A chemical production process involves a reaction step followed by the step of separating or refining products from by-products. This process often uses distillation and also commonly uses absorption, extraction, or crystallization. Meanwhile, distillation columns are used in component separation for petroleum. Also, component separation is performed in natural gas refining. The component separation principle in such a separation/refining step is based on phase equilibrium relationship. Vapor-liquid equilibrium (VLE) for distillation or absorption, liquid-liquid equilibrium (LLE) for extraction, and solid-liquid equilibrium (SLE) for crystallization serve as central factors determining separation limits or energies required for separation. Thus, rational apparatus or operational design or process step selection cannot be performed without the determination of precise phase equilibrium relationship.
Although the phase equilibrium relationship is exceedingly important for chemical, petroleum, or natural gas refining as described above, a problem that is faced by using the phase equilibrium relationship in such a chemical production or petroleum or natural gas refining step is the imprecise prediction of the phase equilibrium relationship. This is because non-ideality in the liquid phase (i.e., the property in which the mixture is no longer regarded as a single component due to the value of activity coefficient deviated from 1) cannot be predicted. Fundamental problems that are encountered in trying to solve this problem are the unpredictable intermolecular interaction strength and molecular orientation of non-ideal solutions. This blocks theoretical progress. There also remains a practical issue: unknown phase equilibrium relationship hinders a production process from being determined or an apparatus from being designed. Taking distillation column design as an example, the number of contact stages or the height of a packed column cannot be determined. Under the circumstances, the phase equilibrium relationship is actually measured for a system that is subject to separation, and correlated with operating parameters, and the resulting correlation is used in design calculation. Although a lot of vapor-liquid equilibrium data (see e.g., Non-patent Document 1), liquid-liquid equilibrium data (see e.g., Non-patent Document 2), or solid-liquid equilibrium data has been reported for use in the design of the separator, every such data is actually measured values. The actually measured values, however, are accompanied by errors (statistical errors responsible for variations and systematic errors responsible for imprecision). In fact, since precise phase equilibrium relationship is unknown, apparatus design is performed in expectation of large safety factors in order to secure product properties (e.g., purity) or productivity (i.e., yield). Such large safety factors are significantly economically inefficient because chemical production uses large-scale apparatus. Hereinafter, thermodynamic relationship satisfied by true values free from such errors will be described first.
When phases I and II reach equilibrium upon contact, the equilibrium relationship of components constituting these phases with component i is given by the following equation (see e.g., Non-patent Document 3):[Formula 1](fi)I=(fi)II  (1)
(Method for Correlation of Low-Pressure Binary Vapor-Liquid Equilibrium Data)
First, assuming that liquid and vapor phases consisting of components 1 and 2 are in equilibrium at a low pressure, the equation (1) is specifically represented as follows for this low-pressure binary vapor-liquid equilibrium (VLE) (see e.g., Non-patent Document 3):[Formula 2]Py1=γ1x1p1s  (2)[Formula 3]Py2=γ2x2p2s  (3)wherein P represents a system pressure; y1 and y2 represent the mole fractions of components 1 and 2, respectively, in the vapor phase; x1 and x2 represent the mole fractions of components 1 and 2, respectively, in the liquid phase; γ1 and γ2 represent the activity coefficients of components 1 and 2, respectively, in the liquid phase; p1s and p2s represent the vapor pressures of components 1 and 2, respectively, at a system temperature T. The activity coefficients are introduced for representing non-ideality in the liquid phase. γ1 (activity coefficient of component i)=1 holds for ideal solutions. Many activity coefficient equations have been proposed for indicating non-ideality in the liquid phase. For example, the equation (4) or (5) (the Margules equation) is often used as the activity coefficient equation. Other known activity coefficient equations include van Laar, UNIQUAC, NRTL, Wilson, and Redlich-Kister equations.[Formula 4]ln γ1=x22[A+2(B−A)x1]  (4)[Formula 5]ln γ2=x12[B+2(A−B)x2]  (5)wherein x1 represents the mole fraction of component 1 in the liquid phase, and x2 represents the mole fraction of component 2 in the liquid phase. Binary parameters A and B are given by the following equations (6) and (7):[Formula 6]A=ln γ1∞  (6)[Formula 7]B=ln γ2∞  (7)wherein γ1∞ and γ2∞ represent the infinite dilution activity coefficients of components 1 and 2, respectively, in the liquid phase. Specifically, the binary parameters A and B are expressed in terms of the infinite dilution activity coefficients γ1∞ and γ2∞ by considering x1→0 and x2→0 as limits in the equations (4) and (5).
