Thermal Transpiration in Microsphere Membranes
Marcus Young, Yen Lin Han, E.P. Muntz, G. Shiflett
USC AME Department
Andrew Ketsdever
Air Force Research Laboratory
Amanda Green
Jet Propulsion Laboratory, California Institute of Technology
Abstract. Self-assembled glass microsphere membranes as an alternative transpiration membrane for application in a
Knudsen Compressor are discussed. A performance model is constructed and used to compare the performance of glass
microsphere membranes to silicon aerogel membranes for this application. An initial experimental Knudsen Compressor
stage based on glass microsphere membranes has been designed and experimentally tested. Preliminary performance
results show a discrepancy between the predicted and observed pressure differences produced by the single stage.
Several possible explanations for the discrepancy are discussed. Two variations of a proposed design for a Knudsen
Compressor employing a microsphere transpiration membrane are discussed. It is concluded that beds of glass
microspheres may be attractive candidates for transpiration membrane materials over the entire pressure range of
operation for a micro-scale vacuum pump, 10mTorr to 760 Torr.
1. INTRODUCTION
1.1) Background
Knudsen Compressors have proven viable candidates for both micro-scale gas roughing pumps [1] (760 Torr to
10 mTorr) and micro-scale gas compressors (1atm to ~ 10 atm) [2]. The thermal transpiration membrane is the
flow-driving component of the Knudsen Compressor and is therefore a critical component of the overall design.
Transpiration membrane materials must have thermal conductivities on the order of tens of mW/(m*K) to be viable
from power consumption considerations. Initial experimental versions of the Knudsen Compressors operated near
atmospheric pressure and used 520µm thick sheets of low density silicon aerogel as the transpiration membrane [3].
Aerogel was chosen for its high porosity (>95%) and low thermal conductivity (17 mW/(m*K) at atmospheric
pressure). More recent results from volume and power optimization studies indicate that it is most efficient to
operate the Knudsen Compressor such that the gas flow through the transpiration membrane has a Knudsen number
of roughly one [4]. Optimum pore diameters at the practical low-pressure pumping limit (10 mTorr) are on the
order of 0.5mm indicating that the previously used low-density silicon aerogels, having pore diameters of 20 nm, are
inappropriate for application at low pressures. One option for a low-pressure transpiration membrane is to etch
optimally sized capillaries through the aerogel with the diameters depending on the operational pressure [1].
Another possibility, discussed here, is to use a membrane of glass microspheres as the thermal transpiration
membrane.
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
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1.2) Knudsen Compressor Description
The Knudsen Compressor, first suggested by Pham-Van-Diep et al [5] and demonstrated by Vargo et al [6,7], is
a modern version of the original multistage thermal transpiration compressor described by Knudsen [8,9] in 1910. A
series of individual stages are used to provide the required pressure ratio in a Knudsen Compressor. A single stage
of a Knudsen Compressor is shown schematically in Fig. 1. Also shown in Fig. 1 are the temperature, T, and
pressure, p, variations along the stage.
(i-1)th
Stage
(i+1)th
Stage
ith Stage
Lx,i
LX,i
Capillary
radius Lr,i
x
T
LR,i
Connector
Section
Capillary
Section
∆Ti/2
Tavg
∆Ti/2
TL,i
P
∆pC,i
pi
κi∆pmax,i
(pi)eff
(pi-1)Eff
FIGURE 1. Illustrative stage of a Knudsen Compressor [3]
Each stage has a capillary section where the temperature increases, causing a pressure increase due to the rarefied
gas dynamic phenomenon of thermal creep or transpiration. The capillary membrane is followed by a connector
section with a significantly larger radius than the individual capillaries. The connector section operates with the gas
flow closer to the continuum regime so that the pressure is approximately constant while the temperature is lowered
to TL,i, the gas temperature at the beginning of the stage. Each capillary section is reminiscent of the single thermal
transpiration element used by Reynolds [10] in his experiments reported in 1879. The compressor's potential
applicability has been significantly expanded since Knudsen’s 1910 investigations by the recent, serendipitous
availability of small capillary membranes in materials with extremely low thermal conductivity.
1.3) Previous Aerogel Transpiration Membrane Results
Experimental MEMS Knudsen Compressor stages based on aerogel transpiration membranes previously have
been built and tested. [3] A schematic of a single stage and a typical performance result are shown in Fig. 2.
FIGURE 2. Single Stage Experimental Knudsen Compressor and Typical Pump Down Result [3]
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A 520 µm thick disc of low density silicon aerogel was used for the transpiration membrane. The aerogel disc
edges were sealed with Torr Seal® epoxy. The aerogel was sandwiched between two silicon thermal guards with
Deep Reactive Ion Etched (DRIE) holes through them. One of the thermal guards had a thin film resistive heater
deposited on it. Current was driven through the heater to establish the temperature difference and drive the pump.
The silicon thermal guards were anodically bonded to pyrex connector sections. Proof of concept has been
