The Laliberté–Cooper density model: Self-consistency and a new method of parameterization

Available online at www.sciencedirect.com

Fluid Phase Equilibria 266 (2008) 14–20

The Lalibert´e–Cooper density model: Self-consistency and a new method of parameterization Jacob G. Reynolds a,∗ , Robert Carter b a

b

URS-Washington Division, 723 The Parkway, Richland, WA 99352, United States Energy Solutions, 2345 Stevens Drive, Suite 240, Richland, WA 99354, United States

Received 2 November 2007; received in revised form 4 January 2008; accepted 4 January 2008 Available online 1 February 2008

Abstract The recently published Lalibert´e–Cooper model predicts the density of multi-component aqueous electrolyte solutions as a function of composition using coefficients derived from single electrolyte solutions. In this model, an individual electrolyte is a cation and anion pair, and it has not been clear if the model results depend on the user’s choice of how the cations and anions are paired together in multi-ion mixtures. The present paper investigates the self-consistency of the Lalibert´e–Cooper model when cations and anions are paired in different ways. The densities of a number of mixtures of monovalent cations and anions were modeled. The resulting densities were insensitive to the way the cations and anions were paired up to the 5th significant figure in the density for all but the most concentrated solutions. These results confirm that the Lalibert´e–Cooper model is self-consistent. This conclusion affords the opportunity to derive model coefficients for unmeasured electrolytes when coefficients for other electrolytes are available that contain the cation and anion in the electrolyte of interest. An example was performed where it was shown that the model coefficients for KAl(OH)4 could be calculated solely from the coefficients for NaAl(OH)4 , NaOH, and KOH without directly fitting the model to experimental data. The results were confirmed by comparing the calculated densities to published experimental data. The model was able to accurately predict experimentally determined densities (R2 > 0.99) using model coefficients derived without using any experimental data from solutions containing KAl(OH)4 . © 2008 Elsevier B.V. All rights reserved. Keywords: Density; Lalibert´e–Cooper; Electrolyte solution

volume of water, m3 /kg. The electrolyte specific volume for one electrolyte is given by Eq. (2),

1. Introduction A number of models have been developed to predict the densities of multi-component aqueous solutions as a function of composition [1–7]. A promising new model is one developed by Lalibert´e and Cooper [6], here denoted as the Lalibert´e–Cooper model and given by Eq. (1), ρm =

1 wH2 O v¯ H2 O +



(1)

wi v¯ app,i

i

where ρm is the solution density, kg/m3 , wi is the mass fraction of electrolyte i, v¯ app,i is the electrolyte specific volume, m3 /kg, wH2 O is the water mass fraction, and v¯ H2 O is the specific ∗

Corresponding author. Tel.: +1 509 371 3842. E-mail addresses: [email protected] [email protected] (R. Carter).

(J.G.

Reynolds),

0378-3812/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2008.01.017

v¯ app,i =

wi + c 2 + c 3 t (c0 wi + c1 ) e(0.000001(t+c4 )

(2)

2)

where c0 to c4 are empirical constants. The coefficients c0 and c1 are in kg/m3 , c2 is dimensionless, c3 is in ◦ C−1 , c4 is in ◦ C, and t is in ◦ C. When the solution mixture contains more than one electrolyte, Lalibert´e and Cooper [6] suggest a modified form of Eq. (2), where the total electrolyte concentration (1 − wH2 O ) is used instead of just the electrolyte concentration in question, wi . This modification gave much greater accuracy in density prediction and is given by Eq. (3). v¯ app,i =

(1 − wH2 O ) + c2 + c3 t (c0 (1 − wH2 O ) + c1 ) e(0.000001(t+c4 )

2)

(3)

This equation reduces to Eq. (2) for a solution of just one electrolyte. The specific volume of water was calculated from

J.G. Reynolds, R. Carter / Fluid Phase Equilibria 266 (2008) 14–20

the inverse of the following correlation given in Lalibert´e and Cooper [6]: (((((−2.8054253×10−10 t + 1.0556302×10−7 )t

