Consistency of Vapor Pressure Equations at the Critical Point

Dec 10, 2015 - IUFFyM, Universidad de Salamanca, 37008 Salamanca, Spain. ABSTRACT: In this work, it is shown that the most common approximations for ...
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Consistency of Vapor Pressure Equations at the Critical Point Santiago Velasco,†,‡ Maria J. Santos,† and Juan A. White*,†,‡ †

Departamento de Física Aplicada, Universidad de Salamanca, 37008 Salamanca, Spain IUFFyM, Universidad de Salamanca, 37008 Salamanca, Spain

‡

ABSTRACT: In this work, it is shown that the most common approximations for the vapor-pressure equation of a fluid yield an inappropriate behavior for the second temperature derivative of the saturation pressure in the vicinity of the critical point. This fact gives rise to an unphysical divergent behavior for the change of the specific heat at vaporization (Δvcv) near the critical point. An expression for the vapor-pressure equation with the appropriate critical behavior is proposed in terms of a truncated Renormalization Group (RG) expansion. As an example, it is shown that the well-known Wagner and Pruss vapor-pressure equation and the IAPWS-95 result for water yield an incorrect critical behavior for Δvcv whereas the RG equation, in agreement with the results of a crossover equation of state, provides the appropriate critical behavior. An application to the Ambrose-Walton vapor pressure equation is also presented. water.3 In Section 5, an application is made for the AmbroseWalton vapor-pressure equation4 which is known to give accurate results for most simple fluids. We conclude with a brief summary of the main results of this work.

1. INTRODUCTION The vapor pressure equation for a pure fluid, pσ = pσ(T), is defined as the functional relation that can be established between the saturation pressure pσ and the temperature T at liquid−vapor coexistence. Two exact thermodynamic relations are of fundamental interest in the study of vapor pressures: (i) the well-known Clapeyron-Clausius equation pσ′ (T ) ≡

dpσ dT

=

Δv h T Δv v

2. CRITICAL BEHAVIOR OF THE VAPOR PRESSURE AND RELATED QUANTITIES Near the critical point, the asymptotic scaling description of fluid criticality provided by the renormalization group theory yields the following expansion for the reduced vapor pressure, pr = pσ/pc,5,6

(1)

that expresses the first temperature derivative of pσ in terms of the specific enthalpy of vaporization Δvh and the specific volume change Δvv = vg − vl, where vg and vl are the vapor (g) and liquid (l) specific volumes at saturation, and (ii) the Yang− Yang equation,1 pσ″(T ) ≡

d2pσ dT 2

=

Δ v cv T Δv v

⎡ pr = 1 + A 0τ + A1τ 2 + Aτ 2 − α⎢1 + ⎢⎣

i

i=1

⎥⎦

(3)

and for the change of the reduced density, ρr = ρ/ρc = vc/v,6 ⎡ Δ v ρr = Bτ β ⎢1 + ⎢⎣

(2)

that links the second temperature derivative of the saturation pressure to Δvv and the specific heat change Δvcv = cgv2 − clv2, where cgv2 and clv2 are the two-phase vapor and liquid specific heat capacities at constant volume. In spite of the above exact thermodynamic relations, only approximate results are known for the vapor pressure of a fluid. The goal of this paper is to show that these approximate results for the vapor pressure can give rise to consistency problems in the critical region when used in combination with the exact eqs 1 and 2. This paper is structured as follows: Section 2 presents the critical behavior prescribed by the renormalization group (RG) theory for the vapor pressure and the other quantities considered in eqs 1 and 2. In Section 3, it is shown that the most commonly used approximate expressions for the vapor pressure can give rise to consistency problems near the critical point. An expression for the vapor pressure equation in the vicinity of the critical point is proposed in Section 3 that yields the appropriate critical behavior from an estimation of the Riedel point of the fluid.2 In Section 4, we apply the findings of Section 3 to the Wagner and Pruss vapor-pressure equation for © XXXX American Chemical Society

