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An excess Gibbs free energy based model to calculate viscosity of multicomponent liquid mixtures
International Journal of Greenhouse Gas Control 42 (2015) 494–501
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International Journal of Greenhouse Gas Control journal homepage: www.elsevier.com/locate/ijggc
An excess Gibbs free energy based model to calculate viscosity of multicomponent liquid mixtures Diego D.D. Pinto a,∗ , Hallvard F. Svendsen b a b
PROCEDE Group B.V., PO Box 328, 7500 AH Enschede, The Netherlands Department of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 20 July 2015 Received in revised form 31 August 2015 Accepted 2 September 2015 Keywords: Viscosity Excess Gibbs Liquid NRTL CO2 capture
a b s t r a c t Solution densities and viscosities are important parameters for the design and simulation of absorption processes. Accurate models are needed and in this work, a new model for calculating the liquid viscosity of mixtures is presented. The model uses an analogy to excess Gibbs energy models to account for the deviation from a simple mixing rule based on the pure component viscosities. In this work, we chose the functional form of the NRTL model to represent the excess Gibbs energy and the resulting model is referred to as NRTL-DVIS. Eleven systems (eight binaries and three ternary) were chosen for testing the accuracy of the model. The ternary systems were built from the optimized binaries and pure component systems. With few adjustable parameters, the NRTL-DVIS model represented the tested systems with good accuracy. With few exceptions the calculated total deviation (AARD) was within 3.5%. The NRTL-DVIS model shows better accuracy than other models proposed in the literature. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Densities and viscosities of solutions are especially important when designing and simulating processes. In particular viscosity is important in processes where mass transfer is involved (e.g., absorbers and desorbers). Moreover, equipments such as pumps and heat exchangers are better modeled when the physical properties of the system are accurately calculated (Fu et al., 2012; Weiland et al., 1998). Several properties of a system can be directly correlated to its viscosity. For instance, Versteeg et al. (1996) show that the diffusivity of alkanolamines can be estimated from viscosity with a modified Stokes–Einstein relation. Hence, a good representation of the viscosity of the solution is crucial. Proposed correlations for liquid viscosity available in the literature, unfortunately, do not share the same theoretical basis as for gas viscosity (Poling et al., 2000). Therefore, it is desirable to estimate the liquid viscosity from experimental data whenever available. Many correlations are available in the literature to calculate the viscosity of both pure liquids and mixtures of liquid components. A number of proposed models use the pure liquid viscosity as a starting point and apply a mixing rule to calculate the viscosity
∗ Corresponding author. E-mail address: [email protected] (D.D.D. Pinto). http://dx.doi.org/10.1016/j.ijggc.2015.09.003 1750-5836/© 2015 Elsevier Ltd. All rights reserved.
of the mixture. In these cases, it is required that pure component liquid viscosities are known at given temperatures and pressures. However, for solutions where the viscosity of at least one of the components is not known, the applicability of these types of equations is not straightforward. Aqueous solutions of hydroxide salts are examples of solutions where the liquid viscosity of one component (in this case, the hydroxide) is not known for a wide range of temperatures. These solutions are of interest for CO2 capture processes (see Yoo et al., 2013; Mahmoudkhani and Keith, 2009; Stolaroff et al., 2008). When the pure viscosity of one (or more components) is not available, models which use a reference viscosity are usually applied (see, for instance, Först et al., 2002; Mathlouthi and Reiser, 1994; Vand, 1948). An extensive review on this type of viscosity models is found in Longinotti and Corti (2008). Alternatively, one can still use models which require the pure viscosity of the components by, for instance, setting the unknown viscosity to a constant value. In this work, a new model for calculating the viscosity of multicomponent liquid mixtures is presented. The model is based on the fact that excess viscosities show similar behaviour as that observed for excess Gibbs energy. Hence, the functional forms of the many existing models capable of representing excess Gibbs energy can be used for representing excess viscosities. In particular, the functional form of the NRTL model was chosen in this work. By setting few binary interaction parameters, the model is able to accurately represent the viscosity of liquid mixtures. Eight binary systems and
D.D.D. Pinto, H.F. Svendsen / International Journal of Greenhouse Gas Control 42 (2015) 494–501
three ternary systems were tested to verify the accuracy of the model. Although the model has the same functional form as that of excess Gibbs energy models, it is important to stress that the fitted interaction parameters are not connected at all to the NRTL model and should not be used for activity calculations. Moreover, it is important to check if, at the desired conditions of composition, temperature and pressure, the mixture is a single phase liquid solution. 2. Liquid viscosity correlations Several viscosity correlations are available in the literature. For a pure component in its liquid state, the Andrade type of equation (Andrade, 1930) is well accepted and is implemented in many process simulators. One of the forms of the Andrade equation is given in Eq. (1). Here ◦ is the dynamic viscosity of the liquid, T is the temperature of the system and A, B and C are the adjustable parameters of the model. ln (◦ ) = A +
B + C · ln (T ) T
(1)
For multicomponent systems, the Grunberg–Nissan model the (Grunberg and Nissan, 1949) can be used. This model expands Arrhenius proposition adding adjustable parameters ˛i,j that are fitted to experimental data. Eq. (2) shows the multicomponent form of the Grunberg–Nissan model. In Eqs. (2), (3) and (7), c denotes the concentration basis (mol or mass fraction). ln () =
NC
NC NC
ci · ln ◦i +
i=1
i=1
ci cj ˛i,j
(2)
j=1 j= / i
The Grunberg–Nissan model, however, is not recommended for aqueous solutions (Poling et al., 2000) and several modifications of the Grunberg–Nissan model have been presented in the literature in order to improve its accuracy. For instance, Marczak et al. (2012) introduced the “kinetic” mol fraction concept where the mol fraction scale is redefined, taking into account the associative degree of the components present in the solution. By modifying the scale of the mol fraction in this way, the authors were able to improve the accuracy of the model for aqueous solutions. The model shown in Eq. (3) was proposed by Song etal.(2003). The correlation uses symmetric kij and antisymmetric lij binary interaction parameters allowing the total of adjustable number parameters to be reduced. The term ln ij is called the cross term binary for the viscosities and is defined as in Eq. (4) so that lim ln ij = 0. This model is implemented in the commercial
kij = aij + lij = mij +
ln =
NC
ci ln ◦i +
i=1
NC NC i=1 j>1
kij ci cj ln ij +
NC i=1
⎡ ⎤3 NC 1/3 ⎦ ci ⎣ cj lij ln ij j= / i
(3)
ln ij =
ln i − ln j 2
(4)
T nij T
, kij = kji
(5)
, lij = −lji
(6)
The above presented models require previous knowledge of the components pure viscosities which are usually correlated by temperature functions (as in the Andrade equation). Teng et al. (1994), on the other hand, successfully used a model where the viscosity of aqueous amine solutions was described based on the pure water viscosity and a polynomial function on the amine concentration (Eq. (7)). The model was later applied by Chowdhury et al. (2010) for the same purpose. The authors, however, regressed a set of parameters (pi ) for each desired temperature. ln = ln ◦H
2
O+
P
pi c i
(7)
i=1
Pinto (2014) applied a temperature dependency in the polynomial terms, as shown in Eq. (8), so that with one set of parameters the model would calculate the viscosity of a system at any temperature. pi = ai +
bi T
(8)
The drawback of the approach presented in Pinto (2014) is that the pure component viscosity was not calculated with the same accuracy as before (using one set of parameters for each temperature) although the calculated values were very close to the measurements. Moreover, the number of parameters in the models based on Teng’s approach (Teng, 1994) needed for correlating a system was usually higher than for other models and dependent on the order of the polynomial (P) chosen for modeling the system. In addition, the models based on Teng et al. (1994) approach are only valid for binary systems and the expansion to a multicomponent form would largely increase the number of adjustable parameters in the model. In the next section a new model for calculating liquid mixture viscosities is presented. The model uses the functional form of an excess Gibbs energy model to account for the excess viscosity, i.e., the deviation between the measured viscosity and that calculated using a simple mixing rule and the pure component viscosities. 3. The functional form of excess Gibbs energy models The viscosity of most liquid mixtures cannot be explained by a simple mixing rule as described by Eq. (9).
i →j
process simulator Aspen Plus, and has been shown to be able to represent the viscosity of a number of systems. This model is referred to as the “Aspen liquid mixture viscosity model” in the Aspen Plus process simulator software. To account for calculations at different temperatures, the authors suggested that the binary adjustable parameters kij and lij are given a temperature dependency as shown in Eqs. (5) and (6).
bij
495
ln =
NC
xi ln ◦i
(9)
i=1
However, this simple mixing rule is the basis for several available models, as well as for the proposed model. The difference between the left- and right-hand sides of Eq. (9), except for pure components, is non-zero for most systems. In Fig. 1, the shape of this difference is shown for the H2 O-MDEA system. This shape resembles the excess Gibbs energy function as represented, for example by the Wilson (Wilson, 1964), the NRTL (Renon and Prausnitz, 1968) and the UNIQUAC (Abrams and Prausnitz, 1975) models. Based on this, we propose an excess viscosity model which uses the functional form of an excess Gibbs energy model to account for the deviation from the simple mixing rule as shown in Eq. (10). The viscosity of the liquid mixtures can then be calculated by the simple mixing rule (Eq. (9)) corrected by an excess viscosity model, multiplied by a factor (R). This factor allows the excess viscosity model
496
D.D.D. Pinto, H.F. Svendsen / International Journal of Greenhouse Gas Control 42 (2015) 494–501 Table 1 Systems studied in this work.
