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Calculation of solid-liquid equilibria in aqueous solutions of nitrate salts using an extended UNIQUAC equation
Chemical Engineering .Scipncc, Vol. 41, No. 5, pp. 1197-1202. Printed in Great Britain.
1986.
s3.00 + 0.00 ooo9-2509/S6 Pergamon Press Ltd.
CALCULATION OF SOLID-LIQUID EQUILIBRIA IN AQUEOUS SOLUTIONS OF NITRATE SALTS USING AN EXTENDED UNIQUAC EQUATION BO SANDER, PETER RASMUSSEN and AAGE FREDENSLUND Instituttet for Kemiteknik, The Technical University of Denmark, DK-2800 Lyngby, Denmark (Received
3 December
1984 accepted 22 July
1985)
Abstract-An extended UNIQUAC-Debye-Hiickel model is used to predict solid-liquid equilibria for aqueous solutions of nitrate salts with the cations K+, big*+ and Ca2 + . In addition to a model for activity coefficients in mixed electrolyte solutions, the method requires knowledge of the solubility products of all possible solid phases. Furthermore, some ternary solid-liquid equilibrium data are required. It is shown that the extended UNIQUAC equation can be applied to the calculation of solid-liquid equilibria in the K+-Mg”-Ca ‘+-NOT-HZ0 system with good results. A new algorithm for calculation of solid-liquid equilibrium in ionic solutions has been proposed. The number and amount of phases in equilibrium are found by minimization of the total Gibbs function of the system.
IONi
INTRODUCTION
The calculation of solid-liquid equilibria (SLE) is an important area of application of electrolyte solution models. Examples of previous work on salt solubility calculations are given in [l-3]. SLE in the K+-Mg *+-Ca*+-NO;-Hz0 system has not been modelled by any of these methods. In this work we investigate the application of the extended UNIQUAC equation as described in [4] to this system. An attempt was made to include salt solubilities in nitric acid-water-nitrate salt systems, but it has not been possible to correlate VLE and SLE data simultaneously for these systems. The calculation of solid-liquid equilibria requires a good model for the activity coefficients in saturated systems. Furthermore, the number and amount of phases in equilibrium at a given total composition and temperature must be known. An algorithm for this purpose is described.
THERMODYNAMICS
OF SOLID-LIQUID
EQUILIBRIUM
Consider an aqueous solution of NI ionic species in equilibrium with NS solid phases. It is assumed that no solid solutions are formed, i.e. all solid phases have a composition of known stoichiometry. The number of chemical species in the system is NSP = 1+ NI + NS. Index numbers for the species are defined as follows: HsO: 1, ions:i = 2, _ . _ , NI+ 1, solid phases:s = NI NSP. For each of the NS solid phases we +2,..., have an equilibrium of the type: NI+l ~I320
+
C i=2
(ION,&,
h-1
*
(solid);
(H20),1,
r =
. . , NS
for ion i. vt, and vi, are of, respectively,
(1)
water
ing to chemical equilibrium r. The equilibrium composition of the system for a given temperature and pressure is determined by requiring that the total Gibbs function is at a minimum. The total Gibbs function of the system, nG, is given by: NI+I nG
=
nIpI
=
nI
+
c i=2
NSP wI+
(GP +RTlnal)+
c s=NI+2
NI+1
1
RA
(2)
ra,(G:+RTlr@
i=2
NSP +
c s-NI+2
(3)
%G,”
where nL and pk (k = 1, i. s) are the mole number and chemical potential of species k; G “iand G : are standard Gibbs functions for, respectively, water and ion i (unsymmetric convention 151) a, = x*y,. a, = xiy,+, and G,” is the molar Gibbs function for solid phase s. For each of the NS chemical equilibria we introduce a reaction coordinate, c,, so that the mole number of species k is expressed as: n, = a; -
2 Vb<,, r= I
k=l,...,NSP
where nE is the initial mole number of species matrix notation, this can be written: II = IlO-Et
n i=2
1,.
formula
coefficients
and ion i in solid phase s (s = r + 1 + NI) correspond-
NI+I VJONi
is the chemical
the stoichiometric
(4) k.
In (5)
where q and no are (NSP x 1) vectors, ‘j is an (NSP x NS) matrix, and C is an (NS x 1) vector. ‘E is of the 1197
B. SAHDER~~
1198
al.
Equation (9) may now be written as:
form:
nG/RT = constant +
VI2
. . .
vc.*
. . .
v,,
. . .
NI+1 +
C
t=2
NS
c & In K, + n: ln al c= 1
npln*-
y & r=r
vi,lnai
Nlfl +
v=
vas2
. . .
‘dNS
v*.*
. . .
va-NS
0 -1
:::
0
...
(6)
v= -1
0 A necessary condition for a minimum in the total Gibbs function is that the gradient vector is equal to 0:
-1
1
-1
0
[
n;G:+Nzlni’Gf+
(
NE
i=2
C s-NI+2
function
C
nphruI
i=2 NI+l
-
=
F ,=I
e.(
v&a
+
constant -kT
F r=1
C i=2
<,
vi&a,
(8) >
v,,G; c
NI+t +
C i=2
v*Gi’-GT+NI+l >
+nflna,
+
m+i
C
n4lnat
a=2
-,zl
C.(vblnal
+Tz2’
(9)
vhha,).
The thermodynamic solubility product for solid phase (r + NI + I), corresponding to chemical equilibrium r, is defined as: InK, =
Gz+N,+I -vl,Gi
irnil i=2
vtiG:
,p2
up-‘=
K,,
r =
1,. . . , NS
(14)
or n
(xjyF)(“*)
=
K,,
r =
1,. . . , NS. (15)
Equation (14) is the classical formulation of the chemical equilibrium problem Below we will describe an algorithm for SLE calculations which is based on minimization of the total Gibbs function as expressed in eq. (11).
