Short Notes #3

 The following is a summary of reactor design equations discussed in previous entries:

PVT DIAGRAMS FOR PURE SUBSTANCES

 The Temperature-Specific Volume Diagram

To obtain this diagram we repeat the process discussed in the last blog entry (Click Here) at different pressure values. The resulting curves for water look as shown below:

It can be observed that with increasing pressure the horizontal line connecting saturated liquid and saturated vapor states becomes shorter. The reason is that as pressure increases, the specific volume of saturated liquid increases and the specific volume of the saturated vapor decreases. At P = 22.09MPa, the horizontal line between the saturated liquid and vapor states shrinks to a point at which the constant pressure line forms an inflection point with a slope = 0. This point is referred to as the critical point.

At a critical point, the saturated liquid and saturated vapor states are identical and the temperature, pressure and specific volume of a substance at this point are called the critical temperature, critical pressure and critical volume, respectively. When the pressure is above the critical pressure, a liquid and vapor phase of a pure substance does not exist in equilibrium.

The saturated liquid states can be connected by a line called the saturated liquid line while saturated vapor states are connected by a line called the saturated vapor line. These two lines meet each other at the critical point, forming a dome as shown:

All the subcooled liquid states are located in the region to the left of the saturated liquid line ad is referred to as the subcooled liquid region.

All the superheated vapor states are located to the right of the saturated vapor line and this is called the superheated vapor region.

In these two regions locates outside the dome, a pure substance exists either in liquid or vapor phase (single phase)

The region under the dome is called the saturated liquid-vapor mixture region where the liquid and vapor phases are in equilibrium.

The Pressure-Specific Volume Diagram

For a pure substance, the Pressure-Specific Volume diagram is similar to that of Temperature-Specific Volume diagram, however, the isotherms (constant temperature) lines have a downward trend as can be seen below:

The Pressure-Temperature (P-T) Diagram

Pure substances can exist as solids, liquids or as a vapor. The P-T diagram is a graphical method of showing the effects of pressure and temperature on the phases of a pure substance. It is referred to as the phase diagram with the three phases separated from one another by three lines as shown:

The curve that separates the solid and vapor phases is called the sublimation curve, and along it the solid and vapor phases are in equilibrium. The slope of the sublimation curve gives the rate of change of sublimation pressure of the solid with temperature.

The curve that separates the solid and liquid phases is called the fusion (or melting) curve, and along it the solid and liquid phases are in equilibrium. Its slope gives the rate of change of melting or freezing of solid with temperature. The fusion curve has a positive slope for most substances but water has a negative slope.

The curve that separates the liquid and vapor phases is called the vaporization curve, and along it the vapor and liquid phases are in equilibrium. Its slope gives the rate of change of vapor pressure of liquid with temperature. This curve ends at the critical temperature and pressure of the substance.

At temperatures and pressures higher than the critical values, substances are called supercritical fluids i.e., they exist in the fluid (or supercritical) region. They possess both the gaseous properties (viscosity, diffusivity, surface tension) of being able to easily diffuse into substances, and the liquid property (density) of being able to dissolve substances.

When P < P_c, a substance in the gaseous state is called either a gas (T > T_c) or a vapor (T < T_c). Under isothermal conditions, while a vapor can be liquefied by exerting pressure, a gas cannot be liquefied regardless of what pressure is applied to it. That is, a pure gas cannot be liquefied at temperatures above its critical temperature no matter what pressure is applied to it.

On the phase diagram, the point where the solid, liquid, and vapor phases coexist in equilibrium is called the triple point. This is where the liquid-vapor (vapor pressure curve), solid-liquid (fusion or melting curve), and solid-vapor (sublimation pressure curve) coexistence curves intersect. The number of degrees of freedom at the triple point is zero.

Since the fusion curve generally has a very steep slope, the triple point temperature for most substances is close to their melting (or freezing) temperature at atmospheric pressure and this is known as the normal melting (or freezing) point.

 

Phase Change of a Pure Substance

A pure substance is a substance that has a fixed chemical composition throughout. It can exist in more than one phase, but the chemical composition must be the same in all phases. An example of this is ice-liquid water mixture and liquid water-steam mixture.

Scenario: Let us consider a system consisting 1 Kg of liquid water in a piston cylinder apparatus as shown below. Assume that the ambient pressure and the piston weight maintain the cylinders pressure at 0.125MPa with the initial temperature at 25°C.

At the initial conditions (P = 0.125MPa, T = 25°C), water is a called subcooled liquid i.e., it will not vaporize if heat is transferred to the system. 

When heat is transferred to the water, its temperature increases significantly and the specific volume increases slightly while pressure remains constant. 

When T = 105.99°C, the additional heat transfer results in the water boiling and a phase change occurs. A liquid that is about to boil is referred to as a saturated liquid. 

When the liquid is vaporizing, its pressure and temperature remain constant but its specific volume increases. At this point it is called a saturated liquid-vapour mixture where both liquid and vapour phases coexist in equilibrium. 

When all liquid vapourizes, only vapour exists in the cylinder and is called saturated vapour. Any heat loss from a saturated vapour leads to condensation. 

An increase in heat into a saturated vapour leads to an increase in both temperature and specific volume and its called a superheated vapour.

  

This whole water heating process at a constant pressure can be represented on a T - Ṽ diagram as shown below:

• State 1 is the initial subcooled liquid state

• State 2 is the saturated liquid state. A saturated liquid is ready to boil with the addition of heat, and T = 105.99°C represents the boiling point temperature.

• State 4 represents a saturated vapor. A saturated vapor is ready to condense with the removal of heat, and T = 105.99°C represents the dew point temperature.

• The horizontal line joining states 2 and 4 represents an isobaric and isothermal process where phase change from liquid to vapor, or vice versa, occurs. During phase change the liquid and vapor phases are in equilibrium with each other and, as a result, both temperature and pressure remain constant.

• The line joining the states 4 and 5 represents the process in which the steam is superheated at constant pressure.

At any given pressure, the temperature at which a pure substance boils is referred to as the saturation temperature (T^sat). While at any given temperature, the pressure at which a pure substance boils is referred to as the vapour or saturation pressure (P^vap). For a pure substance, there is a definite relation between the vapour pressure and the saturation temperature which results in a vapour pressure curve shown below. The boiling of a pure component starts when the P^vap = Ambient Pressure. This explains why water boils at a temperature less than 100°C at a mountain top.