Microfilter Simulations and Scaling Laws
David R. Mott, Elaine S. Oran, and Carolyn R. Kaplan
Laboratory for Computational Physics and Fluid Dynamics
U. 5. Naval Research Laboratory, Washington, B.C. 20375
Abstract. This work presents DSMC simulations used to quantify the effect of Knudsen number on flows
through filters with micron-scale holes. The empirical scaling laws currently used to predict pressure drop
as a function of flow conditions and filter geometry are based on experiments and calculations within the
continuum regime. We illustrate that noncontinuum effects are significant for filters designed to capture
biowarfare agents and pathogens of current interest. We also suggest a scaling based on Knudsen number
for correcting the classic scaling laws to include these effects.
I
INTRODUCTION
Biological agents and other airborne contaminants may be isolated by a filtering process in which air is pulled
through grates with micron-sized holes. Filters with holes that are smaller than the target particles are used
so that these particles do not pass through the filter and can be gathered at the filter surface. Such filtering
processes require that large amounts of air be filtered since the concentrations of these agents can be minute.
Unfortunately, filtering such large volumes of air becomes costly since the power requirement for filtering is the
product of the volume flow rate and the pressure drop across the filter that induces this flowrate. Therefore,
filters must be designed to minimize this pressure drop to enhance efficiency.
Conventional filters appear in a variety of practical applications, from isolating particulates in a gas sample
to controlling turbulence levels in a wind tunnel. Researchers have long tried to predict filter performance
based on some global geometric properties and flowfield conditions [1,2]. These studies led to "scaling laws," or
algebraic expressions which indicate how the pressure drop across a screen or filter scales with these parameters.
Most recently, experimental and computational studies have been conducted in the continuum regime for
filters with circular holes with diameters from 5 to 12 microns [3,4], which is well outside the parameter range
of previous work. These studies yielded empirical relationships between pressure drop and flow rate in terms
of two geometric parameters and one flow parameter: the ratio of hole area to filter area (which is called the
opening factor, /?), the ratio of filter thickness to hole diameter, t/d, and the Reynolds number, Re, given by
The density, velocity, and viscosity of the inflow are given by /?2-n, [/,-„ and /^-n, respectively. This definition of
Re includes the blockage effect of the filter by scaling [/,-„ by /?. Using the effective flow through the filter holes
in the definition for Re improves the correlation of results for various hole geometries and flow conditions [2].
In these recent studies, initial computational results did not agree well with experiment [3]. However, closer
inspection of the filters revealed discrepancies between the intended hole geometry (which was used in the
computations) and that which was actually fabricated. Calculations performed using a more accurate hole
geometry agreed well with experiment, and led to the expression [4]
(10/Ee + 0.22)
(2)
for the nondimensional pressure drop, K, across the filter. Equation (2) indicates that K is inversely proportional to the square of the opening factor, linear in t/d, and linear in I/Re.
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
2001 American Institute of Physics 0-7354-0025-3
480
Equation (2) was developed using solutions to the Navier-Stokes equations and then compared to experimental results. The calculations included noncontinuum effects in the form of a slip boundary condition at the
filter surface, but these effects were minor for the cases tested. Rather than include an additional parameter
such as the Knudsen number, Kn, in the scaling law, these effects have been incorporated into the values of
the constants in Eq. (2). Filters for biodetection, however, will require holes that are even smaller than those
tested in these studies. As hole diameters are reduced to 1 micron or less, Knudsen-number effects will be
substantial, and a continuum Navier-Stokes solver will not provide accurate predictions. Furthermore, these
effects must be included explicitly in the scaling law.
This paper describes current two-dimensional DSMC calculations used to quantify the relationship between
power consumption and flow rate for filters in the high Knudsen number regime. These simulations led to a
refinement in the scaling laws based on Knudsen number which expands the range in which these expressions
accurately predict filter performance.
II
NUMERICAL APPROACH
Two-dimensional DSMC calculations were first performed for a "baseline" case described in Fig. 1, and
then parameter studies were performed by varying the flow conditions are geometry relative to this case. The
Symmetry Boundary
1.5 Jim
-7
t = 3 jim
/ Diffuse Wall at
Local Temperature
Symmetry Boundary
FIGURE 1. Baseline geometry.
baseline case consisted of a filter three microns thick with one micron holes spaced four microns apart. These
dimensions give t/d = 3 and /3 = 1/4. For the current studies, we fix /3 and t/d so that variation in the results
is limited to Re effects and Kn effects. The domain shown in Fig. 1 exploits planes of symmetry along the
centerline of a filter hole and along the boundary between adjacent holes to reduce the computational expense
of each calculation. The baseline inflow conditions are air at roughly sea-level temperature and density, flowing
at 4 m/s towards the filter from the left. The Knudsen number for this baseline case, defined as
Kn =
~~ characteristic length
d
in terms of the mean free path of the inflow, A« n , is 0.0576.
The code used for these calculations is a modified version of Bird's DSMC2.FOR [5]. The modified code
allows the user to include an arbitrary number of surfaces within the domain and impose an adiabatic, diffusereflection condition at the surface. The filter is then modeled as a step-shaped blockage as shown in Fig. 1. In
order to control Re from case to case, the mass flux through the domain was specified by the inflow conditions.