RESULTS
>
NI+l
>
NI+I upr)
n:G.”
vi,G:
+n;lnar+
VA,”
which implies that at equilibrium
(7)
i-2
NSP +
c
(13)
1=2
r-Nx+2
vI,G”I+
R:. ,Fl “(
= 0
Nl-b1
NI+l --
V (nG/RT)
(xlyl)(“lr)
Using eqs (4) and (6), the total Gibbs divided by RT may be written: nG/RT=kT
>
b
6 0 1 20
1 0 1
(11)
vblnui
Differentiation of nG/RT with respect to c, and application of the Gibbs-D&rem equation gives:
c’ and c” indicate different cations and a’ and a” indicate different anions. The lower (NS x NS) matrix is equal to the negative unity matrix. As an example, consider the system HzO, K+, Mg2+, NO:, KNOs, Mg(NOs h - 6H20 for which L is given by:
0
C 1=2
RT.
(10)
AND
DISCUSSION
The prediction of solid-liquid equilibria requires a model for activity coefficients in mixed electrolyte solutions and knowledge of the thermodynamic solubility products of all possible solid phases in the system considered. For representation of SLE in the K+-Mg2+-Ca2+-NO;-H20 system, we will apply the extended UNIQUAC equation as described in [4] with model parameters estimated from binary activity coefficient data. These parameters are presented in the tables in [4]_ From knowledge of the solubilities of single salts in aqueous solutions, values of the thermodynamic solubiity product can be calculated using eq. (15) with activity coefficients calculated from the model. Phase diagrams of aqueous solutions of two or more salts can then be predicted. However, it is not possible to give reliable predictions of ionic activity coefficients in mixed electrolyte solutions from binary information alone. Hence the predicted phase diagrams will not be satisfactory. In [4], u;j, ~1,. S$,,,, and Sb.nIO parameters for cationstion interactions were assigned values of 2500 K, 0,O K and 0, respectively. The representation of ternary SLE data can be significantly improved if the parameters for interactions between cations of
1199
Solid-liquid equilibriain aqueous solutionsof nitratesalts different types are adjusted to the ternary experimental SLE data. Adjustment of these parameters does not influence calculated single salt activity coefficients. For a series of experimental ternary SLE data points at constant temperature with equilibrium between the same solid phase and aqueous solutions of varying composition, eq. (14) is valid in each data point. This fact is the basis for defining the following objective function for parameter estimation: NI+l
vIrlnoI+ F
=
C
vsInai
i=2
s;~.~.t,~
2
NI+1
vl,lnar+ c
vtlnoi
and
c
(16) >>ref
i=2
-(
where c,
1
indicate summa tions over,
respecti;el;f’) solid OFties, temperatures and data points. Subscript r indicates a reference data point, which is here taken as the binary solubifity of the considered salt. Single salt solubihty data used for calculation of thermodynamic solubility products as a function of
temperature have been obtained from the compilation by Linke and Seidell [6-j. The calculated values are represented by the expression: In K, = a, + 9/( T/K) + c, In (T/K)
(17)
a;, 6, and c, parameters are presented in Table 1 along with the temperature ranges of the experimental points. The table includes parameters for the two double salts KNOa _Ca(NOp)2. 3HZO and KN09. SCa(NO& . 10H20. K-values for these salts have been obtained from the same ternary data that has been used for estimation of parameters (see below). Experimental and calculated binary solubility diagrams above 20°C for the three salts KN03, Mg(NOJ)r and CF@IO~)~ in mixtures with water are sho’wn in Figs 1-3. The KNOs-Hz0 diagram is simple solid KNOa. The with only one phase, Mg(NOJ b-Hz0 and Ca(N03 )2-Hz0 diagrams are more complicated due to the occurrence of hydrates Mg(NO& .6Hi?O, Mg(NO,)z *2H,O, Ca(NOJ)2. 4H20, Ca(NO& .3&O and Ca(NO& . 2H,O. All hydrates, except Ca(NO,), . 2H20, are congruently melting with melting points of 89.5, 129,
Table 1. Parametersfor calculationof the thermodynamicsolubigtyproduct on mole fraction basis as a function of temperature by eq. (17) Solid phase KNOs Mg(NO3b-6HzO Mg(NO&.2HzO CWNO3 )1 *4HxO C~OVO&.~HZO ‘3AN03)2-2H20
WNOJ )2 KNOS.Ca(N0~),.3H20 KNO~~5Ca(N0,)a-10H~0
a, 177.49 -487.71 - 1052.0 -3.229 - 10.38 -4.659 - 10.05 2.592 -25.87
b,X 10-b
c,
-1.2157 2.0964 5.0482 -0.1285 0.1390 0.0175 0.320 - 0.5633 -0.3386
Temperature range (“C)
-25.482 72351 155.898 %
25-140 25-89.5 6&129 2S42.7 40-51 45-54 25-80 25-M 35-50
0.0 0.0 0.0 0.0
100
60 60
P
z -
60-
40
20
-
0
50
I’
Fig. 1. Experimental and calculated solubility diagram for the KNOS-Hz0 system. Solid phase: KNOJ. Experimental data: Linke and Seidell [6].
Mg(NO,),
(wt.%) ! 51
1
52
Fig. 2. Experimental and calculated solubility diagram for the system. solid phases: Mg(NOA-Hz0 MgPO& .6H20 @I A Mg(NG )x. 2HsO (~2) and Mg(NO,)z. Experimental data: Linke and Seidcll [6].
1200
Et. SANDER et al.
