Reciprocating Compressors

RECIPROCATING COMPRESSORS There are various compressor designs: Rotary vane; Centrifugal & Axial flow (typically used on gas turbines); Lobe (Roots blowers), and Reciprocating. The main advantages of the reciprocating compressor are that it can achieve high pressure ratios (but at comparatively low mass flow rates) and is relatively cheap. It is a piston and cylinder device with (automatic) spring controlled inlet and exhaust valves. Delivery is usually to a receiver. The receiver is effectively a store of energy used to drive (eg) compressed air tools.

Inlet

Delivery

BDC TDC ¬ Swept vol. ®

Clearance vol.

Receiver

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Reciprocating compressors usually compress air but are also used in refrigeration where they compress a superheated vapour (to which the gas laws strictly do not apply). In order to be practical there is a clearance between the piston crown and the top of the cylinder. Air 'trapped' in this clearance volume is never delivered, it expands as the piston moves back and limits the volume of fresh air which can be induced to a value less than the swept volume. The induced volume flow is an important purchasing parameter. It is called the "Free Air Delivery" (FAD), and it measures the capacity of a compressor in terms of the air flow it can handle. It is normally measured at standard sea level (SSL) atmospheric conditions and allows the capacities (size) of compressors to be compared. N.B. The induced mass per cycle must equal the delivered mass per cycle (continuity!), although the induced and delivered volumes will be different.

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Cycle Analysis The cycle may be analysed as two non-flow (compression and expansion) processes and two flow processes (delivery and induction) PROCESS

GROSS WORK

1®2

Compression

p2V2 - p1V1 n-1

2®3

Delivery

p2(V2-V3)

3®4

Expansion

4®1

Induction

p4V4 - p3V3 n-1 p1(V4-V1)

Note that we assume polytropic compression and expansion. This is because some degree of cooling is usually attempted for reasons we shall see later. If no cooling were attempted n becomes g. On p-V co-ordinates:

pressure (kPa)

1000 900 800 700

3

2

600 500 400 300 200 100 0

4 0

0.2

1 0.4

0.6

0.8

Volume (litres)

1

1.2 3

The work per cycle is given by: å gross work p4V4 - p3V3+ p1(V4-V1) work per cycle = p2V2 - p1V1 + p2(V2-V3) + n-1 n-1 p4V4 - p1V1 + p1(V4-V1)+ p2V2 - p3V3 + p2(V2-V3) n-1 n-1 but p1=p4 & p2=p3 =

work per cycle =

p1(V4-V1) p2(V2-V3) p2(V2-V3) + p 1(V4-V1) + + n-1 n-1

= p1(V4-V1) {1+ 1 } + p2(V2-V3){1+ 1 } n-1 n-1 but \

mass delivered = mass induced p2(V2-V3) = RT2

p1(V1-V4) RT1

p2(V2-V3) = p1(V1-V4) T2 T1 n work per cycle = p1(V1-V4) { n-1 } [ T2 -1] T1 for a polytropic process :

T2 = ( T1

n-1 p2 ) n p1

=

n-1 n rp

Noting that (V1-V4) is the induced volume (Vind), and p1 is the inlet pressure (pin) we may re-arrange and write: n-1

work per cycle = n pin Vind { rp n -1} n-1 NB Power required = work per cycle x cycles per sec 4

Volumetric Efficiency We have already noted that the induced volume is less than the swept volume. To enable this effect to be evaluated we define volumetric efficiency (hvol) as: hvol =

Induced volume Swept volume

= V1-V4 Vs but

n p3V3

=

n p4V4

\

V4 = V3 rp

1 n

V3 is the clearance volume (Vc), and V1 = Vc + Vs hvol = Vc + Vs Vs

\

hvol

1 Vc rp n

1 V c =1( rp n - 1 ) Vs

The reference conditions (p & T) at which the volumetric efficiency is measured should always be quoted (it would normally be SSL conditions). [The concept of hvol applies also to reciprocating engines.]

hvol

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Vc = 0.05 Vs n = 1.27

0

10

20

30

40

rp

50 5

Volumetric Efficiency referred to SSL conditions. In testing a compressor, the measured induced volume flow will be that of the actual test inlet conditions. It is unlikely that these inlet conditions will be SSL. We therefore need to refer our results to SSL conditions.

(Measured) . Inlet >> Vi Ti

SSL Ts Ps

Pi

The mass flow of gas must be the same both at SSL (s) conditions and at Inlet (i) conditions. . . ms = mi . . p i V i psVs = RTi RTs . Vs

=

pi Ts ps Ti

. Vi

. . sides by V dividing both swept

hvol(SSL) =

pi Ts ps Ti

hvol(inlet) 6

Compressor Efficiency If we plot the specific work (kJ/kg delivered) against the polytropic index n we obtain: n-1

w = n RTin { rp n -1} n-1 250 rp=8

200 w kJ/kg

150

rp=4

100 50 0 1.1

1.2

Polytropic

n

1.3

1.4

Adiabatic compression

Isothermal compression

1

It is clear that the closer the compression is to isothermal the less work is required. The savings are greater at higher pressure ratios (eg above 18% @ rp=4 and 27% @ rp=8) It is also apparent that isothermal compression represents the ideal minimum work input for a compression process (where cooling is feasible).

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We can therefore define compressor efficiency as: hiso =

Isothermal work per cycle Actual work per cycle

If we recalculate the work input assuming isothermal compression [ W12 = p1V1 ln(p1/p2) etc] it is found that: hiso =

ln rp n-1 n { r p n -1} n-1

Note that this efficiency is known as the isothermal efficiency. The degree of cooling possible during a single stage compression process tends to be limited. It improves at low speeds but this limits compressor capacity. One way of improving efficiency, especially at higher compression ratios and speeds, is to go to multistage compression with cooling of the gas between each stage.

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Multistage compression To avoid unacceptable reductions in compressor capacity (RPM and volumetric efficiency) and to minimise power input with high compression ratios, multistaging with inter-cooling is used. The number of stages will normally be between two and four. intercoolers > out

in >

Stage 1

Stage 2

Stage 3

Each stage may be treated as a separate compressor, however, with multistaging, all will normally rotate at the same speed. The volumetric efficiency of the compressor as a whole is determined by the first stage. If the inter-cooling is such that the inlet temperature to the following stage(s) is the same as the inlet temperature to the first stage we have ideal (or perfect) inter-cooling. Since the work input to any stage is dependent on the pressure ratio across it, it should be possible to minimise the total work input by the correct choice of compression ratio across each stage.

9

Optimum stage pressure ratio Assume we have two stages of compression with ideal intercooling and the same index of compression (and expansion) 'n' in each stage. Total Work per cycle n-1

n-1

= n ép1in V1ind { rp1 n -1} + p2in V2ind { rp2 n -1}ù n-1 ë û Since the mass induced by the first stage must be equal to the mass induced by the second stage: p1in V1ind p2in V2ind = R Tin R Tin n-1

n-1 n -1} ù

\ Total Work per cycle = n p1in V1ind é{ rp1 n -1} + { rp2 n-1 ë

û

If p1 is the inlet pressure and p2 the final delivery pressure, let pi = the inter-stage pressure: then

rp1 = pi p1

rp2 = p2 pi

&

If we substitute the above in the expression for Total Work and differentiate wrt pi, we can find pi for minimum Total Work. ½

whence

pi = [p1 p2] or

rp1 = rp2

½ p 2 =( ) p1

We could extend the same method to N stages with the result that, for minimum work input, the pressure ratio across each stage must be the same and equal to the Nth root of the overall pressure ratio. rp(opt) = rp(overall)

1 N

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