Analysis of a Two Wrap Meso Scale Scroll Pump
Eric J. Moore*, E. Phillip Muntz*, Francis Erye†, Nosang Myung†, Otto Orient†,
Kirill Shcheglov†, and Dean Wiberg†
*Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089
†
Micro Devices Laboratory, Jet Propulsion Laboratory, Pasadena, CA 91109
Abstract. The scroll pump is an interesting positive displacement pump. One scroll in the form of an Archimedes spiral moves
with respect to another, similarly shaped stationary scroll, forming a peristaltic pumping action. The moving scroll traces an
orbital path but is maintained at a constant angular orientation. Pockets of gas are forced along the fixed scroll from its
periphery, eventually reaching the center where the gas is discharged. A model of a multi-wrap scroll pump was created and
applied to predict pumping performance. Meso-scale scroll pumps have been proposed for use as roughing pumps in mobile,
sampling mass spectrometer systems. The main objective of the present analysis is to obtain estimates of a scroll pump’s
performance, taking into account the effect of manufacturing tolerances, in order to determine if the meso scale scroll pump will
meet the necessarily small power and volume requirements associated with mobile, sampling mass spectrometer systems. The
analysis involves developing the governing equations for the pump in terms of several operating parameters, taking into account
the leaks to and from the trapped gasses as they are displaced to the discharge port. The power and volume required for pumping
tasks is also obtained in terms of the operating parameters and pump size. Performance evaluations such as power and volume
per unit of pumped gas upflow are obtained.
Nomenclature
a - radius of an Archimedes spiral (m)
b - real positive number (m)
CLE - end leak conductance (l/s)
CLSC - edge leak conductance (l/s)
d - gap distance between the scrolls (m)
fD - operating frequency (Hz)
FN - normal force (N)
H - height of the scroll (m)
k - Boltzmann’s constant (J/K)
L - scroll segment length (m)
m - mass of a gas molecule (kg)
n - number density (#/m3)
nE - exhaust number density (#/m3)
nI - inlet number density (#/m3)
N - number of scroll wraps
NT(θ) - number density flow (#/s)
P(φ) - pressure ratio at φ
PE - exhaust pressure ratio
PG - geometric pressure ratio
Q - power for pump (W)
r - orbit radius (m)
R - radius of the scroll (m)
T - temperature of the gas (K)
vf - linear slip velocity (m/s)
VT(φ) - tapped volume at φ (l)
VTI - inlet trapped volume (l)
α - transmission probability
δ - thickness of the scroll walls (m)
φ - orbit angle (rad)
µk - coefficient of kinetic friction
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
1033
1. Introduction
In this paper an analysis of a meso scale scroll pump is presented. The idea behind the scroll pump is not new and
dates back to 1905 when it was invented by Leon Creux [1]. At the time, technology was not sufficiently advanced
to allow the scroll to be machined to the required tolerances for satisfactory operation. It was not until the 1970s
that the commercial market revived the idea [1]. In a scroll pump one scroll remains stationary, while the other
scroll orbits in a circular path without changing it’s angular orientation. During that motion, pockets of gas are
trapped and moved toward the center of the scroll as a result of the orbital motion. At the center of the scroll there is
an exhaust port through which high pressure gas exits the pump. An Archimedes spiral is used to determine the
shape of the fixed scroll. There are numerous ways to derive the shape of the orbiting scroll. Perhaps the easiest is
to use an Archimedes spiral scroll rotated 180° out of phase relative to the fixed scroll. The pump being studied has
been proposed as a roughing pump for a turbo molecular pump in the vacuum system for a mobile, sampling mass
spectrometer.
2.1 Pump Characterization
The geometric pressure ratio of a scroll pump is defined as the ratio of the volume of the initial entrapment of the
gas at the pump’s inlet pressure to the volume of the trapped gas just before it is discharged. For a scroll pump of a
few wraps the geometric pressure ratio is on the order of two. Note that the exhaust pressure can be significantly
higher than indicated by the geometric pressure ratio, since after the trapped gas is exposed to the exhaust pressure it
is generally significantly compressed, but the peristaltic action continues and results in a large fraction of the trapped
gases being forced out of the pump at the prevailing exhaust pressure. As the pocket of gas moves, there is a
backflow from the exhaust pressure to the inlet pressure. This is defined as the leakage of the pump.
Consider a simplified analysis of a two wrap scroll pump, seen in figure 1 below.
FIGURE 1: Two Wrap Scroll Pump
A wrap is defined as a 2π radian change obtained by tracing the scroll element along its length, where the angle is
defined by the line through a point on the scroll and the origin of the scroll. The scrolls in figure 1 actually go
through a 5π radian angular change when they are traced (the additional portion of the scroll in the center is
commonly ignored when designating the number of “wraps” for a scroll pump). The intake port is located outside
the scrolls while the discharge port is located at the center of the scrolls in figure 1. The intake volume (assuming
the scroll is “just closed” when φ = 0) is VTI = VT (φ=0) and the inlet number density is nI. The angle φ is defined as
the angle through which the contact line between the scrolls rotates, beginning when φ = 0 (as illustrated in figure 1).
The scroll pump completes one cycle when φ is 360°. The side views in figure 1 show how the two scrolls seal
together. Each scroll is mounted permanently on a base. The base of one scroll mates with the free edge of the
other scroll forming sliding seals.
There are two types of leaks in the pump. The first and more dominant leak is the leak between the two scrolls
(end leak). As one scroll orbits a gap exists between the wall of the moving scroll and the wall of the fixed scroll
due to manufacturing tolerances. The second type of leak originates where the free edge of one scroll contacts the
base of the other scroll. If the free edge of one scroll does not mate perfectly with the base of the other scroll, a leak
occurs. For simplicity, the two compression chambers can be combined and analyzed as a uniform pressure
annulus. This is possible because both chambers that form one annulus experience equal, although time varying,
pressure differences at their ends and along their edges at all times from their initial trapping to exhaust at high
pressure. The leaks in and out of the annulus can be written as:
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(
−1
(
−1
− f D (n(π ) − nI )C LSC f D + (n(π ) − nI )C LE f D
+ f D (nE − n(π ) )CLSC f D + (nE − n(π ) )CLE f D
−1
−1
)
)
(Out)
(1)
(In)
Here, fD is the operating frequency, n(φ) is the number density at a position φ of the trapped gas in the compression
cycle, nE is the exit number density, CLE and CLSC are the conductance’s of the end leaks and the free edge leaks
respectively.
The leakage is described in more detail in a later section. With the total leakage known, the pumping speed can be
derived. First, the ideal pumping speed (i.e. no leakage) is the intake volume multiplied by the frequency of orbit of
the pump multiplied by two (there are two pockets of gas that are simultaneously trapped each cycle to form the
annulus described above). Assuming isothermal compression, the actual pumping speed can be written as:
 n(π ) 
− 1(C LE (π ) + C LSC (π ) )
S p = 2 f DVTI − 
 nI

