## Graphical Representations Of The Equation Of State Of Water

Water is the prime example of a condensable substance, meaning that in many practical situations it can exist as a vapor or a liquid {and less frequently as a solid). Water is a goldmine of detailed thermodynamic data and applications of thermodynamic theory. Van Ness (1983) and Potter and Somerton (1993) provide very readable expositions of this topic with many illustrative problems.

The equations of state of water, both volumetric and thermal, exhibit features of the-gas equations of state described in Sections 2.2 to 2.4 the condensed substances EOS of Section 2.5. However, because water vapor is often used at high pressures in many processes (e.g., a steam power plant), and because the water molecule is inherently fairly nonideal, the ideal gas law is usually a poor approximation of the EOS of steam. Similarly, liquid water is quite compressible compared to most solids (see Table 2.1), so the simple condensed-phase EOS of Equation (2.18b) is of marginal use when applied to water.

### 2.6.1 Thermodynamic Surfaces

The EOS of water, like those of many other pure substances, expresses a property, pressure, for example, as a function of two other properties, temperature and volume. Thus, the EOS is given by the function p(T,v), which forms a surface in three-dimensional space. This type of graphical representation of the p-v-T properties of water is shown in Figure 2.8.

Represented on this thermodynamic surface are regions where water exists as a single phase, as two coexisting phases, or all three phases simultaneously. The light surfaces in Figure 2.8 depict the single-phase gas, liquid, and solid states. The shaded surfaces represent two-phase regions, the most important being the liquid-vapor portion. The lines labeled with letters are isotherms, or cuts through the surface at constant temperatures.

The "triple line" in Figure 2.8 represents the unique three-phase state of water. The designation of a triple line may appear to be a contradiction of the usual FIGURE 2.8 p-v-T surface of a condensable substance such as water.

terminology of a triple point introduced in Section 1.12. The explanation lies in the meaning of the volume coordinate axis in Figure 2.8. While there is no ambiguity about the pressure and temperature axes, the volume axis represents the mass-fraction-weighted average of the specific volumes of the phases present. Only in the single-phase regions does the volume denote that of a well-defined phase. In the two-phase liquid-vapor region, for example, the points along the volume axis are defined by:

where x = mass fraction of vapor in the mixture (also called the quality) vg = specific volume of the steam (gas) phase vf = specific volume of the liquid (fluid) phase

2.6.2 Two-Dimensional Projections of the EOS Surface

Better insight into the EOS of water is obtained by the projections on the three coordinate planes than by the three-dimensional thermodynamic surface. Figure 2.9 shows the p-v, the F-v, and the p-T projections of Figure 2.8.

The most commonly-used projection, the heavy bell-shaped curve in the F-v diagram (middle graph), is the envelope of the two-phase vapor-liquid region. To   FIGURE 2.9 Projections of the equation of state of water (above the triple line in the f-vgraph).

the left of this zone is the compressed-liquid single-phase region, and to the right of the envelope is the superheated-vapor region (labeled "gas" in the diagram). The left-hand and right-hand boundaries of the two-phase envelope are the locus of states of the saturated liquid and the saturated vapor, respectively. The maximum of the envelope is the critical point of water. At temperatures greater than that at this point, there is no distinction between liquid and vapor. Problem 2.9 is an example of analyzing a process using the T-v diagram.

The other lines in the T-v projection of Figure 2.9 are isobars for pressures ranging from 0.1 to 40 MPa. The slopes of the isobars in the compressed liquid region are all approximately the same and equal to the coefficient of thermal expansion of the liquid. In this range, Equation (2.18b) is a fair approximation of the EOS. In analogous fashion, isobars such as G-H are reasonably well described by the ideal gas law in the vicinity of H. However, near the saturation curve at G, a nonideal gas EOS such as the Van der Waals equation (Equation [2.3]) is required for an accurate analytic description of the p-v-T behavior of water vapor.

The isobars in the two-phase region of the T-v diagram of Figure 2.9 are horizontal. This means that the temperature is a function of pressure only, or equivalently, pressure is a function of temperature only. This p-T relationship in the two-phase region describes the vapor pressure of water. The p-T relationship is independent of the average volume of the mixture, which simply reflects the relative quantities of liquid and vapor, but does not affect the vapor pressure. For a one-component system with two phases present, the phase rule, Equation (1.21), permits only one degree of freedom. That is, only one thermodynamic property need be specified in order to fix all of the others.

Following the 0.1 MPa isobar in the T-v projection in Figure 2.9 illustrates the evolution of water as its average volume increases due to vaporization. Along the segment A-B, water is a single-phase liquid.* At point B, the liquid is just saturated with respect to the vapor at point C. Along the horizontal segment B-C, both temperature (100°C) and pressure (0.1 MPa) are constant. Moving from B to C represents vaporization of the liquid, terminating with the saturated vapor at C. At an intermediate point such as Z, the mass fraction of vapor in the mixture (i.e.,the quality of the mixture) can be calculated from the definition of the average volume given by Equation (2.17). Here vf and vs are, respectively, the specific volumes of the saturated liquid at B and the saturated vapor at C. The average mixture volume v corresponds to the abcissa value at point Z. The ratio of the line segments B-Z and B-C gives the quality at point Z:

BC vg-vf vg-vf

Equation (2.21) is an example of the lever rule applied to phase diagrams. The name lever rule arises from the determination of the fraction of a phase in a two-phase mixture (in this case the quality x) as the ratio of the lengths of horizontal line segments on the diagram.

From point C in the T-v projection, the 0.1 MPa isobar continues into the superheated (gas) single-phase zone. The p-v-T surface here is the same as that described above for the G-H isobar.

The p-T projection in Figure 2.9 (right-hand panel) appears simpler than the other two views because isochores (constant-volume lines) have not been superimposed on it. In addition, the two-phase envelopes, prominent features of the other projections, are viewed on edge, and so are collapsed in the p-T view to curved lines. Similarly, the triple line in the T-v projection regains its proper status as the triple point in the p-T projection. The simpler appearance of the p-T projection is due to the elimination of the volume as a represented variable in the diagram.

The line labeled L/V in the p-T projection gives the temperature dependence of the vapor pressure of liquid water, also called the vaporization curve. The S/V line is the sublimation curve. It represents the equilibrium pressure of water vapor over ice. The S/L line is called the melting line and gives the combinations of pressure and temperature for which solid and liquid water coexist at equilibrium. Wat»* is unusual among pure substances because its melting temperature is lowered as the pressure is increased. Most other substances behave in the opposite way; their melting

* The line AB in the T-v plot of Figure 2.9 is hidden in the saturated-liquid curve.

curves tilt to the right from the triple point rather than to the left. The unusual behavior of water is due to the higher density of the liquid compared to that of the solid.

Detailed analyses of the vaporization curve and the melting line are presented in Chapter 5. 