loo
80
e #
60
40
20
0
SO t 1 I I ca(No,), (wt.W ;, A2:,
loo
Fig. 3. Experimental and calculatedsolubihty diagram for
the Ca(NOl)2-Hz0 system.Solid phases:C&NO& .4H20 (s,h WNO.)Z*~HZO (s~h ~WNO,)Z*~HZO (~3) and Ca(lUO~)t. Experimentaldata: Linke and Seidell [6]. 42.7 and Sl.l”C, respectively. Both melting points and cutectic points (equilibrium between two solid phases and one liquid phase) are represented with good accuracy by the model. The good representation of both branches of the Mg(NOJh - 6HaO solubility curve confirms, that the model of [4,5] is applicable at very high molalities. If we look at two points at a given temperature with one point located on each branch, eq. (15) must be valid in each point. The predicted location of the two branches is therefore dependent on the concentration dependence of the activity coefficients. By application of the Gibbs-Duhem equation it can be shown, that the quantity on the left-hand side of eqs (14) and (15) always has a maximum at a liquid phase concentration corresponding to the stoichiometric composition of the solid phase. The congruent melting point temperature is the point where this maximum value is equal to the thermodynamic solubility product. Cation-cation interaction parameters have been estimated from ternary SLE data by minimization of the objective function defined in eq. (16). A list of
references for the experimental data used for parameter estimation is given in Table 2. The estimated parameters are presented in Table 3. The a&j parameters for K+-Ca2+ and Mgz+-Ca2+ interactions have retained the previously assigned values of 2500 K. This shows, that in these cases the “like-ion repulsion” principle is applicable also in mixed electrolyte solutions. The experimental and calculated phase diagram for the system KNOs-Mg(NOJ)2-H20 at 25°C is shown in Fig. 4. The solubility curves for the two salts Mg(NOJ)2 - 6Hz0 and KNOJ intersect at the eutonic point, where one liquid phase is in equilibrium with two solid phases. The phase diagrams at 50 and 75°C are represented with comparable accuracy, except in the neighbourhood of the eutonic point at 75°C. The model predicts that the eutonic point with solid phases Mg(NOJ)2 - 4H20 and KNOs has disappeared at this temperature; this is not in agreement with the experimental data. It is, however, correct that the eutonic point disappears at a temperature somewhat below the congruent melting point of Mg(N03)2 - 6HaO at 89.5%.
KNO,
(wt.-W
Fig. 4. Experimental and calculated solid-liquid phase for the KNOI-Mg(NOI)z-H,O system at 25°C. Experimentaldata: Linke and Seidell[6]. 1, Liquid; 2,
diagram
liquid + Mg(NOJ)*. 6H,O; 3, liquid + KNO,; + Mg(NOa)2.6H,O + KNOa.
Table 2. Experimental ternary SLE data sets used for parameter estimation Temperature (“C)
System
25.5475 25.35,50 20 50,60,70,80
Table 3. Cation-tion i K+ Kf Mg= +
Ref.
Linke and Seidell [6] Flatt and Bocherens [7] Silcock [8] Rutkova [9]
interaction parameters
i
uPi (K)
u:, x 10’
&Ii,1.120(K)
dij*HI0 x lo2
Mg= + Ca2’ Ca’+
1219.1 2500 2500
-7715 0.0 0.0
-483 -3680 459
1347 5917 3565
4,
liquid
Solid-liquid equilibriain aqueous solutions of nitratesalts The experimental and calculated phase diagram for the system KNO~-Ca(NOs)l-HsO at 25°C is shown in Fig. 5. The solubllity curves for the two salts Ca(NOa)z -4HzO and KNOJ intersect the solubility curve for the double salt KNOs - Ca(NOs)* - 3H20. Two eutonic points are therefore present in this phase diagram. The solid phase KNO, - 5Ca(NOs)z * 10HzO is not formed at 25°C but starts to appear at about 30°C. This behaviour is correctly predicted by the model. Table 4 shows the experimental and calculated eutonic points at 25, 35 and 50°C. The representation of the MglNO3 )2Ca(NOa)a-H,O system is less satisfactory than that for the other two systems. The experimental observation that the solid phase Ca(NO& * 3H30 appears in the phase diagram at 20°C is not predicted by the model. However, in the concentration range where wt% Ca(N03)2 is less than 30 or wt Y0 Mg(NOs)r is less than 10, the predictions are reliable.
1201
The predicted phase behaviour of the quaternary system KNOs-Mg(NO&Ca(NO~)r-Hz0 is rather complex with several points of equilibrium between three solid phases and one liquid phase. As an example, at 25°C the three solid phases KNOa, Mg(N03)2 - 6HrO and KN0,*Ca(NOJ),*3H,0 are in equilibrium with a liquid phase of the following composition: 9.4 wt % KNOp. 23.7 wt y0 Mg(NO& and 26.5 wt o/0 CalNO,),. No experimental data are available for the quaternary system, and hence the predictions cannot he confkmed. In [4] we compared calculated vapour pressures in the KNO~Xh(NOB)2-Hz0 system at 25°C with an experimental data set. With parameters based on binary information, the root-mean-squared deviation (RMSD) between experimental and calculated vapour pressures is 5.9%. A repetition of this calculation including the ternary parameters estimated from SLE data shows a signifkant improvement of the predicted vapour pressures. The RMSD is reduced to 2.7 %. This result indicates that the extended UNIQUAC equation can represent VLE and SLE simultaneously in highly concentrated mixed salt solutions.
CONCLUSION It baa been shown that the extended UNIQUAC
KNO,
equation can be applied to the calculation of solid-liquid equilibria in the K+-Mg2+-Ca2+NO;-H20 system. A good representation of the ternary phase diagrams can be obtained if interaction parameters between cations of different types are fitted to the experimental data. An exception is those parts of the Mg(NOs)2~(N0,)2-H20 diagrams which are highly concentrated in both salts. An algorithm for the calculation of solid-liquid equilibrium in ionic solutions is given in Appendix A. The number and amount of phases in equilibrium are found by minimization of the total Gibbs function of the system. An extension of the algorithm to include mixed solvents and/or chemical reactions in the liquid phase is straightforward.
(wt.%)
Fig. 5. Experimental and calculated solid-liquid phase diagram for the KNOa-Ca(NO&-Hz0 system at 25°C. data: Flatt and Bocherens Experimental 171. 1. liquid 3, Liquid; 2, liquid + Ca(NO,), .4&O; + KNO, *Ca(NO,)z .3H,O; 4, liquid + KNOs; 5, liquid 6. liquid + Ca(NO,)z -4HrO + KNOa . Ca(NOs)x .3H,O; + KNOs . Ca(NOs)x .3HrO + KNOa.
Table 4. Experimental and calculatedeutonic points in the KNOs-Ca(NOa)2-Hz0 Solid phases+
Temperature (“C)
system
Liquid phase composition (wt %)
Experimental KNOo
G@O,),
calculated KNO,
c@lO,),
1+4 4+5
25 25
10.8 23.0
54.8 44.3
8.8 22.7
56.7 43.6
1+3 3+4 4+5
3: 35
9.0 11.7 26.7
60.2 59.1 45.0
5.2 11.0 26.4
61.8 59.2 44.4
2+3 3+4 4+5
50 50 50
0.6 23.9 33.1
72.8 55.0 47.0
0.2 18.5 31.4
73.2 57.8 46.4
Ca(NOA.4H20; +1, KNO,-Ca(N0&-3Hz0;
Ca(NOs)r-3HzO; 2, 5. KNOo.
