LESSON - 19 p-V DIAGRAM, T-s DIAGRAM, p-T DIAGRAM, TRIPLE POINT AND CRITICAL POINT

 

19.1.  p-V DIAGRAM

19.1.1.  p-V diagram for water (solid-liquid-vapor region)

If we heat ice at different vapor pressures and note down the corresponding change in volumes, the saturation state points for solid, liquid and vapor (state from which a change of phase may occur without change of pressure and temperature) for different pressures may be obtained on a p-V diagram. 

By joining the saturated solid states at various pressures, a saturated solid line ‘ECA’ is obtained. Similarly, by joining all the saturated liquid states with respect to solidification and by joining all the saturated liquid states with respect to vaporization, saturated liquid lines ‘FG’ and ‘HG’ are obtained. Finally, by joining all the vapor states at various pressures saturated vapor line ‘HB’ is obtained and a phase equilibrium diagram of water on p-v co-ordinates will be formed as shown in Fig. 19.1.

Fig. 19.1. Phase equilibrium diagram of water on p-v co-ordinates.

The horizontal portion ‘AB’ of constant pressure or temperature in Fig. 19.1 represents the transition from saturated solid directly into saturated vapor called sublimation. There is obviously one such line ‘CGD’ in this figure, the part ‘CG’ of which is the boundary between the liquid-vapor region (L+V) and the solid-vapor region (S+V) and the remaining part ‘GD’ of which is boundary between the solid-liquid region (S+L) and the solid-vapor region (S+V). This ‘CGD’ line is called the triple point. Triple point is the only point at which three phases of a pure substance coexist. In the case of ordinary  water , the triple point  is at  a pressure of 4.58 mm of Hg and a temperature of 0.01^OC, and the line extends from a volume of 1 cm³/g (saturated liquid) to a volume of 206000 cm³/g (saturated vapor)

19.1.2. p-V diagram for water (Liquid-vapour region only)

Liquid is generally the working fluid in power cycles, therefore interest is often centered to the liquid-vapor region only. Fig. 19.2 shows p-v diagram for water and other pure substance indicating only liquid and vapor phases.

Introduce 1g of water at 94°C into a 2 litres vessel/system at 1 atmospheric pressure. Start evacuating the vessel to a pressure less than 1 atmosphere until the water in the vessel evaporates completely to a condition of superheated/unsaturated vapor. This condition of superheated steam is represented by the point ‘A’ on the p-v diagram in Fig. 19.2.

Now, if the superheated vapor in the vessel at state ‘A’ is compressed slowly and isothermally, the pressure will rise until there is saturated vapor at the point ‘B’. If the compression is continued, condensation takes place, the pressure remaining constant so long as the temperature remains constant. The straight line ‘BC’ represents the isothermal isobaric condensation of water vapor, the constant pressure being called the vapor pressure. At any point between ‘B’ and ‘C’, water and steam are in equilibrium; at the point ‘C’ there is only liquid water, or saturated liquid. Since a very large increase of pressure is needed to compress liquid water, the line ‘CD’ is almost vertical. At any point on the line ‘CD’ the water is said to be in the liquid phase, at any point on ‘AB’ in the vapor phase, and at any point on ‘BC’ there is equilibrium between the liquid and the vapor phases. ‘ABCD’ is a typical isotherm of a pure substance on a p-v diagram.

Fig. 19.2. Isotherms of a pure substance

Just like water at 94°C, at other temperatures the isotherms are of similar character, as shown in Fig. 19.2. It is seen that the lines representing equilibrium between liquid and vapor phases, or vaporization lines, get shorter as the temperature rises until a certain temperature is reached, the critical temperature. The isotherm at the critical temperature is called the critical isotherm, and the point that represents the limit of the vaporization lines is called the critical point. It is seen that the critical point is a point of inflection on the critical isotherm. The pressure and volume at critical point are known as the critical pressure and the critical volume respectively.

All points at which the liquid is saturated lie on liquid saturation curve, and all points representing saturated vapor lie on the vapor saturation curve. The two saturation curves denoted by dotted lines meet at the critical point. Above the critical point the isotherms are continuous curves. At large volumes and low pressures, they approach equilateral hyperbolas, i.e., the isotherms of an ideal gas.