The pressure drop across the filter is then measured at the end of each simulation. Particles entering from the
downstream side of the domain were sampled from a prescribed distribution. This condition overspecifies the
subsonic outflow, and close to the exit plane the solution adjusts in an attempt to resolve the inconsistency. By
placing the downstream boundary sufficiently far away from the filter, we prevent this effect from corrupting
the calculation near the filter.
Ill
RESULTS AND DISCUSSION
Contour plots of the x- velocity and the density for the baseline case are shown in Figs. 2 and 3. The velocity
plot shows how the flow accelerates through the hole in order to maintain mass flux, reaching a maximum speed
along the hole centerline around 24 m/s. The flow also exhibits a distinct boundary-layer structure as it passes
481
y (µm)
y (µm)
U: 000
U:
U:
22
22
44
66
10 12
12 14
18 20
20 22
22 24
24
10
12
14 16
16 18
888 10
00
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-10
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-5-5
00
(µm)
xx(µm)
55
10
10
FIGURE2.2.2.Contours
Contoursofofofx-velocity
x-velocityfor
forthe
thebaseline
baselinecase
case
FIGURE
FIGURE
Contours
x-velocity
for
the
baseline
case
y (µm)
y (µm)
DENSITY: 1.06
1.061.07
1.071.08
1.081.09
1.091.10
1.101.11
1.111.12
1.121.13
1.131.14
1.141.15
1.151.16
1.161.17
1.17
1.18
DENSITY:
DENSITY:
1.06
1.07
1.08
1.09
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.171.18
1.18
2
2
00
-2-2
-10
-10
-5
-5
-5
000
x(µm)
(µm)
xx(^im)
555
10
10
10
FIGURE3.3.3.Density
Densitycontours
contoursfor
forthe
thebaseline
baselinecase.
case.
FIGURE
FIGURE
Density
contours
for
the
baseline
case.
throughthe
thechannel.
channel. The
Thedensity
densityplot
plotindicates
indicatesaaaslight
slightincrease
increaseinin
inthe
thedensity
densityasas
asthe
the
lter
approached.
through
through
the
channel.
The
density
plot
indicates
slight
increase
the
density
thelter
filterisis
isapproached.
approached.
This\ram
\rameect"
eect"isisisproduced
producedby
byour
ourforcing
forcingthe
theow
owinininatatataaaspecied
speciedmass
massux.
ux.This
Thisincrease
increaseofof
ofaround
around
2%
This
This
"ram
effect"
produced
by
our
forcing
the
flow
specified
mass
flux.
This
increase
around2%
2%
in
the
density
is
followed
by
an
expansion
through
the
lter
that
gives
a
nal
density
value
approximately
10%
inin the
the density
density isis followed
followed by
by an
an expansion
expansionthrough
throughthe
the lter
filter that
thatgives
givesaanal
finaldensity
densityvalue
valueapproximately
approximately10%
10%
belowthat
thatofofofthe
theincoming
incomingow.
ow.
below
below
that
the
incoming
flow.
Plots
of
the
pressure,
x-velocity,
andthe
thedensity
densityare
aregiven
giveninininFig.
Fig.4.4.4.The
Thecurves
curvesfor
forthe
thedensity
densityand
andx-velocity
x-velocity
Plots
of
the
pressure,
x-velocity,
Plots of the pressure, x-velocity,and
and
the
density
are
given
Fig.
The
curves
for
the
density
and
x-velocity
mirror
the
contour
plots
above.
The
noise
seen
in
these
curves
is
due
to
the
disparity
between
the
thermal
mirror
the
contour
plots
above.
The
noise
seen
in
these
curves
is
due
to
the
disparity
between
the
mirror the contour plots above. The noise seen in these curves is due to the disparity between thethermal
thermal
velocity
of
the
particles
in
the
simulation,
which
is
on
the
order
of
500
m/s,
and
the
mean
ow
velocity.
Even
velocity
of
the
particles
in
the
simulation,
which
is
on
the
order
of
500
m/s,
and
the
mean
ow
velocity.
velocity of the particles in the simulation, which is on the order of 500 m/s, and the mean flow velocity.Even
Even
with
this
noise,
however,
the
plateaus
in
pressure
in
front
of
and
behind
the
lter
are
easy
to
identify,
and
the
with
this
noise,
however,
the
plateaus
in
pressure
in
front
of
and
behind
the
lter
are
easy
to
identify,
and
with this noise, however, the plateaus in pressure in front of and behind the filter are easy to identify, andthe
the
dierence
in
these
levels
is
the
value
of
p
that
our
scaling
laws
attempt
to
predict.
dierence
p that
difference in
in these
these levels
levels isis the
the value
value of
ofAp
that our
ourscaling
scalinglaws
lawsattempt
attempttotopredict.
predict.
Theresults
resultsofofofthe
thebaseline
baselinecase
case and
andseveral
several other
othercases
cases obtained
obtainedbyby
byvarying
varying the
theow
ow conditions
and
the
scale
The
The results
the baseline
caseand
severalother
casesobtained
varyingthe
flowconditions
conditionsand
andthe
thescale
scale
the
geometry
are
included
in
Fig.