Fig. 1. Difference between the left- and right-hand sides of Eq. (9) for H2 O-MDEA system. Experimental data from Teng et al. (1994) at: ( ) 25 ◦ C, ( ) 40 ◦ C and (•) 60 ◦ C.
to reach the desired corrective values while working with parameters of the same magnitude as those used to calculate excess Gibbs energies. In this work it was optimized with the H2 O-MDEA system data and kept constant at 6.48803 for all other studied systems.
ln =
NC
xi ln ◦i (T ) + R · E (T, x)
(10)
i=1
In this work, the functional form of the NRTL model is used to calculate the excess viscosity term (E ). Nevertheless, any other functional form capable of representing excess Gibbs energy is expected to be able to represent the viscosity of liquid mixtures with the same accuracy. For ease of identification, the model will be referred to as the NRTL-DVIS model. The excess viscosity is calculated according to Eqs. (11)–(13). The non-randomness parameters ˛ij were tested at 0.1, 0.2 and 0.3 for all systems. With few exceptions, the best results were obtained for a value of 0.2. If necessary this parameter may be included in the optimization routine. However, in this work the only adjustable parameters in the model are the binary energy parameters ij which are assumed to have a temperature dependency as indicated in Eq. (13). E
=
NC
NC xi
G x j=1 ji ji j
NC
G x k=1 ki k
i=1
Gij = exp −˛ij ij ij = aij +
bij T
(11)
#
System
1 2 3 4 5 6 7 8 9 10 11
H2 O-MEA H2 O-MDEA H2 O-DEA H2 O-TEA H2 O-AMP H2 O-PZ H2 O-MEA-MDEA H2 O-MEA-AMP H2 O-MDEA-DEA H2 O-NaCl H2 O-Ethanol
Component
CAS
MEA MDEA DEA TEA AMP PZ NaCl Ethanol
141-43-5 105-59-9 111-42-2 102-71-6 124-68-5 110-85-0 7647-14-5 64-17-5
respectively by Eqs. (15) and (16), were used to measure the deviations between model and experimental viscosities.
exp 2 NE i − calc i FObj. = exp calc i
i=1
ii = 0
calc 100 i − i AARD (%) = exp NE i
(15)
MAD (mPa s) = max
(16)
NE
i=1
exp
exp − calc i
i
5. Results In this section, 11 systems were chosen to be modeled using the NRTL-DVIS model. All the studied systems are aqueous solutions and the knowledge of the viscosity of pure water is required. The correlation given in Bingham and Jackson (1918) (Eqs. (17) and (18)) was used to calculate the viscosity of pure water where is the fluidity of water in P−1 , ◦H O is the viscosity of pure water in 2 mPa.s and T is the temperature in K. The studied systems are given in Table 1.
(12)
= 2.1482 (T − 281.585) + ,
(14)
· i
8078.4 + (T − 281.585)
(13)
4. Optimization procedure The parameters of the proposed model were estimated to fit the experimental viscosity data for some selected systems. The particle swarm optimization (Kennedy and Eberhart, 1995) algorithm was used in this work. As previously done in Pinto et al. (2014), the lbest topology was chosen with the inertia factor (ω = 0.7298) and the acceleration coefficients (1 and 2 = 1.49618) (Poli et al., 2007). The particles were generated within the interval [−3, 3] and [−3000, 3000] for the adjustable parameters aij and bij , respectively. An objective function (Eq. 14), which weighs equally to the experimental data points (Weiland et al., 1993), was used in the optimization procedure. The average absolute relative deviation (AARD) and the maximum absolute deviation (MAD) given,
2
− 120 (17)
◦H
2O
=
100
(18)
The model was able to calculate the viscosity of the aqueous solutions with excellent accuracy as shown in Table 4. Except for a few systems, the model correlates the experimental data with less than 3 % deviation (AARD). Nevertheless, the total AARD was never higher than 5%. 5.1. Aqueous alkanolamine solutions Aqueous solutions of alkanolamines are widely employed for removing acid gases from various gas streams, both at low and high pressure. Several laboratory and pilot tests can be found for CO2
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497
Table 2 Parameters for the Andrade equation.
MEA MDEA DEA TEA AMP PZ
A
B
C
−149.6008 −170.8208 −271.0938 −342.1746 −339.8066 1.0000
10241.350 12186.230 18423.324 21723.859 21713.806 0.000
20.7372 23.5683 37.8468 48.3915 47.7361 0.0000
post combustion capture (Pinto et al., 2014; Brúder et al., 2012; Tobiesen et al., 2007). For simulation and design, accurate representation of the viscosity of the solution is required. Some ternary systems were chosen due to the large amount of data available and a sequential optimization procedure (SOP) was applied. In this procedure, firstly the smallest subsystem is optimized and its parameters carried to the larger subsystems. The optimization is continued until the complete system is optimized, as exemplified in Pinto et al. (2013) for the eNRTL model. Monteiro et al. (2013) and Frailie et al. (2011) also used the SOP approach.
Fig. 2. Viscosity of the H2 O-MEA system. ( ) NRTL-DVIS and experimental data: ( ) (Maham et al., 2002), ( ) (Amundsen et al., 2009) and ( ) (Hartono et al., ◦ ) NRTL-DVIS and experimental data: ( ) (Maham et al., 2002), 2014) at 25 C. ( ) NRTL( ) (Amundsen et al., 2009) and ( ) (Hartono et al., 2014) at 40 ◦ C. ( DVIS and experimental data: ( ) (Maham et al., 2002), ( ) (Amundsen et al., 2009) and ( ) (Hartono et al., 2014) at 80 ◦ C.
5.1.1. Pure component liquid viscosity The viscosity of the pure amines was calculated with the Andrade equation (Eq. (1)) and the optimized parameters are given in Table 2. The Andrade equation is able to accurately predict the viscosity of the pure amines. The calculated deviations are shown in Table 3. 5.1.2. Binary systems Following the SOP, the parameters for the binary systems (H2 O-MEA, -MDEA, -DEA, -TEA, -AMP and -PZ) are estimated. Unfortunately, no data for the binary amine-amine system were found in the consulted literature. Figs. 2–6 show that the model is able to accurately represent the viscosity of the binary systems for the whole range of concentrations. The calculated deviations are given in Table 4. For a comparison with the new model, the parameters for the Aspen model (Song et al., 2003) were also optimized for the H2 O-MEA system. With the optimized parameters (kH2 O,MEA ) and (lH2 O,MEA ) given in Eqs. (19)–(20), the Aspen model calculates the viscosity of the H2 O–MEA system with a total AARD of 2.86%. From Table 4 it is seen that the NRTL-DVIS model is able to correlate the viscosity of the H2 O–MEA with lower deviations. Therefore,
Fig. 3. Viscosity of the H2 O-MDEA system. ( ) NRTL-DVIS and experimental ) NRTL-DVIS and experimental data: data: ( ) (Teng et al., 1994) at 25 ◦ C. ( ( ) (Teng et al., 1994), ( ) (Bernal-García et al., 2004) and ( ) (Chowdhury et al., ) NRTL-DVIS and experimental data: ( ) (Bernal-García et al., 2010) at 40 ◦ C. ( 2004) at 90 ◦ C.
Table 3 Calculated deviations for the pure amine viscosity. Amine
Trange (K)
# Points
AARD (%)
MAD (mPa s)
Ref.
