Thermodynamic Processes and Equations!
A process is a change in the state of a gas as a result of flow of energy. During this flow a change takes place in properties of the substance such as pressure, volume, temperature and also the energy quantities such as internal energy, heat and work.
The five important processes commonly dealt with in engineering are as under: 1. Constant Volume Process or Isochoric Process 2. Constant Pressure Process or Isobaric Process 3. Constant Temperature Process or Isothermal Process 4. Adiabatic Process or Isentropic Process 5. Polytropic Process 6. Hyperbolic Expansion 7. Free Expansion 8. Throttling Process.
The graphical representation of these processes which show simultaneous values of pressure and volume are called curves and series of these curves are plotted in such a manner that they form a diagram which is known as P-V diagram.
In our further discussion the suffix 1 indicates the condition of the gas at the beginning of the process and the suffix 2 indicates the condition of the gas at the end of the process. The mass of the gas will be denoted by m. Thus P1, V1 and will be the pressure, volume and temperature of certain quantity of a gas at the beginning of the process and will be the pressure, volume and temperature of the same quantity of a gas at the end of the process.
If the process is of expansion, then the ratio V2/V1 is known as the ratio of expansion. If the process is of compression, then the ratio V1/V2 is known as the ratio of compression. Generally, the ratio of expansion or compression is denoted by the letter “r”.
1. Constant Volume Process:
(a) Representation on P-V and T-ɸ diagram:
2. Constant Pressure Process:
Isothermal process is one in which the temperature of the working substance remains the same. In this process the temperature remains constant during the process. This process follows Boyle’s law.
(a) Representation on P-V and T-ɸ diagram (fig. 2-16):
Since the temperature remains constant during this process, by Boyle’s law,
PV = constant.
This leads us to the fact that the curve showing relation between P and V on P-V diagram (fig. 2-16) is a rectangular hyperbola. The diagram will be rectangle.
An isothermal process on the temperature entropy diagram is represented by the horizontal line ab as shown in fig. 2-16. The cross-hatched area under the line ab represents the total heat added.
In this process the gas neither receives nor gives out heat. We may imagine that the gas is contained in a cylinder made of ideal non-conducting material. The gas can then be expanded or compressed but no heat can either be given to or taken from the gas.
Students very frequently misunderstand the meaning of isentropic (adiabatic) expansion or compression. During expansion, the temperature falls and there is loss of internal energy but no heat is lost in the form of heat from the system. Fig. 2-17 shows a frictionless totally isolated cylinder and piston mechanism of heat engine.
In the isentropic process, the gas do no loose any heat to the surrounding and is completely isolated system by use of ideal insulating materials. If the friction is present in the mechanism then such a process is called Adiabatic process and the isentropic process is a frictionless adiabatic process.
By first law of thermodynamics as applied to non-flow process,
heat supplied = change in internal energy + work done.
Let us consider unit mass of a gas.
Let dQ denote a small quantity of heat taken in by unit mass of a perfect gas and dT and dv being the resulting small increments of temperature and volume respectively; then according to first law of thermodynamics,
Thus, we get in equation which is followed by a gas when it expands or compresses isentropically.
(b) Representation on P-V and T-ɸ Diagram:
The isentropic process on P-V diagram is represented as shown in fig. 2-18. The T-ɸ diagram shows the entropy remains constant.
As usual, we get
change in internal energy = U2 – U1 = m x Cv (T2 – T1).
Another way of determination of change in internal energy is very common in isentropic operation.
By first law of thermodynamics as applied to non-flow process,
heat supplied = change in internal energy + work done; but heat supplied is zero.
∴ Change in internal energy = – work done.
Thus, we get an important relation in an isentropic process. This relation can be stated as “Change in internal energy is numerically equal to work done”. When the work is done by the gas, it loses internal energy and it gains internal energy when the work is done on the gas.
During an isentropic operation the transferred heat is zero.
Δ Q= 0.
Change in enthalpy = H2 – H1 = m x Cp (T2 – T1).
The change in entropy is 0 because dQ, = 0.
dɸ = ɸ2 – ɸ1 = 0.
As for frictionless isentropic process the heat added or removed will be zero, the change in entropy will also be zero. Hence, we get ɸ1= ɸ2; in other words we say that during frictionless isentropic process the entropy remains constant and its representation on a temperature entropy diagram will be a vertical straight line ab as shown in fig. 2-18.
The frictionless isentropic process is called an isentropic process, since the entropy remains constant. It is seen from fig. 2-18 that no area exists under the vertical line representing isentropic operation.
5. Polytropic Process:
Many processes which occur in practice can be described approximately by an equation of the form PVn = constant, where n is a constant. Such a process is called a polytropic process.
(a) Relation for Polytropic Processes:
Each of the four processes which have been
previously considered is a special case of a polytropic process. The main difference in equations of isentropic and polytropic process is that if we replace γ by n in the relations of isentropic operation, we get relation for polytropic processes.
When these values are plotted on P-V diagram as shown in fig. 2-19, it can be seen that they form a family of curves.
6. Hyperbolic Expansion:
When a gas expands or compresses in such a manner that the product of pressure and volume remains constant during the whole of the expansion or compression, the expansion is known as hyperbolic expansion because such a process on the P-V diagram will give a rectangular hyperbola. Hence a hyperbolic expansion is one which follows the law, pressure × volume — constant.
∴ PV = constant.
For perfect gases this operation is the same as the isothermal operation. Fig. 2-22 shows PV diagram for hyperbolic expansion.
When a fluid is allowed to expand suddenly into a vacuum chamber through an orifice of large dimensions, the free expansion of a gas occurs. During this operation no heat has been supplied or rejected and no external work has been done. Putting these zero values in First Law of Thermodynamics as applied to non-flow process, ΔQ= ΔU + ΔW we get ΔU = 0 i.e. U1 = U2.
Thus internal energies before and after free expansion are equal. Fig. 2-23 shows free expansion.
The throttling process is an expansion process in which the pressure reduces after expansion and the velocity is negligible. A throttling expansion occurs when a gas or a vapour is expanded through an aperture of minute dimensions such as a throat or a slightly opened valve.
During the throttling process the expanding fluid is forced through the aperture by its pressure but the hole is so narrow that the frictional resistance between the fluid and the wall reduces the fluid velocity to a negligible amount; as a result, the fluid escapes with a small amount of kinetic energy.
Due to friction kinetic energy reappears as heat and the gas is raised to its initial temperature. During a throttling process no heat is supplied or rejected, no external work is done and in the case of a perfect gas there is no alteration in temperature.
When a fluid expands through a throttle valve or a constricted orifice, the enthalpy before the throttling valve is equal to the enthalpy after throttling. This does not mean, however, that the throttling process is constant enthalpy process. Here the properties before and after the process are defined. Fig. 2-24 shows throttling process.
H1 = H2.
In this process the enthalpy remains constant. The enthalpy at condition 1 is equal to enthalpy at condition 2.