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Thermodynamic properties of the LiCl–H2O system at vapor–liquid equilibrium from 273 K to 400 K
international journal of refrigeration 31 (2008) 287–303
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Thermodynamic properties of the LiCl–H2O system at vapor–liquid equilibrium from 273 K to 400 K J. Pa´tek*, J. Klomfar Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Dolejsˇkova 5, CZ 182 00 Prague 8, Czech Republic
article info
abstract
Article history:
A formulation of the thermodynamic properties of the LiCl–H2O system at vapor–liquid
Received 1 December 2006
equilibrium is presented in the form of separate Gibbs energy equations for the vapor
Received in revised
and solution phases. Explicit thermodynamically consistent equations for density, isobaric
form 27 March 2007
heat capacity, enthalpy, entropy, enthalpy of dilution and osmotic coefficient are given.
Accepted 7 May 2007
The pressure at vapor–liquid equilibrium is calculated from the condition of phase equilib-
Published online 18 May 2007
rium. The description of the properties is valid from 273 K or from the crystallization line up to 400 K in temperatures and for solution composition from 0 to 50 wt% of LiCl in the
Keywords:
solution, which is the region covewred by available experimental data. The thermody-
Thermodynamic property
namic properties of the gas phase are approximated by the properties of pure water vapor
Aqueous solution
computed from the IAPWS formulation 1995. The Gibbs energy equation for solution is
Water
based upon a body of experimental data that have been critically assessed. Within the
Lithium chloride
present study, 136 experimental works have been collected containing a total of 2921
Equilibrium
data points on various thermodynamic properties of the LiCl–H2O solutions. Gaps in the data-
Vapour
base are shown to give experimenters orientation for future research. The uncertainties
Liquid
associated with correlation are estimated to be 0.4% for density, 2.0% for pressure
Calculation
and 2.4% for isobaric heat capacity. The uncertainty in values of enthalpy is estimated
Pressure
to be less than 10 kJ kg1 and less than 0.03 kJ kg1 K1 for entropy. Values of the partic-
Equation of state
ular properties generated by the representative equations are provided to assist with the confirmation of computer implementation of the calculation procedure. ª 2007 Elsevier Ltd and IIR. All rights reserved.
Proprie´te´s thermodnamiques d’un syste`me au LiCl-H2O en e´quilibre vapeur-liquide entre 273 K et 400 K Mots cle´s : Proprie´te´ thermodynamique ; Solution aqueuse ; Eau ; Chlorure de lithium ; E´quilibre ; Vapeur ; Liquide ; Calcul ; Pression ; E´quation d’e´tat
* Corresponding author. Tel.: þ420 266053153; fax: þ420 28584695. E-mail address: [email protected] (J. Pa´tek). 0140-7007/$ – see front matter ª 2007 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2007.05.003
288
international journal of refrigeration 31 (2008) 287–303
Nomenclature cp g g hw h M m mw N n n u
p p P f Dhdil s T s v w x
1.
molar isobaric heat capacity (J mol1 K1) molar Gibbs free energy (J mol1) activity coefficient partial enthalpy of water in solution (J mol1) molar enthalpy (J mol1) molar mass (kg mol1) mass (kg); in Table 7: molality (mol kg1) chemical potential of water in the solution (J mol1) number of points amount of substance (mol) number of moles of particles that are formed when one mole of salt is dissolved chemical potential of water vapor except for the term logarithmic in the pressure (mgw ¼ RT ln( p/pc) þ u) dimensionless pressure variable (p ¼ p/pc) pressure (Pa) polynomial term (P ¼ asnpm for pure water and P ¼ asnpmxk for solution) osmotic coefficient molar enthalpy of dilution (J mol1) molar entropy (J mol1 K1) temperature (K) dimensionless temperature variable (s ¼ Tc/T ) molar volume (m3 mol1) mass fraction (w ¼ mLiCl/(mLiCl þ mw)) molar fraction (x ¼ nLiCl/(nLiCl þ nw))
Introduction
Although significant improvements are being made in predicting thermodynamic properties of pure fluids and mixtures using theory based methods, more accurate equations of state for calculating properties over a wide range of pressures and temperatures are to be developed by correlation of selected experimental data. For the development of such empirical
x 9 MRD RMSD
transformed composition variable (x ¼ [x/(1 x)](1/2)) molar density (mol m3) mean relative deviation, % ðMRD ¼ ð102 =NÞ P ðzexp =zcal 1ÞÞ root mean square relative deviation, % ðRMSD ¼ P 102 ½ð1=NÞ ðzexp =zcal 1Þ2 1=2 Þ
Subscripts c at the critical point cal calculated e excess quantity exp experimental LiCl lithium chloride p isobaric w water s solute Superscripts g vapor states of pure substance l liquid states of pure substance v vapor (gaseous phase of the binary system) 0 infinitely dilute standard states Physical R MLiCl MH2 O Tc pc rc
constants 8.31451 J mol1 K1 0.04932 kg mol1 0.018015268 kg mol1 647.096 K 22.064 106 Pa 17,874 mol m3 (322 kg m3)
formulations, the application of linear optimization procedures and nonlinear multiproperty fitting algorithm is stateof-art (Span, 2000). Modern equations of state are usually explicit in the Helmholtz energy as a function of density and temperature (Span, 2000). As far as binary mixtures are concerned, Tillner-Roth and Friend (1998), for example, have published a formulation of the thermodynamic properties of the
Table 1 – Coefficients and exponents of Eqs. (37)–(41) i
0 1 2 3 4 5 6 7 8 9 10 11 12
Liquid phase
Vapor phase
ni
mi
ai
ni
mi
ai
– – 1 0 1 2 3 4 1 0 1 2 3
– – 0 0 0 0 0 0 1 1 1 1 1
0.0 5.88137 101 9.18938 8.33307 101 1.84228 102 1.70534 102 1.05457 102 2.94856 101 1.96926 101 8.68008 101 2.25497 2.60177 1.17349
– – 1 0 3 6 7 8 15 – – – –
– – 0 0 0 1 1 1 3 – – – –
1.0 3.98432 6.68909 8.78439 5.18911 102 6.25248 101 5.07144 101 1.24364 101 4.49013 101 – – – –
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Table 2 – Maximum differences of values of enthalpy, entropy, molar volume and isobaric thermal capacity calculated from Eqs. (38)–(41), respectively, from the IAPWS 95 formulation Quantity 1
Dh (J mol ) Ds (J mol1 K1) Dv (%) Dcp (%)
Liquidus
Vapor
0.6 0.0015 0.01 0.04
10 0.02 0.8 0.5
ammonia–water system in the form of Helmholtz free energy. Experimental data covering single-phase region are indispensable for reasonable application of that approach. Data on the properties of the LiCl–H2O concerning the states outside the vapor–liquid equilibrium are scarce and incomplete (Brown et al., 2004; Majer et al., 1989, 1991; Pepinov et al., 1989; Urusova, 1971; Ohling and Schneider, 1979; Gates and Wood, 1985; Ellis, 1966). Two separate Gibbs energy functions for liquid and gas phase provide an alternative form of the thermodynamically consistent description of the vapor–liquid equilibrium region. Yuan and Herold (2005) and Kim and
Table 3 – Property values calculated from Eqs. (37)–(41) for validation of computer programs P (Pa)
9 (mol m3)
cp (J mol1 K1)
Liquid phase 275 300 400
698.46145 3536.8953 245,882.35
55,503.3 55,316.3 52,035.3
75.912 75.314 76.666
139.75 2028.3 9600.8
0.50970 7.08295 28.8458
0.4164 96.571 1937.5
Gas phase 275 300 400
698.46145 3536.8953 245,882.35
0.305671 1.420484 76.10544
33.847 34.344 40.022
45,115.5 45,937.1 48,913.6
164.058 153.446 127.128
0.4164 96.571 1937.5
T (K)
h (J mol1)
s (J mol1 K1)
m (J mol1)
Table 4 – Sources of data on p–T–x relation of LiCl–H2O solutions at vapor–liquid equilibrium Author(s) and year
Range of values Temperature (K)
Mass fraction (wt%)
315–373 273 273 293 244–374 298 293 333–453 291 275–383 303–343 293 299–348 278–338 303–394 333 398 348 423 256–303 323–423 298–323 283–313 303–373 398 374–387 341–376
7–22 8–30 2–35 0.4–4 – 0.4–3 15–44 – 3–42 – 5–51 4–43 9–44 – 11–47 4–34 4 7–37 31–42 – 2–11 24–43 – 13–44 10–35 5–25 10–30
Tamman (1885) Dieterici (1891) Tower (1908) Lovelace et al. (1923) Hu¨ttig and Reuscher (1924)a Pearce and Nelson (1932) Gibson and Adams (1933) Applebey et al. (1934)a Lannung (1934) Gokcen (1951)a Johnson and Molstad (1951) Kangro and Groeneveld (1962) Schlu¨nder (1963) Acheson (1965)a Uemura et al. (1965) Broul et al. (1969) Lindsay and Liu (1971) Sada et al. (1975) Fedorov et al. (1976) Adamcova´ and Benesˇ (1977)a Campbell and Bhatnagar (1979) Khripun et al. (1986) Apelblat (1993)a Chaudhari and Patil (2002) Safarov et al. (2003) Vercher et al. (2004) Kola´rˇ et al. (2005) a Vapor pressures of saturated solutions.
