15Parameter Estimation in NonlinearThermodynamic Models:Activity Coefficients
Activity coefficient models offer an alternative approach to equations ofstate for the calculation of fugacities in liquid solutions (Prausnitz et al. 1986; Tas-sios, 1993). These models are also mechanistic and contain adjustable parametersto enhance their correlational ability. The parameters are estimated by matchingthe thermodynamic model to available equilibrium data. In this chapter, we con-sider the estimation of parameters in activity coefficient models for electrolyte andnon-electrolyte solutions.
15.1 ELECTROLYTE SOLUTIONS
We consider Pitzer's model for the calculation of activity coefficients inaqueous electrolyte solutions (Pitzer, 1991). It is the most widely used thermody-namic model for electrolyte solutions.
15.1.1 Pitzer's Model Parameters for Aqueous Na2SiO3 Solutions
Osmotic coefficient data measured by Park (Park and Englezos, 1998; Park,1999) are used for the estimation of the model parameters. There are 16 osmoticcoefficient data available for the Na2SiO3 aqueous solution. The data are given inTable 15.1. Based on these measurements the following parameters in Pitzer's
Parameter Estimation in Activity Coefficients Thermodynamic Models 269
activity coefficient model can be calculated: p(0), P(1>, and C9 for Na2SiO3. Thethree binary parameters may be determined by minimizing the following leastsquares objective function (Park and Englezos, 1998).
c a l c - e x P 2
(15.1)
where q>calc is the calculated and <pcxp is the measured osmotic coefficient and <59 isthe uncertainty.
Table 15.1 Osmotic Coefficient Data for the Aqueous Na2SiO3 Solution
wherez is the chargeM denotes a cationX denotes an anionf is equal to -A«//2/(l +bl"2)A,p is the Debye-Huckel osmotic coefficient parameterm is the molality of solute
1 -i x"1 ~>is the ionic strength ( I = — > m,zf )2'wi
v is the number of ions produced by 1 mole of the solute, andb is a universal parameter with the value of 1 .2 (kg.mol)"2
The parameters P(0)MX, P ( I )MX, C\ix are tabulated binary parameters specific
to the electrolyte MX. The P(2V« is a parameter to account for the ion pairing ef-fect of 2-2 electrolytes. When either cation M or anion X is univalent, oci = 2.0.For 2-2, or higher valence pairs, «i = 1.4. The constant cc2 is equal to 12. The pa-rameter vector to be estimated is k=[p(0), p(1), C* ]T.
15.1.2 Pitzer's Model Parameters for Aqueous Na2SiO3-NaOH Solutions
There are 26 experimental osmotic coefficient data and they are given inTable 15.2 (Park and Englezos, 1999; Park, 1999). Two sets of the binary pa-rameters for the NaOH and Na2SiO3 systems and two mixing parameters,
2- and T , 9 are required to model this system. The binary+ - M } 3
parameters for the Na2SiO3 solution were obtained previously and those for theNaOH system, p(0)
Na0n = 0.0864, P(1)NaOH = 0.253, and C^OM = 0.0044 at 298.15
K are available in the literature. The remaining two Pitzer's mixing parameters,9 „.„, and T , ~> were determined by the least squares optimizationOH SiO,-- Na+OI-rSiO^ J M F
method using the 26 osmotic coefficient data of Table 1 5.2.The Gauss-Newton method may be used to minimize the following LS ob-
Parameter Estimation in Activity Coefficients Thermodynanlic Models 271
where cpexp is the measured osmotic coefficient, a9 is the uncertainty computedusing the error propagation law (Park, 1 999) and (pcalc is calculated by the follow-ing equation,
<pcalc = } + ( 2 / m i ) [ f < P l + cm a(B? a +ZC c a)
where
m is the molality of the solute
f is equal to -A/'^l +bI1/2)
A,p is the Debye-Hilckel osmotic coefficient parameter
b is a universal parameter with the value of 1.2 (kg.mol)"2
I is the ionic strength ( I = — > irhz2 )s v 24- ' 'i
c is a cation
a is an anion
B'cpa isequalto P^'+Pca exp(-a,I1/2)+p^ exp(-a2I l / 2)
are tabulated mixing parameters specific to the cation-cation or anion-anion pairs. The ^f^ is also tabulated mixing parameter specific to thecation-anion-anion or anion-cation-cation pairs.