ρ H2 O =

−4.6170461 × 10−5 )t − 0.0079870401)t +16.945176)t+999.83952) 1+0.01687985t

(4)

Some of the salient features of the Lalibert´e–Cooper model (Eqs. (1) through (4)) are: 1. The model appears to be accurate, judging by the accuracy that the model predicts multi-component solutions presented by Lalibert´e and Cooper [6]. A companion paper has confirmed that the Lalibert´e–Cooper model accurately predicts the densities of solution mixtures of eight different electrolytes at total ionic strengths between 3 and 8 mole\l [8]. 2. The model does not require rigorous thermodynamic speciation of the ions in aqueous solution. This enables the model to be employed in process flowsheet models without speciation calculations. 3. The density (or volume) is directly determined from composition, and composition is usually measured directly in chemical plants or tracked in process flowsheets. 4. The model coefficients for multi-component electrolyte solutions can be derived from individual electrolyte solution data, and there is a copious amount of data available for single electrolyte solutions [9]. 5. Model coefficients for many industrially important electrolytes have already been determined and published by Lalibert´e and Cooper [6]. In order for the model to be of practical use in industry, it must be robust. Lalibert´e and Cooper [6] have tested their model in three key areas of robustness. First, they have shown that the model is accurate at reproducing the data used to derive the coefficients. They have shown that the model predicts data that was not used to develop the coefficients. In fact, they were able to accurately predict the density of 27 multi-component solutions with Eq. (3) using coefficients that were derived solely by fitting Eq. (2) to single electrolyte solutions. Finally, they were able to show that the model extrapolated reasonably well. These three conditions are important features of a robust model [10].

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Another model criterion that should be imposed on the model is self-consistency. The model predicts the density of mixed electrolyte solutions from coefficients derived from individual electrolyte solutions. When a solution mixture contains two electrolytes with a common anion or common cation, the model coefficients used are unambiguous. When the mixture is a blend of electrolytes with two different cations and two different anions, the choice is more ambiguous. For instance, a mixture of K+ , Na+ , Cl− and NO3 − , could be modeled as a mixture KCl and NaNO3 or KNO3 and NaCl. If the model is self-consistent, it will not matter how the cations and anions are paired. Nonetheless, it has not been clear that the requirement of self-consistency is built into the Lalibert´e–Cooper model. The purpose of this paper is to demonstrate that the Lalibert´e–Cooper model is relatively self-consistent. When this has been established, it will be shown that this self-consistency allows for a new method of model parameterization without directly fitting Eq. (2) to data. The model coefficients can be determined by fitting Eq. (3) to simulated data calculated using coefficients for other electrolytes that contain the cation and anion of the electrolyte of interest.

2. Modeling approach The purpose of this study is to determine if the model coefficients published by Lalibert´e and Cooper [6] are self-consistent. This study will focus on the salts of monovalent cations that have the same anions as other monovalent cations in the database. The electrolytes in question and their model coefficients from Lalibert´e and Cooper [6] are shown in Table 1. Three temperatures were investigated, 25, 50, and 75 ◦ C. The densities of mixtures of two cations with two anions were calculated using Eqs. (1), (3), and (4) using the model coefficients in Table 1. In each mixture, each cation was added at 50 mole% of the cations and each anion was added at 50 mole% of the anions. Eq. (3) uses mass percent, so the mole percentages were converted to mass percent of the solution. With a 50 mole% mixture for both cations and anions, either cation can be completely paired with either anion in the calculation. For instance, in a mixture of Na+ , K+ , OH− , Cl− the density can be modeled using either the coefficients for NaCl and KOH or using the coefficients for NaOH and KCl. The densities were calculated using both combinations

Table 1 Model coefficients used and maximum mass fraction investigated, from [6] Electrolyte

c0

c1

c2

c3

c4

Maximum mass fraction of validity

HCl HNO3 KCl KNO3 KOH NaCl NaNO3 NaOH NH4 Cl NH4 NO3 NaAl(OH)4 a

−80.061 12.993 −0.46782 7.5436 194.85 0.00433 49.209 385.55 6.56150 1379.3 −258.681

