Variable thickness scroll compressor performance analysis--Part I: Geometric and thermodynamic modeling

Abstract

In order to investigate the performance of variable thickness scroll compressors, a detail mathematical modeling based on energy and mass balances is established in this two-part. In part I, the geometric modeling and thermodynamic modeling are developed. The profile based on circle involute, high order curve, and arc is built up using the base line method. The volume of working chambers from suction to discharge is defined. Thereafter, the evolution and derivative of the working chamber volume with respect to the orbiting angle are discussed. The energy and the mass balance for working chamber are described. Suction gas heating, radial and flank leakage, heat transfer between the working fluid, scroll wraps and plates are considered in the thermodynamic modeling. The established geometric modeling and thermodynamic modeling can provide better understanding of the variable thickness scroll compressor working process. The dynamical modeling and model validation are reported in part II.

Full text

Original article
Variable thickness scroll compressor
performance analysis—Part I: Geometric
and thermodynamic modeling
Peng Bin
1,2
, Vincent Lemort
3
, Arnaud Legros
3
,
Zhang Hongsheng
1,2
and Gong Haifeng
4
Abstract
In order to investigate the performance of variable thickness scroll compressors, a detail mathematical modeling based
on energy and mass balances is established in this two-part. In part I, the geometric modeling and thermodynamic
modeling are developed. The profile based on circle involute, high order curve, and arc is built up using the base line
method. The volume of working chambers from suction to discharge is defined. Thereafter, the evolution and derivative
of the working chamber volume with respect to the orbiting angle are discussed. The energy and the mass balance for
working chamber are described. Suction gas heating, radial and flank leakage, heat transfer between the working fluid,
scroll wraps and plates are considered in the thermodynamic modeling. The established geometric modeling and
thermodynamic modeling can provide better understanding of the variable thickness scroll compressor working process.
The dynamical modeling and model validation are reported in part II.
Keywords
Scroll compressor, variable thickness, geometric modeling, thermodynamic modeling, mathematical modeling
Date received: 7 September 2015; accepted: 1 March 2016
Introduction
In recent years, protecting environment and
reducing energy consumption become two major
issues for the human being. Many researchers concen-
trate on lessening machinery energy consumption,
cutting down driving force spoilage, and gaining
better efficiency in order to economize energy sources.
As a kind of highly efficient positive displacement
machine, scroll compressor has many advantages
such as simple structure, low noise, high reliability,
low vibration, light weight, and small size
compared to other types of compressors. It is becom-
ing popular and widely used in refrigeration,
air-conditioning, various kinds of gas compression,
pressurized pump products, etc. At present with the
wide application of the scroll compressor and its out-
standing advantages have attracted the attentions of a
lot of users.
1
The industrial refrigeration systems and pneumatic
gas sources need higher compression ratios in the
future. The constant thickness scroll compressors
have to add turns in order to achieve higher compres-
sion ratios. And more turns would add substantial
complexity to the manufacturing process. For a
given total area of a circle, increasing the turns of
the scroll first decreases the suction volume of gas,
makes the compression take place rather slowly,
increases the rate of leakage, and finally decreases
the efficiency of the scroll compressor. The variable
thickness scroll air compressor achieves a high com-
pression ratio with less profile turns compared to con-
ventional scroll compressors.
2
There are at least two possible ways to change the
compressor thickness: the first one is to change the
geometry of the scrolls, and the second one is to
change the orbit of the motion. Usually researches
only consider the first one because the second one is
difficult to realize for scroll compressors. In variable
thickness scroll compressors research, some literatures
Proc IMechE Part E:
J Process Mechanical Engineering
2017, Vol. 231(4) 633–640
!IMechE 2016
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1
School of Mechanical and Electronical Engineering, Lanzhou University
of Technology, Lanzhou, China
2
Wenzhou Pump&Valve Engin, Research Institute, Lanzhou University
of Technology, Wenzhou, China
3
University of Liege, Energy Systems Research Unit, Liege, Belgium
4
College of Mechanical Engineering, Chongqing University of
Technology, Chongqing, China
Corresponding author:
Bin Peng, School of Mechanical and Electronical Engineering, Lanzhou
University of Technology, Lanzhou 730050, China.
Email: pengb2000@163.com

