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

p-v and t-s diagram reciprocating compressor

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 p1= Suction pressure (pressure before compression)
v1 = Suction volume
T1= Suction temperature
p2, v2, T2 are the corresponding pressure, volume, and temperature after compression.
r is compression ratio (p1/p2)

Work done during isothermal compression

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

W1= p1 v1

BC1 represent compression of fluid when piston moving from bottom dead center to top dead center. When the pressure inside the cylinder reaches p2, the discharge valve opens. Further movement piston towards the top dead center cause the compressed air to discharge. C1D represent discharging of fluid.

Work done during compression is W2 = Area of BC1C1'B'
work equation
Work done during discharge W3= Area C1DA'C1'
W3= p2v2
Work done by the compressor during the one complete cycle of operation is equal to W= W3+W2-W1
work equation
Since  p1v1 = p2v2
work eqaution
But p1v1 = mRT1 and
pv equation

Work done during polytropic compression (PVn = constant)

The work done during the compression is equal to area under BCC'B'
work equation
Work done W= W3+W2-W1
work equation
For polytropic compression p1v1n = p2v2n, n is ploytropic index
pv equation for polytropic
Put the value of v2/v1 in equation of work done
work done equation
since
pt equation for polytropic
The above equation will become
work equation

Work done during isentropic compression

The curve BC2 Shows isentropic compression. The equation for work done during isentropic compression is similar to that of during polytropic compression.
work equation

Here γ is isentropic index.
since
isentropic index equation
and
work equation
Here cp and care specific heats
work equation



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