255.42 −23.579 4.30800 26.38800 407.31 0.06471 94.737 753.47 89.772 1124.4 851.308

118.42 −3.6070 2.3780 1.2396 0.14542 1.01660 0.77927 −0.10938 4.9024 0.65598 48.50073

1.0164 0.0079416 0.022044 0.011656 0.002040 0.014624 0.0075451 0.0006953 −0.016574 0.0014106 −0.15934

2619.5 −2427.1 2714.0 2214.0 1180.9 3315.6 1819.2 542.88 −2089.3 176.41 −2266.55

0.4000 0.80110 0.2800 0.24000 0.5964 0.26031 0.46820 0.700000 0.40000 0.78740 0.2793

a

From [11].

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J.G. Reynolds, R. Carter / Fluid Phase Equilibria 266 (2008) 14–20

of cations and anions. The residuals of the densities calculated using the two combinations of cation–anion pairing were taken to determine how large the discrepancy was for each mixture. A residual is defined as the absolute magnitude of the density of the mixture calculated when paired the first way minus the density of the mixture calculated using the second cation–anion pairing. If a model is self-consistent, the result should be the same regardless of how the cations and anions are paired and the residuals would be approximately zero. In the modeling, the mole ratio of dissolved ions was held constant but the total moles of dissolved electrolytes varied from 0.01 to a maximum specific to each electrolyte mixture. The maximum range investigated was based on the valid range for the coefficients reported by Lalibert´e and Cooper [6] and is shown in the last column of Table 1. When Lalibert´e and Cooper [6] developed the coefficients by fitting Eq. (2) to experimental data, the experimental data covered only a limited (though usually large) concentration range. The coefficients are less accurate outside this range [6]. For each ion mixture investigated in the present study, the density was calculated up to the maximum concentration of the most limiting electrolyte investigated by Lalibert´e and Cooper [6]. The limiting electrolyte was frequently the heaviest electrolyte in the mixture because a 1:1 mole ratio of two electrolytes results in more than a 1:1 mass ratio of the heavier electrolyte. Equal molar solutions were used because this maximizes the difference between the two different pairings. Consider the modeling of the density of a solution containing ions A, B, X and Y as AX and BY or AY and BX. If there was more AX than BY in the solutions, then some of the A and the X would have to be modeled as AX even if it were preferred to model the system as AY and BX. The maximum difference between the two pairings is an equal molar solution because there is no electrolyte that has to be used in both pairings.

(4.5 × 10−1 kg/m3 , Table 2) at the highest concentration. This specific mixture had the highest electrolyte concentration of all electrolyte pairs studied, going up to 40 wt% NH4 Cl and only 26.6 wt% water. For this mixture, and most others, the residuals increased with increasing concentration of dissolved electrolytes. Thus, this high residual primarily results from of the high electrolyte concentration. The residuals for the HCl–NH4 NO3 system increased with increasing temperature, but the residuals did not increase with increasing temperature for all mixtures. The second highest residual observed for a mixture was for the K+ , Na+ , Cl− , OH− system modeled as either KCl and NaOH or KOH and NaCl. The densities calculated for the sample with the highest residual were 1953.236 and 1953.308 kg/m3 , which equal a difference of 7.2 × 10−2 kg/m3 between the two ways the cations and anions were paired. This equates to a relative error of only 0.0037%. For most ion mixtures investigated in Table 2, the difference was considerably better than even this small error (Table 2). Hence, it can be concluded that the model is selfconsistent and that the choice of how anions and cations are paired together makes little difference on the calculated result. Lalibert´e and Cooper [6] used a limited (though large) range of electrolyte concentrations. The residual results shown in Table 2 are only those results where the mixture was below the maximum electrolyte mass fraction listed in the last column of Table 1. These mass fractions of validity listed in Table 1 are the maximum electrolyte mass fraction used by Lalibert´e and Cooper [6] to parameterize the model for the given electrolyte. The residuals were frequently worse than 7.2 × 10−2 kg/m3 when the mass fraction of one of the electrolytes exceeded that mass fraction range. Therefore, the model is best used within the range of validity. The largest residual for any given mixture and temperature was usually, though not always, at the highest total dissolved electrolyte concentration used in the simulation.

