, as R1 , had been chosen such that the complex salt S4crystallized first, then the intersection of the plane of i4and El with either the curve GJ or the curve J H would have been involved. To find which of these curves is actually intersected, it is sufficient to compute the triple scalar product,
-.....-
~
[RIJS41=
1
5l0
2zo
5Q0
j1
j2
j3
154
1E4
aE4
i
~
(5)
j
If the vector set is right-handed, [gl%4] > 0 and the intersection is on the curve JH. If the vector set is left-handed, < 0 and the intersection is on the curve JG. The vanishing of the determinant means that R 1 , J, S4 arc coplanar. The calculation of this triple scalar product should not be regarded as a time-consuming side calculation of embarrassing necessity, since for estimations this same determinant enters into the quantitative computations. Suppose, in fact, that the above determinant is positive. The phase rule, equation 1, states that F=n-s-1=4-2-1=1
[ji%i]
4
4
4
so that the curve J H can be expressed as a function of one variable. 21 =:
jl(t)
22
= jdt)
23
= j3(0
(6)
The point of intersection of the characteristic plane with the rurvr JH, the first crystallization end point, is given by setting 2l0 1
fi(t)
520
fdt)
x30
f3(Q
1 ~
E
0
(7)
154 zE4 at4 I and solving the resultant equation for t . This, together with equation 6, gives the end point, which is all that is required to determine the amount of water to be evaporated and the crystal yield to be expected, thus completing the problem. Thus from equation 3
MATHEMATICAL ANALYSIS OF FRACTIONAL CRYSTALLIZATION
581
It is interesting to note that it is just the vanishing of expression 7 that assures the equality of the different forms of the solution for u 4 . The amount of water to be evaporated is given by p =
lOOO(1 - A) -
04w4
The second crystallization elid point is given by the experimental data, i.e., point G, H, or J, and since these involve zero degrees of freedom, a material balance will yield all of the desired information directly. If the solution is taken to complete dryness, three solid phases remaining, it is natural to inquire which of the three possible sets of three solids will appear, as indicated by the points G, H, and J. A material balance answers this question. Thus, for three generalized solids such a material balance yields
+
X1,Xi
+ XS,XS+
~ 2 ~ x 2
+
~4~x41000H20
=
XZ ZEI X3 SEI x4 ,EI(HZO)UI
‘ d 1
+ d G Xz atr X4 ,t2(HzOLZ + I t s XZ X3 X4 ,tsOW))u, + PHZO and equating coefficients of the same ion = + + = + + = ab+ + ,E!
zts
‘J&
210
0 2 1t2
01 161
at8
0 3 153
r20
‘J1 2 E l
‘J2 2 b
0 3 2t3
~3~
01
‘JZ 3€2
0 3 aE3
Considering this set as unknown in ~1 tities yields:
‘J1
z t z X3
, u2 , and 0 3 , solution for these quan-
=
0 3
=
where
i
3t:
at2
3t3
If each determinant is reflected about its principal diagonal, the expansions present themselves, not so much as algebraic quantities, but as triple scalar products of vectors which already hare been met. Thus
The nature of the problem requires that the amount of any solid obtained be positive. This has the vector interpretation that K must lie inside the
582
H. J. GARBER AND A. W. GOODMAN
i
510
230
XZO
I
'I j l ( t ) j2(t) j a w = 0
I
23
154
354
254
= fdt) =
(7)
j 3
+ dha -
j3)
1 and now plays the rBle of t. Thus equation 7 with this where 0 i7 linear assumption gives
I
1
510
x20
x30
MATHEMATICAL ANALYSIS OF FRACTIONAL CRYSTALLIZATION
583
which, together with equation 8, gives the first crystallization end point. Note that the numerator is a determinant which has been previously computed (equation 5) for the purpose of determining the curve being intersected. Thus, in this case the numerator should be intrinsically positive, and the denominator intrinsically negative and of such a value that 7 lies between 0 and +l. Various cases for the sign and magnitude of 7 which might arise have definite interpretations. If 7 < 0, the line of intersection is actually either GJ or FJ, while if 7 > 1, the line of intersection is actually GH or HE. In either case the assumption that S4 is the solid which precipitates out must be reexamined. If the equation set 8 is used in expression 3, the resulting set %lo= u 4 1I.4
a. =
~4 254
Zso = ~4 8i4
+ XI.$ + dhl - jdl + XLh + d h -~ jdl + XLia + - jdl
may be viewed as linear in the unknowns u 4 , A, and AT.
x= where
D
=
154
2E4
aE4
jl
j2
ja
hi
hz
ha
Thus
584
€€.