⎤

∞

∑ aiτ λ ⎥

⎤

∞

∑ biτ λ ′⎥ i

i=1

⎥⎦

(4)

where the coefficients of the expansions are fluid-dependent quantities, τ = 1 − Tr, with Tr = T/Tc, α is the critical exponent of the specific heat, β is the critical exponent of the coexistence curve, and the λi’s and λ′i ’s are positive leading correction-toscaling exponents. Eq 3 gives rise to a finite slope of the vapor pressure curve at the critical point, ⎛ dpr ⎞ ⎟ = −A 0 ⎜ ⎝ dTr ⎠c

(5)

and prescribes that the second temperature derivative of the vapor pressure curve diverges at the critical point as Received: September 24, 2015 Revised: December 9, 2015 Accepted: December 10, 2015

A

DOI: 10.1021/acs.iecr.5b03577 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research d2pr dTr 2

using a RG compatible vapor pressure equation near the critical point instead of eq 10. The method consists of the use of a truncated version of eq 3,

∼ τ −α (6)

On the other hand, eq 4 implies that the jump of the specific volume at the critical point vanishes following the power law Δv v ∼ τ β

pr = 1 + A 0τ + A1τ 2 + Aτ 2 − α

where α = 0.109 and the coefficients A0, A1, and A are obtained from a reference point near the critical region: the Riedel point.2 Recently,2 we have shown that the so-called Riedel function, defined as9

(7)

Using eqs 1, 5, and 7, the specific enthalpy of vaporization must become zero at the critical point with the asymptotic behavior Δv h ∼ τ β

(8)

while using eqs 2, 6, and 7, the change of the two-phase heat capacity at constant volume must behave at the critical point according to the power law Δ v cv ∼ τ β − α

αR (Tr) ≡

r

and

1 − τR p′ pr,R r,R

(17)

(18)

A 0τR + A1τR 2 + AτR 2 − α = pr,R − 1

(11)

A 0 + 2A1τR + (2 − α)AτR 1 − α = − 2A1 + (2 − α)(1 − α)AτR −α =

(12)

Since Δvcv must tend to zero as one approaches the critical point, one has that the chosen value for t2 must satisfy: t2 > 2 − β. In other words, the occurrence of both the divergence of the second temperature derivative of the vapor pressure curve and the asymptotic behavior of Δvcv in the proximities of the critical point implies that the exponent t2 in eq 10 must verify 2 − β < t2 < 2

(16)

with p′r,R = p′r (Tr,R). From eq 14, using eqs 17 and 18, one obtains

(10)

The use of eqs 2 and 10 together with the scaling law (eq7) for the change of the specific volume at vaporization leads to Δ v cv ∼ τ β+ t2 − 2

(15)

r,R

αR̅ = αR (Tr,R ) =

i

t 2 < 2 < t3 < ... < tn

dTr

Tr p′(Tr) pr r

=

where τR = 1 − Tr,R, pr,R = pr(τR) and

where the ci’s are substance-dependent coefficients and the exponents ti satisfy t1 = 1

d ln Tr

d ln pr

Taking into account eq 15, from eq 16 one has pr,R ″ = pr″(Tr,R ) = pr,R αR̅ (αR̅ − 1) (1 − τR )2

n i=1

= Tr

⎡ dαR ⎤ =0 ⎥ ⎢ ⎣ dTr ⎦T = T

3. SIMPLE APPROXIMATION FOR THE VAPOR PRESSURE EQUATION IN THE CRITICAL REGION The reduced vapor pressure equations most frequently used in the literature for a large number of fluids have the form:7,8

∑ ciτ t

d ln pr

presents a minimum at a reduced temperature, Tr,R, that for most fluids occurs in the range of 0.95−0.98,2 just below the point of minimum curvature for the vapor pressure curve that can be considered as a reference point for the critical region.10 The fact that the Riedel function αR(Tr), defined by eq 15, goes through a minimum at a reduced temperature Tr,R is mathematically expressed by the relation

(9)