(2)
where VTI is the initial trapped volume, VT(φ) is the volume of the trapped gas at a specific point φ of the trapping
cycle and n(π)/nI is the number density ratio halfway through one cycle of the scroll pump.. Isothermal compression
is a valid assumption since the compression ratio is small, at least until the exposure of the trapped gas to the
exhaust pressure, and for meso scale pumps the flow is quite rarefied in the critical range of inlet pressures below 10
torr. The number density after one cycle can be found from the number of molecules flowing in and out of the
trapped volume at one cycle per unit time divided by the volume flow per unit time at one cycle, as seen in equation
(3).
n(2π ) =
N T (2π )
2 f DVT (2π )
(3)
N T (2π ) = 2nI f DVTI − [(n(π ) − nI )C L ] + [(nE − n(π ) )C L ]
The number density ratio after one cycle in terms of the initial trapped volume, the number density ratio nE/nI, and
the leakages is given in equation (4).

 nE
1
n( 2π )
n(π ) 

=
−2
2 f DVTI + C L 1 +
2 f DVT ( 2π ) 
nI
nI
n I 

(4)
Assuming a linear relationship between φ and n(φ), the number density ratio at half a cycle can be approximated by:
n(2π ) − nI
2
n( 2π ) 2n(π )
=
−1
nI
nI
n(π ) = nI +
(5a)
(5b)
Substituting equation 4 into equation 5b gives:

 n  
C L 1 + E  

nI  
 P (2π ) + 
+1
G

4 f DVT (2π )
n(π ) 1 
= 

CL
nI
2

1+


2 f DVT (2π )