MEA
298.15–353.15 298.15–353.15 298.15–353.15
5 5 10
2.61 1.14 1.88
0.75 0.30 0.75
Maham et al. (2002) Amundsen et al. (2009) Total
MDEA
298.15–353.15 313.15–363.15 303.15–323.15 298.15–363.15
5 4 5 14
0.45 0.62 0.55 0.53
0.88 0.36 0.41 0.88
Teng et al. (1994) Bernal-García et al. (2004) Chowdhury et al. (2010) Total
DEA
298.15–353.15 303.15–323.15 298.15–353.15
6 5 11
0.71 0.56 0.64
2.82 2.31 2.82
Teng et al. (1994) Chowdhury et al. (2010) Total
TEA
298.15–353.15 303.15–353.15 298.15–353.15
5 11 16
1.94 0.92 1.24
3.75 3.01 3.75
Maham et al. (2002) Ko et al. (2001) Total
AMP
313.15–343.15 303.15–353.15 303.15–353.15
4 6 10
1.49 1.14 1.28
0.43 0.63 0.63
Henni et al. (2003) Li and Lie (1994) Total
498
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Fig. 4. Viscosity of the H2 O-DEA system. ( ) NRTL-DVIS and experimental ) NRTLdata: ( ) (Teng et al., 1994) and ( ) (Spasojevic´ et al., 2013) at 25 ◦ C. ( DVIS and experimental data: ( ) (Teng et al., 1994), ( ) (Chowdhury et al., 2010) ◦ ) NRTL-DVIS and experimental data: and ( ) (Spasojevic´ et al., 2013) at 40 C. ( ( ) (Teng et al., 1994) and ( ) (Spasojevic´ et al., 2013) at 60 ◦ C.
Fig. 7. Viscosity of the H2 O-PZ system. ( ) NRTL-DVIS and experimental from: ( ) (Derks et al., 2005), ( ) (Murshid et al., 2011) and ( ) (Samanta and ) NRTL-DVIS and experimental data: ( ) Bandyopadhyay, 2006) at 25 ◦ C. ( (Derks et al., 2005), ( ) (Murshid et al., 2011), ( ) (Liu et al., 2012) and ( ) (Samanta ◦ ) NRTL-DVIS and experimental data: ( ) and Bandyopadhyay, 2006) at 40 C. ( (Murshid et al., 2011) and ( ) (Samanta and Bandyopadhyay, 2006) at 60 ◦ C.
From Fig. 2 a small discrepancy between the sources for the pure viscosity at 25 ◦ C is seen. At ambient conditions, pure PZ is in its solid state. No data were found for the viscosity of pure PZ measured above its melting point. Therefore, the viscosity of pure PZ was assumed to be constant and set to 0. Moreover, aqueous solutions of PZ present solubility limits. The viscosity data found for aqueous PZ solutions were only measured in the diluted range. However, this range covers the CO2 capture operational range for PZ solvents. Fig. 7 shows that the model is able to represent the viscosity of aqueous solution of PZ. It is also possible to see in the figure a small discrepancy between the sources. Fig. 5. Viscosity of the H2 O-TEA system. ( ) NRTL-DVIS and experimental data: ( ) (Maham et al., 2002) at 25 ◦ C. ( ) NRTL-DVIS and experimental data: ) NRTL-DVIS and ( ) (Maham et al., 2002) and ( ) (Ko et al., 2001) at 40 ◦ C. ( experimental data: ( ) (Maham et al., 2002) and ( ) (Ko et al., 2001) at 60 ◦ C.
only the results from the NRTL-DVIS model will be discussed in this section. kH2 O,MEA = 4.2916 +
43.6297 T
lH2 O,MEA = −5.5117 +
777.1026 T
(19) (20)
Fig. 6. Viscosity of the H2 O-AMP system. ( ) NRTL-DVIS and experimental ) NRTL-DVIS and experimental data: from: ( ) (Henni et al., 2003) at 25 ◦ C. ( ) NRTL-DVIS and ( ) (Henni et al., 2003) and ( ) (Li and Lie, 1994) at 40 ◦ C. ( experimental data: ( ) (Henni et al., 2003) and ( ) (Li and Lie, 1994) at 60 ◦ C.
5.1.3. Ternary systems Once the parameters for calculating the liquid viscosity of the pure components and the binary systems are optimized, the remaining parameters for a ternary system can be fitted to experimental data. The NRTL-DVIS model is able to accurately correlate the tested ternary systems (H2 O-MEA-MDEA, H2 O-MEA-AMP and H2 O-MDEA-DEA) as shown in Figs. 8–10. The calculated deviations are given in Tables 4 and 5 shows the NRTL-DVIS model parameters. These examples show the applicability of the NRTL-DVIS model for multicomponent liquid mixtures. Additionally, by using the SOP, it is guaranteed that the ternary system will always regress to the
Fig. 8. Deviations for the H2 O-MEA-MDEA system. Experimental data: ( ) (Li and Lie, 1994), ( ) (Mandal et al., 2003) and ( ) (Fu et al., 2012).
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499
Table 4 Calculated deviations for the liquid mixtures viscosity. System
Trange (K)
# Points
AARD (%)
MAD (mPa s)
Ref.
H2 O-MEA
298.15–353.15 298.15–353.15 293.15–353.15 293.15–353.15
60 30 26 116
1.78 1.99 1.44 1.77
0.76 0.48 0.07 0.76
Maham et al. (2002) Amundsen et al. (2009) Hartono et al. (2014) Total
H2 O-MDEA
298.15–353.15 313.15–363.15 303.15–323.15 298.15–363.15
70 28 60 158
2.72 2.54 1.82 2.35
5.73 0.72 1.25 5.73
Teng et al. (1994) Bernal-García et al. (2004) Chowdhury et al. (2010) Total
H2 O-DEA
298.15–353.15 303.15–323.15 298.15–343.15 298.15–353.15
60 55 50 165
2.73 1.99 2.12 2.31
15.84 9.65 0.06 15.83
Teng et al. (1994) Chowdhury et al. (2010) Spasojevic´ et al. (2013) Total
H2 O-TEA
298.15–353.15 303.15–313.15 298.15–353.15
65 18 83
3.86 1.02 3.07
18.21 3.01 18.21
Maham et al. (2002) Ko et al. (2001) Total
H2 O-AMP
298.15–343.15 296.75–349.85 303.15–353.15 293.15–323.15 298.15–353.15
66 12 12 7 97
5.28 1.76 2.91 8.28 4.71
12.61 0.06 0.63 0.77 12.61
Henni et al. (2003) Xu et al. (1991) Li and Lie (1994) Mandal et al. (2003) Total
H2 O-PZ
293.15–323.15 298.15–333.15 283.15–323.15 298.15–333.15 283.15–333.15
19 32 25 32 108
3.32 5.89 2.77 1.40 3.38
0.12 0.11 0.05 0.02 0.12
Derks et al. (2005) Murshid et al. (2011) Liu et al. (2012) Samanta and Bandyopadhyay (2006) Total
H2 O-MEA-MDEA
303.15–353.15 293.15–323.15 293.15–343.15 293.15–343.15
36 42 96 174
1.47 2.42 6.70 4.59
0.05 0.17 0.82 0.82
Li and Lie (1994) Mandal et al. (2003) Fu et al. (2012) Total
H2 O-MEA-AMP
303.15–353.15 293.15–323.15 293.15–353.15
36 42 78
2.31 3.90 3.17
0.19 0.62 0.62
Li and Lie (1994) Mandal et al. (2003) Total
H2 O-MDEA-DEA
293.15–373.15 293.15–323.15 293.15–373.15
20 42 62
2.11 3.27 2.90
0.73 0.23 0.73
Rinker et al. (1994) Mandal et al. (2003) Total
H2 O-Ethanol
293.15–303.15 292.15 292.15–303.15
31 13 44
4.58 4.07 4.43
0.18 0.26 0.26
González et al. (2007) Koohyar et al. (2012) Total
H2 O-NaCl
293.15–373.15
204
0.73
0.07
binary systems with the same accuracy as for the regressed binary systems in case the system is depleted of one of the components. 5.2. Non-amine systems The model can also calculate the viscosities of other liquid solutions than aqueous amine solutions. As an example, the aqueous ethanol solution is represented by the NRTL-DVIS model.
Kestin et al. (1981)
Experimental data from González et al. (2007) and Koohyar et al. (2012) were used in the optimization procedure. The parameters of the Andrade equation were optimized to calculate the viscosity of pure ethanol. Eq. (21) represents the viscosity of pure ethanol (mPa s) at a given temperature (K). ln EtOH = −231.1861 −
11734.199 − 33.6819 · ln (T ) T
(21)
Table 5 Parameters for the NRTL-DVIS model. Component
Pair value
i
j
aij
aji
bij
bji
˛ij
H2 O H2 O H2 O H2 O H2 O H2 O MEA MEA MDEA H2 O H2 O
MEA MDEA DEA TEA AMP PZ MDEA AMP DEA Ethanol NaCl
2.18950 0.32194 3.56622 2.22927 0.26558 −31.95187 −1.95269 −13.25264 13.84543 −12.29712 −2.77753
−2.36085 −2.85327 −3.90934 −2.95869 −2.49559 −148.60569 104.51276 712.89558 −6.21575 1.77837 17.71668
−30.14092 1179.30750 365.23495 563.61046 971.24906 9841.72047 274.5111 4205.40984 344.25523 4763.645 129.42804
471.76056 521.44416 509.90036 567.69868 429.15545 52759.768 −30162.821 −205243.927 1042.3131 −1027.575 −78.87626
0.2 0.2 0.1 0.2 0.2 0.1 0.2 0.1 0.1 0.2 0.1
500
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Fig. 9. Deviations for the H2 O-MEA-AMP system. Experimental data: ( Lie, 1994) and ( ) (Mandal et al., 2003).