Taking into account that Δvcv must vanish at the critical point, eq 9 indicates that β must be greater than α, as it is the case for the 3-dimensional Ising universality class where α ≈ 0.109 and β ≈ 0.326. If instead of considering the accepted RG behavior of the preceding equations one resorts to approximations, a crucial test of the validity of the approximations in the critical zone is given by the fulfillment of the inequality β > α > 0. In the next section, we shall show that the usual prescriptions for the vapor pressure based on a truncated expansion of ln pr can lead to consistency problems in the critical region because this inequality is not fulfilled.2

1 ln pr = Tr

(14)

(19)

pr,R 1 − τR pr,R

(1 − τR )2

αR̅

(20)

αR̅ (αR̅ − 1) (21)

a set of algebraic equations from which one can obtain A0, A1, and A in terms of τR, pr,R, α̅ R, and the critical exponent α, A0 =

(13)

where β is the critical exponent choice for the change of specific volume (or density) at vaporization. The most common values used in the literature for the exponent t2 in eq 10 are t2 = 1.25 and t2 = 1.5. The condition provided by eq 13 shows that these values are not consistent with the customary choices for the exponent β, ranging in the interval [0.32, 0.40]. Therefore, such vapor pressure equations can not be used for obtaining Δvcv near the critical point. In this work, we propose a simple method for solving this situation by

A1 =

C2 − (3 − α)(1 − pr,R − C1) (1 − α)τR

−

1 − pr,R τR

(22)

C2 − (2 − α)(1 − pr,R − C1) ατR 2

(23)

and A=

2(1 − pr,R − C1) − C2 α(1 − α)τR 2 − α

(24)

with B

DOI: 10.1021/acs.iecr.5b03577 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research C1 =

pr,R τR 1 − τR

αR̅

and

C2 =

pr,R τR 2 (1 − τR )2

αR̅ (αR̅ − 1) (25)

4. AN EXAMPLE: THE WAGNER AND PRUSS VAPOR PRESSURE EQUATION OF ORDINARY WATER Numerous correlations for the vapor pressure of water can be found in the literature, but the internationally accepted expression of the form of eq 10 is the one proposed by Saul and Wagner in 198711 and converted to ITS-90 temperatures by Wagner and Pruss in 1993,3 1 ln pr = [c1τ + c 2τ 3/2 + c3τ 3 + c4τ 7/2 + c5τ 4 + c6τ15/2] Tr

Figure 1. Plot of the second temperature derivative of the reduced vapor pressure p″r vs the reduced temperature Tr for water. The solid line represents eq 32, which is based on a truncated expansion, the dashed line represents the IAPWS-95 result obtained from RefProp 9.1,12 the dotted line is obtained from the Wagner and Pruss equation, eq 26, and the dot-dashed line is the crossover result of Kiselev and Friend.13 The inset shows the behavior of pr″ near the critical point.

(26)

with c1 = −7.85951783, c2 = 1.84408259, c3 = −11.7866497, c4 = 22.6807411, c5 = −15.9618719, and c6 = 1.80122502. From eq 26, the following value for the slope of the vapor pressure curve at the critical point (τ = 0) is obtained: ⎛ d ln pr ⎞ ⎛ dpr ⎞ ⎟ = −c1 = 7.85951783 ⎟ =⎜ ⎜ ⎝ dTr ⎠c ⎝ dTr ⎠c

scaling law given by eq 28. In 1999, in order to avoid the problems of IAPWS-95 with the physically incorrect limiting behavior of some caloric properties when approaching the critical point, Kiselev and Friend developed a crossover formalism for the equation of state of water.13 The second temperature derivative of the vapor pressure curve obtained from this crossover formalism is also plotted in Figure 1, showing an asymptotic behavior clearly different from that of eq 28 (see the inset). Taking into account eqs 15 and 26, the Riedel function associated with the Wagner and Pruss vapor pressure equation presents a minimum at a reduced temperature Tr,R = 0.975829274 (τR = 0.024170726), with a reduced pressure pr,R = 0.828863766 and a value of the Riedel function α̅ R = 7.63151925. With these data and using eqs 22 and 25 with α = 0.109, we obtain the following values for the coefficients of the vapor pressure (eq 14): A0 = −7.82607134, A1 = −25.1770953, and A = 37.3428070. The vapor pressure equation and its first and second temperature derivatives that we propose here for water are, respectively,

