Using cubic equations of state for a single-component fluid, we compute pseudocritical loci where the isobaric heat capacity is a relative maximum at constant pressure, or at constant temperature. These two loci, called the Widom line and the characteristic isobaric inflection curve (CIIC), are quite different from each other, as we show using the van der Waals equation, based on which the two loci admit a closed-form representation in the (P, T) plane. Similarly, the Redlichâ  Kwong (RK) equation leads to a closed-form representation for the CIIC in the (T,v) plane. With the Soaveâ  Redlichâ  Kwong (SRK) and the Pengâ  Robinson (PR) equations we find almost coincident predictions for the above-mentioned pseudocritical loci; furthermore, comparing our results with a correlation obtained by regression of experimental data for CO2and water shows that the increased complexity of the SRK and PR equations (as compared to RK) allows improved agreement with the experimental data

Ambrosini, W.

Lamorgese, A.

Mauri, R.

English

Archivio della Ricerca - Università di Pisa

Widom Line Prediction by theSoave-Redlich-Kwong and Peng-RobinsonEquations of StateA. Lamorgese∗, W. Ambrosini, R. MauriDepartment of Civil and Industrial Engineering,Laboratory of Multiphase Reactive Flows,Università di Pisa, I-56126 Pisa, ItalyAbstractFor a pure component, we compute the characteristic isobaric inflectioncurve (a.k.a. the Widom line) for the Soave-Redlich-Kwong (SRK) and Peng-Robinson (PR) equations of state. Incidentally, we also show that using theRedlich-Kwong (RK) equation of state leads to a closed-form representation ofthe Widom line in the (T, v) plane. The SRK and the PR equations of statelead to almost coincident predictions for the Widom line; furthermore, compar-ing our numerical results with a correlation in reduced coordinates obtained byregression of experimental cp data for CO2 and water shows that the increasedcomplexity of the SRK and PR equations (as compared to RK) allows improvedagreement with the experimental data.Keywords: Characteristic curves; Cubic equations of state; Thermodynamics.1. IntroductionAs is well known, the physical properties of a pure substance that enters itssupercritical region by heating or cooling are strong functions of temperatureand pressure, which can greatly affect heat transfer rates in practical applica-tions. For example, in a supercritical constant pressure process, all physicalproperties of the substance show abrupt (but continuous) variations (as its be-havior changes from liquid-like to vapor-like) within a narrow temperature rangeacross the pseudocritical temperature, at which the isobaric heat capacity of the∗Email: andrea.lamorgese@unipi.it.Preprint submitted to Elsevier March 30, 20170.8 0.9 1 1.1 1.2 1.3 1.4 1.5051015202530354045Trc p/RPr = 1.1Pr = 1.2Pr = 1.5Pr = 1.8Figure 1: Isobaric heat capacity for a van der Waals fluid as a function of reduced temper-ature at different supercritical pressures (Pr = P/Pc shown in the legend being the reducedpressure).substance has a relative maximum, while its density, thermal conductivity, anddynamic viscosity decrease drastically. In particular, as the pressure increasesfurther in the supercritical range, the relative maxima in the isobaric heat capac-ity decrease and move to higher temperatures. Collectively, the temperaturesand pressures corresponding to the relative maxima in the cp are called thepseudocritical (or Widom) line. However, at lower pressures (in particular atthe critical pressure), abrupt (discontinuous) physical property variations char-acterize the substance in the neighborhood of its critical temperature, since anyproperty which can be represented as a second-order derivative of a thermo-dynamic potential (such as, e.g., the isobaric heat capacity and the isothermaland isentropic compressibilities) will blow up at the critical point in a first-ordertransition.Note that even the simplest (i.e., the van der Waals) EoS model that al-lows for a liquid-vapor transition can reproduce qualitative agreement with theaforementioned pseudocritical behavior of a pure substance. In fact, starting2from the van der Waals EoS in reduced coordinates,(Pr +3v2r)(vr −13)=83Tr, (1)where Pr ≡ P/Pc, Tr ≡ T/Tc, vr ≡ v/vc (where the c subscript denotes acritical value), and noting that the specific isochoric heat capacity for a classicalvan der Waals fluid is a constant, cv = 3R/2 (with R denoting the universalgas constant), it is easily seen that the isobaric heat capacity takes the