A Novel Theory for Predicting Critical Constants and the α Function for

Oct 8, 2010 - The cubic equation of state (CEoS) requires critical temperature (Tc), critical pressure (Pc), and the alpha (α) function for all compo...
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Ind. Eng. Chem. Res. 2010, 49, 11801–11808

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A Novel Theory for Predicting Critical Constants and the r Function for Pure Polymers Chorng H. Twu* Tainan UniVersity of Technology, 529 Chung Cheng Road, Yung Kang, Tainan 71002, Taiwan

The cubic equation of state (CEoS) requires critical temperature (Tc), critical pressure (Pc), and the alpha (R) function for all components including polymers before it can be used for the prediction of phase behavior and thermodynamic properties for the system. For nonpolymers or conventional components, Tc, Pc, and R usually are available in databanks. However, this is not the case for polymers since their critical point is not measurable and they lack vapor pressures for the derivation of the R function. As a result, the task of finding CEoS critical constants as well as the R function for polymers becomes extremely challenging in the case of polymeric systems. A rigorous theory requiring only the liquid density data of the pure polymer is successfully developed in this work for the accurate prediction of the critical temperature, the critical pressure, and the R function for the pure polymer for use in all types of cubic equations of state. The proposed theory extends the application of cubic equations of state to polymer systems in a very simple fashion. Introduction Accurate modeling of polymer manufacturing processes requires thermodynamic models which are able to deal with phase equilibria with large size differences between polymer and solvent molecules as well as strong deviations from ideal behavior over wide ranges of operating conditions. In reality, polymer systems contain polymer and solvent components. The phase equilibrium in polymer/solvent systems is solved by equating the fugacities of each component in system at equilibrium. When we deal with polymer systems, polymer cubic equations of state have to perform accurately for polymers as well as for solvent components. Our recent efforts1–10 toward the development of a unified excess Helmholtz energy (AE) mixing rule and a new cubic equation of state TST (Twu-SimTassone) for highly nonideal and associated systems in predicting vapor-liquid equilibria (VLE) have been highly successful. The advanced cubic equation of state has also been successfully extended to the prediction of solid-liquid equilibria (SLE).11 At this time, we intend to extend the advanced cubic equation of state to the polymer system so that CEoS can be applied to describe polymer/solvent systems over a broad range of temperatures and pressures in a consistent and unified framework. There are two parameters a and b in the cubic equation of state to be determined. Generally, there are two ways to evaluate them. One is to fit the parameters to experimental data, usually the vapor pressure and liquid density. The other is to derive the parameters from the critical constraints. Since cubic equations of state do not represent PVT behavior well, particularly near the critical region, the prediction of liquid densities from a cubic equation of state produces large errors. This is one of the inherent limitations of any cubic equation of state. Consequently, the major drawback of forcing the equation-of-state parameters to fit the liquid density not only fails to satisfy the critical constraints but also sacrifices the ability of the prediction of more desired K-value property from an equation of state. Furthermore, this procedure leads to an overestimation of critical temperature and critical pressure and has a detrimental effect on the calculation of the derived thermodynamic properties such as enthalpy and entropy which depend on derivative (∂V∂T)P * To whom correspondence should be addressed. Telephone: 8866-2422603. Fax: 886-6-2433812. E-mail: [email protected].

for their values. Since the enthalpy prediction is used to calculate heat duties, the result from this procedure will affect the operation of process plant. Although the liquid densities predicted from CEoS are badly in error, the fugacity predicted from the cubic equation of state for the liquid and vapor phases is quite accurate. Accordingly, using the equality of pure polymer fugacities in the liquid and vapor is expected to give an accurate estimate for the parameters needed in the cubic equation of state. On the other hand, using the liquid density of the pure polymer directly in cubic equation of state will be expected to create some problems in the prediction of phase behavior and thermodynamic properties. The use of critical constraints in determining the cubic equation of state parameters is a very unique feature. Since most noncubic equations of state are complex, it is almost impossible to obtain analytical expressions explicitly for their parameters from the critical constants. The cubic equation of state is simple enough in its functional form to allow its parameters to be determined analytically from the critical properties. This unique feature not only satisfies the critical constraints of the CEoS but also enhances the prediction of K-values, especially at or near the critical point. In the application of cubic equation of state to low-molecular-weight components, the pure component parameters a and b are calculated from the critical constraints using the critical temperature and critical pressure for the components. Since the parameter a takes into account the intermolecular forces among system molecules, the function of the parameter a should be temperature-dependent. The temperature-dependent alpha (R) function in the parameter a is determined from the experimental vapor pressure of pure liquid in the temperature ranging from the triple point to the critical point. The liquid volume data are not used in the derivation of the R function because of the inherent limitation of the cubic equation of state in the prediction of liquid volume. For polymers, the critical parameters are not available since the critical point is not measurable for polymers. Their vapor pressures are also too low to be measured. Therefore, in spite of the important advances recently in CEoS, how to derive the CEoS parameters a and b for the pure polymer becomes one of the most challenging developments in extending cubic equations of state to polymer system.

