Air-Standard Analysis of Otto Cycle: An Ideal Power Cycle for Petrol Engine

An ideal power cycle that studies the petrol engine under the cold air-standard assumptions (as articulated in general in our previous article) is called the Otto Cycle. It is shown in the P-V and T-s diagrams below.

PV Diagram of Otto Cycle
TS Diagram of Otto Cycle

An Otto cycle consists of four internally reversible processes, which are articulated below:

  1. Process 1-2: Isentropic Compression of the fixed mass of air when the piston moves from the bottom dead center to the top dead center.
  2. Process 2-3: Isochoric Heat is the addition of the fixed mass of air from an external source when the piston is at the top dead center. It signifies the ignition of the air-fuel mixture in the real petrol engine and the subsequent rapid burning.
  3. Process 3-4: Isentropic expansion during the power stroke.
  4. Process 4-1: Isochoric Heat Rejection from the hot air when the piston is at the bottom dead center. It represents the end of the complete thermodynamic power cycle of an ideal petrol engine.

In the ideal Otto cycle, heat is approximated to be added at constant volume. But why? It is because the combustion process in the real petrol engine is rapid, localized, and fully instantaneous everywhere in the available volume with a few degrees of crankshaft rotation. Therefore, the simulation of the combustion process in the ideal petrol engine is closely approximated with heat addition at constant volume.

Along the same lines, one can ask: Why is heat rejection at a constant volume? It can be responded by relating it to the existence of the real-time constant volume of the combustion chamber when the piston is at the top dead centre when burned gases leave the opened exhaust valve/s during the exhaust process.

Likewise, isentropic compression and expansion of air in the ideal diesel cycle imply that both compression and expansion are adiabatic and fully reversible processes that represent the accurate thermodynamic behaviour of the ideal gas.

Interpretation of the P-V and T-s Diagram of an Air-Standard Otto Cycle

The area under the P-V and T-s diagram shows the work done and net heat supplied during the cycle, respectively.

P-V Diagram: As shown on the P-V plot, the area under curve 1-2-a-b-1 represents the work done per unit of mass during the compression process, whereas the area under curve 3-4-b-a-3 shows the work done per unit of mass during the expansion process.

The enclosed area of the P-V diagram shows the net work done per unit of mass.

T-s Diagram: As shown on the T-s plot, the area under the curve, 2-3-a-b-2, shows the heat supplied per unit of mass, whereas the area below the curve, 1-4-a-b-1, shows the net heat rejected per unit of mass.

The enclosed area of the T-s diagram shows the net heat supplied per unit of mass.

Since the Otto cycle works in a closed system and there are no changes in the kinetic and potential energies of the air, one can write the energy balance for the Otto cycle as below:

\left(q_{in}-q_{out}\right)+\left(w_{in}-w_{out}\right)=\triangle u\;\;\;\;\;\;\;\;\left(kJ/kg\right)

Because the heat transfer processes take place at constant volume, one can state that no work is involved during heat transfer processes. So, the above equation takes the following expression.

q_{in}-q_{out}=\;\triangle u

Heat addition and heat rejection for the Otto cycle can thus be calculated as follows:

q_{in}=u_3-u_2=c_v\left(T_3-T_2\right)
q_{out}=u_4-u_1=c_v\left(T_4-T_1\right)

Now, the thermal efficiency of an ideal petrol engine cycle or an Otto cycle is calculated as follows:

n_{otto\;cycle}=\frac{W_{net}}{q_{in}}=\frac{q_{in}-q_{out}}{q_{in}}=1-\frac{q_{out}}{q_{in}}=1-\frac{T_4-T_1}{T_3-T_2}=1-\frac{T_1\left({\displaystyle\frac{T_4}{T_1}}-1\right)}{T_2\left({\displaystyle\frac{T_3}{T_2}}-1\right)}

By repeating the assumptions of the cold air-standard cycle for an Otto cycle, one can state that for first, processes 1-2 and 3-4 are isentropic, and for second, v2=v3 and v4=v1. With these idealizations, one can write with one’s previous knowledge that:

\frac{T_1}{T_2}=\left(\frac{V_2}{V_1}\right)^{k-1}=\left(\frac{V_3}{V_4}\right)^{k-1}=\frac{T_4}{T_3}

Putting these values in the above expression’s final form and solving it yields:

\eta_{otto\;cycle}=1-\frac{T_1}{T_2}=1-\frac1{r^{k-1}}

Where;

r=\frac{v_{max}}{v_{min}}=\frac{v_1}{v_2}

and k is called the specific heat ratio and is defined as cp/cv

Analysis

The final expression of the thermal efficiency of an Otto cycle is given below:

\eta_{otto\;cycle}=1-\frac1{r^{k-1}}

The above expression shows that the thermal efficiency of an ideal petrol engine under the cold air standard assumptions is the function of the compression ratio and the specific heat ratio of the working fluid.

Upon closer examination, it is discernible that by increasing both the compression ratio of the engine and the specific heat ratio of the working fluid, the thermal efficiency of an Otto cycle tends to increase. Experiments have shown that in the real case, variation in thermal efficiency is directly related to the values of c and k.

For example, as shown in the T-s diagram above, when the compression ratio increases for an Otto cycle, the closed loop changes its path from 1-2-3-4-1 to 1-2՝-3՝-4-1. It shows that in the new path, due to the higher temperature of heat addition than in the older path, the thermal efficiency of the Otto cycle would increase. In contrast, the temperature of heat rejection would be the same in both cases.

Will the increase in the compression ratio increase the thermal efficiency of an Otto cycle indefinitely? No! Because the possibility of autoignition is always there, it acts as an upper limit for the maximum value of the compression ratio.

The undifferentiated and careless selection of compression ratio for the SI engine results in the generation of shock waves caused by autoignition (or detonation), which damages the engine body along with massive heat loss.

Therefore, fuels that are chemically synced with tetraethyl lead are desirable as they are more resistant to autoignition and, therefore, workable at higher compression ratios. The use of unleaded gasoline addresses rising environmental concerns today; it limits the compression ratio up to 9.

In the case of compression ratio engines, such as diesel engines, the higher compression ratio is a more viable option because only air is to be compressed. Lastly, in spark ignition engines, fuels with higher volatility are desired.