# Air-standard Power Cycles

From Uni Study Guides

**UNDER CONSTRUCTION**

This is a topic from Thermodynamics:

## Introduction

A basic understanding of various forms of thermodynamic cycles is necessary for solving thermodynamic problems of a practical nature.

## The Compression Ratio

For certain piston cycles, where there is a swept and clearance volume, the compression ratio is defined as:

**r**_{v}= (V_{swept}+ V_{clearance}) / V_{clearance}

## The Air-Standard "Otto" Cycle

The cycle is as follows:

- 1→2: Isentropic compression
- 2→3: Constant volume heat addition
- 3→4: Isentropic expansion
- 4→1: Constant volume head rejection

The Otto cycle on a PV diagram:

Analysis:

- The volume in states 2 and 3 is the 'clearance' volume, such that the piston is almost at the top of the cylinder
- The volume in states 1 and 4 is the 'swept' volume , such that the piston is as far down the cylinder as it reaches during the cycle
- The work done for the isochoric processes (2→3 and 4→1) is equal to zero, and so by the first law: :::Q
_{H}= m(u_{3/1}- u_{2/4}) = mC_{V}(T_{3/1}- T_{2/4}) - Recalling the definition of engine efficiency:

- η
_{th}= W_{net}/ Q_{in}= (Q_{H}- Q_{L}) / Q_{H}, - η
_{th}= (mC_{V}(T_{3}- T_{2}) - mC_{V}(T_{4}- T_{1}))/mC_{V}(T_{3}- T_{2}) = ((T_{3}- T_{2}) - (T_{4}- T_{1})) / (T_{3}- T_{2})

- η

- And so:

**η**_{th}= 1 - ((T_{4}- T_{1}) / (T_{3}- T_{2})

- Meanwhile for the isentropic processes:

- T
_{2}/ T_{1}= (v_{1}/v_{2})^{k-1}and T_{3}/ T_{4}= (v_{2}/v_{3})^{k-1}, and so **T**_{2}/ T_{1}= T_{3}/ T_{4}

- T

- The compression ratio here would be given by:

- r
_{v}= v_{1}/v_{2}

- r

*And thus the expression for efficiency can be rewritten:*

**η**_{th}= 1 - T_{1}/ T_{2}= 1 - r_{v}^{1-k}

- ...we can see therefore, that increasing the compression ratio is best for efficiency.