Number of data total/used 96/70 4/0 5/0 10/0 22/0 25/25 8/8 19/0 20/20 16/0 58/58 18/18 75/0 13/0 56/0 17/17 1/1 6/0 2/0 34/0 30/30 12/12 7/0 72/72 3/0 6/6 9/9
Deviation (%) MRD
RMSD
0.09 9.46 17.59 2.04 – 1.28 2.19 – 0.34 – 1.35 0.22 0.76 – 1.31 0.07 0.02 13.11 1.37 – 0.23 0.47 – 0.43 20.78 0.48 0.07
0.51 12.11 36.03 2.33 – 2.07 2.56 – 0.53 – 2.01 0.37 3.72 – 6.82 1.25 0.02 19.09 1.38 – 0.45 1.86 – 0.86 25.93 0.57 0.78
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Table 5 – Sources of data on 9–T–x relation of LiCl–H2O solutions at vapor–liquid equilibrium Author(s) and year
Kremers (1856) Gerlach (1859) Fouque´ (1867) Kohlrausch and Grotrian (1875) Sprung (1876) Kohlrausch (1879) Kuschel (1881) Bender (1883) Kohlrausch (1885) Ro¨ntgen and Schneider (1886) Bender (1887) Engel (1888) Valson (1890) Wagner (1890) Jahn (1891) Abegg (1893) Perkin (1894) Lemoine (1897) Conroy (1899) Linebarger (1899) Forchheimer (1900) Hosking (1904) Che´neveau (1907) Green (1908) Tower (1908) Guerdikova (1910) Baxter et al. (1911) Washburn and MacInnes (1911) Grufki (1913) Lu¨bben (1913) Sachanov (1913) Baxter and Wallace (1916) Henderson and Kellogg (1916) Alfimoff (1917) de Block (1925) Hu¨ttig and Keller (1925) Sugden (1926) Fontell (1927) Hu¨ttig and Ku¨kenthal (1928) Kohner (1928) Palitzsch (1928) Schreiner (1928) Jones and Bradshaw (1932) Applebey et al. (1934) Lanman and Mair (1934) Scott et al. (1934) Gibson (1935) Scott and Bridger (1935) Nickels and Allmand (1937) Guillaume (1946) Stratmann (1948) Rodnyanskii and Galinger (1955) Campbell and Kartzmark (1956) Kapustinskii et al. (1960) Hasaba et al. (1964) Lengyel et al. (1964) Bogatykh and Evnovitch (1965) Vaslow (1966) Millero and Drost-Hansen (1968) Desnoyers et al. (1969) Ostroff et al. (1969) Vaslow (1969) Postnikov (1970)
Range of values Temperature (K)
Mass fraction (wt%)
292 288 273–285 289–293 283 288–291 286–294 288 291 291 288 273 288 298 293 289 288–293 273 291 298 288 273–373 292–303 291–298 273 298 298 273 291 291 298 273–373 373 273–350 289 293 291 279–343 293 298 298 291 298 274–429 298 308 298 308 298 293 291–333 298–623 298 298 278–363 288–308 283–373 298 293–313 298 298 278–308 293–353
5–38 10–43 8–28 5–10 8–27 3–40 0.2–25 11–14 4 3–7 4–20 40.5 4 0.5–4 5–11 4–8.5 25–50 4–43 2–4 9–40 4–36 0.02–37 4–43 2–41 2–35 10–43 0.6–12 0.1–4 2–14 2–14 7–46 1.2–45 0.1–10.8 7–47 4–42 0.4–33 0.4–6 3–44 0.4–34 0.4–28 8–45 2–34 0.1–11 41–58 2–8 4–42 4.5–39 28–41 0.4–15 6.6–32 4–14.5 4–11 10 2–16 11-46 32–45 32–46 0.1–12 0.5 0.3–3 4–19 0.2–11 –
Number of data total/used 6/6 5/5 4/0 2/2 3/3 4/4 9/0 4/0 1/1 2/0 6/0 2/0 1/1 4/0 2/0 2/2 6/0 6/4 2/2 4/2 5/5 96/96 14/12 16/16 5/5 7/0 12/10 9/9 4/0 4/2 4/2 27/27 7/0 18/11 5/4 11/6 5/0 24/24 11/4 8/8 7/7 7/7 12/10 15/5 3/3 8/8 8/8 3/3 12/0 2/0 24/0 18/7 1/1 9/9 40/34 48/48 98/93 27/27 21/21 8/0 9/9 34/34 7/0
Deviation (%) MRD
RMSD
0.11 0.08 3.51 0.00 0.00 0.01 0.13 0.47 0.07 0.21 0.38 0.79 0.03 0.34 0.22 0.10 3.35 0.06 0.04 0.06 0.12 0.01 0.00 0.04 0.04 0.90 0.03 0.02 0.21 0.04 0.02 0.01 5.07 0.11 0.03 0.05 0.10 0.03 0.03 0.00 0.01 0.03 0.03 0.04 0.06 0.00 0.03 0.06 0.35 0.01 0.22 0.04 0.24 0.00 0.03 0.04 0.02 0.01 0.01 0.87 0.11 0.05 –
0.12 0.10 4.72 0.01 0.04 0.09 0.13 0.49 0.07 0.33 0.59 0.79 0.03 0.34 0.23 0.17 3.40 0.12 0.05 0.06 0.18 0.09 0.17 0.08 0.09 0.99 0.05 0.05 0.32 0.05 0.04 0.12 7.09 0.16 0.09 0.06 0.11 0.09 0.03 0.02 0.09 0.05 0.04 0.06 0.06 0.06 0.06 0.10 0.35 0.17 0.29 0.07 0.24 0.02 0.21 0.06 0.18 0.02 0.01 1.00 0.13 0.06 –
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Table 5 – (continued) Author(s) and year
Mashovets et al. (1971) Millero et al. (1977) Isono (1980) Gates and Wood (1985) Tanaka and Tamamushi (1991) Wimby and Berntsson (1994) Apelblat and Manzurola (2001) Safarov et al. (2003) Vercher et al. (2003)
Range of values Temperature (K)
Mass fraction (wt%)
213–273 298 288–328 298 288–328 290–343 278–338 398 288–318
5–35 0.5–4.5 0.2–30 0.2–17 0.2–46 10–45 0.4–4 10–35 4.5–34
Infante Ferreira (2006) developed a description of this form for the LiBr–H2O solutions. No thermodynamically consistent description of the properties of lithium chloride–water system is available in open literature. The LiCl–H2O system is one of the working-pairs for absorption cycles studied as an alternative to the most common working fluids – H2O–NH3 and LiBr–H2O. In practical computations, separate equations are often preferred for describing particular thermodynamic properties of interest, though such equations are thermodynamically consistent only approximately in the numerical sense. There exist a number of studies that provide description of various binary systems in the form of a set of separate equations for pressure, density, isobaric thermal capacity, enthalpy, and entropy of the system at vapor–liquid equilibrium. The historical Haltenberger’s method (1939) is used to compute values of enthalpy and entropy of the system from experimental data. In the case of LiCl–H2O solutions, Chaudhari and Patil (2002) have published their enthalpy in the form of polynomial valid up to 50 wt% and 393 K. Conde (2004) has published separate equations for various thermophysical properties of the system including equilibrium pressure, density, isobaric thermal capacity and differential enthalpy of dilution. The aim of the present study was to provide a formulation of the thermodynamic properties of LiCl–H2O system in the vapor–liquid equilibrium states valid from 273 K or from the crystallization temperature (whichever is greater) up to 400 K in temperatures and from pure water up to 50 wt% in compositions. The formulation should be based on all available experimental data and it should approximate the correlated data within their experimental uncertainties. The property description should be computationally effective, i.e., explicit, simple and easily implementable. These are indispensable properties for the description to find its users in practical calculations. The formulation has the form of two separate equations for Gibbs energy of the liquid and vapor phase so that the equilibrium p–T–x relation can be established from the condition of phase equilibrium. The gas phase of the system can be considered as pure steam as the vapor pressure of the LiCl is quite negligible to the vapor pressure of H2O. All vapor phase properties are therefore considered as functions of temperature and pressure only, independent on the solute mole fraction y which is considered to be identically equal to zero. The description of the thermodynamic properties of the pure water based on
Number of data total/used 26/0 9/9 70/70 8/8 113/113 52/45 183/183 3/0 36/36
Deviation (%) MRD
RMSD
0.28 0.01 0.04 0.01 0.04 0.04 0.01 0.61 0.09
0.32 0.01 0.06 0.04 0.12 0.07 0.01 0.75 0.12
approximation of the IAPWS 1995 formulation (Wagner and Pruß, 2002) is given in Section 4. Thus the main task is to develop a description of the molar Gibbs energy of the LiCl–H2O solutions. In the present paper, partial molar quantities of components in a mixture and molar quantities of pure substance related to water or solute are distinguished by subscript w or s, respectively, e.g., mw, mvs . The quantities related to the vapor phase of the mixture are distinguished by superscript v, e.g., mvw. The quantities related to the solution which occur the most often in equations have no superscript letter, to make the notation more readable, i.e., g, h, mw, ms. The phase is indicated by superscript g for gas and by l for the liquid phase, in the case of a property of a pure substance to distinguish it from that of mixture, e.g., glw, ggw. All solution properties are considered as functions of temperature, pressure and composition expressed as the mole fraction x of the solute in the solution. It should be mentioned that the molar Gibbs energy of pure water is identical to its chemical potential, i.e., ggw h mgw and glw h mlw. Specific quantities and mass fraction w used in common computational practice are related to the molar quantities by the following relations: xMLiCl ; xMLiCl þ ð1 xÞMw
(1)
w=MLiCl ; w=MLiCl þ ð1 wÞ=Mw
(2)
w¼
x¼
hðwÞ 1 hðxÞ ¼ ; J kg1 xMLiCl þ ð1 xÞMw J mol1
(3)
and similarly for density, entropy and isobaric heat capacity.
2. Development of the Gibbs energy equation for the solution 2.1. Determination of the Gibbs energy values from experimental data Values of the molar Gibbs energy g(T, p, x) of the solution with composition given by the molar fraction x of the solute are defined through the derivatives of the Gibbs energy and by a reference value chosen by convention in an arbitrary reference
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Table 6 – Sources of data on cp–T–x relation of LiCl–H2O solutions at vapor–liquid equilibrium Author(s) and year
Tucker (1915) Jauch (1921) Richards and Ro¨we (1921) Lange and Du¨rr (1926) Gucker and Schminke (1932) Bennewitz and Kratz (1936) Kapustinskii et al. (1960) Uemura et al. (1965) Ru¨terjans et al. (1969) Fortier et al. (1974)
Range of values Temperature (K)
Mass fraction (wt%)
272–314 291 293 299 298 293 298 283–383 303–403 298
10–41 2–14.5 0.6–8.5 2–46 0.2–9.5 0.2–4 2–16 8–48 2–7 0.2–4
state. Experimental data on the thermodynamic properties of the system such as molar volume v, molar isobaric heat capacity cp, differential enthalpy of dilution Dhdil, chemical potential mw of the water and ms of the solute provide, in general, redundant information on the derivatives of the molar Gibbs energy in accordance with the standard thermodynamic relations:
vg vp
¼ v;
12/7 5/5 5/5 36/36 17/17 6/6 9/9 28/13 33/0 11/0
v2 g vT2
T2 x2
Number of data total/used
Deviation (%) MRD
RMSD
0.03 0.38 0.17 0.24 0.27 0.04 0.66 0.89 5.82 0.21
1.25 0.59 0.26 0.28 0.36 0.05 0.73 1.66 6.45 0.43
cp ¼ ; T p;x
(5)
v2 g glw ¼ Dhdil : vTvx Tx p
(6)
The chemical potentials of the water and salt in solution are related to the molar Gibbs energy by the equations:
(4)
T;x
Table 7 – Sources of data on f–T–m and g–T–m relation of LiCl–H2O solutions at vapor–liquid equilibrium Author(s) and year
Range of values 1
Temperature (K)
Molality (mol kg )
f(T, m) Robinson (1945) Kangro and Groeneveld (1962) Lindsay and Liu (1971) Hamer and Wu (1972) Gibbard and Scatchard (1973) Holmes and Mesmer (1981) Pan (1981) Holmes and Mesmer (1983) Davis et al. (1985) Davis et al. (1986) Guendouzi et al. (2001)
298.15 298.15 398–548 298.15 298–373 383–473 298.15 498–523 318.15 323.15 298.15
0.1–20.0 1.0–18.0 0.9934–1.0395 0.001–19.219 1.0208–18.585 0.5–6.0 0.0001–0.1 0.6841–6.3164 0.8117–4.2057 0.3–4.5 0.2–6.0
g(T, m) Noyes and MacInnes (1920) MacInnes and Beattie (1920) Scatchard (1925) ˚ kerlo¨f (1926) Harned and A Harned (1929) Pearce and Nelson (1932) Scatchard and Prentiss (1933) Robinson and Sinclair (1934) Robinson and Harned (1941) Robinson (1945) Stokes and Robinson (1948) Lengyel et al. (1960) Caramazza (1963) Momicchioli et al. (1970) Hamer and Wu (1972) Holmes and Mesmer (1981) Pan (1981) Guendouzi et al. (2001)
298.15 298.15 298.15 298.15 298.15 298.15 268–273 298.15 298.15 298.15 298.15 283–307 273–323 251–273 298.15 383–473 298.15 298.15
0.001–3.0 0.001–3.0 0.001–1.0 0.01–3.0 0.01–4.0 0.1–11.1 0.001–1.1 0.1–3.0 0.1–3.0 0.1–20.0 0.1–2.0 0.695–17.73 0.1–6.0 0.0199–3.1623 0.001–19.219 0.5–6.0 0.0001–0.1 0.2–6.0
Number of data total/used
Deviation (%) MRD
RMSD
51/0 12/10 7/1 42/0 54/52 28/0 21/21 67/0 18/0 19/0 14/0
0.44 0.29 0.04 0.28 0.02 0.04 0.15 – 0.23 0.04 0.31
0.60 0.39 0.04 0.60 0.32 0.36 0.17 – 0.40 0.30 0.76
13/0 8/0 14/0 6/0 12/0 20/0 17/0 10/0 10/0 51/0 8/0 27/0 65/0 23/0 42/0 28/0 21/0 14/0
1.46 1.22 1.56 1.15 0.61 43.29 – 0.35 0.05 0.16 0.82 1.31 0.64 – 0.41 3.90 0.90 0.33
1.89 1.82 1.74 1.68 2.41 59.31 – 0.71 0.63 1.11 0.84 2.41 5.06 – 1.84 3.93 0.97 1.04
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Table 8 – Coefficients and exponents of Eq. (17) i
ni
mi
ki
ai
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
0 1 2 3 0 1 2 0 1 2 0 2 3 2 0 0 2 2 1 2 0 1 2
0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1
0 0 0 0 0 0 0 1 2 3 4 4 6 7 10 11 11 12 1 1 2 2 2
1.20915 101 7.05750 1.87439 1.73372 101 5.77828 101 6.52315 101 1.45691 101 1.01324 101 2.86194 101 3.16725 101 3.78901 101 5.29680 101 3.37911 9.89664 101 6.01495 102 6.71921 102 3.42347 102 2.90822 102 1.34713 101 8.68565 102 1.08902 8.20282 101 1.26456 101
vg gx ¼ mw ; vx T;p
(7)
vg ¼ ms : gþ 1x vx T;p
(8)
Values on the chemical potentials are usually given through the (practical) osmotic coefficient f of the solvent and the mean activity coefficient g of the solute defined on a molality basis by the relations (Pitzer, 1998, p. 246): x ; 1x
(9)
xg ; 1x
(10)
mw mlw ¼ nRTf ms m0s ¼ nRTln
where m0s (m0s h g0s ) is the chemical potential of the solute in the standard state. The infinitely dilute standard state is used for the solute (Pitzer, 1998). The factor n in Eqs. (9) and (10) is equal to the number of moles of the particles that are formed when one mole of salt is dissolved (Pitzer, 1998). The factor n is equal to 2 for LiCl. When the gas phase of the system at the
Fig. 1 – Percentage deviations between the primary experimental data of the pressure p of vapor–liquid equilibrium and values calculated from Eq. (34). (@) Tamman (1885), (B) Pearce and Nelson (1932), (4) Gibson and Adams (1933), (9) Lannung (1934), (3) Johnson and Molstad (1951), (6) Kangro and Groeneveld (1962), (,) Broul et al. (1969), (>) Lindsay and Liu (1971), (D) Campbell and Bhatnagar (1979), (7) Khripun et al. (1986), ( ) Chaudhari and Patil (2002), (1) Vercher et al. (2004), (8) Kola´rˇ et al. (2005).