(27)

while the second temperature derivative of the vapor pressure curve diverges at the critical point as d2pr dTr 2

∼ τ −0.5 (28)

i.e., with a critical exponent α = 0.5 instead of the value α ≈ 0.109 prescribed by the RG theory. This value of α is due to the exponent t2 = 3/2 in eq 26 which implies that the condition in eq 13 is not fulfilled for a typical value of β ∼ 0.32−0.40. As we will see in what follows, this fact will lead to an inappropriate behavior of Δvcv in the vicinity of the critical point. In 1995, the International Association for the Properties of Water and Steam (IAPWS) adopted a new formulation for the thermodynamic properties of water based on a fundamental equation for the specific Helmholtz free energy split into a part that represents the properties of the ideal gas and a part that takes into account the residual fluid behavior.14 Then, the vapor pressure curve was obtained by applying the Maxwell criterion for the phase-equilibrium condition. The IAPWS-95 vapor pressure data are also reported by the NIST RefProp9.1 program.12 The differences between the results of eq 26 for the vapor pressure of water and those obtained from the IAPWS-95 formulation are very small.14 The main differences appear in the critical region. For example, the slope of the vapor pressure curve at the critical point obtained from extrapolation of the results given by eq 1 using IAPWS-95 data obtained from RefProp9.1 is ⎛ dpr ⎞ ⎟ = 7.8379 ⎜ ⎝ dTr ⎠c

pr = 1 − 7.82607134τ − 25.1770953τ 2 + 37.3428070τ 1.891 (30)

pr′ ≡

dpr dTr

= 7.82607134 + 50.3541906τ − 70.6152480τ 0.891 (31)

and pr″ ≡

d2pr dTr2

= −50.3541906 + 62.9181860τ −0.109 (32)

These equations yield for pr, p′r , and p″r the same values of those obtained from the Wagner and Pruss equation (eq 26), at the Riedel temperature Tr,R = 0.975829274. The relative differences between eqs 26 and 30 are less than 0.0007% from Tr,R to 1. The relative differences between eq 31 and p′r obtained from eq 26 are less than 0.03% except at the critical point where the slope given by eq 31 is

(29)

i.e., 0.3% below the value given by eq 27. The second temperature derivative of the vapor pressure curve given by eq 26 and the result obtained from the IAPWS-95 formulation are plotted in Figure 1 for the reduced temperature range of 0.975−1 (630.919−647.096 K). Although the main differences arise very close to the critical point, as it is shown in the inset of Figure 1, both results diverge at the critical point following the

⎛ dpr ⎞ ⎟ = 7.82607134 ⎜ ⎝ dTr ⎠c

(33)

that is, 0.4% below the value given by eq 27 and 0.1% below the value given by eq 29. The second temperature derivative given C

DOI: 10.1021/acs.iecr.5b03577 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research by eq 32 diverges asymptotically at the critical point following the RG scaling law, and it is plotted in Figure 1 in order to be compared with the Wagner and Pruss, the IAPWS-95, and the crossover results. In this figure, one can see how the results of eq 32 go from the value given by Wagner and Pruss and the IAPWS-95 formulation at Tr,R to the same asymptotic behavior as the one obtained from the crossover vapor pressure curve at the critical point (see the inset). Furthermore, Wagner and Pruss proposed the following expression for the reduced saturated-liquid density of ordinary water: ρrl = 1 + d1τ1/3 + d 2τ 2/3 + d3τ 5/3 + d4τ16/3 + d5τ 43/3 + d6τ110/3

Figure 2. Plot of the reduced specific heat change Δvcv* = (Tc/ pcvc)Δvcv = TrΔvvrpr″ vs the reduced temperature Tr for water. The dashed line is obtained from the RefProp 9.112 results for Δvvr and pr″. The other lines are obtained using Δvvr determined from eqs 34 and 35 and pr″ calculated from eq 32 (solid line), from the Wagner and Pruss equation (eq 26, dotted line), and from the crossover result of Kiselev and Friend13 (dot-dashed line).

