INTRODUCTION
Fractional crystallization is a separation technique employed to produce a
wide variety of materials, including production of potassium nitrate, potassium
chloride, potassium sulfate, sodium sulfate, boric acid, adipic acid. The technique
consists of sequences of heating, cooling, evaporation, dilution, solventingout,
saltingout and solidliquid separation. Fractional crystallization effects
separations by manipulating the relative solubilities of the components in a
solution by dilution, evaporation, dissolution, stream combination and temperature
change (Ng, 1991; AlHarahsheh and
AlItawi, 2005; Mansouri and Ghader, 2009).
Many articles have been published concerning the synthesis of separation processes
to obtain salts from mixtures of salts. These studies can be classified into
two categories: those that use a phase diagram to identify the types of processes
that may be applied to specific types of solidliquid equilibrium behavior (Ng,
1991; Cisternas and Rudd, 1993; Dey
and Ng, 1995; Berry and Ng, 1996) and those that
use a network flow model between a priori determined thermodynamic states that
allow selection of an optimal structure (Cisternas and Swaney,
1998; Cisternas, 1999; Cisternas
et al., 2001).
When two salts have no ions in common, a double decomposition can occur in
which a pair of salts is reconstituted into another pair. This phenomenon is
known as reciprocal salt pairing, or metathetical salt systems. This type of
system is of interest for the production of various chemical compounds, for
example, the production of potassium nitrate beginning with sodium nitrate and
potassium chloride. Purdon and Slator (1946) describes
precisely how to present such phase equilibrium data in a diagram. These authors
also explained how to present evaporation paths by means of phase diagrams.
A simple description can be found by Mullin (2001). Fitch
(1970) discussed how the phase diagram could be used to represent processes
for fractional crystallization, including the treatment of reciprocal salt pairs.
However, he did not address the problem of synthesizing the process flowsheet.
Ng (1991) presents a synthesis method for separation
of solids based on selective crystallization and dissolution. The method can
be applied to mixtures of solids that do not form double salts and have solubilities
that are monotonically increasing or decreasing functions of temperature. Cisternas
and Rudd (1993) presented a systematic procedure for identification of alternative
process designs for fractional crystallization from solution. They searched
designs that involve low rates of recycle, evaporation and dilution. Dey
and Ng (1995) present guideline for the design of fractional crystallization
processes to separate two and threesolute mixtures. By using solvent addition/removal,
stream combination and cooling/heating these processes bypass regions of multiple
saturation in the phase diagram and recover solute. Design equations are formulated
and the constraints on the design variables are identified. Effect of changes
in the design variables on recycle flows is discussed and the cost of fractional
crystallization separation is estimated.
Berry and Ng (1996) gave a method for the separation
of salts in the pure, solid state for reciprocal salt pair systems. They identified
three classes of separations that could be used to separate simple salts from
a solution. These separation classes could be used as guidelines for the conceptual
design of separation processes for these systems. However, this procedure is
limited, as it is common that metathetical salt systems can have complex behaviors
that do not compare with the examples studied by Berry and Ng, or specific problem
conditions may apply, such as the existence of more than one feed. Also, certain
flow pattern alternatives, such as stream splitting or mixing, may be difficult
to envision within the phase diagram. Cisternas and Swaney
(1998) presented a method to synthesize process flow sheets for separations
of mixtures by fractional crystallization. Using equilibrium data for a candidate
set of potential operating point temperatures, a network flow model is constructed
to represent the set of potential separation flow sheet structures that can
result. By employing specified approaches to multiple saturation point conditions,
linear network constraints are obtained. Solution of the network flow model
shows the optimal mass flow pattern between the candidate equilibrium states
and from this the corresponding process flow sheet is deduced. The method as
presented is generally applicable to problems with two salts and one or more
solvents, including systems forming one or more multiple salts or hydrates.
Then, Cisternas (1999) presented network flow model
for synthesizing crystallizationbased separations for multicomponent systems.
The construction of the network flow is based on the identification of feasible
thermodynamic states. In other words, the synthesis approach proposed is based
upon a flow network superstructure that has embedded streams between several
key states. Optimizing this network through a nonlinear programming formulation,
solution networks are derived that can be transformed to feasible flow sheets.
The method allows consideration of several operation temperatures, complex solidliquid
equilibrium behavior and several multi component feeds and products. For systems
with two solutes a linear programming model is obtained, while for systems with
three or more solutes a nonlinear programming model is obtained. The relative
composition diagram is proposed to determine feasible operation points.
Cisternas et al. (2001) presented a methodology
for the synthesis of fractional crystallization processes with heat integration.
The methodology is based on the construction of three networks. The first network
is based on the identification of feasible thermodynamic states. Using equilibrium
data for a candidate set of potential operation point temperatures, a network
flow model is constructed to represent the set of potential separation flow
sheet structures that can result. In this network the nodes correspond to multiple
saturation points, solute intermediate, process feeds and end products. The
second network is used to represent the variety of tasks that can be performed
at each multiple saturation point. Multiple saturation nodes can be used for
different tasks depending on the characteristic of the input and output streams.
These tasks include cooling crystallization, evaporative crystallization, reactive
crystallization, dissolution and leaching. This multiple task condition for
each equilibrium state is modeled using disjunctive programming and then converted
into a mixed integer program. Heat integration is included using a heat exchanger
network which can be regarded as a transhipment problem.
Thomsen et al. (1998) simulated and optimized
fractional crystallization processes, including reciprocal salt pairs. The analysis
of alternatives that can be obtained by simulation is limited, however, because
a very large number of processing possibilities exists. Cisternas
et al. (2003) presented a method for the design of separation schemes
based on fractional crystallization for systems formed of reciprocal salt pairs.
The method is based on a network model with flows between thermodynamic states.
Using the phase diagram, a method is developed for identification of a set of
solidliquid equilibrium points that are believed to include the best prospects
for operating points. Then, based on these points, only the feasible physical
interconnections are identified, in order to reduce the number of arcs in the
flow network. A mathematical model is developed based on the network flow model
to find the optimal flow pattern for the separation of the reciprocal salt pairs.
A method for determining the desired process flow sheet for fractional crystallization
processes is presented by Cisternas et al. (2004)
that explicitly incorporates the decisions and effects of cake washing operations.
The synthesis model is comprised of coupled networks of four types: the thermodynamic
state network, corresponding task networks, the heat integration network and
the cake wash network. Once the representation is specified, the problem is
formulated and easily solved as a MILP. The sensitivity of the optimal cost
to changes in the values specified for product impurity levels and the level
of residual liquor retained in the cake also is shown. Previous studies present
either process design for fractional crystallization or calculation of material
balance by mathematical programming and a convenient way and detailed presentation
of material balance is not presented.
In this study a convenient method for material balance calculation of reciprocal salt pairs is introduced. All streams in the process are mixtures of ions in water because all salts decompose in water completely. By this method, we obtain ionic balance in process equipment. The resulting ionic flow rates are transformed to flow rates of salts which are reported as results of material balance.
DESCRIPTION OF PROCESS STUDIED
A reciprocal salt pair is formed by dissolving two salts that do not have common
ions in a solvent. Through a metathesis reaction, a total of four simple salts
may precipitate from this solution which is a mixture of ions in solution. For
example, in the reaction between NaNO_{3} and KCl, KNO_{3} and
NaCl can be produced by the following reaction:
NaNO_{3}+KCl→KNO_{3}+NaCl 
Quaternary ionic solutions with two anions and two cations can be represented
in a three dimensional diagram with the anion charge fraction as the xcoordinate,
the cations charge fraction as the ycoordinate and the mole ratio of water
as the zcoordinate. The system can also be represented on a waterfree basis
as a twodimensional diagram named Janecke projection. The quaternary diagram
represents mole compositions of different solutions on a dry basis. For the
system (Na^{+}, K^{+}, Cl¯,)
in water the definitions are (n is number of moles):
In the Janecke projection z coordinate is depicted in separate diagram (Mullin,
2001).

