Air or Gas Compressors: A steady-state, steady flow machine that is usedto compressed air or gas to final pressure exceeding 241.25 Kpa gage.Types of Compressor:

1. Centrifugal Compressors: For low pressure and high capacity applications.2. Rotary Compressors: For medium pressure and low capacity

application.3. Reciprocating Compressors: For high pressure and low capacity

application.Uses of compressed air:

1. Operation of small engines 2. Pneumatic tools 3. Air hoists 4. Industrial cleaning by air blast 5. Tire inflation 6. Paint Spraying

7. Air lifting of liquids8. Manufacture of plastics and other industrial products

9. To supply air in mine tunnels 10. Other specialized industrial applicationsAnalysis of Centrifugal and Rotary Type

1 Suction

2 Discharge

W (Work)

AssumptionKE = 0 PE = 0

For a compressor, work is done on the system W = h - Q 1. For Isentropic Compression: PVk = C Q = 0

W = h

P

V

P2

P1

PVk = C

Where:m – mass flow rate in kg/sec Cp – constant pressure specific heat in KJ/kg-C or KJ/kg-K

1kRk

Cp

111

k1k

1

211

k1k

1

2

1

2

12P

mRTVP

1PP

1kVkP

W

PP

TT

TTmCΔhW ; 0Q

2. Polytropic compression: PVn = C

111

n1n

1

211

Vn

n1n

1

2

1

2

12n12P

mRTVP where; 1PP

1nVnP

W

n1nk

CC ; PP

TT

TTmCQ ; TTmCh ; QΔhW

P

V

P2

P1

PVn = C

3. Isothermal Compression: PV = C

111

1

211

mRTVPPP

lnVPW

-QWtherefore 0h ; QΔhW

Analysis of Reciprocating Type Compressor (Piston-in-cylinder type):

piston

Valves

cylinder Piston rod

P

V

P2

P1

PV = C

Pressure-Volume Diagram (PV)

HE – head endCE – Crank endL – length of strokeP1 – suction pressureP2 – discharge pressureV1’ – volume flow rate at intakeVD – displacement volumeCVD – clearance volumeCVD = V3

L

HE CE

P

V1

23

4

P2

P1

V1’

VDCVD

VD

1. Isentropic Compression: PVk = C

11'1

k1k

1

21'1

k1k

1

2

1

2

12P

mRTVP

1PP

1kVkP

W

PP

TT

TTmCΔhW ; 0Q

Where:V1’ – volume flow rate at intake, m3/secm – mass flow rate corresponding V1’

P1 – suction pressure, KpaP2 – discharge pressure, KpaT1 – suction temperature, KT2 – discharge temperature, KW – work, KW

2. Polytropic Compression: PVn = C

n-1n-k

CCmRTVP1PP

1nVnP

WPP

TT

TTmCh ; TT mCQ Q;-ΔhW

vn11'1

n1n

1

21'1n

1n

1

2

1

2

12P12n

; ; ;

3. Isothermal Compression: PV = C

11'1

1

21'1

mRTVPPP

lnVPW

-QWtherefore 0h ; QΔhW

Percent Clearance: Ratio of the clearance volume to the displacement volume.

D3

D

3

CVV

100% xVV

C

Note: For compressor design values of C ranges from 3 to 10 percent.

Pressure Ratio: Ratio of the discharge pressure to suction pressure.

1

2p P

P=r

Volumetric Efficiency: Ratio of the volume flow rate at intake to the displacement volume.

% 100 xVV

=ηD

1'v

1. For Isentropic Compression and Expansion process: PVk = C

k1

1

2v P

PCC1η

2. For Polytropic Compression and Expansion process: PVn = C

n1

1

2v P

PCC1η

2. For Isothermal Compression and Expansion process: PV = C

1

2v P

PCC1η

Actual Volumetric Efficiency: Ratio of the actual volume of air drawn in by the compressor to the displacement volume.

100% xVV

ηD

ava

For an air compressor handling ambient air where pressure drop and heating of air occurs due to fluid friction and irreversibilities of fluid flow, less amount of air is being drawn by the cylinder. The actual volumetric efficiency is:

TT

PP

ηη1

O

O

1vva

Where:PO – ambient air pressure in KpaTO – ambient air temperature in K

Displacement Volume: Volume of air occupying the highest stroke L ofthe piston within the cylinder. The length of stroke L is the dis-tance from the HE (head end) to the CE (crank end).

1. For Single Acting: For single acting compressor, the piston works on one side (at head end only) of the cylinder.

secm

4(60)

Nn'LDV

32

Dπ

2. For Double Acting: For double acting compressor, the piston works on both sides (at head end & crank end) of the cylinder.

a. Without considering the volume of the piston rod.

secm

4(60)

Nn'LDV

32

D2π

b. Considering the volume of the piston rod.

secm

d-2D4(60)LNn'

V3

22D

π

Where:D – diameter of piston in metersd – diameter of piston rod in metersN – no. of RPMn’ – no. of cylinders

Piston Speed: It is the linear speed of the piston.

secm

602LN

PS

minm 2LNPS

Compressor Performance Factor:

1. Compression Efficiency: Ratio of Ideal Work to Indicated Work.

100% xWork Indicated

Work Idealηcn

2. Mechanical Efficiency: Ratio of Indicated Work to Brake or Shaft Work.

100% xWorkShaft or Brake

Work Indicatedηm

3. Compressor Efficiency: Ratio of Ideal Work to Brake or Shaft Work.

mcnc

c

η ηη

100% xWorkShaft or Brake

Work Idealη

MULTISTAGE COMPRESSION:Multi staging is simply the compression of air or gas in two or more cylinders in place of a single cylinder compressor. It is used in reciprocating compressors when pressure of 300 KPa and above are desired, in order to: 1) Save power 2) Limit the gas discharge temperature 3) Limit the pressure differential per cylinder 4) Prevent vaporization of lubricating oil and to prevent its ignition if the tem- perature becomes too high.It is a common practice for multi-staging to cool the air or gas between stages of compression in an intercooler, and it is this cooling that affects considerable saving in power.

