Thermodynamic Properties of Fluids from Tao-Mason Equation of State: Results for Liquid, Vapor and Supercritical Regions

Thermodynamic Properties of Fluids from Tao-Mason Equation of State: Results for Liquid, Vapor and Supercritical

Regions

M. M. Papari ¹, J. Moghadasi ², and M. Kiani ¹

1Department of Chemistry, Shiraz University of Technology, Shiraz 71555-313, Iran

2Department of Chemistry, Shiraz University, Shiraz 71454, Iran papari@sutech.ac.ir

Introduction

Equations of state (EOS) are important tool in chemical engineering designs and assumed an expanding role in the study of the phase equilibria of fluids and fluid mixtures.

However, most of them suffer from the fact that they fail to represent properties accurately in a vast range of temperatures and pressures. The present work assesses the performance of a molecular based equation of state known as Tao-Mason (TM) in predicting thermodynamic properties of fluids in a very wide range of temperatures encompassing 100K<T<1100K and pressures ranging from zero to 3200 bar. The fifteen fluids including argon (Ar), Krypton (Kr), xenon (Xe), nitrogen (N2), oxygen (O2), carbon dioxide (CO2), methane(CH4), ethane (C2H6), propane(C3H8), normal butane (n-C4H10), isobutene (i- C4H10), ethen (C2H4), benzene (C6H6), toluene (C7H8) and 1,1,1,2 tetra fluoroethane (R134a) are selected and compared with literature data. The calculations cover the ranges from the dilute vapor or gas to the highly compressed liquid and supercritical region. The thermodynamic properties are the vapor and liquid densities, the vapor pressure, the internal energy, the enthalpy, the entropy, the heat capacity at constant pressure and constant volume, the Joule-Thomson coefficient, and the speed of sound. It is found that the overall agreement with literature in all phases especially the vapor/gas phase is remarkable. Furthermore, the Zeno line regularity can be well represented by TM EOS.

The TM Equation of State

In 1994, Tao and Mason calculated a perturbation correction term for the effect of attractive forces and combined with Song and Mason equation of state [1] to present an improved equation of state (TM EOS) [2]. The final form of the TM EOS is:

2

T T T

P

= + − α ρ + α (T )ρ

+ − ² (e^{kTc / T} − A₂ ) (8)

ρkT 1 (B₂ (T ) (T ))

1 − ρb(T )λ A₁ (α (T ) B₂ (T ))b(T )ρ

1 + 1.8( ρb(T ))⁴

Where A₁ = 0.143

A = 1.64 + 2.65[e( κ−1.093) −1] (9)

κ = 1.093 + 0.26[(ω + 0.002)^{1/ 2} + 4.50(ω + 0.002)] (10)

In Eqs. (9) and (10) ω is the Pitzer acentric factor.

The features of TM EOS are remarkable: 1) the critical region is described less accurately using TM equation of state 2) the entire fluid equation of state and its vapor- liquid phase boundaries can be calculated from the intermolecular potential energies plus a few liquid densities 3) if the potential is not known, measurements of the second virial coefficients as a function of temperature can be used instead 4) in the absence of any such measurements, one can use the principle of corresponding state to predict the second virial coefficients. 5) This equation can predict vapor pressures and orthobaric densities with reasonable accuracies from the Maxwell equal area construction.

The second virial coefficient has a main role in the TM equation of state. Among all procedures to determine this property, macroscopic corresponding states correlations by which B2 (T) can be estimated from two arbitrary constants Tc and Pc and acentric factor ω have been found wide applicability. In the present study, we have used correlation equations for second virial coefficient proposed by Tsonopoulos [3]:

(Pc/RTc) B2 =ƒ ⁽⁰⁾ (Tr) +mƒ ⁽¹⁾ (Tr) (11)

Where Tr =T/Tc and the group RTc/Pc serves as a pseudo critical volume .The functions ƒ⁽⁰⁾ and ƒ⁽¹⁾ are:

ƒ ⁽⁰⁾ (T_r ) = 0.1445 − 0.330 − 0.1385 − 0.0121 − 0.000607

, (12)

2 3 8

r r r r

ƒ ⁽¹⁾ (T_r ) = 0.0637 + 0.331

− 0.423

0.008 − (13)

T

r r r

Other parameters in TM equation of state (a (T) and b (T)) can be formulated in

terms of the Boyle temperature TB and the Boyle volume VB, defined by B2(TB)=0 and VB=TB B2'(TB), where B2'=dB2/dT. The dimensionless quantities a/VB and b/VB are almost universal functions of the dimensionless temperature T/TB7,8, and can be calculated with sufficient accuracy from the following formulas based on a (12, 6) potential:

a(T)/VB=a1exp(-c1(T/TB)+ a2[1-exp(-c2/(T/TB)^1/4] (14) b(T)/VB=a1[1-c1(T/TB)]exp(-c1(T/TB)+ a2 {1-[1+c2/(4(T/TB) ^1/4)]exp(-c2/(T/TB) ^1/4}(15) where the constant a1, a2, c1, and c2 are -0.0648, 1.8067, 2.6038, 0.9726, respectively.