10.1021/ie101474d  2010 American Chemical Society Published on Web 10/08/2010

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Since the saturated liquid volumes of polymer are available in literature,12,13 most researchers use a regression method to simultaneously fit CEoS parameters to the experimental liquid density data of polymer. However, as mentioned already, forcing the equation-of-state parameters to fit the liquid density not only fails to satisfy the critical constraints but also sacrifices the ability of the prediction of more desired phase equilibrium from an equation of state. Kontogeorgis et al.14 fitted the two parameters a and b to the experimental volume of the polymer at two different temperatures. Fitting two liquid density data at two different temperatures to solve for the two parameters a and b results in a parameter a that is temperature-independent. This method ignores the inherent limitations of any cubic equation of state in correlating liquid density data, overlooks the temperaturedependent R function, and neglects the critical constraints. Therefore, it will not be surprising to find that the method predicts rather poor volumetric behavior at high pressures and unrealistically high vapor pressures for the pure polymer. The latter causes problems in the erroneous prediction of finite polymer solubility in the vapor phase. Orbey and Sandler15 tried to prevent the polymer from appearing in the vapor phase by assuming a very low fixed vapor pressure to pure polymer and then fitted the two parameters a and b of the CEoS to volumetric data at the temperature range of interest by assuming that the polymer vapor pressure is equal to 10-7 MPa. For each molecular weight, a different set of parameters is obtained, which makes the method time-consuming. However, even when assigned a very low fixed vapor pressure of 10-7 MPa to the polymer, the predicted vapor pressures are rather high, especially at high temperatures. The predicted volumetric behavior is rather poor also. Kalospiros and Tassios16 made parameter b temperaturedependent by introducing a parameter called the thermal expansion coefficient. After the b parameter was calculated, they fitted the parameter a to the experimental volume of the polymer at zero pressure at different temperatures. The resulting a parameter is temperature-dependent in terms of three empirical parameters. Since the parameter b of the Kalospiros-Tassios method is temperature-dependent, it has been observed that any temperature dependence whatsoever in the covolume term of van der Waals type equations will result in anomalies in the predicted thermodynamic properties of fluids at extremely high pressures. Although they were led to good results and low vapor pressure for the pure polymer, the necessity of properties such as the glass transition temperature of the polymer, its volume at this temperature, and its thermal expansion coefficient in the glassy state makes the method impractical. Orbey et al.17 adopted a mix of Orbey-Sandler15 and Kontogeorgis14 methods. The final empirical form for the parameters of the pure polymer is a quite complicated function, a function of degree of polymerization, mass-based polymer specific volume, polydispersity index, and number-average molecular weight. Since the functions of the parameters are so complicated, it will be very difficult for engineers to apply their functions to calculate the phase behavior for the polymer systems. Orbey et al.18 eventually abandoned their complicated functions developed in their previous work and proposed another way to estimate the critical constants for polymers as a function of degree of polymerization, the critical temperature and pressure of monomer, segment molecular weight, and number average molecular weight of polymer. Unfortunately, their approach was