have been released. Bush et al.
3
applied general con-
jugacy relations to define a scroll geometry that fills a
circular periphery and makes better use of available
space for displacement. Wang et al.
4
presented the
theory of variable thickness scroll compressor profiles
and its shape optimization methodology based on
functional theory. Wang and Li
5
analyzed the mesh-
ing characteristic and generative process of one thick-
ness wrap profile for twin wrap scroll machinery.
Gravesen and Henriksen
6
introduced a new method
to calculate scroll geometry by deriving each scroll
curve from the radius of curvature parameterized
with involute angle. Shaffer and Groll
7
used a derived
control volume approach to solve the pocket volume
with parametric representation of all scroll geometry.
Liu et al.
8,9
developed a geometric model of variable
radii scroll profiles and got optimum solution using
Computer Aided Engineering (CAE). There are a lot
of researches on mathematical models of constant
thickness scroll compressors. According to different
refrigerations, working conditions and structures, the
geometric characteristics, heat transfer characteristics,
leakage characteristics, and dynamic characteristics
about constant thickness scroll compressors based on
circle involute have been investigated.
10–31
However,
for variable thickness scroll compressor, the detail
mathematical model and quantitative analysis about
the scroll compressor working chamber have not
been published so far. It is worthwhile to get more
comprehensive understanding on the various aspects
of its working process in order to increase the accuracy
of simulation and performance prediction. In this two-
part, effort has been made to present in detail inner
working process of variable thickness scroll compres-
sor. This research is expected to provide an insight to
understand the qualitative and quantitative character-
istics of the working process of variable thickness scroll
compressor. So this study develops a novel scroll com-
pressor with a combination scroll profile, which is
based on high order curve, circle involute, and arc
and presents a comprehensive simulation model that
predicts the performance of a novel scroll compressor.
An experimental campaign is also conducted to meas-
ure compressor performance and validate the mathem-
atical modeling. The study is divided into two parts to
illustrate. In part I, a detail geometric modeling and a
thermodynamic modeling are presented. In part II, a
dynamic modeling is introduced, performance analysis
and some experiment results are given and analyzed.
Finally, the conclusions of the research are summarized
in this two-part.
Geometric modeling
Profile equation
The profile of variable thickness scroll compressor is
built up using the base line method. The base line
combines circle involute, high order curve, and arc.
1
Base line equation
x¼Rgcos ’þRssin ’
y¼Rgsin ’Rscos ’
ð1Þ
High order curve
Rg¼C0þC1ð’=2ÞþC2ð’=2Þ2þC3ð’=2Þ3
Rs¼C1þ2C2ð’=2Þþ3C3ð’=2Þ2ð2Þ
Circle involute
Rg¼a,Rs¼a’ð3Þ
Non-working arc
x¼kcos ’þd
y¼ksin ’
ð4Þ
Twin-circular arcs modification profile
x¼eicos ’þdi
y¼eisin ’þci
ð5Þ
Figure 1 is the profile of variable thickness scroll com-
pressor and the twin-circular arc modification of
scroll wrap. The a is the connection point of arcs, b
and e are the connection points of arc and circle invo-
lute, f and c are the connection points of circle invo-
lute and high order curve, g is the connection point of
non-working arc and high order curve, d and h are the
end points of profiles. _
ab and _
ae are twin-circular arcs’
modification profile, _
ef and _
bc are circle involutes, _
fg
and _
cd are high order curves, _
gh is non–working
arc.
1,32
Volume calculation of various chambers
The control volume goes through six steps from
the suction until the discharge. Figure 2 repre-
sents the different steps during the gas compres-
sion process. The basic parameters of the
variable thickness scroll compressor are listed in
Table 1.
(a) Suction process, 0452. The chamber is made
up of high order curve. The working chamber
volume can be calculated by
V¼hRor Lhþ½Rghð’eÞRgh ð’sÞ

ð6Þ
(b) Compressor process, 2452þ. The cham-
ber is made up of high order curve. The equation
of working chamber volume can be expressed
with equation (6).
(c) Compression process, 2þ452þx
(where x¼2:51l). The chamber is
made up of high order curve and circle
634 Proc IMechE Part E: J Process Mechanical Engineering 231(4)

involute. The working chamber volume can be
calculated by
V¼hRor LhþLcþ½Rgcð’eÞRgh ð’sÞ

ð7Þ
(d) Compression process, 2þx454þ. The
chamber is made up of high order curve, circle
involute, and arc. The working chamber volume
can be calculated by
V¼hRor LhþLcþLaþ½Rgað’eÞRgh ð’sÞ