(34)

with d1 = 1.99274064, d2 = 1.09965342, d3 = −0.510839303, d4 = −1.75493479, d5 = −45.5170352, and d6 = −6.74694450 × 105, while the equation for the reduced saturated-vapor density of ordinary water has the form ρrv = exp[d1′τ1/3 + d 2′τ 2/3 + d3′τ 4/3 + d4′τ 9/3 + d5′τ 37/6 + d6′τ 71/6]

(35)

water is only relevant in the vicinities of the critical point, as shown in Figure 2. In this respect, it is worth mentioning that neither the Wagner and Pruss vapor-pressure equation nor the IAPWS-95 equation of state were designed to give the appropriate critical behavior. For other fluids, it is quite common to use approximations for the vapor-pressure and the saturated densities that are not consistent in the critical zone (see, e.g., refs 15 and 16). Of course, out of the critical zone, such approximations are perfectly valid but one should be aware of potential pitfalls near the critical point as we will see in the next section.

with d 1′ = −2.031500240, d 2′ = −2.68302940, d 3′ = −5.38626492, d′4 = −17.2991605, d′5 = −44.7586581, and d′6 = −63.9201063. From eqs 26, 34, and 35, one has t2 = 1.5 and β = 1/3 so that t2 < 2 − β and the condition in eq 13 is not verified. Therefore, taking into account eq 12, the change of the specific heat at vaporization calculated from eqs 26, 34, and 35 diverges asymptotically at the critical point as Δ v cv ∼ τ −1/6

(36)

In other words, the Wagner and Pruss equation (eq 26) is not consistent with eqs 34 and 35 at the critical point. Furthermore, eqs 26, 34, and 35 are not thermodynamically consistent with IAPWS-95,14 in spite of the fact that they are auxiliary equations for calculating properties along the liquid−vapor phase boundary in the IAPWS-95 formulation.14 However, the change of the specific heat at vaporization calculated from eq 2, eqs 34 and 35, and the second temperature derivative of the vapor pressure proposed in the present work, eq 32, vanishes at the critical point following the power law Δ v cv ∼ τ 0.224

5. APPLICATION TO THE AMBROSE-WALTON VAPOR-PRESSURE EQUATION One of the most accurate vapor-pressure equations of the form given by eq 10 is the Ambrose-Walton (AW) equation:4 ln pr =

1 AW (c1 (ω)τ + c 2AW (ω)τ 3/2 + c3AW (ω)τ 5/2 + c4AW (ω)τ 5) Tr

(38)

This equation was formulated within an extended corresponding states principle so that the fluid-dependent coefficients cAW i are functions of the acentric factor ω.17,18 More concretely, the coefficients cAW in the Ambrose-Walton equation are quadratic i functions of ω:4

(37)

in close agreement with the RG power law given by eq 9. The behavior of Δvcv with the temperature near the critical point is shown in Figure 2, where we have plotted the reduced specific heat change for water Δvcv* = ΔvcvTc/(pcvc) = TrΔvvr(d2pr/dTr2) vs Tr calculated from different prescriptions for the vapor pressure equation. At the Riedel reduced temperature Tr,R = 0.975829274, both the result obtained from eq 32 (solid line) and the Wagner and Pruss result (obtained from eq 26) (dotted line) for Δvc*v coincide and are very close to the RefProp 9.112 (IAPWS-95) result (dashed line) and slightly different from the crossover result obtained from the Kiselev and Friend13 equation of state (dot-dashed line). However, very close to the critical point, one can observe how the IAPWS-95 and the Wagner and Pruss results yield a diverging behavior for Δvcv whereas the results of the eq 32 and the crossover equation show the appropriate behavior, vanishing with the same asymptotic law. In should be remarked here that the lack of consistency between the approximations given by eq 26 for the vapor pressure and by eqs 34 and 35 for the saturated densities of

