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Purdue e-Pubs
International Compressor Engineering Conference
School of Mechanical Engineering
1990
Utilization of Volumetric Displacement in
Reciprocating Compressors
J. A. McGovern
Trinity College
Follow this and additional works at: http://docs.lib.purdue.edu/icec
McGovern, J. A., "Utilization of Volumetric Displacement in Reciprocating Compressors" (1990). International Compressor Engineering
Conference. Paper 713.
http://docs.lib.purdue.edu/icec/713
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llriLIZATION OF VOLUMimUC DISPLACEMENT
IN RBCIPROOATING ru!PRESSORS
J. A. McGovern
Department of Mechanical and Manufacturing Engineering
Trinity College, Dublin, Ireland
ABSTRACT
The main factors affecting utilization of volumetric displacement of
reciprocating compressors are analysed. A coherent system of parameters to
quantify these is formulated. Reference is made to conventional parameters
and to the work of other authors. Some new parameters are introduced and
int~rrelationships are described.
The overall_purpose of the theory is to
apportion shortcomings in the volumetric efficiency of reciprocating
compressors to specific causes, on the basis of test measurements. The
measurement problems which would arise in applying the theory fully are not
addressed.
THE PROCESSES OF THE RECIPROCATING CCMPRESSOR CYCLE
Fig. 1 illustrates the cycle of a reciprocating compressor as a diagram
of cylinder pressure versus volume. The horizontal lines through points 4 and
1 and through points 3 and 2 represent the mean values of the fluctuating
pressures within the suction and discharge pipes respectively.
v
Fig. 1 An indicator diagram as might be measured experimentally for a
reciprocating compressor, showing overpressure and underpressure
caused by valve throttling.
VOLUMETRIC EFFICIENCY
The volumetric efficiency, which is also known as the displacement
utilization efficiency, quantifies the utilization of the volumetric
displacement of the piston and is defined as follows:
.
mv
~
sw
254
(1)
where,
volumetric efficiency, or, displacement utilization efficiency
r)v
m
N
v
v
.•w
=
=
actual external mass flow rate produced by the compressor
number of cycles per unit time
(compressor speed)x(number of cylinders)
swept volume per cylinder
=
specific volume of the fluid at the suction condition
The volumetric efficiency can be determined experimentally
of the mass flow rate and the compressor speed.
fro~n
measurements
FAC'l"ttiS WHICH AFFECT DISPLACEMENT llriLIZATION
Kleinert and Najork [ 1] presented an expression for the volumetric
efficiency as the product of four efficiency terms which they described.
These terms quantified different influences on the overall utilization of
displacement. Pandeya and Soedel [2] had previously described an approach
wherein the losses in IIB.IIS flow rate were edditive.
In the following sections the interrelationships between various
influences on displacement utilization are re-examined, taking account of
earlier work, and it is shown how the quantifying parameters can be combined
to approximate the actual volumetric efficiency.
RB-EXPANSION OF '!HE CLEARANCE MASS
The fluid in the clearance volume when the piston is at the top dead
centre position is re-expandecJ. in the initial part of the induction stroke.
The volume occupied by this 'clearsnce IIB.IIS 1 of refrigerant at the suction
pressure reduces the volume of fresh cha.rae taken into the cylinder
correspondingly.
The fundamental expression for the volumetric efficiency of a compressor
which is ideal in every respect except that it haa a clearance ratio greater
than zero haa the fon11
(2)
where
A.
=
r
=
c
v~
v
3
=
clearance volumetric efficiency
clearance ratio
= clearance volume
swept volume
specific volume of the fluid at the end of the re-expansion
process
specific volume of the fluid at the start of the
re-expansion process
For an actual compressor it is proposed that the clearance volumetric
efficiency should be evaluated using the specific volume values corresponding
to points 4 and 3 on the ac,;·,L•'iil indicator diagrsm as represented in Fig. 1 •
It should be emphasized that the ratio of the specific volumes is not
necessarily the same as the ratio of the actual volumes on the indicator
diagrsm, due to leakage effects.