Fig. 1: 
Phase diagram of system ((Na ^{+}, K ^{+}, Cl¯, )
H _{2}O at 20°C calculated by extended UNIQUAC model. Numbers
on intersections and four corners represent moles of H _{2}O/total
moles of ions 
The phase diagram may be considered as visual tool which can be employed to design fractional crystallization processes, for example, information on which salt can be encountered in a given solution as a function of temperature as well as to display the process paths for such processes.
Figure 1 shows the Janecke projection of phase diagram representing
an aqueous system formed by monovalent salts Na^{+}, K^{+},
Cl¯,
and H_{2}O at 20°C. The coordinates of one point in the diagram
are given by the ratio of the concentration of K^{+} to the total cations
concentration (K^{+} + Na^{+}), the ratio of the concentration
of
to total concentration of anions (Cl¯ + )
and by the moles of solvent per total moles of ions. Numbers on intersections
and four corners of Fig. 1 represent moles of H_{2}O/total
moles of ions at equilibrium points calculated by extended UNIQUAC model. The
points A, C, H and J are the saturated solutions of pure salts dissolved in
water. The double saturation points (B, D, G, I) represent solutions saturated
with two salts. Points E and F represent triple saturation points of the quaternary
system. At a given temperature, the double saturation lines divide the Janecke
projection in four single saturation regions, one for each component. Regions
formed by the ABED, BCGFE, DEFIH and FGJI borders represent saturation surfaces
of the KCl, KNO_{3}, NaCl and NaNO_{3} salts, respectively.
The IF, GF, DE, BE and EF borders represent conditions under which the solution
is saturated by two salts.
Table 1: 
Some predictions of extended UNIQUAC model for saturation
lines of Janecke projection at 20°C that are used in Fig.
1 