Table 15.2 Osmotic Coefficient Data for the Aqueous Na2SiOrNaOH Solution
Parameter Estimation in Activity Coefficients Thermodynamic Models 273
The parameters '0jj(I) and 0'i/I) represent the effects of asymmetricalmixing. These values are significant only for 3-1 or higher electrolytes (Pitzer,1975). The 2-dimensional parameter vector to be estimated is simply
,2-f.k=[9L OH S Na+OH SiO3
15.1.3 Numerical Results
First the values for the parameter vector k=[p(0), p(1), C* ]T were obtained byusing the Na2SiO3 data and minimizing Equation 15.1. The estimated parametervalues are shown in Table 15.3 together with their standard deviation.
Subsequently, the parameter vector k=[6M J F L OH~SiOrT
Na+OH~SiO|~was
estimated by using the data for the Na2SiO3-NaOH system and minimizing Equa-tion 15.3. The estimated parameter values and their standard errors of estimate arealso given in Table 15.3. It is noted that for the minimization of Equation 15.3knowledge of the binary parameters forNaOH is needed. These parameter valuesare available in the literature (Park and Englezos, 1998).
Table 15.3 Calculated Pitzer's Model Parameters forNa2SiO3 and Na2SiO3-NaOH Systems
Parameter Value
P(0) = 0.0577
P(1) = 2.8965
C9 = 0.00977
0 _ . 2_ = -0.2703
T , = 0.0233Na + OH SiO^
Standard Deviation
0.0039
0.0559
0.00176
0.0384
0.0095
Source: Park and Englezos (1998).
The calculated osmotic coefficients using Pitzer's parameters were com-pared with the experimentally obtained values and found to have an average per-cent error of 0.33 for Na2SiO3 and 1.74 for the Na2SiO3-NaOH system respec-tively (Park, 1999; Park and Englezos, 1998). Figure 15.1 shows the experimentaland calculated osmotic coefficients of Na2SiO3 and Figure 15.2 those for theNa2SiO3-NaOH system respectively. As seen the agreement between calculated
and experimental values is excellent. There are some minor inflections on the cal-culated curves at molalities below 0.1 mol/kg H2O. Similar inflections were alsoobserved in other systems in the literature (Park and Englezos, 1998). It is notedthat it is known that the isopiestic method does not give reliable results below 0.1mol/kg H2O.
Park has also obtained osmotic coefficient data for the aqueous solutions ofNaOH-NaCl- NaAl(OH)4 at 25°C employing the isopiestic method (Park andEnglezos, 1999; Park, 1999). The solutions were prepared by dissolvingA1C13-6H2O in aqueous NaOH solutions. The osmotic coefficient data were thenused to evaluate the unknown Pitzer's binary and mixing parameters for theNaOH-NaCl-NaAl(OH)4-H2O system. The binary Pitzer's parameters, P(0), P(1),and C9, for NaAl(OH)4 were found to be -0.0083, 0.0710, and 0.00184 respec-tively. These binary parameters were obtained from the data on the ternary systembecause it was not possible to prepare a single (NaAl(OH)4) solution.
1 .0
ido<J
>^N+->O
0.9 -
0.8 -
0.70 .0 0 .5 1 .0 1 .5 2 .0 2 .5
Molality (mol/kg H2O)
Figure 15.1 Calculated and experimental osmotic coefficients for \'a2SiO3The line represents the calculated values.
15.2 NON-ELECTROLYTE SOLUTIONS
Activity coefficient models are functions of temperature, composition and toa very small extent pressure. They offer the possibility of expressing the fugacity
Parameter Estimation in Activity Coefficients Thermodynamic Models 275
of a chemical j, f j ' , in a liquid solution as follows (Prausnitz et al. 1986; Tassios,1993)
f j L = x J ? J f J (15-5)
where Xj is the mole fraction, YJ is the activity coefficient and fj is the fugacity ofchemical species j.
1.0
ao"o
ooo
o
0.9 -
0.8 -
0.0 0.5 1.0 1.5 2.0Molality (mol/fcg H2O)
Figure 15.2 Calculated and experimental osmotic coefficients for the Na2SiO3
-NaOH system. The line represents the calculated values.
Several activity coefficient models are available for industrial use. They arepresented extensively in the thermodynamics literature (Prausnitz et al., 1986).Here we will give the equations for the activity coefficients of each component ina binary mixture. These equations can be used to regress binary parameters frombinary experimental vapor-liquid equilibrium data.