not successful because the correlations predict the critical temperature of polymer to be lower than that of monomer and the critical pressure of polymer to be higher than that of monomer in many cases. The prediction from their correlations is obviously contrary to reality. Orbey et al.19 ultimately abandoned all their previous approaches15,17,18 and suggested the use of a common set of critical constants Pc ) 10 bar and Tc ) 1800 K for all polymers, which are independent of the polymer MW. The consequence of selecting one arbitrary fixed set of critical constants for all polymers leads to the existence of polymers in the vapor since the predicted vapor pressures are high, especially at high temperatures. The predicted volume of the polymer is very poor for high MW polymers. Using a fixed value for Pc and Tc for all polymers implies the critical constants are not related to the attractive forces and size differences among polymer molecules, which is not realistic for design. So far, researchers have not been able to propose a rigorous theory to obtain the two parameters of CEoS for pure polymers. Louli and Tassios20 evaluated the polymer parameters a and b of a CEoS by fitting them to pure polymer PVT data and assuming that a/MW and b/MW are independent of MW. Kontogeorgis et al.21 applied the same methodology as Kontogeorgis et al.14 by estimating the two parameters based on volumetric data at low pressures. Arce and Aznar22 regressed the pure component parameters of CEoS by fitting pure component PVT data. Voutsas et al.23 obtained the attractive and covolume parameters of the CEoS for the polymers from Louli and Tassios.20 Ji24 estimated the parameters from the experimental PVT data of polymer melts. Arce et al.25 and Arce and Aznar26 obtained the pure polymer parameters a and b for the CEoS by fitting available pure liquid PVT data and assuming that parameters a/MW and b/MW are independent of MW. Arce’s methodology is the same as that of Louli and Tassios.20 Chen et al.27 obtained the pure polymer parameters of CEoS by fitting the available PVT data. Arce and Aznar28 obtained the pure component parameters for each polymer by fitting liquid pure component PVT data over a pressure range. As described above, these researchers just simply fitted the parameters to volumetric data. Forcing the pure polymer CEoS parameters a and b to fit pure polymer PVT data will result in inconsistency in predicting properties. This was demonstrated by rather poor prediction of volumetric behavior and unrealistic prediction of high vapor pressure. In extending a CEoS to polymers, it is important to keep in mind that the equation of state and their parameters must retain their consistency and robustness in predicting phase behavior and thermodynamic properties for systems including polymer and nonpolymer components. Since the only PVT data available for pure polymers are liquid volumes, we present a rigorous theory in this work to show how to derive the CEoS parameters and maintain their consistency from using only the pure polymer liquid density data for the accurate prediction of pure polymer critical temperature, critical pressure, and alpha (R) function for use in all types of cubic equations of state. Theory for Deriving Pure Polymer CEoS Parameters The TST cubic equation of state was developed by Twu et al.1 to allow better prediction of liquid densities for heavy hydrocarbons and polar components as well as accurate prediction of vapor pressure of all components in DIPPR databanks.29 The TST cubic equation of state is represented by the following equation:

Ind. Eng. Chem. Res., Vol. 49, No. 22, 2010

a RT V-b (V + 3b)(V - 0.5b)

P)

(1)

where P is the pressure, T is the absolute temperature, V is the molar volume, a is the attraction parameter, and b is the covolume of molecules. The values of a and b at the critical temperature are found by setting the first and second derivatives of pressure with respect to volume equal to zero at the critical point, resulting in

{(

ln(f ) ) -1 - ln

( RTb ) - ln(V* - 1) - (w -1 u) RT1 ba × V* + w ln( V* + u )

(12)

ac ) 0.470507R2T2c /Pc

(2)

bc ) 0.0740740RTc/Pc

(3)

ln

Zc ) 0.296296

(4)

a RT V-b (V + ub)(V + wb)

(5)

where the constants u and w, which are equation-of-statedependent, are u ) 3 and w ) -0.5 for the TST equation. At a given temperature and pressure, the liquid and vapor volumes can be calculated from eq 5. However, at low pressure, the liquid volumes are adequately assumed to be independent of the pressure since the liquid volume is not sensitive to the variation of pressure. The liquid volume VL of the pure polymer is then proposed to be solved from the cubic equation of state by neglecting the vapor pressure of the polymer and selecting the smaller root: V*L )

The fugacity of a pure polymer at zero pressure, which can be derived from eq 5, is expressed by the following:

Equation 12 can be rewritten in a reduced form by including the system pressure P to get the expression of the fugacity coefficient as

where subscript c denotes the critical point. The values of Zc from the SRK and PR equations are both larger than 0.3, whereas that from the TST equation is slightly below it and is closer to the true value. Equation 1 can be rewritten in general form as: P)

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1 a* -u-w 2 b*

) [(u + w - b*a* )

2

(

- 4 uw +

a* b*

( Pf ) ) -1 - ln b* - ln(V* - 1) - (w -1 u) b*a* ln( V*V* ++ wu ) (13)