Results and discussion

We have computed all thermodynamic properties of fifteen selected fluids in vapor, liquid, and supercritical regions using Tao and Mason (TM) EOS. Absolute average deviations (AAD) of all thermodynamic properties of fluids for 1248 data points from the literature data [4] have been obtained. Typically, the outcome of computations for vapor phase as well as the range of temperatures and pressures considered here has been gathered in Table 1.

Furthermore, Figure 1 demonstrates the deviation of the calculated vapor pressure from those give in literature. As the figure shows the deviations lies within 5%.

We conclude by pointing out that this EOS is capable of predicting thermodynamic properties of fluids with remarkable accuracies. Besides, the attractiveness of the TM EOS lies in its simplicity of calculation, its statistical mechanical foundation, and its performance in prediction of liquid thermodynamic properties.

6

4

2

0

-2

-4

-6

-8

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

TlTc

Figure 1.

xenon benzene

krypton

butane

Deviation(%)

toIuene

nitrogen

argon

ethene isobutane

oxygen methane propane ethane CO2 R134a

i X Table 1.

Ar Kr Xe N2

O2 CO2

CH4

C2H6

C3H8

C₄H₁₀ i-C4H10

C2H4

C6H6

C7H8

R134a

36 100-700 46 130-800 34 200-800 21 100-1000 29 120-400 35 240-1100 29 120-600 33 160-600 35 220-600 28 320-589 22 300-570 28 200-400 24 400-600 20 500-570 28 300-450

N

0-1000 0-1000 0-1000 0-2000 0-800 0-3200 0.-2000 0-700 0-1000

0-690 0-350 0-400 0-780 0-1000

0-700

0.07 0.13 0.12 0.05 0.1 0.26 0.12 0.3 1.15

0.7 0.46 0.24 0.37 0.72 0.48

0.45 1.13 1.18 0.12 1.14 0.42 0.35 0.16 0.44 0.55 0.14 0.22 0.33 0.34 0.33

0.35 0.95 0.99 0.08 0.97 0.38 0.32 0.18 0.43 0.69 0.14 0.18 0.31 0.37 0.29

0.54 1.28 1.44 0.14 1.36 2.61 0.52 0.14 1.79 1.45 0.11 0.19 0.28 0.31 0.44

0.94 1.92 1.48 0.21 0.61 1.71 1.51 1.05 1.17 1.07 0.65 1.31 1.19 0.36 0.99

0.72 1.58 1.46 0.16 1.05 0.9 0.72 1.05 1.3 2.47 0.56 1.81 1.96 0.62 0.9

0.42 0.3 0.27 0.21 0.16 0.55 0.56 1.23 0.53 4.92 1.67 1.36 3.53 1.99 1.81

10.32 0.21 10.58

- 10.33

9.8 14.22

8.14 6.46 6.79 7.92 7.75 12.66

5.26 6.64

∑ (

^{Xexp . − calc.} i

)

^{/ Xexp ×100}

Fluid Number of Points Range of temperature(K) (bar) Range of pressure Density Internal energy Enthalpy Entropy Cv Cp Speed of sound Joule-Thamson

i

a AAD = ⁱ⁼¹

N

References

1. Song, Y. and Mason, E. A., J. Chem. Phys. 91, 7840 (1989) 7840-7853.

2. Tao, F. M. and Mason, E. A., J. Chem. Phys. 100, 9075 (1994) 9075-9087.

3. Tsonopoulos, C., Fluid Phase Equilib. 211, 35 (2003).

4. E. W. Lemmon, M. O. Mclinden, D. G. Friend: "Thermophysical properties of fluid systems" in NIST chemistry webbook, NIST Standard Reference Database Number 69, eds. W. G. Mallard and P. J. Linstrom November (2005), Nat. Inst. Stand. Tech.

Gaithersburg, MD20899 http://webbook.nist.gov