form(Johnston, 2014):cpR−cvR= ZcTr(∂vr∂Tr)Pr(∂Pr∂Tr)vr=4Trv3r4Trv3r − (3vr − 1)2. (2)In Fig. 1, plots of this quantity as a function of (reduced) temperature at dif-ferent supercritical pressures confirm that the relative maxima in the cp moveto higher temperatures as the pressure is increased further in the supercriticalrange. In fact, a straightforward calculation shows that these relative maximaare located for a van der Waals fluid atPr = 4Tr − 3, (and Tr ≥ 1) (3)in the (Pr , Tr) plane, or, equivalently, at vr = 1 (and Tr ≥ 1) in the (Tr, vr)plane.To date, although the Widom line has been studied in simple model systems,such as, e.g., a van der Waals or a Lennard-Jones fluid (Brazhkin and Ryzhov,2011; May and Mausbach, 2012), a systematic discussion of the Widom line aspredicted by the classical cubic EoS models, i.e., the van der Waals, Redlich-Kwong, Soave-Redlich-Kwong, and Peng-Robinson EoS has not been presentedin the literature heretofore and is the principal result reported herein.2. Parametric Construction of Characteristic CurvesSince our main objective in this paper is to discuss predictions of the Widomline based on different cubic EoS models, we first show that an assumption ofRedlich-Kwong fluid properties leads to a closed form representation of theWidom line in the (T, v) plane. We begin by noting that the pseudocritical linecan be defined thermodynamically by the condition:(∂cp∂P)T= −T(∂2v∂T 2)P= 0, (4)3where we have considered that cp =(∂h∂T)Pin addition to(∂h∂P)T= v−T(∂v∂T)P.Below, we will determine the Widom line by looking at the zero level set of(∂2T∂v2)P. Before switching to reduced coordinates Pr ≡ P/Pc, Tr ≡ T/Tc,vr ≡ v/vc (where the c subscript denotes a critical value), we note that thefollowing two-parameter EoS (Schmidt and Wenzel, 1980)P =RTv − b− aα(Tr)(v + δ1b)(v + δ2b), (5)with a and b denoting temperature-independent constants (while δ1,2 are pa-rameters), comprises all four EoS models (i.e., van der Waals, Redlich-Kwong,Soave-Redlich-Kwong, and Peng-Robinson) discussed herein. Now, in reducedcoordinates, Eq. (5) becomesPr =Trx− Ωb− Ωaα(Tr)(x+ δ1Ωb)(x+ δ2Ωb), (6)where x ≡ Zcvr (Zc being the critical compressibility factor). With the SRKand PR EoS, Ωa and Ωb are defined based on the following relations (Smithet al., 2005):a = Ωa(RTc)2Pc, and b = ΩbRTcPc. (7)Note that with the RK EoS the definition for Ωa is with Tc raised to the 5/2power (instead of T 2c ) in the first of these relations. For example, with the vander Waals EoS we have: δ1 = δ2 = 0, Zc = 3/8, Ωa = 27/64, and Ωb = 1/8,and α(Tr) ≡ 1. On the other hand, RK fluid properties are obtained by lettingδ1 = 0, δ2 = 1, Zc = 1/3, Ωa = 1/[9(21/3 − 1)], Ωb = 13 (21/3 − 1), andα(Tr) ≡ T−1/2r . These last coefficients are still valid with the SRK EoS exceptthat the α-function introduced by Soave (1972) incorporates a dependence onthe Pitzer acentric factor, i.e.,α(Tr) =[1 +m(ω)(1− T 1/2r )]2, (8)where m(ω) = 0.48+1.574ω−0.176ω2. Finally, PR fluid properties are obtainedby letting δ1 = 1 +√2, δ2 = 1 −√2, Zc = 0.3074013, Ωa = 0.45723552,Ωb = 0.077796074. The α-function for the PR EoS is the same as that inEq. (8), but with a slightly different prescription for m(ω), i.e.,m(ω) = 0.37464 + 1.54226ω− 0.26992ω2. (9)4Before turning to the principal result discussed herein, which relates to the SRKand PR EoS models, we note that the pseudocritical line in the (Tr, x) planecorresponding to a choice of RK fluid properties can be written down in closedform after solving a quadratic equation, 0 = a0(x)+a1(x)T3/2r +a2(x)T3r , wherea0(x) =Ω3a410x2 + 10Ωbx+ 3Ω2bx5(x+Ωb)5, (10)a1(x) =Ω2a2−5x4 − 10Ωbx3 + 5Ω2bx2 + 4Ω3bx+ 2Ω4bx4(x− Ωb)3(x+Ωb)4, (11)a2(x) =Ωa415x4 − 30Ωbx3 − 9Ω2bx2 + 12Ω3bx+ 8Ω4bx3(x− Ωb)4(x+Ωb)3. (12)After some easy algebra, the result is:Tr ={Ωa(x− Ωb)x(x +Ωb)(15x4 − 30Ωbx3 − 9Ω2bx2 + 12Ω3bx+ 8Ω4b)×[5x4 + 10Ωbx3 − 5Ω2bx2 − 4Ω3bx− 2Ω4b + (5x2 − Ωbx− 2Ω2b)×√−5x4 + 20Ωbx3 + 2Ω2bx2 − 8Ω3bx− 5Ω4b]}2/3. (13)This locus is shown as the dot-dashed curve in Fig. 2. From this figure it alsoappears that predictions for the Widom line based on SRK and PR EoS areessentially identical for Tr values less than about 1.75; for larger values of Tr,however, the two curves are distinct. In addition, the RK prediction for theWidom line shows a significant departure from the Widom line (vr = 1, Tr ≥ 1)for the van der Waals EoS; finally, there is a significant discrepancy betweenthe Widom lines for the SRK and PR EoS models as opposed to that for theRK EoS. Similar considerations apply to Fig. 3, showing