3.
KNOa-SCa(NOa
)z. lOH,O;
1202
B. SANDER~~ REFERENCES
Pitzer K_. J. Phys. Chem. 1973 77 268. [:I Meissnu H. P. and Tester J. W., Ind. Engng Chew Proc. Des. Dev. 1972 II 128. Crux J.-L. and Renon I-I.. A.1.Ch.E. J. 1978 24 817. [:I Sander B.. Rasmussen P. and Fredenslund A., Cttem. Engng Sci. 1986 41 1171. CSI Sander B., Fredenslund A. and Rasmussen P., ‘Chem. Engng Sci. 1986 41 1185. II61Linke W. F. and Seidell A., Sofubilities of Inorganic and Metal-Organic Compounds, Vols. I and II. American Chemical !3acicty. Washington 1965. Flatt R. and Bocherens P.. Helv. Chim. Acta 1962 IS 187. 1: SibocL H. L, Solubilities of tnorgcrnie od Organic Cornpounds, Vols. l-3. Pcrgamon Press, oxford 1979. [91 Rutkova V. M., Mokslo Da&u Rinkiniai (L.&t. Zemes Ukio Akurf.) 1976 1 82. cw Smith W. R., Computational aspects of chcmic+l equilibrium in complex systems. In Thewetical Chemistry. Advances and Perspectives. Vol. 5, p_ 115, 1980. Cl11 Michelsen M. L., SEP 8205, Instituttet for Kemiteknik 1982. Cl21 Fletcher R., Practical Methods of Op&dzation. Vol. 1. John Wiley 1980.
APPENDIX
A. AN ALGORlTHM SOLID-LIQUID
FOR CALCUL.ATlON EQUILIBRIUM
OF
A reliable nwzrical method for calculation of solid-liquid equilibrium is a necest~~ requirement for an adequate application of a thermodynamic model for solid-liquid systems, such as the one described in the previous section. As discussed by Smith [lo], the chemical equilibrium problem may be formulated ti several difllerent ways. The various formulations result from the choice of either mole numbers, n, or reaction coordinates. <, as independent variables, and from whether the prdblem is formulated aS a minimization of the total Gibbs function or as an attempt to solve a set of non-linear equations resulting from the chemical equilibrium criteria. As noted by Michelsen (ll], the minimization formulation has the advantage that by a proper choice of minimization method, the solution can be found regardless of the quality of the initial estimate_ This is achieved by assuring that each new iterate corresponds to‘s lower total Gibbs function than the previous one. In a procedure for solving a set of non-linear equations, on the other hand, a poor initial estimate may lead to divergence of the iterations. Since safety is one of our primary aims for an SLE algorithm, the minimization formulation has been chosen for implementation. The total Gibbs function may be minimized by using the set of mole numbers II as independent variables. In this case, the function defined in eq. (3) is minimized subject to a set of linear constraints, resulting from mass balances over each element in the system, and to a set of non-negativity constraints, resulting from the condition of non-negative mole numbers of all species. This is a “non-stoichiometric” type of algorithm in the notation of Smith [lo]. A “stoichiometric” algorithm is obtained by introducing the set of reaction coordinates, C,. and rewriting the total Gibbs function as in eq. (11). The total Gibbs function is . . . ed using e as the vector of independent variables. This is an unc&strained minimization problem, except that the non-negativity constraints must still be satis5ed. Since the stoichiometric formulation involves fewer variables and no
al.
linear constraints, this is the most suitable choice for dur purpose. The mathematical formulation of the chemical equilibrium problem at constant temperature is then: ynF(e)=
E ,‘l
C;lnK,+n~lnal(~> NI+l
+
c i-t
nrLnrrt(t)-
“c” ,=a
C(v1rIne(t))
NI-I-1 +
subjectton;-?
,Fz
.= 1
vl.ln41(t)
vti&20,
(18)
k=l.....NSP.
Partial derivatives of F (e) are given by: (El
\dC,/c+.
=lnK,-
Nlfl vI,lnai
-
x
v,lnu‘.
(19)
I=1
A main problem in the calculation of solid-liquid equilibrium is that the number and types of solid phases at equilibrium are not known in advance, This problem is solved by using a stepwise procedure, where Gibbs function minimization and stability analysis of the obtained solution is used alternately. This leads to the following algorithm:
(1) The temperature and total composition of the system are speci5cd. The thermodynamic solubility products of all possible solid phases in the system are calculated. (2) no is chosen so that C = 0 corresponds to one liquid phase and no solid phases. This implies that only positive values of the elements in C are allowed. (3) A stability analysis is performed. Partial derivatives of F with re+cct to C, [eq. (19)] are calculated for all solid phases. If any of these derivatives arc negative, the total Gibbs function can be lowered by allowing the corresponding solid phase to precipitate, and the system is therefore unstable. If all derivatives are non-negative, the liquid phase is stable. (4) If the sysiem is unstable, it is split into one liquid phase and one solid phase. We choose the solid phase which has the largest negative value of the derivative. The equilibrium composition is found by minimization of the total Gibbs function [eq. (18)]. (5) The equilibrium phases arc tested for stability as described in point (3). If the system is unstable, it is split into one more solid phase, and the total Gibbs function is minimixed. This procedure is repeated until the stable solution has been found. It may happen that a solid phase, which is not present in the final solution, is introduced into the calculations. This is detected in the minimization procedure when C, for a solid phase becomes negative in the iterations. The solid phase in question can then be eliminated from the search. A quasi-Newton method has been chosen for the minimixation problem. This type of method is similar to Newton’s method with line search, except that the inverse Hessian matrix is approximated by a symmetric positive definite matrix a which is updated in each iteration_ Only 5rst derivatives are required. It can be seen from eqs (18) and (19) that 5rst derivatives may be evaluated without any additional computational cost when the value of the objective function has been calculated. The unity matrix is used as an initial estimate of If, and n is updated by application of the Davidson-Fletcher-Powell formula. A description of the numerical method can be found in [ 123 and will not be given here.