19.2. T-s DIAGRAM FOR A PURE SUBSTANCE (CO₂)

The T-s diagram for a substance (CO₂) is shown in Fig. 19.3. The curve from A to F is a typical isobar representing a series of reversible isobaric processes in which solid is transformed finally into vapor. Thus,

AB - isobaric heating of solid to its melting point;  BC - isobaric isothermal melting; CD - isobaric heating of liquid to its boiling point;  DE - isobaric isothermal vaporization;           EF - isobaric heating of vapor (superheating). The area under the line BC represents the heat of fusion at the particular temperature, and the area under the line DE represents the heat of vaporization. Similarly, the heat of sublimation is represented by the area under any sublimation line. It is obvious from the diagram that heat of vaporization decreases as the temperature rises and becomes zero at the critical point and also that the heat of sublimation is equal to the sum of heat of fusion and the heat of vaporization at the triple point.

Fig. 19.3. T-s diagram for CO2. The two dashed lines  bounding the solid-liquid region are a guess

The T-s diagram for the liquid, liquid-vapor, and vapor regions of water and benzene are shown in Fig. 19.4 and Fig. 19.5, respectively. T-s diagram of water also show isobars, lines of constant quality, and lines of constant superheat.

During reversibly and adiabatic expansion (isentropic process) from point ‘A’ to point ‘B’:

• The saturated steam becomes wet (Fig. 19.4).

• The saturated benzene becomes unsaturated/superheated (Fig. 19.5).

Hence, the quality of pure substance after reversibly and adiabatic expansion (isentropic process) is different for all pure substances and it depends upon the type of pure substance expanding.

 

Fig. 19.4. T-s diagram for wet and for superheated steam.

Fig. 19.5. T-s diagram for benzene.

19.3. p-T DIAGRAM FOR WATER If the pure substance is heated at low pressure until its triple point (pressure and temperature at which three phases of a pure substance coexist) is reached and while heating the vapor pressure of a solid is measured at various temperatures and then plotted on a p-T diagram, shown in Fig. 19.6. These plotted points represent the coexistence of solid and vapor and the line through these points is called sublimation curve. If the pure substance at triple point is further heated until the critical point is reached and while heating the vapor pressure of a liquid is measured at various temperatures and then plotted on a p-T diagram, the results will appear as shown in Fig. 19.6. The results of these plotted points represent the coexistence of liquid and vapor and the line through these points is called vaporization curve.

Fig. 19.6. p-T diagram for a substance such as water.

On the other hand, if the substance at the triple point is compressed until there is no vapor left on the resulting mixture of solid and liquid phase and the pressure on the resulting mixture of solid and liquid is increased further, the temperature will have to be changed for equilibrium to exist between the solid and the liquid. Measurements of these pressures and temperatures give rise to a third curve on the p-T diagram, starting at the triple point and continuing indefinitely. This is fusion curve.

• The points on the sublimation curve represent the coexistence of solid and vapor.

• The points on the vaporization curve represent the coexistence of liquid and vapor.

• The points on the fusion curve represent the coexistence of liquid and solid.

In the particular case of water, the sublimation curve is called the frost line, the vaporization curve is called the steam line, and the fusion curve is called the ice line.

The slopes of the sublimation and the vaporization curves for all substances are positive. The slope of the fusion curve, however, may be positive or negative. The fusion curve of most substances has a positive slope. Water is one of the important exceptions. Any substance, such as water, which expands upon freezing, has a fusion curve with a negative slope (represented by solid line in Figure 19.6), where as opposite is true for substance such as CO₂, which contracts upon freezing i.e. the substance which contracts upon freezing, has a fusion curve with a positive slope (represented by dotted line in Figure 19.6). In other words, in the case of water, the freezing temperature decreases with an increase in pressure while for CO₂, the freezing temperature increases as the pressure increases.

19.4. TRIPLE POINT

The pressure and temperature at which three phases of a pure substance coexist is called triple point. The triple point is merely the point of intersection of the sublimation and vaporization curves, It has been found that on a ‘p-T’ diagram the triple point is represented by a point (Fig. 19.6) and on a ‘p-v’ diagram it is a line (Fig. 19.3), and on a ‘u-v’ diagram it is a triangle. In the case of ordinary water, the triple point is at a pressure of 4.58 mm Hg and a temperature of 0.01^OC.