If eq 13 is applied to the liquid phase, V* is the reduced liquid volume at zero pressure. Similarly, if eq 13 is applied to the vapor phase, V* is the reduced vapor volume at zero pressure. The approach used here eliminates the need for finding the equilibrium pressure for the system to solve for the liquid and/ or vapor volume. Applying eq 13 to the liquid phase of polymer, the liquid fugacity coefficient of a pure polymer f L/P at zero pressure is:

()

ln

a* fL 1 × ) -1 - ln b* - ln(V*L - 1) P (w - u) b* V*L + w ln V*L + u

(

fL ) fV

1/2

where V* ) V/b in eq 6 is the reduced liquid volume at zero pressure. Most of the methods proposed in the literature to derive the CEoS parameters a and b for the pure polymer use the liquid density of the pure polymer in eq 6, except that the critical constraints are not applied to the parameters a and b. Equation 6 represents the analytical solution of the cubic equation of state for liquid density at zero pressure for temperatures up to the normal boiling point. The parameters a* and b* in eq 6 are defined as: a* ) Pa/R2T2

(7)

b* ) Pb/RT

(8)

(14)

The superscript L denotes that the properties are for the liquid phase. The reduced liquid volume V*L at zero pressure is directly obtained from eq 6. For a pure polymer system in vapor-liquid equilibrium, the fugacities of liquid and vapor are equal:

)] }

(6)

)

(15)

Equation 15 can be rewritten in terms of fugacity coefficient,

( ) ( )

ln

fL fV ) ln S S P P

(16)

where PS is the saturated pressure of the pure polymer. At the saturated condition, the system pressure P is the saturated pressure PS. At zero or very low saturated pressure, the saturated vapor fugacity coefficient is unity. Equation 16 becomes

( ) ( )

ln

fL fV ) ln ) ln(1) ) 0 PS PS

(17)

Substituting eq 14 into eq 17 results in The equation-of-state parameter a is generally expressed in terms of a temperature dependence alpha (R) function as the following, a ) Rac

(9)

For components, the R(T) function is determined from their vapor pressures. Equations 7 and 8 can be rewritten in terms of critical constants. For the TST cubic equation of state, eqs 7 and 8 become a* ) R(0.470507P*/T*2)

(10)

b* ) 0.0740740P*/T*

(11)

where T* ) T/Tc and P* ) P/Pc.

-1 - ln b* - ln(V*L - 1) -

(

)

a* V*L + w 1 ln )0 (w - u) b* V*L + u (18)

Equation 18 is the rigorous theory developed in this work to use only pure polymer liquid density in the equality of fugacities in the liquid and vapor at equilibrium at zero pressure. The methodology eliminates the need for solving vapor volume at zero pressure for the system. As a result of using zero-pressure equality of fugacities in deriving eq 18, one of the most important advantages of this methodology is that the parameters derived from this methodology will avoid any prediction of finite polymer solubility in the vapor phase.

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Equation 18 will be applied to derive the pure polymer CEoS parameters a and b from the pure polymer liquid density data to extend the application of CEoS to the polymer system. As shown in eq 18, the CEoS parameters a and b are a function of only zero-pressure liquid volume. The CEoS parameters a* and b* in eq 18 are associated with eqs 2 and 3, which mean that the critical constraints are applied to a* and b*. The b parameter derived from our proposed method of using zero-pressure equality of fugacities is temperature independent. Liquid Volume Translation As mentioned earlier, the use of critical constraints in determining the cubic equation of state parameters is a very unique feature for CEoS. The critical constraints on the parameters also enhance the accuracy of the cubic equation of state in the prediction of K-values, especially at or near the critical point. The critical constraints, however, result in a constant value of critical compressibility factor Zc for all components. The value of Zc of real fluids is known from experiment to be generally smaller than that predicted from the cubic equation of state. As a result, the predicted liquid densities differ considerably from their experimental values. This is one of the inherent limitations of any cubic equation of state. Fortunately, the underprediction of liquid density from cubic equations of state can be corrected by the method of volume translation without affecting any K-value calculation. Twu and Chan30 have successfully developed a rigorous, universal, but simple volume-translation methodology for the accurate prediction of the liquid density from cubic equations of state for components without introducing any regressed parameters. This technology of volume-translated CEoS will now be applied to the polymer system for the correction of liquid volume. The desired volume correction, c, is the difference between saturated liquid volume calculated from any cubic equation of state, Vs,CEoS, and the experimental saturated liquid volume, Vs,exp: c ) Vs,CEoS - Vs,exp