ð8Þ
(e) Discharge process, 4þ454þx.The
chamber is made up of circle involute and arc.
The working chamber volume can be calculated by
V¼1
2ha2
3½ð3Þ3ð2þÞ3
þðþþÞ3ðþÞ3
a2ð4Þ2sm
sm¼1
6a2½ðþþÞ3ðþÞ3
þ1
2a21
2lðR2r2Þð9Þ
(f) Discharge process, 4þx456þ. The
chamber is made up of arcs. The working cham-
ber volume can be calculated by
V¼1
2hðR2r2Þðtsin tÞ,t¼6:51ð10Þ
Figure 3(a) is the evolution of the working chamber
volume with respect to the orbiting angle. The a, b, c,
d, e, and f are chamber volume value for different
orbiting angle. It can be seen that the volume
increases in suction process until it reaches the max-
imum and decreases again until the suction chamber is
closed, the suction process is over. As compression
volume decreases, temperature and pressure increase.
The compression chambers change into discharge
chambers at discharge angle. As the discharge process
ends, one entire suction-compression-discharge pro-
cess finished. Figure 3(b) is the derivative of the work-
ing volume with respect to the orbiting angle. It shows
that scroll compressor chamber volume changes in
different process. During suction process the deriva-
tive increases first then drops. When suction chamber
reaches the maximum, the derivative is 0. After that
the derivative is negative. It means the volume begins
to decrease until the end of the discharge process.
Thermodynamic modeling
Basic equations
The change of temperature, mass, and pressure in
each compression chamber with respect to orbiting
angle can be calculated from the first law of thermo-
dynamics (equation (11)) for an open control volume
in conjunction with the equation of states (equations
(13) and (14)) and the mass balance (equation (15)).
The change of the gas temperature with respect to 
can be written as
14
dT
d¼1
mcv
Tdp
dT

v
dV
dv
!
_
min _
mout
ðÞ

X_
min
!ðhhinÞþ
_
Q
!
8
>
>
<
>
>
:
9
>
>
=
>
>
;
ð11Þ
!¼d
dt ð12Þ
Figure 1. Variable thickness scroll compressor. (a) Profile of variable thickness scroll compressor and (b) twin-circular arc modi-
fication of scroll wrap.
Bin et al. 635

p¼pðT,vÞð13Þ
For air, the specific heat ratio cpcan be obtained
from the following relation
33
dh ¼cpdT,
cp¼
1:9327T41010 7:9999T3107
þ1:1407T21030:4489Tþ1057:5
!
ð14Þ
dm
d¼X_
min
!X_
mout
!ð15Þ
Suction gas mass flow
The suction gas mass flow rate _
mis calculated using
the flow equation for isentropic flow of a compressible
ideal gas, corrected by a flow factor
34
_
m¼ Asffiffiffiffiffiffiffiffiffiffiffiffi
2phh
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1
pl
ph

2

pl
ph

þ1

"#
v
u
u
tð16Þ
This flow is restricted by a critical pressure ratio
pl=phfor choked flow conditions.
Heat transfer model
The compression process in a scroll compressor is
achieved in pockets between the scrolls. The efficiency
of the compressor depends significantly on the convect-
ive heat transfer between the scrolls and gas. Therefore,
the accurate understanding of the inside physical con-
dition of gas that is influenced by the convective heat
transfer with scrolls, is very important to correctly ana-
lyze the compression process in scroll compressor.
Heat transfer can occur at several locations in a
scroll compressor because of the different temperatures
of ambient, shell, gas, scrolls. Heating during the com-
pression process causes the compression to go farther
away from the ideal isentropic compression process.
Heat transfer during suction and discharge. The outlet
temperature Ts,oof gas can be calculated by
35
Ts,o¼Tpipe ðTpipe TiÞexp dpLphc
_
mcp

ð17Þ
Figure 2. The gas compression process. (a)0452, (b) 2452þ, (c)2þ452þx, (d)2þx454þ, (e)
4þ454þx, and (f) 4þx456þ.
Table 1. The basic parameters of the variable thickness scroll
compressor (combination profile).
Parameters Value
Original angle 0.993
Base circle radius a2.25 mm
Scroll height h20 mm
Orbiting radius Ror 2.6 mm
Radius of big circular arc R4.4279 mm
Radius of small circular arc r1.8279 mm
Constant of wall thickness C032.1464
Constant of wall thickness C111.821
Constant of wall thickness C22.1066
Constant of wall thickness C30.0868
Control parameter of wall thickness k46.075
Control parameter of wall thickness d3.375
Internal volume ratio 4.5
Delivery 4 m3=h
636 Proc IMechE Part E: J Process Mechanical Engineering 231(4)

The heat flow rate from the pipe to the gas _
Qpipe is
given by
_
Qpipe ¼_
mcpðTpipe TiÞ1exp dpLphc
_
mcp

ð18Þ
which is evaluated for both the inlet and outlet paths
of the compressor.
Heat transfer with scrolls. As the gas is compressed, it
experiences heat transfer from the scrolls and plates.
Due to this heat transfer, the gas expands and thus
increases its pressure, which results in a higher com-
pression work rate. In order to account for these
losses, the heat transfer process needs to be modeled
and incorporated into the compression process model.
The correlation hcm for the spiral heat exchanger is
as follows
36–38
hcm ¼0:023 
Def
Re0:8Pr0:41þ1:77 Def
Raver