2 Stage Compressor without pressure drop in the intercooler:

1 2 3 4Suction DischargeQx

Intercooler

1st stage 2nd stage

For an ideal multistage compressor, with perfect inter-cooling and minimum work, the cylinder were properly designed so that: a) the work at each stage are equal b) the air in the intercooler is cooled back to the initial temperature c) no pressure drop occurs in the intercooler

Work of 1st stage cylinder (W1): Assuming Polytropic compression on both stages.

11'1

n1n

1

21'11

mRTVP

1PP

1nVnP

W

Work of 2nd stage cylinder (W2): Assuming Polytropic compression on both stages.

33'3

n1n

3

43'32

mRTVP

1PP

1nVnP

W

P

V

P4

P1

Px

1

4

32

5

6

7

8

PVn = C

W1

W2

S

T

4

3

2

1

P4 Px P1

Qx

For perfect inter-cooling and minimum work:W1 = W2

T1 = T3

W = W1 + W2

W = 2W1

P2 = P3 = Px

thereforeP1V1’ = P3V3’

21

1

4

1

2

41x

32x3

4

1

2

PP

PP

PP P

then ; P P Pbut ; PP

PP

Where: Px – optimum intercooler pressure or interstage pressure

1PP

1nV2nP

W

1PP

1nV2nP

W

2n1n

1

41'1

n1n

1

21'1

Then the work W for an ideal 2-stage compressor is:

Heat losses calculation:1. Heat loss during compression at 1st stage cylinder

Q1 = mCn(T2 – T1)2. Heat loss during compression at 2nd stage cylinder

Q2 = mCn(T4 – T3)3. Heat loss in the intercooler

Qx = mCp(T2 – T3)

2 Stage Compressor with pressure drop in the intercooler:

1 2 3 4Suction DischargeQx

Intercooler

1st stage 2nd stage

With pressure drop in the intercooler:T1 T3 and P2 P3

W = W1 + W2

P1V1’ P3V3’

1PP

1nVnP

W

1PP

1nVnP

W

n1n

3

43'32

n1n

1

21'11

P

V

P4

P1

P3

1

4

32

5

6 7

8

PVn = C

W1

W2

S

T

4

3

2

1

P4 P1

Qx

P2

P2 P3

3 Stage Compressor without pressure drop in the intercooler:

1 2 3 4

Suction Discharge

Qx

LP Intercooler

1st stage 2nd stage 3rd stage

5 6Qy

HP Intercooler

S

T

4

3

2

1

P6 Px P1

Qx

P

V

P6

P1

Px

1

4

3

2

5

67

12

PVn = C

W1

W2

Py 9

10

11

8

W3

Py

5

6

Qy

For perfect inter-cooling and minimum work:T1 = T3 = T5 Px = P2 = P3 W1 = W2 = W3 Py = P4 = P5

W = 3W1P1V1’ = P3V3’ = P5V5’

mRT1 = mRT3 = mRT5

Therefore: rP1 = rP2 = rP3

Work for each stage:

1PP

1nVnP

Wn

1n

1

21'11

1st Stage:

1PP

1nVnP

Wn

1n

3

43'32

2nd Stage:

3rd Stage:

1PP

1nVnP

Wn

1n

5

65'53

3 261y

36

21x

y

6

x

y

1

x

5

6

3

4

1

2

PPP ; PPP

hence

PP

P

P

PP

or PP

PP

PP

Intercooler Pressures:

Total Work:

W = 3W1

11'1

3n1n

1

61'1

mRTVP

1PP

1nV3nP

W

Heat Losses during compression:Q1 = mCn(T2 – T1)Q2 = mCn(T4 – T3)Q3 = mCn(T6 – T5)

Heat loss in the LP and HP intercoolers:LP Intercooler

Qx = mCp(T2 – T3)HP Intercooler

Qy = mCp(T4 – T5)Note:1. For isentropic compression and expansion process, no heat loss during compression.2. For isothermal compression and expansion process, the loss during

compression is equivalent to the compression work, and no intercooleris required.

For multistage compression with minimum work and perfect inter-cooling and no pressure drop that occurs in the inter-coolers between stages, the following conditions apply: 1. the work at each stage are equal 2. the pressure ratio between stages are equal 3. the air temperature in the inter-coolers are cooled to the original temperature T1

4. the total work W is equal to

1PP

1nVSnP

W2S

1n

1

2S'

11Where: s – is the number of stages.

Note: For multistage compressor with pressure drop in the intercoolers the equation of W above cannot be applied. The total work is equal to the sum of the work for each stage that is computed separately.