Article pubs.acs.org/IECR
Equation-of-State Modeling of Solid−Liquid−Gas Equilibrium of CO2 Binary Mixtures Ilias K. Nikolaidis,†,‡,⊥ Georgios C. Boulougouris,†,§ Loukas D. Peristeras,† and Ioannis G. Economou*,†,‡ †
National Center for Scientific Research “Demokritos”, Institute of Nanoscience and Nanotechnology, Molecular Thermodynamics and Modeling of Materials Laboratory, 15310 Aghia Paraskevi, Attikis, Greece ‡ Texas A&M University at Qatar, Chemical Engineering Program, PO Box 23874, Education City, Doha, Qatar § Democritus University of Thrace, Department of Molecular Biology and Genetics, 68100 Dragana, Alexandroupolis, Greece ⊥ National Technical University of Athens, School of Chemical Engineering, 9 Heroon Polytechniou Street, Zografou Campus, 15780 Athens, Greece S Supporting Information *
ABSTRACT: Thermodynamic models of different complexity have been used to model the solid−fluid equilibrium (SFE) of pure carbon dioxide (CO2) and binary mixtures of CO2 with other compounds. The models used to describe the solid phase include an empirical correlation fitted on experimental data at SFE conditions, a model based on thermodynamic integration, and a solid-phase equation of state (EoS) developed for pure CO2. These models have been coupled with Peng−Robinson, Soave−Redlich−Kwong, and Perturbed Chain− Statistical Associating Fluid Theory EoS and have been applied to mixtures of CO2 with N2, H2, and CH4, where CO2 forms the solid phase, and also to mixtures of CO2 and ethylene with naphthalene and phenanthrene, where the heavier compound forms the solid. Very good agreement with experimental data has been achieved in all cases, both with no binary interaction parameter and with an interaction parameter fitted to vapor−liquid equilibrium data.
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INTRODUCTION Fossil fuels are currently the most widely used sources for power and heat generation. They are also used in heavy industrial manufacturing operations. The extensive consumption of fossil fuels contributes significantly to the increased levels of greenhouse gases in the atmosphere, which subsequently leads to environmental problems such as global warming. The most important greenhouse gas, in terms of quantity and impact, is carbon dioxide (CO2). As the global energy demand has increased, CO2 levels have risen significantly, from the preindustrial levels of 280 ppm to 403 ppm in January 2016.1 Moreover, fossil fuels will continue to play an important role in power and heat production and also be used in large industrial operations in the foreseeable future.2−5 Unless major measures are taken for the reduction of CO2 emissions, CO2 concentration is projected to rise even more over the next 25 years because global demands for energy are anticipated to increase.6 A significant amount of research has been conducted for the development of new technologies that aim to reduce the levels of CO2 in the atmosphere. The most mature technology today is carbon capture and sequestration (CCS), which is the process of capturing CO2 from the flue gas of a large point source (typically a power plant), transporting it to a sequestration site, and then depositing it to a geological formation, which can be a saline aquifer or a depleted oil well. The CCS process can be divided into three main parts: CO2 capture, transport, and storage. There are © XXXX American Chemical Society
several different methodologies for the capture process, including precombustion, postcombustion, and oxy-combustion process. These processes result in streams that contain different gases such as N2, CH4, O2, Ar, SO2, H2S, and H24,7,8 at various concentrations. An important part of the CCS process is the transportation of the CO2-rich stream from the capture plant to the sequestration site. In most cases, transporting CO2 via pipelines is the most cost-effective way of transport.7 Preliminary conceptual design, detailed design, simulation, and optimization of the transport process require, among other things, accurate knowledge of the physical properties of the chemical system involved as a function of temperature, pressure, and composition. Quite often, the system exists in more than one phase (i.e., liquid, vapor and/or solid), and as a result, process design calculations have to take into account the phase equilibrium conditions and also the composition of the relevant phases and the respective physical property values. The two challenges that arise are the accurate prediction or correlation of the physical properties of the system and the conditions of instability, where the system is going to split into two or more coexisting phases. Vapor−liquid equilibrium (VLE) of CO2 mixtures has attracted most of the attention both in terms of experimental Received: February 18, 2016 Revised: April 19, 2016 Accepted: May 5, 2016
A
DOI: 10.1021/acs.iecr.6b00669 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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of CO2 mixtures with other compounds. These models include an empirical correlation model, a model based on thermodynamic integration, and a solid-phase EoS for pure CO2. The different models are coupled with three fluid-phase EoS (PR, SRK, and PC-SAFT), and the performance of each combined model is evaluated for various binary mixtures. In total, seven different models are examined (i.e., the empirical correlation and the thermodynamic integration models each one coupled with SRK, PR, and PC-SAFT EoS and an EoS developed for solids coupled with an EoS developed for fluids). The solid-phase EoS used for this purpose was the EoS proposed by Jäger and Span,25 although other appropriate EoS, such as the one proposed by Trusler,26 could be used. For the case of fluid EoS, PC-SAFT was selected. Here again, any other cubic or higher-order EoS could be also used. Finally, to examine the general validity of the proposed scheme, mixtures where the solid forming component is not CO2 are examined.