The activity coefficients are given by the following equations
=-/M(XJ + A 1 2 x 2 ) + x 2A 12 A 21
X ] + A 1 2 x 2 A 2 1 x 1 + x 2(15.6a)
• x l A 12 A 21X 1 + A 1 2 x 2 A 2 1 x , + x 2
(15.6b)
The adjustable parameters are related to pure component molar volumes andto characteristic energy differences as following
V2
vl RT(15.7a)
A vlA 2 i =—Lexp -v2
(15.7b)
where V) and v2 are the liquid molar volumes of components 1 and 2 and the X'sare energies of interaction between the molecules designated in the subscripts. Thetemperature dependence of the quantities (A,12-A,n) and (^12-^22) can be neglectedwithout serious error.
15.2.2 The Three-Parameter NRTL Model
Renon used the concept of local composition to develop a non-random, two-liquid (NRTL) three parameter (a]2, 112, ^21) equations given below (Prausnitz etal., 1986).
Parameter Estimation in Activity Coefficients Thermodynamic Models 277
G12 = exp(-a,2T12) (15. 8c)
G21 = ex/X-a12T21) (15.8d)
^2=^ (15.8e)
_ §21 -gn msn*21 - —— —— —— (15. »t)K 1
The parameter gy is an energy parameter characteristic of the i-j interaction.The parameter a\2 is related to the non-randomness in the mixture. The NRTLmodel contains three parameters which are independent of temperature and com-position. However, experience has shown that for a large number of binary sys-tems the parameter a,2 varies from about 0.20 to 0.47. Typically, the value of 0.3is set.
15.2.3 The Two-Parameter UNIQUAC Model
The Universal Quasichemical (UNIQUAC) is a two-parameter (i^, T 2 j )model based on statistical mechanical theory. Activity coefficients are obtained by
(15.9a)
I n y 2 = l n —— + — q 2 I n — ̂ - + (p , ( l 2 - — l , ) -q 2 ln(9'22 2 cp2
Segment or volume fractions, <p, and area fractions, 9 and 9' , are given by
x , r, x „ r,(15.9e)
x i r, + x -, r2 x | r, + x 2 r,
(15.9f)x , q , + x 2 q 2 ~ x , q , + x 2 q 2
-. 62 =——^^ , (15.9g)x ,q , + X 2 q 2 x , q , + x 2 q 2
Parameters r, q and q'are pure component molecular-structure constantsdepending on molecular size and external surface areas. For fluids other than wa-ter or lower alcohols, q = q ' .
For each binary mixture there are two adjustable parameters, T [2 and T21.These in turn, are given in terms of characteristic energies Au ]2 =u]2-u22 andAu21=u2|-Un given by
RT RT(IS.lOa)
RT RT
Characteristic energies, Au12 and Au2i are often only weakly dependent ontemperature. The UNIQUAC equation is applicable to a wide variety of nonelec-trolyte liquid mixtures containing nonpolar or polar fluids such as hydrocarbons,alcohols, nitriles, ketones, aldehydes, organic acids, etc. and water, including par-tially miscible mixtures. The main advantages are its relative simplicity using onlytwo adjustable parameters and its wide range of applicability.
15.2.4 Parameter Estimation: The Objective Function
According to Tassios (1993) a suitable objective function to be minimized insuch cases is the following
Parameter Estimation in Activity Coefficients Thermodynamic Models 279
(15.11)v >
This is equivalent to assuming that the standard error in the measurement ofYJ is proportional to its value.
Experimental values for the activity coefficients for components 1 and 2 areobtained from the vapor-liquid equilibrium data. During an experiment, the fol-lowing information is obtained: Pressure (P), temperature (T), liquid phase molefraction (xi and x2=l-x1) and vapor phase mole fraction (V] and y2=l-yO-
The activity coefficients are evaluated from the above phase equilibriumdata by procedures widely available in the thermodynamics literature (Tassios,1993; Prausnitz et al. 1986). Since the objective in this book is parameterestimation we will provide evaluated values of the activity coefficients based on
the phase equilibrium data and we will call these values experimental. These yexp
values can then be employed in Equation 15.11.Alternatively, one may use implicit LS estimation, e.g., minimize Equation
14.23 where liquid phase fugacities are computed by Equation 15.5 whereasvapor phase fugacities are computed by an EoS or any other available method(Prausnitz et al., 1986).
