Thermodynamic Analysis of Solid–Liquid Phase ... - ACS Publications

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Thermodynamic Analysis of Solid
Liquid Phase Equilibria of Nitrate Salts Scott M. Davison* and Amy C. Sun Chemical and Biological Systems, Sandia National Laboratories, Albuquerque, New Mexico 87185, United States ABSTRACT: In this work, we analyze solid
liquid phase equilibria of molten nitrate salt mixtures. Molten salts are used as heat transfer fluids within concentrated solar power systems. Further understanding of the thermophysical properties of the salt solutions is integral to designing the newest generation of solar power systems. We make use of classical thermodynamics to quickly model the phase equilibrium of mixtures of nitrate salts. This modeling work can serve as a complement to existing experimental efforts in identifying appropriate multicomponent salt mixtures for solar power applications. We present phase calculations of ternary and quaternary mixtures of LiNO3, NaNO3, KNO3, and CsNO3 modeled using the Wilson equation for liquid phase activity coefficients and binary solid
liquid equilibrium data.

’ INTRODUCTION Solutions of molten salts are commonly used as working fluids in concentrated solar power (CSP) applications.1,2 Mixtures of salts are used to take advantage of the range of temperature over which they are liquid and favorable heat capacity for utilizing solar energy and storage.3 A typical commercial salt mixture currently used in CSP applications is HITEC solar salt, a mix of NaNO3 and KNO3.4 This simple two salt mixture is widely used and meets basic requirements necessary for a typical CSP tower application. With new solar technologies being developed and updates to existing tower requirements, it is desirable to study additional mixtures of salts to produce an improved working fluid. The working fluid property under investigation in this Article is the solid
liquid transition temperature. A key determinant for assessing CSP applications, especially in distributed situations like a parabolic trough installation, is the solid
liquid transition temperature. The salt mixture is no longer useful if it solidifies anywhere in its path through the solar power system. Preliminary experiments indicate that incorporating additional salts, such as LiNO3 or nitrites, to the existing mixture can lower the solidification temperature.5,6 Experimental determination of the solidification temperatures of a wider range of mixtures is time-consuming and costly. Utilizing a numerical tool to estimate the solidification temperature would allow the study of a wide range of salt mixtures in a timely fashion and provide candidate mixtures for testing. A numerical thermodynamic approach was used to explore the solid
liquid phase equilibria of mixtures of nitrate salts. A macroscopic, thermodynamic methodology was chosen as an expedient complement to either a molecular dynamics numerical method or an experimental exploration. A wide range of mixing models used to represent the behavior of nonideal mixtures can be found in Prausnitz et al.7 Possible mixing models include the van Laar equation,8 the Margules equation,9 the Wilson equation,10 the nonrandom, two liquid (NRTL) equation,11 the UNIQUAC equation,12 and the quasi-chemical or lattice theory equation.13 The work in this Article is based upon using the Wilson equation14 to calculate solid
liquid phase equilibria. Wilson’s formulation r 2011 American Chemical Society

was chosen because it contains only two fitting parameters, is easily adaptable to mixtures beyond two components, and had previously been successfully applied to mixtures of alkali halide salts.15,16

’ THEORY The thermodynamic theory used to estimate solid
liquid phase equilibrium of nitrate salts is based upon the Wilson equation for the excess Gibbs energy of mixing.14 The mixtures of nitrate salts in this study are nonaqueous, so the mixture components remain neutral and nondissociating throughout the calculations. The primary focus of the study lies in determining the solid
liquid phase equilibrium. The solid and liquid phase equilibrium condition at constant temperature and pressure is the equality of fugacities for each component in the mixture, that is: liquid

fisolid ¼ fi

ð1Þ

The solid is assumed to consist of pure solute. The liquid fugacity at equilibrium conditions is further broken down into a product of the reference state fugacity, f0i , the activity coefficient, γi, and mole fraction, xi: liquid

fisolid ¼ fi

¼ γi xi fi0

ð2Þ

Here, we define the reference state as the pure solute liquid at its subcooled temperature. The ratio of fugacities for a solute is also related to the free energy change, Δgi, during the solid
liquid equilibrium: Δgi ¼ RT ln

fi0 fisolid

¼ RT ln

fi0 1 0 ¼ RT ln γi xi γi xi fi

ð3Þ

Here, R is the universal gas constant and T is the temperature. Received: March 22, 2011 Accepted: September 21, 2011 Revised: September 7, 2011 Published: October 26, 2011 12617