vapor–liquid equilibrium can be considered as a vapor of the pure solvent, which is the present case, the equilibrium pressure p(T, x) enters the equilibrium condition of the form: mw ðT; p; xÞ ¼ mgw ½T; pðT; xÞ:
(11)
To calculate values of the molar Gibbs energy g(T, p, x) of the solution, the values of the right-hand side of Eqs. (4)–(8) and of Eq. (11) should be expressed from experimental data.
Table 9 – Property values calculated from Eqs. (18)–(26) for validation of computer programs x 0.05 0.05 0.1 0.1 0.2 0.3 0.3
T (K)
p (Pa)
9 (mol m3)
H (J mol1)
275 400 300 400 275 350 400
609.80381 216,992.40 2369.4499 174,068.99 144.89490 5105.0296 40,415.042
55,395.5 52,535.6 54,664.0 52,775.2 54,054.8 51,488.6 50,301.2
62.662 8685.8 1515.8 8124.5 840.62 8358.5 11,365.4
S cp (J mol1 K1) (J mol1 K1) 1.63868 27.8383 7.08674 26.0841 0.00299 15.0865 23.1153
70.206 71.104 66.084 66.863 66.297 60.254 60.140
mw (J mol1)
ms (J mol1)
hdil (J mol1)
f
ln g
310.602 2342.19 1094.32 3059.19 3595.59 6836.97 7867.66
4364.55 4488.22 3747.36 4441.14 18,589.6 26,213.6 25,422.2
44.833 162.60 462.52 800.28 3475.6 6975.7 7574.7
1.28876 1.15426 1.17998 1.51577 3.14470 2.44095 2.07885
0.18788 0.15365 1.02366 0.44166 3.64926 3.17387 2.24631
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Fig. 2 – (a) Percentage deviations between the primary experimental data of the density 9 of LiCl–H2O solutions and values calculated from Eq. (22). (9) Gerlach (1859), (C) Abegg (1893), (r) Conroy (1899), (,) Forchheimer (1900), (8) Green (1908), (6) Baxter et al. (1911), (7) Baxter and Wallace (1916), (B) Alfimoff (1917), (:) de Block (1925), (>) Fontell (1927), (-) Applebey et al. (1934), ( ) Gibson and Adams (1933), (;) Campbell and Kartzmark (1956), (D) Bogatykh and Evnovitch (1965), (3) Apelblat and Manzurola (2001); (b) (,) Kremers (1856), (:) Lemoine (1897), (=) Linebarger (1899), (6) Hosking (1904), (;) Che´neveau (1907), (7) Hu ¨ ttig and Keller (1925), (9) Hu ¨ ttig and Ku ¨ kenthal (1928), (>) Scott et al. (1934), (r) Scott and Bridger (1935), (D) Hasaba et al. (1964), (3) Lengyel et al. (1964), (-) Ostroff et al. (1969), (8) Isono (1980), (B) Gates and Wood (1985); (c) (4) Kohlrausch and Grotrian (1875), (5) Kohlrausch (1879), (B) Kohlrausch (1885), (:) Lu ¨ bben (1913), (9) Sachanov (1913), (7) Kohner (1928), (-) Palitzsch (1928), (8) Schreiner (1928), (D) Jones and Bradshaw (1932), (,) Lanman and Mair (1934), (6) Kapustinskii et al. (1960), (>) Millero and Drost-Hansen (1968), (3) Millero et al. (1977); (d) (,) Sprung (1876), (-) Valson (1890), (6) Tower (1908), (3) Washburn and MacInnes (1911), (=) Rodnyanskii and Galinger (1955), (9) Vaslow Vaslow (1966), (8) Vaslow (1969), (D) Tanaka and Tamamushi (1991), (>) Wimby and Berntsson (1994), (B) Vercher et al. (2004).
Based on the thermodynamic theory of electrolyte solutions, molar Gibbs energy of the solution can be expressed in the following general form (Pitzer, 1998, p. 247):
where ki ¼ 0 for the terms of the solute Gibbs energy g0s of the infinitely dilute standard state. The square root of the molar ratio:
i h x 1 þ ge : g ¼ 1 x glw þ xg0s þ nxRT ln 1x
x¼
(12)
The Gibbs energy of pure liquid water glw and the third term of the right-hand side of Eq. (12) representing the ideal mixing contribution are known functions. The excess Gibbs energy ge and the Gibbs energy g0s ( g0s h m0s ) of the solute at the infinitely dilute standard state (Pitzer, 1998, p. 246), should be determined from experimental data on thermodynamic properties of the solution. In the present work, the potentials ge and g0s are approximated by polynomials with terms Pi of the following general form: Pi ¼ ai
Tc T
ni mi p x ki =2 ; pc 1x
(13)
x 1=2 ; 1x
(14)
has been selected as a transformed composition variable x based on the Debye–Hu¨ckel limiting low. To make the notation more concise, let the coefficients ai of the fitting polynomials for g0s and ge be defined so that: g0s ¼ RT
N0 X
(15)
Pi ;
i¼1
with ki ¼ 0 for i N0 and,
ge ¼ xRT
N X i¼N0 þ1
Pi :
(16)
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295
Fig. 2 – (continued).
In this way, the solute standard property and the excess property are included in one polynomial and the approximation function for the molar Gibbs energy is expressed as: " # N x l 1X 1þ Pi : (17) g ¼ 1 x gw þ nxRT ln 1x n i¼1 Definitions (15) and (16) imply that the numerical factor n has been included into the coefficients of the fitting polynomial. As a result n does not enter explicitly the expressions for the solution enthalpy, entropy, molar volume, isobaric heat capacity and differential enthalpy of dilution, which are the quantities most often used in practical computations. Based on Eq. (17) for the Gibbs energy the standard thermodynamic relations give the following expressions for other thermodynamic quantities of the liquid phase of the system: ! N x 1X ki 1þ Pi ; mw T; p; x ¼ mlw nRT 1x n i¼1 2 " # N x 1 X ki ms T; p; x ¼ m0s þ nRT ln þ Pi ; 1þ 1x n i¼N0 þ1 2
N X n i Pi ; h T; p; x ¼ 1 x hlw þ xRT
(18)
N RT X v T;p;x ¼ 1 x vlw þ x mi Pi ; p i¼1 N X ni ðni 1ÞPi ; cp T;p;x ¼ 1 x clp;w xR
(22)
(23)
i¼1 N x X ki n i Pi ; Dhdil T;p;x ¼ RT 1 x i¼1 2
(24)
N 1X ki Pi ; f T;p;x ¼ 1 þ n i¼1 2
(25)
N 1 X ki Pi : 1þ ln g T;p;x ¼ n i¼N0 þ1 2
(26)
The equations for the pure water properties mlw, hlw, slw, vlw, and clp, w are given in Section 4. As ki ¼ 0 for i N0, the lower limit i ¼ 1 in the sums contained in Eqs. (18), (24) and (25) can be changed to i ¼ N0 þ 1. The excess properties are given by the terms of the fitting polynomial with i > N0. For example the excess volume is expressed as:
(19)
N RT X mi Pi : ve T; p; x ¼ x p i¼N0 þ1
(20)
2.2.
(21)
The molar Gibbs energy g(T, p, x) of the solution is determined by the relations (4), (5) and (7) up to an arbitrary function g0(T, x) of the form:
(27)
Reference states
i¼1
" # N x 1X 1 s T;p;x ¼ 1 x slw nxR ln ðni 1ÞPi ; 1x n i¼1
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Fig. 3 – Percentage deviations between the primary experimental data on the isobaric heat capacity cp of LiCl–H2O solutions and values calculated from Eq. (23). (@) Tucker (1915), (,) Jauch (1921), (>) Richards and Ro¨we (1921), ( ) Lange and Du ¨ rr (1926), (D) Gucker and Schminke (1932), (B) Bennewitz and Kratz (1936), (6) Kapustinskii et al. (1960), (4) Uemura et al. (1965).
Tc ; g0 T; x ¼ xRT a1 þ a2 T
(28)
which has no effect on the values of enthalpy and entropy in the pure water limit. In the present study, the coefficients a1 and a2 have been adjusted in accordance with common practice so that the enthalpy and entropy of the solution take their zero values at composition w ¼ 0.3 at temperature 273.15 K (Chaudhari and Patil, 2002). In the present description of the thermodynamic properties of the LiCl–H2O system, the reference values of enthalpy and entropy of pure water are adjusted consistently with the reference state of IAPWS 95 formulation (see Section 4 of the present paper). Since the 5th International Conference on the Properties of Steam in London in 1956, the specific internal energy and specific entropy of the saturated liquid at the triple point have been arbitrarily set equal to zero. This yields for the specific enthalpy of the saturated liquid at the triple point the value of 0.611782 J kg1 (Wagner and Pruß, 2002, p. 429). The number of the significant figures of the coefficients ai of the resultant fitting polynomial has been limited to six in the present study. As a result, the reference values of enthalpy and entropy in the reference states are only approximated. The deviations from exact reference values are small in
Fig. 4 – Percentage deviations between the primary experimental data of the enthalpy of dilution Dhdil of LiCl– H2O solutions and values calculated from Eq. (24). (D) Lange and Du ¨ rr (1926), (B) Johnson and Molstad (1951), ( ) Uemura et al. (1965).
comparison with the enthalpy and entropy of the phase change so that there is no substantial effect on the calculated value of the equilibrium pressure.
2.3. Fitting of the functional form for the Gibbs energy to the data If a certain functional form FðT; p; x; aÞ has been selected for the Gibbs energy, experimental data on thermodynamic properties related through thermodynamic relations to it or its derivatives can be used to determine the unknown coefficients ai (expressed as the vector a) by minimizing the following sum of squares (Wagner and Pruß, 2002): c2 ¼
Mj h J i2 X X ð jÞ ð jÞ zexp;i zcal ðTexp;i ; pi ; xexp;i ; aÞ ; fj j¼1
(29)
i¼1
where Mj is the number of used data points for the jth j) is the ith experimental value for the property property, z(exp,i (j) and zcal,i is the value for the property calculated from the equation for F with the parameter vector a at Texp,i and xexp,i. The weighting factors fj are introduced to equalize the effect of various properties on the regression matrix. The optimum set of regressors for the approximation
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Fig. 5 – Percentage deviations between the primary experimental data of the osmotic coefficient f of LiCl–H2O solutions and values calculated from Eq. (25). (,) Kangro and Groeneveld (1962), (>) Lindsay and Liu (1971), (6) Gibbard and Scatchard (1973), (3) Pan (1981).
function was selected using a step regression method (de Reuck and Armstrong, 1979). Thus, the right-hand sides of j) Eqs. (4)–(8) and (11) represent the experimental values z(exp (j) of the property z , j ¼ 1–6. The left hand sides of Eqs. (4)– j) of the (8) and (11) represent the calculated values z(calc respective properties. At the initial step of the fitting procedure used here, the equilibrium property data available as usual in dependence on the temperature and solution composition (i.e., v(T, x), cp(T, x), etc.) should be completed by corresponding equilibrium values of pressure p(T, x). A separate equation fitted to experimental p–T–x data was used to this purpose. As soon as the first version of the Gibbs energy equation (12) is established, the values of pressure p(T, x) for the next iteration step are computed from the condition of vapor–liquid equilibrium as described in the next section. In each step, a new set of optimal regressors is selected and their corresponding coefficients ai are computed.