) (Li and
Fig. 12. Viscosity for aqueous solution of NaCl at 20, 40 and 80 ◦ C (from top to ) NRTL-DVIS and Experimental data: (䊐) (Kestin et al., 1981) bottom). (
in the measurements. Nevertheless, the model is able to calculate the viscosity of aqueous solutions of ethanol with reasonably good accuracy. The deviations are reported in Table 4 and the parameters are given in Table 5. 5.3. Systems containing solid pure components
Fig. 10. Deviations for the H2 O-MDEA-DEA system. Experimental data: ( ) (Rinker et al., 1994) and ( ) (Mandal et al., 2003).
From Fig. 11 it is seen that the highest deviations occur between 0.3 and 0.4 ethanol mol fractions. An experimental variation between the two used independent sources is also observed. Koohyar et al. (2012) measured viscosity of aqueous ethanol solutions at 19 ◦ C. As these measurements are at a temperature 1 ◦ C lower than the ones measured by González et al. (2007) it was expected that the viscosities at lower temperature were greater than the ones measured at higher temperature. This behavior is not observed for the whole range of composition when comparing these two sources and reflects the experimental uncertainties
Although the model requires the pure component viscosity to calculate the viscosity of the solution, it is still possible to use the NRTL-DVIS model when the viscosity of one of the components is not available. For this type of systems, we recommend setting the A, B and C parameters in the Andrade equation to 1, 0 and 0, respectively and to optimize the NRTL-DVIS parameters, as done with PZ. Aqueous solutions of NaCl are an example of this kind of system and was used to test the NRTL-DVIS model. Experimental data from Kestin et al. (1981) were used to optimize the model parameters. The parameters are given in Table 5 and the calculated deviations are presented in Table 4. The model is able to correlate the experimental viscosity of aqueous solutions of NaCl with very low deviation as seen in Fig. 12. 6. Conclusions The NRTL-DVIS model uses the NRTL model structure to correlate the excess viscosity of liquid mixtures. The NRTL model is relatively simple and presents only few binary adjustable parameters which are fitted to experimental data. The binary interaction parameters are assumed to have a simple temperature dependency to account for calculations at different temperatures. The NRTL-DVIS model is able to correlate the viscosity of liquid mixtures with very good accuracy. The model is also able to predict the viscosity of aqueous solutions, for which models like the Grunberg–Nissan are not suitable, for the whole range of composition and at different temperatures. The model was able to represent the viscosity of tested cases with higher accuracy than other models available in the literature. It is important to stress that the parameters used in the NRTL-DVIS model do not have any relationship with the thermodynamic NRTL model. References
Fig. 11. Viscosity for aqueous solution of Ethanol at 20, 25 and 30 ◦ C (from top to ) NRTL-DVIS and Experimental data from: (䊐) (González et al., bottom). ( 2007) and (◦) (Koohyar et al., 2012) at 19 ◦ C.
Abrams, D.S., Prausnitz, J.M., 1975. Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems. AIChE J. 21, 116–128. Amundsen, T.G., Øi, L.E., Eimer, D.A., 2009. Density and viscosity of monoethanolamine + water + carbon dioxide from (25 to 80) ◦ C. J. Chem. Eng. Data 54, 3096–3100. Andrade, E.N.C., 1930. The viscosity of liquids. Nature 125, 309–310.
D.D.D. Pinto, H.F. Svendsen / International Journal of Greenhouse Gas Control 42 (2015) 494–501 Bernal-García, J.M., Galicia-Luna, L.A., Hall, K.R., Ramos-Estrada, M., Iglesias-Silva, G.A., 2004. Viscosities for aqueous solutions of N-methyldiethanolamine from 313.15 to 363.15 K. J. Chem. Eng. Data 49, 864–866. Bingham, E.C., Jackson, R.F., 1918. Standard substances for the calibration of viscometers. Bull. Bureau Standards 14, 59–86. Brúder, P., Owrang, F., Svendsen, H.F., 2012. Pilot study-CO2 capture into aqueous solutions of 3-methylaminopropylamine (MAPA) activated dimethyl-monoethanolamine (DMMEA). Int. J. Greenh. Gas Control 11, 98–109. Chowdhury, F.I., Akhtar, S., Saleh, M.A., 2010. Viscosities and excess viscosities of aqueous solutions of some diethanolamines. J. Mol. Liq. 155, 1–7. Derks, P.W., Hogendoorn, K.J., Versteeg, G.F., 2005. Solubility of N2O in and density, viscosity, and surface tension of aqueous piperazine solutions. J. Chem. Eng. Data 50, 1947–1950. Först, P., Werner, F., Delgado, A., 2002. On the pressure dependence of the viscosity of aqueous sugar solutions. Rheol. Acta 41, 369–374. Frailie, P., Plaza, J., Wagener, D.V., Rochelle, G.T., 2011. Modeling piperazine thermodynamics. Energy Procedia 4, 35–42, 10th International Conference on Greenhouse Gas Control Technologies. Fu, D., Chen, L., Qin, L., 2012. Experiment and model for the viscosity of carbonated MDEA-MEA aqueous solutions. Fluid Phase Equilib. 319, 42–47. González, B., Calvar, N., Gómez, E., Domínguez, A., 2007. Density, dynamic viscosity, and derived properties of binary mixtures of methanol or ethanol with water, ethyl acetate, and methyl acetate at T = (293.15, 298.15, and 303.15) K. J. Chem. Thermodyn. 39, 1578–1588. Grunberg, L., Nissan, A.H., 1949. Mixture law for viscosity. Nature 164, 799–800. Hartono, A., Mba, E.O., Svendsen, H.F., 2014. Physical properties of partially CO2 loaded aqueous monoethanolamine (mea). J. Chem. Eng. Data 59, 1808–1816. Henni, A., Hromek, J.J., Tontiwachwuthikul, P., Chakma, A., 2003. Volumetric properties and viscosities for aqueous AMP solutions from 25 ◦ c to 70 ◦ c. J. Chem. Eng. Data 48, 551–556. Kennedy, J., Eberhart, R., 1995. Particle swarm optimization, Neural Networks, 1995. In: Proceedings, IEEE International Conference on, pp. 1942–1948. Kestin, J., Khalifa, H.E., Correia, R.J., 1981. Tables of the dynamic and kinematic viscosity of aqueous NaCl solutions in the temperature range 20-150 ◦ C and the pressure range 0. 1-35 MPa. J. Phys. Chem. Ref. Data 10, 71–88. Ko, J.-J., Tsai, T.-C., Lin, C.-Y., Wang, H.-M., Li, M.-H., 2001. Diffusivity of nitrous oxide in aqueous alkanolamine solutions. J. Chem. Eng. Data 46, 160–165. Koohyar, F., Kiani, F., Sharifi, S., Sharifirad, M., Rahmanpour, S.H., 2012. Study on the change of refractive index on mixing, excess molar volume and viscosity deviation for aqueous solution of methanol, ethanol, ethylene glycol, 1-propanol and 1, 2, 3-propantriol at t = 292.15 k and atmospheric pressure. Res. J. Appl. Sci., Eng. Technol. 4, 3095–3101. Li, M.-H., Lie, Y.-C., 1994. Densities and viscosities of solutions of monoethanolamine + N-methyldiethanolamine + water and monoethanolamine + 2-amino-2-methyl-1-propanol + water. J. Chem. Eng. Data 39, 444–447. Liu, J., Wang, S., Hartono, A., Svendsen, H.F., Chen, C., 2012. Solubility of N2O in and density and viscosity of aqueous solutions of piperazine, ammonia, and their mixtures from (283.15 to 323.15) K. J. Chem. Eng. Data 57, 2387–2393. Longinotti, M.P., Corti, H.R., 2008. Viscosity of concentrated sucrose and trehalose aqueous solutions including the supercooled regime. J. Phys. Chem. Ref. Data 37, 1503–1515. Maham, Y., Liew, C.-N., Mather, A., 2002. Viscosities and excess properties of aqueous solutions of ethanolamines from 25 to 80 ◦ C. J. Solution Chem. 31, 743–756. Mahmoudkhani, M., Keith, D.W., 2009. Low-energy sodium hydroxide recovery for CO2 capture from atmospheric air-thermodynamic analysis. Int. J. Greenh. Gas Control 3, 376–384. Mandal, B.P., Kundu, M., Bandyopadhyay, S.S., 2003. Density and viscosity of aqueous solutions of (n-methyldiethanolamine + monoethanolamine), (n-methyldiethanolamine + diethanolamine), (2-amino-2-methyl-1-propanol + monoethanolamine), and (2-amino-2-methyl-1-propanol + diethanolamine). J. Chem. Eng. Data 48, 703–707. Marczak, W., Adamczyk, N., Lezniak, M., 2012. Viscosity of associated mixtures approximated by the grunberg-nissan model. Int. J. Thermophys. 33, 680–691.