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The free energy change during a solid
liquid transition of a pure component is measurable in a laboratory. The free energy change during phase transition depends on the heat of fusion, Δfushi, the isobaric heat capacity change between solid and liquid phases, ΔCp,i, and the triple point temperature, Tt. Following Prausnitz et al.7 and Knapp et al.,17 the change in Gibbs energy can be written:
 Z T ΔC Z T Δf us hi T Tt p,i
1
T dT þ Δgi ¼ ΔCp,i dT Tt T Tt T Tt ð4Þ Simplifications to eq 4 can be made when the heat capacity change is negligible18 or when the temperature dependence of the heat capacity change is small.7 The molten salts of interest in this Article have a negligible change in heat capacity between the solid and liquid phases. The two terms containing ΔCp,i are neglected in this work due to the negligible magnitude of these terms. Second, the melting point temperature (Tm) and the heat of fusion at the melting temperature are substituted for the properties at the triple point given the small differences among properties at these two points. The two expressions for the free energy change during transition can be combined to associate pure component phase transition (eq 4) to the solubility of the same component in a multicomponent mixture (eq 3):
 1 Δf us hi Tm
1 ¼ ln ð5Þ γi xi RTm T Thermodynamic Method
Binary System. The activity coefficient, γi, and solubility, xi, are unknowns in the combined free energy change expression. The Wilson equation14 was employed herein to calculate the activity coefficients needed for mixtures of nitrate salts. The Wilson equation to calculate a component’s activity coefficient is 1 0

ln γk ¼
lnð

m

∑

j¼1

xj Λkj Þ þ 1


C B B xi Λik C C Bm A i¼1 @ xj Λij m

∑

∑

ð6Þ

j¼1

where γk is the activity coefficient of component k, xi is the mole fraction of component i, and Λij and Λji are defined as the interaction coefficients of i
j and j
i pairs. Binary interaction coefficients represent relative affinities of dissimilar molecules and can be defined as functions of molar volume, ν, and pairwise interaction energy, λ:7 ! νj λij
λii Λij  exp
νi RT ! ð7Þ λji
λjj νi Λji  exp
νj RT The terms in eq 7 can be used in two methods for fitting the numerical prediction to experimental results. In one scheme, the two parameters would be the quantities: λij
λii and λji
λjj. Using the numerators in eq 7 as adjustable parameters, the interaction coefficients (Λij and Λji) become temperature dependent. The other scheme is to use the interaction coefficients (Λij and Λji) as fitting parameters. This numerical scheme is

Figure 1. Schematic illustration of the quasi-binary approach. The binary phase boundaries are calculated along a line of constant r, where r = (a/b) = (c/d) = (x2)/(x3). The complete ternary space is mapped by varying the value of r three times, once with each component acting as the free component.

