Solve problems using data extracted from thermodynamic property tables

In this section we are concerned with the interpretation and extraction of data on thermodynamic properties of working fluids listed in tables as arranged by Rogers and Mayhew.

These tables, commonly known as ‘steam tables’, give values for the properties of steam and refrigerants over an extensive range of pressures and temperatures.

The ability to understand and extract data from the tables extends into the solution of problems in this section and also later when the steady flow energy equation is dealt with.

Before examining the tables, however, definitions need to be attached to specific thermodynamic quantities listed.

Specific volume

This is the volume occupied per unit mass (1 kg) of a substance and is identified by the symbol v.

i.e $v=\frac{\text{VOLUME}}{\text{MASS}}=\frac{V}{m}$ 

It is measured in m³kg^-1

Thermodynamic tables give the specific volume of dry saturated steam at a particular pressure under the heading .

For example:

SPECIFIC VOLUME OF DRY SATURATED STEAM AT 1.4 bar = 1.236m³kg^-1

For superheated steam the specific volume is read against the symbol v for different pressures and temperatures.

For example

SPECIFIC VOLUME OF SUPERHEATED STEAM AT 6 bar and 250°C = 0.3940m³kg^-1

Internal energy

A fluid may be defined as a substance or a mixture of substances in the liquid or gaseous state. All fluids consist of large numbers of molecules that move in random directions at high speeds. Each molecule possesses a minute amount of kinetic energy and the total kinetic energy possessed by all the molecules is known as the internal energy of the fluid.

When heat energy, which is a transient form of energy, is transferred to a fluid, the temperature and molecular activity of the fluid increases. These increases result in a corresponding increase in the store of internal energy within the fluid.

As a result of experimental work on this subject, Joule concluded that the internal energy of a fluid is a function of temperature only and is independent of pressure and volume (Joule’s Law).

For there to be a change in the internal energy of a fluid there must be a change in temperature.

The symbol used for the total internal energy in a fluid is U and its unit is the Joule (J).

Generally, the internal energy of a fluid is quoted as per unit mass (per kg). This quantity is known as specific internal energy and takes the symbol u.

The unit for specific internal energy is the Joule per Kilogram (J kg^-1).

Thermodynamic tables give three values for the specific internal energy of steam as shown:

Flow or Displacement Energy

Any volume of fluid entering or leaving a system must displace an equal volume ahead of itself in order to enter or leave the system as the case may be.  

If we let the mass of fluid between X and Y in the figure below have a total Volume V₁. For flow to occur, this volume must be displaced by an equal volume from outside the system. If the pressure in the fluid is p₁, then the work done on the fluid inside the system by the incoming fluid = force × distance the fluid is displaced.

© A.Henderson, UHI

Because force is pressure times cross sectional area, and work done is force times distance, that means work done is pressure times area times distance:

$F=pA$ 

$W=Fd$ 

$\text{∴ }W=pAd$ 

But because cross sectional area times distance is equal to the volume of fluid displaced, that means work done is equal to cross sectional area times volume of fluid displaced:

$W=p_1{V}_1$ 

Or, in specific terms (work done per kilogram), the work done on the system can be calculated by:

$W=p_1{v}_1$ 

This is variously called flow energy, displacement energy or pressure energy depending on who is speaking.

At entry energy is received by the system.

At exit energy is lost by the system.

Enthalpy

In steady-flow thermodynamic systems, internal energy and flow energy are present in the moving fluid. Accordingly, it is convenient to combine these energies into a single energy quantity known as enthalpy, thus

Total Enthalpy = Internal Energy + Flow Energy

The symbol for enthalpy is H

H=U+pV

The unit for enthalpy is the Joule.

When considering 1 kg of working fluid, then:

Specific enthalpy h takes the unit The Joule Per kg (J kg^-1).

Thermodynamic property tables give four values for the specific enthalpy of steam as listed below:

• h_f is specific enthalpy of saturated water
• h_fg is specific enthalpy of evaporation
• h_g is specific enthalpy of dry saturated steam
• h is specific enthalpy of superheated steam

The Formation of Steam

Consider a quantity of water initially at 0°C being heated in a vessel fitted with a movable piston such that a constant atmospheric pressure can be maintained in the vessel. If the water is heated until it has all been converted to steam then the temperature/time graph would look like this:

© A.Henderson, UHI

During the stage A to B sensible heat energy flows to the water accompanied by a rise of temperature. At B the water boils at a temperature referred to as saturation temperature. This temperature depends on the pressure in the vessel and is 100°C at atmospheric pressure. The energy required to produce this temperature rise is called the ‘liquid enthalpy’.

During stage B to C steam is being formed whilst the temperature remains constant and the contents of the vessel will be a mixture of water and steam known as ‘wet steam’. At point C the steam will have received all the heat energy required to convert the water completely to dry steam. The energy required to produce the total change from all water to all dry steam is called the ‘enthalpy of evaporation’.

When completely dry saturated steam has been formed at saturation temperature, further transfer of heat energy will produce superheated steam which will be accompanied by a rise in temperature. The amount of heat energy in the superheat phase is called the ‘superheat enthalpy’.

Steam, therefore, can exist in three states: wet, dry, or superheated. Values for specific enthalpy, specific internal energy, and specific volume may be obtained directly from thermodynamic property tables for dry and superheated steam. For wet steam, it is necessary to know the degree of ‘dryness’, or the dryness fraction, of the steam before the various properties can be calculated.

Dryness Fraction

The degree of dryness, or dryness fraction, of steam is that proportion of a given mass of water which has been evaporated to form steam.

The dryness fraction may have any value from 0 (corresponding to boiling water) to 1 (corresponding to dry saturated steam). For example, steam with a dryness fraction 0.6 means that for each kg of water, 0.6 will be steam and 0.4 kg of saturated liquid.

The symbol x is used to represent dryness fraction:

$\text{Dryness fraction, }x=\frac{\text{Mass of dry steam}}{\text{Total mass of Steam and Moisture}}$