# Thermodynamic Cycles

Prepared by:

Er. Ashok Subedi, Mechanical Engineer (Institute of Engineering)

Dr. Durga Bastakoti (Ph.D. in Mechanical Engineering)

Mostly the equipment which is employed for the conversion of energy operated on the cyclic process either for producing the output work by supplying the heat from the fuel combustion or for providing the effect of heating or cooling by delivering the work input via the electrical power.

## Classification of cycles

Cycles are classified into different groups based on a different basis. Some are explained below;

1. According to power: Based on the power, cycles are categorized into power and refrigeration cycles.
2. Power cycle: If the cycle provides the power to the surrounding when the cycle is being executed then the cycle is known as the power cycle. Brayton cycle, Otto cycle, diesel cycle, Rankine cycle etc. are the typical example of the power cycle. A heat engine is operated on the power cycle. So the network is always positive for the power cycle.
3. Refrigeration cycle: If power is required for executing the cycle, then the cycle is said to be the refrigeration cycle. Vapour compression refrigeration cycle is a typical example. Heap pump and refrigeration is operated on the refrigeration cycle, so the network is always negative for the refrigeration cycle.
4. According to working medium/ substance: Based on this, cycles are categorized as gas ad vapour cycle.
5. Gas cycle: If the working substance of the cycle remains in the gaseous state, then the cycle is said to be a gas cycle. Brayton cycle, Otto cycle, Diesel cycle are the typical example
6. Vapour cycle: If the working substance undergoes the change of phase, i.e. from liquid to vapour then the cycle is said to be vapour cycle. Rankine cycle is one of the common examples of vapour cycle.
7. According to the location of combustion: This method of classification is relevant to power cycle only. Based on this approach, there are internal and external combustion cycles.
8. Internal combustion cycles: If the combustion occurs inside the components of the system, then the cycle is said to be an internal combustion cycle. In this cycle, the fuel and the working substance comes in contact. Otto and Diesel’s cycles are common examples.
9. External combustion cycles: If the combustion occurs outside the component of the system, then the cycle is said to be external combustion cycles. In this cycle, the fuel and working substance do not come into contact. Brayton and Rankine’s cycles are common examples.

## Air Standard Analysis

The idealized analysis of the cycle where the air is considered as the working substance is said to be air standard analysis. Different assumptions are made during this approach which is listed as below;

• Mass of air during the cycle is fixed
• Process of expansion and compression are isentropic
• Heat addition process is considered instead of the combustion process for the external source.
• Heat rejection process is considered instead of the exhaust process
• Air properties remain constant

### Air Standard Internal combustion cycle:

It is the idealized model of the petrol and the diesel engines.

### Operation of Four-stroke engine:

The operation of the four-stroke engine completes in four different strokes, namely suction, compression, power and exhaust strokes.

### Operation of Four stroke petrol engine:

Petrol engines are also known as the spark ignition engine as the fuel is ignited by means of the electric spark which is produced by the spark plug. The operations of the petro engine are explained below;

1. Suction stroke: During this stroke, the piston moves from the BDC to the TDC, and the mixture of air and petrol enters the cylinder via the inlet valve which is open at the suction stroke whereas the outlet valve is closed.
2. Compression stroke: Before starting the compression stroke, the inlet valve is closed, and the mixture of air and petrol is compressed isentropically when the piston moves from the BDC to TDC. During this process, pressure and temperature increases while the volume decreased.
3. Power stroke: It is also known as the expansion stroke. At the end of the compression stroke, the spark plug creates the electric spark, and the mixture burns at high pressure and temperature. Because of the burning impact, piston expands isentropically from TDC to BDC.
4. Exhaust stroke: During this stroke, the exhaust valve is opened, and the burnt product inside the cylinder flows out of the exhaust valve to the surrounding.

During the compression and the power strokes, the inlet and exhaust valves remain closed.