measurements and modeling using equations of state (EoS). Coquelet et al.9 measured the VLE of the CO2−Ar binary mixture at six different isotherms from 233.32 to 299.21 K and pressures up to 14 MPa and modeled the data using the Peng−Robinson (PR) EoS. Li and Yan10 modeled the VLE of CO2 with various impurities, using different cubic EoS and concluded that Soave− Redlich−Kwong (SRK)11 and PR12 EoS are the most accurate ones. Diamantonis et al.13 evaluated the performance of cubic, statistical associating fluid theory (SAFT)14,15 and perturbed chainSAFT (PC-SAFT)16 EoS using the standard van der Waals one fluid theory (vdW 1f) mixing rules in predicting and correlating the VLE of different binary CO2 mixtures with impurities, based on isothermal experimental data reported in the literature. The authors also performed liquid density calculations and concluded that PC-SAFT is on average the most accurate model for VLE prediction, but when a binary interaction parameter (BIP) was used, the accuracy of all models was comparable. Chapoy et al.17 reported new experimental VLE data for the CO2−H2S mixture from 258.41 to 313.02 K in a pressure range of 1.0−5.5 MPa. The authors have shown that cubic EoS are able to provide a satisfactory description of the phase behavior when appropriate BIPs are used. Experimental measurements for the CO2−SO2 and CO2−NO mixtures at different isotherms were also performed by Coquelet et al.18 Moreover, the VLE of two ternary mixtures (CO2−O2−Ar and CO2−SO2−O2) were measured in the same work, and the PR EoS was used to model the phase behavior and to calculate the critical locus of the two binary mixtures. Recently,19 a new setup for VLE measurements of CO2-rich mixtures was presented, and new isothermal VLE data were determined for the binary mixture CO2−N2 at four different isotherms from 223 to 303 K. The authors modeled the new data using higher-order EoS (GERG-2008/EOS-CG) and an EoS/GE model. The constant composition VLE phase envelopes of binary CO2 mixtures with N2, O2, Ar, and CH4 were experimentally determined by Ahmad et al.,20 who used the SRK, the GERG-2008, and a group contribution EoS (EOS-CG) to model the phase behavior. Blanco et al.21,22 performed experimental measurements for the determination of the constant composition VLE phase envelopes of CO2 mixtures with CO and CH4 at different compositions. In these works, experimental measurements were performed to obtain new pressure−density−temperature data for the two mixtures. The PR, PC-SAFT, and GERG EoS were used to model the VLE of CO2−CO mixture, whereas only GERG was used to model CO2−CH4 mixture. Furthermore, the VLE of CO2−O2−Ar−N2 mixture was experimentally determined by Chapoy et al.23 Although a significant amount of research has been conducted for the experimental determination and modeling of the VLE of CO2-rich mixtures with the compounds mentioned previously, relatively little work has been performed to measure and predict the solid−fluid equilibrium (SFE) of CO2 mixtures, which is critical to the design and operation of CO2 pipelines and storage facilities. Hazard assessment studies associated with CO2 transport include scenarios of accidental releases with sharp expansion, where solid−vapor (SVE) as well as solid−liquid (SLE) equilibria may occur.24 CO2 exhibits a relatively high Joule−Thomson