demonstrated for up to three stages in series. Operation was investigated from atmospheric pressure down to
roughly one hundred Torr for several different working gases [3]. Fig. 2 also includes a typical pressure difference
verses heater power result for a variety of operational gases.
A comparison of the characteristics for a Knudsen Compressor cascade based on a 2% Carbon doped low density
silicon aerogel and one based on a similar membrane with optimally sized holes etched through it is shown in Table
1. Aerogel MEMS processing techniques, including pattern etching, have recently been demonstrated [11]. The
cascades are designed to operate from 10mTorr to 100 mTorr, but because of manufacturing concerns, the stage
dimensions were held constant for each of the cascades. A temperature difference of 100K was assumed across the
aerogel transpiration membrane. The membranes are 500 µm thick and have a diameter of 1cm. The connector
section also had a diameter of 1cm. and a length of 4cm. The power efficiency is defined as the energy required to
pump a single molecule through the specified pressure range. The volume efficiency is the pump volume per
pumped molecule.
TABLE 1. Low Pressure 10m Torr to 100m Torr Base Cascade Analysis Results
Low Density 2% Carbon Doped Si
Low Density 2% Carbon Doped Si
Aerogel (base case)
Aerogel with Optimally Sized Holes
Number of Stages
24
34
Pressure Ratio
10
10
Flow Rate (#/s)
6E13
3E16
Power Efficiency (J/mol)
2.5E-14
7.E-17
Volume Efficiency (m3/mol)
1.8E18
6.7E20
A relatively large number of stages, 24, is required to produce the design pressure ratio of 10 for the base design
case employing low density silicon aerogel due to the inefficiency of the Knudsen Compressor near the lower
pressure limit with this initial design. Etching optimally sized holes in the aerogel decreases the pressure rise
efficiency of each stage due to the decrease in the Knudsen number, but increases the conductance of the membrane
by several orders of magnitude, providing an increase for both the power and volume efficiency of greater than an
order of magnitude. Low-density aerogel membranes with optimally sized holes through them have predicted
volume and power efficiencies that are attractive despite the increase in the number of stages required to achieve a
particular pressure ratio.
2. GLASS MICROSPHERE BEDS
Three evaluation parameters can be used to determine the effectiveness of a candidate transpiration membrane
material: effective pore diameter, gas conductance or porosity, and thermal conductivity. For optimal operation, the
effective pore diameter of the membrane must be sized such that the gas flow in the pores has a Knudsen number of
roughly one [4]. The transpiration membranes must have sufficient porosity to allow efficient mass flow through
the membrane. Materials with sufficiently low thermal conductivities (10s of mW/mK) are required to minimize the
power required to maintain the temperature difference across the transpiration membrane since it is the temperature
difference that drives the pump.
2.1) Sphere Bed Geometries
Different sphere bed geometries can be realized depending on the manufacturing procedure for the microsphere
membrane. The different geometries have different mean pore diameters, porosities, and thermal conductivities.
The simple cubic (SC) and face centered cubic (FCC) pore geometries shown in Fig. 3 represent the entire range of
porosities for ordered sphere arrangements [12].
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SC
FCC
FIGURE 3. Unit Cell for Simple Cubic (SC) and Face Centered Cubic (FCC) Microsphere Bed Geometries [12]
(c) 2000, this material is used by permission of John Wiley & Sons, Inc.
The first pattern, simple cubic, represents the highest porosity, mp = 47.64%, among the ordered packing
patterns. Each side of the unit SC cell has the length of the glass microsphere diameter. The face centered cubic
pattern has the lowest porosity among the non-random packing patterns. Each side of the unit cubic cell has the
length of 2 times the glass microsphere diameter.
The equivalent pore diameter, obtained from geometric considerations, for beds of microspheres is given by
2 mp
δ avg =
d gr
3 1− m p
(1)
Where dgr is the sphere diameter. This provides an average pore diameter of δavg = 0.607dgr for the SC arrangement
and δavg = 0.234dgr for the FCC arrangement. Microspheres are commercially available with diameters of 20nm to
above 1mm [13]. This indicates that microsphere membranes can be made with effective pore diameters from
roughly 10nm to greater than 1 mm. For air at a mean temperature of 350K this corresponds to an efficiently
operating transpiration membrane from roughly 10 atm (for an effective pore diameter of 10nm) to below 100mTorr
(for an effective pore diameter of 0.5mm). It also appears that glass microsphere transpiration membranes have
sufficient porosity, between roughly 25% and 50%, to allow efficient mass flow through the membrane.
2.2) Sphere Bed Thermal Conductivity Model
The thermal conductivity through a bed of spheres has been estimated previously by Kaganer for use in granular
insulation layers [14]. His model assumes that the particles of the granular materials are hard spheres with a
constant diameter. Planes passing through the centers of the spheres perpendicular to the outward normal of the
membrane are assumed to be isothermal. Ignoring radiation, Kaganer expressed the gaseous and solid fraction of
the thermal conductivity of a bed of spheres surrounded by a continuum gas as
 5.8(1 − m ) 2  1
κ ref  
k
p