Not all the combinations are possible and thus no
conditions exist in the phase diagram of Fig. 1 under which
KCl and NaNO_{3} can be cocrystallized. These two salts are termed incompatible salts, while the NaClKNO_{3}
pair, which cocrystallize at border EF, are termed compatible salts. Table
1 shows a few predictions of extended UNIQUAC model of saturation lines
in Janecke projection at 20°C which are used in Fig. 1.
In the phase diagram as presented in Fig. 1 more than one
isotherm must be shown in order to design processes for separation by fractional
crystallization. For this reason, it is more common to present this type of
system by the Janecke projection. In addition to 20°C, case of Fig.
1, the phase diagram at 100°C is also included in Fig.
2. It is possible to design various processing alternatives between these
two temperatures. Assume that the production of KNO_{3} (and thus NaCl)
is desired beginning with KCl and NaNO_{3}. By adding or removing water,
any mixture of KCl and NaNO_{3} can be brought to a position in the
diagram, where only one salt precipitates. By regulating temperature and adding
or evaporating water, a cycle for fractionating the two salts can be found.
One possible process is indicated in Fig. 2 by points O, K,
L and M. Any mixture of KCl and NaNO_{3} lies on the line connecting
them. Process feed is formed by mixing equimolar mixture of KCl and NaNO_{3}
at point O. When mixing two streams represented by two different points in a
quaternary diagram, the resulting stream will have a composition lying on a
line between the two points representing the streams.