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Mathlouthi, M., Reiser, P., 1994. Sucrose: Properties and Applications. Springer US. Monteiro, J.G.-S., Pinto, D.D., Zaidy, S.A., Hartono, A., Svendsen, H.F., 2013. VLE data and modelling of aqueous N,N-diethylethanolamine (DEEA) solutions. Int. J. Greenh. Gas Control 19, 432–440. Murshid, G., Shariff, A.M., Keong, L.K., Bustam, M.A., 2011. Physical properties of aqueous solutions of piperazine and (2-amino-2-methyl-1-propanol + piperazine) from (298.15 to 333.15) K. J. Chem. Eng. Data 56, 2660–2663. Pinto, D.D.D., Monteiro, J.G.M.-S., Bersås, A., Haug-Warberg, T., Svendsen, H.F., 2013. eNRTL parameter fitting procedure for blended amine systems: MDEA-PZ case study. Energy Procedia 37, 1613–1620, GHGT-11. Pinto, D.D.D., Monteiro, J.G.M.-S., Johnsen, B., Svendsen, H.F., Knuutila, H., 2014. Density measurements and modelling of loaded and unloaded aqueous solutions of MDEA (N-methyldiethanolamine), DMEA (N,N-dimethylethanolamine), DEEA (diethylethanolamine) and MAPA (N-methyl-1,3-diaminopropane). Int. J. Greenh. Gas Control 25, 173–185. Pinto, D.D.D., Knuutila, H., Fytianos, G., Haugen, G., Mejdell, T., Svendsen, H.F., 2014. CO2 post combustion capture with a phase change solvent. pilot plant campaign. Int. J. Greenh. Gas Control 31, 153–164. Pinto, D.D.D., 2014. CO2 Capture Solvents: Modeling and Experimental Characterization. Norwegian University of Science and Technology, Department of Chemical Engineering. Poli, R., Kennedy, J., Blackwell, T., 2007. Particle swarm optimization. Swarm Intelligence 1, 33–57. Poling, B., Prausnitz, J., Connell, J., 2000. The Properties of Gases and Liquids. McGraw Hill Professional, McGraw-Hill Education. Renon, H., Prausnitz, J.M., 1968. Local compositions in thermodynamic excess functions for liquid mixtures. AIChE J. 14, 135–144. Rinker, E.B., Oelschlager, D.W., Colussi, A.T., Henry, K.R., Sandall, O.C., 1994. Viscosity, Density, and Surface Tension of Binary Mixtures of Water and N-Methyldiethanolamine and Water and Diethanolamine and Tertiary Mixtures of These Amines with Water over the Temperature Range 20-100 ◦ C. J. Chem. Eng. Data 39, 392–395. Samanta, A., Bandyopadhyay, S.S., 2006. Density and viscosity of aqueous solutions of piperazine and (2-amino-2-methyl-1-propanol + piperazine) from 298 to 333 K. J. Chem. Eng. Data 51, 467–470. Song, Y., Mathias, P.M., Tremblay, D., Chen, C.-C., 2003. Liquid viscosity model for polymer solutions and mixtures. Ind. Eng. Chem. Res. 42, 2415–2422. ˇ ´ V.D., Serbanovi ´ S.P., Djordjevic, ´ B.D., Kijevˇcanin, M.L., 2013. Densities, Spasojevic, c, viscosities, and refractive indices of aqueous alkanolamine solutions as potential carbon dioxide removal reagents. J. Chem. Eng. Data 58, 84–92. Stolaroff, J.K., Keith, D.W., Lowry, G.V., 2008. Carbon dioxide capture from atmospheric air using sodium hydroxide spray. Environ. Sci. Technol. 42, 2728–2735. Teng, T.T., Maham, Y., Hepler, L.G., Mather, A.E., 1994. Viscosity of aqueous solutions of N-methyldiethanolamine and of diethanolamine. J. Chem. Eng. Data 39, 290–293. Tobiesen, F.A., Svendsen, H.F., Juliussen, O., 2007. Experimental validation of a rigorous absorber model for CO2 postcombustion capture. AIChE J. 53, 846–865. Vand, V., 1948. Viscosity of solutions and suspensions. i. theory. J. Phys. Colloid Chem. 52, 277–299. Versteeg, G.F., Dijck, L.A.J.V., Swaaij, W.P.M.V., 1996. On the kinetics between CO2 and alkanolamines both in aqueous and non-aqueous solutions. an overview. Chem. Eng. Commun. 144, 113–158. Weiland, R.H., Chakravarty, T., Mather, A.E., 1993. Solubility of carbon dioxide and hydrogen sulfide in aqueous alkanolamines. Ind. Eng. Chem. Res. 32, 1419–1430. Weiland, R.H., Dingman, J.C., Cronin, D.B., Browning, G.J., 1998. Density and viscosity of some partially carbonated aqueous alkanolamine solutions and their blends. J. Chem. Eng. Data 43, 378–382. Wilson, G.M., 1964. Vapor-liquid equilibrium. XI. a new expression for the excess free energy of mixing. J. Am. Chem. Soc. 86, 127–130. Xu, S., Otto, F.D., Mather, A.E., 1991. Physical properties of aqueous AMP solutions. J. Chem. Eng. Data 36, 71–75. Yoo, M., Han, S.-J., Wee, J.-H., 2013. Carbon dioxide capture capacity of sodium hydroxide aqueous solution. J. Environ. Manage. 114, 512–519.
Contents lists available at ScienceDirect
International Journal of Greenhouse Gas Control journal homepage: www.elsevier.com/locate/ijggc
An excess Gibbs free energy based model to calculate viscosity of multicomponent liquid mixtures Diego D.D. Pinto a,∗ , Hallvard F. Svendsen b a b
PROCEDE Group B.V., PO Box 328, 7500 AH Enschede, The Netherlands Department of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 20 July 2015 Received in revised form 31 August 2015 Accepted 2 September 2015 Keywords: Viscosity Excess Gibbs Liquid NRTL CO2 capture
a b s t r a c t Solution densities and viscosities are important parameters for the design and simulation of absorption processes. Accurate models are needed and in this work, a new model for calculating the liquid viscosity of mixtures is presented. The model uses an analogy to excess Gibbs energy models to account for the deviation from a simple mixing rule based on the pure component viscosities. In this work, we chose the functional form of the NRTL model to represent the excess Gibbs energy and the resulting model is referred to as NRTL-DVIS. Eleven systems (eight binaries and three ternary) were chosen for testing the accuracy of the model. The ternary systems were built from the optimized binaries and pure component systems. With few adjustable parameters, the NRTL-DVIS model represented the tested systems with good accuracy. With few exceptions the calculated total deviation (AARD) was within 3.5%. The NRTL-DVIS model shows better accuracy than other models proposed in the literature. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Densities and viscosities of solutions are especially important when designing and simulating processes. In particular viscosity is important in processes where mass transfer is involved (e.g., absorbers and desorbers). Moreover, equipments such as pumps and heat exchangers are better modeled when the physical properties of the system are accurately calculated (Fu et al., 2012; Weiland et al., 1998). Several properties of a system can be directly correlated to its viscosity. For instance, Versteeg et al. (1996) show that the diffusivity of alkanolamines can be estimated from viscosity with a modified Stokes–Einstein relation. Hence, a good representation of the viscosity of the solution is crucial. Proposed correlations for liquid viscosity available in the literature, unfortunately, do not share the same theoretical basis as for gas viscosity (Poling et al., 2000). Therefore, it is desirable to estimate the liquid viscosity from experimental data whenever available. Many correlations are available in the literature to calculate the viscosity of both pure liquids and mixtures of liquid components. A number of proposed models use the pure liquid viscosity as a starting point and apply a mixing rule to calculate the viscosity
∗ Corresponding author. E-mail address: [email protected] (D.D.D. Pinto). http://dx.doi.org/10.1016/j.ijggc.2015.09.003 1750-5836/© 2015 Elsevier Ltd. All rights reserved.
