An Investigation of The Volumetric. Efficiency of A Roots Blower
An Investigation of The Volumetric. Efficiency of A Roots Blower
An Investigation of The Volumetric. Efficiency of A Roots Blower
Purdue e-Pubs
International Compressor Engineering Conference
1974
B. W. Imrie
University of Leeds
B. N. Cole
University of Leeds
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B.W. Imrie ,
Lectu rer,
Dept. of Mech anical Engin eering ,
Unive rsity of Leeds .
INTRODUCTION
A Roots blowe r is a form of posit ive displa
cemen t
Review
Attem pts have been made to estim ate the amoun
t of
intern al leaka ge flow and, in gener al, accur ate
resul ts have not been achie ved witho ut resor ting
to
empir ical metho ds.
Schop per (l) consi dered a
blowe r with rotor s of equal lengt h and diame
ter, and
calcu lated the leaka ge flow from the cleara nce
area,
the press ure differ ence and an empir ical coeff
icien t
which was a funct ion of the press ure differ ence.
Winte r (2) consi dered the total leaka ge as flow
throu gh a nozzl e and calcu lated the flow using
the
cleara nce area as the nozzl e throa t area.
The flow
was assum ed to be subso nic below an overa ll press
ure
ratio of 1.9 and sonic above it.
Comp arison of
calcu lated and exper iment al value s of volum etric
effici ency for press ure ratio s up to 1.7, showe
d
that leaka ge was over- estim ated below a certa
in
235
THEOR.E'l'ICAL CONSIDERATIONS
The geometry of Roots blower rotors is of basic
importance for the theoretical Quantity of air
delivered and the corresponding volumetric
This paper considers the Roots blower
efficiency.
with two-lobed involute rotors and the theoretical
rotor profile is designed for no clearance between
There should be only one point of conthe rotors.
tact for a given position of the rotor and the
locus of this point of contact should be continuous.
If these conditions are not fulfilled, then air will
be trapped between the rotors and transferred from
the high pressure to the low pressure side of the
The air delivered from the cell volume
blower.
(between the rotor and casing) to the high pressure
side is at low pressure (temperature) conditions,
whereas the trapped (or 'carry-over') volume
returned to the low pressure side is at high
In effect,
pressure (temperature) conditions.
this increases the 'blockage factor' of the rotor.
The blockage factor is a convenient non-dimensional
parameter which can be used in Roots blower
It is defined as the ratio of
geometry analysis.
the rotor cross-sectional area to the area of a
circle with diameter eQual to the rotor diameter
and the volume of air delivered per cycle is
Thus for
proportional to (1- blockage factor}.
maximum delivery the blockage factor should be as
low as possible.
The analysis of Roots blower geometry develops a
basic analysis by Castle (6) for involute (and
In the calculation of
cycloid) rotor profiles.
the area for END leakage between the ends of the
rotors and the casing, the length (i} varies with
This length is
the position angle of rotation, s.
assumed to be normal to the direction of leakage
The length was first approximated
flow (Fig.l).
to two straight lines, i.e. (Tl S + STz}; then to
four straight lines, i.e. (T1W1 + W1S + SWz + W2T2),
Assuming that the inlet and outlet ports start in
the same plane as the rotor centres, then there is
a point of discontinuity at 6 = 0 i.e. when the
For this
rotors are mutually perpendicular.
position the length is (T3S + SWz + W2T 2 ) and would
These
also be valid for a casing with cus.ps.
lengths divided by the rotor diameter were computed
for different values of a.
d2
p
0
tc,
fL
CT
r
p
rt'
L
CL ..
'r
L
p
pLu
)J
(),
'5:
pu2
CT
is
CL
lS
is
is
rotor length
.9.,
is
ph
is
pt
is
Po
is
Lip
is
Red is
is
is
i.s
lS
)J
is
viscosity of air
is
r
T
p
0
pu2
ph
. "-j
ph
CL ,..
CT
L
' (Re X-)'
d
d
r ' r
L
PR,
p
p
is
wiT0
where CE
n.lS
pu2
236
isentropic work
Roots work
e(y-l)/y_l
e -
EXPERIMENTAL WORK
A standar d Roots blower, 5 inches long by 5 inch
centres (8.1 inch tip dia.) was modifie d for static
leakage tests.
The metal headpla te was replace d
by a perspex copy so that the positio ns of the
rotors could be examine d easily.
The gears were
left exposed and a pin used to lock the gears (40
teeth - 9 increme nts in angular displace ment S)
and therefo re the rotors in positio n.
Shims were
used to vary the end plate clearan ces which were
divided equally between the two ends by setscrew
adjustm ent.
Lobe/lo be and tip)cas ing clearan ces
were also measure d, for each rotor positio n, with
feeler gauges.
All clearan ces were accessi ble
from either the high pressur e or low pressur e side
and each clearan ce was measure d two or three times
along the length of the blower and an average value
taken.
The princip al tests carried out meas~ed:(a) leakage through all interna l clearan ces, with
end clearan ces of 0.016, 0.021 and 0.027 inches
equally divided between the two ends;
(b) leakage through interna l clearan ces with lobe/
lobe clearan ces blocked and end "clearan ce of
0.021 inches equally divided ;
(c) leakage along the shafts with an end clearan ce
of 0.021 inches equally divided .
Compres sed air was supplie d through 2 in dia.
copper tube to the test blower and control led by a
pressur e regulat or and valve upstream and by a
valve downstr eam of the blower, to regulat e the
pressur e ratio across the blower.
The interna l
'leakag e' air was then measure d using an orifice
plate and corner tapping s to BS 1042:19 64 (10)
specifi cations .
Pressur e measure ments (ph)
Qpstream and (pt) downstr eam of the blower were
made by manome ters at tapping s in the 2 in dia.
tubes 2 feet from the ports.
Lobe/lo be clearan ces
were blocked effecti vely by rolling a thin sheet of
rubber between the lobes of the rotors.
Results
Tests (a) and (b) were used togethe r to estimat e
the leakage through the differe nt clearan ces for
differe nt angular positio ns and,wit h (nondimensi onal) leakag e: (1- n ),value s of volume tric
efficien cy (n ) were obtaine dvfor pressur e ratios
e = 1.2, 1.3,vl. 4, 1.45, 1.5 and 1.6.
A typical
set of results is shown in Fig.3 for 8 : 1.6,
extrapo lated to zero end plate clearan ce for each
positio n of the rotors.
The differe nce between
the extrapo lation value of volume tric efficien cy
and the actual volume tric efficien cy, can be
ascribe d to END clearan ce leakage .
The differe nce
between the values of nv for an end clearan ce of
0.021 in and for the case with the lobes blocked is
a measure of the LOBE leakage .
Average values for
all angular positio ns of the rotors were calcula ted.
237
(m)
(m)
(m)
(m)
0.325
0.407
0.468
0.294
o.4o6
0.506
0.580
0.373
o.470
0.588
0.676
0.439
o.499
0.626
0.721
0.470
0.528
0.661
0.765
0.498
0.583
0.729
0.849
0.554
0.125
0.200
0.343
0.282
0.113
0.012
0.162
0.244
0.418
0.344
0.133
0.029
0.182
0.288
0.494
0.407
0.149
0.032
0.189
0.310
0.532
0.437
0.156
0.032
0.196
0.332
0.569
0.465
0.163
0.034
0.207
0.376
0.642
0.522
0.175
0.032
EL-16
EL-21
EL-27
L-T2l+L -E21
(e) L-ID+L-EO
L-El6
L-E27
L-E2l
L-L21
L-T21
12
13
14
145
15
16
and Macinnes (12) to give the temperature distribution around the rotors and casing.
The maximum
difference between the trailing and leading portions
of the rotor was almost 10C.
Temperature indicating paints used on the rotors of a 5 x 5 blower
running normally showed a temperature difference of
approximately soc between the trailing and leading
portions of the rotor (11).
TIP
0.009
0.0920
LOBE
0.012
0.0585
END
0.016
0.2004
796
0.181
796
0.115
796
0.394
0.032
141
0(10 3 )
0.156
0.310
1081
672
0 (10 4 ) 0 (5x10 3 )
COMMENT
Further development of experimental techniques for
evaluating leakage flows through internal clearances
under static conditions would provide more insight
into the mechanisms of leakage flow together with
improved empirical measurements of coefficients and
indices.
This information would enable more
effective models of dynamic performance to be
established thereby improving the design and
development of Roots blowers.
ACKNOWLEDGEMENTS
The work reported formed part of a research programme undertaken by the University of Leeds on
behalf of Messrs. w.c. Holmes & Co., Ltd, of
Huddersfield, whose support is gratefully acknowledged by the authors.
The assistance provided by
Mr. Bernard Hunter, University Design Engineer, and
by Dr. Susan Bloor, Computational Assistant, is also
greatly appreciated.
REFERENCES
(1) SCHOPFER, K. 1Das Roots-GeblMse als
Ladungsverdichter an Mercedes-Benzmotoren 1 ,
ATZ 38, (1935)
(2) WINTER, E.F. 'The Effect of Running Clearances
on the Performance of Roots-type Superchargers' RAE Rept.Eng.4129.(1945)
(3) ERTL, H. 'Die Auslegung von Roots-Geblasen,
Konstruktion 5 No.3,(1953)
(.4) PATTERSON, J. and RITCHIE, J.B. 'Roots Blower
Performance' Int.J.Mech.Sci., 11,(1969)
(5) RITCHIE, J.B. and PATTERSON, J. 'Geometry and
Leakage Aspects of Involute Rotors for the
Roots Blower' Proc.Instn.Mech.Engrs., 183
Pt.l, No.36,(1968-69)
(6) CASTLE, P. Private Communication from W.C.
Holmes & Co. Ltd.
(7) HUNTLEY, H.E. Dimensional Analysis, MacDonald
(1952)
(8) COLE, B.N., GROVES, J.F. and IMRIE, B.W.
'Performance Characteristics of Roots Blower
Systems' Conference on Industrial Reciprocating and Rotary Compressors, Instn.Mech.
Engrs., (October, 1970)
(9) HIERSCH, F.A. 'Proposed expressions for Roots
supercharger design and efficiencies'
Trans.Am.Soc.Mech.Engrs. (November, 1943)
(10) B.S.l042:1964, Part 1 'Methods for the
measurement of fluid flow in pipes'
(11) GROVES, J.F. Private Communication from W.C.
Holmes & Co. Ltd.
(12) SHILLABEER, R.N., THOMAS, M.A. and MaciNNES,
L.A. solving Temperature Distributions in
Practical Engineering Problems' I.E.E.
Colloquium Digest No.l967/12 Applications of
Computers to Field Analysis
at (Tddn' pd).
Leakage mass is xnanmi.
Simplifying assumptions included constant inlet and outlet
conditions, average values of parameters used for
each revolution, instantaneous mixing of leakage air
and temperature gradients around the rotors and cas~
ing and across the casing were neglected.
An iterative process was used to determine blower
conditions after each revolution and the computer
program was run to 1.5 x 10 5 revolutions for a
5 x 5 blower with end clearance = 0.008 inches,
lobe clearance = 0.012 inches, atmospheric pressure
= 14.7 lbf/in 2 and atmospheric temperature= 20C,
for (a) 8 = 1.6, N = 1500 rev/min (b) 8 = 2.0,
N = 2000 rev/min.
A graph of temperature of
delivered air, rotors and casing is shown in Fig.5.
The analysis was developed by the lumped parameter
network technique as described by Shillabeer, Thomas
238
Outlet
Inlet
Air delivered
I Tddn Pd )
(1 - x n) arfl'l i
A
L---
I(Tddn Pd)
I
I
(l,l
L..
(l,l
~ hcan
Ill---~
EAsco
Rotors
'
<(
IO'iin Piin )
hran
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--H ea t flow
--- -A ir flow
h - hea t tran sfer
coe ff icit: nt
As- surf ace area
ICT;n
I
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239
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240
1-0
0-9
0-8
0-7
'~
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~ ~
~
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c:-
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-~
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Pressure ratio, 9
1-5
241
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242
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