Widom lines from theaforementioned EoS models in the (Pr , Tr) plane. Also shown is a correlationobtained by regression of experimental cp data for CO2 and water from Smithet al. (2013). In fact, based on their analysis of experimental data, these authorsclaim that the corresponding states principle can be used to estimate the pseu-docritical line of a pure substance. We assume that this conclusion is accurate;however, there is still a noticeable difference in Fig. 3 between the Widom linesfor the SRK and PR EoS models and that from the Smith et al. (2013) corre-lation. At least in part, such differences can be attributed to the temperaturedependence in the α-function for the SRK (or PR) EoS (Soave, 1972; Peng andRobinson, 1976) being markedly different from that for the RK EoS. Since the50.4 1 70.511.522.53vrTrvdW−WLRK−WLSRK−WLPR−WLRK−BinodalRK−SpinodalvdW−BinodalvdW−SpinodalFigure 2: Widom lines in the (Tr , vr) plane from the Peng-Robinson, Soave-Redlich-Kwong,Redlich-Kwong, and van der Waals EoS models. Also shown are the Redlich-Kwong and vander Waals binodal and spinodal curves.Soave (1972) α-function passes the consistency test advocated by Le Guennecet al. (2016), it can be used at supercritical conditions (for values of the acen-tric factor in certain allowable ranges) up to a temperature T ∗r = [1+m−1(ω)]2,where it attains a relative minimum. However, as can be seen from Fig. 3, inour case (i.e., with both the SRK and PR EoS models) the α-function never at-tains its minimum in the supercritical range (for the values of the acentric factorω = 0 and ω = 0.34 considered herein), so the oft-quoted issue with the Soave(1972) α-function that it might stop decreasing in the supercritical range asthe reduced temperature increases (Mahmoodi and Sedigh, 2016; Gasem et al.,2001; Neau et al., 2009a,b) is not relevant here.AcknowledgementsFinancial support provided by MIUR (Grant no. PGR10DN9YV) is grate-fully acknowledged.ReferencesBrazhkin, V., Ryzhov, V., 2011. Van der Waals supercritical fluid: Exact for-mulas for special lines. J. Chem. Phys. 135 (8), 084503.60 1 2 3 4 502468101214161820TrPrSRKPRRKvdWSmith et al.Figure 3: Widom lines in the (Pr , Tr) plane from the Peng-Robinson, Soave-Redlich-Kwong,Redlich-Kwong, and van der Waals EoS models. Also shown (dotted) is the correlation (ob-tained by regression of experimental cp data for CO2 and water) by Smith et al. (2013).Gasem, K. A. M., Gao, W., Pan, Z., Robinson, R., 2001. A modified temperaturedependence for the Peng–Robinson equation of state. Fluid Phase Equilib.181 (1), 113–25.Johnston, D. C., 2014. Advances in Thermodynamics of the van der WaalsFluid. Morgan & Claypool Publishers, San Rafael, CA (USA).Le Guennec, Y., Lasala, S., Privat, R., Jaubert, J.-N., 2016. A consistency testfor α-functions of cubic equations of state. Fluid Phase Equilib. 427, 513–38.Mahmoodi, P., Sedigh, M., 2016. Soave alpha function at supercritical temper-atures. J. Supercrit. Fluids 112, 22–36.May, H.-O., Mausbach, P., 2012. Riemannian geometry study of vapor-liquid phase equilibria and supercritical behavior of the Lennard-Jones fluid.Phys. Rev. E 85 (3), 031201.Neau, E., Hernández-Garduza, O., Escandell, J., Nicolas, C., Raspo, I., 2009a.The Soave, Twu and Boston–Mathias alpha functions in cubic equations of7state: Part I. Theoretical analysis of their variations according to tempera-ture. Fluid Phase Equilib. 276 (2), 87–93.Neau, E., Raspo, I., Escandell, J., Nicolas, C., Hernández-Garduza, O., 2009b.The Soave, Twu and Boston–Mathias alpha functions in cubic equations ofstate. Part II. Modeling of thermodynamic properties of pure compounds.Fluid Phase Equilib. 276 (2), 156–64.Peng, D.-Y., Robinson, D. B., 1976. A new two-constant equation of state. Ind.Eng. Chem. Fundam. 15 (1), 59–64.Schmidt, G., Wenzel, H., 1980. A modified van der Waals type equation of state.Chem. Eng. Sci. 35 (7), 1503–12.Smith, J. M., Van Ness, H. C., Abbott, M. M., 2005. Introduction to ChemicalEngineering Thermodynamics, 7th Edition. McGraw-Hill, Boston.Smith, R., Inomata, H., Peters, C., 2013. Introduction to Supercritical Fluids.Elsevier, Amsterdam.Soave, G., 1972. Equilibrium constants from a modified Redlich-Kwong equationof state. Chem. Eng. Sci. 27 (6), 1197–203.8

Widom line prediction by the Soave-Redlich-Kwong and Peng-Robinson equations of state

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Widom line prediction by the Soave-Redlich-Kwong and Peng-Robinson equations of state

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