c1AW (ω) = −5.97616 − 5.03365ω − 0.64771ω 2 c 2AW (ω) =

1.29874 + 1.11505ω + 2.41539ω 2

c3AW (ω) = −0.60394 − 5.41217ω − 4.26979ω 2

(39)

c4AW (ω) = −1.06841 − 7.46628ω + 3.25259ω 2

Therefore, the critical behavior of the second temperature derivative of the AW vapor-pressure equation is given by eq 28, with a critical exponent α = 0.5 like in the case of the Wagner and Pruss equation for water. Very recently,19 we have presented a formulation of the Watson expression for the temperature dependence of the enthalpy of vaporization20,21 in terms of the acentric factor:19 Δ v h(τ ; ω) =

RTc (7.2729 + 10.4962ω + 0.6061ω 2)τ 0.38 M (40)

D

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Industrial & Engineering Chemistry Research where R is the gas constant and M is the molecular mass of the fluid. Comparing this expression with the asymptotic behavior given by eq 8, one obtains a critical exponent β = 0.38 which is smaller than the value α = 0.5 of the AW equation. As previously commented, this fact can lead to consistency issues in the critical zone when eqs 38 and 40 are used together. In particular, eqs 1 and 2 allow one to determine the exact relation, TcΔ v cv =

pr″ pr′

Δv h

Figure 4. Plot of the reduced specific heat change Δvc*v = (Tc/ pcvc)Δvcv vs the reduced temperature Tr for argon (acentric factor ω = −0.0022). The symbols are RefProp 9.112 results. The lines are obtained from eq 41, using the Watson equation (eq 40) and either the AW vapor pressure equation (eq 38; dashed line) or the truncated RG expansion equation (eq 14; solid line) with the coefficients given in eq 42. The dotted line indicates the location of the reduced Riedel temperature TAW r,R = 0.976156 for argon.

(41)

that, when using eqs 38 and 40, leads to the unphysical critical behavior Δvcv ∼ τ−0.12. The approximation for the vapor pressure equation presented in Section 3 provides a simple way to remedy the lack of consistency between the AW eq 38 and the Watson eq 40. Taking into account eqs 15 and 26, for each value of the acentric factor ω, the Riedel function associated with the AW vapor pressure equation presents a minimum at a reduced AW temperature TAW r,R (ω), with a reduced pressure pr,R (ω) and a AW value of the Riedel function α̅R (ω). Then, from eqs 22 and 25 with α = 0.109, for each value of ω, one can numerically obtain AW the coefficients of the vapor pressure in eq 14 to be AAW 0 , A1 , AW and AAW . Figure 3 shows a plot of the coefficients A for 2 i

pressure presented in eq 14 with the coefficients given in eq 42. The results obtained from RefProp 9.112 are also plotted for comparison. As one can observe in Figure 4, from the reduced Riedel temperature TAW r,R = 0.976156 up to Tr ∼ 0.995, the results of the AW vapor-pressure equation and of the truncated RG expansion are very close, with good agreement with RefProp 9.1 data. However, near the critical point, for Tr > 0.995, the differences between the AW and the RG results become increasingly larger, especially very close to the critical point where the AW equation gives rise to a divergent behavior for Δvcv (due to its lack of consistency with the Watson equation, eq 40) while the results of RefProp 9.1 and of the truncated RG expansion still show a good agreement. A similar behavior is obtained for propane (Figure 5), but in this case, the agreement between the AW results and RefProp 9.1 data is better, except, of course, for the divergent behavior of the AW results near the critical point.