Specifically, P-x data and P-y data for the methanol (1)-water (2) system at a temperature of 323.15 K are shown in FIG. 1 taking the correlation of VLE data as an example.
In FIG. 1, the open circle (∘) denotes actually measured values of P-x1 relationships (see Part 1, p. 56 of Non-patent Document 1), and the filled circle (●) denotes actually measured values of P-y1 relationships (see Part 1, p. 56 of Non-patent Document 1).
The system pressure P is given by the following equation (8) from the sum of the equations (2) and (3) by considering y1+y2=1:[Formula 8]P=+γ1x1p1s+γ2x2p2s  (8)
The binary parameters A and B were determined using the method of least squares to most represent actually measured P-x relationships. From the values of A and B (A=0.6506 and B=0.5204) thus determined, P-x relationships were calculated according to the Margules equation and indicated by the solid line (-) in FIG. 1.
Also, P-y1 relationships were calculated according to the following equation (9) using the binary parameters A and B representing P-x relationships and indicated by the dotted line ( . . . ) in FIG. 1.[Formula 9]y1=γ1x1p1s/P  (9)
Likewise, binary parameters A and B correlating constant-pressure binary vapor-liquid equilibrium data can also be determined. In addition to the Margules equation, other activity coefficient equations, such as van Laar, Wohl, UNIQUAC, NRTL, and Wilson equations, may be used in the correlation (see Non-patent Document 3). Non-patent Document 1 discloses examples of correlation of binary vapor-liquid equilibrium data for more than 12500 systems. Also, this literature provides binary parameters most representing constant-temperature P-x data and constant-pressure T-x data using each activity coefficient equation. Further, this literature provides binary parameters most representing constant-temperature P-x-y data and constant-pressure T-x-y data using vapor-phase mole fraction data. These binary parameters can be converted to A and B that provide infinite dilution activity coefficients. Even if binary parameters representing P-x data are determined using the activity coefficient equation, correlation results do not always produce true values due to measurement errors (statistical errors and systematic errors) contained in the data. Depending on VLE data, there may exist an activity coefficient equation that is capable of correlation with particularly high accuracy. The precision of VLE data, however, is not directly related to correlation errors unless experimental errors are removed. A method for determining binary parameters by excluding measurement errors from VLE data has not yet been found, though the vapor-liquid equilibrium relationship can be determined from the given two binary parameters, A and B.
(Method for Correlation of Binary Liquid-Liquid Equilibrium Data)
When liquid phases I and II consisting of a binary system are in liquid-liquid equilibrium, the equation (1) is specifically represented as follows (see Non-patent Document 3):
                    [                  Formula          ⁢                                          ⁢          10                ]                                                                                  (                                          γ                i                            ⁢                              x                i                                      )                    I                =                              (                                          γ                i                            ⁢                              x                i                                      )                    II                                    (        10        )                                [                  Formula          ⁢                                          ⁢          11                ]                                                                                                ∑                              i                =                1                            2                        ⁢                                          (                                  x                  i                                )                            I                                =          1                ,                                            ∑                              i                =                1                            2                        ⁢                                          (                                  x                  i                                )                            II                                =          1                                    (        11        )            wherein γi represents the activity coefficient of component i in each liquid phase I or II, and xi represents the mole fraction of component i in each liquid phase.
For data correlation, parameters A and B can be determined according to the equations (4), (5), and (10) from binary liquid-liquid equilibrium data (mutual solubility data) at one temperature, for example, in the case of using the Margules equation. Unfortunately, a method for accurately correlating relationships between the parameter A and temperature over a wide temperature range has not yet been found. This holds true for relationships between the parameter B and temperature.