J . GARBER AND A. W. GOODMAN
THE ?%-COMPONENT SYSTEM
If water is evaporated isothermally from an n-component system consisting of n ions and water, a material balance for the first crystallization end point gives:
Equating coefficients of the same ion X i , there are obtained n pendent equations of the form Zio
=
01
ib
- 1 inde-
+ XZ,
(14)
For the incipient formation of a second solid phase, equation 1 gives F = n - s - l = n - 2 - 1 = n - 3 or there exist n thus
- 1 independent functions of n - 3 independent variables,
-
= f + ( t i , tL , la, * L a ) Using these in the above set, a set of n - 1 independent equations
+
x,, = 01 151 X f d t l , t 2 , t 3 , ' * ' tn-3) (15) t,-3, u l , and X, is obtained. The in the n - 1 unknowns t l , t 2 , t a , solution of this set gives the desired information. For the second crystallization end point the set of equations 15 becomes %to
and for the
=
~1 s t 1
+
~2 i t 2
+
Xfi(t1
Q
ts,
*
crystallization end point the set of n
~ t h
tm-4)
(16)
- 1 equations is
When all of the water acting as solvent has been removed, the solution will pass to some invariable point a t which n - 1 solids are in equilibrium or the equa.tions 17 become
The number of moles of any particular solid can be found by applying Cramer's rule for non-homogeneous linear equations to equation 18. Only in the case of equation 18 is it possible to display the numerical answer, because in that case the unknown functions involved in equations 15, 16, and 17 do not enter. 1
The assumption of linearity for the n-component system Consider the n-component system having u invariable points at which n - 1 solids are in equilibrium, and a set of invariable points for which n - 2 of the solids are the same. At this point two obstacles appear
MATHEMATICAL ANALYSIS OF FRACTIONAL
CRYSTALLIZATION
585
which cannot be overcome completely in a generalized discussion. First, given any n - 2 solids, there may not exist even one invariable point a t which these n - 2 solids are in equilibrium, and secondly, assuming that such points do exist, there is no way, a t present, of predicting the number of points for any given n - 2 solids. However, for any pair of points A and B having the same n - 2 solids in equilibrium, the phase rule, equation l, predicts that F=n-s-l=n(n - 2) - I = 1 or a relationship including the points A and B exists, depending on one independent variable. The assumption of linearity imposes the form
+
xi = ai @(Si - ai) O S P 6 1 where 6 is the independent variable. If another invariable point C exist with n - 2 of its solids the same as A, but only n - 3 solids the same as B, for the n - 2 solid equilibrium
+ Y(Ci - ai)
O S y j l But between A, B, and C, n - 3 solids are the same, and since the phase rule predicts two degrees of freedom, the linear assumption for the function giving the points of n - 3 solid equilibrium takes bhe form xi = ai
+
xi ai -t p ( b i - ai) Y ( C i - ai) Xote, however, that whenever two or more independent variables are involved, @,y, . . . , no limits can be set for the range of values of these variables. The generalization is evident. For two-solid equilibrium where the number of independent variables p, y, 6, . . . K is n - 3 and there are n - 2 invariable points, =3
5,
= ai
+ B(bi - ai) + y(Ci - ai) +
* *
+
K(kj
- ai)
A material balance for the first crystallization end point, gives the equation set 15 and, using the assumption of linearity for the two-solid equilibrium, xio
= vi
it1
+ X[ai + @(bi - + y ( c i - ai) + + ~ ( k -i ai)] ai)
Considering the unknowns as ul, A, A@, . . and XK, n - 1 in number, the solution of the set of non-homogeneous linear equations can be written x2,) Z Q 0
al bl
a2
as
bz
b3
-
-
-
xn-1,)
an-1 bn-1
586
H. J. GARBER AND A. W. GOODMAN
where
D =
Similarly, for the second crystallization end point, n equations of the form xi0
=
UI
ib
+
uz ifi
+
- 1 independent
hi
arise, where now the linear assumption is X i
= bi
it1
+
+ r(C
