# Work Done By Reciprocating Compressor without Clearance Volume (Single Stage, Single Acting) With PV and TS Diagram

Read: Working of reciprocating compressor

The working of reciprocating compressor includes three operations, suction, compression, and discharge of compressed fluid. thus there are three work is included in a cycle of reciprocating compressor, work of piston during the suction of fluid/ refrigerant, work of piston during the compression of fluid as well as work during the discharge of compressed fluid. Mathematically the work done by the reciprocating compressor is equal to the work done by compressor during compression and discharge minus the work done during the suction of fluid.

Consider a single stage, single acting reciprocating compressor without clearance volume. The following figure shows the PV and TS diagram of this compressor. The compression process may be isentropic, polytropic, or isothermal

Let p

_{1}= Suction pressure (pressure before compression)

v

_{1}= Suction volume

T

_{1}= Suction temperature

p

_{2}, v

_{2}, T

_{2}are the corresponding pressure, volume, and temperature after compression.

r is compression ratio (p

_{1}/p

_{2})

### Work done during isothermal compression

The line AB represents suction of fluid, area under AB (ie ABB'A') represent work done during the suction processW

_{1}= p

_{1}v

_{1}

_{}BC

_{1}represent compression of fluid when piston moving from bottom dead center to top dead center. When the pressure inside the cylinder reaches p

_{2}, the discharge valve opens. Further movement piston towards the top dead center cause the compressed air to discharge. C

_{1}D represent discharging of fluid.

Work done during compression is W

_{2}= Area of BC

_{1}C

_{1}'B'

Work done during discharge W

_{3}= Area C_{1}DA'C_{1}'
W

Work done by the compressor during the one complete cycle of operation is equal to
W= W_{3}= p_{2}v_{2}_{3}+W

_{2}-W

_{1}

_{1}v

_{1}= p

_{2}v

_{2}

_{1}v

_{1}= mRT

_{1}and

###
Work done during polytropic compression (PV^{n} = constant)

The work done during the compression is equal to area under BCC'B'_{3}+W

_{2}-W

_{1}

_{1}v

_{1}

^{n}= p

_{2}v

_{2}

^{n}, n is ploytropic index

_{2}/v

_{1}in equation of work done

### Work done during isentropic compression

The curve BC_{2}Shows isentropic compression. The equation for work done during isentropic compression is similar to that of during polytropic compression.

Here γ is isentropic index.

since

andsince

_{p}and c

_{v }are specific heats