1986.
s3.00 + 0.00 ooo9-2509/S6 Pergamon Press Ltd.
CALCULATION OF SOLID-LIQUID EQUILIBRIA IN AQUEOUS SOLUTIONS OF NITRATE SALTS USING AN EXTENDED UNIQUAC EQUATION BO SANDER, PETER RASMUSSEN and AAGE FREDENSLUND Instituttet for Kemiteknik, The Technical University of Denmark, DK-2800 Lyngby, Denmark (Received
3 December
1984 accepted 22 July
1985)
Abstract-An extended UNIQUAC-Debye-Hiickel model is used to predict solid-liquid equilibria for aqueous solutions of nitrate salts with the cations K+, big*+ and Ca2 + . In addition to a model for activity coefficients in mixed electrolyte solutions, the method requires knowledge of the solubility products of all possible solid phases. Furthermore, some ternary solid-liquid equilibrium data are required. It is shown that the extended UNIQUAC equation can be applied to the calculation of solid-liquid equilibria in the K+-Mg”-Ca ‘+-NOT-HZ0 system with good results. A new algorithm for calculation of solid-liquid equilibrium in ionic solutions has been proposed. The number and amount of phases in equilibrium are found by minimization of the total Gibbs function of the system.
IONi
INTRODUCTION
The calculation of solid-liquid equilibria (SLE) is an important area of application of electrolyte solution models. Examples of previous work on salt solubility calculations are given in [l-3]. SLE in the K+-Mg *+-Ca*+-NO;-Hz0 system has not been modelled by any of these methods. In this work we investigate the application of the extended UNIQUAC equation as described in [4] to this system. An attempt was made to include salt solubilities in nitric acid-water-nitrate salt systems, but it has not been possible to correlate VLE and SLE data simultaneously for these systems. The calculation of solid-liquid equilibria requires a good model for the activity coefficients in saturated systems. Furthermore, the number and amount of phases in equilibrium at a given total composition and temperature must be known. An algorithm for this purpose is described.
THERMODYNAMICS
OF SOLID-LIQUID
EQUILIBRIUM
Consider an aqueous solution of NI ionic species in equilibrium with NS solid phases. It is assumed that no solid solutions are formed, i.e. all solid phases have a composition of known stoichiometry. The number of chemical species in the system is NSP = 1+ NI + NS. Index numbers for the species are defined as follows: HsO: 1, ions:i = 2, _ . _ , NI+ 1, solid phases:s = NI NSP. For each of the NS solid phases we +2,..., have an equilibrium of the type: NI+l ~I320
+
C i=2
(ION,&,
h-1
*
(solid);
(H20),1,
r =
. . , NS
for ion i. vt, and vi, are of, respectively,
(1)
water
ing to chemical equilibrium r. The equilibrium composition of the system for a given temperature and pressure is determined by requiring that the total Gibbs function is at a minimum. The total Gibbs function of the system, nG, is given by: NI+I nG
=
nIpI
=
nI
+
c i=2
NSP wI+
(GP +RTlnal)+
c s=NI+2
NI+1
1
RA
(2)
ra,(G:+RTlr@
i=2
NSP +
c s-NI+2
(3)
%G,”
where nL and pk (k = 1, i. s) are the mole number and chemical potential of species k; G “iand G : are standard Gibbs functions for, respectively, water and ion i (unsymmetric convention 151) a, = x*y,. a, = xiy,+, and G,” is the molar Gibbs function for solid phase s. For each of the NS chemical equilibria we introduce a reaction coordinate, c,, so that the mole number of species k is expressed as: n, = a; -
2 Vb<,, r= I
k=l,...,NSP
where nE is the initial mole number of species matrix notation, this can be written: II = IlO-Et
n i=2
1,.
formula
coefficients
and ion i in solid phase s (s = r + 1 + NI) correspond-
NI+I VJONi
is the chemical
the stoichiometric
(4) k.
In (5)
where q and no are (NSP x 1) vectors, ‘j is an (NSP x NS) matrix, and C is an (NS x 1) vector. ‘E is of the 1197
B. SAHDER~~
1198
al.
Equation (9) may now be written as:
form:
nG/RT = constant +
VI2
. . .
vc.*
. . .
v,,
. . .
NI+1 +
C
t=2
NS
c & In K, + n: ln al c= 1
npln*-
y & r=r
vi,lnai
Nlfl +
v=
vas2
. . .
‘dNS
v*.*
. . .
va-NS
0 -1
:::
0
...
(6)
v= -1
0 A necessary condition for a minimum in the total Gibbs function is that the gradient vector is equal to 0:
-1
1
-1
0
[
n;G:+Nzlni’Gf+
(
NE
i=2
C s-NI+2
function
C
nphruI
i=2 NI+l
-
=
F ,=I
e.(
v&a
+
constant -kT
F r=1
C i=2
<,
vi&a,
(8) >
v,,G; c
NI+t +
C i=2
v*Gi’-GT+NI+l >
+nflna,
+
m+i
C
n4lnat
a=2
-,zl
C.(vblnal
+Tz2’
(9)
vhha,).
The thermodynamic solubility product for solid phase (r + NI + I), corresponding to chemical equilibrium r, is defined as: InK, =
Gz+N,+I -vl,Gi
irnil i=2
vtiG:
,p2
up-‘=
K,,
r =
1,. . . , NS
(14)
or n
(xjyF)(“*)
=
K,,
r =
1,. . . , NS. (15)
Equation (14) is the classical formulation of the chemical equilibrium problem Below we will describe an algorithm for SLE calculations which is based on minimization of the total Gibbs function as expressed in eq. (11).
RESULTS
>
NI+l
>
NI+I upr)
n:G.”
vi,G:
+n;lnar+
VA,”
which implies that at equilibrium
(7)
i-2
NSP +
c
(13)
1=2
r-Nx+2
vI,G”I+
R:. ,Fl “(
= 0
Nl-b1
NI+l --
V (nG/RT)
(xlyl)(“lr)
Using eqs (4) and (6), the total Gibbs divided by RT may be written: nG/RT=kT
>
b
6 0 1 20
1 0 1
(11)
vblnui
Differentiation of nG/RT with respect to c, and application of the Gibbs-D&rem equation gives:
c’ and c” indicate different cations and a’ and a” indicate different anions. The lower (NS x NS) matrix is equal to the negative unity matrix. As an example, consider the system HzO, K+, Mg2+, NO:, KNOs, Mg(NOs h - 6H20 for which L is given by:
0
C 1=2
RT.