expansion coefficient, and a pipeline depressurization will lead to rapid cooling, which can reach very low temperatures. As a result, solid formation can be expected.24 Taking this into account, it is easily understood that the formation of dry ice resulting from SFE can largely affect the safety of CCS facilities during equipment depressurization, process shutdown, or other process upsets. In this work, solid-phase thermodynamic models of different complexity are applied to model the SFE of pure CO2 and also that
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SOLID-PHASE MODELS Empirical Correlation Model. Calculation of the SFE of a multicomponent mixture requires the equilibration of the chemical potentials of a solid-forming compound in the two coexisting phases (S: solid phase, F: fluid phase) at the same temperature and pressure: μiS (T , P) = μi F (T , P , x F)
(1)
where μSi stands for the chemical potential of the solid former in a pure solid phase and μFi is the chemical potential of the same
compound in the coexisting fluid phase of molar composition characterized by the vector xF. If the ideal gas reference state is used to calculate the chemical potential for both phases, then eq 1 can be substituted by the equation of fugacities:27 S F fi ̂ (T , P) = fi ̂ (T , P , x F)
(2)
This leads subsequently to the expression: ⎤ ⎡ vS sat ⎢ 0i (P − P0sati (T ))⎥ P0sati (T ) φ0̂ sat ( T , P ) exp 0i i ⎦ ⎣ RT = x iF φî F(T , P , x F) P
(3)
Psat 0i (T) is
where the saturation pressure of the pure solid former at sat temperature T, φ̂ sat 0i (T, P0i ) is the fugacity coefficient of the pure F F solid former at temperature T and pressure Psat 0i , φ̂ i (T, P, x ) is the fugacity coefficient of the solid former in the fluid mixture of molar composition xF at temperature T and pressure P, and vS0i is the temperature- and pressure-independent pure solid molar volume. To use eq 3 for SFE calculations, it is necessary to couple a fluid-phase EoS with a model that provides the saturation pressure of the solid former. This solid-phase model can be an empirical correlation fitted to experimental data under SLE or SVE conditions. In this work, two correlations for solid CO2 taken from Span and Wagner28 were used. Thermodynamic Integration Model. An alternative way to model SFE was proposed by Seiler et al.29 Here, in the case of SLE, the reference state is the pure, subcooled melt at system temperature and standard pressure (P+). The pure superheated sublime reference state at system temperature and standard pressure can be used to model the SVE. The standard pressure is chosen by taking into account the existence of measured (or accurately calculated) caloric data at this reference state. B
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30 where ΔhSV is the enthalpy of 0i (26 300 J/mol for CO2) sublimation at sublimation temperature TSV 0i (194.5 K for CO2), vS0i is the pure solid-former solid molar volume (29.069 cm3/mol for CO2),31 and ΔcSV P,0i* is the difference of the molar, isobaric heat capacities between the hypothetical superheated sublime and the solid (−23.611 J/(mol K) for CO2). The solid-phase heat capacity value for CO2 is taken from DIPPR.31 The vapor-phase heat capacity is calculated from PC-SAFT EoS because there are no available experimental data. The ideal gas heat capacity is