## Operation of Four-stroke Diesel Engine

Diesel engine is also known as CI engine because diesel is exposed to the high temperature of the air at the end of the compression stroke which is greater than the self-ignition temperature of the diesel because of which the diesel gets self-burnt. The operation of the diesel engine is explained below;

1. Suction stroke: During this stroke, the piston moves from TDC to BDC, and the air enters into the engine cylinder via the inlet valve while the exhaust valve remains closed.
2. Compression stroke: Before starting the compression stroke, the inlet valve is closed, and the air is compressed isentropically when the piston moves from the BDC to TDC. During this process, pressure and temperature increases while the volume decreased.
3. Power stroke: It is also known as the expansion stroke. At the end of the compression stroke, the fuel injector injects diesel into the compressed air for the particular interval where the temperature of the air is greater than the self-ignition of the diesel because of which the diesel gets burns itself. Because of the burning impact, the piston expands isentropically from TDC to BDC.
4. Exhaust stroke: During this stroke, the exhaust valve is opened, and the burnt product inside the cylinder flows out of the exhaust valve to the surrounding.

## Air standard Otto cycle

It is the idealized model of the petrol engine which consists of two isentropic and two isochoric processes.

### Process 1-2: Isentropic compression

During this process, the piston will be at BDC, and the working substance gets compressed when the piston moves from BDC to TDC under the isentropic condition. At this process, the pressure and temperature increase while the volume decreases, and the entropy remains constant.

### Process 2-3: Isochoric Heat addition

A spark plug produces an electric spark at the end of the compression stroke where the petrol, i.e. fuel gest burn instantaneously so heat addition is considered. During this process, pressure, temperature and entropy increases and volume remain constant as it is an isochoric process.

### Process 3-4: Isentropic Expansion

During this process piston expand because of the burning impact and moves from TDC to BDC. Pressure and temperature decrease while the volume increases and the entropy remains constant.

### Process 4-1: Isochoric Heat Rejection

During this process, the exhaust valve open and the exhaust gas moved out of the cylinder thud rejecting the heat to the surrounding. Temperature, pressure and entropy decrease while the volume remains constant.

## Efficiency

The efficiency of the IC cycle is determined using the formulae:

η=1-q_L/q_H                .........    (1)

Where,

qL = heat rejected per kilogram of air per cycle

qH = heat added per kilogram of air per cycle

Heat addition during the process 2-3 is given as;

qH = q23 = CV (T3 – T2)… … …(2)

Heat rejection during the process 4-1 is given as;

qL = q41 = CV (T4 – T1)… … …(3)

Using these value, the efficiency equation becomes as;

η=1-[(T_4-T_1)/(T_3-T_2 )]    .........             (4)

To determine the efficiency of the air standard Otto cycle, it is necessary to know the temperature of each state.

From the process 1-2 and 3-4, the following relations can be used;

As we know for isochoric process V4 = V1 and V3 = V2 so the above equation becomes as;

Further, when equating the equations we get,

Using the componendo and dividend above equation becomes as;

Now using this relation in the efficiency equation we get,

Further in terms of volume ratio, the efficiency can be expressed as;

Where,

r=V_1/V_2 =compression ratio

The above equation illustrates that the efficiency has a direct relation with the compression ratio, i.e. efficiency increases with an increase in compression ratio. To get the high efficiency, it is desired to have high compression ratio however the high-pressure ratio causes the high temperature, and there is the restriction in real engine cylinder which can withstand the maximum temperature or pressure. In the real engine, the compression ratio lies between 8 and 12.

## An expression for Compression Ratio

To determine the expression for compression ratio firstly, we need to understand some terms related to the piston-cylinder, as shown in the figure. Stroke length is the distance between the TDC and BDC and denoted by LS, and the volume occupied in this space is called stroke volume (VS).

Similarly, the distance between the TDC and the head of the engine is called clearance space, and the volume occupied by this space is called clearance volume (VC). Then the compression ratio is given as;

Where the volume of stroke is as;

Air standard Diesel cycle

It is the idealized cycle for the diesel engine. Like Otto cycle, it also consists of four processes, namely isentropic compression, isobaric heat addition, isentropic expansion, and isochoric heat rejection.