(19)

Spencer and Danner31 found that the most accurate and simplest means of predicting the effect of temperature on the saturated liquid densities is by the Rackett equation. Hence, the experimental saturated liquid volume, Vs,exp, in eq 19 is proposed to be replaced by the Rackett equation, Vs,RA: c ) Vs,CEoS - Vs,RA

(20)

The Rackett equation of Spencer and Danner29 is written in the following form: Vs,RA )

( )

RTc [1+(1-T*)2/7] Z Pc RA

(21)

Here ZRA is the Rackett parameter and is a particular constant for the Rackett equation and T* ) T/Tc is the reduced temperature. The value of ZRA is unique to each component, and its value is determined from the saturated liquid density data of the component and can be found in many literature reports. The methodology developed by Twu and Chan30 offers a convenient way to select any reference temperature for the volume correction. They pointed out that the accuracy of volume translation depends primarily on the selected reference temperature, not the reference component. They found that volume

Table 1. Liquid Density of n-Eicosane at Saturated Vapor Pressure from DIPPR29 T (K)

liquid density (kmol/m3)

vapor pressure (bar)

310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 390.00 400.00 450.00 500.00

2.7466 2.7246 2.7024 2.6800 2.6575 2.6347 2.6118 2.5887 2.5653 2.5418 2.4205 2.2919

1.2563 × 10-7 3.9109 × 10-7 1.1234 × 10-6 2.9998 × 10-6 7.4958 × 10-6 1.7629 × 10-5 3.9229 × 10-5 8.2973 × 10-5 1.6750 × 10-4 3.2396 × 10-4 5.1253 × 10-3 4.0926 × 10-2

correction at T* ) 0.5 is accurate in correcting the liquid volume prediction from the cubic equation of state and will be used in this work. If T* ) 0.5 is picked, we have c)

( )

RTc * 1.82034 (Vs,CEoS | T*)0.5 - ZRA ) Pc

(22)

After finding the volume correction c from eq 22, the CEoS saturated liquid volume Vs,CEoS can be calculated from eq 19 by adding the volume correction c to the given experimental saturated liquid volume Vs,exp. This value of Vs,CEoS can then be used in either eq 6 or 18 to find the parameters a and b for the polymer. Verification of the Developed Theory The accuracy of predicting vapor pressure from any CEoS depends entirely on whether the value of the R(T) function is properly derived or not. For the accurate description of the phase behavior for the system, the R(T) function has to be regressed from the vapor pressure data of pure component. However, the polymer’s vapor pressures are usually not available, but liquid density data are available. Therefore, the R(T) function of polymer has to be derived from the available liquid density data of the pure polymer. If the developed theory in this work is truly rigorous and consistent, the derived R(T) function either from vapor pressure or liquid density data should make no difference. A long-chain hydrocarbon polymer, n-eicosane (C20H42), is used as an example here to verify the theory. The critical temperature, critical pressure, and Rackett parameter of n-eicosane from DIPPR29 are 768.00 K, 11.60 bar, and 0.2329, respectively. The liquid density data of n-eicosane at the saturated vapor pressure also from DIPPR29 are given in Table 1. There are two ways to use liquid density to get the R function for n-eicosane. One is to use liquid density in zero-pressure equation of state, eq 6, to derive the R function for n-eicosane. The other is the theory developed in this work to use liquid density in the equality of fugacities at zero pressure, eq 18, to derive the R function. Equation 6 is the conventional method proposed by most of researchers in the literature to fit CEoS parameters. Equation 18 is the rigorous theory developed in this work for the same purpose. The R(T) functions derived from eqs 6 and 18 using the liquid density data will then be compared with the accurate R(T) function derived from the vapor pressure data. The accurate values of the R(T) function for n-eicosane obtained from the regression of its vapor pressure are shown in Table 2a and 2b.

