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Journal of the Chilean Chemical Society

versão On-line ISSN 0717-9707

J. Chil. Chem. Soc. v.51 n.4 Concepción dez. 2006

http://dx.doi.org/10.4067/S0717-97072006000400011 

 

J. Chil. Chi. Soc., 51, N°.4 (2006), p.1044-1048

 

LIQUID-VAPOR EQUILIBRIUM IN THE CRITICAL REGION

 

DR. MIGUEL N. ALVAREZ1,DRA. ELIZABETH J. LAM2, MARÍA E. GALVEZ3

1 Department of Chemical Engineering, Faculty of Engineering, Universidad de Antofagasta, Campus Coloso s/n, Antofagasta, Chile, malvarez@uantof.cl
2 Department of Chemical Engineering, Faculty of Engineering and Geological Sciences, Universidad Católica del Norte, Avda. Angamos 0601, Antofagasta, Chile, elam@ucn.cl
3 Department of Chemical Engineering, Faculty of Engineering and Geological Sciences, Universidad Católica del Norte, Avda. Angamos 0601, Antofagasta, Chile, mgalvez@ucn.cl


ABSTRACT

In the various models existing for the determination of thermodynamic properties, it is common to find references to the difficulties in calculations carried out in the critical region. This situation presents even more problems in relation to calculations related to the saturation zone in the critical region. Experimental measurements in this region also present serious difficulties becauseexperimentalinformation relative to this isscarce. This study presents an analysis of the liquid-vapor equilibrium in the critical region based on the IAPWS-IF97 model, which was determined withexperimental data for water.

Keywords: Thermodynamic properties, líquid-vapor equilibrium, crítical region


INTRODUCTION

The calculation of thermodynamic properties of the liquid-vapor equilibrium in the critical region requires adapted procedures which are based on the behavior of these properties in this region. An example of this situation becomes evident upon reviewing the IAPWS model for water, which is one of the most refined models which exists for determining thermodynamic properties. In effect, this model covers a range of pressures from zero to 100 MPa, and a temperature range between 273.15 K and 2273.15 K , and divides this range into five different zones. For three of these zones, the model is based on equationsfor the Gibbs free energy as a function of the pressure and temperature, which relates pSat and TSat in the saturation zone, and one equation which relates the Helmholz free energyto the density and temperature in the critical region.

2. MODEL FOR THE LIQUID-VAPOR EQUILIBRIUM IN THE CRITICAL REGION

The IAPWS-IF97 model [1, 2, 5] for the saturation line for water represents the entire liquid-vapor saturation from the triple point to the critical point. This model is able to use an equation for the explicit calculation of the saturation pressure when the temperature is known or even an equation for the explicit calculation of the saturation temperature if the pressure is known. Also, an equation is available which relates the Helmholz free energy with the density and temperature in the critical region. By an adequate combination of these equations it is possible to determine the thermodynamic properties ofsaturated water in the critical region.

2.1 Equation for the saturation pressure (pSat)

The equation for the saturation pressure pSat as a function oftemperature is:

The values of ni are included in Table 1. Equation (1) completely covers the range of the liquid-vapor saturation curve from the temperature at the triple point to the critical point, which is:

 

2.2 Equation for the saturation temperature (TSat)

The equation for the saturation temperature TSat as a function ofpressure is:

The values of ni are included in Table 1. Equation (7) completely covers the range of the liquid-vapor saturation curve from the pressure at the triple point to the critical point , which is:


2.3 Equations for the determination of thermodynamic properties in the critical region.

The IAPWS-IF97 model [1, 2, 5] divides the range of application into five different regions. Of these, one is that ofliquid-vapor saturation , whose equations correspond to those presented in sections 2.1 and 2.2.Three other regions exist, for which the model is based on the Gibbs free energy as a function of pressure and temperature. For the critical and supercritical regions the equation differes from the other regions in that it is an equation for the Helmholtz free energyf as a function of density and temperature.

ni, Ii, Ji= IAPWS coefficients and exponents included in Table 2.


