This document discusses positive displacement machines, specifically reciprocating and rotary compressors. It describes the basic functions and types of positive displacement machines, including reciprocating compressors which have low flow rates but high pressure ratios, and rotary compressors which have continuous flow but lower pressure ratios. The document provides analysis of compressor performance in terms of indicated work, actual power input, efficiency, and volumetric efficiency. It also covers topics such as multi-stage compression, intercooling, and the operation of rotary machines like roots blowers and vane compressors.

1. Positive Displacement Machines

2. Reciprocating and rotary machines
 Function of a compressor (gas/often air)
 Efficient machine – minimum input of mechanical work
 Types
 Reciprocating positive displacement machines (pulsating in action limiting flow rate)
 Rotary positive displacement machines (continuous action, small, light, simple)
 Reciprocating – low mass rate of flow + high pressure ratios (500 bar +)
 Rotary – high mass rate of flow + low pressure ratios – scavenging and supercharging
of engines, exhausting, vacuum pumping (9 bar)
 Forms – single stage or – multi stage
 Air or water cooling
 Properties at inlet and outlet are average values taken over the cycle

3. Reciprocating compressors
 Negligible clearance
volume
 Working fluid – perfect
gas
 Cycle – one crankshaft
revolution
 Operation of valves
 Constant temp, pressure,
zero heat exchange, mass
increase/decrease
 𝑇2- law of compression –
reversible polytropic
𝑝𝑉𝑛 = 𝑐𝑜𝑛𝑠𝑡.

4. Analysis of performance
 Reversible polytropic 𝑝𝑉𝑛
= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
 Indicated work done on the gas per cycle = area abcd
 Work input=
𝑛
𝑛−1
(𝑝2𝑉𝑏 − 𝑝1𝑉
𝑎)
 𝑝2𝑉𝑏 = 𝑚𝑅𝑇2 ; 𝑝1𝑉
𝑎 = 𝑚𝑅𝑇1
 Work input per cycle =
𝑛
𝑛−1
𝑚𝑅(𝑇2 − 𝑇1)
 𝑚 - rate of mass flow thus indicated power
 Work done on air = work input per cycle X number of cycles
 𝑇2 = 𝑇1
𝑝2
𝑝1
(𝑛−1) 𝑛
 Actual power input to compressor > indicated power – coz friction
 Shaft power = indicated power + friction power
 Compressor mechanical efficiency=
𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑑 𝑝𝑜𝑤𝑒𝑟
𝑠ℎ𝑎𝑓𝑡 𝑝𝑜𝑤𝑒𝑟
 Input power=
𝑠ℎ𝑎𝑓𝑡 𝑝𝑜𝑤𝑒𝑟
𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑜𝑓 𝑚𝑜𝑡𝑜𝑟 𝑎𝑛𝑑 𝑑𝑟𝑖𝑣𝑒
Mcconkey Pg. 63

5. Other expressions for indicated work
 Work input per cycle =
𝑛
𝑛−1
𝑚𝑅(𝑇2 − 𝑇1) ; 𝑝2𝑉𝑏 = 𝑚𝑅𝑇2 ; 𝑝1𝑉
𝑎 = 𝑚𝑅𝑇1

6. The condition for minimum work
 𝑝𝑉 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 − 𝑖𝑠𝑜𝑡ℎ𝑒𝑟𝑚𝑎𝑙
 𝑝𝑉𝛾
= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 − 𝑖𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐
 Isothermal efficiency
=
𝑖𝑠𝑜𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑤𝑜𝑟𝑘
𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑑 𝑤𝑜𝑟𝑘
 Air/water cooling
 n = 1.2 – 1.3 Mcconkey Pg.57

7. Reciprocating compressors including clearance
 Clearance – mechanical freedom to working parts
 Clearance volume - 6%, 2%, 30 – 35%
 Clearance reduces induced volume at 𝑝1 and 𝑇1 from 𝑉
𝑠 to (𝑉
𝑎 − 𝑉𝑑)
 The mass delivered per unit time 𝑚𝑏 − 𝑚𝑐 = 𝑚𝑎 − 𝑚𝑑
 Indicated power =
𝑛
𝑛−1
𝑚𝑅(𝑇2 − 𝑇1); 𝑚 = 𝑚𝑎 − 𝑚𝑑 - mass induced per unit time
 Double acting – mass delivered per unit time

8. Volumetric efficiency
 𝜂𝑣 - The mass of gas delivered, divided by the mass of gas which
would fill the swept volume at the free air conditions of pressure
and temperature.
 𝜂𝑣 =
𝑉𝑎−𝑉𝑑
𝑉𝑠
 Note: Conditions of p and T in cylinder are during induction are
identical with those of the free air.

9. Multistage compression
 Delivery temp 𝑇2 increases with increase in pressure
ratio
 Increase in pressure ratio decreases volumetric
efficiency
 FAD increase got by increasing pressure ratio
 Isothermal compression got with cooling
 Ideal intermediate pressure – value giving same
pressure ratio for each stage
 Complete intercooling

10. Rotary machines
 Continuous rotary action – smaller for a given flow than reciprocating machines
 No cooling
 Conditions adiabatic
 Examples
 Roots blower
 Vane type

11. Roots blower
 3 and 4 lobe versions
 Spacing to reduce friction
 Leakage that reduces efficiency due to increased
pressure ratio
 Work done per cycle= 𝑝2 − 𝑝1 𝑉 –(per rev X 4)
 Power input= 𝑝2 − 𝑝1 𝑉
𝑠
 Power input=
𝛾
𝛾−1
𝑝1𝑉
𝑠
𝑝2
𝑝1
(𝛾−1) 𝛾
- isentropic
 Roots efficiency=
𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 𝑖𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐𝑎𝑙𝑙𝑦
𝑎𝑐𝑡𝑢𝑎𝑙 𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒
 r – pressure ratio 𝑝2 𝑝1
 Increase in pressure ratio decreases efficiency
 Lots of imperfections, scavenging + supercharging

12. Vane type
 Rotor mounted eccentrically
 Non-metallic blades
 Compression due to decreasing volume
 Further compression from back-flow of air from receiver
 Uncooled machine being isentropic
 Vane type requires less work input compared to roots blower
 Work input = A + B
 Lubrication is important – blades + casing

13. Vacuum pumps
 Produce a vacuum or do scavenge a vessel
 Presence of condensable vapours affects pump
efficiency
 Affects seals and lubricating properties of oils

14. Ref
 T. D. Eastop and A. Mcconkey. Applied Thermodynamics for Engineering Technologists. 5th
Edition
NOTE:
 The term "polytropic" was originally coined to describe any reversible process on
any open or closed system of gas or vapor which involves both heat and work
transfer, such that a specified combination of properties were maintained constant
throughout the process.
 The polytropic process can describe gas expansion and compression, which include
heat transfer.
 It obeys the relation: 𝑝𝑉𝑛= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