Ind. Eng. Chem. Res., Vol. 49, No. 22, 2010 Table 2. Comparison of the Derived r(T) Values of n-Eicosane a. Comparison of the Derived R(T) Values of n-Eicosane from eq 6 Using Liquid Density Data with Those from Vapor Pressure Data

T (K) 310.00 320.00 330.00 AAD%

R(T) from vapor pressure data

R(T) from eq 6 using liquid density data

2.4830 2.4260 2.3714

5.7910 4.4700 3.6921

Devi% 133.23 84.25 55.69 91.06

b. Comparison of the Derived R(T) Values of n-Eicosane from eq 18 Using Liquid Density Data with Those from Vapor Pressure Data

T (K) 310.00 320.00 330.00 AAD%

R(T) from vapor pressure data

R(T) from eq 18 Using liquid density data

2.4830 2.4260 2.3714

2.4863 2.4239 2.3680

Devi% 0.13 -0.09 -0.14 0.12

For the purpose of examining the derived R(T) function of n-eicosane from eqs 6 and 18, the critical temperature, critical pressure, and Rackett parameter given by DIPPR for n-eicosane are used directly in these two equations. As shown in Table 1, the first three data points at 310, 320, and 330 K are at very low vapor pressures. Therefore, to simulate the pure polymer system, these three points are selected in the calculation. Listed below is the procedure for using eq 6, the cubic equation of state at zero-pressure, to derive the R(T) function for n-eicosane: 1. Given an experimental liquid volume Vs,exp of n-eicosane at a specific temperature, for example, T ) 310 K. 2. Calculate the volume correction c at T* ) 0.5 from eq 22. 3. The CEoS saturated liquid volume Vs,CEoS is then computed from eq 19 by adding the volume correction c to the given experimental saturated liquid volume Vs,exp. 4. The parameter b* is computed from eq 11 using the saturated vapor pressure PS of n-eicosane at T. 5. Convert the CEoS saturated liquid volume Vs,CEoS to reduced form V*L using the following: V*L ) Vs,CEoS/b ) (PSVs,CEoS/RT)/(PSb/RT) ) ZL/b*

(23) 6. The CEoS liquid volume V*L and b* are then used in eq 6 to find the value of a*. 7. The value of the R(T) function is computed from a* based on eq 10. 8. Repeat this procedure for another temperature. The procedures of using eq 18 to derive the R(T) function for n-eicosane are the same as those described above for eq 6, except that eq 6 in step 6 is replaced by eq 18. To evaluate eq 6 and eq 18, the derived R(T) functions from both equations using liquid density data will be compared with that from vapor pressure data. The values of the R(T) function for n-eicosane derived from eq 6 and eq 18 are reported in Table 2a and Table 2b, respectively. As stated before, the accurate R(T) function must be determined from the vapor pressures of the system of interest to guarantee the accurate prediction of phase behavior. If the R(T) function derived from the developed methodology using the liquid density data can reproduce the result from the vapor pressure data, the developed methodology is validated. Examining the results as shown in Table 2a and 2b, it is not surprising to find that the R(T) function derived from the method

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of eq 6 is significantly different from that of eq 18. The R(T) value from eq 6 is about twice larger than that from eq 18. When comparing with the R(T) function derived from the vapor pressures, the average absolute deviation percent (AAD%) of the R(T) value derived from eq 6 is as high as 91.06%. The huge deviation in the R(T) values from eq 6 will result in serious errors in the prediction of the phase equilibrium and thermodynamic property for the system. On the other hand, it is quite interesting to observe the predicted R(T) values from eq 18, as shown in Table 2b, are almost exactly the same as those from the vapor pressure data. The AAD% of the R(T) value from eq 18 is only 0.12%. This result proves that the accurate R(T) function can also be derived from the liquid density data if the equality of fugacities in the liquid and vapor at zero pressure as well as the liquid volume translation are imposed. This is the first report to illustrate that the R(T) function can be derived accurately from using only the liquid density data. Note that the values of ac and bc given by eqs 2 and 3 only provide values for a and b at the critical temperature. The critical constraints of eqs 2 and 3 provide no guidance as to what the temperature dependence might be. We found the derived alpha function from liquid density data for the polymer is a strong function of temperature. We also found that the alpha function of the polymer can be described by a very simple function for the temperature range of interest as the following: R)

1 T*L

(24)