3. Residual results

4. New method for model parameterization

The residuals of the densities of each aqueous solution ion mixture calculated with the cations and anions paired two different ways are shown in Table 2. The largest residual observed was for the H+ , NH4 + , Cl− , NO3 − mixture, modeled as either HCl and NH4 NO3 or NH4 Cl and HNO3 , at 75 ◦ C

The previous section demonstrated that the model parameters presented by Lalibert´e and Cooper [6] are relatively self-consistent and insensitive to the manner that cations and anions are paired. In this section, it will be demonstrated that model parameters (c0 –c4 ) can be determined for elec-

Table 2 Maximum residuals observed for ion mixtures calculated as two different sets of electrolytes Ion mixture

Limiting electrolyte mass fraction

Maximum mass fraction from [6]

Largest density residual observed at 25, 50 and 75 ◦ C (kg/m3 )

K+ , Na+ , Cl− , OH− K+ , Na+ , Cl− , NO3 − NH4 + , Na+ , Cl− , NO3 − H+ , Na+ , Cl− , NO3 − H+ , K+ , Cl− , NO3 − H+ , NH4 + , Cl− , NO3 − K+ , NH4 + , Cl− , NO3 − K+ , Na+ , OH− , NO3 −

NaCl KNO3 NaCl NaCl KNO3 NH4 Cl KNO3 KNO3

0.26 0.24 0.26 0.26 0.24 0.40 0.24 0.24

7.0 × 10−2 , 7.1 × 10−2 , 7.2 × 10−2 3.2 × 10−3 , 2.9 × 10−3 , 6.8 × 10−4 2.1 × 10−2 , 3.9 × 10−3 , 1.2 × 10−3 3.0 × 10−2 , 1.5 × 10−2 , 4.2 × 10−2 4.0 × 10−3 , 3.1 × 10−3 , 1.8 × 10−3 2.8 × 10−1 , 3.7 × 10−1 , 4.5 × 10−1 4.0 × 10−3 , 5.4 × 10−3 , 4.1 × 10−3 1.7 × 10−2 , 1.6 × 10−2 , 1.4 × 10−2

J.G. Reynolds, R. Carter / Fluid Phase Equilibria 266 (2008) 14–20

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Table 3 Characteristics of experimental data for the KOH–KAl(OH)4 –H2 O system Source referencea

Densities in kg/m3 (minimum, maximum)a

Temperatures (◦ C)

[12] [13]

1092.8, 1515.2 1053, 1535

25, 50, 75, 25

KOH mass fraction range (minimum, maximum)a

KAl(OH)4 mass fraction range (minimum, maximum)a

0.1166, 0.4773 0.0200, 0.4203 0.0300, 0.2962 0.0095, 0.3581

H2 O mass fraction range (minimum, maximum)a 0.4038, 0.8634 0.4205, 0.9347

Total number of data pointsa 30 24

a The samples with the highest KOH concentration at each temperature for both data sets were not used because the model did not fit these data points well. The ranges and number of data points reported here are the ranges of the data actually used for model parameterization and validation.