The clearance volumetric efficiency of an actual compressor may also
wri ttm as the following well Imown expression
255
be
(3)
were
p•
=
mean pressure in suction pipe
mean pressure in discharge pipe
pd
n
=
p::.lytropic index for the re-exparmion process
log(p ) - log(p )
3
n
log(v
4
4
) -
log(v
3
)
(4)
It should be noted that l. depends on the value of the p::.lytropic index
for the re-expansion process, n, and decreases with decreasins n. For an
ideal pa a reversible adiabatic re-expanaion process lmuld. have an index, n,
equal to the ratio of the specific heats, J. A value of unity IOOuld. be
elqleeted for isothermal re-exparmion: this would involve heat transfer to the
gas durin& the process. The clearance volumetric efficiency, l., thus includes
the effect of heat transfer to the fluid. during re-expanaion.
Using the above expressions, ( 2) , or ( 3) with ( 4) , the clearance
volumetric efficiency is precisely defined in terms of the actual specific
volume values at p::.ints 3 and 4 on the indicator diagram.
The Indicated Volumetric Efficiency
This parameter is calculated fran measurements on an indicator diagram
such as that shown in Fig. 1.
TJYI
=
=
=
v1 - v4
v
...
(5)
indicated volumetric efficiency
volumes at the p::.ints of suction pressure equalization,
from the indicator diagram, Fig. 1.
The indicated volumetric efficiency is less than the clearance volumetric
efficiency for two main reasons:
1•
The volume at the end of the re-expansion stroke ( p::.int 4 , Fig. 1 ) rllBY be
greater than that in the case of an ideal compressor with· the same
clearance ratio and p::.lytropic index (based on pressure and specific
voluae) , due to factors such as leakage or backflow through the discharge
valve during re-expansion.
2.
Due to underpressure within the cylinder wen the piston reaches the
bottom dead centre position, the induction process continues Wile the
piston begins to move back up the cylinder and so the volume within the
cylinder is less than the sum of the clearance and swept volunes wen
pressure equalization occurs (point 1 , Fig. 1 l •
To quantify the first effect the 're-expansion loss ratio' is defined:
256
=
v...
vel -
+
v~
...
X -
(6)
v
where
vel =
clearance volume per cylinder
This loss ratio quantifies the difference between the volume increase of
the actual re-expansio n process and that of an ideal compressor (having the
same polytropic index for re-expansio n, the same clearance ratio and the same
value of :1.) as a proportion of the swept volume. It quantifies part of the
suction leakage effects which are described later.
To quantify the second effect described above, the 'under-pres sure loss
ratio' is defined:
V +V
-V
=
••
t
cl
...
v
(7)
The indicated volumetric efficiency is related to the clearance
volumetric efficiency by the loss terms defined in equations ( 6) and ( 7 l .
(8)
Various other effects, some of which are described in this paper, cause
the actual volumetric efficiency to be less than the indicated volumetric
efficiency .
Throttling at the Suction Valve
As well as causing underpress ure at the bottolu dead centre position, as
already mentioned, this reduces the suction fluid density. Costagliol a
analysed these effects [ 3 ( 1950) 1 and presented and explained the derivation
of an expression equivalent to equation ( 9) for volumetric efficiency , based
on an indicator diaaram such as that in Fia. 1.
'l,.t
=
J- 1
'1.-
J
YI
I
v1 [ 1 -
(9)
v~
where
'lvst
l
p
V
=
volumetric efficiency taking account of suction throttling
ratio of specific heats
pressure within cylinder
volume enclosed by the cylinder head, cylinder, and piston
Note: In Costagliol a's original paper the terms in brackets in equation (9)
were incorrectly printed in a fo= equivalent to (p/p, -1).
The second term in Costagliol a' s equation ( 9) takes account of the
irreversib le suction throttling process which has the overall effect of
increasing the temperatur e and reducing the density at point 1 in the diagram,
T.he equation can also be written in the following form:
=
11 vi
w
_ .L.=..l -v---ol
l
,.P,
0
257
(10)
where
W•
i
=
=
indicated suction M:Jrk
lower crosshatched area in Fig. 1.