(10)
AND
DISCUSSION
The prediction of solid-liquid equilibria requires a model for activity coefficients in mixed electrolyte solutions and knowledge of the thermodynamic solubility products of all possible solid phases in the system considered. For representation of SLE in the K+-Mg2+-Ca2+-NO;-H20 system, we will apply the extended UNIQUAC equation as described in [4] with model parameters estimated from binary activity coefficient data. These parameters are presented in the tables in [4]_ From knowledge of the solubilities of single salts in aqueous solutions, values of the thermodynamic solubiity product can be calculated using eq. (15) with activity coefficients calculated from the model. Phase diagrams of aqueous solutions of two or more salts can then be predicted. However, it is not possible to give reliable predictions of ionic activity coefficients in mixed electrolyte solutions from binary information alone. Hence the predicted phase diagrams will not be satisfactory. In [4], u;j, ~1,. S$,,,, and Sb.nIO parameters for cationstion interactions were assigned values of 2500 K, 0,O K and 0, respectively. The representation of ternary SLE data can be significantly improved if the parameters for interactions between cations of
1199
Solid-liquid equilibriain aqueous solutionsof nitratesalts different types are adjusted to the ternary experimental SLE data. Adjustment of these parameters does not influence calculated single salt activity coefficients. For a series of experimental ternary SLE data points at constant temperature with equilibrium between the same solid phase and aqueous solutions of varying composition, eq. (14) is valid in each data point. This fact is the basis for defining the following objective function for parameter estimation: NI+l
vIrlnoI+ F
=
C
vsInai
i=2
s;~.~.t,~
2
NI+1
vl,lnar+ c
vtlnoi
and
c
(16) >>ref
i=2
-(
where c,
1
indicate summa tions over,
respecti;el;f’) solid OFties, temperatures and data points. Subscript r indicates a reference data point, which is here taken as the binary solubifity of the considered salt. Single salt solubihty data used for calculation of thermodynamic solubility products as a function of
temperature have been obtained from the compilation by Linke and Seidell [6-j. The calculated values are represented by the expression: In K, = a, + 9/( T/K) + c, In (T/K)
(17)
a;, 6, and c, parameters are presented in Table 1 along with the temperature ranges of the experimental points. The table includes parameters for the two double salts KNOa _Ca(NOp)2. 3HZO and KN09. SCa(NO& . 10H20. K-values for these salts have been obtained from the same ternary data that has been used for estimation of parameters (see below). Experimental and calculated binary solubility diagrams above 20°C for the three salts KN03, Mg(NOJ)r and CF@IO~)~ in mixtures with water are sho’wn in Figs 1-3. The KNOs-Hz0 diagram is simple solid KNOa. The with only one phase, Mg(NOJ b-Hz0 and Ca(N03 )2-Hz0 diagrams are more complicated due to the occurrence of hydrates Mg(NO& .6Hi?O, Mg(NO,)z *2H,O, Ca(NOJ)2. 4H20, Ca(NO& .3&O and Ca(NO& . 2H,O. All hydrates, except Ca(NO,), . 2H20, are congruently melting with melting points of 89.5, 129,
Table 1. Parametersfor calculationof the thermodynamicsolubigtyproduct on mole fraction basis as a function of temperature by eq. (17) Solid phase KNOs Mg(NO3b-6HzO Mg(NO&.2HzO CWNO3 )1 *4HxO C~OVO&.~HZO ‘3AN03)2-2H20
WNOJ )2 KNOS.Ca(N0~),.3H20 KNO~~5Ca(N0,)a-10H~0
a, 177.49 -487.71 - 1052.0 -3.229 - 10.38 -4.659 - 10.05 2.592 -25.87
b,X 10-b
c,
-1.2157 2.0964 5.0482 -0.1285 0.1390 0.0175 0.320 - 0.5633 -0.3386
Temperature range (“C)
-25.482 72351 155.898 %
25-140 25-89.5 6&129 2S42.7 40-51 45-54 25-80 25-M 35-50
0.0 0.0 0.0 0.0
100
60 60
P
z -
60-
40
20
-
0
50
I’
Fig. 1. Experimental and calculated solubility diagram for the KNOS-Hz0 system. Solid phase: KNOJ. Experimental data: Linke and Seidell [6].
Mg(NO,),
(wt.%) ! 51
1
52
Fig. 2. Experimental and calculated solubility diagram for the system. solid phases: Mg(NOA-Hz0 MgPO& .6H20 @I A Mg(NG )x. 2HsO (~2) and Mg(NO,)z. Experimental data: Linke and Seidcll [6].
1200
Et. SANDER et al.