Figure 3. Coefficients of the vapor pressure (eq 14) obtained from the AW vapor pressure (eq 38) as a function of the acentric factor ω. The (circles), AAW (squares), and symbols are numerical results for AAW 0 1 AW A2 (diamonds). The lines are the polynomial fits (eq 42) to the numerical results.

several values of ω. Since it is not possible to obtain analytical results for the AAW i , in order to obtain an expression of practical application, we have performed polynomial fits to the numerical data plotted in Figure 3, obtaining:

Figure 5. Same caption as in Figure 4 for propane, with ω = 0.1521 and TAW r,R = 0.977326.

A 0AW (ω) = −5.94308 − 5.03415ω − 0.53713ω 2 − 0.173558ω3 + 0.0570099ω4 A1AW (ω) = −13.4191 − 20.1186ω − 40.522ω 2 − 50.6873ω3 + 12.9613ω4 A 2AW (ω) =

(42)

6. SUMMARY To conclude, in this paper, we have pointed out the importance of ensuring consistency of the approximate vapor-pressure equations near the critical point. This consistency is obtained when the critical exponent β of the change of specific volume is larger than the critical exponent α of the specific heat since in this situation the change of the specific heat at vaporization vanishes at the critical point with a power law τβ−α. Since the second derivative of the vapor-pressure equation diverges at the critical point with an exponent α and the value of β ranges in the interval [0.32, 0.40], for applications near the critical point,

20.7278 + 32.8052ω + 42.3338ω 2 + 38.099ω3 − 9.7028ω4

The results of these fits are also plotted in Figure 3 for the sake of comparison. Figure 4 shows the results of the reduced specific heat change Δvc*v = (Tc/pcvc)Δvcv for argon obtained from the exact relation given by eq 41, using the Watson equation (eq 40) for the enthalpy of vaporization and either the AW vapor-pressure equation (eq 38) or the truncated RG expansion for the vapor E

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Industrial & Engineering Chemistry Research

(6) Kim, Y. C.; Fisher, M. E.; Orkoulas, G. Asymmetric fluid criticality. I. Scaling with pressure mixing. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 67, 061506. (7) Wagner, W. New vapour pressure measurements for argon and nitrogen and a new method for establishing rational vapour pressure equations. Cryogenics 1973, 13, 470−482. (8) Ambrose, D. The correlation and estimation of vapour pressures IV. Observations on Wagner’s method of fitting equations to vapour pressures. J. Chem. Thermodyn. 1986, 18, 45−51. (9) Riedel, L. Eine neue universelle Dampfdruckformel Untersuchungen über eine Erweiterung des Theorems der übereinstimmenden Zustände. Teil I. Chem. Ing. Tech. 1954, 26, 83−89. (10) Velasco, S.; Santos, M. J.; White, J. A.; Srinivasan, K. The curvature of the liquid-vapor reduced pressure curve and its relation with the critical region. J. Chem. Thermodyn. 2013, 60, 41−45. (11) Saul, A.; Wagner, W. International equations for the saturation properties of ordinary water substance. J. Phys. Chem. Ref. Data 1987, 16, 893−901. (12) Lemmon, E. W.; Huber, M. L.; McLinden, M. O. NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 9.1; National Institute of Standards and Technology, Standard Reference Data Program: Gaithersburg, MD, 2013. (13) Kiselev, S.; Friend, D. Revision of a multiparameter equation of state to improve the representation in the critical region: application to water. Fluid Phase Equilib. 1999, 155, 33−55. (14) Wagner, W.; Pruss, A. The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. J. Phys. Chem. Ref. Data 2002, 31, 387−535. (15) Duschek, W.; Kleinrahm, R.; Wagner, W. Measurement and correlation of the (pressure, density, temperature) relation of carbon dioxide II. Saturated-liquid and saturated-vapour densities and the vapour pressure along the entire coexistence curve. J. Chem. Thermodyn. 1990, 22, 841−864. (16) Chen, Q.; Hong, R.; Chen, G. Gaseous {PVT} properties of ethyl fluoride. Fluid Phase Equilib. 2005, 237, 111−116. (17) Pitzer, K. S. The Volumetric and Thermodynamic Properties of Fluids. I. Theoretical Basis and Virial Coefficients. J. Am. Chem. Soc. 1955, 77, 3427−3433. (18) Pitzer, K. S.; Lippmann, D. Z.; Curl, R. F.; Huggins, C. M.; Petersen, D. E. The Volumetric and Thermodynamic Properties of Fluids. II. Compressibility Factor, Vapor Pressure and Entropy of Vaporization. J. Am. Chem. Soc. 1955, 77, 3433−3440. (19) Velasco, S.; Santos, M.; White, J. Extended corresponding states expressions for the changes in enthalpy, compressibility factor and constant-volume heat capacity at vaporization. J. Chem. Thermodyn. 2015, 85, 68−76. (20) Watson, K. Prediction of critical temperatures and heats of vaporization. Ind. Eng. Chem. 1931, 23, 360−364. (21) Román, F. L.; White, J. A.; Velasco, S.; Mulero, A. On the Universal Behavior of Some Thermodynamic Properties Along the Whole Liquid-Vapor Coexistence Curve. J. Chem. Phys. 2005, 123, 124512−1−6.