(2)
ln s − 1 −
+ 1
k = kg 
 κ ref
 
κ ref
2

k
g

 

where kg is the thermal conductivity of the gas, ks is the thermal conductivity of the bulk solid material, and
κ ref = 1 − k g k s ,. A similar expression is obtained for a bed of microspheres surrounded by a transitional gas by
replacing
k g with
kg
1+ 2 β
where β = 9γ −5 2 (γ +1) ⋅ 2 −α α , γ =
Cp
Cv
o
λ
(3)
δ avg
and α is the accommodation coefficient. Klein [15] considered radiation
scattering on uniform diameter spheres to get the following expression for the fraction of the thermal conductivity
due to radiation,
4d gr σ ε
(4)
T3
kr =
s 2−ε
where σ is the Stefan–Boltzmann constant, ε is the effective packed-bed emittance, s is the average solid fraction.
The total thermal conductivity is then given by
kt = k + kr
(5)
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2.3) Sphere Bed Gas Conductance Models
A capillary tube model [16] has been used to model the gas flow through a packed bed of microspheres. The pores
in the microsphere bed are modeled as an array of capillary tubes with uniform diameter. For a tube with a small
L/D the conductance can be effectively modeled by treating it as an aperture and long tube in series as shown by
−1
 1
1 

C st = 
+

 C lt C a 
The large L/D gas conductance in the transitional flow regime is given by [4]
(6)
2πr 3 kTavg
(7)
Qp
lc
8m
where r is the pore radius, lc is the pore length, Tavg is the average temperature, m is the particle mass and Qp is the
pressure flow coefficient [4]. Lafferty uses a curve fit to model the gas conductance for transitional flow through an
aperture by [17]
1


−
(8)
C a = C ma + (Ci − C ma )1 − 1.05 2 Kn 




where Kn is the Knudsen Number, the conductance through a free-molecule aperture is given by
C lt =
C ma = πr 2
kTavg
(9)
2πm
The conductance through an aperture in isentropic flow is given by
1/ 2
1
1−γ 

kTavg Pu − γ  2γ 
C i = πr
Kp 
1 − K p γ 
(10)

m ∆P
 γ − 1 


in which Pu is the higher pressure ∆P is the pressure difference and Kp is the pressure ratio. When modeling a
porous membrane as an array of tubes it is also necessary to determine the number of tubes because the gas
conductance through the membrane depends linearly on the number of tubes and is given by
Am p
N=
(11)
πr 2
Where A is the cross sectional area of the bed. The total conductance of the bed of spheres is equal to N times the
conductance of a single capillary tube.
2
2.4) Membrane Thermal Transpiration Model
The Knudsen Compressor performance model [4] for aerogel membranes can be used to analyze the thermal
transpiration properties of the glass microsphere membrane. The pressure difference maintained across the ends of a
tube due to thermal transpiration can be given by [4]
∆T Qt
κ
∆P = Pavg
(12)
Tavg Q p
where Pavg is the average pressure for the stage, Qt is the transpiration flow coefficient, Qp is the pressure return flow
coefficient, and κ is the fraction of the maximum no upflow pressure difference that is realized. The mass flow is
given by [4]
m
∆T Qt 