Using equations 2 and 6, the pumping speed can be written as:
1035
(6)


 n 
C L 1 + E 


nI 
 P (2π ) + 
+1 
 G

2 f DVT (2π )
− 1C L
S p = 2 f DVTI − 
C
L


2+


f DVT (2π )




(7)
Where C L = C LSC (π ) + C LE (π ) is the conductance of the total leakage of the pump. Note that the number density
ratio nE / nI is equal to PE due to the assumption of isothermal compression. The inlet number density of a perfect
gas can be found by:
nI =
pI
kTI
(8)
where pI is the inlet pressure, TI is the inlet temperature and k is Boltzmann’s constant. The maximum pumping
speed (Spmax) can be obtained by setting the pressure ratio PE = 1 [3].
S P max
CL


 PG (2π ) + f V (2π ) + 1 
D T
= 2 f DVTI − 
− 1C L
CL


2
+


f DVT (2π )


(9)
Where PG = VTI/VT(2π) is the geometric compression ratio. For the pump of interest (two wraps) PG is the ratio of
the volume of the initial trapped gas to the volume of the trapped gas after one cycle (φ = 2π) of the orbital motion.
The maximum pressure ratio (Pmax) can be found by setting the pumping speed (equation 6) to zero [3].
2f V
Pmax =  D TI
 CL
 4 f DVT (2π )
1 
 +1

+1+
CL
PG (2π ) 