15.3 PROBLEMS
A number of problems formulated with data from the literature are givennext as exercises. In addition, to the objective function given by Equation 15.11the reader who is familiar with thermodynamic computations may explore the useof implicit objective functions based on fugacity calculations.
15.3.1 Osmotic Coefficients for Aqueous Solutions of KC1 Obtained by theIsopiestic Method
Thiessen and Wilson (1987) presented a modified isopiestic apparatus andobtained osmotic coefficient data for KC1 solutions using NaCl as reference solu-tion. The data are given in Table 15.4. Subsequently, they employed Pitzer'smethod to correlate the data. They obtained the following values for three Pitzer'smodel parameters: p^ = 0.05041176, p^ = 0.195522, C^x = 0.001355442 .
Using a constant error for the measurement of the osmotic coefficient, esti-mate Pitzer's parameters as well as the standard error of the parameter estimates byminimizing the objective function given by Equation 15.1 and compare the resultswith the reported parameters.
Table 15.4 Osmotic Coefficients for Aqueous KCl Solutions
MolalityofKCl
0.09872
0.098930.5274
0.96341.0431.1571.9292.9194.148
Osmotic Coefficient (q>)
0.9325
0.9265
0.89460.89440.89810.90090.9120
0.93510.9675
Source: Thiessen and Wilson (1987).
15.3.2 Osmotic Coefficients for Aqueous Solutions of High-Purity NiCl2
Rard (1992) reported the results of isopiestic vapor-pressure measurementsfor the aqueous solution of high-purity NiCl2 solution form 1.4382 to 5.7199mol/kg at 298.1510.005 K. Based on these measurements he calculated the os-motic coefficient of aqueous NiCl2 solutions. He also evaluated other data fromthe literature and finally presented a set of smoothed osmotic coefficient and ac-tivity of water data (see Table IV in original reference).
Rard also employed Pitzer's electrolyte activity coefficient model to correlatethe data. It was found that the quality of the fit depended on the range of molalitiesthat were used. In particular, the fit was very good when the molalities were lessthan 3 mol/kg.
Estimate Pitzer's electrolyte activity coefficient model by minimizing the ob-jective function given by Equation 15.1 and using the following osmotic coeffi-cient data from Rard (1992) given in Table 15.5. First, use the data for molalitiesless than 3 mol/kg and then all the data together. Compare your estimated valueswith those reported by Rard (1992). Use a constant value for a,p in Equation 15.1.
15.3.3 The Benzene (1 )-i-Propyl Alcohol (2) System
Calculate the binary parameters for the UNIQUAC equation by using thevapour-liquid equilibrium data for benzene(l)-i-propyl alcohol (2) at 760 mmHg(Tassios, 1993). The following values for other UNIQUAC parameters are avail-able from Tassios (1993): r,=3.19, q,=2.40, r2=2.78, q2=2.51. The data are given inTable 15.6.
Tassios (1993) also reported the following parameter estimates
Au,R
= -231.5 (15.12a)
Au-R
= 10.6 (15.12b)
The objective function to be minimized is given by Equation 15.11. The experi-mental values for the activity coefficients are also given in Table 15.5.
15.3.4 Vapor-Liquid Equilibria of Coal-Derived Liquids: Binary Systemswith Tetralin
Blanco et al. (1994) presented VLB data at 26.66±0.03 kPa for binary sys-tems of tetralin with p-xylene, g-picoline, piperidine, and pyridine. The data forthe pyridine (l)-tetralin (2) binary are given in Table 15.7.
Blanco et al. have also correlated the results with the van Laar, Wilson,NRTL and UNIQUAC activity coefficient models and found all of them able todescribe the observed phase behavior. The value of the parameter a12 in the NRTLmodel was set equal to 0.3. The estimated parameters were reported in Table 10 ofthe above reference. Using the data of Table 15.7 estimate the binary parametersin the Wislon, NRTL and UNIQUAC models. The objective function to be mini-mized is given by Equation 15.11.
Monton and Llopis (1994) presented VLB data at 6.66 and 26.66 kPa for bi-nary systems of ethylbenzene with m-xylene and o-xylene. The accuracy of thetemperature measurement was 0.1 K and that of the pressure was 0.01 kPa. Thestandard deviations of the measured mole fractions were less than 0.001. The dataat 26.66 for the ethylbenzene (1) - o-Xylene (2) are given in Table 15.8 and theobjective is to estimate the NRTL and ITNIQUAC parameters based on these data.
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