of the mixture. In these cases, it is required that pure component liquid viscosities are known at given temperatures and pressures. However, for solutions where the viscosity of at least one of the components is not known, the applicability of these types of equations is not straightforward. Aqueous solutions of hydroxide salts are examples of solutions where the liquid viscosity of one component (in this case, the hydroxide) is not known for a wide range of temperatures. These solutions are of interest for CO2 capture processes (see Yoo et al., 2013; Mahmoudkhani and Keith, 2009; Stolaroff et al., 2008). When the pure viscosity of one (or more components) is not available, models which use a reference viscosity are usually applied (see, for instance, Först et al., 2002; Mathlouthi and Reiser, 1994; Vand, 1948). An extensive review on this type of viscosity models is found in Longinotti and Corti (2008). Alternatively, one can still use models which require the pure viscosity of the components by, for instance, setting the unknown viscosity to a constant value. In this work, a new model for calculating the viscosity of multicomponent liquid mixtures is presented. The model is based on the fact that excess viscosities show similar behaviour as that observed for excess Gibbs energy. Hence, the functional forms of the many existing models capable of representing excess Gibbs energy can be used for representing excess viscosities. In particular, the functional form of the NRTL model was chosen in this work. By setting few binary interaction parameters, the model is able to accurately represent the viscosity of liquid mixtures. Eight binary systems and
D.D.D. Pinto, H.F. Svendsen / International Journal of Greenhouse Gas Control 42 (2015) 494–501
three ternary systems were tested to verify the accuracy of the model. Although the model has the same functional form as that of excess Gibbs energy models, it is important to stress that the fitted interaction parameters are not connected at all to the NRTL model and should not be used for activity calculations. Moreover, it is important to check if, at the desired conditions of composition, temperature and pressure, the mixture is a single phase liquid solution. 2. Liquid viscosity correlations Several viscosity correlations are available in the literature. For a pure component in its liquid state, the Andrade type of equation (Andrade, 1930) is well accepted and is implemented in many process simulators. One of the forms of the Andrade equation is given in Eq. (1). Here ◦ is the dynamic viscosity of the liquid, T is the temperature of the system and A, B and C are the adjustable parameters of the model. ln (◦ ) = A +
B + C · ln (T ) T
(1)
For multicomponent systems, the Grunberg–Nissan model the (Grunberg and Nissan, 1949) can be used. This model expands Arrhenius proposition adding adjustable parameters ˛i,j that are fitted to experimental data. Eq. (2) shows the multicomponent form of the Grunberg–Nissan model. In Eqs. (2), (3) and (7), c denotes the concentration basis (mol or mass fraction). ln () =
NC
NC NC
ci · ln ◦i +
i=1
i=1
ci cj ˛i,j
(2)
j=1 j= / i
The Grunberg–Nissan model, however, is not recommended for aqueous solutions (Poling et al., 2000) and several modifications of the Grunberg–Nissan model have been presented in the literature in order to improve its accuracy. For instance, Marczak et al. (2012) introduced the “kinetic” mol fraction concept where the mol fraction scale is redefined, taking into account the associative degree of the components present in the solution. By modifying the scale of the mol fraction in this way, the authors were able to improve the accuracy of the model for aqueous solutions. The model shown in Eq. (3) was proposed by Song etal.(2003). The correlation uses symmetric kij and antisymmetric lij binary interaction parameters allowing the total of adjustable number parameters to be reduced. The term ln ij is called the cross term binary for the viscosities and is defined as in Eq. (4) so that lim ln ij = 0. This model is implemented in the commercial
kij = aij + lij = mij +
ln =
NC
ci ln ◦i +
i=1
NC NC i=1 j>1
kij ci cj ln ij +
NC i=1
⎡ ⎤3 NC 1/3 ⎦ ci ⎣ cj lij ln ij j= / i
(3)
ln ij =
ln i − ln j 2
(4)
T nij T
, kij = kji
(5)
, lij = −lji
(6)
The above presented models require previous knowledge of the components pure viscosities which are usually correlated by temperature functions (as in the Andrade equation). Teng et al. (1994), on the other hand, successfully used a model where the viscosity of aqueous amine solutions was described based on the pure water viscosity and a polynomial function on the amine concentration (Eq. (7)). The model was later applied by Chowdhury et al. (2010) for the same purpose. The authors, however, regressed a set of parameters (pi ) for each desired temperature. ln = ln ◦H
2
O+
P
pi c i
(7)
i=1
Pinto (2014) applied a temperature dependency in the polynomial terms, as shown in Eq. (8), so that with one set of parameters the model would calculate the viscosity of a system at any temperature. pi = ai +
bi T
(8)
The drawback of the approach presented in Pinto (2014) is that the pure component viscosity was not calculated with the same accuracy as before (using one set of parameters for each temperature) although the calculated values were very close to the measurements. Moreover, the number of parameters in the models based on Teng’s approach (Teng, 1994) needed for correlating a system was usually higher than for other models and dependent on the order of the polynomial (P) chosen for modeling the system. In addition, the models based on Teng et al. (1994) approach are only valid for binary systems and the expansion to a multicomponent form would largely increase the number of adjustable parameters in the model. In the next section a new model for calculating liquid mixture viscosities is presented. The model uses the functional form of an excess Gibbs energy model to account for the excess viscosity, i.e., the deviation between the measured viscosity and that calculated using a simple mixing rule and the pure component viscosities. 3. The functional form of excess Gibbs energy models The viscosity of most liquid mixtures cannot be explained by a simple mixing rule as described by Eq. (9).
i →j
process simulator Aspen Plus, and has been shown to be able to represent the viscosity of a number of systems. This model is referred to as the “Aspen liquid mixture viscosity model” in the Aspen Plus process simulator software. To account for calculations at different temperatures, the authors suggested that the binary adjustable parameters kij and lij are given a temperature dependency as shown in Eqs. (5) and (6).
bij
495
ln =
NC
xi ln ◦i
(9)
i=1
However, this simple mixing rule is the basis for several available models, as well as for the proposed model. The difference between the left- and right-hand sides of Eq. (9), except for pure components, is non-zero for most systems. In Fig. 1, the shape of this difference is shown for the H2 O-MDEA system. This shape resembles the excess Gibbs energy function as represented, for example by the Wilson (Wilson, 1964), the NRTL (Renon and Prausnitz, 1968) and the UNIQUAC (Abrams and Prausnitz, 1975) models. Based on this, we propose an excess viscosity model which uses the functional form of an excess Gibbs energy model to account for the deviation from the simple mixing rule as shown in Eq. (10). The viscosity of the liquid mixtures can then be calculated by the simple mixing rule (Eq. (9)) corrected by an excess viscosity model, multiplied by a factor (R). This factor allows the excess viscosity model
496
D.D.D. Pinto, H.F. Svendsen / International Journal of Greenhouse Gas Control 42 (2015) 494–501 Table 1 Systems studied in this work.
Fig. 1. Difference between the left- and right-hand sides of Eq. (9) for H2 O-MDEA system. Experimental data from Teng et al. (1994) at: ( ) 25 ◦ C, ( ) 40 ◦ C and (•) 60 ◦ C.
to reach the desired corrective values while working with parameters of the same magnitude as those used to calculate excess Gibbs energies. In this work it was optimized with the H2 O-MDEA system data and kept constant at 6.48803 for all other studied systems.
ln =
NC
xi ln ◦i (T ) + R · E (T, x)
(10)
i=1
In this work, the functional form of the NRTL model is used to calculate the excess viscosity term (E ). Nevertheless, any other functional form capable of representing excess Gibbs energy is expected to be able to represent the viscosity of liquid mixtures with the same accuracy. For ease of identification, the model will be referred to as the NRTL-DVIS model. The excess viscosity is calculated according to Eqs. (11)–(13). The non-randomness parameters ˛ij were tested at 0.1, 0.2 and 0.3 for all systems. With few exceptions, the best results were obtained for a value of 0.2. If necessary this parameter may be included in the optimization routine. However, in this work the only adjustable parameters in the model are the binary energy parameters ij which are assumed to have a temperature dependency as indicated in Eq. (13). E
=
NC
NC xi
G x j=1 ji ji j
NC
G x k=1 ki k
i=1
Gij = exp −˛ij ij ij = aij +
bij T
(11)
#
System
1 2 3 4 5 6 7 8 9 10 11
H2 O-MEA H2 O-MDEA H2 O-DEA H2 O-TEA H2 O-AMP H2 O-PZ H2 O-MEA-MDEA H2 O-MEA-AMP H2 O-MDEA-DEA H2 O-NaCl H2 O-Ethanol
Component
CAS
MEA MDEA DEA TEA AMP PZ NaCl Ethanol
141-43-5 105-59-9 111-42-2 102-71-6 124-68-5 110-85-0 7647-14-5 64-17-5
respectively by Eqs. (15) and (16), were used to measure the deviations between model and experimental viscosities.
exp 2 NE i − calc i FObj. = exp calc i
i=1
ii = 0
calc 100 i − i AARD (%) = exp NE i
(15)
MAD (mPa s) = max
(16)
NE
i=1
exp
exp − calc i
i
5. Results In this section, 11 systems were chosen to be modeled using the NRTL-DVIS model. All the studied systems are aqueous solutions and the knowledge of the viscosity of pure water is required. The correlation given in Bingham and Jackson (1918) (Eqs. (17) and (18)) was used to calculate the viscosity of pure water where is the fluidity of water in P−1 , ◦H O is the viscosity of pure water in 2 mPa.s and T is the temperature in K. The studied systems are given in Table 1.
(12)
= 2.1482 (T − 281.585) + ,
(14)
· i
8078.4 + (T − 281.585)
(13)
4. Optimization procedure The parameters of the proposed model were estimated to fit the experimental viscosity data for some selected systems. The particle swarm optimization (Kennedy and Eberhart, 1995) algorithm was used in this work. As previously done in Pinto et al. (2014), the lbest topology was chosen with the inertia factor (ω = 0.7298) and the acceleration coefficients (1 and 2 = 1.49618) (Poli et al., 2007). The particles were generated within the interval [−3, 3] and [−3000, 3000] for the adjustable parameters aij and bij , respectively. An objective function (Eq. 14), which weighs equally to the experimental data points (Weiland et al., 1993), was used in the optimization procedure. The average absolute relative deviation (AARD) and the maximum absolute deviation (MAD) given,
2
− 120 (17)
◦H
2O
=
100
(18)
The model was able to calculate the viscosity of the aqueous solutions with excellent accuracy as shown in Table 4. Except for a few systems, the model correlates the experimental data with less than 3 % deviation (AARD). Nevertheless, the total AARD was never higher than 5%. 5.1. Aqueous alkanolamine solutions Aqueous solutions of alkanolamines are widely employed for removing acid gases from various gas streams, both at low and high pressure. Several laboratory and pilot tests can be found for CO2
D.D.D. Pinto, H.F. Svendsen / International Journal of Greenhouse Gas Control 42 (2015) 494–501
497
Table 2 Parameters for the Andrade equation.
MEA MDEA DEA TEA AMP PZ
A
B
C
−149.6008 −170.8208 −271.0938 −342.1746 −339.8066 1.0000
10241.350 12186.230 18423.324 21723.859 21713.806 0.000
20.7372 23.5683 37.8468 48.3915 47.7361 0.0000
post combustion capture (Pinto et al., 2014; Brúder et al., 2012; Tobiesen et al., 2007). For simulation and design, accurate representation of the viscosity of the solution is required. Some ternary systems were chosen due to the large amount of data available and a sequential optimization procedure (SOP) was applied. In this procedure, firstly the smallest subsystem is optimized and its parameters carried to the larger subsystems. The optimization is continued until the complete system is optimized, as exemplified in Pinto et al. (2013) for the eNRTL model. Monteiro et al. (2013) and Frailie et al. (2011) also used the SOP approach.
Fig. 2. Viscosity of the H2 O-MEA system. ( ) NRTL-DVIS and experimental data: ( ) (Maham et al., 2002), ( ) (Amundsen et al., 2009) and ( ) (Hartono et al., ◦ ) NRTL-DVIS and experimental data: ( ) (Maham et al., 2002), 2014) at 25 C. ( ) NRTL( ) (Amundsen et al., 2009) and ( ) (Hartono et al., 2014) at 40 ◦ C. ( DVIS and experimental data: ( ) (Maham et al., 2002), ( ) (Amundsen et al., 2009) and ( ) (Hartono et al., 2014) at 80 ◦ C.
5.1.1. Pure component liquid viscosity The viscosity of the pure amines was calculated with the Andrade equation (Eq. (1)) and the optimized parameters are given in Table 2. The Andrade equation is able to accurately predict the viscosity of the pure amines. The calculated deviations are shown in Table 3. 5.1.2. Binary systems Following the SOP, the parameters for the binary systems (H2 O-MEA, -MDEA, -DEA, -TEA, -AMP and -PZ) are estimated. Unfortunately, no data for the binary amine-amine system were found in the consulted literature. Figs. 2–6 show that the model is able to accurately represent the viscosity of the binary systems for the whole range of concentrations. The calculated deviations are given in Table 4. For a comparison with the new model, the parameters for the Aspen model (Song et al., 2003) were also optimized for the H2 O-MEA system. With the optimized parameters (kH2 O,MEA ) and (lH2 O,MEA ) given in Eqs. (19)–(20), the Aspen model calculates the viscosity of the H2 O–MEA system with a total AARD of 2.86%. From Table 4 it is seen that the NRTL-DVIS model is able to correlate the viscosity of the H2 O–MEA with lower deviations. Therefore,
Fig. 3. Viscosity of the H2 O-MDEA system. ( ) NRTL-DVIS and experimental ) NRTL-DVIS and experimental data: data: ( ) (Teng et al., 1994) at 25 ◦ C. ( ( ) (Teng et al., 1994), ( ) (Bernal-García et al., 2004) and ( ) (Chowdhury et al., ) NRTL-DVIS and experimental data: ( ) (Bernal-García et al., 2010) at 40 ◦ C. ( 2004) at 90 ◦ C.
Table 3 Calculated deviations for the pure amine viscosity. Amine
Trange (K)
# Points
AARD (%)
MAD (mPa s)
Ref.
MEA
298.15–353.15 298.15–353.15 298.15–353.15
5 5 10
2.61 1.14 1.88
0.75 0.30 0.75
Maham et al. (2002) Amundsen et al. (2009) Total
MDEA
298.15–353.15 313.15–363.15 303.15–323.15 298.15–363.15
5 4 5 14
0.45 0.62 0.55 0.53
0.88 0.36 0.41 0.88
Teng et al. (1994) Bernal-García et al. (2004) Chowdhury et al. (2010) Total
DEA
298.15–353.15 303.15–323.15 298.15–353.15
6 5 11
0.71 0.56 0.64
2.82 2.31 2.82
Teng et al. (1994) Chowdhury et al. (2010) Total
TEA
298.15–353.15 303.15–353.15 298.15–353.15
5 11 16
1.94 0.92 1.24
3.75 3.01 3.75
Maham et al. (2002) Ko et al. (2001) Total
AMP
313.15–343.15 303.15–353.15 303.15–353.15
4 6 10
1.49 1.14 1.28
0.43 0.63 0.63
Henni et al. (2003) Li and Lie (1994) Total
498
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Fig. 4. Viscosity of the H2 O-DEA system. ( ) NRTL-DVIS and experimental ) NRTLdata: ( ) (Teng et al., 1994) and ( ) (Spasojevic´ et al., 2013) at 25 ◦ C. ( DVIS and experimental data: ( ) (Teng et al., 1994), ( ) (Chowdhury et al., 2010) ◦ ) NRTL-DVIS and experimental data: and ( ) (Spasojevic´ et al., 2013) at 40 C. ( ( ) (Teng et al., 1994) and ( ) (Spasojevic´ et al., 2013) at 60 ◦ C.
Fig. 7. Viscosity of the H2 O-PZ system. ( ) NRTL-DVIS and experimental from: ( ) (Derks et al., 2005), ( ) (Murshid et al., 2011) and ( ) (Samanta and ) NRTL-DVIS and experimental data: ( ) Bandyopadhyay, 2006) at 25 ◦ C. ( (Derks et al., 2005), ( ) (Murshid et al., 2011), ( ) (Liu et al., 2012) and ( ) (Samanta ◦ ) NRTL-DVIS and experimental data: ( ) and Bandyopadhyay, 2006) at 40 C. ( (Murshid et al., 2011) and ( ) (Samanta and Bandyopadhyay, 2006) at 60 ◦ C.
From Fig. 2 a small discrepancy between the sources for the pure viscosity at 25 ◦ C is seen. At ambient conditions, pure PZ is in its solid state. No data were found for the viscosity of pure PZ measured above its melting point. Therefore, the viscosity of pure PZ was assumed to be constant and set to 0. Moreover, aqueous solutions of PZ present solubility limits. The viscosity data found for aqueous PZ solutions were only measured in the diluted range. However, this range covers the CO2 capture operational range for PZ solvents. Fig. 7 shows that the model is able to represent the viscosity of aqueous solution of PZ. It is also possible to see in the figure a small discrepancy between the sources. Fig. 5. Viscosity of the H2 O-TEA system. ( ) NRTL-DVIS and experimental data: ( ) (Maham et al., 2002) at 25 ◦ C. ( ) NRTL-DVIS and experimental data: ) NRTL-DVIS and ( ) (Maham et al., 2002) and ( ) (Ko et al., 2001) at 40 ◦ C. ( experimental data: ( ) (Maham et al., 2002) and ( ) (Ko et al., 2001) at 60 ◦ C.
only the results from the NRTL-DVIS model will be discussed in this section. kH2 O,MEA = 4.2916 +
43.6297 T
lH2 O,MEA = −5.5117 +
777.1026 T
(19) (20)
Fig. 6. Viscosity of the H2 O-AMP system. ( ) NRTL-DVIS and experimental ) NRTL-DVIS and experimental data: from: ( ) (Henni et al., 2003) at 25 ◦ C. ( ) NRTL-DVIS and ( ) (Henni et al., 2003) and ( ) (Li and Lie, 1994) at 40 ◦ C. ( experimental data: ( ) (Henni et al., 2003) and ( ) (Li and Lie, 1994) at 60 ◦ C.