1be right hand term in equation ( 10) represents the loss in volumetric
efficiency due to suction throttling:
Rlot
where
R
=
(11)
volumetric efficiency loss due to suction throttling.
lot
In order to illustrate the likely siQlificsnce of the right hand term in
equation ( 10), typical values from CC~QPreSSor tests are given below.
For
compressor speed
suction pressure
ratio of specific heats (R-12)
=
600 r.p.m.
=
=
2.19 1::ara
1.18
typical test results are:
nvi
..
=
w.
0.865
6.30 J
36.33 J
=
=
0.026
0.865 - 0.026
=
0.839
Throttlina at the suction valve depends not only on the geometry and flow
area versus lift characteristics of the valve port and reed, but, also on the
stiffness and dynamic characteristics of the reed. Furthermore, if there is a
preload on the valve reed, holding it in position, or if there is any tendency
for it to adhere to the seat, there may be a delay from the time when pressure
equalization occurs across .the valve to the mcment when the reed begins to
lift. This effect too influences the efficiency loss due to suction valve
throttling.
Heat Transfer to the Suction Vapour
Heat transfer which occurs to the fluid after it leaves the suction pipe
and before the suction valve closes on the induction stroke further reduces
the fluid density at point 1 in Fig, 1. Gosney [4 ( 1951), 5 ( 1953)] extended
the type of analysis carried out by Costa&liola to include the effect of heat
transfer to the suction fluid, equation ( 12) • A full derivation of the
equation, based on the approach he used, is given in [6],
(12)
where
=
Q
:
volumetric efficiency taking account of suction fluid
throttling and heat pick up
heat transfer to the fluid per cycle from the point where it
leaves the suction pipe to closure of the suction valve
Equation ( 12) is the same as equation ( 10) in the special case where the
amount of heat transfer is zero. The term representing the loss in volumetric
258
efficie ncy due to suction heat transf er can be separa
ted out as follow s:
_Q_ y-1
v••P.
where
R
lq
=
--1-
(13)
volume tric efficie ncy loss due to suctio n heat transf
er
In order to estima te 1J
, using equatio n ( 12) , the amount of heat
vst.q
transf er to the fluid must be determi nec:l. 'This
could
to the indica tor diaaram, the mean fluid temper ature be done, if, in additio n
enterin g the suction
valve and the mean fluid temper atures within the
cylind er at points 1 and 4 in
Fig. 1 were availa ble.
Heat tnmsf er occurs to the suction fluid in tl«l
stages . 'The first st.ase
occurs within the suction plenum before the fluid
enters the suction valve.
It can be evalua ted if the mean temper ature of the
fluid enterin g the suctio n
valve is measur ed.
(14)
For an ideal gas,
(15)
where
Q
sp
mE
h
c
p
T
=
=
=
heat transf er to fluid in the suction plenum per
cycle
the II88B transfe r'l'ed extern ally per cycle
specif ic enthalp y
specif ic heat capaci ty of fluid at consta nt pressu
re
absolu te temper ature
subscr ipts:
s
indica tes a proper ty mean value in the suction pipe
sv
indica tes a proper ty mean value at entry to the
suction valve
The second stage of heat transf er to the suction
cylind er, from point 4 to point 1 of the cycle (Fig. fluid occurs within the
1)•
Q
=
Q
sp
+
Q
cyl
( 16)
heat transfe r to the fluid within the cylind er betwee
n points 4
and 1 of the cycle
Assunti.ng T
, T
and T v are known, the first step in the evalua
1
4
tion of
0
is to determ ine the temper ature, T 1a, which would
exist at point 1 if
there were no heat transf er during the induct ion
proces s and if the same
indica tor diagram apPlied . The term 'adiab atic
suctio n temper ature after
induct ion' is used to denote this quanti ty. Wherea
s the throttl ing proces s
itself produc es no temper ature clumae , if ideal
gas behavi our is assume d, a
temper ature increa se would come about due to compre
ssion as the pressu re of
the induce d charge increas ed adiaba tically to the
suction pressu re at point 1.
A deri va.tion of equatio n ( 17) for the adiaba tic
suction temper ature after
inducti on is given in Append ix A.
Qcyl
259
T la
=
--T
----------------------w.i
1V-(T- +
1
where
T 13
=
V
T
1
ov
(17)
1
- - l - - P.V
ov
4
-
1
4
adiabatic suction temperature after induction
The quantity of heat transfer to the fluid within the cylinder from the
start of the induction process to the start of the compression process is
given by equation (18) (a derivation is given in Appendix B).
=
_J_ p
l - 1
VT[-1- -_1 )
Tta
o t sv
Tt
(18)
Thus, a procedure has been presented whereby T 1 a , Q and thence 11 v • t q can
be dete:mined from an indicator diagram with the corresponding mean gas
temperatures at the points of suction pressure equalization ( 4 and 1 in
Fig. 1 ) and the mean temperature of the fluid entering the suction valve.
BACKFLOW AND LEAKAGE
Due to late closure of the suction valve, fluid may flow back into the
suction plenum. Also, leakage past the piston and past the suction valve when
it is seated causes a loss in 11888 fram the cylinder to the suction plenum
durina compression, duri.nir the discharae process, and during re-expsnsion.
(19)
where
mb I•
m
1
=
=
backflow throuah the suction valve and leakage from the cylinder
to the suction plenum per cycle
mass
induced
The mass induced is defined as follows:
=
m1 - m4
(20)
of fluid present within the cylinder at the point of suction
pressure equalization nearest bottom dead centre (point 1, Fig. 1)
ma.ss of fluid within the cylinder at the point of suction pressure
equalization nearest top dead centre (point 4, Fig. ll
DDSS
The 'suction leakage and backflow ratio' can be defined as follows:
Ill
I)
bls
blo
(21)
The 'suction side sealing efficiency' can be defined as
=
1 -
I)
bl•
(22)
The volumetric efficiency allowing for valve throttling, heat transfer to the
auction fluid and leakage is given by
260
(23)
where
r)valql
=
volumet ric efficien cy taking account of suction fluid
throttlin g, heat transfer and. leaksae
anmR FAC'l'C&S
Throttli na of the fluid enterina the suction plem.m chamber,
fran the
inlet pipe where the suction pressure is JlleaSUl'ed, will reduce
voluoetr ic
efficien cy. This effect is included in the equation s already
listed •
In refriger ant COJUpreSsors there ._y also be a loss of
refriger ant from
the cylinder with lubricat ing oil which bypasses the piston,
as the solubili ty
of refriger ant in the oil can be hilb at the condi tiona encounte
red within the
cylinder . This is in fact !lllOther form of leakage and. would
be included in
the ratio v
•
bls
The possibi lity that cyclic solubili ty of refriger ant in lubricat
ing oil
may reduce the volumet ric efficien cy of reciproc ating refriger
ant compres sors
has been suggeste d in referenc es [1] and [6]. The expressi
ons presente d in
this paper could be axl.ified to take account of any such effect.
CXNCLUSIONS
The processe s which occur within a reciproc ating compres sor
and
particul arly the factors which affect its displace ment utilizat
ion have been
describe d. These factors have been discusse d, with referenc
e to the
literatu re, and. paramete rs have been defined to assist in their
quantifi cation.
The expressi ons given in this paper which follow on from the
work of
Costa,Kl iola, are based on the assUIIpti on that the fluid can
be regarded as an
ideal gas. These expressi ons do not strictly apply to real
fluids,
particul arly refriger ants, which have rather ccmplica ted empirica
l equation s
of state, or to refriger ant/oil mixtures . Equivale nt expressi
ons to those
describe d in this paper can be formulat ed for real gases, vapours,
or even
refriger ant/oil mixtures , but, cannot all be written as relative
ly simple
expressi ons in tenns of a sme.ll number of measurab le paramete
rs. The ideal
gas assumpti on is, however, a useful tool in identify ing
and quantify ing, if a
little crudely, the main factors which influenc e the utilizat
ion of volumet ric
displace ment.
!nevi tably, generali zations and approxim ations muat be made
in practice
and in theory, due to the limitati ons of test measurem ents,
and in order to
separate out and roughly quantify effects which, in reality,
overlap, interact
and are not totally distinct . The theory is put forward as a
set of tools for
the detailed analysis of displace ment utilizat ion in reciproc
ating
compres sors.
ACKNOWLEOOEMENT
The author greatly apprecia tes and acknowle dges the helpful
discussi ons
on the subject matter of this paper which he has had with his
colleagu e,
Professo r W.G. Scaife. A Thomas Andrew Coamon Grant from the
Institut ion of
Mechani cal Engineer s to~ atterdin g the conferen ce is grateful
ly
scJmowle dged.
261
1.
2.
3.
4.
5.
6.
REF'ERENCES
Kleinert, H.J. aild Najork, H. 1986. "Energetic and volumetric
characteristics for the uniform valuation of gas and refrigeration
compressors". Proceedings of the 1986 International Compressor
~ineering Conference - at Purdue, edited by Hamilton, James F. and
Cohen, Raymond, West Lafayette, Indiana.: Purdue University. vol. I,
pp.210-227.
Pand.eya, Prakash and Soedel, Werner 1978. "A generalized approach tmerds
compressor perfonBnCe analysis". Prcceedinis of the 1978 Purdue
Compressor Technology Conference, edited by Hamilton, James F. , West
Lafayette, Indiana.: Purdue University, pp.l35-143.
Costagliola, Michael. 1950. "'The theory of spring-loacled valves for
reciprocating compressors" • ASME Transactions, J. Appl. Mech. , Dec. , 17,
4, pp.415-420.
Gosney, W.B. 1951. "'!he effect of wiredrawing and cylinder heating on the
volumetric efficiency of compressors". Modem Refria:eration, vol. IV, no.
638, pp.128-131.
Gosney, W.B. 1953. "An analysis of the factors affecting performance of
small CQ~~~J~ressors". Proceed.inas of the Institute of Refria:eration
1952/53, pp.185-216.
McGovem, J.A. 1988, On Refrigerant Compressors, PhD Thesis, 'The
Uni.versi ty of Dublin, Trinity College.
APPENDIX A
DERIVATION OF THE ADIABATIC SUCTION TlH'ERATURE AFI'ER INDUCTION
In the definition of the adiabatic suction temperature after induction it
is assumed that the actual indicator diagram applies for a hypothetical
adiabatic induction process,
Application of the energy equation to the closed system bounded by the
cylinder heacl, cylinder, and piston for the induction process between points 4
and 1 on the indicator diagram shown in Fig, l results in the following
expression:
Q
cy I
=
cp
T
T
=
+ Wc i
= m1c pT 1
- m4 c p T 4 - (m 1 - m4 )c p Tsv
heat transfer to the fluid within the cylinder during the
induction process
specific heat capacity at constant pressure
absolute temperature
the mean absolute temperature of the fluid entering the suction
valve
ov
From equation (24), if there is no heat transfer, Qcyl
-
where
T1a
m
1a
(24)
=
=
m4 c p T4
..
w_
= 0 and
+
the adiabatic suction temperature after induction
the IIIII.SS within the cylinder at point 1 on the indicator
diaa:ram after adiabatic induction
Note: Both T ta and mta are hypothetical rather than actual quantities.
262
(25)
But m = pVlf(r and so
c
Po
'Where
R
=
v1Rp
-
p
a
v4
c
p
~
specific gas constant
Also c /R = J1 ( J
p
P,
- 1) and
J
y-=-T
(V
1 -
vt
4
T;
_]__
P. J - 1
oi
=T +
4
= v4
Th
+
W
V)
v4
~
1a
therefore
+
v1
T
- v4
-
vt
v1 - v4
T
w•I
TOYPO
OY
OY
w
-
v
• i
TOYpo
T
-
18
l
v4
T;
_1
[
T
ov
J - 1
--J-
J - 1
--J-
1
=
F~
OY
(26)
APPENDIX B
EVALUATION OF THE HEAT TRANSFER 1'0 THE FLUID
WITHIN THE CYLINDER DURING INDUCTION
equations (24) and (25)
Qeyl =
Butm
--------------------------T
m c T
1 p 1
-
m
c T
ta p 1a
-
(m
1
- m
la
)c T
p ov
pV/RT and so
=
___1_ p V T
l
- 1 •
263
t
sv
(-1.
T - ..L}
T
h
1
(27)