trolytes solely from model coefficients available for other electrolytes containing the anion and cation of interest. The KOH, KAl(OH)4 , NaAl(OH)4 electrolytes will be used for this purpose. Coefficients for KAl(OH)4 will be determined from published coefficients for KOH and NaAl(OH)4 without direct fitting to experimental data. Model results using these KAl(OH)4 coefficients will then be compared against experimentally determined densities from the KOH–KAl(OH)4 –H2 O system [12,13]. While the purpose of this study is to demonstrate that KAl(OH)4 coefficients can be determined from KOH and NaAl(OH)4 coefficients without direct use of experimental data, the coefficients will also be determined directly from the data so that they can be compared to the coefficients determined by simulation. KAl(OH)4 is only stable in water in the presence of KOH. Consequently, the coefficients for KAl(OH)4 are determined from the data published by Mashovets et al. [12,13] for the KOH–KAl(OH)4 –H2 O system, summarized in Table 3, using Eqs. (1), (3), and (4). The model coefficients used for KOH were from Lalibert´e and Cooper [6] (see also Table 1). The c0 through c4 values for KAl(OH)4 in Eq. (3) were determined iteratively by minimizing the sum of square error of the density result calculated in Eq. (1). The initial values for c0 to c4 (1, 1, 1, 0.0025, 1500) recommend by Lalibert´e and Cooper [6] were used at the start of the iterative process for the KAl(OH)4 coefficients. The regression results shown in Table 4. The R2 value (calculated per [14]) for the Mashovets et al. [12] dataset was 0.997 and was 0.999 for the Marshovets et al. [13] dataset. For brevity, this set of coefficients derived from experimentally determined data will be denoted the “experimental coefficients”.

Table 4 Model coefficients for KAl(OH)4 determined in this study Parameter

Determined by regressing experimental data (experimental coefficients)

Determined by regressing simulated densities (simulated coefficients)

wmin wmax tmin ( ◦ C) tmax ( ◦ C) c0 c1 c2 c3 c4

0.0096 0.4203 25 75 181.4549 104.2742 0.406813 −0.00229 −1537.55

0.0691 0.2691 25 75 200.807 74.8365 0.21964 0.002252 1522.25

A second set of KAl(OH)4 coefficients was developed from simulation without any direct fitting to real experimental data from solutions containing KAl(OH)4 (Table 4). A set of solution densities was calculated at 10 ◦ C intervals between 25 and 75 ◦ C for mixtures of K+ , OH− , Na+ , and Al(OH)4 − using the Lalibert´e–Cooper model. The coefficients used for KOH were from Lalibert´e and Cooper [6] and the coefficients used for NaAl(OH)4 were from Reynolds and Carter [11]. Each simulated solution had identical molarities of NaAl(OH)4 and KOH, however the total solute mole fractions (NaAl(OH)4 + KOH) were varied in the dataset at 0.0100 electrolyte mole fraction intervals between a low mole fraction of 0.0100 and a high mole fraction of 0.1000. The Lalibert´e–Cooper model uses mass fraction rather than mole fraction, and the electrolyte mole fraction 0.1000 corresponds to a maximum mass fraction for NaAl(OH)4 of 0.2367 in the data set. In turn, a mass fraction of 0.2367 for NaAl(OH)4 is just under the 0.2793 maximum NaAl(OH)4 mass fraction used for coefficient development by Reynolds and Carter [11]. The complete set of simulated densities is shown in Table 5. The solution compositions in Table 5 were then recast as a mixture of NaOH and KAl(OH)4 rather than KOH and NaAl(OH)4 . A set of coefficients for KAl(OH)4 were developed from the simulated densities by iteratively varying the c0 through c4 coefficients in Eq. (3) until the minimum of the squared error between the simulated densities in Table 5 and the densities predicted with the new c0 through c4 coefficients was obtained. The partial specific volume for NaOH was calculated in this regression using Eq. (3) and the c0 through c4 coefficients published by Lalibert´e and Cooper [6] (also shown in Table 1). The resulting c0 through c4 coefficients for KAl(OH)4 determined from the simulated densities are shown in Table 4. The maximum KAl(OH)4 mass fraction in the simulated density dataset was 0.2691 (Table 5). The overall R2 value for the regression of the simulated densities was 0.999. For brevity, this set of coefficients derived from simulated densities will be denoted the “simulated coefficients”.