### Process 1-2: Isentropic compression

During this process, the piston will be at BDC, and the cylinder contains air inside it. The air inside the cylinder gets compressed when the piston moves from BDC to TDC under the isentropic condition. At this process, the pressure and temperature increase while the volume decreases, and the entropy remains constant.

### Process 2-3: Isobaric Heat addition

At the end of the compression stroke, air temperature increases and become more than the self-ignition temperature of the diesel. Similarly, the fuel injector injects the diesel into the cylinder, and diesel gets burnt itself. However, the diesel did not get burn completely, and the piston moves from TDC to BDC simultaneously with the addition of heat. During this process, the pressure remains constant, while the temperature, volume and entropy increases.

### Process 3-4: Isentropic Expansion

During this process piston expand further because of the burning impact. Pressure and temperature decrease while the volume increases and the entropy remains constant.

### Process 4-1: Isochoric Heat Rejection

During this process, the exhaust valve opens and the exhaust gas moved out of the cylinder thud rejecting the heat to the surrounding. Temperature, pressure, and entropy decrease while the volume remains constant.

## Efficiency of Air standard Diesel cycle

The efficiency of the diesel cycle is given as;

η=1-(q_L/q_H)              .........      (15)

Where,

qL = heat rejected per kilogram of air per cycle

qH =  heat added per kilogram of air per cycle

Heat addition during the process 2-3 is given as;

Heat rejection during the process 4-1 is given as;

Using these two values in the efficiency equation, we get,

From the process 1-2, applying the relation of temperature and volume, we get,

Where,

 r=V_1/V_2 =compression ratio

From the process 2-3, applying the relation of temperature and volume, we get,

T_3/T_2 =V_3/V_2 =α         ...........     (20)

Further form process 3-4, applying the temperature-volume relation we get,

Substituting V4= V1, compression ratio and cut-off ratio in the equation (21) we get,

Further multiplying the equations (19), (20) and (22) we get,

Then substituting the equation (19), (20) and (23) in efficiency equation, we get,

## Mean effective pressure:

It is the parameter that is used to compare the various IC cycles, which are defined as the constant pressure magnitude which would produce the same work as that produced by the actual pressure varying cycle. Mathematically it is given as;

## Brayton cycle:

It is the idealized cycle for the power cycle, including the gas turbine. The process of this cycle is divided into four different stages, namely two isentropic and two isobaric.

### Process 1-2: Isentropic compression

Air having low temperature and low pressure is supplied at the heat exchanger with low temperature to the compressor where the air gest compressed and delivered to the heat exchanger having high temperature. During this process, the pressure and temperature of the working substance increases, volume decreases and entropy of the system remain constant.

### Process 2-3: Isobaric Heat addition

The external source is used for supplying the heat to the heat exchanger having high temperature where the working substance is heated at the constant pressure. During this process, specific volume, entropy and temperature all increases.

### Process 3-4: Isentropic expansion

The air having the high pressure and temperature leaves the heat exchanger with high temperature and then supplied to the turbine where the turbine uses this air for producing the work by consuming the energy of the air at the inlet of the turbine. During the process, pressure and temperature decreases while the specific volume increases further, and entropy remain constant.

### Process 4-1: Isobaric Heat Rejection

Air leaving the turbine is entered into the heat exchanger with the low temperature where heat is rejected to the sink with low temperature such that the initial stage of the system is restored. This process occurs at constant pressure where the specific volume, temperature, and entropy of the air decreases.