(10)
2.2 Leakage
In order to determine the actual pumping speed, an expression for the leakage must be found. For much of the
range of pressures in the pumps, the leaks can be modeled as being in free molecular flow (mean free path is greater
than a characteristic dimension of the leak). In order for the scroll pump to be effective, the average gap distance
between the scrolls must be very small (on the order of a micron). Consider a generic scroll pump leak as seen in
figure 2.
FIGURE 2: Simple Leak
The coordinate system defined in the figure will be used for all the leaks in the system. The x direction corresponds
to the thickness of the scrolls, the y direction corresponds to the length of contact, and the z direction corresponds to
the gap distance between the scrolls. The edge leak will be analyzed first. The gap between one scroll and the other
scroll base can be viewed as a large flat plate. The large flat plate is semi-infinite along the contact length and in the
1036
flow direction the thickness of the scrolls is much greater than the gap distance between the scroll and the scroll
base. The equation below is for the free molecular conductance of an aperture, traditionally given as a volume flow.
C=A
kT
A 8kT
=
2πm 4 πm
(15)
Where A is the cross sectional area of the aperture, k is Boltmann’s constant, T is the gas temperature, m is the
molecular mass of the gas, and ((8kT/πm)1/2 is the mean thermal speed of the gas molecules. Since the edge leaks
are not an aperture, there is only a finite probability that once a molecule has entered the aperture it will leak out of
the system. The transmission probability α, is the probability that a molecule, once it has entered the inlet aperture,
will travel through the gap. The transmission probability is found using an approximation for short slits or large flat
plates (in the x direction) [4].
α=
δ z   2δ x 1 
− 
ln
δ x   δ z 2 
(16)
δz is the gap distance (z direction), and δx is the thickness of the scroll walls (x direction). This approximation can
be used if the length of the slit (scroll segment length at the top and bottom) and the width of the slit (thickness of
the scroll) is much larger than the gap between the base of one scroll and the free edge of the other scroll. The net
conductance of the edge leak is
kT  n A 
1 − 
2πm  nB 
C LSC (π ) = α LSC ASC
(17)
where nA and nB are the number densities outside and inside the annulus respectively. The majority of the edge
leaks are from the scrolls segments exposed to the inlet pressure. The edge leak for the scroll segment exposed to
exhaust pressure increases the pressure at 2π which then leaks to the inlet pressure surrounding the scroll pump.
The end leak can be modeled in the same way as the edge leak. The transmission probability is different than that
of the edge leak since the contact length is the same order as the height of the scrolls. The transmission probability
is greater for the end leak than for the edge leak. The net conductance of the end leak is
C LE (π ) = α LE AE
kT  n A 
1 − 
2πm  nB 
(18)
For comparison a continuum leak is shown below.
C L = 2α L ASC
p 
kT 
1 − A 
pB 
2πm 
(19)
2.3 Power Requirements
Power is a critically important factor in determining the usefulness of small scale mobile devices. The overall
system needs to be small including the power source. The power required presented here is a very basic estimate.
Simply put, it is the power required to move one scroll, i.e. the power required to overcome friction and gravity.
For frictional and fabrication purposes the scrolls are made of nickel while the base of the scrolls are made of
sapphire. The coefficient of kinetic friction between sapphire and nickel is low. The normal force required to keep
the two scrolls together can be approximated as:
FN = pE AE
Where pE is the exhaust pressure and AE is the area through which the force is exerted. The frictional force is:
1037
(20)
(21)
F f = µ k FN
One scroll is orbiting quickly rotationally relative to the fixed scroll. The dimensions of the scrolls are relatively
small therefore the linear velocity is small compared to the rotational velocity. With the frictional force and the
linear velocity, the power required by the moving scroll can be found; remembering that the moving scroll is
orbiting but maintaining a constant attitude
.
Q = F f v f = µ k pE AE f D 2πr
(22)
where r is the orbit radius. The weight of the scrolls is neglected since they are small compared to the normal force
needed to hold the scrolls together. The above analysis is oversimplified due to the complex nature of friction and
frictional forces. In the case where the walls of the scrolls touch each other, another sliding frictional force is
created. Since the length of contact and the relative speeds between the scrolls are small, the force is insignificant
compared to the force created by the scroll edge and the scroll base. If the walls were to touch, the lifetime of the
scroll pump would be diminished.
2.4 Staging
The above derivations were for a single stage scroll pump. To increase the overall pressure ratio, a multistage
system may be required. Assuming there are no leaks between the stages, the mass flow through each stage must be
the same for time independent flow. The conservation of mass flow equation is:
n I S p1 = n1− 2 S p 2 = n2−3 S p 3 = n( n −1) − n S pn
(23)
The above equation results in a system of n-1 equations and n-1 unknowns, where n is the number of stages. Thus,
for a three stage scroll pump system, the equations become
n I S p1 (n1−2 / n I ) = n1−2 S p 2 (n2−3 / n1−2 )
n1−2 S p 2 (n1−2 / n I ) = n2−3 S p 3 (n3−4 / n2−3 )
(24)