5. Model parameterization discussion Inspecting the model coefficients in Table 4 indicates that the experimental and simulated coefficients are different from each other. Reynolds and Carter [11] determined that a small amount of data could have a large effect on the magnitude of the Lalibert´e–Cooper model coefficients, but have little effect on the model predictability. They attributed this to the functional form of Eq. (3), where both the process- and mass fraction-based

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J.G. Reynolds, R. Carter / Fluid Phase Equilibria 266 (2008) 14–20

Table 5 Simulated solutions used to develop the simulated coefficients T

NaAl(OH)4 , mole fraction

KOH, mole fraction

H2 O, mole fraction

NaAl(OH)4 , mass fraction

KOH, mass fraction

H2 O, mass fraction

Density (kg/m3 )

25 25 25 25 25 25 25 25 25 35 35 35 35 35 35 35 35 35 45 45 45 45 45 45 45 45 45 55 55 55 55 55 55 55 55 55 65 65 65 65 65 65 65 65 65 75 75 75 75 75 75 75 75 75

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

0.980 0.970 0.960 0.950 0.940 0.930 0.920 0.910 0.900 0.980 0.970 0.960 0.950 0.940 0.930 0.920 0.910 0.900 0.980 0.970 0.960 0.950 0.940 0.930 0.920 0.910 0.900 0.980 0.970 0.960 0.950 0.940 0.930 0.920 0.910 0.900 0.980 0.970 0.960 0.950 0.940 0.930 0.920 0.910 0.900 0.980 0.970 0.960 0.950 0.940 0.930 0.920 0.910 0.900

0.0608 0.0881 0.1136 0.1374 0.1597 0.1807 0.2005 0.2191 0.2367 0.0608 0.0881 0.1136 0.1374 0.1597 0.1807 0.2005 0.2191 0.2367 0.0608 0.0881 0.1136 0.1374 0.1597 0.1807 0.2005 0.2191 0.2367 0.0608 0.0881 0.1136 0.1374 0.1597 0.1807 0.2005 0.2191 0.2367 0.0608 0.0881 0.1136 0.1374 0.1597 0.1807 0.2005 0.2191 0.2367 0.0608 0.0881 0.1136 0.1374 0.1597 0.1807 0.2005 0.2191 0.2367

0.0289 0.0419 0.0540 0.0653 0.0760 0.0859 0.0953 0.1042 0.1126 0.0289 0.0419 0.0540 0.0653 0.0760 0.0859 0.0953 0.1042 0.1126 0.0289 0.0419 0.0540 0.0653 0.0760 0.0859 0.0953 0.1042 0.1126 0.0289 0.0419 0.0540 0.0653 0.0760 0.0859 0.0953 0.1042 0.1126 0.0289 0.0419 0.0540 0.0653 0.0760 0.0859 0.0953 0.1042 0.1126 0.0289 0.0419 0.0540 0.0653 0.0760 0.0859 0.0953 0.1042 0.1126

0.9102 0.8700 0.8324 0.7973 0.7643 0.7333 0.7042 0.6767 0.6507 0.9102 0.8700 0.8324 0.7973 0.7643 0.7333 0.7042 0.6767 0.6507 0.9102 0.8700 0.8324 0.7973 0.7643 0.7333 0.7042 0.6767 0.6507 0.9102 0.8700 0.8324 0.7973 0.7643 0.7333 0.7042 0.6767 0.6507 0.9102 0.8700 0.8324 0.7973 0.7643 0.7333 0.7042 0.6767 0.6507 0.9102 0.8700 0.8324 0.7973 0.7643 0.7333 0.7042 0.6767 0.6507

1066.380 1098.474 1129.141 1158.529 1186.755 1213.912 1240.082 1265.330 1289.719 1062.278 1094.062 1124.527 1153.799 1181.977 1209.145 1235.374 1260.724 1285.250 1057.437 1088.936 1119.220 1148.393 1176.541 1203.735 1230.037 1255.502 1280.178 1051.964 1083.203 1113.326 1142.418 1170.55 1197.783 1224.172 1249.764 1274.602 1045.938 1076.939 1106.920 1135.946 1164.077 1191.362 1217.848 1243.577 1268.587 1039.411 1070.197 1100.054 1129.030 1157.170 1184.518 1211.111 1236.985 1262.174

model coefficients appear on both top and bottom of the quotient. Putting the mass fraction and process variables in both the numerator and denominator makes the model more flexible [15], which may contribute to the success of the Lalibert´e–Cooper model. The important criterion is that the coefficients as a set can be used to accurately model the data.