Non-patent Document 2 discloses a lot of binary mutual solubility data. Values of binary parameters were determined according to the UNIQUAC activity coefficient equation for each mutual solubility data. As specific examples, binary liquid-liquid equilibrium data for the 1-butanol (1)-water (2) system is shown in FIG. 2(a), and binary liquid-liquid equilibrium data for the 2-butanone (1)-water (2) system is shown in FIG. 2(b). A method for favorably predicting a method for correlation of the temperature dependence of such mutual solubility has not yet been found. In FIGS. 2(a) and 2(b), the open circle (∘) denotes actually measured values of (x1)2 (see Non-patent Document 2), and the filled circle (●) denotes actually measured values of (x1)1 (see Non-patent Document 2). In this context, (x1)1 represents the mole fraction of component 1 in liquid phase I, and (x1)2 represents the mole fraction of component 1 in liquid phase II. The ordinate denotes a temperature (T−273.15 K).
(Method for Correlation of High-Pressure Binary Vapor-Liquid Equilibrium Data)
In one approach, the equation (1) is specifically represented as follows for high-pressure vapor-liquid equilibrium data (see Non-patent Document 3):[Formula 12]φiVyi=φiLxi  (12)wherein φiV represents the fugacity coefficient of component i in the vapor-phase mixture; φiL represents the fugacity coefficient of component i in the liquid-phase mixture; xi represents the mole fraction of component i in the liquid phase; and yi represents the mole fraction of component i in the vapor phase. The fugacity coefficients are used in the high-pressure system to represent not only non-ideality in the vapor phase, but also non-ideality in the liquid phase. The fugacity coefficients can be expressed using the equation of state and mixing rules for pure components, albeit in a complicated manner. The mixing rules include system-dependent interaction parameters. The method used involves adjusting values of the interaction parameters to simultaneously correlate high-pressure P-x data and P-y data. The interaction parameters thus obtained by correlation are further used in the prediction of phase equilibrium relationship. Such a method for correlation of high-pressure phase equilibrium has the disadvantages that: i) the calculation of fugacity coefficients is complicated; ii) interaction parameters do not exhibit correlation as a function of operating parameters (temperature, pressure, etc.) or factors characteristic of the system (polarizability, etc.); and iii) some regions including the vicinity of critical points are observed with significantly low correlation accuracy.
Secondly, the high-pressure vapor-liquid equilibrium is represented by expressing non-ideality in the liquid phase in terms of activity coefficients and given as follows (see Non-patent Document 3):
                    [                  Formula          ⁢                                          ⁢          13                ]                                                                                  ϕ            iV                    ⁢                      Py            i                          =                              γ            i                          (                              P                ⁢                                                                  ⁢                a                            )                                ⁢                      x            i                    ⁢                      p            is                    ⁢          exp          ⁢                                                    v                iL                            ⁡                              (                                  P                  -                                      P                    ⁢                                                                                  ⁢                    a                                                  )                                      RT                    ⁢          exp          ⁢                                                    v                i                o                            ⁡                              (                                  P                  -                                      p                    is                                                  )                                      RT                                              (        13        )            wherein γi(Pa) represents the activity coefficient of component i at a system temperature T and a pressure Pa in the liquid phase; viL represents the partial molar volume of component i in the liquid phase; vio represents the molar volume of pure liquid i; and R represents a gas constant. In this context, φiV represents the fugacity coefficient of component i in the vapor-phase mixture; xi represents the mole fraction of component i in the liquid phase; yi represents the mole fraction of component i in the vapor phase; P represents a system pressure; and Pis represents the vapor pressure of component i at a system temperature T.
The equation (13) expresses non-ideality in the vapor phase in terms of fugacity coefficients and thus incorporates three disadvantages described above in their entirety. Although non-ideality in the liquid phase is also expressed in terms of activity coefficients, this is not practical due to insufficient physical data (e.g., partial molar volume) necessary for determining exponential terms on the right side representing pressure dependence. The correlation of high-pressure vapor-liquid equilibrium data is much more complicated than that of low-pressure data. For apparatus design, there is no choice but to adopt a method using data obtained in a practical operating range.