loo
80
e #
60
40
20
0
SO t 1 I I ca(No,), (wt.W ;, A2:,
loo
Fig. 3. Experimental and calculatedsolubihty diagram for
the Ca(NOl)2-Hz0 system.Solid phases:C&NO& .4H20 (s,h WNO.)Z*~HZO (s~h ~WNO,)Z*~HZO (~3) and Ca(lUO~)t. Experimentaldata: Linke and Seidell [6]. 42.7 and Sl.l”C, respectively. Both melting points and cutectic points (equilibrium between two solid phases and one liquid phase) are represented with good accuracy by the model. The good representation of both branches of the Mg(NOJh - 6HaO solubility curve confirms, that the model of [4,5] is applicable at very high molalities. If we look at two points at a given temperature with one point located on each branch, eq. (15) must be valid in each point. The predicted location of the two branches is therefore dependent on the concentration dependence of the activity coefficients. By application of the Gibbs-Duhem equation it can be shown, that the quantity on the left-hand side of eqs (14) and (15) always has a maximum at a liquid phase concentration corresponding to the stoichiometric composition of the solid phase. The congruent melting point temperature is the point where this maximum value is equal to the thermodynamic solubility product. Cation-cation interaction parameters have been estimated from ternary SLE data by minimization of the objective function defined in eq. (16). A list of
references for the experimental data used for parameter estimation is given in Table 2. The estimated parameters are presented in Table 3. The a&j parameters for K+-Ca2+ and Mgz+-Ca2+ interactions have retained the previously assigned values of 2500 K. This shows, that in these cases the “like-ion repulsion” principle is applicable also in mixed electrolyte solutions. The experimental and calculated phase diagram for the system KNOs-Mg(NOJ)2-H20 at 25°C is shown in Fig. 4. The solubility curves for the two salts Mg(NOJ)2 - 6Hz0 and KNOJ intersect at the eutonic point, where one liquid phase is in equilibrium with two solid phases. The phase diagrams at 50 and 75°C are represented with comparable accuracy, except in the neighbourhood of the eutonic point at 75°C. The model predicts that the eutonic point with solid phases Mg(NOJ)2 - 4H20 and KNOs has disappeared at this temperature; this is not in agreement with the experimental data. It is, however, correct that the eutonic point disappears at a temperature somewhat below the congruent melting point of Mg(N03)2 - 6HaO at 89.5%.
KNO,
(wt.-W
Fig. 4. Experimental and calculated solid-liquid phase for the KNOI-Mg(NOI)z-H,O system at 25°C. Experimentaldata: Linke and Seidell[6]. 1, Liquid; 2,
diagram
liquid + Mg(NOJ)*. 6H,O; 3, liquid + KNO,; + Mg(NOa)2.6H,O + KNOa.
Table 2. Experimental ternary SLE data sets used for parameter estimation Temperature (“C)
System
25.5475 25.35,50 20 50,60,70,80
Table 3. Cation-tion i K+ Kf Mg= +
Ref.
Linke and Seidell [6] Flatt and Bocherens [7] Silcock [8] Rutkova [9]
interaction parameters
i
uPi (K)
u:, x 10’
&Ii,1.120(K)
dij*HI0 x lo2
Mg= + Ca2’ Ca’+
1219.1 2500 2500
-7715 0.0 0.0
-483 -3680 459
1347 5917 3565
4,
liquid
Solid-liquid equilibriain aqueous solutions of nitratesalts The experimental and calculated phase diagram for the system KNO~-Ca(NOs)l-HsO at 25°C is shown in Fig. 5. The solubllity curves for the two salts Ca(NOa)z -4HzO and KNOJ intersect the solubility curve for the double salt KNOs - Ca(NOs)* - 3H20. Two eutonic points are therefore present in this phase diagram. The solid phase KNO, - 5Ca(NOs)z * 10HzO is not formed at 25°C but starts to appear at about 30°C. This behaviour is correctly predicted by the model. Table 4 shows the experimental and calculated eutonic points at 25, 35 and 50°C. The representation of the MglNO3 )2Ca(NOa)a-H,O system is less satisfactory than that for the other two systems. The experimental observation that the solid phase Ca(NO& * 3H30 appears in the phase diagram at 20°C is not predicted by the model. However, in the concentration range where wt% Ca(N03)2 is less than 30 or wt Y0 Mg(NOs)r is less than 10, the predictions are reliable.
1201
The predicted phase behaviour of the quaternary system KNOs-Mg(NO&Ca(NO~)r-Hz0 is rather complex with several points of equilibrium between three solid phases and one liquid phase. As an example, at 25°C the three solid phases KNOa, Mg(N03)2 - 6HrO and KN0,*Ca(NOJ),*3H,0 are in equilibrium with a liquid phase of the following composition: 9.4 wt % KNOp. 23.7 wt y0 Mg(NO& and 26.5 wt o/0 CalNO,),. No experimental data are available for the quaternary system, and hence the predictions cannot he confkmed. In [4] we compared calculated vapour pressures in the KNO~Xh(NOB)2-Hz0 system at 25°C with an experimental data set. With parameters based on binary information, the root-mean-squared deviation (RMSD) between experimental and calculated vapour pressures is 5.9%. A repetition of this calculation including the ternary parameters estimated from SLE data shows a signifkant improvement of the predicted vapour pressures. The RMSD is reduced to 2.7 %. This result indicates that the extended UNIQUAC equation can represent VLE and SLE simultaneously in highly concentrated mixed salt solutions.
CONCLUSION It baa been shown that the extended UNIQUAC
KNO,
equation can be applied to the calculation of solid-liquid equilibria in the K+-Mg2+-Ca2+NO;-H20 system. A good representation of the ternary phase diagrams can be obtained if interaction parameters between cations of different types are fitted to the experimental data. An exception is those parts of the Mg(NOs)2~(N0,)2-H20 diagrams which are highly concentrated in both salts. An algorithm for the calculation of solid-liquid equilibrium in ionic solutions is given in Appendix A. The number and amount of phases in equilibrium are found by minimization of the total Gibbs function of the system. An extension of the algorithm to include mixed solvents and/or chemical reactions in the liquid phase is straightforward.
(wt.%)
Fig. 5. Experimental and calculated solid-liquid phase diagram for the KNOa-Ca(NO&-Hz0 system at 25°C. data: Flatt and Bocherens Experimental 171. 1. liquid 3, Liquid; 2, liquid + Ca(NO,), .4&O; + KNO, *Ca(NO,)z .3H,O; 4, liquid + KNOs; 5, liquid 6. liquid + Ca(NO,)z -4HrO + KNOa . Ca(NOs)x .3H,O; + KNOs . Ca(NOs)x .3HrO + KNOa.
Table 4. Experimental and calculatedeutonic points in the KNOs-Ca(NOa)2-Hz0 Solid phases+
Temperature (“C)
system
Liquid phase composition (wt %)
Experimental KNOo
G@O,),
calculated KNO,
c@lO,),
1+4 4+5
25 25
10.8 23.0
54.8 44.3
8.8 22.7
56.7 43.6
1+3 3+4 4+5
3: 35
9.0 11.7 26.7
60.2 59.1 45.0
5.2 11.0 26.4
61.8 59.2 44.4
2+3 3+4 4+5
50 50 50
0.6 23.9 33.1
72.8 55.0 47.0
0.2 18.5 31.4
73.2 57.8 46.4
Ca(NOA.4H20; +1, KNO,-Ca(N0&-3Hz0;
Ca(NOs)r-3HzO; 2, 5. KNOo.