Fig. 1 shows the accuracy of the model at fitting the Mashovets et al. [12] data using the experimental coefficients. Fig. 2 shows the fit of the Mashovets et al. [13] data. As seen from these figures, the Lalibert´e–Cooper model can accurately predict the densities that were used to derive the model coefficients. This is consistent with the large R2 of the fit (>0.99) mentioned

J.G. Reynolds, R. Carter / Fluid Phase Equilibria 266 (2008) 14–20

Fig. 1. Comparing measured vs. predicted density values for the reference [12] dataset using the experimental coefficients.

earlier. Reynolds et al. [8,11] found that the Lalibert´e–Cooper model was able to predict the densities of sodium salt solutions for data not included during model parameterization. What is of interest for this paper, however, is that the model can use coefficients derived from electrolyte solutions that do not even contain KAl(OH)4 and still accurately predict the densities of KAl(OH)4 -containing solutions. Figs. 3 and 4 show the prediction of the Mashovetes et al. [12,13] data using the KAl(OH)4 c0 through c4 coefficients derived from simulated densities. Comparing Figs. 1 and 2 with Figs. 3 and 4 indicates that the simulated coefficients are nearly as good at predicting densities as the experimental coefficients. The R2 of the fit of the Mashovetes et al. [12] data was 0.991 and it was 0.996 for Mashovetes et al. [13] data using simula-

19

Fig. 4. Comparing measured vs. predicted density values for the reference [13] dataset using the simulated coefficients.

ted coefficients, nearly as high as the R2 from the experimental coefficients. Figs. 5 and 6 directly compares the residuals (measured minus predicted densities) for the experimental and simulated coefficients. As can be seen from these figures, the model can closely predict all of the experimental data from references [12] and [13]. The very worst results were off be less than 20 kg/m3 regardless of which sets of coefficients were used. Thus the accuracy of the model depends little on whether the model coefficients are derived from experimental data or from simulated density

Fig. 5. Comparison of the residuals for the experimental and simulated coefficients for the reference [12] dataset. Fig. 2. Comparing measured vs. predicted density values for the reference [13] dataset using the experimental coefficients.

Fig. 3. Comparing measured versus predicted density values for the reference [12] dataset using the simulated coefficients.

Fig. 6. Comparison of the residuals for the experimental and simulated coefficients for the reference [13] dataset.

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J.G. Reynolds, R. Carter / Fluid Phase Equilibria 266 (2008) 14–20

data calculated from other electrolytes that contain the same cations and anions as the target electrolyte. This is valuable because there are many electrolytes for which data is unavailable, and this method of parameterization can be useful for those systems. Nonetheless, from inspecting Fig. 6, it can be recommended that the experimental coefficients be used, because the residuals of the experimental coefficients tended to be closer to zero. Likewise, there seemed to be a slight bias in the simulated coefficients, were these coefficients tended to under-predict the density for the reference [13] data. Nonetheless, the majority of the data points had an error less than 1% regardless of which coefficients were used. Perhaps the biggest advantage of using the experimental coefficients in this example is that they cover a larger KAl(OH)4 mass fraction range. The simulated coefficients were fit to a maximum KAl(OH)4 mass fraction of only 0.2691. This range was limited by the valid mass fraction range of the NaAl(OH)4 coefficients used to derive the KAl(OH)4 coefficients. In contrast, the experimental coefficients are valid up to a KAl(OH)4 mass fraction of 0.4203, a much higher concentration. Given that Eq. (3) uses the mass fraction of water rather than the electrolyte mass fraction, it is the lower limit of the water mass fraction that is most limiting in multi-electrolyte systems such as the KOH–KAl(OH)4 –H2 O. The minimum mass fraction of water used to fit the simulated coefficients was 0.65 (Table 5), whereas many of the experimental data points from references [12] and [13] had water mass fractions below this value (which is why there are less residuals shown for the simulated coefficients in Figs. 5 and 6). The residuals for the simulated coefficients were worse than those shown in Figs. 5 and 6 for the samples when water mass fraction was less than 0.65, though still less than 35 kg/m3 even for densities greater than 1300 kg/m3 . Experimental data is not available for all electrolyte systems, and the high accuracy of the simulated coefficients indicates an opportunity to model the densities of those systems in the absence of experimental data. 6. Conclusion This study has shown that the Lalibert´e–Cooper model is relatively self-consistent because it is insensitive to the way cations and anions are paired into electrolytes. The largest discrepancy for different cation and anion combinations amongst eight ion mixtures investigated was at the 4th significant figure of the density for the NH4 NO3 –HCl–H2 O system. The sample with the largest discrepancy was also the most concentrated of all