M& =
C st  Pavg
(13)
(1 − κ )
kTavg
Tavg Q p 

2.5) Performance Predictions for Glass Microsphere Membranes
A comparison of the performance of a single stage of three different transpiration membrane materials operating
over the identified mean pressure range is shown in Fig. 4. The three membrane materials are low density Si
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aerogel, low density Si aerogel with optimally sized pores (varying with mean pressure) etched through it, and a bed
of microspheres sized to provide optimal pore dimensions (varying with mean pressure).
1.E-17
Q&
N&
(J / mol )
Low Density Si Aerogel
microsphere (nonoptimal)
microsphere (optimal)
etched aerogel (nonoptimal)
etched aerogel (optimal)
1.E-18
1.E-19
0.01
0.1
1
P(torr)
10
100
1000
FIGURE 4. Energy Efficiency for Candidate Membrane Materials
As shown in Fig. 4 the unetched low-density aerogel is naturally optimized for operation at pressures above 1
atmosphere, but is not competitive over the entire pressure range, as the pressure is decreased, the low density Si
aerogel becomes too inefficient for operation. Aerogel MEMS processes are limited to feature scales greater than
roughly 10µm and etched hole diameters are unpractical above 1mm so the available pore diameters for this design
can range between 10µm and 1mm. This indicates that there is a limited pressure range where fully optimized pore
diameters are available. Outside of this range the power efficiency is calculated by using the “best available” pore
diameter or 1mm for low pressures and 10 µm for higher pressures. Microspheres with diameters above 1mm are
available, but diameters this large lead to impractical designs. A limit of 1mm is therefore placed on the largest
microsphere diameter for this study. At lower pressures 1mm is used as the microsphere diameter. Glass
microsphere membranes have a wide pressure range of potential application (10 mTorr to above 1 atm) and appear
to be efficient enough to warrant further investigation.
3. MICROSPHERE BED EXPERIMENTS
An experimental single stage Knudsen Compressor based on a glass microsphere transpiration membrane has
been constructed to test the thermal transpiration properties of the glass microsphere beds and is shown, in false
color, in Fig. 5. Fig. 5 also shows an image of the experimental setup for the tests. The glass microspheres used in
this investigation had a diameter of 1mm. The microspheres were housed between two aluminum thermal guards
with arrays of 0.5mm diameter holes conventionally machined through them. A 1mm thick Teflon® sealing ring
surrounded the microsphere bed.
FIGURE 5. Single Stage Knudsen Compressor Based on Glass Microspheres
The device was placed in a vacuum chamber that was evacuated to a pressure below 0.1 mTorr and then filled with
varying pressures of pure gases. Helium was used for the results described here. A temperature difference was
maintained across the microsphere bed by resistively heating the hot thermal guard and placing the evacuation
chamber in thermal contact with the vacuum chamber. The temperature difference produced a steady-state pressure
difference between the evacuation chamber and the surroundings. This pressure difference was measured by a
differential baratron. The vacuum chamber pressure was slowly decreased allowing the measurement of the
differential pressure dependence on the mean pressure. The temperature difference was measured using
thermocouples placed on the hot and cold thermal guards. Thermo-molecular pressure differences, or TMPD, as
given by (∆P/P)/(∆T/T) [18] were measured for different operating conditions. Fig. 6. shows a typical experimental
result for helium compared to analytical predictions made from the model described above.
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1
Experiment
Model
TMPD
0.1
0.01
0.001
0.1
1
Pressure (Torr)
10
FIGURE 6. Comparison of Experimental and Analytical Single Stage Performance
There is an order of magnitude difference in the analytically predicted TMPD and the experimentally measured
TMPD. Uncertainties in the differential pressure measurement were no more than 25% (at a mean pressure of
0.1Torr) and uncertainties from the thermocouples were no more than 5% of the temperature difference indicating
that measurement errors do not account for the discrepancy in TMPD. Three possible explanations for the
discrepancy require further investigation. One effect unaccounted for in the analytical performance model is that the
thermal guards may not be performing well at the low pressures. The purpose of the thermal guards is to maintain
the temperature difference in the gas and to contain the glass microspheres. In this design the thermal guard has
capillary diameters that are similar to the mean diameter of the glass microsphere bed and an L/D of 2. Therefore
the gas traveling through the thermal guard holes may not have had the desired accommodating collision with the
walls. This is particularly problematic for the cold gas in the surrounding vacuum chamber entering through the hot
thermal guard, since a reverse thermal transpiration may be caused that will counteract the effectiveness of the
pump. Another possible explanation for the discrepancy is that it is difficult to obtain uniform thermal contact
between the outer layer of microspheres and the hot thermal guard. This causes the effective temperature difference
to deviate from the measured temperature difference. Finally, it is difficult to estimate the equivalent pore diameter
