**Mixed/Dual Cycle – Process and its Derivation:**

Mixed/Dual cycle is a combination of Otto cycle and Diesel cycle, where heat addition takes place partly at constant volume and partly at constant pressure. In fact Otto cycle and Diesel cycle are special cases of Dual cycle. Both Otto cycle (constant volume heat addition) and Diesel cycle (constant pressure heat addition) are over simplistic and unrealistic. In actual case, combustion takes place neither at constant volume (time required for chemical reaction), nor at constant pressure (rapid uncontrolled combustion). Dual cycle is used to model the combustion process and consists of the following operation.

**Process 1 – 2 – Adiabatic compression**

In this process, the piston moves from BDC to TDC, the air or air-fuel mixture undergoes adiabatic compression. The volume decreases from V_{1} to V_{2}, pressure rises from P_{1} to P_{2}, temperature rises from T_{1} to T_{2} and the entropy remains constant (isentropic process) S_{1} = S_{2}.

**Process 2 – 3 – Addition of heat at constant volume**

The piston remains at the TDC for a moment, it is the time required for chemical reaction. Heat is added at constant volume V_{2 }= V_{3}. Pressure increases from P_{2} to P_{3}. Temperature increases from T_{2} to T_{3}, entropy increases from S_{2} to S_{3}.

**Process 3 – 4 – Addition of heat at constant pressure**

The piston remains at the TDC position, heat is added at constant pressure P_{3} = P_{4}, the volume increases from V_{3} to V_{4}. The temperature increases from T_{3} to T_{4} and the entropy also increases from S_{3} to S_{4}.

**Process 4 – 5 – Adiabatic expansion**

The piston moves from TDC to BDC position. Due to the combustion the mixture undergoes adiabatic expansion, the volume increases from V_{4} to V_{5}, pressure decreases from P_{4} to P_{5}, temperature decreases from T_{4} to T_{5} and the entropy remains same (isentropic process) S_{4} = S_{5}.

**Process 5 – 1 – Heat rejection at constant volume **

The piston remains at the BDC position for a moment. Heat is rejected at constant volume. V_{5} = V_{1}. The pressure decreases from P_{5} to P_{1} and temperature decreases from T_{5} to T_{1}. Entropy also decreases from S_{5} to S_{1}.

**Air standard efficiency:**

Compression ratio

In process 1 – 2, adiabatic compression,

In process 2 – 3 constant volume heating

Substituting T_{2} from equation (2)

In the process 4 – 5 – adiabatic expansion

In constant pressure heating process

Substituting the value of T_{3} from equation (3)

Substituting the value of T_{4} from equation (5) to equation (4) we get

Substituting the value of T_{2}, T_{3}, T_{4} and T_{5} in equation (1) we get