calculated using a DIPPR correlation, and the residual part is calculated using PC-SAFT. To use eq 8, the physical properties of the solid-forming compound are needed at the standard pressure P+(0.1 MPa for CO2). Inherent to the model described by eq 8 are the following assumptions that must be emphasized: • The solid phase consists of only one component. • The solid-phase volume is pressure- and temperatureindependent. • The difference in the isobaric heat capacities is temperature-independent. Gibbs Free Energy Equation of State for Solid CO2. Jäger and Span25 proposed a new EoS that describes the thermodynamic behavior of solid CO2 that is based on the Gibbs free energy. It is an empirical model that is explicit in the Gibbs free energy by using a fundamental expression for it and is fitted to experimental data (heat capacity, molar volume, expansion coefficient, and compressibility) of solid CO2. The Gibbs free energy can be written as
On the basis of these reference states, the chemical potential of the solid-forming component i in each phase is given by29 μiS (T , P) = μ0Si (T , P) + RT ln
x Si φiS(T , P , x S) P φ0Si(T , P) P
μi F (T , P , x F) = μ0Fi *(T , P) + RT ln
(4)
x iF φi F(T , P , x F) P φ0Fi *(T , P) P (5)
where * refers to the reference state. At equilibrium, the chemical potential of each component must satisfy eq 1. Thus, the equation that describes the solubility of a solid component in the fluid phase is x iF =
x Si φiS φ0Fi * φ0Si
φi
F
⎡ 1 ⎤ exp⎢ − (μ F *(T , P) − μ0Si (T , P))⎥ ⎣ RT 0i ⎦ (6)
The difference of the standard state chemical potentials of the pure substance in eq 6 is calculated as a Gibbs free energy change by applying a thermodynamic cycle29 similar to the one by Prausnitz et al.27 but also taking into account the pressure dependence. The expression that applies to the SLE proposed by Seiler et al.29 is x iL
=
φ0Li * φi
L
⎡ (v S − v L *)(P+ − P) Δh0SLi 0i − exp⎢ − 0i RT RT ⎢⎣
SL * ⎛ ΔcPSL,0i* T0SLi ⎤ T ⎞ ΔcP ,0i SL ⎥ ⎜1 − SL ⎟ + ( T 0i − T ) − ln RT R T ⎥⎦ T 0i ⎠ ⎝
g (P , T ) = h0 − Ts0 +
(7)
−T
difference of the molar, isobaric heat capacities between the hypothetical subcooled melt and the solid. In eq 7 the fugacity coefficients are calculated by an appropriate
25
φiS φ0Si
φ0Vi * φi V −
−
R
⎤ T0SV i ⎥ T ⎥⎦
T
cP(T , P0) dT + T
∫P
P
v (P , T ) d P
0
Jäger and Span used appropriate functional forms for the heat capacity, the thermal expansion coefficient, and the partial derivative of the molar volume with respect to pressure so that these quantities could be accurately fitted to experimental data. The final equation for the Gibbs free energy is
= 1.
⎧ ⎛ ϑ2 + g 2 ⎞ ⎪ g 4 ⎟ = g0 + g1Δϑ + g2Δϑ2 + g3⎨ln⎜⎜ 2 ⎟ ⎪ RT0 + 1 g ⎩ ⎝ 4 ⎠ −
⎫ ⎡ ⎛ϑ⎞ ⎛ 1 ⎞ ⎤⎪ 2ϑ ⎢ arctan⎜⎜ ⎟⎟ − arctan⎜⎜ ⎟⎟⎥⎬ g4 ⎢⎣ ⎝ g4 ⎠ ⎝ g4 ⎠⎥⎦⎪ ⎭
⎧ ⎛ ϑ2 + g 2 ⎞ ⎡ ⎛ϑ⎞ ⎪ 2ϑ ⎢ 6 ⎟ ⎜⎜ ⎟⎟ + g5⎨ln⎜⎜ − arctan 2 ⎟ ⎪ g6 ⎢⎣ ⎝ g6 ⎠ ⎩ ⎝ 1 + g6 ⎠ ⎫ ⎛ 1 ⎞ ⎤⎪ − arctan⎜⎜ ⎟⎟⎥⎬ + g7Δπ[e fα (ϑ) + K (ϑ) g8] ⎝ g6 ⎠⎥⎦⎪ ⎭ + g9 K (ϑ) [(π + g10)(n − 1)/ n − (1 + g10)(n − 1)/ n ]
SV * ⎛ Δh0SV T ⎞ ΔcP ,0i SV i ⎜1 − SV ⎟ + ( T 0i − T ) RT ⎝ RT T 0i ⎠
ln
cP(T , P0) dT
(9)
⎡ ⎡ φ V *(T , P) P ⎤ (v S )(P+ − P) ⎥ − ln⎢ V0*i exp⎢ − 0i ⎢⎣ RT ⎢⎣ φ0i (T , P+) P+ ⎥⎦
ΔcPSV,0*i
∫T
0