The frequency of each stage or the size of each stage can be changed. When the operating frequency is changed for
just one stage, the mass flow through that stage is different from the other stages. The system adapts by increasing
the pressure between the stages, thus decreasing the pressure ratio and equalizing the mass flow for each stage. The
same adaptation occurs when the size of one stage is different from the size of the other stages. The effect of
different operating frequencies is discussed in the next section.
3. Computational Analysis
A computer program was written to model one and multiple stage scroll pumps. The program requires the user to
input the number of wraps of the scrolls, radius of the scrolls, the thickness of the scroll walls, and the height of the
scrolls. The program assumes that the two scrolls are identical (i.e. they have the same manufacturing tolerances).
The scroll pump of interest is two wraps, approximately one centimeter in diameter, and operating at 100 Hz.
Figure 1 shows the cycle of the scroll pump. As the green scroll orbits relative to the red scroll, the yellow pockets
of gas is moved along the scrolls and discharged at the center of the scrolls. Figure 3 shows the pumping speed
versus average gap distance for a one stage scroll pump at 50 torr inlet pressure. The end leak gaps and the edge
leak gaps are equal. After about one micron the pumping speed drops dramatically. When the gap distances
become greater than about three microns the pump is useless.
1038
14
Pumping Speed (ml/s)
12
10
8
6
4
2
0
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
Average gap distance (microns)
FIGURE 3: Pumping Speed vs. Gap Thickness @ 50 Torr Inlet Pressure (Meso Scale, 1 Micron, 100 Hz)
Figure 4 shows one, two, and three stage scroll pumps with an average gap distance of one micron. The two stage
scroll pump can reach an ultimate pressure of about 100 millitorr, while the three stage scroll pump can pump to 10
millitorr. In order to achieve the ultimate pressures needed for a mobile mass spectrometer system, a three stage
scroll pump will be necessary. More stages will require much more power while only decreasing the ultimate
pressure of the system by a small amount.
14
Pumping Speed (ml/s)
12
10
8
One Stage
Two Stages
6
Three Stages
4
2
0
0.001
0.01
0.1
1
10
100
1000
Inlet Pressure (torr)
FIGURE 4: Pumping Speed vs. Gap Thickness @ 50 Torr Inlet Pressure (Meso Scale, 1 Micron, 100 Hz)
Energy required to pump a molecule for one, two, and three stages is shown in Figure 5. As the number of
molecules decrease, the energy required to pump one molecule increases exponentially. When the pump reaches its
ultimate pressure the energy asymptotes to infinity. At this pressure the pumping speed reaches zero but energy is
still being used by the system (to orbit the scroll).
Energy per Molecule (J/molecule)
1.00E-13
1.00E-14
1.00E-15
1.00E-16
1 Stage
1.00E-17
2 Stages
3 Stages
1.00E-18
1.00E-19
1.00E-20
1.00E-21
1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06
Pressure Ratio
FIGURE 5: Energy required to pump one molecule (Meso Scale, 1 Micron, 100 Hz)
Table 1 is a comparison of one, two, and three stage scroll pumps. The three stage scroll pump has a higher
volume efficiency than the two or one stage pump, but it requires much more power to operate. The only advantage
a three stage pump has over a one stage is the ability to reach much lower pressures.
1039
# of stages
1 stage
2 stages
3 stages
Energy efficiency
6.74e-17
9.53e-15
3.86e-14
1.39e-24
9.77e-23
2.65e-22
22.07
0.024
0.0037
2.5
4.8
7.2
(J/molecule)
Volume efficiency
(m3/#/s)
Ultimate Pressure
(torr)
Operating Power
(Watt)
TABLE 1: Comparison of one, two, and three stage scroll pump
To validate the accuracy of the computer model an off-the-shelf scroll pump was tested. The scroll pump was
loaned to USC by Synergy Vacuum Inc. The critical dimensions of the scrolls were measured and imported to the
computer model. The model results were then compared to the pumping performance provided in literature by
Synergy Vacuum and independent experimental tests conducted at USC.
Experimental
Spmax
Computational
Computational
Computational
(gap: one
micron)
(gap: three
microns)
(gap: ten
microns)
4.267
4.303
4.289
4.238
1.85e-2
7.95e-3
7.26e-2
10
(l/s)
Pressure
(torr)
TABLE 2: Off the shelf scroll pump
Table 2 compares the computer model predictions with the actual performance of the large scroll pump. The actual
gap distances cannot be measured accurately, therefore the model calculations were performed with various average
gap distances. The pumping speeds for the model calculations are slightly higher than the experimental results.
Overall, the computer model predicts (within a reasonable accuracy) the pumping speed and ultimate pressure of the
large scroll pump. Now that the model has been validated, a meso scale pump will be tested and compared with the
model.
REFERENCES
1. Gravensen, J., and Henriksen, C.,”The Geometry of the Scroll Compressor”, in Society for Industrial and Applied
Mathematics, Vol. 43, No. 1, 113-126 (2001).
2. Gravensen, J., Henriksen, C., and Howell, P., Danfoss: Scroll Optimization, in 32nd European Study Group with Industry,
Department of Mathematics, Technical
University of Denmark, 1998, pp. 3-35.
3. Muntz, E.P., and Vargo, S.E., “Microscale Vacuum Pumps,” in The MEMS Handbook, edited by M. Gad-el-Hak, John Wiley
& Sons, Inc., New York, 2002, pp. 1-27.
4. Lafferty, J., Foundations of Vacuum Science and Technology, John Wiley & Sons, New York, 1998.
5.
DeWitt, D.P., and Incropera, F.P., Introduction to Heat Transfer, John Wiley & Sons, New York, 1996.
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