## Thermal Efficiency for Atkinson Cycle

In general the **thermal efficiency**, *η***_{th}**, of any heat engine is defined as the ratio of the work it does,

**W**, to the heat input at the high temperature, Q

_{H}.

The **thermal efficiency**, *η***_{th}**, represents the fraction of

**heat**,

**Q**

**, that is converted**

_{H}**to work**. Since energy is conserved according to the

**first law of thermodynamics**and energy cannot be be converted to work completely, the heat input, Q

_{H}, must equal the work done, W, plus the heat that must be dissipated as

**waste heat Q**

**into the environment. Therefore we can rewrite the formula for thermal efficiency as:**

_{C}The heat absorbed occurs during combustion of fuel-air mixture, when the spark occurs, roughly at constant volume. Since during an isochoric process there is no work done by or on the system, the **first law of thermodynamics** dictates *∆U = ∆Q. *

Therefore the heat added and rejected are given by:

Q_{add} = mc_{v} (T_{3} – T_{2})

Q_{out} = mc_{p} (T_{4} – T_{1})

Substituting these expressions for the heat added and rejected in the expression for thermal efficiency yields:

Furthermore, it can be derived that in terms of:

- the ratio V
_{1}/V_{2}, which is known as the compression ratio – CR - the ratio V
_{4}/V_{3}, which is known as the expansion ratio – ER. **κ = c**_{p}**/c**_{v}

The expression for thermal efficiency using these characteristics is:

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