The values of A and B must be known for the prediction of phase equilibrium using activity coefficients shown above. Since excess partial molar free energies are obtained by multiplying logarithmic values of the infinite dilution activity coefficients represented by the equations (6) and (7) by RT, ln γ1∞ and ln γ2∞ are naturally predicted to be monotone functions of 1/T, as in many thermodynamic rules. Non-patent Document 1 has reported 60 vapor-liquid equilibrium data sets for the methanol (1)-water (2) system and also reported values of Margules binary parameters A and B representing these VLE data sets. Thus, a plot of A=ln γ1∞ and B=ln γ2∞ for the methanol (1)-water (2) system against 1/T is shown in FIG. 3. For low-pressure data, ln γ1∞ and ln γ2∞ must be single functions (single-valued functions) of temperature. As is evident from FIG. 3, the data sets are too variable to fix a representative line. This holds true for ln γ1∞ and ln γ2∞ determined using other activity coefficient equations, such as the UNIQUAC equation. Also, plots of ln γ1∞ and ln γ2∞ for other binary systems against IT produce similar results. This fact that ln γ1∞ and ln γ2∞ representing non-ideality in the liquid phase cannot be represented as single functions of temperature is the greatest reason for hampering the highly accurate correlation of vapor-liquid equilibrium data and the prediction of equilibrium relationship. In spite of more than 100 years since the introduction of the Margules equation, the accurate method for correlation of binary parameters remains undeveloped.
By contrast, a group contribution method, such as the UNIFAC or ASOG method, involves dividing molecules in the phase equilibrium-forming system into atomic groups such as CH2 and OH, determining the rates of contribution of the atomic groups to binary parameters from a large number of binary phase equilibrium data, and predicting phase equilibrium using the rates. This method, albeit simple, exhibits significantly low prediction accuracy and is thus used only for grasping the outline of phase equilibrium. Still, measurement data must be used for detailed design.
Recently, the present inventor has found an accurate method for correlation of phase equilibrium data and filed patent applications (Japanese Patent Application Nos. 2010-58632 (Patent Document 1) and 2010-112357 (Patent Document 2)). This correlation utilizes the property in which phase equilibrium data highly accurately converges to the thermodynamic consistency line represented by the equation (14):
                    [                  Formula          ⁢                                          ⁢          14                ]                                                            β        =                              F                                                        B                -                A                                                            =                      aP            b                                              (        14        )            wherein A and B represent binary parameters; P represents a system pressure; a and b represent constants specific for the binary system; β represents a polarity exclusion factor; and F represents deviation from A=B in the one-parameter Margules equation. F is defined according to the following equation:
                    [                  Formula          ⁢                                          ⁢          15                ]                                                            F        =                              100            n                    ⁢                                    ∑                              k                =                1                            n                        ⁢                                                                                              y                                                                  1                        ⁢                        k                                            ,                      smooth                                                        -                                      y                                                                  1                        ⁢                        k                                            ,                                              M                        ⁢                                                                                                  ⁢                        1                                                                                                              y                                                            1                      ⁢                      k                                        ,                                          M                      ⁢                                                                                          ⁢                      1                                                                                                                                            (        15        )            wherein y1k,smooth represents a value of the mole fraction y1 of component 1 in the vapor phase with respect to x1k (value of x1 in the k-th part of the liquid-phase mole fraction x1 of component 1 divided into n equal parts (e.g., n=40) between 0 and 1). The value of y1k,smooth is determined using the equations (2) to (5) and (9) from binary parameters A and B determined to represent P-x relationships. y1k,M1 represents a value of y1 calculated using one parameter obtained by the correlation of P-x relationships using the one-parameter Margules equation that satisfies A=B (Patent Document 1). Alternatively, the value of y1k,M1 may be calculated more simply according to the one-parameter Margules equations shown below and the equations (2) and (3) using E determined according to the following equation (16) using Margules equation incorporating one parameter E (=A=B) in activity coefficients in the equation (8):[Formula 16]P=ex22Ex1p1s+ex12Ex2p2s  (16)(Patent Document 2 and Non-patent Document 4). E is defined according to the following one-parameter Margules equations wherein x1 and x2 represent the mole fractions of components 1 and 2, respectively, in the liquid phase; P represents a system pressure; and p1s and p2s represent the vapor pressures of components 1 and 2, respectively, at a system temperature T:[Formula 17]ln γ1=x22E  (17)[Formula 18]ln γ2=x12E  (18)wherein γ1 and γ2 represent the activity coefficients of components 1 and 2, respectively.
The equation (14) indicates relationships for constant-pressure data. For constant-temperature data, an average vapor pressure ps,ave obtained according to the following equation can be used instead of P:
                    [                  Formula          ⁢                                          ⁢          19                ]                                                                      p                      s            ,            ave                          =                                            p                              1                ⁢                s                                      +                          p                              2                ⁢                s                                              2                                    (        19        )            
The thermodynamic consistency line of constant-temperature data agrees with that of constant-pressure data (Patent Document 2 and Non-patent Document 4). This property relating to VLE data, i.e., the property in which constant-pressure data and constant-temperature data exhibit the same thermodynamic consistency lines, has been found for the first time by the invention described in Japanese Patent Application No. 2010-112357 (Patent Document 2). The correlation based on the equation (14) is highly accurate. Taking the 60 data sets for the methanol (1)-water (2) binary system described above as an example, correlation errors derived from the equation (14) are merely 0.25% for 18 constant-temperature data sets and 0.55% for 42 constant-pressure data sets, demonstrating the exceedingly high convergence of β vs. pressure relationships compared with the low convergence of A and B shown in FIG. 3. Many examples of thermodynamic consistency lines are shown in Non-patent Document 4.
The values of two binary parameters, A and B, must be known as functions of temperature or pressure in order to predict binary vapor-liquid equilibrium relationship using activity coefficients. In the prediction of phase equilibrium relationship using the thermodynamic consistency line obtained in the invention described in the patent application mentioned above, the equilibrium relationship cannot be determined directly because only one relationship of A and B (only the thermodynamic consistency line) is known. Thus, it is required to find highly converged correlations of A and B independent of the equation (14). For this purpose, relationships between the following deviation D from the Gibbs-Duhem equation and experimental errors can be used (Patent Document 1):[Formula 20]D=2(B−A)Δx12  (20)wherein D represents deviation from the Gibbs-Duhem equation, and Δx1 represents an experimental error appearing in the mole fraction of component 1 in the liquid phase. The equation (20) is an approximation of the Gibbs-Duhem (GD) equation obtained by applying the Margules equation to the activity coefficient incorporated in the GD equation under isothermal and isobaric conditions and applying the central difference scheme to the derivative of ln γi with respect to change in x1.
According to the Gibbs-Duhem equation, D=0 in the following equation:
                    [                  Formula          ⁢                                          ⁢          21                ]                                                            D        =                                            x              1                        ⁢                                          d                ⁢                                                                  ⁢                ln                ⁢                                                                  ⁢                                  γ                  1                                                            d                ⁢                                                                  ⁢                                  x                  1                                                              +                                    (                              1                -                                  x                  1                                            )                        ⁢                                          d                ⁢                                                                  ⁢                ln                ⁢                                                                  ⁢                                  γ                  2                                                            d                ⁢                                                                  ⁢                                  x                  1                                                                                        (        21        )            wherein x1 represents the mole fraction of component 1, and γ1 and γ2 represent the activity coefficients of components 1 and 2, respectively.
The values of activity coefficients representing binary vapor-liquid equilibrium data must satisfy the Gibbs-Duhem equation, i.e., D=0, under constant-temperature and constant-pressure conditions. Since the values of B and A representing VLE data are different from each other, the equation (20) shows that the GD equation is not satisfied unless experimental errors are removed. This means that precise phase equilibrium relationship does not hold in Δx1≠0. Since a method for eliminating experimental errors has not been found previously, a method for prediction of phase equilibrium has not been established. The method for eliminating experimental errors is to obtain converged correlations using a lot of phase equilibrium data which are obtained in different measurement apparatuses, measurement methods, or measurers. The equation (14) provides linear relationships that hold for data differing in apparatuses, methods, or measurers. Thus, consistent relationships that satisfy the GD equation without errors are shown on this line.
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