The R(T) function is a temperature-dependent function which takes into account the attractive forces between molecules. This R(T) value will decrease with increasing temperature and become unity at the critical point. The real gas behavior approaching that of an ideal gas at high temperatures requires that R goes to a finite value as the temperature becomes infinite. After the verification of the developed equality of liquid and vapor fugacity method (eq 18), the methodology is ready for deriving Tc, Pc, and L parameters from liquid density data of the pure polymer to extend the equation of state for application to polymer systems. There are three unknowns (Tc, Pc, and L parameter) in eqs 10, 11, 18, and 24. At least three experimental liquid density data of the pure polymer are needed. Listed below is the procedure to solve for Tc, Pc, and L parameters for the pure polymer from using only the liquid density data: 1. Given three experimental liquid volumes of the pure polymer Vs,exp at their temperatures T. 2. The saturated vapor pressure of the pure polymer is assumed to be a very low fixed value of 10-6 bar. 3. Assume initial values for the three unknowns Tc, Pc, and L. 4. Determine the Rackett parameter ZRA from eq 21 using the experimental liquid volumes of the pure polymer. 5. Calculate the volume correction c at T* ) 0.5 from eq 22. 6. The CEoS saturated liquid volume Vs,CEoS is then calculated from eq 19 by adding the volume correction c to the given experimental saturated liquid volume Vs,exp. 7. The value of b* is computed from eq 11 using the fixed saturated vapor pressure of 10-6 bar. 8. The value of the R(T) function is computed from eq 24. 9. The value of a* is computed from eq 10 using the fixed saturated vapor pressure of 10-6 bar. 10. Using the values of a* and b* in eq 18 to solve for the CEoS liquid volume V*L ) Vs,CEoS/b.

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Table 3. Comparison of the Derived Tc, Pc, and r(T) Values of n-Eicosane

* 2 aseg ) Rac,seg ) R(Tseg )(0.470507R2Tc,seg /Pc,seg)

(28)

bseg ) bc,seg ) 0.0740740RTc,seg/Pc,seg

(29)

a. Critical Temperature (Tc), Critical Pressure (Pc), and L Parameter Derived from Liquid Density Data of n-Eicosane Using eq 18 Tc(K)

Pc(bar)

DIPPR

this work

DIPPR

this work

L

768.00

769.94

11.60

11.41

0.987506

where Tc,seg and Pc,seg are the critical temperature and pressure of the segment, respectively, and Tseg* ) T/Tc,seg. Combining eqs 26-29 gives

b. Comparison of the R(T) Values Derived from Liquid Density Data of n-Eicosane with Those from Vapor Pressure Data

T (K) 310.00 320.00 330.00 AAD%

R(T) from vapor pressure data

R(T) from liquid density data Using eq 24

2.4830 2.4260 2.3714

2.4495 2.3739 2.3028

Devi% -1.35 -2.15 -2.89 2.13

11. Convert the calculated reduced form V*L from step 10 to a dimensional volume Vs,CEoS by using eq 23. The value of Vs,CEoS determined in this step is the calculated liquid volume. 12. Repeat this procedure for the other two liquid density data. 13. Calculation is iterated until the calculated liquid volumes Vs,CEoS from step 11 match the given data from step 6 for all three liquid volumes. The derived critical temperature (Tc), critical pressure (Pc), and the L parameter for n-eicosane from liquid density data are reported in Table 3a. The calculated R(T) values from the derived Tc and L using eq 24 are shown in Table 3b. These predicted parameters will be compared with the critical temperature and critical pressure of n-eicosane given in DIPPR. As shown in Table 3a, our method predicts accurately the critical temperature and critical pressure for n-eicosane using only the liquid density data. The values of R(T) from two methods (one uses liquid density data and the other uses vapor pressures) are compared in Table 3b. There is about 2% difference between them, but this slight difference is immaterial because, based on the theory developed in this work, this new set of three parameters (Tc, Pc, and L) will predict very low vapor pressures for the system. These accurate results verify the theory developed in this work. The number of segment r in the polymer is calculated from the number average molecular weight (MW) of the polymer, Mn, divided by the MW of the base molecule, Mseg, r)

Mn Mseg

(25)

When a polymer molecule consists of r segments, the molecular parameters a and b of the polymer are related to the segment parameters by32 a ) r2aseg

(26)

b ) rbseg

(27)

where a and b represent pure polymer parameters on a molecular basis, aseg and bseg are the pure polymer parameters on a segment basis, and r is the number of segments per molecule. Similar to the molecular parameters a and b, the segment parameters aseg and bseg are related to the segment critical temperature Tc,seg and the segment critical pressure Pc,seg by:

2 a ) r2R(T*eg)(0.470507R2Tc,seg /Pc,seg)

(30)

b ) r(0.0740740RTc,seg/Pc,seg)

(31)

The number of segments per molecule r is calculated from eq 25. After giving the number of segments r, there are three unknowns (Tc,seg, Pc,seg, and Lseg parameter) in eqs 18, 24, 30, and 31. The procedures to find these three unknowns for segment are the same as those described in the previous case for the polymer molecule. We continue to use n-eicosane (C20H42) as an example, but the number of segments r is set to be 2 in this case. The derived segment critical temperature (Tc,seg), segment critical pressure (Pc,seg), and segment Lseg parameter from liquid density data of n-eicosane are reported in Table 4. The critical temperature and critical pressure of n-decane (C10H22) are also shown in Table 4 for reference. Before discussing the result given in Table 4, note that two molecules of n-decane will not simply form one molecule of n-eicosane because the number of hydrogen atom in n-eicosane (C20H42) is simply not twice of that of n-decane (C10H22). One would not expect the values of critical constants of n-eicosane to be divided by a factor of 2 to become the critical constants of the segment either. However, the segment critical temperature is expected to be lower than that of the polymer molecule and the segment critical pressure to be higher than that of the polymer molecule. The result shown in Table 4 indicates that this indeed is the correct trend. Note that the values of critical temperature and critical pressure for the segment are quite close to those of n-decane. Again, using these segment parameters or the molecular parameters will give the same result and avoid any prediction of finite polymer solubility in the vapor phase. Conclusions The cubic equation of state needs critical constants as input parameters before it can be applied to predict thermophysical properties. However, a polymer lacks critical point and vapor pressure data, and only liquid density data are available. We demonstrate that forcing the cubic equation-of-state parameters to fit the liquid density results in a huge deviation in the R(T) function, which will affect seriously the vapor pressure and phase behavior predictions for the polymer system. A rigorous and thermodynamically consistent methodology has been developed for the accurate prediction of critical temperature, critical pressure, and alpha (R) function for the pure polymer using only the liquid density data of the pure polymer. Using liquid density in the equality of pure polymer fugacities in the liquid and vapor at zero pressure allows the prediction of very accurate CEoS parameters a Table 4. Segment Critical Temperature (Tc,seg), Segment Critical Pressure (Pc,seg), and Segment Lseg Parameter Derived from Liquid Density Data of n-Eicosane Tc,seg (K)

Pc,seg (bar)

no. of segment

segment

n-decane

segment

n-decane

Lseg

2

623.18

617.70

18.44

21.10

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and b for a polymer. The parameters predicted from this method are ready to be used in cubic equations of state for the application to polymer systems in a consistent way. The methodology eliminates the need for solving vapor volume and equilibrium pressure for the system by equating the fugacities of a polymer in the liquid and vapor phases at equilibrium at zero pressure. Since the concept of the zeropressure equality of fugacities is developed, the parameters derived from this methodology will avoid any prediction of finite polymer solubility in the vapor phase. The developed methodology in this work has successfully extended the application of cubic equations of state to polymer systems in a very simple fashion. Acknowledgment I am thankful for the financial support of the National Science Council of Taiwan for the Project No. NSC97-2221E-165-001. Nomenclature. a, b ) cubic equation of state parameters a*, b* ) reduced parameters of a and b c ) volume correction f ) fugacity M ) molecular weight (MW) P ) pressure R ) gas constant T ) temperature u, w ) cubic equation of state constants V ) molar volume V* ) reduced liquid volume at saturated pressure Z ) compressibility factor ZRA ) Rackett compressibility factor AbbreViations CEoS ) cubic equation of state Devi% ) [(calculated value - experimental value)/experimental value] × 100% PR ) refers to the Peng-Robinson equation of state SRK ) refers to the Soave-Redlich-Kwong equation of state TST ) refers to the Twu-Sim-Tassone equation Greek letters R ) alpha function in the CEoS parameter a Subscripts c ) critical property r ) number of segment s ) saturated property seg ) segment Superscripts L ) liquid phase or parameter in eq 24 V ) vapor phase s ) saturated property * ) reduced property

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ReceiVed for reView July 10, 2010 ReVised manuscript receiVed September 15, 2010 Accepted September 16, 2010 IE101474D