A GIBBS ENERGY EQUATION FOR LiBr/H2O SOLUTIONS

A GIBBS ENERGY EQUATION FOR LiBr/H2O SOLUTIONS
D.S. Kim*
C.A. Infante Ferreira
Engineering Thermodynamics, Mechanical Engineering, TU Delft
Mekelweg 2, 2628 CD Delft, the Netherlands
Abstract
A Gibbs energy equation has been developed for aqueous LiBr solutions in the concentration range
from 0 to 70wt.% and the temperatures from 0 to 210oC with the equilibrium pressures ranging from
74Pa to 1MPa. Osmotic coefficients were determined using 405 experimental solution density and 496
vapor pressure data points from the literature. A function for the enthalpy of infinitely dilute solution
was determined using experimental heat capacity data. The resulting enthalpy, heat capacity and
entropy values are compared with literature values and the comparative analysis results are critically
evaluated. The results of this study would be useful especially for the systems where the solution
undergoes large concentration- or pressure differences such as low-temperature driven multi-stage
solar cycles and power cycles where the solutions would be exposed to high-pressure environments.
Introduction
The aqueous LiBr solutions have been popular in absorption refrigeration industry for a long time due
to the outstanding thermodynamic characteristics. And their use seems not to be limited to the
conventional applications but extends well beyond. As more attention is being paid to rational use of
energy, more researchers are trying to exploit unconventional working ranges with advanced
absorption cycles (e.g., Kang et al[1], Kojima et al[2]). The need for a study capable of accurately
describing the properties of the solutions in wide working ranges has been growing (e.g., Hellmann
and Grossman[3]).
Among the early studies on the thermodynamics of the solution, the most prominent is probably
Löwer[4] in 1960. It was the first complete study to present practically all thermophysical properties of
the solution. The applicable range of the study is, however, limited by maximum solution temperature
of 130oC.
The most well-known work is, however, probably McNeely[5] in 1979. He presented dew
temperatures and solution enthalpy data for wide ranges of temperature and concentration using
Dühring’s rule and the Haltenburger’s method[6]. It has been successful in industry, even though it
shows some questionable trends above 64wt.% as the author reported.
In 1987, Herold and Moran[7] have successfully reproduced McNeely’s data[5] using the Gibbs
energy equation with a modified Debye-Hückel model. Using a statistical method, they determined
coefficients of a Gibbs energy equation from a very limited amount of known data at the time.
In 1994, Feuerecker et al[8] carried out a study based on their own experimental data in the
concentration range from 40~75wt% and temperatures up to 190oC. Their vapor pressures agree well
with McNeely[5] up to 60wt.%, but significant differences were reported in the high-concentration
region. Their lower concentration limit of application range is 40wt.% and the heat of dilution
calculated from their dew temperature equation shows somewhat large deviations from the
measurement of Lange and Schwartz[9].
In 2000, Chua et al[10] developed a set of equations for the solutions in the concentration range from 0
to 75wt% and temperature range from 0 to 190oC. For a dew temperature equation, they deliberately
curve-fitted the Dühring gradients and intercepts of McNeely[5] for solutions below 60wt% and those
of Feuerecker et al[8] above this concentration, resulting in two 20th-order polynomial functions with
40 constants. But it is questionable if Dühring’s rule is applicable to such a wide concentration and
pressure ranges and if the choice of LiBr weight fraction as a fitting parameter is justifiable.
One year later in 2001, Kaita[11] suggested a set of new equations mainly for triple-effect machines
in high-pressure environments, having supplemented the vapor pressures of Feuerecker et al[8] with
the high-pressure data from Lenard et al[12] and the low-temperature data from McNeely[5]. His
results are valid in the limited concentration range from 40 to 65wt.% and temperatures from
1
20oC(40oC for entropy) to 210oC with the equilibrium pressure up to 1MPa. He introduced a 2nd-order
temperature term in his dew temperature equation to cover the wide pressure range. But since his
enthalpy calculation did not include any information of his dew temperature equation, the heat of
dilution from his enthalpy differs substantially from the one from his dew temperature equation. And
he used the doubtful part of heat of solution from McNeely[5] in the high concentration region.
As described above, all of the earlier studies were either limited to narrow working ranges or failed to
provide simple and accurate descriptions for wide working ranges of the solution. Based on the
observed need for a new description of extended solution property fields, this study intends to provide
a Gibbs energy equation for aqueous LiBr solutions in the concentration range from 0 to 70wt.% and
the temperature range from 0 to 210oC with the pressure ranging from 74Pa to 1MPa.
Thermodynamic equations for binary electrolyte solutions
Unlike the mixture models for non-electrolyte solutions, the description of an electrolyte solution
requires a hypothetical reference fluid called ‘infinitely dilute solution’ because the pure solute is a
solid substance at standard states. In a electrolyte system, concentration of a solute is casually
expressed in molality m which is customarily defined as ‘the number of mol solute per kg solvent’. But
it is redefined here as ‘the number of kmol solute per kg solvent’ for convenience and its relations with
other concentrations are given in Eq.(1). In the following, the subscript 1 denotes LiBr and the
subscript 2 denotes water and complete dissociation of the solute is assumed(υ=2).
m=
x1w
x1
=
(1 − x1 ) M 2 (1 − x1w ) M 1
(1)
x1 and x1w are the stoichiometric molar fraction and the weight fraction of LiBr respectively. Based on
the thermodynamic theories of electrolyte solutions, molar Gibbs energy of an electrolyte solution can
be expressed as follows (e.g., Ruiter[13]).
⎡ ⎛m⎞ ⎤
G l (T , p , x1 ) = x1G1∞ (T , p ) + (1 − x1 )G2l (T , p ) + x1υ RT ⎢ln ⎜ ⎟ − 1⎥ + G E (T , p ,m )
⎣ ⎝ mD ⎠ ⎦
(2)
The 1st term is a contribution of the infinitely dilute solution and the 2nd terms is that of pure water.
The 3rd term is the Gibbs energy generation in an ideal mixing process, where mo is standard molality
(mo=0.001 kmol/kg solvent). And the last term is the excess Gibbs energy GE by which a real solution
differs from ideal one. It is separately given in Eq.(3).
G E (T , p , x1 ) = x1υ RT ⎡⎣ln γ ± + (1 − φ ) ⎤⎦
(3)
φ and γ± are called osmotic- and mean ionic activity coefficient respectively. They are the measures of
non-idealistic behavior of molecules in a real solution. Since only the steam exists in the vapor phase,
the osmotic coefficient φ has the relationship with the states of pure water as in Eq.(4). It is clear that
φ serves as a VLE(Vapor-Liquid-Equilibrium parameter) of the system. And from the definition of
‘partial property’, the Gibbs-Duhem relation of Eq.(5) applies between φ and γ±.
1
φ=
RTυ mM 2
p*
∫ (V
g
2
− V2l )dp
ln γ ± = φ − 1 + ∫
(4)
m
0
p
(φ − 1)
dm
m
(5)
Therefore, once φ is determined as a function of concentration from Eq.(4), γ± follows from Eq.(5)
and so GE in Eq.(3) can be fully described.
Differentiations of Eq.(2) give the rest of the solution properties as follows. They also apply to the
corresponding excess properties.
2
⎡ ∂ (G l / RT ) ⎤
H l = − RT 2 ⎢
⎥
∂T
⎣
⎦ p,x
(6)
⎛ ∂G l ⎞
Sl = −⎜
⎟
⎝ ∂T ⎠ p , x
(7)
⎛ ∂G l ⎞
Vl =⎜
⎟
⎝ ∂p ⎠T , x
(8)
⎛ ∂ 2G l ⎞
Cp l = −T ⎜
2 ⎟
⎝ ∂T ⎠ p , x
(9)
Determination of osmotic coefficient
φ is a function of pressure p, temperature T and concentration m. φ may be assumed to have such a
functional form as Eq.(10).
φ = φ ′(T , m) + φ ′′(T , p, m)
(10)
The pressure-dependent term φ″ can be determined when solution densities are known. And then, φ′
can be determined from Eq.(4) using equilibrium vapor pressures.
Inserting Eq.(2) into Eq.(8) gives a pressure derivative of the molar Gibbs energy of the solution, i.e. a
molar volume of the solution, as follows.
V(lT , p ,m ) = x1V1(∞T , p ) + (1 − x1 )V2(l T , p ) + V(TE , p ,m )
(11)
V1∞ and V2l are the molar volume of the infinitely dilute solution and pure water respectively. The last
term VE is the excess volume, a pressure derivative of GE.
⎡⎛ ∂ ln γ ± ⎞
⎛ ∂φ ⎞ ⎤
V E = x1υ RT ⎢⎜
−
⎟
⎜ ⎟ ⎥
⎢⎣⎝ ∂p ⎠T , x ⎝ ∂p ⎠T , x ⎥⎦
(12)
V1∞ and VE in Eq.(11) can be determined from experimental density data. The experimental solution
density from the International Critical Table[14] (hereafter ICT), Löwer[4], Lee et al[15] and the
specific volume of water from Schmidt[16] were used. Neglecting the pressure dependence of solution
and water volume, Eq.(11) could be expressed as a function of temperature and concentration by
2
2
V l = x1 RT b0 + (1 − x1 )V2l + x1 RT ∑ bi mi / 2
where bi = ∑ bijT − j
i =1
(13)
j =0
The coefficients bij are given in Table 2. V1∞ and VE in Eq.(11) can be identified in comparison with
Eq.(13). The m1/2 as a fitting parameter in Eq.(13) was found to be the best, which is also supported by
the theory of Debye-Hückel[17]. Eq.(13) reproduces the solution density from ICT[14], Löwer[4] and
Lee et al[15] within 0.1%, 0.2% and 0.3% of average deviation respectively.
2
Lower
ICT
Lee et al
0
150oC
(ρexp-ρcal)/ρexp(%)
-2
2
0
100oC
-2
2
0
t=50oC
-2
0
0.2
0.4
0.6
0.8
LiBr weight fraction
Figure 1. Calculated and experimental solution density
3
From Eq.(5), (12) and (13), the pressure derivative of φ is given by Eq.(14). And then the pressuredependent term φ″ in Eq.(10) may be expressed by Eq.(15)
⎛ ∂φ ⎞
1 2
=
i ⋅ bi mi / 2
∑
⎜ ⎟
∂
2
υ
p
i =1
⎝ ⎠T , x
φ ′′ =
(14)
p 2
∑ i ⋅ bi mi / 2
2υ i =1
(15)
The pressure-independent term φ′ in Eq.(10) is then determined from Eq.(4). From Eq.(4), (10) and
(15), φ′ is given by
p*
1
φ′ =
RTυ mM 2
g
l
∫ (V2 − V2 )dp −
p
p 2
i ⋅ bi mi / 2
∑
2υ i =1
(16)
For ease of calculation, the following function has been developed for the term Vv2-Vl2 in Eq.(16)
based on the steam data from Schmidt[16].
V2g -V2l =
RT
tanh [α - β ln( p) ]
P
(17)
where α = 11.375 − 3.859 × 103 T −1 + 5.132 × 105 T −2 and β = 0.86 − 1.958 ×102 T −1 + 2.314 × 104 T −2 . Since
V2l is negligibly small, the term tanh[α-βln(p)] can be used as the compressibility factor with
maximum error of 0.11% in steam volume up to 1,200kPa and 270oC. The saturated steam pressure p*
from Schmidt[16] and Perry et al[18] and total 6 sets of vapor pressure data were used to calculate φ′
from Eq.(16). A brief summary of the data sources and fitting results is given in Table 1.
Table 1. Equilibrium vapor pressure data sources and fitting results for φ′
Data source
Löwer[4]
ICT[14]
McNeely[5]
Feuerecker et al[8]
Iyoki and Uemura[19]
Lenard et al[12]
t(oC)
0~130
0~100
0~180
45~190
101~180
125~211
LiBr wt.%
0~70
0~45
45~64
40.4~76
38.9~70.3
43.8~65.2
No. of data
185
36
131
80
40
24
Avg. dev.(%)
1.7
0.7
1.6
1.9
4.3
4.6
The vapor pressures from ICT[14] are exceptionally higher than the others above solution
concentrations of 45wt.% and they were disregarded because they are likely in error as McNeely[5]
and Koehler et al[20] suggested. The heat of dilution curves of McNeely[5] show maximum values
between 64wt.% and the crystallization points. As he mentioned, since it is doubtful if his pressure
gradients are correct in the region, his vapor pressure data above 64wt.% were disregarded. The data
below 45wt.% were also disregarded because it was found identical to ICT[14] in that region.
φ′ in Eq.(16) was calculated for each vapor pressure data and the results were fitted with a 6th-order
polynomial function of m1/2. This fitting function was used in Eq.(10) with Eq.(15) to express φ as
6
φ = 1 + ∑ (ai +
i =1
ibi
p )mi / 2
2υ
2
where ai = ∑ aijT − j
(18)
j =0
The coefficients are listed in Table 2. Some examples of the calculated φ′ and the fitting curves are
shown in Fig.2.
Since φ″ is negligibly small, for the purpose of pressure calculations, φ′ can be safely used instead of
φ. This avoids iterations of the following pressure equation for the converged solution of p.
p =exp{β −1[α - ln (θ + θ 2 - 1)]} where θ = cosh(α -β ln p* ) ⋅ exp(φυ mM 2 β )
4
(19)
6
Lower
ICT
McNeely
Feuerecker et al
Iyoki & Uemura
Jetter et al
this study
4
t=25 oC
Hamer & Wu
Robinson & McCoach
this study
o
t=20 C
5
100 oC
4
φ'
φ / ln γ
3
180 oC
2
3
φ
2
1
1
lnγ
0
-1
0
0.04
0.08
0.12
0.16
0.2
0
0.04
m1/2
0.08
m
Figure 2. Calculated φ′ and the fitting curves
0.12
1/2
Figure 3. φ and lnγ± at 25oC
With φ determined from Eq.(18), lnγ± and GE are given by Eq.(5) and (3) respectively as follows.
6
2
ib
ln γ ± = ∑ (1 + )(ai + i p )mi / 2
i
2υ
i =1
6
ib p ⎤
⎡2
G E = x1υ RT ∑ ⎢ (ai + i ) ⎥mi / 2
2υ ⎦
i =1 ⎣ i
(20)
(21)
The experimental φ and lnγ± for solutions at 25oC from Hamer and Wu[21] and Robinson and
McCoach[22] are shown with Eq.(18) and (20) in Fig.3.
Enthalpy of solution
The molar enthalpy of the solution is given by Eq.(2) and (6) as follows.
H (lT , p ,m ) = x1 H1(∞T , p ) + (1 − x1 ) H 2(l T , p ) + H (ET , p ,m )
(22)
Using the water data from Schmidt[16], the molar enthalpy of pure water H2l can be calculated by
2
2
⎡
⎛ ∂V l ⎞ ⎤
H = H + ∫ Cp dT + ∫ ⎢V2l − T ⎜ 2 ⎟ ⎥dp , where Cp2l = R ∑ d jT j , V2l = R ∑ e jT j
j =0
j =0
⎝ ∂T ⎠ p ⎥⎦
TD
pD* ⎢
⎣
p
T
l
2
l
2D
l
2
(23)
The coefficients dj and ej are listed in Table 2.
And from Eq.(3) and (6), the excess enthalpy HE is given by
6
⎡ ∂ ln γ ± ∂φ ⎤
2 ∂ai
i ∂bi
2
υ
(
−
=
−
+
H E = − x1υ RT 2 ⎢
x
RT
p )mi / 2
∑
1
⎥
∂
∂
∂
∂
2
υ
T
T
i
T
T
i =1
⎣
⎦
(24)
The enthalpy of infinitely dilute solution H1∞ can be determined using either experimental ‘first heat of
solution’ or heat capacity of the solutions. Since the experimental first heat of solutions are, however,
not available for many temperatures, the heat capacity data of a reference solution were used to
determine H1∞. As recommended by Jetter et al[23], the heat capacity data for 60wt.% solution from
Lower[4], Feuerecker et al[8], Jetter et al[23] and Rockenfeller[24] were used. But it turned out that
any choice of the reference concentration between 50 and 60wt.% would have given virtually identical
results. The enthalpy values have been set to zero for pure water and 50wt.% solution at 0oC. The
resulting H1∞ is given by Eq.(25) whose coefficients are listed in Table 2.
⎡ ⎛ 1 1 ⎞ c1 ⎛ 1
⎛ ∂V1∞ ⎞ ⎤
1 ⎞ c2 ⎛ 1
1 ⎞⎤ ⎡ ∞
*
H = H − R ⎢c0 ⎜ − ⎟ + ⎜ 2 − 2 ⎟ + ⎜ 3 − 3 ⎟ ⎥ + ⎢V1 − T ⎜
⎟ ⎥ ( p − pD )
TD ⎠ 3 ⎝ T TD ⎠ ⎦ ⎣⎢
⎝ ∂T ⎠ p ⎦⎥
⎣ ⎝ T TD ⎠ 2 ⎝ T
∞
1
∞
1D
5
(25)
The results agree relatively well with Feuerecker et al[8] and Kaita[11] as shown in Fig.4. The major
reason of the discrepancies is the difference in the heat of dilution in Fig.6. The heat of dilution from
the two studies show relatively large deviations from Lange and Schwartz[9]. Particularly, Kaita[11]’s
dew temperature and enthalpy equation are inconsistent in terms of heat of dilution as shown in Fig.6
In Fig.4, the results of McNeely[5], Herold and Moran[7] and Chua et al[10] show similar trends in
comparison with this study, all becoming smaller as temperature increases and resembling a rotated S
shape. This similarity originates from that fact that they were based on the vapor pressures from
McNeely[5]. McNeely[6]’s doubt about his dew temperature gradient in high concentration region is
confirmed in Fig.5. His heat capacity curves are almost flat for solutions between 40 and 60wt.%
above 100oC, which is contradictory to the measurements of Feuerecker et al[8], Jetter et al[23] and
Rockenfeller[24], and even decrease with increasing solution temperature for higher concentrations.
0%
4.3
20
180oC
4.1
10%
3.9
3.7
3.5
-20
20%
3.3
3.1
20
130oC
2.9
3.1
30%
2.9
Cp(kJ/K kg solution)
hlit-hcal(kJ/kg solution)
0
-20
20
100oC
0
-20
20
2.7
2.5
2.7
40%
2.5
2.3
2.1
2.4
50%
2.2
2
60oC
1.8
0
2.2
60%
2
-20
1.8
1.6
20
t=20oC
2
70%
1.8
0
1.6
1.4
-20
0
0
0.2
0.4
0.6
20
40
60
80
100
120
140
160
180
200
t(oC)
0.8
LiBr weight fraction
Figure 5. Calculated and experimental heat
capacities (Löwer[4]: ○, Jetter et al[23]: △,
Rockenfeller[24]: □, McNeely[5]: ----,
Feuerecker et al[8]: __ _ __ , this study: ____)
Figure 4. Comparison of enthalpy (Löwer[4]:
○, Chua et al[10]: △, McNeely[5]: □,
Feuerecker et al[8]: ●, Herold & Moran[7]: ■,
Kaita[11]: ▲)
As for the dilute solutions below 140oC, the enthalpy of this study positions between Löwer[4] and
McNeely[5] as shown in Fig.4 primarily because the dew temperature gradients of this study have
values between those of the two studies. For the solutions above 140oC, the temperature gradients of
the heat capacity curves from this study show better consistency with those of pure water and
measurements from Jetter et al[23]and Rockenfeller[24] (see Fig.5).
t=25 oC
Lange & Schwartz
McNeely
Feuerecker
Lower
Kaita
Kaita (enthalpy)
this study
Hd(kJ/kg water)
400
300
200
100
0
0.35
0.4
0.45
0.5
0.55
0.6
0.65
LiBr weight fraction
Figure 6. Calculated and experimental heat of dilution
6
Entropy of solution
The molar entropy of solution is given by Eq.(2) and (7) as follows.
⎡ ⎛m⎞ ⎤
S(lT , p ,m ) = x1S1(∞T , p ) + (1 − x1 ) S2(l T , p ) − x1υ R ⎢ln ⎜ ⎟ − 1⎥ + S(ET , p ,m )
⎣ ⎝ mD ⎠ ⎦
(26)
And from Eq.(3) and (7), the excess enthalpy SE is given by
SE =
6
ib
∂a
2⎡
H E − GE
i ∂bi ⎤ i / 2
= − x1υ R ∑ ⎢ ai + i p + T ( i +
p) m
T
∂T 2υ ∂T ⎥⎦
2υ
i =1 i ⎣
(27)
The molar entropy of pure water S2l was calculated with the heat capacity and volume of water from
Eq.(23). And S1∞ could be also calculated in the same way with the heat capacity determined from the
enthalpy of the infinitely dilute solution in Eq.(25) and the volume from Eq.(13). The entropy values
have been set to zero for pure water and 50wt.% solution at 0oC.
The results were compared with those of Löwer[4], Feuerecker et al[8], Chua et al[10], Kaita[11] and
Koehler et al[20]. Except for the Koehler et al[20] who didn’t give enthalpy data, the other results
show trends similar to those for the enthalpy. All studies agreed within the error band of ±30J/kgK
except for Löwer[4] at high concentrations.
Conclusion
A Gibbs energy equation has been successfully developed for aqueous LiBr solutions. The developed
osmotic coefficient equation can reproduce the original experimental solution density and vapor
pressures data within 0.3% and 5%. The resulting enthalpy equation was evaluated in comparison with
experimental heat of dilution and heat capacity data from the literature and the agreement was found
excellent. The discrepancies with other studies were analyzed and the possible causes were identified.
The analysis suggests that the Dühring’s rule is not appropriate to describe the solutions in a wide
pressure or concentration range. It is supported by the fact that the high-pressure data from Lenard et
al[12] cannot be reproduced accurately without inclusion of the temperature dependence of the dew
temperature gradient, which is why Kaita[11] used a 2nd-order dew temperature equation and Chua et
al[10] simply disregarded Lenard et al[12]. Also the heat of dilution of Lange and Schwartz[9] in high
concentration region cannot be reproduced by the Dühring equation of Feuerecker et al[8]. The Gibbs
energy model used in this study has been proved to be very flexible and accurate for the description of
LiBr/H2O solutions
Summary of equations
G2l and GE are given in Eq.(2) and (3). G1∞ and G2l are summarized below.
R
Cp1∞
+ V1∞ ( p − pD* ) , where Cp1∞ = 2
T
T
TD
T
T
G1∞ = H1∞D − TS1∞D + ∫ Cp1∞ dT − T ∫
TD
T
2
cj
∑T
j =0
j
2
, V1∞ = RT ∑
j =0
b0j
Tj
(28)
T
2
2
Cp2l
dT + V2l ( p − pD* ) , where Cp2l = R ∑ d jT j , V2l = R ∑ e jT j
T
j =0
j =0
TD
G2l = H 2l D − TS 2l D + ∫ Cp2l − T ∫
TD
Table 2. Coefficients and constants for Gibbs energy equation
a1j
a2j
a3j
a4j
a5j
a6j
0
-6.26858E+01
+1.70300E+03
-2.98958E+04
+2.45983E+05
-8.78572E+05
+1.06577E+06
1
+2.80465E+04
-8.40657E+05
+2.07935E+07
-2.01762E+08
+7.94891E+08
-1.05602E+09
7
2
-3.80422E+06
+1.52545E+08
-4.59904E+09
+5.43648E+10
-2.62248E+11
+4.41273E+11
(29)
b0j
b1j
b2j
ci
di
ei
H1o∞
S1o∞
To
-4.418E-05
+3.079E-04
-4.081E-04
-1.917E+04
+1.197E+01
+2.674E-03
-5459.607
28.061
273.15
+3.115E-02
-1.863E-01
+2.161E-01
-5.898E+08
-1.831E-02
-3.920E-06
H2ol
S2ol
P o*
-4.361E+00
+2.739E+01
-2.518E+01
0
+2.871E-05
+7.534E-09
0
0
0.6108
Acknowledgements
This work was supported by the Dutch Organization for Energy and Environment(NOVEM).
Cp
G
Gi
H
M
m
P
R
S
T
V
x
Greek
γ±
φ
Nomenclature
υ
dissociation number (2 for LiBr)
molar heat capacity (kJ/K kmol)
molar Gibbs energy (kJ/kmol)
Superscripts
chemical potential(kJ/kmol species i)
d
dilution
molar enthalpy (kJ/kmol)
∞
property
of infinitely dilute solution
molar mass (kg/kmol)
*
saturation
state of pure solvent
molality (kmol solute/kg solvent)
l
liquid
phase
pressure (kPa)
g
vapor phase
universal gas constant (kJ/K kmol)
E
excess property
molar entropy (kJ/K kmol)
Subscripts
temperature (K)
1
solute
molar volume (m3/kmol)
concentration
2
solvent
ο
reference, standard state
w
weight
mean ionic activity coefficient
osmotic coefficient
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