it is important to consider vapor-pressure equations that lead to a value of α smaller than the chosen value for β. The usual approximations for the vapor-pressure equation of a given fluid do not fulfill the inequality β > α. Of course, far enough from the critical zone, this is irrelevant, but near the critical point, this lack of consistency leads to an unphysical divergent behavior for the change of the specific heat at vaporization. On the basis of a truncated version of the RG expansion of the reduced vapor pressure and using the Riedel point as a reference, we have obtained a simple expression for the vapor-pressure equation near the critical point with the appropriate asymptotic behavior. As an example, we have considered the Wagner and Pruss vapor-pressure equation3 for water obtaining that this equation does not fulfill the requirement β > α, and therefore, it gives rise to a divergent behavior of Δvcv with the temperature near the critical point. The same behavior has been obtained for the IAPWS-95 results14 provided by the RefProp9.1 program12 for water. However, using our vapor-pressure equation near the critical point, we have obtained the appropriate asymptotic behavior of Δvcv, in close agreement with the results of the crossover equation of Kiselev and Friend.13 In this context, and from a practical viewpoint, it seems plausible to consider a piecewise vapor-pressure equation in which the Wagner and Pruss equation is used for temperatures below the Riedel temperature whereas the truncated expansion is used for temperatures above the Riedel temperature up to the critical point. Finally, we have considered the Ambrose-Walton vaporpressure equation.4 In this case, a truncated RG expansion with the correct critical behavior has been obtained in terms of the acentric factor ω. An application of the obtained results to the specific heat change Δvcv of argon and propane shows good agreement with RefProp 9.112 data.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are thankful for the financial support from the Ministerio de Economı ́a y Competitividad of Spain under Grant ENE201340644-R.

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REFERENCES

(1) Yang, C. N.; Yang, C. P. Critical Point in Liquid-Gas Transitions. Phys. Rev. Lett. 1964, 13, 303−305. (2) Velasco, S.; White, J. A.; Srinivasan, K.; Dutta, P. Waring and Riedel Functions for the Liquid-Vapor Coexistence Curve. Ind. Eng. Chem. Res. 2012, 51, 3197−3202. (3) Wagner, W.; Pruss, A. International Equations for the Saturation Properties of Ordinary Water Substance. Revised According to the International Temperature Scale of 1990. Addendum to J. Phys. Chem. Ref. Data 16, 893 (1987). J. Phys. Chem. Ref. Data 1993, 22, 783−787. (4) Ambrose, D.; Walton, J. Vapour Pressures up to their Critical Temperatures of Normal Alkanes and 1-Alkanols. Pure Appl. Chem. 1989, 61, 1395−1403. (5) Sengers, J. M. H. L.; Greer, S. C. Thermodynamic anomalies near the critical point of steam. Int. J. Heat Mass Transfer 1972, 15, 1865− 1886. F

DOI: 10.1021/acs.iecr.5b03577 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX