A reciprocating compressor takes in a gas and delivers it at a higher pressure through the cyclic action of pistons in cylinders. There are two main types - single-acting and double-acting. The compression process can follow different thermodynamic paths like isothermal, polytropic, or isentropic on a pressure-volume or temperature-entropy diagram. Isothermal compression provides the minimum work and highest efficiency. The indicated power and efficiency of a reciprocating compressor depends on parameters like mass flow rate, inlet and outlet pressures and temperatures, and the compression process path.

1. 12
PogitiveDisplacementMachines
The functionof a compressoris to takea definitequantityof fluid (usuallya
gas,and most oftenair) and deliverit at a requiredpressure.The mostefficient
machineisonewhichwill accomplishthiswith theminimuminput of mechanical
work. Both reciprocatingand rotary positivedisplacementmachinesare used
for a varietyof purposes.On the basisof performancea generaldistinctioncan
be madebetweenthe two typesby definingthe reciprocatingtype as having
the characteristicsof a low massrateof flow and high-pressureratios,and the
rotary type as having a high massrate of ffow and low-pressureratios. The
pressurerangeof atmosphericto about9 bar is commonto both types.
Somerotary machinesaresuitableonly for low-prcssureratio work, andare
applied to the scavengingand superchargingof engines,and the various
applicationsof exhaustingand vacuum pumping. For pressuresabove9 bar
the vane-typerotary machinecan be usedto supply boost pressures,but for
sustainedhigh-pressurework up to 500bar and above,for specialpurposes,
the reciprocatingtype is used.
Both basic types exist in many difrerent forms each having its own
characteristics.They may be singleor multistage,and haveeitherair or water
cooling.The reciprocatingmachineis pulsatingin action which limits the rate
at which fluid can bedelivered,but the rotary machineis continuousin action
and doesnot havethis disadvantage.The rotary machinesare smallerin size
for a given flow, lighter in weight and mechanicallysimpler than their
reciprocatingcounterparts.The treatmentand scopeof the following sections
is fundamental and is not exhaustive.Many compressorsare designedto
overcome the deficienciesof the basic machines and to satisfy spccid
requirements.For descriptionsof thesemachinestheexcellentliteraturesupplied
by the manufacturersconcernedshouldbe consulted.
For a compressorwhich operatesin a cyclic or pulsatingmanner,suchrs
a reciprocatingcompressor,the propertiesat inlet and outlet arc thc averiNgc
valuestaken over the cycle.Alternativelythe boundary of the control vslrrrrp
is chosensuchthat statesI and 2 areconstantwith time,the positioossdcrld
beingremotefrom the pulsatingdisturbance.

2. br*-drrltm*nrn-
rt l?-r
Singlc-acting(a) and
doublc-acting(b)
rcciprocating air
comprcssors
&
72.1 Reciprocating comprossors
Typical reciprocatingcompressorcylinder arrangem€ntsare shom i
Fig.I2.I (a)and(b).Themechanisminvolvedisthebasicpiston,connecting-m{,
FE IZ
volume
'rccipro
comprc
clearan
Rctiivcr...+-pressurc -;>
Induction E Atmospheric
+- pressurre
He8E
o'
To reccivcr
or next stage
Double-acting
compressoror stage
"tll!,,and
cylinderarrangement.Initially the clearancevolumein thecylindcr
will be considerednegligible.Also the working fluid wilt be assumedto bc e
perfectgas.The cycle takesone revolution of the crankshaftfor compkrir
and the basicindicatordiagramis shownin Fig. 12.2.
Thevalvesemployedin mostair compressorsaredesignedto giveautometb
action. They are of the spring-loadedtype operatedby a small differere i
pressureacrossthem, the light spring pressuregiving a rapid closingactir

3. try-.lzz Pressurc-
volumediagramfor a
rcciprocating
compressorwith
clcaranceneglected
let frclrrtr;n
The lift of the valvg to give th€ required.air0ow should,be assmall aspossiblc
and shouldoperatewithout shock.
In Fig. 12.2the line d:-a representsthe induction stroke.The massin the
cylinder increasesfrom zero at d to that requiredto fill the cylinder at a. In
the idealcasethc tcmperatureis constantat T1for this processand thereis no
:heatexchangewith the surroundings.Induction commdnceswhcnthe pressure
differenceacoss the valve is sufficientto opcn it. Line abc representsthe
compressionand delivery stroke, As the piston begrns'its return stroke the
pressurein thecylinderrisesandclosestheinletvalve.Thepressurerisecontinues
with the returning piston as shownby line ab until the pressureis:reachedat
which the deliveryvalveopens(a valuedecidedby the valveand the pressure
in the receiver).The deliverytakesplaceas shown by the line bc, which is a
processat constanttemperatureT2,constantpressurep7, zQtoheatexchangg
and decreasingmass.At the end of this strokethe:cycleis repeated.The value
of the deliverytemperature?r dependsupon the law of compressionbetween
a and b, which in turn dependsupon the heatexchangewith the surroundings
during this process.It may be assumedthat the generalform of compression
is the reversiblepolytropic (i.e.pV' = constant).
The net work donein the cycleis givenby the4reaof thep-V diagramand
is the work done on the gas.
Indicatedwork done on the gasper cycle
: afeaabcd
: area abef* area bcOe- area ad0f
/El
F
tEnr
rr*
I
I *-r*i
h:-5c :fffi
rec ro bt

4. I
I
turir d-pLcrnfit mac{rincr
Le.
Fig.123 Compression
processonap-udiagram
Example12.1
Solution
M
Usingequation(3.24)for areaabef"
workinput:W#9 t pzva
: (pzvb-p,n 1(-1_ * , )
n-l /
Workinput:(pzvo- "' I +n- I
Prv")-
nlf-
=n .QzVa- Pr4)
From equation(2.6)we canwrite
plVr: mRT, and p.rVo= mRT2
wheren is the massinducedand deliveredpercycle.Then
Work input per cycle- n
: mR(T,- T,l
n-l
properties(i.e.p againstu).
l.0l3xlx105
Thedeliverytemperatureisgivenby theequation{3.29),
/ ' ( a - t Y a
i.e. T": TI2l- .p,/
A single-stagereciprocatingcompressortakes I m3 of air per minute at
l.0l3bar and l5'c and deliversit at 7 bar. Assumingthat the law of
compressionis pyr'ts : constant,and that clearaneeis nigli!;ible,calculate
the indicatedpower.
Massdeliveredpermin,n : P:!t
R?i
(12.1)
(r2.2)
work doneon theair per unit time is equarto the work doneper cycletimes
the numberof cyclesper unit time.The rateof massflow is moreoftenused
than the massper cycle;if the rate of massffow is given the symbor rh,and
replacesm in equation(12.2),then theequationgivei the rateit whichwork
is doneon the aiq or theindicated.po*ei.
The working ffuid changesstatebetweena and b in Fig. 12.2,fromp1and
Tt to pz and Tr, the changebeingshownin Fig. 12.3,which is a diagiamof
287x 288
:1.226kglrnin
whereT, : 15* 273:288K.
Detiverytemp.,rz : rr(f)" "'" : zas(#)(1'3'-rvr'!5
:475.4K

5. T
r l l l f
r12.2)
dc tirncs
lfcn uscd
lnad
ich qort
I p, aod
gmm of
jnute at
r lan of
calculatc
Theactualpowerinput to thecompressorislargerthanthc indicatcdpower,
due to the work necessaryto overcomethe lossesdue to friction' ac.
12.1 ReciprocetingcomprolaoF
Fromequation(12.2)
Indicatedpower: -
n-;p1
Tz- Ttl
n-l
wherem is themassflow rate,
1.35x 1.226x 787x (475.4- 28t)
l.e. Indicatedpow€r=
l03x(1.35-l)x6O
:4.238kW
i.e. Shaftpower= indicatedpower + friction power
The mechanicalefficiencyof the machineis givenby
CompressormechanicalefficiencY:
indicatedpower
shaftpower
Input power=
shaftpower
(12.5)
efficiencyofmotor anddrive
If the compressorof Exaniple12.1is to be drivenat 300rev/min and is a
single-acting,single-cylindermachine,calculatethe cylinder bore required,
assumingaitrot<i to boreratio of 1.5/1.Calculatethe powerof themotor
requiredlo drivethecompressorif themechanicalefficiencyof thecompressor
is gS9/.and that of the motor transmissionis 90%'
Volumedealtwith per unit timeat inlet: I m3/min
therefore
Volumedrawnin percycle=
*
: 0'00333m3/cycle
i.e. Cylindervolume: 0.00333m3
therefore
n
d'L = o.oo333
4
whered is the bore and Lthe stroke,
i.e. la'fij x d): o.oo333
4
{G
(r2.3)
(114)
To determinethe power input requiredthe efficiencyof the driving motor
must be takeninto account,in addition to the mechanicalefficiency.Then
Example12.2
Solution

6. Hthlr dbplcrmcrrt mecfrinc
therefore
d3:0.00283m3
i.e. Cylinderbore= 141.4mm
Powerinput to thecompresro,:
4,'.'l
= 4.ggkw'
0.85
therefore
4.gg
Motor power=
6:
5.54kW
Proceedingfromequation(lz.2l, otherexpressionsfor thcindicatedwork
canbederived,i.e.
Indicatedpower:
fi**1
Tz- Tr)=
fi*^rr(?
- t)
Alsofromequation(3.29)
T2 ( Pr{n-rttn
T: P'/
Therefore
Indicatedpower:-!-,;,p7r{(g)t"-t"'- ,} 1rr.uyn-l 'tptz
)
or Indicatedpower:
*o,o{(f)"-""
- ,} e2.7)
wheretzis thevolumeinducedperunit time.
Fig. 12.4 l
compressio
on a p-o di
Eral
The condition for minimum work
The work done on the gasis given by the areaof the indicator diagram,and
the work donewill bea minimum whenthe areaof thediagramis a minimum.
The_heightof the diagram is fixed by the requiredpt rrui. ratio (whenp, is
lrr{): and the length of the line da is fixed by the cylinder volume,which is
itselffixedby therequiredinductionofgas.Theonly prooesswhichcaninfluene
the areaof thediagramis the line ab.The positiontakenby this line is decidcd
by the valueof theindexn; Fig. r2.4showsthe limits of thi possiblcprooesscs.
Line abt is accordingto the law pll: constant(i.e.isothermar)
Line ab2is accordingto the law pyr = constant(i.e.isentropic)
Both processesare reversible.
-
Isothermalcompressionisthemostdesirableprocessbetweenaandb,giving
the minimum work to be done on the gas.ihis meansthat in an rcrnl
1{16

7. rork
Jra0i
[o',
I
Ld
F.
f : t r
!
Fg. 12.4 Possible
compressionprocesses
on a p-v diagram
Example12.3
indicatedworkpercycle: p2V6,lnPf,
= PrVrhA
Pt
: mRThU
(12"8)
(l2.e)
(lzt0)
isothermalemAaryd fu
12.1 RcciwocdilrO compr=.t
p
Pz
plz] = const.
pZ'= const.
pZ : const.
compressorthe gastemp€raturemust be kept ascloseaspossibleto its initial
value,andameansofcoolingthegasisalwaysprovided,eitherbyairor bywater.
The indicatedwork done when the gasis compressedisothermallyis givcn
by the areaablcd.
Area ablcd : ateaabref+ areabrc0e- areaad0f
Areaablef: pzVa,6& lfrom equation(3.9))'
Pr
i.e. indicatedwork perclcle= p2V6,ln!2* PtVt,- PrV"
P t
Alsop1Vr: p276,,sincethe processab1is isothermal,therefore
Pt
Whenm and %in equations(12.8)and(12.9)arethemassandvolumeinduccd
per unit time, then theseequationsgivethe isothermalpower.
lsothermal efficiency
By definition,basedon the indicator diagram
isothermalwork
,tv43
EI
Isothermalefficiency:
Using the data of Example
compressor.
indicatedwork
l2.l calculatethe
t3

8. Fig.125 Isothermal,
polytropic, and
iscntropiccomPression
processeson a T-s
diagram
Solution From equation(12.9)
Isothermalpower: rhRTlrr.U
P t
t.225x0.287x288
,,ln
7
1.01360
: 3.265kW
FromExample.l2.l,
Indicatgdpower:4.238 kW
Thereforeusingequation(12.10)above
Isothermalefficiencv:3.?91:0.77 ot 77oh'
4.238
Theleastdesirableformofcompressionin reciprocatingcompressorsisthat
givenby theisentropicprocess(seeFig.12.4).Theactualformofcompression
will usuallybe onebetweenthesetwo limits.Thethreeprocessesareshown
representedon a T-s diagramin Fig' 12.5:
1-2' representsisothermalcompression
I-2" representsisentropiccompression
1-2 representscompressionaccordingto a lawputr= constant
Thevalueof n is usuallybetween1.2and 1.3for a reciprocatingair compressor'
The main methodusedfor coolingtheair is by surroundingthe cylinderby a
waterjacket and designingfor the bestratio of surfaceareato volumeof the
cylinder.
12.2 Reciprocating comprossors including clearance
Clearanceis necessaryin a compressorto give mechanicalfreedomto ttrc
working parts and allow the necessaryspacefor valveoperations.
Figure 12.6showsthe ideal indicatordiagramwith the clearancevolumc
included.For good-qualitymachinestheclearancevolumeis about6% of the
sweptvolume,and with a sleeve-valvemachineit can be as low as2%, but
machineswith clearancesof 30-35Yoarealsocommon'
F!S. l2f
indicator r
reciprocati
comprEsg
clearance
Fig. 12.?
and re-cx
massesol
reciproca
compress
tm

9. )
I
3
Fig. f2.6 Ideal
indicator diagrarnfor a
reciprocating
compressorwith
clearance
Fig. 12.7 Compression
and re-expansionof
massesof gasin a
reciprocating
compressor
12.2 Reciprcceting corprrt -c|rfi -ro
When the deliverystroke bc is completedthe clearancevolume 7" is full of
gas at pressurep, and temperatureTz. As the piston proceedson the next
induction stroke the air expandsbehind it until the pressurept is reached.
Ideally assoonasthepressurereachespr, the induction offresh gaswill begin
andcontinueto theendof thisstrokeat a.Thegasisthencompressedaccording
to the law pV' : C, and deliverybeginsat b ascontrolled by the valves.The
effectof clearanceis to reducethe inducedvolume at p, and I from ( to
(V - Vi. The massesof gasat the four principal pointsaresuchthat m" : flt
andrh,: nra.The massdeliveredpef unit timeis givenby (mu'- fr"I, whichis
equalto that induced,givenby (n" - fri.The propertiesof the working fluid
changein processesa-b and c-d asshownin Fig. 12.7.
Referringto Fig. 12.6the indicatedwork done is given by the areaof the
p-V diagram.
Indicatedwork : areaabcd
= areaabef- areacefd
Then,usingequation(12.2)
Indicatedpower= -!-rrr^n(Tz - Tr)-
r,nrn(T,
- 1i)
,n I
/t?

10. Hin dbphcrncrt machincl
Example12.4
410
i.e. Indicatedpower:
;lR(m"
- rt,dl(T2- Ttl
:fr**1r,_r,) (r2.11)
wherem is the massinducedper unit time : Qh,- rhi.
A comparisonof equations(12.11) and(12.2)showsthat they areidentical.
The work doneon compressingthe massof gasn" (or mu)on compression"
a-b, is returnedwhen the gasexpandsfrom c to d. Hencethe work doneper
unit massof air deliveredis unafrectedby the sizeof the clearancevolume.
Other expressionscan be derivedas before.From equation(12.7)
Indicatedpower:
no,t{(f)"
"'" - ,}
Also, if therearelcycles per unit time, then we have:
Fig. l2t I
volumedia
Examplel!
V: f V"- Yal
therefore
Indicatedpower:
;o,f( %- n ){(
?)"
"'" - r}
(t2.12)
(12.13)
Themassdeliveredperunit timecanbeincreasedbydesigningthemachinc
toh doubleacting,i.e.gasisdealtwithonbothsidesofthepiiton,theinduction
strokefor onesidebeingthecompressionstrokefor theother(seeFig.12.1).
A single-stage,double-actingair compressoris requiredto deliver 14m3 of
air per minute measuredat 1.013barand 15"C. The deliverypressurcb
7 bar and the speed300rev/min. Take the clearancevolumeas 5% of tb
swept volume with a compressionand re-expansionindex of n = lJ.
Calculatethe sweptvolume of the cylinder,the delivery temperaturgrd
the indicatedpower.
Solution Referringto Fig.12.8
Sweptvolume* (V^- V"): V"
and Clearancevolume,V": 0.05V"
i.e. V,- 1.05V,
Usingequation(12.12)for a double-actingmachine
Volumeinducedpercycle,(V,- Yi =
300x2
- 0.0233m3/cycle
(cyclesper minute = revolutionsper min{te x cyclesper revolutioo}
t4

11. -
12.2 Rcciprocating comprur.on including clcaruncc
Fig. l2.E Pressure-
volumediagramfor
Example12.4
L l l t
icaf
rim.
r Psf
B
l-4-l
:o.osn(ft)""
therefore
(Y,- Y) = 1.05r,- 0.221V':0.0233m3/cycle
therefore
I'.:
oio=2=3=3= 0.0281m3/cycle'
0.829
i.e. Sweptvolumeof compressor= 0.0281m3
DeliverYteml - /P'{r-
t)h
)''T2: tlffi/ fromequation(3'29)
Tr: 15*273:288K
/ t ( 1 ' 3 - l Y l . 3
T":2881 I'
r.013/
-450K
V
L l l t ,^:*(?),,.
i.e. Va:0.221V,
Lll I
lio€
Doc
Ll'
tr of
lc rt
t thc
: ! - i
ihc
and
i.e.
therefore
Deliverytemp.= 177'C
Usingequation(12.7)
Indicatedpower
:ft',r{(f,)"","_,}

12. II
Pcritir Cieisnqn mactrinoc
1.3
=_x
0.3
1 . 0 1 3x 1 0 5x l 4 ( / 7 ( r . 3 -r l ir . 3 )
rot*o it,*"/
- rlkw
l.e. Indicatedpower: 57.6kW
The approachusedfor a particularproblemdependson how the dataare
statedand thequantitiesevaluatedduringthesoluiion.In someproblemsit is
betterto evaluatern and r, andthenuseequation(l2.ll) forihe indicated
Power;e.g.in Example12.4above,I hasbeencalculated,andthemassinduced
is givenby
l.0l3x14xl05
0.287x288x103
Then,usingequation(12.11)
= 17.16kglmin
Indicatedpower:
fr*^,Tz
- TJ
1.3x 17.16x 0.287(450-288)
0.3 x 60
: 57.6kW (asbefore)
The diagramspreviouslyshown(e.g.Fig. 12.8)areidealdiagrams.An actual
indicator diagramis similar to the ideal exceptfor the induction and delivery
processeswhich aremodifiedby a valveaction.This is shownin Fig. 12.9.Thc
wavinessof the lines d-a and b-c is due to valve bounce.Automatic valvcs
are in generaluse(seeFig. l2.l), and theseare lessdefinite in ssflsn rhen
cam-operatedvalves;they also give more throttling of the gas.The inductioo
stroked-a is a mixing process,theinducedair mixing with that in thecyliodcr_
Vofumetric efficiency, Iv
It hasbeenshownthat one of the efectsof clearanccis to rrb t *d
volume to a value lessthan that of the sweptvolumc,Tli n th h r
m=
Ftg; 12.9 Actual
indicator diagram for a
reciprocating
qomprssor
412
Fig. l2-t
diagram
reciproc
compres

13. t
lb
E
E
12.2 Rociprocating comprolsorr Includlng clcarlncc
requiredinductionthecylindersizemustbeincreasedoverthatcalculatedon
theassumptionofzeroclearance.Thevolumetricefficiencyisdefinedasfollows:
4, : themassofgasdelivered,dividedbythemassofgaswhichwould
fill the sweptvolumeat thefreeair conditionsof pressureand
tempcrature (t2.14')
or
4" : the volumeof gasdeliveredmeasuredat thefreeair pressureand
temperature,divided by the sweptvolumeof the cylinder(12.15)
The volumeof air dealt with per unit time by an air compressoris quotedas
the freeair delivery(FAD), and is the rate of volumeffow delivered,measured
at the pressureand temperatureof the atmospherein which the machineis
situated.
Equations(12.14)and (12.15)canbe shownto beidentical,i.e.if the FAD
per cycleis 14at p and T then the massdeliveredper cycleis
pV
m:-
RT
The massrequiredto fill the sweptvolume,V, al p and T is givenby
pU,
t'=rr
Thereforeby equation(12.14),
mpVRTV
n.:-=:-X-:-," ms RT pV" U,
The volumetric efficiencycan be obtained from the indicator diagram.
Referringto Fig. 12.10
Volume induced: U,- Va: V"+ V"- Va
fl3
p
Pt
Fig. 12.10 Indicator
diagramfor a
reciprocating
comPressor

14. Pctthr dlrplaccmcnt mechlnc
Ftl
volu
Exan
andusingequation(3.25)
therefore
2:(?)'''ievo=r(r)"
:v3-""{(fi)''-'}
ie 4.:!_fr{&),,._,}
v"v,
(r2t6)
(rzr7l
r r./ ,, ( Prt'nVolumeinduced= I/,+ V.- r";)
Henceusingequation(12.15),
4 r :
V,- Vo_V,- V"{(prlp)tt,- r)
It is important to note that this definition of volumetricefficiencyii only
consistentwith that of equations(lz.l4) and(t2.15) if thecondition. of press.rr"
and temperaturein the cylinder during the induction strokeareidenticalvith
thoseof the freeair. In fact the gaswil be heatedby the cylinder wal| aod
thgrewill bea reductionin pressuredueto thepressuredrop iequiredto ind,.oe
the gasinto the cylinder againstthe resistanceto flow. Thise modificationsto
the.ideal caserequire a more careful application of the formulae prcno'oy
derived.
For example,when the FAD per cycleis denotedby v atp and T thca
^ = Pv=Pt(v:- vi :
. RT R?i
i.e. FAD/cycle,V=(V,- Vn++
I r P
wherep, and I are the suctionconditions.
A single-stagc,double-actingair compressorhas a FAD of t4 rr,,r
measuredat l.0l 3 barand I 5"c. Thepressureandtemperaturein rhcqr-
during induction are 0.95bar and li'c. the deliveryprcssurch 7 b d
the.index of compressionand expansion,n, is equai io 1.3.Crb t
i"lr:or{
powerrequiredandthevolumetribefficiency.Thcdt nc*
is SVoof the sweptvolume.
Thep V dragramis shownin Fig. 12.11
Massdeliveredper unit ti^", ,h : Pt
Rr
Example12.5
Solution
414

15. v.-ll u ,l v
:0.05y, ' | |
wherethe FAD is V at p and T
l,e. m:
1.013x 14x l0s
0.287x 288x
122 Brlprocdng coff,Frrr hffil -ns
I
fit, f2.f f Pncssure-
volumcdiagramfor
Erample12.5
T r b
,..ll/1 3 - const
305K
1
.o
o
g
0.95
l0I
: 17'16kglmin
whereT=15*273=288K.
Tr: Tr(!2"-t''" fromequation(3.2g)- 'prl
/ ? ( 1 ' 3 - r Y r ' 3
i.e. Iz:305"1+l =483.6K
0.95l
whered :32 * 273: 305K.
Fromequation(12.11)
Indicatedpor".r=
;l
fiR(72- T)
1.3x 17.16x 0.287(483.6- 305)
0.3x 60
:63.5 kW
As before
/ r ^ l / '
vd=v"lv-2l|
pr /
/ 7 1 / 1 ' 3
i.e. ya:0.054(
_ | :0.054 x 7.3680'76e
1u'95/
:0,232V"
therefore
V.- Yo: Yt- A.232y..:LASY.- O.232V,:0.818y.
.n5

16. Po.ld[ dfrpbcrncm nec|rlnc
Usingequation(l2.l7)
FAD/cycle=(V,-"r;?
i.e. FAD/cycle= 0.8182,. 39
"
-995-
305 1.01t:
0'724V,
Thenfromequation(l2.l5)
'":f":ry: o'724ot72'4Yo
Notethat if thevolumetricefficiencyin theaboveexampleis evaluatedusing
equation12.16then
4"=|-3{P"'-,}:, -o'l:n,{(a)""-,}v"Ln/ ) v" [0.95l
'
J
:0.818 or 81.8%
There is a considerabledifferencebetweenthe two values,sincethe lattcr
answerignoresthe differencein temperatureand pressurebetweenthefrcc eir
conditionsand the suctionconditions.
| 2.3 Multistago compression
It is shownin section12.1that theconditionfor minimumwork is that tb
compressionprocessshouldbe isothermal.In generalthe temperaturerfu
compressionis givenby equation(3.29),Tz= Tr(pzlhf'-ry'. Theddir:ry
temperatureincreaseswith thepressureratio.Further,fromequation(lzl6i
F[. tl.l
the volu
of incrtr
delivery
Fig.l11
volume
two-sta
?":l
it can be seenthat as the pressureratio increasesthe volumetric
"tir:itdecreases.This is illustratedin Fig. 12.12.
For compressionfrom pr to pz the cycleis abcd and the FAD pcr ct*.
V"- Vaifor compressionfrom pt to pt the cycleis ab,c,d,and tb FAI) a,
cycleis v.* va'i for compressionfrom pt to p+ the cycleis ab'c.f dL
FAD percycleis Y.- Vr'. Thereforefor a requiredFAD thecy{i&-d
haveto increaseasthe pressureratio increases.
The volumetricefficiencycanbe improvedby carryirg oot rbq-r
in two stages.After the first stageof compressionthe n"a i Fa ar e
smallercylinder in which the gasis compressedto thc reqid H ;1j113"
If themachinehastwo stages,the gaswill bedeliveredlt t a drii -.1,"
but it could be deliveredto a third cylinder for highcr FrrG din Tb
cylinders of the successivestagesare proportioned to La t|c rot' hc of grc
deliveredfrom the previousstage.
-ft{&)"'-'}
416

17. Juernt
I hncr
lec r.n
lI
II
I
b tbc
I dtcr
blirwf
la16
Fig12.12 Effecton
the volumetricefficiency
of increasingthe
dclivery pressure
Fig.12.13 Pressure-
volumediagramfor
two-stagecompression
|2f ffirrt
!'*l | ', .l
The indicator diagram for a two-stagemachineis shown in Fig. 12.13.In
this diagramit is assumedthat the deliveryprocessfrom the first or LP stage
and the induction processof the secondor HP stageareat the samapressure.
The ideal isothermalcompressioncan only be obtainedif ideal cooling is
continuous. This is difficult to obtain during normal compression.With
multistagecompressionthe opportunity presentsitselffor the gasto be-eoolod
asit is beingtransferredfrom one cylinder to the next,by passingit through
an intercooler.If intercoolingis complete,the gaswill enterthe secondstrgp
at the sametemperatureat which it enteredthe first stage.The savingin wort
obtainedby intercoolingis shownby the shadedareain Fig. 12.14and thc
diagramof the plant is shownin Fig. 12,15.The two indicator diagramsabod
and a'b'c'd' are shownwith a commonpressure,p1.This doesnot oacurin I
real machine as there is a small pressuredrop betweenthe cylindcrs.An
after-coolercan be fitted after the deliveryprocessto cool the gas.
{17
!fdc s
)D r*'
F thc
||rould
lcssr;n
into a
Essure-
r stagc,
n Thc
rdgas

18. p
Pz
I
I
btiw dbplrecaut mrchincs
FL. l2.l{ Efrectof
intcrooolingon the
comprcssionwork
Fig. 12.15 Plan
showingintercooling
betweencompressor
stages
Fig. 12.16 T*s
diagramshowing
intercoolingand
aftercooling
418
Fis.
volu
shor
Exa
r(fr)"-"'"o=.(fr)'"-"'t,I
The deliverytemperaturesfrom the two stagesaregivenby
and Tz:
respectively.This assumesthat the gasis cooledin theintercoolerbackto the
inlettemperature,andiscalledcompleteintercooling.To calculatetheindicatcd
powertheequations(12.1l)or (12.13)canbeappliedto eachstagescpantcly
and the results added together. Two-stage compressionwith omplac
intercoolingand after-cooling,and equal pressureratios in each stagF,b
representedon a T-s diagramin Fig. 12.16.
d ' a ' b
First or
LP stage
cycleabc

19. Example12.6
Solution
Fig.12.17 Pressure-
volumediagram
showingbothstagesfor
Example12.7
12t Ht3rf-on
In a single-acting,two-stagereciprocatingair comprc$Ed'+5 rs d ri pcr
minute are compressedfrom 1.013bar and 15'C througb r prcrrc ntb
of 9 to I .Both stageshavethesamepressureratio,andthclar of crycdo
andexpansionin both stagesispV t'3 : constant.If intercoolingisoqLrc'
calculatethe indicated power and the cylinder swept volumcs 13quilGd'
Assumethat the clearancevolumesof both stagesare5o/oof thcir rcspAiw
sweptvolumesand that the compressorruns at 300rev/min'
The two indicatordiagramsare shownsuperimposedin Fig. 12.17.Thc LP
stagecycleis abcdand the HP cycleis a'b'c'd''
p
Pt
Now p, :9Pr, alsop1/p1= PzlPotherefore
P?: P,,Pr:9P?
therefore
pilpr: J9 : 3
Usingequation(3.29)
:: (+)t"-"'" thereforeT' :3tr'r-rvr'r
?i P' /
whereTl:15*273:2SsK,andfisthetemperatureoftheairenteringthc
intercooler,
i.e. 4: 288x 1.289: 371K
Now as n, th, andthe temperaturedifrerenceare the samefor both stageg
thenthework donein eachstageisthesame.Thereforeusingequation(I 2'l I )
Totalindicatedpower:2
" *ftR$t-
T)
2 x 1.3x 4.5x 0.287(37t- 288)
0.3x 60
: 15.5kW
419
back to thc
bc indicatcd
p separately
h complete
ch stage.is

20. Poritivo dirplaccorent machinea
'.-l l- ,' | ,
Fig. l2.IE Pressure-
volumediagramfor LP
stagefor Example12.6
The massinducedper cycleis
4.5
IOO
: 0.OtSkg/cycle
Thismassis passedthrougheachstagein turn.
For the LP cylinder,referringto Fig. l2.lg,
,, ,r mRT, 0.015x 287x 288
v^- vd:- :0.0122mtlcycle
pt 1.013x 105
Usingequation( 12.t6)
V"-V^ V(/n,t'^ )
4,:#: l _ii{ a I * l l: I _0.05(3rirr_l)
v, l,.(p,) )
therefore
4,:1-0'066:0.934
Then
v^:
v'- va. 0'0122
'
0'934
=
0934
: o'0131m'/cYcle
i.e. Sweptvolumeof LP cylinder:0.0131m3
FortheHP stage,a massof0.015kg/cycleisdrawnin at l5 'c andapressure
of p,: 3 x 1.013: 3.039bar,therefore
vorumedrawn,n- o'015x 287x 288
3.039x 105
: 0.00406m3/cycle
Usingequation(12.16)fortheHp stage
, , , r r rt,n
I
n.:t-!:l(-l _l>
4(p'/ )
andsinceY"lY"isthesameasfiortheLp stageandalsopzlI,i= p,lprthen4"
is 0.934asabove.Therefore
Sweptvolumeof Hp Stage:
0'00406
: 0.00436m3-
0.934
Note that the clearanceratio is the samein eachcylinder,and the suction
temperaturesare the samesinceintercoolingis complete,thereforethe swept
volumesarein the ratio of thesuctionpressures,
i.e. U":-
0'0131
:0.00436m3 (asabove)r H 3 s

21. t2.3 Hdthryrory-on
The ideal intermediate pressure
The value chosenfor the intermediatepressurep, influenccsthc rort to bc
doneon the gasand its distribution betweenthe stages.The onditirm for thc
work doneto bea minimum will beprovedfor two-stagecompressionbut catt
beextendedto any numberof stages.
Total work : LP stagework + HP stagework.
Thereforeusingequation(12.6)
Totalpower- J*,i,p7r{(q)'"- "" - t }n-t 'tp,/
)
(r2.18)
It is assumedthat intercooling is completeand thereforethe temperatureat
the startof eachstageis [.
i.e. Totarpower:-!=,i,p7,{(q)"-"'n-, *(&)'"-"'" - r} trr.rnln-t
'[p,/
p,/ J
lf pr, Tr, andp, arefixed,thenthe optimumvalueof p, whichmakesthe
power a minimum can be obtainedby equatingd (power)/(dp,)to zero,i.e.
optimum valueof p, when
d l( p,(r-r)/'
dp'it/
*;,nnr,{(fi)'""'"- ,}
i.e.when
+{(1)"- "''ol,-rt,+ pE,r"(t
1'"-"'"- r} : odp,(ptl p,/
-
)
therefore
,-r-rr,,(,
- l)01,,-ry,)-r
+ or-r,,,(l
- n)ol,r-"u"1-r
:0
n /' n J"
therefore
o-,"-,,,(';t)or',
= py-',,"(T)ou-2')/'
therefore
plztn- ltl tn - (p rp rtn
- ttta
therefore
p?= pflz {t120)
or Pi-Pz (lz2r)
Pt Pt
i.e. the pressure ratio is the same for each stage.
-(fi)'"-"'"-rl:o
lhen n,
Frctron
I sr'ept

22. Pooitive dispt.camslt rnachincc
Totalminimumpower:2 x (powerrequiredfor onestage)
:2 rrfrRr,i(A)'"-"'"- ,]
n-l tp'l )
or in termsoftheoverallpressureratiop,I p, ,wehave,usingequation(12.20|
p,_J;; _ la
P r P t V P t
therefore
rotalminimumpower:,,i*{(f)'"-""" -
This can be shownto extendto z stagesgiving in general,
rotalminimumpower= r-!-,antte)" "'""-'] tr2'22)
Pressureratioforeachstage:(fi)"'
'l
(12.23)
Hencethecondition for minimum work is that the pressureratio in eachstage
is the sameand that intercoolingis complete.(Note that in Example12.6the
information givenimpliesminimum work')
Example 12.7 A three-stage,single-actingair compressorrunning in an atmosphereat
t.0t3bar uiO tst tur a freeair deliveryof 2.83m3/min.The suction
pressureand temperatureare0.98bar and 32'C respectively.Calculatethe
indicatedpoweriequired,assumingcompleteintercooling n: 1.3,and that
the machineis designedfor minimum work. The deliverypressureis to be
70bar.
1.013x tos-I?'gt : i.47kg/minSolution Massof air delivercd- 1Y=:
RT
whereT=15*273:288K.
Thcnusingequation(12.221
TotalindicatedPower
287x288
=,{an"te)""'"-,}
=, *
H "
ry.9;ig{(#)"
rY(3x13}
- r}
=24.2kW
Besidesthe benefitsof multistagecompressionalreadydealt with thereare
also mechanicaladvantages.The higher pressuresare confinedto the smaller
422
F?
thrt
rcdt
ooE
intc

23. rz.t l{bt
cylinders and a multicylinder machine has lcss variation in mteritnl.rccd
and requiresa smallerflywheel.
Energy balance tor a two-stage machine with intercoolcr
Referringto Fig. l2.l9,the steady-flowenergyequation(1.10)can be applicd
to the LP stage,the intercooler,and the HP stage,in turn. Changesin kiGic
energyand heightcanbe neglected,i.e.from equation(1.10)
J0r
L22l
tzil
FIB
lo
let
:iln
Irhc
tfrt
!bc
t are
rllcr
Fig. 12.19 Steadyflow
through a two-stage
reciprocating
compressorwith
intercooler
or*1-+e+w:hz++
for the LP stage,for unit massflow rate,
ht * Qt* W1,: ft,
or for massflow rate,nc
rhcrTr+Qt*Wu-rhcrTi
therefore
et: _ {frr_ *co(Tt_?i)}
i.e. Heat rejectedin LP stage: W, - rhco(T,- T)
for the intercooler,for unit massflow rate
h,+ Qr: ht
or for massflow rate.rlr
rhcoTr* Qr: rhcrTl
therefore
2r: -rhco(Tt- Tr)
i.e. Heat rejectedin intercooler= rhcp(T- Tr)
for the HP stage,for unit massflow rate,
ht * Qn* W11:h2
or for massffow rate,m
rhcrT,+ Q, t frr: rhcrT2
(r2.24)
{t2.zsl
tl'4,

24. Poaitivc dirplacomcnt mcchino
Example12.8
Solution
therefore
Aa: - {fr^ * rhcr(Tz_ ?i)}
i.e. Heatrejectedin Hp stage- W"_ nco(Tz_ Tt)
With completeintercoolingas assumedin Fig. 12.19,and thc
designedfor minimumwork,then,fromequation(iZ.ti),
tU.= Wr:;**1Tz * T,)
(l? 16)
comprct3r
Usingthedataof Example12.6determinetherateof heatlossto thecylindct
jacket cooling water and the rate of heatlossto the intercoolcrcircttating
water.
From Example 12.6we have
frr.: Wn: l5^'5
rw
2
and Tz: Tr:371 K
Then,from squation (12.24|
-Ot: ry- rhco(Ti-)
therefore
-o, : t5'5- 4'5x l'005,
71r- )
2
- --
60
-(371 - 288)
i.e. -8,. = 7.7s- 6.26:1.49 kw
'.
Fromequation(12.26)
-Au= W*- nco(Tz- T)
and lYu= Wr" and Tz: T
therefore
Qn: Qt': - l'49kW
i.e. Heatlossfromthecylinderin eachstage= 1.49kW
Fromequation(12.25)
-Qr = rhcr(T- ti) :
4'5x
-l'005x (371* 2gg)
60
i.e. Heatto intercoolercirculatingwater- 6.26kW
Fg.l?.
6rougl
@prc
F3la
Cmprr
ep-rd
. ThequantitiesWyandtr",.t definedby Fig. 12.19,aretheratesof wort
doneon theair.Theactualpowerinputsexceedthisbytheamountsnccaser,
to overcomefrictionalresistanceto themovingpartsof thcmachinc.It canbc
424

25. -
tL26l.
r€ssor
linder
lating
f work
o6sary
can bc
t2-a Hr-h#
assumedthat about 50% of the friction power gocsto incaasiry th u6r
transferredto thecoolingwater,in additionto thc heattransferrcdto th cotth3
waterfrom the air in the cylinder.
72.4 Steady-flow analysis
In section12.2an expressionwasobtained(equation(12.1I )) for theindicatcd
power requiredto take a massflow rate of gas,m,in state I and deliverit at
a higherpressurein state2. This wasdoneby analysingthe internalpr(rc€ss;
of the machine.Another approachis to considerthe compressionprocessas
one of steadyflow, as shownin Fig. 12.20,with the changeof statefrom I
to 2beingachievedbyanon-flowprocessofpolytropiccompression,asindicated
in the propertydiagramof Fig. 12.21.
Fig. 12.20 Steadyffow
througha reciprocating
compressor
Fig. 12.21
Compressionproccsson
a p-u diagram
+ pr,v2,Tz
Thesteady-flowenergyequationforthesystemshownin Fig.12.20,neglecting
changesin potentialand kineticenergyand for unit massflow rate,is
ht+Q+W:hz
therefore
Q+W:hz-hr
or for an elementalprocess
dQ+dW=dh ( a l
tlits

26. Poritivc dirplaccment machinec
426
dQ*du* pdo
Combining(a) and (b) gives
dh*du*pdu+dW
and, by definition, h: u * pu,
substituting
hencedi:dl* pdu* udp, therefore
Assumingthat noheatistransferredoninductionordeliverytheheattransferrcd,
to or from thesystem,takesplaceduring the polytropic nbn-flowcompression
process.The non-ffow equationfor a reversibleproess states
(b)
du+pdu*udp:du*pdu+dW
therefore
d,W: udp
Then
* :
f' udp : arca ltbat inFig. 12.21,
J r
i.e. w-cr,fg (rin.", '''o
Jtprt,
':o*if ou':C)
crbf( ' "t"-,,,"1'" L, - t)0"
'"'
),
: [(-+) p"-tv,ptt^t)12
Ln- 1/' J,
t-n 12
L,J e'J,
=;to"z - Pflr)
i.e. Workinput,W ::t(pzuz _ prar)
and as ptur: R[ and pzDz: R[ then
__$ta_r,l
12.5 Rotary machines
Becauseof the
-continuous rotary action, the rotary positive displacement
machineis smaller for a given flow than its reciprocatingcounterpart.The
machinesin this categoryare generallyuncooledand asihe compiessionis
carriedout at a high ratetheconditions-areapproximatelyadiabatic.Examplis
of this type are:(i) the Rootsblower; 1ii; vanetype.
Fig. 12
blower
rotor
rf
Pr-
I
I,tf
l-
Fig r
volun
Roots

27. F:
r*rrE4
Fcs*E
lacement
parr-Th€
rssion is
Frernpl6
-----l
125 ncrffi
Roots blower
The two-lobetypeis shownin Fig. 12.22,bvtthree-and four-lobcwfsioc rlc
in usefor highei pressureratios. One of the rotors is connectedto tbc &irc
and the ..cond rotor is geardriven from the first. In this way the rotors robtc
in phaseand the profile of the lobesis of cycloidal or in volute forn giviDS
.oir..t mating of the lobesto sealthe delivery sidefrom the inlct sidc' Thb
sealingcontinluesuntil delivery commences.There must be somecl€aramc
betwein the lobesand betweenthe casingand the lobesto reducewear;thb
clearanceforms a leakagepath which has an increasinglyadverseeffecton
efficiencyasthe pressureratio increases.
As eachsideof eachlobe facesits sideof the casinga volume of gas % at
pressurepr, isdisplacedtowardsthedeliverysideat constantpressure.A further
rotation of the rotor opensthis volumeto the receiver,and the gasflowsback
from the receiver,sincethis gas is at a higher pressure.The gasinducedis
.orpr.$.a irreversiblyby that from thereceiverto the pressurep2,and th€n
deliverybegins.Thisprocessiscarriedout fourtimesperrevolutionof thedriving
shaft.
The p-v diagram for this machineis shown in Fig. 12.23,in which the
pressurerisefrom p, to p2isshownasanirreversibleprocessatconstantvolume'
Work donePercYcle- (P2- P)Y
'therefore
Fig.12.22 Roots
blowerwith a two-lobe
rotor
Fig.12.23 Pressure-
volumediagramfor a
Rootsblower
Work done Perrevolution : 4(Pz* P)V
If I/, is the volumedealt with per unit time at p1and Tr, then
PowerinPut = (P, - PrV"
The ideal compressionproc€ssfrom p, to Pz is a reversibleadiabetic
(i.e.isentropic)lrocess.The work doneperminuteidcallyis thusgwo bl
equation(12.7)withn:7'
i.e. Powerinput:J -,r,O{(fi)" "" - ,}
(tL27l
(lz2t)
tw

28. Pcitiyl di+|rrufi m.c.hino.
llt.e,
Then a comparisonmay be madeon the basisof a Rooe efficiency,
i.e. Rootsefficien", - work doneisentropicauy
actualwork done
Rootsefficien", ={vl0
- l)}pti"{(p.lprl'-"' -
v"(pz- pr)
_ ?,{r(r-rry- l}
(y-l)(r-l)
wherer = pr€ssur€ratio,prf pr.
From equation(2.22),we can write
T Co
: i
v-l R
therefore
Rootsefficien.n= 1l{i'': rl,-
R[ (r-l) J
(r2.2e)
For a Roots air blower valuesof pressureratio, r, of 1.2,1.6,and 2 g.vevaluesfor theRootsefficiencyof 0.945,b.g4,and0.76i respectinety.rrr"s" varuesshow that the efficiencydecreasesasthe piessureratio increases.
actual compressionprocessis not quite as simple u, inu, described.
when the displacementvolume z is openedto the aefiveryspacea pressure
wave enterswhich increaseswith the opening and moves'u,'tr,. vetocityofsound.This waveis reflectedfrom the upprou"ttinglobe to the Jelvery space.Thepressureoscillationssetup unsteadyconditioni in trreaeiiueiyspacewhichvary considerablyfrom one designto another.The actualtorqu. and roadingonthe.rotors are higher.thanisiuggestedby thep-v diagrai,and fluctuate
y.r:h ligh
frequency.This fluctuation ir transmittedto the drive and crearerdifficultiesdue to vibrations.This machinehasa numberoi i.p"rr*,iong butis well suitedto suchtasksasthe scavengingand superchargingofIC engincr
Roots blowersare built for_capacitie-sJf fro* d.t+ to
"-6lrTrniq
,ndpressureratios of the order of 2 to l for a single-stage,".rrin"-*o 3 to l fora two-stagemachine._otherdesignshave beenproducedto improve on thcRootsblower'oneof thesebeingthe Biceracompressor,designedby theBritishInternal combustion EngineeringResearchAssociationtnr"cBne l
vanetype
The simplevanetype is shown in Fig. 12.24andconsistsof a rotor mountcdeccentricallyin the body,and supporrcdby ba[- and rolrer-beaii* t thc codcoversof the body. The rotor is srottedto take the bradeswhich are of anon'metallic material,usually fibre or carbon.As eachbladc moveepo$ thcinlet passage,compressionbeginsdue to decreasingvolune betruro thc rotorandcasing.Deliverybeginswith thearrivarofeachbradcat tncoairrry poscagc.
428
Fig. ll
positir
comp
F[.12
volum
vane-t

29. lEi
' rt2lf
Irod]ai
Thtrr-
iroiH
l i l
dq
-
3pth
C
tur-l
rcT
pt ir &t
lir.*d
FE$r
i:r*r
hrl r
Fig 12.24 Vane-type
positivedisplacement
compressor
F.ry.12,25 Pressure-
volumediagramfor a
YAne-typecompressor
Example 12.9
Solution
12.6 nArr rn*r
IEl,
Krq
t n ,
This type of compressiondiffersfrom that of the Rootsblowerin that someor
all of the compressionis obtained beforethe trapped volume is openedto
delivery.Further compressioncanbeobtaineduy ttri back-flowof air from the
receiverwhich occursin an irreversiblemanner.
Thep-v diagramis shownin Fig. rz.2s.v"is theinducedvolumear pressure
pr and temperature[. compressionoccursto the pressurep,, thc idealform
for an uncooledmachinebeingisentropic.At this prissurethl'displacedgasis
openedto thereceiverandgasflowing backfrom thereceiverraisejtheprJrrur"
irr-eversiblyto pr. The work input is given by the sum of the areasA and B,
referringto Fig. 12.25.comparing the areasof Figs 12.23and 12.2sit can be
seenthat for a given airflow and given pressureratio the vanetyp€ requires
lesswork input than the Roots blower.
A rotary slidingvanetwo-stagemachineis shownin Fig. 12.26;inthis typc
the vanesare in contact with the cylinder walls.
compare the work inputs required for a Roots blower and a yane-typc
compressorhavingthesameinducedvolumeof 0.03m3/rev,theinlct prc rc
being 1.013bar and the pressureratio 1.5to l. For the vaoetypc ernc
that internal compressiontakesplacethrough half the pressurerrqla
Pr : 1.013bar

30. Hth,! dbplsmrnt machlncl
F1E",l?lS Rotary
sliding vanetwo-stage
positivc displacement
compr€ssor
430
therefore
Pz: 1.013x 1.5: l'520bar
For the Rootsblower,referringto Fig. 12.23
Work doneperrevolution: (pz- pt)V"
:(l.s2o-1.013)"
l%P
:1.52kJ/rev
For thevanetype
(1.5x 1.013)+1.013
h= *--**-T--
Referringto Fig. 12.25
= 1.266bar
Fi3I
podti
vrcuuWorkrequired: (areaA + areaB)
Nowusingequation(12.7)withn: Y
areaA:fir,*{(fi)" ""-,}
r.4 1.013x lOsx 0.03ff!g)"
-'''n
_ ,l nr/r.":
dJ
, ___
lo3 l!ot3/
,
J
^"
: 0.?0kJlrev
areaB:(pr- p)Yv
whereZ, is givenby equation{3.19),
i.e.
":
r(?)"': o.o,. (i#)"'
-
: 0.0256m3
i.e. areaB= (1.520- t.266')x 102x 0.0256kJ/rev
: 0.65kJ/rev

31. l2.l YUF
therefore
Workrequired:0.70 + 0.65: 1.35kJ/rev
(compared'withtheworkrequiredfor theRootsmachineof 1.52kJ/rw}
Rotary sliding vanecompressorsare usedwith freeair deliveriesof up to
150m3/min and pressureratios up to 8.5to 1. For specialapplicationsard
boosting,pressureratios of the order of 20 to I havebeenobtainedfrom thiq
type.The largermachinesare usuallywater-cooled.
Lubrication is important with vane'typemachinesand is accomplishedby
injectingoil to the vanetips in contactwith thecasing.Somemachines,having
carbonvanes,requireno lubrication.Anotherversionis designedto reducethe
friction betweenvaneand casing.This employsa ffoatingdrum which rotates
betweenthe rotor and casing,and doesnot allow the vanesto makecontact
with the casing.The only movementof the bladesrelativeto thefloatingdrum
is alongthe slots.SeeFig. 12.26.
12.6 Vaeuum pumps
Rotary positive displacementpumps are used to produce a vacuum or to
scavengea vessel.An exampleof this typeof pump is shownin Fig. 12.27.The
rotor is eccentricallymountedin thestator andcarriestwo bladeswhichsweep
thespacebetweentherotor andstator.Thegasbeingexhaustedentersthrough
{31
W,12.27 Rotary
positivedisplacement
Yacuum"pump
I
lrat
- - t l
--l l/--l*l l.-
-t
// // l---
_:l /l // l;
--l /t ;$*o,o.il l:
__l /A -/t/l---
__l v
---'
,/ l_-:1

32. Pocitive dirplacement machines
432
thevacuumconnectionandiscompressedbeforedelirer.r-throughthedischarge
valve.Theefficiencyof suchpumpsis impairedby thepresenceof condensable
vapours,and meansmust be providedto deal with rheserf necessary.The
vapourstendto condensebeforedeliverythroughthedischargeralveandmix
with the sealingoil. The liquid eventuallyevaporatesinto rhe.,a;uums-y-stem
and lowers the vacuum obtainable,as well as impainng liie .eailng and
lubricatingpropertiesof the oil.
12.7 Air motors
Compressedair is usedin a widevarietyof applicationsin industry.For some
purposesair-operatedmotorsarethe mostsuitableformsof power,especially-
wheretherearesafetyrequirementsto bemet asin miningapplications.
Pneumaticbreakers,picks,spades,rammers,vibrators,riveters,etc.form a
rangeof handtoolswhichhavewideapplicationsin constructionalwork.The1.
are light in constructionand suitablefor operationin remotesituationsfor
which otherformsof powertoolsmay not be suitable.The actionrequiredof
such tools,with the associatedsimplicityand robustnessof construction,is
obtainedwith air-operateddesign.
Basicallythecyclein thereciprocatingexpanderis thereverseof that in the
reciprocatingcompressor.Air is suppliedto theair motor from an air receiver
in which the air is at approximatelyambienttemperature.Thereis a pressure
drop in theair linebetweenthereceiverandthemotor.Theair expandsin the
motor cylinderto atmosphericpressurein a mannerwhich is polytropic(i.e
theexpansionisinternallyreversibleandthelawofexpansionispun- eonstanl
wheren < y, and is usuallyabout 1.3).If the air is initiaiiy at ambienr
temperature,then this form of expansionwill bring alnut a reduction
in theair temperatureaslowerpressuresarereached.Thetemperaturesreached
may be sufficientlylow to be belowthe dew-pointof the moisturein the air
(seesection15.2);thismoisturemaybecondensed,andthewaterformedma1-
evenbecooledto itsfreezing-point.Thismayleadto theformationof icein the
cylinderwith theconsequenceof blockedvalves.To preventthisconditionir
may be necessa.ryto preheatthe air to an initial temperaturewhichis h;5r,
enoughto preventtheformationofice.Thisheatingoftheaircausesanincrease
in volumeat thesupplypressureandreducesthedemandfrom thecompressor.
Further,the temperatureat which the heattransferis requiredis low, and a
low-gradesupplyof heator 'wasteheat'may be utilizedlor thepurpose.
A hypotheticalindicatordiagramfor an air motor is shownin Fig. l2i:
In thiscasetheair expandsfrom I to thepressurep, at theendof thestroke
Thereis thena blow-downof air from 2 to 3.Air is exhaustedfrom3 to 4.and
at4 compressionof thetrappedor cushioncir begins.Air at thesupplypressure.
pu,is admittedto thecylinderat the point 5 whereit mixesirreversiblr*rrl
thecushionair.Thepressurein thecylinderis rapidlybroughtup ro therni.:
value,pu.The furthersupplyof air is madeat constantpressurebehindrhe
Fig. 12.28
volumedie
air motor
Ex

33. I
lr etc hc
ll rort-
situanc
m requird
Dnstructili,
!of that is tE
F eu rE(drlt
I * a pressrr
Feands in ft
follrropic (ia.
l'= constelt,
y at ambicc
I a ieduai<rr
rtures reachod
[rre in the air
r formedmay
n of icein thc
b conditionir
rhich is h;sir
Esan increasc
I Compressor.
b low. and a
Purpose.
in Fig. 12.28-
of thestroke.
m3to4.and
pply'pressurq'
crersiblywith
tp ro theinlet
re behindthe
Fig. 12.28 Pressure-
volumediagramfor an
atr motor
Example12,10
Solution
12.7 Airlnoton
movingpistonto the point of cut-offat l. The cut-offratio is givenby
cut-off ratio :
vt - va
Vt-Vu
Theeffectof thecushionair is to givea smoother-runningmotor.Theposition
of thepoint 5 dependson thepoint of initial compression4, andon thelaw of
compressionpYn= constant.The conditionsmay be suchthat the points5
and 6 coincide,
The analysisof sucha diagramis bestcarriedout from basicprinciples,as
illustratedin the followingexample.
Thecylinderof anair motorhasa boreof 63,5mm anda strokeof 114mm.
Thesupplypressureis6.3bar,thesupplytemperature24'C,and theexhaust
pressureis 1.013bar.Theclearancevolumeis 5% of thesweptvolumeand
the cut-offratio is 0.5.The air is compressedby the returningpistonafter
it has travelledthrough 0.95of its stroke.The law of compressionand
expansionis pVr'3 : constant.Calculate:
(i) the temperatureat theendof expansion;
(ii) the indicatedpowerof themotor whichrunsat 300rev/min;
(iii) the air suppliedperminute.
(i) Referringto thecycleof Fig. 12.28
Clearancevolume- Vo: Vs:0.054
Sincethe cut-offratio,( - y)l(V! - Izu),is 0.5,therefore
: 054 + 0.05t/": 0.55Y"
Vz: 4't Vu: 1.954
Now
V, - Vu: 0.95(Vt - Vt) (given)
or Vn- Vt: 0.054
{s
rb
-
!-

34. I
Porhivs dirplaccment machinee
(ii) Now
( v o ' / n | r ' 3
,,:r,i) = ror3(oosa
)
:2.4e4bar
Workoutputpercycle: area1234561
workoutput: Pr(v, - vu)+ (p'vt
-
!.l,vt-
n-l /
-P,(v,-v)-(ry#!)
therefore
Ve,= 0.054+ 0.05Y.: 0.1I/,
P t V : p 2 V n
therefore
also
i.e.
- (1.013x 105x 0.361
105x0.361x10*3
o,:r,(2): u,(m)": 2ltlbar
rz:r,(2)'
'
:2e?(ffi)": 244.6K
Temperatureafterexpansion:244.6 - 273= -28.4"C
t '
and Sweptvolume:
nx63.52xll4
4xlOe
: 0 . 3 6 1 x l 0 - 3 m 3
therefore
Work output per cycle
: (6.3x lOsx 0.361x l0-3 x 0.5)
lOsx0.361xl0-3
*
O:
{(6.3x 0.55)- (2.718x 1.05)}
rb nD
Cherrtcr
mll yag
ntor: (e
F€!4 (b)
(c) eir cq
TGcd
x l0-3 x 0.95)
U2.4e4x 0.05)- (1.013x 0.1))
0.3
i.e. Workoutputpercycle: I 13.7+ 73.5* 34.7- 2.8
:149.7 Jlcycle
Powerdeveloped=
-I1*#
: 0.749kw
(iii) The massinducedper cycleis given by (n'r,- m+).It is necessaryto
determinethe temperatureof theair at 4,whichcanbetakenasequalto that
03
tt
3 0 2
I
.
ol
I
494

35. b
It
P: 7 bar
llt Abffirr
at3.It isassumedthattheairinthecylindcratthepoint2cxpandsircotropi(dt
to theexhaustpressur€.Therefore
t.e.
1.013x 105x 0.0361x l0-3
287x 184.5
6.3x105x0.55x0.361
:0.0691x 10-3kg
x l0-3
287 x 297
= 1.4675x l0-3 kg
therefore
Inducedmasspercycle: (1.467S x l0
- 3)_ (0.0691x l0
- 3)
: 1.398x 10-3kg
i.e. Massflow rateof air supplied: 1.39gx l0-3 x 300: 0.42kg/min
"
=
"(?)1'-"t'
- 144r(#)0'4lr'4 : 184.5K
^n=W,:
^, _PrVr _-
R?i
Air motorscanbe rotaryin actionandarethensimilarin form to their
com_pressorcounterparts,seesection12.5.Figure12.29(al,(b), and(c) show
theformsof theperformancecharacteristicsola small,0.tkw; vanetypeair
motorin termsof power/speed,torque/speed,andair consumpiion/speei.nn
air motorwhichreceivesair froma constintpressuresupplycanbecontrolled
to meettheloadrequirementsby fittinga restrictoreitlir beforeor afterthe
motor. It can be shownby a considerationof a simplifid p-v diagram,
neglectingclearance,thatfittingtherestrictorbeforethemotorrequiresalower
airffowthanfittingit after.Thereadershouldestablishthisforhimselfandalso
showthattheairflowraterequiredisapproximatelyproportionalto thesupply
pressureto themotorfor agivenduty.Figure12.30showstheresultsofatCt
on a smallair motorwhichgivesa Zsvoreductionin air requirementif the
restrictoris on theinletsideto themotor.
ju. rT,
Cheracteristicsof a
rrnrll ya6-1yp" 6t
Eotor: (a) power-
tpocd,(b) torque-spoed,
(c) air consumption-
Toed
2 4 6 8
Sp€ed/103(revlmin)
{ a)
2 4 6 8
Speed/103(revlmin)
(b)
2 4 6 8
Speed/103(revlmin)
(c)
3r
.E6
A <
t
e1o
o
. ! 3
0.3
tJ
b 0.2
'
A
435

36. Pcative displacement machines
Fig. 12$ Test results
for a vane-type air
motor with restrictor
control (no load)
43q
o t
100 r50 200
Speed/( rev/min )
Problems
12.1 Air is to becompressedin a single-stagereciprocatingcompressorfrom 1.013bar and
I 5"C to 7 bar.Calculatetheindicatedpowerrequiredforafreeairdeliveryof0.3m3
/min,
whenthe compressionprocessis asfollows:
(i) isentropic;
(ii) reversibleisothermal;
(iii) polytropic,with n = 1,25.
What will bethe deliverytgmpelalyr.e-ineachcase?
(1.31kW; 0.98kW; 1.20kW; 221.3"C:l5 "C; 150.9"C)
12.2 The compressorof Problem l2.l is to run at l0O0rev/min. If the compressoris
single-actingand hasa stroke/boreratio of 1.2/1,calculatethe boresizerequired.
(68.3mm)
12.3 A single-stage,single-actingair compressorrunning at 1000rev/min deliversair at
25bar.For thispurposetheinductionandfreeair conditionscanbetakenas 1.013bar
and 15'C, and the FAD as 0.25mr/min.The clearancevolumeis 3% of the swept
volumeandthestroke/boreratiois 1.2/1.Calculate:
(i) theboreandstroke;
(ii) the volumetricefficiency;
(iii) theindicatedpower;
(iv) theisothermalefficiency.
Take the indexof compressionand re-expansionas 1.3.
(73.2mm;87.8mm;67,1Voi2kW:67.7YoI
12.1 The compressorof Problem12.3hasactualinductionconditionsof I bar and 40"C,
and the delivery pressureis 25bar. Taking the bore and stroke as calculatedin
Problem12.3.calculatetheFAD referredto 1.013barandI 5'C andtheindicatedpower
required.Calculatealsothe volumetricefficiencyand compareit with that of Problem
l2'3'
{0.226mrlmiH;1.98kW;6t.Zoh;67.i%l
Restrictoron,inlet slde

37. )t3 x: atd
l-1nl mn
I
l: 15o.9'Cl
Dpressor b
quired.
r 6 8 3 m m f
hers air at
x I 0ll bar
f rhc swepr
t$i:.67-7%l
end tlo'C,
*:ulated in
leted power
of Problern
ttt.: 67.7oh
Frotbr
12.5 A single-actingcompressorisrequiredto deliverair at 70bar from anindlrctim prtsnrrc
of I bar,attherateof2.4mr/minmeasuredatfreeairconditionsofl.0l3berrDdt5'C-
The compressionis carried out in two stageswith an ideal intermcdiarcprcsure eld
completeintercooling.Theclearancevolumeis 3% of thesweptvolumein cact c1tlitrdct
and the compressorspeedis 750rev/min.The indexof compressionand re+rplrirn
is I.25for bothcylindersandthetemperatureat theendof theinductionslrolc itract
cylinderis 32'C. The mechanicalefficiencyof thecompressoris 857o.Cdcularc:
(i) theindicatedpowerrequired;
(ii) thesavingin powerovei single-stagecompressionbetweenthesamepressurcs;
(iii) thesweptvolumeof eachcylinder;
tiv) therequiredpoweroutputof thedrivemotor.
(22.74 kW;5.98kW; 0.00396m3,0.000474m3;26.75kW)
12.6 For thecompressorof Problem12.5calculatetheheatrejectedperminuteto thejacket
coolingwaterof eachstage,andtheheatrejectedperminuteto theintercooler.Assume
that 50% of the frictionpowerin eachstageis transferredto thejacketcoolingwater.
{264kJ/min;478kUmin)
.2.7 A single-cylinder,single-actingair compressorof 200mm bore by 250mm strokeis
constructedsothat itsclearancecanbealteredby movingthecylindcrhead,thestroke
beingunaffected.
(a) Usingthedatabelowcalculate:
(i) thefreeair delivery;
(ii) the powerrequiredfrom thedrivemotor.
Data Clearancevolumesetat 700cm3lrotationalspeed,300rev/min;deliverypressure.5 bar;
suctionpressur€and temperature,I bar and 32"C; freeair conditions,1.013bar and
l5'C; indexof compressionandre-expansion,1.25;mechanicalefficiency,8070.
(b) To what minimum valuecan the clearancevolumebe reducedwhenthe delivery
pressureis4.2bar,assumingthatthesamedrivingpowerisavailableandthatthesuction
conditions,speed,valueof index,and mechanicalefficiency,remainunaltered?
(1.68m3/min;7.1kW; 458cm3)
12.8 A single-acting,single-cylinderair compressorrunningat 300rev/minis drivenby an
electricmotor.Usingthedatagivenbelow,andassumingthattheboreis equalto the
stroke.calculate:
(i) thefreeair delivery;
(ii) thevolumetricefficiency;
(iii) theboreandstroke.
Data Air inletconditions,1.013barand15'C; deliverypressure,8 bar;clearancevolume,TTo
of sweptvolume; indexof compressionand re-expansion,1.3;mechanicaleffciencyof
thedrivebetweenmotorandcompressor,8770;motorpoweroutPut,23kW.
(4.47m3/min; 72.7%; 297mm)
12.9 A two-stageair compressorconsistsof threecylindershavingthesameboreandstroke.
Thedeliverypressureis 7 bar andthe FAD is 4.2m3/min.Air is drawnin at 1.013bar,
15"Candan intercoolercoolstheair to 38"C.Theindexof compressionis 1.3for all
threecylinders.Neglectingclearance'calculate:
(i) theintermediatepressure;
(ii) thepowerrequiredto drivethecompressor;
(iii) theisothermalefficiencv.
(2.19bar; 16.2kIY: tL5%)
12.f0 A four-stagecompressorworks betweenlimits of I bar and ll2bar. Thc indgr of
compressionin eachstageis 1.28,the temperatureat the start of comprereionin crh
4{r

38. Positive displacement machines
stageis 32'C, and the intermediatepressuresare so chosenthat the work is dividdl
equallyamongthe stages,Neglectingclearance,calculate:
(i) the temperatureat deliveryfrom eachstage;
(ii) the volumeof freeair deliveredper kilowatt-hourat 1.013bar and 15'c;
(iii) the isothermalefficiency.
(122'C;6.23m3/kWh; 87.6%)
'12.11 A single-cylinder,single-actingreciprocatingair compressorsuppliesa water-cooted
receiverfrom which the air is drawn off for processwork. Taking the polytropic indcr
of compressionand re-expaqsionas 1.3,and usingthe data below,calculate:
(i) the pressurein the receiver;
(ii) the rateofheat rejectionfrom the receiver;
(iii) the volumetricefficiencyof the compressor;
(iv) the requiredpower input to thc compressor.
Data Cylinderbore,200mm;stroke,250mm;rotationalspeed,440rev/min; clearanccvolumc,
57oof sweptvolume;ambientpressureand temperaturein compressorhouse,l.0l bar
and l0"C; averagepr€ssureand temperatureduring the induction stroke, lbar and
20'C; volumeflow rateof air drawn off for processwork, 0'6m3/min at 17"C.
Note: Usea trial-and-errormethodfor part (i).
(5 bar;8.14kW; 83.9%;9.85kW)
12.12 Air at l.0l3bar and 15'c is to be compressedat the rateof 5.6m3/minto l.75bar.
Two machinesare considered:(a) the Roots blower; and (b) a sliding vane rotary
compressor.Compare the powersrequired,assumingfor thc vane typ€ that internal
compressiontakesplacethrough 75% ol the pressurerisebeforedeliverytakesplacg
and that the compressoris an ideal uncooledmachine.
(6.ggkw; 5.71kw)
12.13 Air iscompressedin a two-stag€vane-typecompressorfrom 1.013bar to 8.75bar' Using
the data below,and assumingequal pressureratios in eachstage,that compressionis
completein eachstage,that the rnachineoperatesin an idealmanner,and is uncoolcd
apart from the intercooler,calculate:
(i) the powerrequirod;
(ii) the volumeffow rate measuredat the delivcrypr€ssure.
Data Freeair delivery,42m3/min at 1.013bar and l5'C; intercoolingbetweenstagesis 757.
comPlete. (lE? kw; z.2l m3/miD)
12.14 The following particularsreferto a single-actingair motor: cylinderdiameter380mm;
stroke610mm;speed200rev/min; supplypressureandtcmperatwe6.2bar and 150"C;
back pressure1.03bar; index of expansionand compression1'35;cut-off ratio 0.46;
clearancevolume 2Ao/oof sweptvolume;mechanicalefficiency95%'
Assumingthat the temperatureand pressureof thc air in the clcarancespaceat thc
beginningof admissionare6.2bar and 150"C,calculate:
(i) the air consumPtion;
(ii) the air tcmperatureaftcr blow-down;
(iii) the fraction of stroke travclledby the piston bcforerecomprcssionbegins;
(iv) the shaftpowerdevelopcd.
(0.54kg/s; _ 14.3"c; 0.463;72.9kw)
Reference
l2.l BS1571Testingol PositiveDisplacementCompressorsandExhaustersPartI (1987|
Part II (1984).
tt38