Fig. 2: 
Two isotherms and cyclic process for fractional crystallization
of NaCl (at 100°C) and KNO_{3} (at 20°C) 
The feed is mixed with solution M, producing mixture K. The water quantity
is adjusted to precipitate NaCl and generate solution L at 100°C. The water
quantity and temperature in mixture L is different from mixture M. The solution
L is cooled and the water adjusted in order to crystallize KNO_{3} and
produce solution M, which is recycled to the process and mixed with feed. The
solution from which crystallization takes place can be considered as a mixture
of the solid phase and the liquid phase. Hence, the gross composition of the
solution is on a straight line between the solid salt and the solubility curve
corresponding to the salt. Note that Janecke projection shows ionic composition
of solution in equilibrium with salt. Solution L is in equilibrium with NaCl
salt (at 100°C) and solution M is in equilibrium with KNO_{3} salt
(at 20°C). It is clear that more alternatives exist, but for our aim which
is describing material balance in reciprocal salt pair system only one alternative
(i.e., process OKLM in Fig. 2) is sufficient. It should be
noted that all solutions O, K, L, M are ionic mixture in water.
The problem to be solved can be defined in the following way: given the amount of desired product and the conditions of solidliquid equilibrium at two temperatures, find the flow rate of salts and water in streams of the process. It is clear that the phase diagram at different temperatures must be available to calculate solidliquid equilibrium.
CALCULATIONS OF PHASE DIAGRAM BY EXTENDED UNIQUAC MODEL
Design of fractional crystallization processes requires a detailed knowledge
of the phase equilibria for such systems. It is necessary to have phase diagram
of system K^{+}, Na^{+}, Cl¯,
in water at different temperatures. By use of the phase diagram, ionic composition
and amount of water in solution which is in equilibrium with the crystallized
salt can be calculated. The thermodynamic model used to compute phase equilibrium
is the Extended UNIQUAC model described by Thomsen et
al. (1996) and applied for describing SLE, VLE and thermal properties
of aqueous electrolyte systems. Extended UNIQUAC model comprises a DebyeHuckel
term to describe the long range electrostatic forces and a UNIQUAC term to describe
short range interactions. Hence, the excess Gibbs energy expression is represented
by the sum of three terms: a combinatorial or entropic term and a residual or
enthalpic term, as for nonelectrolyte systems (shortrange contribution) and
an electrostatic term (longrange contribution):
The combinatorial and the residual terms are identical to those used in the traditional UNIQUAC equation. The interaction parameters are considered temperature dependent:
which is also used in the case k = l. The only parameters for the Extended
UNIQUAC model are the UNIQUAC interaction parameters and the volume and surface
area parameters. More information can be found in (Thomsen
et al., 1996; Thomsen and Rasmussen, 1999;
Iliuta et al., 2000). Extended UNIQUAC volume
and surface area parameters and interaction parameters are reported in Table
24 (Iliuta et al., 2000).
Table 2 presents the extended UNIQUAC r and q parameters while
Table 3 and 4 present the and
parameters
for calculating extended UNIQUAC interaction parameters, respectively.
The equilibrium between an aqueous phase and the solid salt K_{κ}Aα·nH_{2}O(s) consisting of κ cations K, α anions A and n water molecules can be described by the equation:
Table 3: 
parameters for calculating extended UNIQUAC interaction energy parameters
(Thomsen, 1997) 

Table 4: 
parameters for calculating extended UNIQUAC interaction energy parameters
(Thomsen, 1997) 