electrolytes in the dataset. All other electrolyte mixtures were self-consistent up to at least the 5th significant figure (Table 2). This self-consistency affords the opportunity to parameterize the model for electrolytes for which no experimental data are available when coefficients are available for other electrolytes containing the same cation and anion. As an example, Lalibert´e–Cooper model coefficients for KAl(OH)4 were determined from the coefficients for KOH, NaAl(OH)4 , and NaOH without fitting the model to any experimentally determined densities from solutions containing KAl(OH)4 . These simulated coefficients were able to predict experimentally determined densities for the KOH–KAl(OH)4 –H2 O system nearly as well as coefficients derived from directly fitting the model to the data. The model was able to predict the densities from references [12] and [13] with an R2 of 0.991 and 0.996, respectively. Acknowledgments The work described in this paper was performed for the United States Department of Energy under contract DE-AC2701RV14136, to support the Hanford Tank Waste Treatment and Immobilization Plant Project. The development of this paper was greatly aided by manuscript reviews and technical discussions with Jeanne Bernards, Dan Reynolds, David Sherwood, and Ven Arakali. Karen Reynolds performed grammatical editing and formatted the document for submission. References [1] J. Padova, J. Chem. Therm. 10 (1978) 143–149. [2] A. Kumar, J. Chem. Eng. Data 31 (1988) 19–20. [3] B.S. Krumgalz, R. Pogorelsky, K.S. Pitzer, J. Solution Chem. 24 (1995) 1025–1038. [4] C.X. Li, S.B. Park, J.S. Kim, H. Lee, Fluid Phase Equilib. 145 (1998) 1–14. [5] Y.F. Hu, J. Solution Chem. 29 (2000) 1229–1236. [6] M. Lalibert´e, W.E. Cooper, J. Chem. Eng. Data 49 (2004) 1141–1151. [7] P.M. Mathias, Ind. Eng. Chem. Res. 43 (2004) 6247–6252. [8] J.G. Reynolds, J.K. Bernards, R. Carter, Waste Management’07. Proceedings Waste Management Symposia Inc., Tucson, AZ, 2007 (published on CD). [9] O. Sohnel, P. Novotny, Densities of Aqueous Solutions of Inorganic Substances, Elsevier, New York, NY, USA, 1985. [10] R.D. Snee, Technometrics 19 (1977) 415–428. [11] J.G. Reynolds, R. Carter, Hydrometallurgy 89 (2007) 233–241. [12] V.P. Mashovets, V.V. Kurochkina, N.V. Penkina, L.V. Puchkov, M.K. Fedorov, J. Appl. Chem. USSR 40 (1967) 2473–2474. [13] V.P. Mashovets, N.V. Penkina, L.V. Puchkov, M.K. Fedorov, J. Appl. Chem. USSR 44 (1971) 24–27. [14] D.W. Marquardt, R.D. Snee, Technometrics 16 (1974) 533–537. [15] W.W. Focke, B. Du Plessis, Ind. Eng. Chem. Res. 43 (2004) 8369–8377.