3.
KNOa-SCa(NOa
)z. lOH,O;
1202
B. SANDER~~ REFERENCES
Pitzer K_. J. Phys. Chem. 1973 77 268. [:I Meissnu H. P. and Tester J. W., Ind. Engng Chew Proc. Des. Dev. 1972 II 128. Crux J.-L. and Renon I-I.. A.1.Ch.E. J. 1978 24 817. [:I Sander B.. Rasmussen P. and Fredenslund A., Cttem. Engng Sci. 1986 41 1171. CSI Sander B., Fredenslund A. and Rasmussen P., ‘Chem. Engng Sci. 1986 41 1185. II61Linke W. F. and Seidell A., Sofubilities of Inorganic and Metal-Organic Compounds, Vols. I and II. American Chemical !3acicty. Washington 1965. Flatt R. and Bocherens P.. Helv. Chim. Acta 1962 IS 187. 1: SibocL H. L, Solubilities of tnorgcrnie od Organic Cornpounds, Vols. l-3. Pcrgamon Press, oxford 1979. [91 Rutkova V. M., Mokslo Da&u Rinkiniai (L.&t. Zemes Ukio Akurf.) 1976 1 82. cw Smith W. R., Computational aspects of chcmic+l equilibrium in complex systems. In Thewetical Chemistry. Advances and Perspectives. Vol. 5, p_ 115, 1980. Cl11 Michelsen M. L., SEP 8205, Instituttet for Kemiteknik 1982. Cl21 Fletcher R., Practical Methods of Op&dzation. Vol. 1. John Wiley 1980.
APPENDIX
A. AN ALGORlTHM SOLID-LIQUID
FOR CALCUL.ATlON EQUILIBRIUM
OF
A reliable nwzrical method for calculation of solid-liquid equilibrium is a necest~~ requirement for an adequate application of a thermodynamic model for solid-liquid systems, such as the one described in the previous section. As discussed by Smith [lo], the chemical equilibrium problem may be formulated ti several difllerent ways. The various formulations result from the choice of either mole numbers, n, or reaction coordinates. <, as independent variables, and from whether the prdblem is formulated aS a minimization of the total Gibbs function or as an attempt to solve a set of non-linear equations resulting from the chemical equilibrium criteria. As noted by Michelsen (ll], the minimization formulation has the advantage that by a proper choice of minimization method, the solution can be found regardless of the quality of the initial estimate_ This is achieved by assuring that each new iterate corresponds to‘s lower total Gibbs function than the previous one. In a procedure for solving a set of non-linear equations, on the other hand, a poor initial estimate may lead to divergence of the iterations. Since safety is one of our primary aims for an SLE algorithm, the minimization formulation has been chosen for implementation. The total Gibbs function may be minimized by using the set of mole numbers II as independent variables. In this case, the function defined in eq. (3) is minimized subject to a set of linear constraints, resulting from mass balances over each element in the system, and to a set of non-negativity constraints, resulting from the condition of non-negative mole numbers of all species. This is a “non-stoichiometric” type of algorithm in the notation of Smith [lo]. A “stoichiometric” algorithm is obtained by introducing the set of reaction coordinates, C,. and rewriting the total Gibbs function as in eq. (11). The total Gibbs function is . . . ed using e as the vector of independent variables. This is an unc&strained minimization problem, except that the non-negativity constraints must still be satis5ed. Since the stoichiometric formulation involves fewer variables and no
al.
linear constraints, this is the most suitable choice for dur purpose. The mathematical formulation of the chemical equilibrium problem at constant temperature is then: ynF(e)=
E ,‘l
C;lnK,+n~lnal(~> NI+l
+
c i-t
nrLnrrt(t)-
“c” ,=a
C(v1rIne(t))
NI-I-1 +
subjectton;-?
,Fz
.= 1
vl.ln41(t)
vti&20,
(18)
k=l.....NSP.
Partial derivatives of F (e) are given by: (El
\dC,/c+.
=lnK,-
Nlfl vI,lnai
-
x
v,lnu‘.
(19)
I=1
A main problem in the calculation of solid-liquid equilibrium is that the number and types of solid phases at equilibrium are not known in advance, This problem is solved by using a stepwise procedure, where Gibbs function minimization and stability analysis of the obtained solution is used alternately. This leads to the following algorithm:
(1) The temperature and total composition of the system are speci5cd. The thermodynamic solubility products of all possible solid phases in the system are calculated. (2) no is chosen so that C = 0 corresponds to one liquid phase and no solid phases. This implies that only positive values of the elements in C are allowed. (3) A stability analysis is performed. Partial derivatives of F with re+cct to C, [eq. (19)] are calculated for all solid phases. If any of these derivatives arc negative, the total Gibbs function can be lowered by allowing the corresponding solid phase to precipitate, and the system is therefore unstable. If all derivatives are non-negative, the liquid phase is stable. (4) If the sysiem is unstable, it is split into one liquid phase and one solid phase. We choose the solid phase which has the largest negative value of the derivative. The equilibrium composition is found by minimization of the total Gibbs function [eq. (18)]. (5) The equilibrium phases arc tested for stability as described in point (3). If the system is unstable, it is split into one more solid phase, and the total Gibbs function is minimixed. This procedure is repeated until the stable solution has been found. It may happen that a solid phase, which is not present in the final solution, is introduced into the calculations. This is detected in the minimization procedure when C, for a solid phase becomes negative in the iterations. The solid phase in question can then be eliminated from the search. A quasi-Newton method has been chosen for the minimixation problem. This type of method is similar to Newton’s method with line search, except that the inverse Hessian matrix is approximated by a symmetric positive definite matrix a which is updated in each iteration_ Only 5rst derivatives are required. It can be seen from eqs (18) and (19) that 5rst derivatives may be evaluated without any additional computational cost when the value of the objective function has been calculated. The unity matrix is used as an initial estimate of If, and n is updated by application of the Davidson-Fletcher-Powell formula. A description of the numerical method can be found in [ 123 and will not be given here.