To use eq 7, the physical properties of the solid-forming compound are needed at the standard pressure P+. Inherent to the model described by eq 7 are the following assumptions: • The solid phase consists of only one component. • The solid volume is pressure- and temperature-independent. • The liquid volume is pressure- and temperature-independent. • The difference in the isobaric heat capacities is temperature-independent. A similar relation can be derived for SVE. In this case, the assumption of pressure-independent vapor volume is unrealistic and can lead to high errors. The corresponding term in the equation can be calculated from the Gibbs energy change using a fluid-phase EoS. The equation now is formulated as x iV =
T
0
SL where ΔhSL 0i is the enthalpy of melting at melting temperature T0i , S L v0i and v0i* are the pure solid former solid molar volume and liquid molar volume at the solid−liquid transition, and ΔcSL P,0i* is the
fluid-phase EoS. In eq 7, it is also assumed that xiS = 1 and
∫T
(10)
where T0 is a reference temperature set equal to 150 K, P0 is a reference pressure set equal to 0.101325 MPa, ϑ = T/T0, Δϑ = ϑ − 1, π = P/P0 and Δπ = π − 1. Equation 10 uses 23 adjustable parameters that are fitted to experimental data. Here, all parameters are kept to the original values proposed by Jäger and Span,
(8) C
DOI: 10.1021/acs.iecr.6b00669 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research except for two, namely, g0 and g1, which have to be retuned for every fluid EoS that is coupled with the solid-phase EoS (eq 10). The purpose of tuning these two parameters for every different fluid EoS is to ensure that the corresponding solid−fluid model is going to be thermodynamically consistent. For more details regarding the parameters of the model, we refer the reader to the original publication of Jäger and Span.25 When different thermodynamic models are used to describe the fluid- and the solid-phase properties of a system, special consideration must be given to the thermodynamic consistency. As a result, any solid−fluid model that is going to be the result of coupling the solid- and fluid-phase EoS has to be adjusted in a way so that all properties are consistent at phase equilibria. A thermodynamic triple point is the point where three phases are in equilibrium simultaneously. For a pure component, only one triple point exists. For mixtures, the degrees of freedom increase, and as a result, there are “triple lines” for binary mixtures, and so on. The pure component triple point and its properties are of great importance to the procedure of tuning the solid−fluid model to be thermodynamically consistent. This point of coexisting phases is used to “anchor” the solid- and fluid-phase models. To make the solid and fluid models thermodynamically consistent, parameters g0 and g1 are adjusted so that the Gibbs free energy of all phases (vapor, liquid, and solid) is the same at the triple point, as suggested by Jäger and Span.25 This is done by solving eqs 11 and 12 with respect to g0 and g1. g S(Ttr , Ptr) = g L (Ttr , Ptr) = g V (Ttr , Ptr)
sS(Ttr , Ptr) = s L(Ttr , Ptr) −
Δhmelt Ttr
of the enthalpy and volume difference between the phases in equilibrium: dP dT
= equil
ΔH T ΔV
(13)
To extend the solid−fluid equilibrium calculations to multicomponent mixtures, eq 3 has to be applied. In the current approach, the pure CO2 SFE pressure at a specified temperature is provided by the Jäger and Span EoS connected with the appropriate fluid-phase EoS.
