THERMODYNAMIC CYCLES CLASS NOTES FOR MECHANICAL ENGINEERING
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On July 31, 2014
THERMODYNAMIC CYCLES CLASS NOTES
FOR MECHANICAL ENGINEERING
A thermodynamic cycle consists of thermodynamic processes in series. As a result, system returns to its initial starting point.
(a) POWER CYCLES (HEAT ENGINE CYCLES) WITH EXTERNAL COMBUSTION
Cycle

12 process
(compression)

Process 2—3
(heat addition)

3—4 process
(Expansion)

Process 4—1
(Heat rejection)

Remarks

CARNOT CYCLE

ISENTROPIC

ISOTHERMAL

ISENTROPIC

ISOTHERMAL

IDEAL HEAT ENGINE CYCLE

BELL COLEMAN CYCLE

ADIABATIC

ISOBARIC

ADIABATIC

ISOBARIC

REVERSED BRAYTON CYCLE

ERICSSON CYCLE

ISOTHERMAL

ISOBARIC

ISOTHERMAL

ISOBARIC

RANKINE CYCLE

ADIABATIC

ISOBARIC

ADIABATIC

ISOBARIC

USED IN STEAM ENGINES

STIRRLING CYCLE

ISOTHERMAL

ISOCHORIC

ISOTHERMAL

ISOCHORIC

USED IN STIRLING ENGINE

(b) POWER CYCLES (HEAT ENGINE CYCLES) WITH INTERNAL COMBUSTION
BRAYTON CYCLE

ADIABATIC

ISOBARIC

ADIABATIC

ISOBARIC

USED IN JET ENGINES

DIESEL CYCLE

ADIABATIC

ISOBARIC

ADIABATIC

ISOCHORIC

USED IN DIESEL ENGINES

OTTO CYCLE

ADIABATIC

ISOCHORIC

ADIABATIC

ISOCHORIC

USED IN PETROL AND GAS ENGINES

© POWER CONSUMPTION CYCLES (REFRIGERATION AND AIR CONDITIONING CYCLES)
CYCLE

COMPRESSION

HEAT REJECTION

EXPANSION

HEAT ABSORPTION

REMARKS

VAPOR COMPRESSION REFRIGERATION CYCLE

ADIABATIC

ISOBARIC (ISOTHERMAL)

ISENTHALPIC

ISOBARIC (ISOTHERMAL)

REVERSED RANKINE CYCLE

VAPOR ABSORPTION REFRIGERATION CYCLE

ADIABATIC

ISOBARIC (ISOTHERMAL)

ISENTHALPIC

ISOBARIC (ISOTHERMAL)

REVERSED RANKINE CYCLE

GAS REFRIGERATION CYCLE

ADIABATIC

ISOBARIC (NONISOTHERMAL)

ADIABATIC

ISOBARIC(NON ISOTHERMAL)

REVERSED BRAYTON CYCLE

Efficiency of a cycle with one or more irreversible processes is less than the efficiency of Carnot cycle. This is due to lesser heat input at high temperature or more heat rejection at lower temperature. Mathematically express it as Clausius inequality.
Clausius Inequality
∮ dQ/T≤0
But ds = dQ/T
The equal sign is applicable only to the ideal cycle (Carnot cycle). In this, integral is equal to net change in entropy in one complete cycle. Therefore change of entropy is zero in an ideal cycle. This inequality is applicable to all heat engines in actual practice.
Prove Clausius Inequality Theorem
∮dQ/T ≤0
Proof
For an heat engine
η = ((Heat supplied at high temp–Heat rejected at low temp))/(Heat supplied at high temp)
= (Q_{H} — Q_{L}) / Q_{H}
For a reversible cycle
η = (T_{H} — T_{L}) / T_{H}
∴ (Q_{H} — Q_{L}) / Q_{H} = (T_{H} — T_{L}) / T_{H}
1– Q_{L}/Q_{H} = 1 — T_{L}/ T_{H}
Q_{L}/Q_{H} = T_{L}/ T_{H}
Q_{L}/ T_{L} = Q_{H}/ T_{H}
Clausius theorem in the integrated form around a reversible cycle becomes
∮dQ/T=0
Efficiency of an real cycle is less than that of a reversible cycle. Therefore the inequality sign comes into existence. Thus it becomes Clausius inequality.
∮dQ/T< 0
Use Van Dar Waals or an empirical equation in a real cycle.
VAN DER WAALS EQUATION
It is a real gas equation. It is not perfect gas equation. In includes two corrections.

First correction for the intermolecular attractions which are weak forces of attractions and repulsion between two molecules of the same substance.
we use p =Kinetic pressure –internal pressure =p’a/v^{2}
p = p’ –a/v^{2}
p’ = p + a/v^{2}
_{ }the value of constant ‘a’ will be different for different gases.’
Constant’ a’ represents the reduction in total pressure due to molecular attractive forces.

Second correction for the finite size of the molecules
We use vb instead v in the perfect equation
Therefore the Final Wan der Waal equation will become (p+a/v^{2})(vb) =RT
Or
p = [RT/(vb)] –a/v^{2}
p =pressure in N/m^{2}
v= molar volume = m^{3}/kg mole
T = Absolute temperature, K
R = Universal gas constant=8.314 k J/kg mole K
a = constant N m^{4}/(kg mole)^{2}
b =constant , m^{3}/kg mole
values of constants a and b will be different for different gases.
a and b are found from the critical properties of a gas under consideration.
a=(27/64) R^{2} T_{c}^{2}/p_{c}
b = RT_{c}/8p_{c}
Predicted values of constants ‘a’ and ‘b’ do not agree when found by different methods. Therefore there is a need to have a Van der Waals Equation free of constants ‘a’ and ‘b’. It was achieved with the use of reduced quantities.
REDUCED QUANTITIES are pressure, volume and temperature.

p_{r}=p/p_{c}

v_{r} = v/v_{c}

T_{r} = T/T_{c}
Put the values in terms of reduced quantities, the Van der Waals equation becomes
(p_{r}+ 3/V_{r}^{2}) (V_{r}—1/3) =(8/3) T_{r}
Now this Van der Waals Equation can be applied to any gas
COMPRESSIBILITY FACTOR ‘Z’ OR COMPRESSION FACTOR
At high pressures and low temperatures there is a big deviation in the ideal gas equation if used. It is because of that the volume of the gas becomes very large under high pressures and low temperatures. This deviation is expressed in terms of compressibility factor Z = pV/RT
Z = 1 for a perfect gas
Z>1 or Z<1 for a real gas.
At low pressures, Z =1 even for a real gas and there is no compressibility factor at low pressures.
This compressibility factor is due to inter molecular attractive and repulsive forces. These forces decrease with the increase of temperature and decrease of pressure. This factor ‘Z’ helps to find the difference between real gas and ideal gas behavior. It is an empirical factor. It is different for different gases. Z>1 at high pressures. The role of Z is more predominant near the phase change condition because of large variations in the resulting volumes.