Equation (14) covers the following temperature ranges:


All the thermodynamic properties can be derived from equation (14) according to the equations listed in Tables 4 and 5.



Starting with the equations of theIAPWS model, calculation can be made of the thermodynamic properties in the critical region.

3. RESULTS

According to equation (17), the lower limit for temperature in equation (14) is T = 623.15 K. For this reason, equation (14) for the liquid -vapor equilibrium in the critical zone is valid in the range:

623.15 K £ T £ Tc (21)

The model must deliver results of the thermodynamic properties for the liquid and for the vapor for this range of temperatures. If we consider the saturation temperature in this range as a datum, and at a point very close to the critical point , then:

Tsat = 625.15 K

Therefore the pressure at saturation can be obtained from equation (1):

Psat = 16.93914 MPa

Equation (14) can be resolved for T = 625.15 Kand different density values, giving the pressure as a result. The calculated values are presented in Figure 1.


This figure 1 shows that for a pressure p = 16.93914 MPa three solutions exist for the density. These results are listed in Table 6.


The behavior shown in Figure 1 is repeated over the range given byequation (21). It is obviously necessary to have a criterion which allows discrimination among correct and incorrect solutions for the liquid-vapor equilibrium. One of these is the phase equilibrium condition based on the Gibbs free energy:

Another alternative for selecting the correct solutions is by using the Maxwell criterion for a phase equilibrium condition given by the following equations:

The results obtained based on equation (14) for equations (25) to (27) are given in Table 7, showing that the correct combination is that given by equations (23) and (24).


The proposed method can be applied in the surroundings of the critical region to obtain the thermodynamic properties of the liquid and saturated steam. The results are in Table 8 and can be compared with obtained values of steam tables.


If the equations and procedures described for the range indicated by (21) are applied, the thermodynamic properties shown in Figures 2, 3 and 4 are obtained.




4. CONCLUSIONS

The IAPWS model separately presents the equations which represent the relation between pSat and TSat in the saturation region and the equations for determining the thermodynamic properties in the critical region.

By correct combination of these equations, it is possible to determine the thermodynamic properties of the liquid-vapor equilibrium in the critical region.

In the critical region, the model which allows calculation of the thermodynamic properties is that for the Helmholtz free energy as a function of the density and temperature. In the saturation zone, once the temperature and pressure are known, the equilibrium properties of the liquid-vapor equilibrium must be calculated in an iterative form.

In the saturation zone of the critical region, the IAWPS model delivers three solutions for a given temperature. The selection of the correct solutions can be carried outbeginning with the phase equilibrium based on the Gibbs free energy, or on Maxwell's equilibrium criterion

The proposed method calculates the thermodynamic properties of the liquid-vapor equilibrium in the critical region with errors non superior to the 0,3%. It has the advantage to determine all the thermodynamic properties from 3 equations and that the iterative methods are simple and easy to obtain their convergence.

ACKNOWLEDGMENTS

The authors thank the "International Association for the Properties of Water and Steam (IAPWS)" for the for the authorization given for the use and dissemination of the IAPWS coefficients and exponents from the IAPWS-IF97 model.

 

REFERENCES

1. Alvarez M., Barbato S., "Calculation of the Thermodynamic Properties of Water Using the IAPWS model", J. Chil. Chem. Soc., 51, Nº 2 (2006).         [ Links ]

2. IAPWS, The International Association for the Properties of Water and Steam, www.iapws.org         [ Links ]

3. Preston-Thomas, H., 1990, "The International Temperature Scale of 1990 (ITS-90)", Metrologia, Vol 27, pp. 3-10         [ Links ]

4. Setzman, U., and Wagner, W., 1989, "A New Method for Optimizing the Structure of Thermodynamic Correlation Equations", Int. J. Thermophysics, Vol 10, pp 1103-1126.         [ Links ]

5. Wagner, W., Cooper, J.R., et al, 2000, "The IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam", Journal of Eng. For Gas Turbines and Power, Vol. 122, pp. 150-182.         [ Links ]

6. Perry, R. H., 1992, "Chemical Engineers" Handbook", McGraw-Hill

 
        [ Links ]

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