simpler to implement but removes the temperature dependence from the fitting terms. Both schemes were implemented and provided no difference in the fit. In this Article, the simpler numerical scheme of fitting the interaction coefficients (Λij and Λji) was used to fit the numerically predicted liquidus line to experimental, binary mixing temperature
mole fraction (T, x) results. Thermodynamic Method
Ternary System. Once the interaction coefficients are determined, through comparison to binary experimental data, they can be utilized to predict the equilibrium properties of ternary or higher order mixtures, an advantage of the Wilson equation. The quasi-binary approach of Nakanishi15 is adopted to extend the binary mixture method to ternary mixtures. The primary reason behind utilizing the quasibinary approach is that it reduces the dimensionality of ternary phase space by defining a pseudocomponent. A pseudocomponent is a fixed ratio of two of the three components in the mixture, designated as r. For a given r, the search for the phase boundary sweeps through a pseudobinary line while varying the composition of the third component from zero to one. Subsequently, a series of pseudobinary lines are calculated to generate a surface representing a ternary phase boundary. This process is repeated for each of the other two components by rotating the component indices to eventually obtain three surfaces across the ternary space. The lines and the point where these surfaces intersect are eutectic compositions in ternary mixtures. The quasi-binary approach provides a convenient numerical scheme to sweep through the possible combinations of components. It also produces numerical prediction data of phase equilibria in a form that is easily interpreted and graphed according to standard phase diagram conventions. The quasi-binary approach is shown schematically in Figure 1. We calculate phase boundaries explicitly using the slope equation for binary eutectic mixtures. Integration of the slope equation is carried out:19 ! ∂xj 1 xj ΔH̅ j j ¼ α ð8Þ g̅ jj RT 2 ∂T p where ΔH̅ j ¼ Δf us hj
Cp, j ðT
Tt Þ
RT ln γαj g̅ 12618

α jj


¼

xj RT



∂μαj ∂xj

! ¼1 þ T,P

∂ ln γαj ∂xj

ð9Þ ! ð10Þ T,P

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Industrial & Engineering Chemistry Research In these equations, ΔH j is the heat of solution of a solid into a regular solution for component j in a multicomponent system, xj is the mole fraction of component j, g αjj is the partial derivative of chemical potential with respect to mole fraction of component j in phase α, and γαj is the activity coefficient of component j in phase α. The activity coefficient is the only unknown in eq 8 and can be calculated using the Wilson equation (eq 6) expanded to multicomponent mixtures. The activity coefficient is calculated directly by eq 6 using the binary interaction coefficients calculated from the binary mixtures. For application in the quasi-binary approach, the activity coefficient expression is expanded and r substituted for the ratio of two of the components. An example for the case of γ1, where r = x2/x3, is: 2 13 0
 6 C 7 6 1
x1 B C7 B x1 Λ11  þ rΛ12 þ Λ13 C7 B
ln γ1 ¼ 1
ln6 4 r þ 1 @ 1
x1 A5 r þ 1
 r þ 1 x1 Λ11 1
x1 

r þ 1 x1 Λ11 þ rΛ12 þ Λ13 1
x1 rΛ21 

r þ 1 x1 Λ21 þ rΛ22 þ Λ23 1
x1 Λ31 

ð11Þ r þ 1 x1 Λ31 þ rΛ32 þ Λ33 1
x1 Equations 8 and 11 are used to sweep over the ternary mixture space and determine the solid
liquid phase equilibrium of the ternary mixture. Thermodynamic Method
Quaternary System. Extending the numerical method from three to four components is analogous to binary extension to ternary; a reduction of dimensionality is required. In a quaternary phase diagram, the four components

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Figure 2. Diagram illustrating the ternary slices taken while holding CsNO3 constant. Each slice is a ternary mixture that is modeled using three sweeps along lines of constant r as in the previous ternary method.

are represented by corners of a pyramid. The quaternary phase space can be resolved by a series of slices where one component is held fixed. Each of these slices represents a ternary mixture and can be swept by three sets of lines of constant r as was done previously for the ternary space. A series of ternary slices at a constant value of the fourth component are taken, and this process is repeated by holding each of the other remaining components. A diagram illustrating a series of slices with CsNO3 held constant is shown in Figure 2. The Wilson equation for activity coefficients (eq 6) is readily scaled to four components, but with four components the equation can not be simplified down to one mole fraction and r. At any point in a quaternary system, the sum of the components is one (x1 + x2 + x3 + x4 = 1). For the designated slice, one component is held constant, given x4 = xfix, x1 + x2 + x3 = 1
xfix. Again, the designated slice is swept in a similar fashion as sweeping a ternary phase space using pseudobinary components. An example of the activity coefficient for the case of γ1 with r = x2/x3 is:



   1
xf ix
x1 1
xf ix
x1 Λ12 þ Λ13 þ xf ix Λ14 ln γ1 ¼ 1
ln x1 Λ11 þ r r þ 1 r þ 1




 1
xf ix
x1 Λ21 r þ 1

 1
xf ix
x1 1
xf ix
x1 Λ22 þ Λ23 þ xf ix Λ24 þ r r þ 1 r þ 1 r

x1 Λ11 

1
xf ix
x1 1
xf ix
x1 Λ12 þ Λ13 þ xf ix Λ14 þ r x1 Λ21 r þ 1 r þ 1
 1
xf ix
x1 Λ31 x Λ r þ 1


 f ix 41

1
xf ix
x1 1
xf ix
x1 1
xf ix
x1 1
xf ix
x1 Λ32 þ Λ33 þ xf ix Λ34 Λ42 þ Λ43 þ xf ix Λ44 þ r x1 Λ41 þ r r þ 1 r þ 1 r þ 1 r þ 1



x1 Λ11


x1 Λ31

ð12Þ The above quaternary version of the Wilson equation (eq 12) is used in conjunction with the slope eq 8 to generate equilibrium curves for each value of constant r. Once the pseudo binary boundary lines are obtained, the ternary boundary surface follows, and then the quaternary boundary volume can be generated. As in the case of the ternary phase diagram, where the eutectic region is represented by lines where the surfaces intersect, the

phase boundary for a quaternary system consists of surfaces that result from the intersection of the four boundary volumes. This is illustrated further in the Results and Discussion.

’ RESULTS AND DISCUSSION We applied the thermodynamic method based on the Wilson equation to resolve solid
liquid phase equilibrium of binary, 12619

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Table 1. Pure Component Properties of Nitrate Salts Δfush (cal/mol)

Tmelt (°C)

LiNO3

5.96

255

NaNO3

3.70

307

KNO3

2.41

337

CsNO3

3.25

407

Table 2. Binary Interaction Coefficients Λij

Λji

LiNO3
NaNO3

2.221

0.5753

LiNO3
KNO3

3.157

2.363

LiNO3
CsNO3

4.281

4.438

NaNO3
KNO3

0.5931

1.021

NaNO3
CsNO3

1.906

1.278

KNO3
CsNO3

1.211

0.5010

Figure 5. Comparison of experimental binary phase diagram23 to calculated phase equilibrium for the LiNO3
CsNO3 system.

Figure 3. Comparison of experimental binary phase diagram22 to calculated phase equilibrium for the LiNO3
NaNO3 system.

Figure 6. Comparison of experimental binary phase diagram24 to calculated phase equilibrium for the NaNO3
KNO3 system.

Figure 4. Comparison of experimental binary phase diagram22 to calculated phase equilibrium for the LiNO3
KNO3 system.

Figure 7. Comparison of experimental binary phase diagram23 to calculated phase equilibrium for the NaNO3
CsNO3 system.

ternary, and quaternary mixtures consisting of LiNO3, NaNO3, KNO3, and CsNO3 salts. Rapid numerical estimation of the

melting point properties would aid in selection of mixtures for further experimental study. Binary interaction coefficients 12620

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(Λij and Λji) are fitted against experimental data to obtain the best numerical estimation of binary mixture phase boundaries. The interaction coefficients are then used to estimate the behavior of ternary and quaternary salt mixtures. Binary Nitrate Salts. The first step in the numerical process is to determine the interaction coefficients that best estimate existing binary mixture results. Pure component properties of the salts are necessary to calculate the estimated binary behavior. The thermodynamic properties needed for the pure salts are the heat of fusion, melting temperature, and change in heat capacity between the solid and liquid phases. For the nitrate salts of interest in this Article, the change in heat capacity was negligible. The values of heat of fusion and melting temperature for the nitrate salts are reported in Table 1.20 The pure component properties are used with eqs 3, 4, and 6 to calculate the liquid phase activity coefficients and molar-based solubilities. The predicted binary mixing curve is fitted with experiment curves by adjusting the binary interaction coefficients. Computation of the concentration
temperature curves is carried out using a FORTRAN program. The best match is determined using an optimization program Dakota.21 Dakota is an open-source software package developed at Sandia National Laboratories used for optimization, uncertainty quantification, and sensitivity analysis. Here, it is used solely to optimize the agreement between the numerical estimation and the experimental results. The quantity that represents the numerical/ experimental agreement is composed of the sum of the squared difference between the temperatures at each concentration where