At equilibrium, the chemical potential (μ) of the solid salt is equal
to the sum of the chemical potentials of the salt’s constituent parts.
The condition for equilibrium therefore is:
The chemical potential of component i is
is the chemical potential of component i in the standard state of component
i, a_{i} is the activity of component i. The equilibrium condition 7
can then be written as:
Where:
is the solubility product of the salt. The numerical value of the solubility product can be calculated by Eq. 9. The composition of the liquid phase can be calculated by Eq. 10. An m salt saturation point has to fulfill the condition 10 for all m salts:
while for all other salts potentially formed by the system:
In Eq. 11 and 12, v_{k,i} is
the stoichiometric coefficient of component i in salt k. Equation
11 is solved by a full NewtonRaphson method using analytical composition
derivatives of the activity coefficients. The m iteration variables are the
amounts of each of the m salts in solution. The calculation of the points in
quaternary solubility diagrams starts with the points where three solid phases
are in equilibrium with a liquid phase. If the solubility products for all other
salts are greater than the activity products of the salts, the solution represents
a point where three solid phases are in equilibrium with a liquid phase. After
that two salt lines are found. When the ends of a two salts line have been found,
the line is approximated by a straight line providing start guess. After all
points and lines in the diagram have been calculated then calculating the amount
of water necessary to satisfy all solubility products begins (Iliuta
et al., 2000). Extended UNIQUAC model has produced good results for
ionic system K^{+}, Na^{+}, Cl¯, in
water. In the other methods as well as this method, fractional crystallization
processes for the separation of salts belonging to reciprocal salt pairs is
presented based on a flow network model constructed between thermodynamic states.
The state nodes represented in the model include feeds, products, intermediate
products and solidliquid equilibrium states. Flows between the states are represented
by arcs within the flow model. A systematic procedure is given for identification
of the solidliquid equilibrium states that could potentially serve as operating
points. This identification is essential to the construction of the flow network
and reduces the problem to a simple mathematical form. The advantages of the
method lie in its capability to consider general process flow pattern, handle
systems forming double salt, deal with several temperatures as potential operation
conditions and handle several feeds and products.
For comparison of extended UNIQUAC results with experimental data see (Thomsen,
1994, 1997). In order to compare and asses the results
of the extended UNIQUAC model, one may also refer to these references (Thomsen,
2005; Faramaezi et al., 2009; Cisternas
et al., 2001).
RESULTS AND DISCUSSION
Calculations of material balance: The problem that is solved in this
paper can be defined as: A mixture of fixed composition is available (feed consists
of KCl, NaNO_{3} and water). Determine the streams flow rate that yields
the desired product KNO_{3} and byproduct NaCl. In this problem two
crystallizers as well as recycle of mother liquor is required. It is necessary
to calculate ionic composition and water to total ions ratio at three points
in the phase diagram, i.e., points K, L and M, before material balance. At point
L, NaCl and at point M, KNO_{3} crystallizes. Temperature of solutions
L and M are 100 and 20°C, respectively. The composition of solutions K,
L and M are calculated by extended UNIQUAC model and are reported in Table
5. In other words, data reported in Table 5 are coordinates
of points K, L and M in Janecke projection. Table 6 shows
mole fraction of K^{+}, Na^{+}, Cl¯, and
H_{2}O in points K, L and M calculated by data of Table
5. For example is
calculated by:
Table 5 Coordinates of chosen points K, L and M in Janecke projection to calculate material balance and ratio of moles of water to total moles of ions.
The process described in phase diagram is shown in Fig. 3.
The solutions K, L and M in Fig. 2 corresponds to streams
2, 4 and 9 in Fig. 3 (streams are shown in squares). In the
first crystallizer, NaCl forms after adjusting amount of water by removing it
as vapor at 100°C. KNO_{3} crystallizes in the second crystallizer
at 20°C. Water is added to feed of second crystallizer to adjust it to amount
of necessary for crystallization of KNO_{3}. The method of depicting
amount of water above Janecke projection and obtaining amount of water required
for adding to feed of KNO_{3} crystallizer is described by Mullin
(2001).
Simple design of crystallization processes can be achieved by trial and error.
Therefore, a procedure for design of the process was developed.
Table 5: 
Coordinates of chosen points K, L and M in Janecke projection
to calculate material balance and ratio of moles of water to total moles
of ions 


Fig. 3: 
KNO_{3} process labeled with total flow rate of streams.
The numbers in squares are stream numbers 
When the amount
of desired product is chosen and temperature of crystallization of KNO_{3} and NaCl is fixed, the solubility diagram shows the boundaries of operating
region.
As depicted in Fig. 3, NaCl is product of first crystallizer.
It means that the amount of ions K^{+} and that
enters and exits from the crystallizer remains constant. Product of second crystallizer
is KNO_{3} and flow rate of ions Na^{+} and Cl¯ in the
feed and effluent streams of this crystallizer does not change. Therefore, with
regard to Fig. 3 we can write mole balance for K^{+},
Na^{+} and water in NaCl crystallizer as:
Streams 2 and 4 in Fig. 3 are mixtures of ions K^{+},
Na^{+}, Cl¯, in
water and F_{2} and F_{3} are total molar flow rate of ions
and water. Reporting composition of F_{2} and F_{3} in terms
of three salts is described by Mullin (2001). F_{4}
is molar flow rate of water evaporated in NaCl crystallizer.
F_{naCl} is molar flow rate of produced crystals of NaCl which is known. x_{K}+_{,2}, x_{Na}+_{,2} and xH_{2}O,2 are mole fraction of K^{+}, Na^{+}, F_{2} and water in F_{2} (solution K). x_{K}+_{,3}, x_{Na}+_{,3} and xH_{2}O,3 are mole fraction of K^{+}, Na^{+} and water in stream F_{3}. Composition of streams F_{2} (solution K) and F_{3} (solution L) and F_{7} (solution M) are reported in Table 6.
The procedure established for calculation of mole balance is:
Table 6: 
Mole fraction of ions and water in points K, L and M calculated
by data in Table 5 

Table 7: 
Total ionic flow rate (sum of flow rates of ions and water)
of streams shown in Fig. 3 

• 
Guess the total molar flow rate of crystallizer feed (F_{2}) 
• 
Calculate F_{3} by Eq. 14 
• 
Calculate F_{NaCl} by Eq. 15 
• 
Compare F_{NaCl} with known amount of it. If agreement does not
exist, guess a new value for F_{2} and repeat the calculations 
• 
Calculate F_{4} by Eq. 16 
Similar calculations can be performed for second crystallizer knowing the value
of produced KNO_{3}, FKNO_{3}. Mole balance of Na^{+},
K^{+} and water in second crystallizer are:
x_{K}+_{,6}, x_{Na}+,_{6} and xH_{2}O,6 are mole fraction of K^{+}, Na^{+} and water in F_{5}.F_{6} has been determined by adding F_{5} to F_{3}. Therefore, F_{7} can be calculated by Eq. 17.
Amount of water added to mixing tank (F_{1}) can be achieved by water mole balance in mixing tank knowing F_{2} and F_{7}:
The results of handy material balance for production of 12.39 kmol h^{1}
KNO_{3} (FKNO_{3} = 12.39) are reported in Table
7 and 8 which is the total ionic flow rate of streams
shown in Fig. 3. Referring to the stoichiometry of reaction,
production of 12.39 kmol h^{1} KNO_{3} requires 12.39 kmol
h^{1} KCl and NaNO_{3}. The results reported in Table
8 are based on ionic flow rate of solutions. In the other words, the results
are flow rate of ions and water in solutions that are calculated by Eq.
1420.
Table 8: 
Flow rate of ions and water in streams shown in Fig.
3 

Table 9: 
Flow rate of salts and water in streams shown in Fig.
3 calculated by data in Table 8 

These data can be transformed to flow rates of
salts as shown in Table 9.
CONCLUSION
For design of KNO_{3} production process it is necessary to determine
material balance. Thermodynamics of electrolyte mixtures is needed to establish
equilibrium composition of process streams. Extended UNIQUAC model was used
to calculate phase diagram of process containing Na^{+}, Cl¯, ions
in water at different temperatures. Hence, composition of streams in process
was calculated. Based on ion conservation in crystallizers, a simple and handy
method was presented to calculate flow rate of all salts and water.
ACKNOWLEDGMENTS
The authors would like to acknowledge the support of Chemical Engineering Department of Shahid Bahonar University of Kerman for their help throughout this project. This project was started in August 2009 and finished in May 2010.
NOMENCLATURE
a_{i} 
: 
Activity of component I 
G^{E} 
: 
Excess Gibbs free energy (Jmol1) 

: 
Shortrange contribution of excess Gibbs free energy (Jmol1) 

: 
Longrange contribution of excess Gibbs free energy (Jmol1) 
F_{i} 
: 
Total molar flow rate of ions and water in stream i (kmolh1) 
K 
: 
Solubility product 
n 
: 
No. of water molecules in salt 
n_{i} 
: 
No. of moles of component I 
q_{i} 
: 
Surface area parameter of component i in extended UNIQUAC 
r_{i} 
: 
Volume parameter of component i in extended UNIQUAC 
R 
: 
Gas constant (Jmol1K1) 
T 
: 
Temperature (K) 
μ_{kl} 
: 
UNIQUAC interaction parameter (K) 
x_{i} 
: 
Mole fraction of component i 
x_{i,j} 
: 
mole fraction of component i in stream j 
Greek letters:
α 
: 
No. of anions in salt 
κ 
: 
No. of cations in salt 
μ_{i} 
: 
Chemical potential of component i (Jmol1) 

: 
Chemical potential of component i at standard state (Jmol1) 
v_{k,i} 
: 
Stoichiometric coefficient of component i in salt k 