for thermal transpiration purposes. The definition of the equivalent pore diameter given above is based purely on
geometric considerations. A broad distribution of pore diameters can exist in the microsphere membrane, with the
upper pore diameter limit being the entire diameter of the membrane. If significant pores are available with
diameters well above the mean pore diameter, the Knudsen Number of the flow will be decreased thereby
decreasing the pressure rise efficiency of the microsphere membrane. It is possible, therefore, that the performance
discrepancy is caused by an unaccounted for fundamental characteristic and/or an inefficient design. Further
investigations are required to determine the relative importance of the possible explanations for the performance
discrepancy.
4. MICROSPHERE BASED KNUDSEN COMPRESSOR DESIGN
Recently a variety of methods for manufacturing synthetic opals, or 3 dimensional arrays of glass microspheres,
have been demonstrated [19,20]. The synthetic opals are used as photonic crystals [21] and as test beds to study the
fundamental physical problem of phase transitions analogous to molecular phase transitions [22]. The most
common microsphere array assembly procedures are solvent evaporation, sedimentation, and electrostatic
interaction. The different methods produce different sizes of arrays, different array geometries, and have varying
amounts of defects. Microsphere self-assembly is a required manufacturing technique for Knudsen Compressors
employing spheres with diameters smaller than the limiting hole dimension for the thermal guards, ≈ 10µm.
Coincidentally, many microsphere self-assembly techniques are found not effective for sphere diameters above 10
µm [20].
Two sample stage cascades have been sized for comparison. The first cascade, shown on the left, operates from
50 Torr to 500 Torr and requires microsphere self-assembly. The second is sized to operate from 50 mTorr to 500
mTorr and does not require microsphere self-assembly since it employs larger diameter microspheres that can be
contained using the thermal guards. Sample designs for a stage of both microsphere based Knudsen Compressor
cascades are shown in Fig. 7.
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Hot Thermal
Guard
Radiant
Heating
FCC Bed of
microspheres
(not shown to scale)
Thin sealant
layer
FCC Bed of
microspheres
(not shown to scale)
Cold Thermal
Guard
FIGURE 7. Sample Microsphere Knudsen Compressor Stages
Glass microspheres with diameters of 0.75 mm and 0.344 µm were chosen as optimal pore diameters for the low
pressure and high pressure cases, respectively. The glass microspheres for the low-pressure case will be self
assembled using the sedimentation technique. The glass microspheres for the high-pressure case will be contained
in all dimensions. Dyed black microspheres are available and will be used to minimize the radiation component of
the thermal conductivity. The temperature difference in both arrangements is provided by radiant heating of the
glass microspheres and the thermal guard. Both microsphere membranes require thin sealing rings on the outside to
stop gas flow through the sides. The low-pressure design requires a hot thermal guard to contain the glass
microspheres. Table 2 lists the performance of the two designs on Nitrogen as estimated from the performance
models discussed above. The mean temperature is 350K and the temperature difference is 100K. For both designs
the capillary membrane diameter and connector section radius is 4mm, the capillary section thickness is 1cm and the
connector section thickness is 3 cm.
TABLE 2. Performance Estimates for Glass Microsphere Based Knudsen Compressor Design
P = 50-500 Torr (self assembled)
P = 50-500 mTorr (not self assembled)
Flowrate (#/s)
6.1E16
4.6E15
Number of Stages
32
46
Power Efficiency (J/mol)
5.4E-17
3.4E-17
The flowrate for the high-pressure case is only one order of magnitude more than for the low-pressure case because
while the pressure is three orders of magnitude higher the gas conductance through the transpiration membrane also
decreases. It has been recently suggested that a power efficiency on the order of 1.E-17 J/mol for a pressure ratio of
10 is sufficient for some purposes [23] indicating that both designs may be suitable.
5. CONCLUSIONS
Microspheres exist in a broad range of materials and with diameters of 20nm and larger. Manufacturing
techniques exist to make beds of glass microspheres over the entire identified pressure range. Initial performance
estimates indicate that beds of glass microspheres can perform nearly as efficiently as low density Si aerogels with
optimally etched hole. Glass microspheres appear to be the most efficient candidate technique at pressures below
roughly 100 mTorr. Results from initial experimental investigations show a discrepancy with the thermo-molecular
pressure difference (TMPD) predicted using the analytical model. Possible explanations for the discrepancy include
fundamental physical considerations and design considerations not included in the computational model indicating
that further investigations are required to determine the true performance capabilities of glass microsphere
membranes in Knudsen Compressors.
ACKNOWLEDGEMENTS
The research described in this paper was partially carried out at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and Space administration.
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