## Efficiency of the Brayton cycle:

The work produced by the turbine per kilogram of air is given as;

The work consumed by the compressor per kilogram of air is given as;

During the condition of steady-state, part of work produced by the turbine is used for running the compressor, so the work delivered to the surrounding is given by,

Then the heat supplied to the air at the heat exchanger with high temperature is;

Then the efficiency of the Brayton cycle is given as;

Further, the above equation (30) can be expressed as;

This equation indicates that for determining the efficiency of the Brayton cycle, the temperature of every stage is necessary. So we can further simplify this equation for reducing the number of needed variables. For this purpose, let us apply the relation of pressure and temperature for the process 1-2 and 3-4,

Replacing P3 = P2 and P4 = P1 into the equation (33) we get,

Now equating the equation (32) and (34) we get;

Then efficiency equation (31) becomes as;

Further, the above equation can be expressed in terms of pressure ratio as below;

Where,

r_p=P_2/P_1 =pressure ratio

The above equation indicates that the Brayton cycle efficiency increases with increase in the pressure ratio.

However, the high-pressure ratio increases the efficiency of the engine, but the high pressure causes the high temperature in reality as the blades of the turbine has the restriction of temperature which they can withstand. In reality, the pressure ratio of the turbine varies from 10 to 16.

## Rankine cycle

It is the idealized cycle for the steam engine, which is the power cycle. Like the Brayton cycle, it also consists of four processes where two are isentropic, and two are isobaric.

### Process 1-2: Isentropic Pumping Process

As it is the idealized cycle of the steam engine, so the working substance is liquid. The saturated liquid with low pressure is supplied to the high-pressure boiler from the low-pressure condenser using the pump where this process is considered as the isentropic process. During pumping, working substance is a liquid state, so its specific volume nearly remains while the pressure and temperature increases and the entropy remain constant.

### Process 2-3: Isobaric Heat addition

The external source is used for supplying the heat to the boiler, which produces the steam by heating the liquid water. The addition of the heat has occurred at the constant pressure where the specific volume, entropy and temperature increases where the steam is considered as a saturated vapour. However it the heat supply rate increase then the steam change into the superheated.

### Process 3-2: Isentropic Expansion

The steam produced by the boiler has high temperature and pressure is supplied to the turbine which produces the work consuming the energy carried by the steam. The process is considered as the isentropic expansion where the pressure and temperature decrease while the specific volume decreases and the entropy remain constant.

### Process 4-1: Isobaric Heat Rejection

During this process, the steam from the exit of the turbine is supplied to the condenser where there is the rejection of the heat from the steam to the surrounding, i.e. cooling water and gets condensed into the saturated liquid, i.e. initial stage. The process is considered to be working at the constant pressure where the temperature remains constant while the specific volume and the entropy of the steam decreases.

## Efficiency of the Rankine Cycle:

Work produced by the steam turbine per kilogram of the steam is;

Worked consumed by the pump per kilogram of the steam is;

During the condition of the steady-state, parts of the work produced by the turbine are used for running the compressor, so the work delivered to the surrounding is;

Similarly, the heat supplied to the steam in the boiler is;

Then the efficiency of the Rankine cycle is;

To determine the efficiency of the Rankine cycle, we need to know the enthalpy of all the stages.

## Vapour Compression Refrigeration cycle

It is the commonly used cycle for heating and cooling of the desired space. Refrigerant is the working substance for this cycle whose major components are shown in the figure:

### Process 1-2

The vapor which exit from the evaporator is supplied to the compressor where it gets compressed using the work input given by the electricity which is considered to be an isentropic process. During this process, pressure, temperature, and enthalpy increases, whereas the entropy of the refrigerant remain constant.

### Process 2-3

Refrigerant with high temperature and pressure is supplied to the condenser from the outlet of the compressor. During this process, the condenser interacts with its surrounding where the refrigerant rejects the heat to the surrounding. The condensation process of the vapor occurs at the constant pressure where there is a decrease in the entropy and enthalpy of the refrigerant.

### Process 3-4

The liquid with the high pressure at the condenser exit is supplied to the throttling valve where the enthalpy remains constant while the pressure and temperature decreases, and the entropy increases.

### Process 4-1

The refrigerant with low temperature and pressure enters into the evaporator from the outlet of the expansion. During this process, the refrigerant takes the heat from the surrounding, and the evaporation of the refrigerant occurs. This process occurs at the constant pressure where the entropy and enthalpy increases.

The COP of the cycle during the heating process is;

The COP of the cycle during the cooling process is;