5.1.3. Ternary systems Once the parameters for calculating the liquid viscosity of the pure components and the binary systems are optimized, the remaining parameters for a ternary system can be fitted to experimental data. The NRTL-DVIS model is able to accurately correlate the tested ternary systems (H2 O-MEA-MDEA, H2 O-MEA-AMP and H2 O-MDEA-DEA) as shown in Figs. 8–10. The calculated deviations are given in Tables 4 and 5 shows the NRTL-DVIS model parameters. These examples show the applicability of the NRTL-DVIS model for multicomponent liquid mixtures. Additionally, by using the SOP, it is guaranteed that the ternary system will always regress to the
Fig. 8. Deviations for the H2 O-MEA-MDEA system. Experimental data: ( ) (Li and Lie, 1994), ( ) (Mandal et al., 2003) and ( ) (Fu et al., 2012).
D.D.D. Pinto, H.F. Svendsen / International Journal of Greenhouse Gas Control 42 (2015) 494–501
499
Table 4 Calculated deviations for the liquid mixtures viscosity. System
Trange (K)
# Points
AARD (%)
MAD (mPa s)
Ref.
H2 O-MEA
298.15–353.15 298.15–353.15 293.15–353.15 293.15–353.15
60 30 26 116
1.78 1.99 1.44 1.77
0.76 0.48 0.07 0.76
Maham et al. (2002) Amundsen et al. (2009) Hartono et al. (2014) Total
H2 O-MDEA
298.15–353.15 313.15–363.15 303.15–323.15 298.15–363.15
70 28 60 158
2.72 2.54 1.82 2.35
5.73 0.72 1.25 5.73
Teng et al. (1994) Bernal-García et al. (2004) Chowdhury et al. (2010) Total
H2 O-DEA
298.15–353.15 303.15–323.15 298.15–343.15 298.15–353.15
60 55 50 165
2.73 1.99 2.12 2.31
15.84 9.65 0.06 15.83
Teng et al. (1994) Chowdhury et al. (2010) Spasojevic´ et al. (2013) Total
H2 O-TEA
298.15–353.15 303.15–313.15 298.15–353.15
65 18 83
3.86 1.02 3.07
18.21 3.01 18.21
Maham et al. (2002) Ko et al. (2001) Total
H2 O-AMP
298.15–343.15 296.75–349.85 303.15–353.15 293.15–323.15 298.15–353.15
66 12 12 7 97
5.28 1.76 2.91 8.28 4.71
12.61 0.06 0.63 0.77 12.61
Henni et al. (2003) Xu et al. (1991) Li and Lie (1994) Mandal et al. (2003) Total
H2 O-PZ
293.15–323.15 298.15–333.15 283.15–323.15 298.15–333.15 283.15–333.15
19 32 25 32 108
3.32 5.89 2.77 1.40 3.38
0.12 0.11 0.05 0.02 0.12
Derks et al. (2005) Murshid et al. (2011) Liu et al. (2012) Samanta and Bandyopadhyay (2006) Total
H2 O-MEA-MDEA
303.15–353.15 293.15–323.15 293.15–343.15 293.15–343.15
36 42 96 174
1.47 2.42 6.70 4.59
0.05 0.17 0.82 0.82
Li and Lie (1994) Mandal et al. (2003) Fu et al. (2012) Total
H2 O-MEA-AMP
303.15–353.15 293.15–323.15 293.15–353.15
36 42 78
2.31 3.90 3.17
0.19 0.62 0.62
Li and Lie (1994) Mandal et al. (2003) Total
H2 O-MDEA-DEA
293.15–373.15 293.15–323.15 293.15–373.15
20 42 62
2.11 3.27 2.90
0.73 0.23 0.73
Rinker et al. (1994) Mandal et al. (2003) Total
H2 O-Ethanol
293.15–303.15 292.15 292.15–303.15
31 13 44
4.58 4.07 4.43
0.18 0.26 0.26
González et al. (2007) Koohyar et al. (2012) Total
H2 O-NaCl
293.15–373.15
204
0.73
0.07
binary systems with the same accuracy as for the regressed binary systems in case the system is depleted of one of the components. 5.2. Non-amine systems The model can also calculate the viscosities of other liquid solutions than aqueous amine solutions. As an example, the aqueous ethanol solution is represented by the NRTL-DVIS model.
Kestin et al. (1981)
Experimental data from González et al. (2007) and Koohyar et al. (2012) were used in the optimization procedure. The parameters of the Andrade equation were optimized to calculate the viscosity of pure ethanol. Eq. (21) represents the viscosity of pure ethanol (mPa s) at a given temperature (K). ln EtOH = −231.1861 −
11734.199 − 33.6819 · ln (T ) T
(21)
Table 5 Parameters for the NRTL-DVIS model. Component
Pair value
i
j
aij
aji
bij
bji
˛ij
H2 O H2 O H2 O H2 O H2 O H2 O MEA MEA MDEA H2 O H2 O
MEA MDEA DEA TEA AMP PZ MDEA AMP DEA Ethanol NaCl
2.18950 0.32194 3.56622 2.22927 0.26558 −31.95187 −1.95269 −13.25264 13.84543 −12.29712 −2.77753
−2.36085 −2.85327 −3.90934 −2.95869 −2.49559 −148.60569 104.51276 712.89558 −6.21575 1.77837 17.71668
−30.14092 1179.30750 365.23495 563.61046 971.24906 9841.72047 274.5111 4205.40984 344.25523 4763.645 129.42804
471.76056 521.44416 509.90036 567.69868 429.15545 52759.768 −30162.821 −205243.927 1042.3131 −1027.575 −78.87626
0.2 0.2 0.1 0.2 0.2 0.1 0.2 0.1 0.1 0.2 0.1
500
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Fig. 9. Deviations for the H2 O-MEA-AMP system. Experimental data: ( Lie, 1994) and ( ) (Mandal et al., 2003).
) (Li and
Fig. 12. Viscosity for aqueous solution of NaCl at 20, 40 and 80 ◦ C (from top to ) NRTL-DVIS and Experimental data: (䊐) (Kestin et al., 1981) bottom). (
in the measurements. Nevertheless, the model is able to calculate the viscosity of aqueous solutions of ethanol with reasonably good accuracy. The deviations are reported in Table 4 and the parameters are given in Table 5. 5.3. Systems containing solid pure components
Fig. 10. Deviations for the H2 O-MDEA-DEA system. Experimental data: ( ) (Rinker et al., 1994) and ( ) (Mandal et al., 2003).
From Fig. 11 it is seen that the highest deviations occur between 0.3 and 0.4 ethanol mol fractions. An experimental variation between the two used independent sources is also observed. Koohyar et al. (2012) measured viscosity of aqueous ethanol solutions at 19 ◦ C. As these measurements are at a temperature 1 ◦ C lower than the ones measured by González et al. (2007) it was expected that the viscosities at lower temperature were greater than the ones measured at higher temperature. This behavior is not observed for the whole range of composition when comparing these two sources and reflects the experimental uncertainties
Although the model requires the pure component viscosity to calculate the viscosity of the solution, it is still possible to use the NRTL-DVIS model when the viscosity of one of the components is not available. For this type of systems, we recommend setting the A, B and C parameters in the Andrade equation to 1, 0 and 0, respectively and to optimize the NRTL-DVIS parameters, as done with PZ. Aqueous solutions of NaCl are an example of this kind of system and was used to test the NRTL-DVIS model. Experimental data from Kestin et al. (1981) were used to optimize the model parameters. The parameters are given in Table 5 and the calculated deviations are presented in Table 4. The model is able to correlate the experimental viscosity of aqueous solutions of NaCl with very low deviation as seen in Fig. 12. 6. Conclusions The NRTL-DVIS model uses the NRTL model structure to correlate the excess viscosity of liquid mixtures. The NRTL model is relatively simple and presents only few binary adjustable parameters which are fitted to experimental data. The binary interaction parameters are assumed to have a simple temperature dependency to account for calculations at different temperatures. The NRTL-DVIS model is able to correlate the viscosity of liquid mixtures with very good accuracy. The model is also able to predict the viscosity of aqueous solutions, for which models like the Grunberg–Nissan are not suitable, for the whole range of composition and at different temperatures. The model was able to represent the viscosity of tested cases with higher accuracy than other models available in the literature. It is important to stress that the parameters used in the NRTL-DVIS model do not have any relationship with the thermodynamic NRTL model. References
Fig. 11. Viscosity for aqueous solution of Ethanol at 20, 25 and 30 ◦ C (from top to ) NRTL-DVIS and Experimental data from: (䊐) (González et al., bottom). ( 2007) and (◦) (Koohyar et al., 2012) at 19 ◦ C.
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