there was experimental data (error1) and the squared difference in the location of the eutectic point (both temperature (error2) and concentration (error3)). The error quantity minimized is a weighted sum of the three individual differences: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi datapoints ðTcalc ðnÞ
Texp ðnÞÞðTcalc ðnÞ
Texp ðnÞÞ error1 ¼ ðTcalc ðnÞÞðTcalc ðnÞÞ n¼1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eutectic eutectic eutectic eutectic ðTcalc
Texp ÞðTcalc
Texp Þ error2 ¼ eutectic eutectic ðTcalc ÞðTcalc Þ ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eutectic eutectic eutectic ðxcalc, 1
xeutectic exp, 1 Þðxcalc, 1
xexp, 1 Þ error3 ¼ eutectic ðxeutectic calc, 1 Þðxcalc, 1 Þ error4 ¼ 2:0error1 þ error2 þ error3

∑

ð13Þ The interaction coefficients that minimized the error quantity were taken to produce the best binary fit. The interaction coefficients that produced the best binary fit are listed in Table 2. The binary curves that are produced using these interaction coefficients are presented in Figures 3
8 and compared to experimental data drawn from the literature.22
25 The agreement between the numerical predictions and the experimental results is very close when simple eutectic behavior is exhibited. The eutectic composition near a 50/50 mixture is observed for binary mixtures of LiNO3 with NaNO3 and KNO3 as well as NaNO3 with KNO3 and CsNO3. In addition to capturing the eutectic composition and temperature, the calculated binary curves match the pure component melting points and agree with the slope of the experimental curves. However, there are certain characteristics of the experimental data that cannot be easily captured when LiNO3 is mixed with KNO3 or CsNO3. These systems display a double eutectic point behavior near the minimum melting point. The Wilson equation, with two fitting parameters, cannot reproduce the region between two eutectic points because this requires a different range of coefficient values. In the LiNO3
KNO3 system, the double eutectic is over a small concentration range, and the base estimation method captures the overall behavior closely. However, this potential nonideal behavior for the LiNO3
CsNO3 system is over a larger concentration range and may be attributed to complexation between LiNO3 and CsNO3 solids. The binary estimation shown in Figure 5 was nevertheless used in higher order mixtures; and the results will be discussed in the next two sections. While a better binary fit can be achieved by incorporating different interaction coefficients in the concentration regime between the eutectic points, this adaptation is difficult to incorporate

Figure 8. Comparison of experimental binary phase diagram25 to calculated phase equilibrium for the KNO3
CsNO3 system.

Table 3. Comparison of the Numerical Binary Eutectic Point to the Experimental Binary Eutectic Pointa composition experimental x1

a

numerical x1

temperature error %

experimental T (°C)

numerical T (°C)

error °C

LiNO3
NaNO3

0.54

0.52

3.7

194

188

LiNO3
KNO3

0.41

0.43

4.6

132

134

2

LiNO3
CsNO3

0.42/0.56

0.49

170/160

132

N/A

NaNO3
KNO3

0.50

0.47

6.0

221

210

11

NaNO3
CsNO3

0.56

0.56

0.0

191

191

0

KNO3
CsNO3

0.60

0.61

2.0

230

231

1

N/A

6

The composition is measured in mole fraction of the first component of the listed binary pair. 12621

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Figure 9. Comparison of (a) experimental27 and (b) calculated phase diagrams of the LiNO3
KNO3
NaNO3 ternary system. Isotherms are in degrees Celsius.

Figure 10. Comparison of (a) experimental28 and (b) calculated phase diagrams of the NaNO3
KNO3
CsNO3 ternary system. Isotherms are in degrees Celsius.

in higher order estimations because this requires additional pairwise parameter estimations for every pair that exhibits more than one eutectic point (i.e., eq 6). In addition, this would be inconsistent with the pure-component boundaries as the composition of one component approaches 0 or 1. In the binary mixtures, excluding the LiNO3
CsNO3 system, the predicted minimum melting temperature and concentration are in agreement with the experimental results. A comparison of the numerical and experimental eutectic points is in Table 3. The calculated binary curves show good agreement with the experimental curves with a maximum difference in composition

of 6% and melt temperature of 11 °C. The range of predicted melt temperatures (231 °C for KNO3
CsNO3 to 134 °C for LiNO3
KNO3) that exists demonstrates wide availability of property variation within the nitrate salts and motivates study of higher order mixtures. Ternary Nitrate Salts. In the absence of experimental data, the Wilson equation is further tested with mixtures involving three salts. In this case, the interaction coefficients are fixed while phase boundaries are resolved followingx the slope equation. The slope equation is obtained by solving the equilibrium condition in differential form.26 12622

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Figure 11. Comparison of (a) experimental29 and (b) calculated phase diagrams of the LiNO3
NaNO3
CsNO3 ternary system. Isotherms are in degrees Celsius.

Table 4. Comparison of the Composition at the Experimental and Numerical Ternary Eutectic Pointa experimental

numerical x1

x2

x3

max error

x1

x2

x3

LiNO3
NaNO3
KNO3

0.38

0.17

0.45 0.36 0.16 0.48

NaNO3
KNO3


0.31

0.37

0.32 0.36 0.34 0.30

16

0.47

0.17

0.36 0.38 0.21 0.41

23

% 6.7

CsNO3 LiNO3
NaNO3
CsNO3 LiNO3
KNO3


N/A

N/A

N/A

0.37 0.36 0.27

N/A

CsNO3 a

The composition is measured in mole fraction of the components in the order listed.

Figure 12. Calculated phase diagram of the LiNO3
KNO3
CsNO3 ternary system. Isotherms are in degrees Celsius. No experimental results are available for comparison.

In the binary case, the eutectic point is the intersection of two solid
liquid equilibrium lines descending from the pure component melting points. The lines are monotonically decreasing from respective pure component melting points until the intersection occurs. Any temperature lower than the eutectic point (the intersecting point) is not a realistic phase boundary. In the ternary system, where an additional degree of freedom is available, the minimum phase equilibrium is located at the intersection of three monotonically decreasing planes from each of the melting points. Each of the planes is formed by calculating the binary curves along lines of constant r as shown in Figure 1. Once the three separate planes are calculated, the data are combined into a ternary phase diagram using a custom program written in

Matlab (Mathworks, Natick, MA). The Matlab program examines the data making up each plane and at points where the planes intersect. The lower temperatures past the intersecting lines are not used. This method produces a unified ternary phase diagram for the mixture. The resulting phase diagrams are shown in Figures 9
12 and compared to experimental results.27
29 In the three systems where experimental results are available, the numerical estimates of ternary phase diagrams correctly capture the phase equilibrium behavior of the salt mixtures. The ternary eutectic points predicted by the numerical method are compared to the published experimental results.27
29 These results are presented in Tables 4 and 5. The LiNO3
NaNO3
KNO3 ternary eutectic point is predicted with similar accuracy as the binary pairs with a difference in composition of 6.7% and melt temperature of 9 °C. The melt temperatures of mixtures of NaNO3
KNO3
CsNO3 and LiNO3
NaNO3
CsNO3 show good agreement with experimental 12623

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Table 5. Comparison of the Temperature at the Experimental and Numerical Ternary Eutectic Point experimental

numerical

error

T (°C)

T (°C)

°C

120 154

129 153

LiNO3
NaNO3
CsNO3

130

133

3

LiNO3
KNO3
CsNO3

N/A

115

N/A

LiNO3
NaNO3
KNO3 NaNO3
KNO3
CsNO3

9 1

results, with differences of 1 and 3 °C. However, the composition of the eutectic point for these two systems differs by an increasing amount, up to 16% for a mixture of NaNO3
KNO3
CsNO3 and up to 23% for a mixture of LiNO3
NaNO3
CsNO3. The larger difference in composition of the LiNO3
NaNO3
CsNO3 mixture results from the nonideal binary boundary (double eutectic) along the LiNO3
CsNO3 side of the ternary diagram. Even with these larger differences in location of the specific eutectic point, the phase diagrams produced are in close agreement with the experimental diagrams, allowing for a reasonable starting point for more in-depth studies of the salt mixtures. It is also apparent (Table 5) that the addition of a third nitrate salt has the potential to lower the melting point of the system. The binary mixtures had melting points ranging from 134 to 231 °C. Two of the ternary mixtures had a similar minimum of around 130 °C, while the LiNO3
KNO3
CsNO3 mixture had a minimum melting temperature of 115 °C. Quaternary Nitrate Salts. While extending the ternary phase diagram to quaternary systems is straightforward, as described in an earlier section, the visualization becomes a challenge. With an additional degree of freedom, the quaternary phase diagram represents the intersections of four volumes, each volume generated from a series of ternary phase planes originated from a vertex. The intersection of two volumes forms a plane, but four planes intersect at a point. Similar to the ternary case, a Matlab program examines the overlap of the volumes and retains the portion with the higher temperatures. The region surrounding the quaternary eutectic point is displayed in Figure 13 using slices where KNO3 is held constant. The diagrams are shaded with colors representing the temperature. In the previous ternary diagrams, isotherms were used. In the quaternary diagram, only portions of the composition space are displayed, and so temperature shading is used because the isotherms would not include sufficient detail in the small composition region. The estimated quaternary eutectic point is at mole fractions of 0.28 LiNO3, 0.14 NaNO3, 0.33 KNO3, and 0.25 CsNO3. The melting point of the quaternary mixture at that point is 100 °C. The melt temperature of a mixture of nitrates salts is reduced by adding further components to the system. A binary mixture melts as low as 134 °C, a ternary mixture could lower the melt tem to 115 °C, and a quaternary mixture reaches 100 °C. While there are no publically available experimental data for the predicted quaternary mixture melting point in this Article, one known quinary mixture melting point has been reported.30 The reported quinary mixture adds Ca(NO3)2 to the quaternary mixture studied herein, and the reported composition is consistent with the findings in this work. The reported experimental quinary composition is compared to the predicted quaternary composition in Table 6. Both the quaternary and the quinary compositions include NaNO3 in minimal amounts and amounts of KNO3 and CsNO3 that are roughly equal. The numerical quaternary prediction contains a larger amount of LiNO3 than the experimental quinary composition,

Figure 13. Illustration of the quaternary eutectic point. (a) Ternary phase diagram with KNO3 held constant at 0.275. (b) Ternary phase diagram with KNO3 held constant at 0.325. (c) Ternary phase diagram with KNO3 held constant at 0.375. The liquidus temperature is shaded in degrees Celsius. 12624

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Table 6. Comparison of the Composition of the Numerically Predicted Quaternary Mixture and the Experimentally Determined Quinary Composition30a component

a

numerical quaternary

experimental quinary

composition, mole %

composition, mole %

LiNO3

28

15

NaNO3

14

10

KNO3

33

30

CsNO3 Ca(NO3)2

25 0

30 15

All quantities are reported in mole %.

although the difference is similar to the amount of Ca(NO3)2 included in the quinary mixture.

’ SUMMARY In this work, a method for estimating the phase equilibrium behavior of mixtures using the Wilson equation has been presented. The method has been applied to binary, ternary, and quaternary mixtures involving LiNO3, NaNO3, KNO3, and CsNO3. The numerical predictions demonstrate reasonable agreement with the existing experimental results for the binary and ternary systems. The prediction of the quaternary mixture melting point provides a promising guide for future experimental work because the predicted melting point is lower than previous mixtures. The method presented can be used to rapidly produce estimates of phase equilibrium behavior to guide experimental study. Furthermore, the predicted behavior can be combined with other mixture property estimates and material costs to develop an optimal salt mixture for a specified application. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

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