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FLUID-PHASE EQUATIONS OF STATE An EoS is the mathematical relation that correlates the pressure (P), the temperature (T), and the molar volume (v) of a pure compound at a thermodynamic equilibrium state. From these three properties, one can be considered as a dependent variable and the other two can be considered independent variables for a single-phase component based on the Gibbs phase rule. This way, the EoS can be solved for one of these variables, whereas the other two are set. In practical problems, the common case is for the equation to be solved for volume (or for density) at constant pressure and temperature; then, all the other properties can be determined using specific thermodynamic relations. The most well-known EoS are the SRK and PR, which belong to the family of cubic EoS and are based on the pioneering work of van der Waals.32 Extension of the two EoS to mixtures requires suitable mixing rules. In this work, the cubic EoS calculations were extended to mixtures via the van der Waals one fluid theory (vdW1f) mixing rules, using only one temperature-independent BIP, kij. A temperature-independent kij is usually preferred so that predictions outside the temperature range of experimental data used to evaluate kij can be reliably calculated. The PC-SAFT EoS was also investigated in this work for SFE calculations. PC-SAFT is a theoretically derived model based on rigorous perturbation theory.16 The basis of this theory has been developed by Wertheim,33−36 who proposed a model for systems with a repulsive core and multiple attractive sites capable of forming chains and closed rings. In his work, Wertheim expanded the Helmholtz energy in a series of integrals of molecular distribution functions and the association potential. He showed that many of these integrals are zero, and as a result, a simplified expression for the Helmholtz energy can be obtained.14 This way, the Helmholtz energy of a fluid can be described as the sum of the Helmholtz energy of a simple reference fluid that is known accurately and a perturbation term, the development of which is the challenging part. In this framework, PC-SAFT EoS is written as a summation of terms corresponding to different types of molecular interactions. In PC-SAFT, the vdW1f theory mixing rules are used in the dispersion term. Moreover, the Lorentz−Berthelot combining rules are used for the segment energy and diameter parameters. As with cubic EoS, a temperature-independent BIP is used in the combining rule for the energy interaction parameter between unlike molecules. The mathematical expressions for all three fluid EoS can be found in the literature11,12,16 and are not given here.
(11)
(12)
The solution of these equations requires certain derivatives of the Gibbs free energy that are not presented here; we refer the reader to the original publication of Jäger and Span25 for more details. The melting enthalpy at the triple point of CO2 is set equal to 8875 J/mol as suggested by Jäger and Span who treated it as an adjustable parameter. The triple point temperature is set equal to the experimental value of 216.58 K. The triple point pressure is predicted by every model as the “intersection” of the solid−vapor and the vapor−liquid saturation curves. In this work, the Jäger and Span EoS was coupled with PC-SAFT EoS, and the resulting values for the g0 and g1 parameters are 7.447399 and −2.19139, respectively. Other fluid-phase EoS can be coupled with the Jäger and Span EoS, but in such cases, different values for g0 and g1 will be needed. The calculation of the equilibrium pressure of pure CO2 at a specified temperature at SLE or SVE conditions is performed by numerically integrating the Clausius−Clapeyron equation. The numerical integration applies a Runge−Kutta fourthorder method, and the enthalpy and volume differences are calculated at each point by numerical integration, using the solid- and the fluid-phase EoS. This ensures that eq 13 is not limited to a narrow range of conditions. The Clausius− Clapeyron equation provides the means to evaluate the change of the phase equilibrium conditions from one equilibrium point to another. The basis behind this equation is that at each phase equilibrium point the Gibbs free energy of each phase is the same; therefore, the differential along the equilibrium phase boundary is zero. On the basis of this, it can be shown that the derivative of the pressure as a function of temperature along the phase equilibrium curve is given by eq 13 and is a function
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MULTIPHASE SOLID−FLUID EQUILIBRIUM CALCULATION In this work, an algorithm for the calculation of the three phase solid−liquid−vapor equilibrium (SLVE) or solid−liquid−gas D
DOI: 10.1021/acs.iecr.6b00669 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Figure 1. Flow diagram for SLVE or SLGE calculation algorithm when solving eqs 15.
equilibrium (SLGE) was applied for binary mixtures. The basic equation that describes the three-phase SLVE at constant temperature and pressure is μiS (T , P) = μi L (T , P , x L) = μi V (T , P , y V )
μiS (T , P) = μi V (T , P , y V ) μi L (T , P , x L) = μi V (T , P , y V )
The Gibbs phase rule for a three-phase binary mixture dictates that there is only one degree of freedom for the thermodynamic state to be uniquely specified. In this work, pressure was set to a specific value. The remaining unknown variables (i.e., T, xL,yV) are determined by combining an isothermal two-fluid-phase flash calculation and a SFE. In the algorithm used, the target function to be solved is
(14)
Equation 14 requires satisfaction of two independent equations, either μiS (T , P) = μi L (T , P , x L) μi L (T , P , x L) = μi V (T , P , y V )
(16)
f = |ws,SFE − ws,flash|
(15)
(17)
where ws,SFE is the mole fraction of the solid former in the fluid phase, either liquid (if eqs 15 are used) or vapor (if eqs 16 are
or E
DOI: 10.1021/acs.iecr.6b00669 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Table 1. Experimental Binary SLGE Data from Literature Modeled in This Work pressure (MPa)
ref
CO2−N2 4.8−13.01 CO2−CH4 97.39−211.56 194.53−215.37 naphthalene−CO2 0.1−23.17 2.2−24.25 10.64−26.35 14.19−25.60 8.84−26.30 0.10−20.00
temperature (K)
ref
CO2−H2 4.3−13.7 37 phenanthrene−CO2 1.40−28.00 42
37 41 38
naphthalene−ethylene 1.1−7.61 48 0.101−17.64 49
42 43 44 45 46 47
Table 2. % AAD between Experimental SLGE Data for the Equilibrium Pressure and Model Calculations for the CO2−N2 Mixture and Corresponding kij Values
Figure 2. Pure CO2 SVE. Comparison of empirical correlation, thermodynamic integration model, and Jäger and Span EoS, coupled with PC-SAFT EoS.
% AADa EoS SRK PR PC-SAFT a
kij
correlation model
thermodynamic integration model
Jäger and Span EoS
0 −0.0172 0 −0.0026 0 0.00575
41.28 (1)b 28.58 20.92 22.48 7.41 4.06
19.76 2.80 2.85 3.53 8.09 5.58
8.07 3.13
Note:
%AAD =
100 NP
NP
∑
Picalculated − Piexperimental Piexperimental
i=1
where NP is the number of experimental data points. bOne data point was not included in the calculation of % AAD.
of the method and failure of the calculation. To remedy this behavior, it is a relatively safe choice to set z1 = z2 = 0.5 as feed composition to the flash calculation. This has the result that the isopleth VL phase envelope will be wider and that there are better chances for the flash not to encounter single-phase conditions. Moreover, to avoid overstepping, a successive underrelaxation scheme is applied to the temperature correction of the form:
Figure 3. Pure CO2 SLE. Comparison of empirical correlation, thermodynamic integration model, and Jäger and Span EoS, coupled with PC-SAFT EoS.
used) given by the iterative solution of a solid−fluid model at constant temperature and pressure, and ws,flash is the mole fraction of the solid former in the vapor or the liquid phase given by the solution of the isothermal flash. The root of eq 17 is determined by setting an initial estimate for the equilibrium temperature, and the value is corrected by applying Newton’s method. When temperature correction is applied, the SFE and the flash are solved again for the new temperature, and the fluid-phase compositions are updated accordingly. When the mole fractions of the solid former in the fluid phase from the two different calculations become equal, solution has been achieved. The derivative of eq 17 with respect to temperature is calculated by using a forward difference scheme. The resulting temperature correction from Newton’s method, especially in the initial iterations, may produce large steps that can lead to conditions where the flash calculation returns one stable phase (rather than liquid and vapor phases at equilibrium) or to conditions very far away from the solution. This results in divergence
T̂
(k + 1)
⎛ df ⎞−1 = T (k) − f (T (k)) ⎜ ⎟ ⎝ dT ⎠
T (k + 1) = qT̂
(k + 1)
+ (1 − q)T (k)
0