1þ8:48 1 exp 5:35StðÞ½ðÞð19Þ
The heat exchange rate _
Qscrolls from the scroll
walls/plates to the gas in any chamber can now be
calculated by an integral method according to the fol-
lowing equation:
14,37,38
_
Qscrolls ¼hcm ZA
Tð’ÞTðk,jÞ½dAð20Þ
Leakage model
During compression process, the pressure difference
of adjacent chambers leads to leakage between the
high and low pressure chambers. There are two dif-
ferent paths for leakage in a scroll compressor. One is
the path that is formed by a gap between the flanks of
the two scrolls and is called flank leakage. Another
path is formed by a gap between the bottom or the
top plates and the scrolls. This kind of leakage is
called radial leakage. The two kinds of leakage are
illustrated in Figure 4. Flank leakage and radial leak-
age account for losses during the compression process
because gas leaking from high pressure regions back
into low pressure regions needs to be re-compressed.
The discharge temperature is increased with increas-
ing leakage. In order to calculate the leakage rate
from a higher pressure chamber to one with low pres-
sure, the width of the flow path needs to be evaluated.
For the flank leakage, the gas flow area Afis
14
Af¼hfð21Þ
For the radial leakage, the gas flow area Aris
14
Ar¼rLrð22Þ
Discharge process
The compression chamber remains closed until the
orbiting angle becomes equal to the discharge angle.
At this angle, the compression chambers begin to
open up to the discharge region and the discharge
process starts. The opening area is so small that the
flow is restricted at first. The pressure in the former
compression chamber is continuously increased.
Compressed gas is discharged gradually through the
opening hole as the change of the orbiting angle.
Because of the profile shape and movement char-
acteristic of scroll compressor, the discharge hole is
opened step by step. Figure 5 shows the discharge
hole over the course of one rotation, the flow area
Adis then calculated by
Ad¼AhAintersection ð23Þ
Figure 3. The evolution and derivative of the working chamber volume with respect to the orbiting angle.
Bin et al. 637

In order to minimize the computational overhead,
the discharge hole free area is calculated for a number
of points per rotation at the beginning of the model
execution. Interpolation is also used to obtain dis-
charge hole blockage areas for orbiting angles in
between the sampled values. Figure 6 shows the
change of flow area for the discharge hole in a
rotation.
Conclusions
1. A new profile based on circle involute, high order
curve, and arc of variable thickness scroll com-
pressor is presented. A detail geometric modeling
is developed. The evolution and derivative of the
working chamber volume with respect to the orbit-
ing angle have been derived.
2. Based on energy and mass conservation equations,
the thermodynamic modeling of variable thickness
scroll compressor is set up.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of
this article.
Funding
The author(s) disclosed receipt of the following financial
support for the research, authorship, and/or publication
of this article: National Natural Science Foundation of
China (Grant No. 51275226), Natural Science Foundation
of ZHEJIANG Province (Grant No. LY12E05010),
Natural Science Foundation of GANSU Province (Grant
No. 1212RJYA010), and Excellent Young Teachers
Program of Chongqing, China.
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Appendix
Notation
Ahtotal area of the discharge hole
Aintersection area of the polygons form the inter-
section of the hole and the tip of the
orbiting scroll
Asarea of the suction chamber opening
cvspecific heat at constant specific
volume
dpdiameter of the pipe
Def hydraulic diameter
ei,di, and ciarc parameters
hgas specific enthalpy of the control
volume
hin gas specific enthalpy flowing into the
control volume
Laarc length of arc base line
Lcarc length of circle involute base line
Lharc length of high order curve base
line
Lplength of the pipe
Lrradial leakage line length
mgas mass
_
min mass flow rate flowing into control
volume
_
mout mass flow rate flowing out of control
volume
pgas pressure
phpressure in the high pressure side
plpressure in the low pressure side
Pr Prandtl number
_
Qheat flow rate flowing into the con-
trol volume
Raver average radius
Re Reynolds number
Rggenerating radius
Rgað’Þgenerating radius of arc at angle ’
Rgcð’Þgenerating radius of circle involute at
angle ’
Rghð’Þgenerating radius of high order curve
at angle ’
Rsswing radius
St Strouhal number
Tigas inlet temperature
Tðk,jÞtemperature of the gas in the kth
chamber at the orbiting angle j
Tpipe temperature of the pipe
specific heat ratio
1modification angle
discharge angle
tcentre angle
conductivity
lcenter angle of modification circular
arc
hdensity of the gas in the high
pressure side
vspecific volume
modification generating angle
!angular speed of compressor shaft
’generating angle
640 Proc IMechE Part E: J Process Mechanical Engineering 231(4)