Introduction to thermodynamics

This section is meant for those readers, who so far have no basic knowledge of  thermodynamics, but who would like to be able to follow the thermodynamic considerations in the following chapters. For further information on thermodynamics we recommend for example [2]

The thermodynamic condition of state of a gas is described by 3 parameters. These  parameters are linked by a function, so that only 2  parameters (eg pressure and temperature)  are necessary to define a condition of state.

Over a large range of conditions this functional link  is given with a good approximation by the simple  characteristic equation of an ideal gas :

Thereby R is  called the characteristic gasconstant. Ris depending on the type of gas.The product of  the characteristic gas constant and the relative molecular mass is  a constant for all gases.

The thermodynamic behaviour of a gas is dependent on its isentropic exponent kappa. The value of kappa is dependent on the number of atoms in the molecule of the gas.

The deviation of a gas from the ideal gas becomes the greater, the nearer the gas condition is to the liquid state. This is then called real gas behaviour.

Order to understand the working principles of compressors the assumption of ideal gas behaviour is sufficient. We  therefore limit ourselves to this behaviour in the following chapters.

Through external influences (on compressors by input of work or extraction of heat)  the gas changes its thermodynamic condition of state.

Changes of state  without any losses are reversible and are described by the polytropic equation. The polytropic exponent n characterises this change of state.

In order to evaluate such changes, parameters for changes of state  for energy and processes are used.  For the evaluation of the the compression of gases the parameters technical work and heat , also the thermal parameters enthalpy and entropy are preferentially used.

For each change of state the law of conservation of energy (first law of thermodynamics) applies. This  law says , that the input  of technical work plus  heat  equals the increase in enthalpy of the gas. When compressing gas and applying cooling, the rise in temperature of the gas is proportional to the difference between the input of work and heat removed.

In order to evaluate energywise changes of state  the following condition of state  diagrams are used :

The p,v-diagram showsthe reduction in volume when compressing from condition 1 to condition 2. This reduction is the greater, the smaller the  polytropic exponent n .The technical work is given as the area below the compression curve and the ordinate.

The T,s-diagram shows the temperature rise when compressing gas. If the gas is being cooled during compression, then n < kappa and the entropy reduces.

When heat is added  (for example through  friction of flow in a turbo compressor rotor ) , then n > kappa and the entropy increases. The amount of heat transferred or removed is given by the area under the curve.

These are  special cases  of the polytropic process and are of particular importance for gas compression :

Isothermal compression ( T = const )  represents a non achieveable limit of a compression with ideal cooling ( the  cooling of the compressor requires a sufficiently high temperature difference to the coolant ).

Isentropic compression  (s = const ) represents the limit of non cooled compression without internal losses.

The table gives the most important parameters for the evaluation of the two changes of state. It follows from the 1. law of thermodynamics, that the same amount of heat has to be removed as technical work was done. With isentropic compression the rise in enthalpy equals the technical work done.

The work done with isothermal compression represents the theoretical minimum of required work and is therefore taken as a base value to compare with  the actual work done .

The technical work done rises with the compression ratio at isentropic ( noncooled )  and isothermal ( ideally cooled) compression. The rise is the greater, the  higher the isentropic exponent.

For safety reasons the outlet temperature  is limited to a maximum of 135 to 200 deg. C , depending on the type of compressor. This  in turn limits  the allowable compression ratio in single stage compression.

However, multistage compression  with interstage cooling  is also used to save energy .

As an example we  examine an ideal two stage compression ( isentropic compression with total isobaric cooling back to suction temperature ).

The p,v  diagram shows the approximation to an isothermal change of state and the reduction in specific work done. The T,s  diagram shows the achieved reduction in outlet temperature.