3.
p–T–x Relation
The p–T–x relation is defined by the conditions of the phase equilibrium given, in general, by equality of chemical potentials of the solvent and solute in coexisting phases:
297
Fig. 6 – Percentage deviations between the data of the activity coefficient g of LiCl–H2O solutions and values calculated from Eq. (26). (<) Noyes and MacInnes (1920), (7) MacInnes and Beattie (1920), (3) Scatchard (1925), ( ) ˚ kerlo¨f (1926), (9) Harned (1929), (,) Robinson Harned and A and Sinclair (1934), (6) Robinson and Harned (1941), (B) Robinson (1945), (>) Stokes and Robinson (1948), (;) Lengyel et al. (1960), (=) Caramazza (1963), (:) Hamer and Wu (1972), (C) Holmes and Mesmer (1981), (D) Pan (1981), (8) Guendouzi et al. (2001).
mw T; p; x ¼ mvw T; p; y ;
(30)
ms T; p; x ¼ mvs T; p; y :
(31)
In the present model, the vapor molar fraction y of the solute is supposed to be equal to zero ( y h 0) and the conditions of equilibrium (30) and (31) reduce to the equation: (32) mw T; p; x ¼ mgw T; p ; where the chemical potential of water in the gas phase mvw(T, p, y) is approximated by the chemical potential of pure water mgw(T, p). The condition (32) defines the equilibrium p–T–x relation. For given values of each two of the variables T, p and x the value of the third one can be computed from the condition. The chemical potential mgw(T, p) of the water in gas phase has the general form of: p þ u T; p ; (33) mgw T; p ¼ RT ln pc where u(T, p) denotes the chemical potential function except for the term logarithmic in pressure (see Eq. (37), Section 4).
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The condition (32) can thus be rewritten into an iteration equation: mw T; pi ; x u T; pi : (34) piþ1 T; x ¼ pc exp RT Eq. (34) is solvable with precision of 0.1 Pa in several iterations starting from an arbitrary initial value of the pressure.
4. Description of the thermodynamic properties of pure water substance Equation for the Gibbs energy ggw(T, p) and glw(T, p) of pure gas and liquid phase water at pressure and temperature of the mixture were developed in the present study, based on data calculated from the IAPWS 95 formulation (Wagner and Pruß, 2002). With the following notation for the dimensionless variables s, p and polynomial terms Pi: s¼
Tc p ; p¼ ; pc T
(35)
Pi ¼ ai sni pmi ;
(36)
the approximation of thermodynamic properties of pure water substance used in the present work can be expressed in a general form common both to liquid and gas phase ( glw h mlw, ggw h mgw): " # N X ml;g ln p þ a ln s þ P T; p ¼ RT a (37) 0 1 i ; w i¼2
hl;g w
"
T; p ¼ RT a1 þ
N X
# (38)
ni Pi ;
i¼2
sl;g w
# N X ðni 1ÞPi ; T; p ¼ R a0 ln p þ a1 1 ln s þ
"
Fig. 7 – Percentage deviations between the rejected experimental data of the pressure p of vapor–liquid equilibrium and values calculated from Eq. (34). (@) Tamman (1885), (;) Dieterici (1891), (-) Tower (1908), (:) Lovelace et al. (1923), (D) Schlu ¨ nder (1963), (3) Uemura et al. (1965), (C) Sada et al. (1975), (<) Fedorov et al. (1976), (r) Safarov et al. (2003).
(39)
i¼2
5.
!
N X RT vl;g mi Pi ; a0 þ w T; p ¼ p i¼2
" # N X ðn n 1ÞP T; p ¼ R a : cl;g 1 i i i p;w
(40)
(41)
i¼2
Table 1 presents the obtained coefficients ai and exponents ni and mi for liquid and gas phase, respectively. The coefficient a0 equal to zero for liquid phase and equal to 1 for gas is introduced only to unify the notation of Eqs. (37)–(41) for both phases. The approximation equations for ggw(T, p) and glw(T, p) are valid at temperatures from 273.16 K to 400 K. The maximum deviations of the calculated values from the IAPWS 95 values are given in Table 2. The accuracy of approximation used in the present study for the liquid phase turned out to be necessary to represent the most precise experimental data on the solution density within their experimental uncertainties. Values generated by Eqs. (37)–(41) are given in Table 3 to facilitate their implementation.
Data selection
Not all the original works provide sufficient comparable evidence on which to base a judgement of the relative merits of the various sets of results. To assess the quality of the particular sets of experimental data, all the data for each property (i.e., p–T–x, 9Tx, and cp–T–x relation) were preliminary fitted simultaneously with weights equal to the reciprocal of the number of points in each set. In this way, the effect of different number of data points contained in particular data sets was canceled. Mean relative (i.e., systematic) deviation of each particular data set provides an idea of how the set is shifted as a whole with respect to the common average values given by the resultant function. In this way, the experimental data were divided into two categories: primary data employed in the development of the correlation and secondary data with excessive systematic deviation and/or excessively scattered, which were used only for comparison purposes. The data sets covering a subregion of the T–x plane covered by no other data set were accepted for correlation in any case. In addition, some isolated points from the accepted data sets were also rejected when their deviation from the preliminary
Fig. 8 – (a) Percentage deviations between the rejected experimental data of the density 9 of LiCl–H2O solutions and values calculated from Eq. (22). (6) Kuschel (1881), (;) Bender (1883), (7) Ro¨ntgen and Schneider (1886), (=) Bender (1887), (r) Engel (1888), (9) Jahn (1891), (5) Lemoine (1897), (1) Linebarger (1899), (8) Che´neveau (1907), (:) Baxter et al. (1911), (>) Grufki (1913), ( ) Lu¨bben (1913), (4) Sachanov (1913), (-) Alfimoff (1917), (@) Jones and Bradshaw (1932), (3) Nickels and Allmand (1937), (,) Guillaume (1946), (B) Hasaba et al. (1964), (C) Bogatykh and Evnovitch (1965), (D) Mashovets et al. (1971), (?) Safarov et al. (2003); (b) (:) Fouque´ (1867), (7) Wagner (1890), (-) Perkin (1894), (B) Guerdikova (1910), (C) Henderson and Kellogg (1916), (r) de Block (1925), (1) Hu ¨ ttig and Keller (1925), (*) Sugden (1926), (6) Hu ¨ ttig and Ku ¨ kenthal (1928), (3) Stratmann (1948), (D) Rodnyanskii and Galinger (1955), (>) Desnoyers et al. (1969), (,) Wimby and Berntsson (1994).
correlation exceeded three times the root mean square deviation. The sources of the assessed sets of experimental data together with their relative root mean square deviation (RMSD) and mean relative deviation (MRD) are listed in Tables 4–7. The temperature, pressure and composition ranges covered by each data set are also indicated here. The number of points used for fitting is given in the table for the primary data while for the secondary data it is equaled to zero. Within the present study, 136 experimental works have been collected containing a total of 2921 data points on various thermodynamic properties of the LiCl–H2O solutions. A body of 1704 of them was selected as the primary data used for fitting of the representative equation.
6.
Fig. 9 – Percentage deviations between the rejected experimental data of the isobaric heat capacity cp of LiCl– H2O solutions and values calculated from Eq. (23). (@) Tucker (1915), (4) Uemura et al. (1965), (-) Ru ¨ terjans et al. (1969), (:) Fortier et al. (1974).
Results
The respective coefficients ai and exponents mi, ni and ki of the Gibbs energy g0s and ge in Eq. (17) are given in Table 8. The number of significant figures given in the coefficients ai is necessary and sufficient to obtain the stated accuracy. Values of the particular properties generated by the resultant representative equations are provided in Table 9 to assist with the confirmation of computer implementation of the calculation procedure.
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Fig. 12 – p–T–x Relation calculated from Eq. (34).
Fig. 10 – Percentage deviations between the rejected data of the osmotic coefficient f of LiCl–H2O solutions and values calculated from Eq. (25). (B) Robinson (1945), (,) Kangro and Groeneveld (1962), (9) Hamer and Wu (1972), (6) Gibbard and Scatchard (1973), (7) Holmes and Mesmer (1981), (3) Davis et al. (1985), (;) Davis et al. (1986), (8) Guendouzi et al. (2001).
Fig. 13 – Density of solution calculated from Eq. (22).
Fig. 11 – Chemical potential mw of water in solution calculated from Eq. (18).
Fig. 14 – Isobaric heat capacity calculated from Eq. (23).
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301
Fig. 18 – Activity coefficients, Eq. (26). Fig. 15 – Enthalpy according to Eq. (20).
Fig. 16 – Entropy according to Eq. (21).
Estimates for the uncertainty of an empirical equation of state must be guided by comparisons with experimental data and the assessment of their uncertainty. In Figs. 1–5 the deviations of the data used in the fitting procedure from the corresponding resultant representative equation are depicted, while in the Figs. 7–10 deviations for the rejected data are plotted. Data on activity coefficient were used only for comparison (Fig. 6). Based on these comparisons the uncertainties associated with correlation are estimated to be 0.4% for molar volume, 2.0% for pressure and 2.4% for isobaric heat capacity. In the case of enthalpy and entropy, comparisons with existing formulations can assist in the assessment. The values of relative uncertainties in pressure, density (molar volume) and isobaric heat capacity for the present formulation of the thermodynamic properties of LiCl–H2O solutions are quite similar to those of the LiBr–H2O system in Pa´tek and Klomfar (2006). Therefore, uncertainty in values of enthalpy can be estimated in accordance with Pa´tek and Klomfar (2006) to be less than
Fig. 17 – Enthalpy of dilution according to Eq. (24).
Fig. 19 – Osmotic coefficients, Eq. (25).
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10 kJ kg1 and for entropy less than 0.03 kJ kg1 K1. With decreasing molar fraction x, all the uncertainties should decrease to those of approximation of pure water properties summarized in Table 2. The resulting thermodynamic properties are depicted in Figs. 11–19 as functions of the solution composition.
7.
Conclusion
The description of the thermodynamic properties of the LiCl– H2O system in the vapor–liquid equilibrium states developed in the present work fulfills most of the requirements given in Section 1. The available experimental data on the properties of the LiCl–H2O system have been compiled and evaluated using a well-substantiated quantitative procedure. The method used to develop the equation for the Gibbs energy makes possible to process all the relevant experimental information symmetrically and the redundancy of the information is thus fully utilized. The formulation has the form of two separate equations for Gibbs energy of liquid and vapor phase, so that the equilibrium p–T–x relation can be established by solution of the equation representing the condition of equilibrium. The code in ANSI C implementing the present formulation of the thermodynamic properties of the LiCl– H2O system is available from the authors on request. The formulation describes well the specific form of the dependency of the osmotic coefficient on composition close to pure water, though at the price that a reasonable extrapolation in the solute molar fraction x beyond the region covered by experimental data (limited from above by x ¼ 0.3 and w ¼ 0.5) is not possible. In contrary, extrapolation at temperatures up to 400 K may be acceptable. Comparisons of available measurements on the thermodynamic properties of the LiCl–H2O solutions have shown that the amount of available experimental data is far less valuable to establish a description of the thermodynamic properties of the system than it might appear at a first glance. Many of the available sets of VLE data are only of limited value, because they show large scatter or systematic deviations when compared to other data. The largest gap in the data is clearly found at temperatures below 298 K, above 373 K and at compositions above 44 wt% (x ¼ 0.25) of LiCl in the solution. Single-phase (compressed liquid) data are quite scarce and fragmentary.
Acknowledgment The work described in this paper has been performed under the research intention no. AV0Z20760514 awarded by the Academy of Sciences of the Czech Republic.
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Thermodynamic properties of the LiCl–H2O system at vapor–liquid equilibrium from 273 K to 400 K J. Pa´tek*, J. Klomfar Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Dolejsˇkova 5, CZ 182 00 Prague 8, Czech Republic
article info
abstract
Article history:
A formulation of the thermodynamic properties of the LiCl–H2O system at vapor–liquid
Received 1 December 2006
equilibrium is presented in the form of separate Gibbs energy equations for the vapor
Received in revised
and solution phases. Explicit thermodynamically consistent equations for density, isobaric
form 27 March 2007
heat capacity, enthalpy, entropy, enthalpy of dilution and osmotic coefficient are given.
Accepted 7 May 2007
The pressure at vapor–liquid equilibrium is calculated from the condition of phase equilib-
Published online 18 May 2007
rium. The description of the properties is valid from 273 K or from the crystallization line up to 400 K in temperatures and for solution composition from 0 to 50 wt% of LiCl in the
Keywords:
solution, which is the region covewred by available experimental data. The thermody-
Thermodynamic property
namic properties of the gas phase are approximated by the properties of pure water vapor
Aqueous solution
computed from the IAPWS formulation 1995. The Gibbs energy equation for solution is
Water
based upon a body of experimental data that have been critically assessed. Within the
Lithium chloride
present study, 136 experimental works have been collected containing a total of 2921
Equilibrium
data points on various thermodynamic properties of the LiCl–H2O solutions. Gaps in the data-
Vapour
base are shown to give experimenters orientation for future research. The uncertainties
Liquid
associated with correlation are estimated to be 0.4% for density, 2.0% for pressure
Calculation
and 2.4% for isobaric heat capacity. The uncertainty in values of enthalpy is estimated
Pressure
to be less than 10 kJ kg1 and less than 0.03 kJ kg1 K1 for entropy. Values of the partic-
Equation of state
ular properties generated by the representative equations are provided to assist with the confirmation of computer implementation of the calculation procedure. ª 2007 Elsevier Ltd and IIR. All rights reserved.
Proprie´te´s thermodnamiques d’un syste`me au LiCl-H2O en e´quilibre vapeur-liquide entre 273 K et 400 K Mots cle´s : Proprie´te´ thermodynamique ; Solution aqueuse ; Eau ; Chlorure de lithium ; E´quilibre ; Vapeur ; Liquide ; Calcul ; Pression ; E´quation d’e´tat
* Corresponding author. Tel.: þ420 266053153; fax: þ420 28584695. E-mail address: [email protected] (J. Pa´tek). 0140-7007/$ – see front matter ª 2007 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2007.05.003
288
international journal of refrigeration 31 (2008) 287–303
Nomenclature cp g g hw h M m mw N n n u
p p P f Dhdil s T s v w x
1.
molar isobaric heat capacity (J mol1 K1) molar Gibbs free energy (J mol1) activity coefficient partial enthalpy of water in solution (J mol1) molar enthalpy (J mol1) molar mass (kg mol1) mass (kg); in Table 7: molality (mol kg1) chemical potential of water in the solution (J mol1) number of points amount of substance (mol) number of moles of particles that are formed when one mole of salt is dissolved chemical potential of water vapor except for the term logarithmic in the pressure (mgw ¼ RT ln( p/pc) þ u) dimensionless pressure variable (p ¼ p/pc) pressure (Pa) polynomial term (P ¼ asnpm for pure water and P ¼ asnpmxk for solution) osmotic coefficient molar enthalpy of dilution (J mol1) molar entropy (J mol1 K1) temperature (K) dimensionless temperature variable (s ¼ Tc/T ) molar volume (m3 mol1) mass fraction (w ¼ mLiCl/(mLiCl þ mw)) molar fraction (x ¼ nLiCl/(nLiCl þ nw))
Introduction
Although significant improvements are being made in predicting thermodynamic properties of pure fluids and mixtures using theory based methods, more accurate equations of state for calculating properties over a wide range of pressures and temperatures are to be developed by correlation of selected experimental data. For the development of such empirical
x 9 MRD RMSD
transformed composition variable (x ¼ [x/(1 x)](1/2)) molar density (mol m3) mean relative deviation, % ðMRD ¼ ð102 =NÞ P ðzexp =zcal 1ÞÞ root mean square relative deviation, % ðRMSD ¼ P 102 ½ð1=NÞ ðzexp =zcal 1Þ2 1=2 Þ
Subscripts c at the critical point cal calculated e excess quantity exp experimental LiCl lithium chloride p isobaric w water s solute Superscripts g vapor states of pure substance l liquid states of pure substance v vapor (gaseous phase of the binary system) 0 infinitely dilute standard states Physical R MLiCl MH2 O Tc pc rc
constants 8.31451 J mol1 K1 0.04932 kg mol1 0.018015268 kg mol1 647.096 K 22.064 106 Pa 17,874 mol m3 (322 kg m3)
formulations, the application of linear optimization procedures and nonlinear multiproperty fitting algorithm is stateof-art (Span, 2000). Modern equations of state are usually explicit in the Helmholtz energy as a function of density and temperature (Span, 2000). As far as binary mixtures are concerned, Tillner-Roth and Friend (1998), for example, have published a formulation of the thermodynamic properties of the
Table 1 – Coefficients and exponents of Eqs. (37)–(41) i
0 1 2 3 4 5 6 7 8 9 10 11 12
Liquid phase
Vapor phase
ni
mi
ai
ni
mi
ai
– – 1 0 1 2 3 4 1 0 1 2 3
– – 0 0 0 0 0 0 1 1 1 1 1
0.0 5.88137 101 9.18938 8.33307 101 1.84228 102 1.70534 102 1.05457 102 2.94856 101 1.96926 101 8.68008 101 2.25497 2.60177 1.17349
– – 1 0 3 6 7 8 15 – – – –
– – 0 0 0 1 1 1 3 – – – –
1.0 3.98432 6.68909 8.78439 5.18911 102 6.25248 101 5.07144 101 1.24364 101 4.49013 101 – – – –
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international journal of refrigeration 31 (2008) 287–303
Table 2 – Maximum differences of values of enthalpy, entropy, molar volume and isobaric thermal capacity calculated from Eqs. (38)–(41), respectively, from the IAPWS 95 formulation Quantity 1
Dh (J mol ) Ds (J mol1 K1) Dv (%) Dcp (%)
Liquidus
Vapor
0.6 0.0015 0.01 0.04
10 0.02 0.8 0.5
ammonia–water system in the form of Helmholtz free energy. Experimental data covering single-phase region are indispensable for reasonable application of that approach. Data on the properties of the LiCl–H2O concerning the states outside the vapor–liquid equilibrium are scarce and incomplete (Brown et al., 2004; Majer et al., 1989, 1991; Pepinov et al., 1989; Urusova, 1971; Ohling and Schneider, 1979; Gates and Wood, 1985; Ellis, 1966). Two separate Gibbs energy functions for liquid and gas phase provide an alternative form of the thermodynamically consistent description of the vapor–liquid equilibrium region. Yuan and Herold (2005) and Kim and
Table 3 – Property values calculated from Eqs. (37)–(41) for validation of computer programs P (Pa)
9 (mol m3)
cp (J mol1 K1)
Liquid phase 275 300 400
698.46145 3536.8953 245,882.35
55,503.3 55,316.3 52,035.3
75.912 75.314 76.666
139.75 2028.3 9600.8
0.50970 7.08295 28.8458
0.4164 96.571 1937.5
Gas phase 275 300 400
698.46145 3536.8953 245,882.35
0.305671 1.420484 76.10544
33.847 34.344 40.022
45,115.5 45,937.1 48,913.6
164.058 153.446 127.128
0.4164 96.571 1937.5
T (K)
h (J mol1)
s (J mol1 K1)
m (J mol1)
Table 4 – Sources of data on p–T–x relation of LiCl–H2O solutions at vapor–liquid equilibrium Author(s) and year
Range of values Temperature (K)
Mass fraction (wt%)
315–373 273 273 293 244–374 298 293 333–453 291 275–383 303–343 293 299–348 278–338 303–394 333 398 348 423 256–303 323–423 298–323 283–313 303–373 398 374–387 341–376
7–22 8–30 2–35 0.4–4 – 0.4–3 15–44 – 3–42 – 5–51 4–43 9–44 – 11–47 4–34 4 7–37 31–42 – 2–11 24–43 – 13–44 10–35 5–25 10–30
Tamman (1885) Dieterici (1891) Tower (1908) Lovelace et al. (1923) Hu¨ttig and Reuscher (1924)a Pearce and Nelson (1932) Gibson and Adams (1933) Applebey et al. (1934)a Lannung (1934) Gokcen (1951)a Johnson and Molstad (1951) Kangro and Groeneveld (1962) Schlu¨nder (1963) Acheson (1965)a Uemura et al. (1965) Broul et al. (1969) Lindsay and Liu (1971) Sada et al. (1975) Fedorov et al. (1976) Adamcova´ and Benesˇ (1977)a Campbell and Bhatnagar (1979) Khripun et al. (1986) Apelblat (1993)a Chaudhari and Patil (2002) Safarov et al. (2003) Vercher et al. (2004) Kola´rˇ et al. (2005) a Vapor pressures of saturated solutions.
Number of data total/used 96/70 4/0 5/0 10/0 22/0 25/25 8/8 19/0 20/20 16/0 58/58 18/18 75/0 13/0 56/0 17/17 1/1 6/0 2/0 34/0 30/30 12/12 7/0 72/72 3/0 6/6 9/9
Deviation (%) MRD
RMSD
0.09 9.46 17.59 2.04 – 1.28 2.19 – 0.34 – 1.35 0.22 0.76 – 1.31 0.07 0.02 13.11 1.37 – 0.23 0.47 – 0.43 20.78 0.48 0.07
0.51 12.11 36.03 2.33 – 2.07 2.56 – 0.53 – 2.01 0.37 3.72 – 6.82 1.25 0.02 19.09 1.38 – 0.45 1.86 – 0.86 25.93 0.57 0.78
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Table 5 – Sources of data on 9–T–x relation of LiCl–H2O solutions at vapor–liquid equilibrium Author(s) and year
Kremers (1856) Gerlach (1859) Fouque´ (1867) Kohlrausch and Grotrian (1875) Sprung (1876) Kohlrausch (1879) Kuschel (1881) Bender (1883) Kohlrausch (1885) Ro¨ntgen and Schneider (1886) Bender (1887) Engel (1888) Valson (1890) Wagner (1890) Jahn (1891) Abegg (1893) Perkin (1894) Lemoine (1897) Conroy (1899) Linebarger (1899) Forchheimer (1900) Hosking (1904) Che´neveau (1907) Green (1908) Tower (1908) Guerdikova (1910) Baxter et al. (1911) Washburn and MacInnes (1911) Grufki (1913) Lu¨bben (1913) Sachanov (1913) Baxter and Wallace (1916) Henderson and Kellogg (1916) Alfimoff (1917) de Block (1925) Hu¨ttig and Keller (1925) Sugden (1926) Fontell (1927) Hu¨ttig and Ku¨kenthal (1928) Kohner (1928) Palitzsch (1928) Schreiner (1928) Jones and Bradshaw (1932) Applebey et al. (1934) Lanman and Mair (1934) Scott et al. (1934) Gibson (1935) Scott and Bridger (1935) Nickels and Allmand (1937) Guillaume (1946) Stratmann (1948) Rodnyanskii and Galinger (1955) Campbell and Kartzmark (1956) Kapustinskii et al. (1960) Hasaba et al. (1964) Lengyel et al. (1964) Bogatykh and Evnovitch (1965) Vaslow (1966) Millero and Drost-Hansen (1968) Desnoyers et al. (1969) Ostroff et al. (1969) Vaslow (1969) Postnikov (1970)
Range of values Temperature (K)
Mass fraction (wt%)
292 288 273–285 289–293 283 288–291 286–294 288 291 291 288 273 288 298 293 289 288–293 273 291 298 288 273–373 292–303 291–298 273 298 298 273 291 291 298 273–373 373 273–350 289 293 291 279–343 293 298 298 291 298 274–429 298 308 298 308 298 293 291–333 298–623 298 298 278–363 288–308 283–373 298 293–313 298 298 278–308 293–353
5–38 10–43 8–28 5–10 8–27 3–40 0.2–25 11–14 4 3–7 4–20 40.5 4 0.5–4 5–11 4–8.5 25–50 4–43 2–4 9–40 4–36 0.02–37 4–43 2–41 2–35 10–43 0.6–12 0.1–4 2–14 2–14 7–46 1.2–45 0.1–10.8 7–47 4–42 0.4–33 0.4–6 3–44 0.4–34 0.4–28 8–45 2–34 0.1–11 41–58 2–8 4–42 4.5–39 28–41 0.4–15 6.6–32 4–14.5 4–11 10 2–16 11-46 32–45 32–46 0.1–12 0.5 0.3–3 4–19 0.2–11 –
Number of data total/used 6/6 5/5 4/0 2/2 3/3 4/4 9/0 4/0 1/1 2/0 6/0 2/0 1/1 4/0 2/0 2/2 6/0 6/4 2/2 4/2 5/5 96/96 14/12 16/16 5/5 7/0 12/10 9/9 4/0 4/2 4/2 27/27 7/0 18/11 5/4 11/6 5/0 24/24 11/4 8/8 7/7 7/7 12/10 15/5 3/3 8/8 8/8 3/3 12/0 2/0 24/0 18/7 1/1 9/9 40/34 48/48 98/93 27/27 21/21 8/0 9/9 34/34 7/0
Deviation (%) MRD
RMSD
0.11 0.08 3.51 0.00 0.00 0.01 0.13 0.47 0.07 0.21 0.38 0.79 0.03 0.34 0.22 0.10 3.35 0.06 0.04 0.06 0.12 0.01 0.00 0.04 0.04 0.90 0.03 0.02 0.21 0.04 0.02 0.01 5.07 0.11 0.03 0.05 0.10 0.03 0.03 0.00 0.01 0.03 0.03 0.04 0.06 0.00 0.03 0.06 0.35 0.01 0.22 0.04 0.24 0.00 0.03 0.04 0.02 0.01 0.01 0.87 0.11 0.05 –
0.12 0.10 4.72 0.01 0.04 0.09 0.13 0.49 0.07 0.33 0.59 0.79 0.03 0.34 0.23 0.17 3.40 0.12 0.05 0.06 0.18 0.09 0.17 0.08 0.09 0.99 0.05 0.05 0.32 0.05 0.04 0.12 7.09 0.16 0.09 0.06 0.11 0.09 0.03 0.02 0.09 0.05 0.04 0.06 0.06 0.06 0.06 0.10 0.35 0.17 0.29 0.07 0.24 0.02 0.21 0.06 0.18 0.02 0.01 1.00 0.13 0.06 –
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Table 5 – (continued) Author(s) and year
Mashovets et al. (1971) Millero et al. (1977) Isono (1980) Gates and Wood (1985) Tanaka and Tamamushi (1991) Wimby and Berntsson (1994) Apelblat and Manzurola (2001) Safarov et al. (2003) Vercher et al. (2003)
Range of values Temperature (K)
Mass fraction (wt%)
213–273 298 288–328 298 288–328 290–343 278–338 398 288–318
5–35 0.5–4.5 0.2–30 0.2–17 0.2–46 10–45 0.4–4 10–35 4.5–34
Infante Ferreira (2006) developed a description of this form for the LiBr–H2O solutions. No thermodynamically consistent description of the properties of lithium chloride–water system is available in open literature. The LiCl–H2O system is one of the working-pairs for absorption cycles studied as an alternative to the most common working fluids – H2O–NH3 and LiBr–H2O. In practical computations, separate equations are often preferred for describing particular thermodynamic properties of interest, though such equations are thermodynamically consistent only approximately in the numerical sense. There exist a number of studies that provide description of various binary systems in the form of a set of separate equations for pressure, density, isobaric thermal capacity, enthalpy, and entropy of the system at vapor–liquid equilibrium. The historical Haltenberger’s method (1939) is used to compute values of enthalpy and entropy of the system from experimental data. In the case of LiCl–H2O solutions, Chaudhari and Patil (2002) have published their enthalpy in the form of polynomial valid up to 50 wt% and 393 K. Conde (2004) has published separate equations for various thermophysical properties of the system including equilibrium pressure, density, isobaric thermal capacity and differential enthalpy of dilution. The aim of the present study was to provide a formulation of the thermodynamic properties of LiCl–H2O system in the vapor–liquid equilibrium states valid from 273 K or from the crystallization temperature (whichever is greater) up to 400 K in temperatures and from pure water up to 50 wt% in compositions. The formulation should be based on all available experimental data and it should approximate the correlated data within their experimental uncertainties. The property description should be computationally effective, i.e., explicit, simple and easily implementable. These are indispensable properties for the description to find its users in practical calculations. The formulation has the form of two separate equations for Gibbs energy of the liquid and vapor phase so that the equilibrium p–T–x relation can be established from the condition of phase equilibrium. The gas phase of the system can be considered as pure steam as the vapor pressure of the LiCl is quite negligible to the vapor pressure of H2O. All vapor phase properties are therefore considered as functions of temperature and pressure only, independent on the solute mole fraction y which is considered to be identically equal to zero. The description of the thermodynamic properties of the pure water based on
Number of data total/used 26/0 9/9 70/70 8/8 113/113 52/45 183/183 3/0 36/36
Deviation (%) MRD
RMSD
0.28 0.01 0.04 0.01 0.04 0.04 0.01 0.61 0.09
0.32 0.01 0.06 0.04 0.12 0.07 0.01 0.75 0.12
approximation of the IAPWS 1995 formulation (Wagner and Pruß, 2002) is given in Section 4. Thus the main task is to develop a description of the molar Gibbs energy of the LiCl–H2O solutions. In the present paper, partial molar quantities of components in a mixture and molar quantities of pure substance related to water or solute are distinguished by subscript w or s, respectively, e.g., mw, mvs . The quantities related to the vapor phase of the mixture are distinguished by superscript v, e.g., mvw. The quantities related to the solution which occur the most often in equations have no superscript letter, to make the notation more readable, i.e., g, h, mw, ms. The phase is indicated by superscript g for gas and by l for the liquid phase, in the case of a property of a pure substance to distinguish it from that of mixture, e.g., glw, ggw. All solution properties are considered as functions of temperature, pressure and composition expressed as the mole fraction x of the solute in the solution. It should be mentioned that the molar Gibbs energy of pure water is identical to its chemical potential, i.e., ggw h mgw and glw h mlw. Specific quantities and mass fraction w used in common computational practice are related to the molar quantities by the following relations: xMLiCl ; xMLiCl þ ð1 xÞMw
(1)
w=MLiCl ; w=MLiCl þ ð1 wÞ=Mw
(2)
w¼
x¼
hðwÞ 1 hðxÞ ¼ ; J kg1 xMLiCl þ ð1 xÞMw J mol1
(3)
and similarly for density, entropy and isobaric heat capacity.
2. Development of the Gibbs energy equation for the solution 2.1. Determination of the Gibbs energy values from experimental data Values of the molar Gibbs energy g(T, p, x) of the solution with composition given by the molar fraction x of the solute are defined through the derivatives of the Gibbs energy and by a reference value chosen by convention in an arbitrary reference
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Table 6 – Sources of data on cp–T–x relation of LiCl–H2O solutions at vapor–liquid equilibrium Author(s) and year
Tucker (1915) Jauch (1921) Richards and Ro¨we (1921) Lange and Du¨rr (1926) Gucker and Schminke (1932) Bennewitz and Kratz (1936) Kapustinskii et al. (1960) Uemura et al. (1965) Ru¨terjans et al. (1969) Fortier et al. (1974)
Range of values Temperature (K)
Mass fraction (wt%)
272–314 291 293 299 298 293 298 283–383 303–403 298
10–41 2–14.5 0.6–8.5 2–46 0.2–9.5 0.2–4 2–16 8–48 2–7 0.2–4
state. Experimental data on the thermodynamic properties of the system such as molar volume v, molar isobaric heat capacity cp, differential enthalpy of dilution Dhdil, chemical potential mw of the water and ms of the solute provide, in general, redundant information on the derivatives of the molar Gibbs energy in accordance with the standard thermodynamic relations:
vg vp
¼ v;
12/7 5/5 5/5 36/36 17/17 6/6 9/9 28/13 33/0 11/0
v2 g vT2
T2 x2
Number of data total/used
Deviation (%) MRD
RMSD
0.03 0.38 0.17 0.24 0.27 0.04 0.66 0.89 5.82 0.21
1.25 0.59 0.26 0.28 0.36 0.05 0.73 1.66 6.45 0.43
cp ¼ ; T p;x
(5)
v2 g glw ¼ Dhdil : vTvx Tx p
(6)
The chemical potentials of the water and salt in solution are related to the molar Gibbs energy by the equations:
(4)
T;x
Table 7 – Sources of data on f–T–m and g–T–m relation of LiCl–H2O solutions at vapor–liquid equilibrium Author(s) and year
Range of values 1
Temperature (K)
Molality (mol kg )
f(T, m) Robinson (1945) Kangro and Groeneveld (1962) Lindsay and Liu (1971) Hamer and Wu (1972) Gibbard and Scatchard (1973) Holmes and Mesmer (1981) Pan (1981) Holmes and Mesmer (1983) Davis et al. (1985) Davis et al. (1986) Guendouzi et al. (2001)
298.15 298.15 398–548 298.15 298–373 383–473 298.15 498–523 318.15 323.15 298.15
0.1–20.0 1.0–18.0 0.9934–1.0395 0.001–19.219 1.0208–18.585 0.5–6.0 0.0001–0.1 0.6841–6.3164 0.8117–4.2057 0.3–4.5 0.2–6.0
g(T, m) Noyes and MacInnes (1920) MacInnes and Beattie (1920) Scatchard (1925) ˚ kerlo¨f (1926) Harned and A Harned (1929) Pearce and Nelson (1932) Scatchard and Prentiss (1933) Robinson and Sinclair (1934) Robinson and Harned (1941) Robinson (1945) Stokes and Robinson (1948) Lengyel et al. (1960) Caramazza (1963) Momicchioli et al. (1970) Hamer and Wu (1972) Holmes and Mesmer (1981) Pan (1981) Guendouzi et al. (2001)
298.15 298.15 298.15 298.15 298.15 298.15 268–273 298.15 298.15 298.15 298.15 283–307 273–323 251–273 298.15 383–473 298.15 298.15
0.001–3.0 0.001–3.0 0.001–1.0 0.01–3.0 0.01–4.0 0.1–11.1 0.001–1.1 0.1–3.0 0.1–3.0 0.1–20.0 0.1–2.0 0.695–17.73 0.1–6.0 0.0199–3.1623 0.001–19.219 0.5–6.0 0.0001–0.1 0.2–6.0
Number of data total/used
Deviation (%) MRD
RMSD
51/0 12/10 7/1 42/0 54/52 28/0 21/21 67/0 18/0 19/0 14/0
0.44 0.29 0.04 0.28 0.02 0.04 0.15 – 0.23 0.04 0.31
0.60 0.39 0.04 0.60 0.32 0.36 0.17 – 0.40 0.30 0.76
13/0 8/0 14/0 6/0 12/0 20/0 17/0 10/0 10/0 51/0 8/0 27/0 65/0 23/0 42/0 28/0 21/0 14/0
1.46 1.22 1.56 1.15 0.61 43.29 – 0.35 0.05 0.16 0.82 1.31 0.64 – 0.41 3.90 0.90 0.33
1.89 1.82 1.74 1.68 2.41 59.31 – 0.71 0.63 1.11 0.84 2.41 5.06 – 1.84 3.93 0.97 1.04
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Table 8 – Coefficients and exponents of Eq. (17) i
ni
mi
ki
ai
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
0 1 2 3 0 1 2 0 1 2 0 2 3 2 0 0 2 2 1 2 0 1 2
0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1
0 0 0 0 0 0 0 1 2 3 4 4 6 7 10 11 11 12 1 1 2 2 2
1.20915 101 7.05750 1.87439 1.73372 101 5.77828 101 6.52315 101 1.45691 101 1.01324 101 2.86194 101 3.16725 101 3.78901 101 5.29680 101 3.37911 9.89664 101 6.01495 102 6.71921 102 3.42347 102 2.90822 102 1.34713 101 8.68565 102 1.08902 8.20282 101 1.26456 101
vg gx ¼ mw ; vx T;p
(7)
vg ¼ ms : gþ 1x vx T;p
(8)
Values on the chemical potentials are usually given through the (practical) osmotic coefficient f of the solvent and the mean activity coefficient g of the solute defined on a molality basis by the relations (Pitzer, 1998, p. 246): x ; 1x
(9)
xg ; 1x
(10)
mw mlw ¼ nRTf ms m0s ¼ nRTln
where m0s (m0s h g0s ) is the chemical potential of the solute in the standard state. The infinitely dilute standard state is used for the solute (Pitzer, 1998). The factor n in Eqs. (9) and (10) is equal to the number of moles of the particles that are formed when one mole of salt is dissolved (Pitzer, 1998). The factor n is equal to 2 for LiCl. When the gas phase of the system at the
Fig. 1 – Percentage deviations between the primary experimental data of the pressure p of vapor–liquid equilibrium and values calculated from Eq. (34). (@) Tamman (1885), (B) Pearce and Nelson (1932), (4) Gibson and Adams (1933), (9) Lannung (1934), (3) Johnson and Molstad (1951), (6) Kangro and Groeneveld (1962), (,) Broul et al. (1969), (>) Lindsay and Liu (1971), (D) Campbell and Bhatnagar (1979), (7) Khripun et al. (1986), ( ) Chaudhari and Patil (2002), (1) Vercher et al. (2004), (8) Kola´rˇ et al. (2005).
vapor–liquid equilibrium can be considered as a vapor of the pure solvent, which is the present case, the equilibrium pressure p(T, x) enters the equilibrium condition of the form: mw ðT; p; xÞ ¼ mgw ½T; pðT; xÞ:
(11)
To calculate values of the molar Gibbs energy g(T, p, x) of the solution, the values of the right-hand side of Eqs. (4)–(8) and of Eq. (11) should be expressed from experimental data.
Table 9 – Property values calculated from Eqs. (18)–(26) for validation of computer programs x 0.05 0.05 0.1 0.1 0.2 0.3 0.3
T (K)
p (Pa)
9 (mol m3)
H (J mol1)
275 400 300 400 275 350 400
609.80381 216,992.40 2369.4499 174,068.99 144.89490 5105.0296 40,415.042
55,395.5 52,535.6 54,664.0 52,775.2 54,054.8 51,488.6 50,301.2
62.662 8685.8 1515.8 8124.5 840.62 8358.5 11,365.4
S cp (J mol1 K1) (J mol1 K1) 1.63868 27.8383 7.08674 26.0841 0.00299 15.0865 23.1153
70.206 71.104 66.084 66.863 66.297 60.254 60.140
mw (J mol1)
ms (J mol1)
hdil (J mol1)
f
ln g
310.602 2342.19 1094.32 3059.19 3595.59 6836.97 7867.66
4364.55 4488.22 3747.36 4441.14 18,589.6 26,213.6 25,422.2
44.833 162.60 462.52 800.28 3475.6 6975.7 7574.7
1.28876 1.15426 1.17998 1.51577 3.14470 2.44095 2.07885
0.18788 0.15365 1.02366 0.44166 3.64926 3.17387 2.24631
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Fig. 2 – (a) Percentage deviations between the primary experimental data of the density 9 of LiCl–H2O solutions and values calculated from Eq. (22). (9) Gerlach (1859), (C) Abegg (1893), (r) Conroy (1899), (,) Forchheimer (1900), (8) Green (1908), (6) Baxter et al. (1911), (7) Baxter and Wallace (1916), (B) Alfimoff (1917), (:) de Block (1925), (>) Fontell (1927), (-) Applebey et al. (1934), ( ) Gibson and Adams (1933), (;) Campbell and Kartzmark (1956), (D) Bogatykh and Evnovitch (1965), (3) Apelblat and Manzurola (2001); (b) (,) Kremers (1856), (:) Lemoine (1897), (=) Linebarger (1899), (6) Hosking (1904), (;) Che´neveau (1907), (7) Hu ¨ ttig and Keller (1925), (9) Hu ¨ ttig and Ku ¨ kenthal (1928), (>) Scott et al. (1934), (r) Scott and Bridger (1935), (D) Hasaba et al. (1964), (3) Lengyel et al. (1964), (-) Ostroff et al. (1969), (8) Isono (1980), (B) Gates and Wood (1985); (c) (4) Kohlrausch and Grotrian (1875), (5) Kohlrausch (1879), (B) Kohlrausch (1885), (:) Lu ¨ bben (1913), (9) Sachanov (1913), (7) Kohner (1928), (-) Palitzsch (1928), (8) Schreiner (1928), (D) Jones and Bradshaw (1932), (,) Lanman and Mair (1934), (6) Kapustinskii et al. (1960), (>) Millero and Drost-Hansen (1968), (3) Millero et al. (1977); (d) (,) Sprung (1876), (-) Valson (1890), (6) Tower (1908), (3) Washburn and MacInnes (1911), (=) Rodnyanskii and Galinger (1955), (9) Vaslow Vaslow (1966), (8) Vaslow (1969), (D) Tanaka and Tamamushi (1991), (>) Wimby and Berntsson (1994), (B) Vercher et al. (2004).
Based on the thermodynamic theory of electrolyte solutions, molar Gibbs energy of the solution can be expressed in the following general form (Pitzer, 1998, p. 247):
where ki ¼ 0 for the terms of the solute Gibbs energy g0s of the infinitely dilute standard state. The square root of the molar ratio:
i h x 1 þ ge : g ¼ 1 x glw þ xg0s þ nxRT ln 1x
x¼
(12)
The Gibbs energy of pure liquid water glw and the third term of the right-hand side of Eq. (12) representing the ideal mixing contribution are known functions. The excess Gibbs energy ge and the Gibbs energy g0s ( g0s h m0s ) of the solute at the infinitely dilute standard state (Pitzer, 1998, p. 246), should be determined from experimental data on thermodynamic properties of the solution. In the present work, the potentials ge and g0s are approximated by polynomials with terms Pi of the following general form: Pi ¼ ai
Tc T
ni mi p x ki =2 ; pc 1x
(13)
x 1=2 ; 1x
(14)
has been selected as a transformed composition variable x based on the Debye–Hu¨ckel limiting low. To make the notation more concise, let the coefficients ai of the fitting polynomials for g0s and ge be defined so that: g0s ¼ RT
N0 X
(15)
Pi ;
i¼1
with ki ¼ 0 for i N0 and,
ge ¼ xRT
N X i¼N0 þ1
Pi :
(16)
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295
Fig. 2 – (continued).
In this way, the solute standard property and the excess property are included in one polynomial and the approximation function for the molar Gibbs energy is expressed as: " # N x l 1X 1þ Pi : (17) g ¼ 1 x gw þ nxRT ln 1x n i¼1 Definitions (15) and (16) imply that the numerical factor n has been included into the coefficients of the fitting polynomial. As a result n does not enter explicitly the expressions for the solution enthalpy, entropy, molar volume, isobaric heat capacity and differential enthalpy of dilution, which are the quantities most often used in practical computations. Based on Eq. (17) for the Gibbs energy the standard thermodynamic relations give the following expressions for other thermodynamic quantities of the liquid phase of the system: ! N x 1X ki 1þ Pi ; mw T; p; x ¼ mlw nRT 1x n i¼1 2 " # N x 1 X ki ms T; p; x ¼ m0s þ nRT ln þ Pi ; 1þ 1x n i¼N0 þ1 2
N X n i Pi ; h T; p; x ¼ 1 x hlw þ xRT
(18)
N RT X v T;p;x ¼ 1 x vlw þ x mi Pi ; p i¼1 N X ni ðni 1ÞPi ; cp T;p;x ¼ 1 x clp;w xR
(22)
(23)
i¼1 N x X ki n i Pi ; Dhdil T;p;x ¼ RT 1 x i¼1 2
(24)
N 1X ki Pi ; f T;p;x ¼ 1 þ n i¼1 2
(25)
N 1 X ki Pi : 1þ ln g T;p;x ¼ n i¼N0 þ1 2
(26)
The equations for the pure water properties mlw, hlw, slw, vlw, and clp, w are given in Section 4. As ki ¼ 0 for i N0, the lower limit i ¼ 1 in the sums contained in Eqs. (18), (24) and (25) can be changed to i ¼ N0 þ 1. The excess properties are given by the terms of the fitting polynomial with i > N0. For example the excess volume is expressed as:
(19)
N RT X mi Pi : ve T; p; x ¼ x p i¼N0 þ1
(20)
2.2.
(21)
The molar Gibbs energy g(T, p, x) of the solution is determined by the relations (4), (5) and (7) up to an arbitrary function g0(T, x) of the form:
(27)
Reference states
i¼1
" # N x 1X 1 s T;p;x ¼ 1 x slw nxR ln ðni 1ÞPi ; 1x n i¼1
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Fig. 3 – Percentage deviations between the primary experimental data on the isobaric heat capacity cp of LiCl–H2O solutions and values calculated from Eq. (23). (@) Tucker (1915), (,) Jauch (1921), (>) Richards and Ro¨we (1921), ( ) Lange and Du ¨ rr (1926), (D) Gucker and Schminke (1932), (B) Bennewitz and Kratz (1936), (6) Kapustinskii et al. (1960), (4) Uemura et al. (1965).
Tc ; g0 T; x ¼ xRT a1 þ a2 T
(28)
which has no effect on the values of enthalpy and entropy in the pure water limit. In the present study, the coefficients a1 and a2 have been adjusted in accordance with common practice so that the enthalpy and entropy of the solution take their zero values at composition w ¼ 0.3 at temperature 273.15 K (Chaudhari and Patil, 2002). In the present description of the thermodynamic properties of the LiCl–H2O system, the reference values of enthalpy and entropy of pure water are adjusted consistently with the reference state of IAPWS 95 formulation (see Section 4 of the present paper). Since the 5th International Conference on the Properties of Steam in London in 1956, the specific internal energy and specific entropy of the saturated liquid at the triple point have been arbitrarily set equal to zero. This yields for the specific enthalpy of the saturated liquid at the triple point the value of 0.611782 J kg1 (Wagner and Pruß, 2002, p. 429). The number of the significant figures of the coefficients ai of the resultant fitting polynomial has been limited to six in the present study. As a result, the reference values of enthalpy and entropy in the reference states are only approximated. The deviations from exact reference values are small in
Fig. 4 – Percentage deviations between the primary experimental data of the enthalpy of dilution Dhdil of LiCl– H2O solutions and values calculated from Eq. (24). (D) Lange and Du ¨ rr (1926), (B) Johnson and Molstad (1951), ( ) Uemura et al. (1965).
comparison with the enthalpy and entropy of the phase change so that there is no substantial effect on the calculated value of the equilibrium pressure.
2.3. Fitting of the functional form for the Gibbs energy to the data If a certain functional form FðT; p; x; aÞ has been selected for the Gibbs energy, experimental data on thermodynamic properties related through thermodynamic relations to it or its derivatives can be used to determine the unknown coefficients ai (expressed as the vector a) by minimizing the following sum of squares (Wagner and Pruß, 2002): c2 ¼
Mj h J i2 X X ð jÞ ð jÞ zexp;i zcal ðTexp;i ; pi ; xexp;i ; aÞ ; fj j¼1
(29)
i¼1
where Mj is the number of used data points for the jth j) is the ith experimental value for the property property, z(exp,i (j) and zcal,i is the value for the property calculated from the equation for F with the parameter vector a at Texp,i and xexp,i. The weighting factors fj are introduced to equalize the effect of various properties on the regression matrix. The optimum set of regressors for the approximation
international journal of refrigeration 31 (2008) 287–303
Fig. 5 – Percentage deviations between the primary experimental data of the osmotic coefficient f of LiCl–H2O solutions and values calculated from Eq. (25). (,) Kangro and Groeneveld (1962), (>) Lindsay and Liu (1971), (6) Gibbard and Scatchard (1973), (3) Pan (1981).
function was selected using a step regression method (de Reuck and Armstrong, 1979). Thus, the right-hand sides of j) Eqs. (4)–(8) and (11) represent the experimental values z(exp (j) of the property z , j ¼ 1–6. The left hand sides of Eqs. (4)– j) of the (8) and (11) represent the calculated values z(calc respective properties. At the initial step of the fitting procedure used here, the equilibrium property data available as usual in dependence on the temperature and solution composition (i.e., v(T, x), cp(T, x), etc.) should be completed by corresponding equilibrium values of pressure p(T, x). A separate equation fitted to experimental p–T–x data was used to this purpose. As soon as the first version of the Gibbs energy equation (12) is established, the values of pressure p(T, x) for the next iteration step are computed from the condition of vapor–liquid equilibrium as described in the next section. In each step, a new set of optimal regressors is selected and their corresponding coefficients ai are computed.
3.
p–T–x Relation
The p–T–x relation is defined by the conditions of the phase equilibrium given, in general, by equality of chemical potentials of the solvent and solute in coexisting phases:
297
Fig. 6 – Percentage deviations between the data of the activity coefficient g of LiCl–H2O solutions and values calculated from Eq. (26). (<) Noyes and MacInnes (1920), (7) MacInnes and Beattie (1920), (3) Scatchard (1925), ( ) ˚ kerlo¨f (1926), (9) Harned (1929), (,) Robinson Harned and A and Sinclair (1934), (6) Robinson and Harned (1941), (B) Robinson (1945), (>) Stokes and Robinson (1948), (;) Lengyel et al. (1960), (=) Caramazza (1963), (:) Hamer and Wu (1972), (C) Holmes and Mesmer (1981), (D) Pan (1981), (8) Guendouzi et al. (2001).
mw T; p; x ¼ mvw T; p; y ;
(30)
ms T; p; x ¼ mvs T; p; y :
(31)
In the present model, the vapor molar fraction y of the solute is supposed to be equal to zero ( y h 0) and the conditions of equilibrium (30) and (31) reduce to the equation: (32) mw T; p; x ¼ mgw T; p ; where the chemical potential of water in the gas phase mvw(T, p, y) is approximated by the chemical potential of pure water mgw(T, p). The condition (32) defines the equilibrium p–T–x relation. For given values of each two of the variables T, p and x the value of the third one can be computed from the condition. The chemical potential mgw(T, p) of the water in gas phase has the general form of: p þ u T; p ; (33) mgw T; p ¼ RT ln pc where u(T, p) denotes the chemical potential function except for the term logarithmic in pressure (see Eq. (37), Section 4).
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The condition (32) can thus be rewritten into an iteration equation: mw T; pi ; x u T; pi : (34) piþ1 T; x ¼ pc exp RT Eq. (34) is solvable with precision of 0.1 Pa in several iterations starting from an arbitrary initial value of the pressure.
4. Description of the thermodynamic properties of pure water substance Equation for the Gibbs energy ggw(T, p) and glw(T, p) of pure gas and liquid phase water at pressure and temperature of the mixture were developed in the present study, based on data calculated from the IAPWS 95 formulation (Wagner and Pruß, 2002). With the following notation for the dimensionless variables s, p and polynomial terms Pi: s¼
Tc p ; p¼ ; pc T
(35)
Pi ¼ ai sni pmi ;
(36)
the approximation of thermodynamic properties of pure water substance used in the present work can be expressed in a general form common both to liquid and gas phase ( glw h mlw, ggw h mgw): " # N X ml;g ln p þ a ln s þ P T; p ¼ RT a (37) 0 1 i ; w i¼2
hl;g w
"
T; p ¼ RT a1 þ
N X
# (38)
ni Pi ;
i¼2
sl;g w
# N X ðni 1ÞPi ; T; p ¼ R a0 ln p þ a1 1 ln s þ
"
Fig. 7 – Percentage deviations between the rejected experimental data of the pressure p of vapor–liquid equilibrium and values calculated from Eq. (34). (@) Tamman (1885), (;) Dieterici (1891), (-) Tower (1908), (:) Lovelace et al. (1923), (D) Schlu ¨ nder (1963), (3) Uemura et al. (1965), (C) Sada et al. (1975), (<) Fedorov et al. (1976), (r) Safarov et al. (2003).
(39)
i¼2
5.
!
N X RT vl;g mi Pi ; a0 þ w T; p ¼ p i¼2
" # N X ðn n 1ÞP T; p ¼ R a : cl;g 1 i i i p;w
(40)
(41)
i¼2
Table 1 presents the obtained coefficients ai and exponents ni and mi for liquid and gas phase, respectively. The coefficient a0 equal to zero for liquid phase and equal to 1 for gas is introduced only to unify the notation of Eqs. (37)–(41) for both phases. The approximation equations for ggw(T, p) and glw(T, p) are valid at temperatures from 273.16 K to 400 K. The maximum deviations of the calculated values from the IAPWS 95 values are given in Table 2. The accuracy of approximation used in the present study for the liquid phase turned out to be necessary to represent the most precise experimental data on the solution density within their experimental uncertainties. Values generated by Eqs. (37)–(41) are given in Table 3 to facilitate their implementation.
Data selection
Not all the original works provide sufficient comparable evidence on which to base a judgement of the relative merits of the various sets of results. To assess the quality of the particular sets of experimental data, all the data for each property (i.e., p–T–x, 9Tx, and cp–T–x relation) were preliminary fitted simultaneously with weights equal to the reciprocal of the number of points in each set. In this way, the effect of different number of data points contained in particular data sets was canceled. Mean relative (i.e., systematic) deviation of each particular data set provides an idea of how the set is shifted as a whole with respect to the common average values given by the resultant function. In this way, the experimental data were divided into two categories: primary data employed in the development of the correlation and secondary data with excessive systematic deviation and/or excessively scattered, which were used only for comparison purposes. The data sets covering a subregion of the T–x plane covered by no other data set were accepted for correlation in any case. In addition, some isolated points from the accepted data sets were also rejected when their deviation from the preliminary
Fig. 8 – (a) Percentage deviations between the rejected experimental data of the density 9 of LiCl–H2O solutions and values calculated from Eq. (22). (6) Kuschel (1881), (;) Bender (1883), (7) Ro¨ntgen and Schneider (1886), (=) Bender (1887), (r) Engel (1888), (9) Jahn (1891), (5) Lemoine (1897), (1) Linebarger (1899), (8) Che´neveau (1907), (:) Baxter et al. (1911), (>) Grufki (1913), ( ) Lu¨bben (1913), (4) Sachanov (1913), (-) Alfimoff (1917), (@) Jones and Bradshaw (1932), (3) Nickels and Allmand (1937), (,) Guillaume (1946), (B) Hasaba et al. (1964), (C) Bogatykh and Evnovitch (1965), (D) Mashovets et al. (1971), (?) Safarov et al. (2003); (b) (:) Fouque´ (1867), (7) Wagner (1890), (-) Perkin (1894), (B) Guerdikova (1910), (C) Henderson and Kellogg (1916), (r) de Block (1925), (1) Hu ¨ ttig and Keller (1925), (*) Sugden (1926), (6) Hu ¨ ttig and Ku ¨ kenthal (1928), (3) Stratmann (1948), (D) Rodnyanskii and Galinger (1955), (>) Desnoyers et al. (1969), (,) Wimby and Berntsson (1994).
correlation exceeded three times the root mean square deviation. The sources of the assessed sets of experimental data together with their relative root mean square deviation (RMSD) and mean relative deviation (MRD) are listed in Tables 4–7. The temperature, pressure and composition ranges covered by each data set are also indicated here. The number of points used for fitting is given in the table for the primary data while for the secondary data it is equaled to zero. Within the present study, 136 experimental works have been collected containing a total of 2921 data points on various thermodynamic properties of the LiCl–H2O solutions. A body of 1704 of them was selected as the primary data used for fitting of the representative equation.
6.
Fig. 9 – Percentage deviations between the rejected experimental data of the isobaric heat capacity cp of LiCl– H2O solutions and values calculated from Eq. (23). (@) Tucker (1915), (4) Uemura et al. (1965), (-) Ru ¨ terjans et al. (1969), (:) Fortier et al. (1974).
Results
The respective coefficients ai and exponents mi, ni and ki of the Gibbs energy g0s and ge in Eq. (17) are given in Table 8. The number of significant figures given in the coefficients ai is necessary and sufficient to obtain the stated accuracy. Values of the particular properties generated by the resultant representative equations are provided in Table 9 to assist with the confirmation of computer implementation of the calculation procedure.
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Fig. 12 – p–T–x Relation calculated from Eq. (34).
Fig. 10 – Percentage deviations between the rejected data of the osmotic coefficient f of LiCl–H2O solutions and values calculated from Eq. (25). (B) Robinson (1945), (,) Kangro and Groeneveld (1962), (9) Hamer and Wu (1972), (6) Gibbard and Scatchard (1973), (7) Holmes and Mesmer (1981), (3) Davis et al. (1985), (;) Davis et al. (1986), (8) Guendouzi et al. (2001).
Fig. 13 – Density of solution calculated from Eq. (22).
Fig. 11 – Chemical potential mw of water in solution calculated from Eq. (18).
Fig. 14 – Isobaric heat capacity calculated from Eq. (23).
international journal of refrigeration 31 (2008) 287–303
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Fig. 18 – Activity coefficients, Eq. (26). Fig. 15 – Enthalpy according to Eq. (20).
Fig. 16 – Entropy according to Eq. (21).
Estimates for the uncertainty of an empirical equation of state must be guided by comparisons with experimental data and the assessment of their uncertainty. In Figs. 1–5 the deviations of the data used in the fitting procedure from the corresponding resultant representative equation are depicted, while in the Figs. 7–10 deviations for the rejected data are plotted. Data on activity coefficient were used only for comparison (Fig. 6). Based on these comparisons the uncertainties associated with correlation are estimated to be 0.4% for molar volume, 2.0% for pressure and 2.4% for isobaric heat capacity. In the case of enthalpy and entropy, comparisons with existing formulations can assist in the assessment. The values of relative uncertainties in pressure, density (molar volume) and isobaric heat capacity for the present formulation of the thermodynamic properties of LiCl–H2O solutions are quite similar to those of the LiBr–H2O system in Pa´tek and Klomfar (2006). Therefore, uncertainty in values of enthalpy can be estimated in accordance with Pa´tek and Klomfar (2006) to be less than
Fig. 17 – Enthalpy of dilution according to Eq. (24).
Fig. 19 – Osmotic coefficients, Eq. (25).
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10 kJ kg1 and for entropy less than 0.03 kJ kg1 K1. With decreasing molar fraction x, all the uncertainties should decrease to those of approximation of pure water properties summarized in Table 2. The resulting thermodynamic properties are depicted in Figs. 11–19 as functions of the solution composition.
7.
Conclusion
The description of the thermodynamic properties of the LiCl– H2O system in the vapor–liquid equilibrium states developed in the present work fulfills most of the requirements given in Section 1. The available experimental data on the properties of the LiCl–H2O system have been compiled and evaluated using a well-substantiated quantitative procedure. The method used to develop the equation for the Gibbs energy makes possible to process all the relevant experimental information symmetrically and the redundancy of the information is thus fully utilized. The formulation has the form of two separate equations for Gibbs energy of liquid and vapor phase, so that the equilibrium p–T–x relation can be established by solution of the equation representing the condition of equilibrium. The code in ANSI C implementing the present formulation of the thermodynamic properties of the LiCl– H2O system is available from the authors on request. The formulation describes well the specific form of the dependency of the osmotic coefficient on composition close to pure water, though at the price that a reasonable extrapolation in the solute molar fraction x beyond the region covered by experimental data (limited from above by x ¼ 0.3 and w ¼ 0.5) is not possible. In contrary, extrapolation at temperatures up to 400 K may be acceptable. Comparisons of available measurements on the thermodynamic properties of the LiCl–H2O solutions have shown that the amount of available experimental data is far less valuable to establish a description of the thermodynamic properties of the system than it might appear at a first glance. Many of the available sets of VLE data are only of limited value, because they show large scatter or systematic deviations when compared to other data. The largest gap in the data is clearly found at temperatures below 298 K, above 373 K and at compositions above 44 wt% (x ¼ 0.25) of LiCl in the solution. Single-phase (compressed liquid) data are quite scarce and fragmentary.
Acknowledgment The work described in this paper has been performed under the research intention no. AV0Z20760514 awarded by the Academy of Sciences of the Czech Republic.
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