5.
Figure
5
also
includes
Yang
et
al.'s
scaling
law
from
Eq.
(2),
as
well
as
ofofofthe
geometry
are
included
in
Fig.
5.
Figure
5
also
includes
Yang
et
al.'s
scaling
law
from
Eq.
(2),
as
well
the geometry are included in Fig. 5. Figure 5 also includes Yang et al.'s scaling law from Eq. (2), as wellasas
some
of
their
experimental
data
[4].
The
experimental
data
is
for
cases
in
the
range
0
:0057Kn
Kn0:00127.
:0127.InIn
some
of
their
experimental
data
[4].
The
experimental
data
is
for
cases
in
the
range
0
:
0057
some of their experimental data [4]. The experimental data is for cases in the range 0.0057 < Kn < 0.0127. In
Fig5,5,squares
squaresindicate
indicatedata
datafor
forconstant
constant Kn
Kn==0:00576
:0576 as denedininEq.
Eq. (3).
(3).These
These points
pointslielie
below
the
curve
Fig
Fig 5,
squares indicate
data for
constant Kn
= 0.0576asasdened
defined in Eq.(3).
Thesepoints
liebelow
belowthe
thecurve
curve
for
Eq.
(2)
by
around
a
factor
of
2,
but
the
experimental
data
lie
below
the
curve
as
well
at
low
Re. .Considering
Considering
for
Eq.
(2)
by
around
a
factor
of
2,
but
the
experimental
data
lie
below
the
curve
as
well
at
low
Re
for Eq. (2) by around a factor of 2, but the experimental data lie below the curve as well at low Re. Considering
thatthe
thecurrent
currentsimulations
simulationsare
aretwo-dimensional
two-dimensionalcompared
compared to the
thethree-dimensional
three-dimensional data
dataofof
ofYang
Yang
et al.,
and
that
that the
current simulations
are two-dimensional
comparedtotothe
three-dimensionaldata
Yangetetal.,
al.,and
and
that
we
are
operating
well
outside
the
parameter
range
in
which
the
empirical
t
was
derived,
agreement
is
that
that we
we are
are operating
operating well
well outside
outside the
the parameter
parameter range
rangeininwhich
whichthe
the empirical
empiricaltfitwas
wasderived,
derived,agreement
agreementisis
good.
A
linear
dependence
of
Konon1=Re
1
=Reisisseen
seen
in
the
Kn==0:00576
:0576data,
data,
which
reinforces
the
form
used
in
good.
A
linear
dependence
of
K
in
the
Kn
which
reinforces
the
form
used
in
good. A linear dependence of K on I/Re is seen in the Kn = 0.0576 data, which reinforces the form used in
Eq.(2)
(2)for
forthe
theRe
Re term.
Eq.
Eq. (2)
for the
Re term.
term.
The
circular
data
pointsinininFig.
Fig. 55correspond
correspondtototoKn
Kn =0.0309,
0.0309,0.117,
0.117, and
and0.233
0.233asas
asone
one travels
travelsaway
away
form
The
circular
data
points
form
The
circular
data
points
Fig. 5indicate
correspond
Kn ==
0.117,and
0.233
onetravels
away
form
the
empirical
curve.
These
points
that
varying
Kn0.0309,
while
holding
the
parameters
in
Eq.
(2)
constant
the
empirical
curve.
These
points
indicate
that
varying
Kn
while
holding
the
parameters
in
Eq.
(2)
constant
the
empirical
curve.
These
points
indicate
that
varying
Kn
while
holding
the
parameters
in
Eq.
(2)
constant
resultsinininsignicant
signicantdierences
dierencesinininKKKthat
thatthe
thescaling
scalinglaw
lawcannot
cannotpredict.
predict. Consider
Considerthe
thecircular
circular
data
point
results
point
results
significant
differences
that
the
scaling
law
cannot
predict.
Consider
the
circulardata
data
point
farthest
from
the
empirical
curve,
for
which
Kn
=
0
:
233
and
Re
=
0
:
68.
If
K
showed
no
dependence
on
Kn, ,
farthest
from
the
empirical
curve,
for
which
Kn
== 00.233
:233 and
Re
==00.68.
:68. IfIfKK showed
nonodependence
ononKn
farthest
from
the
empirical
curve,
for
which
Kn
and
Re
showed
dependence
Kn,
then
this
Kn
=
0
:
233
data
point
would
fall
in
line
with
the
Kn
=
0
:
0576
data.
This
is
clearly
not
the
case;
the
then
this
Kn
== 00.233
:233 data
point
would
fall
ininline
with
the
Kn
==00.0576
:0576 data.
This
isisclearly
not
the
case;
the
then
this
Kn
data
point
would
fall
line
with
the
Kn
data.
This
clearly
not
the
case;
the
valueofofKKinterpolated
interpolatedatatReRe==0:068:68from
fromthe
theKn
Kn==0:00576
:0576data
dataisisapproximately
approximatelythree
threetimes
timesthat
thatgiven
givenbybythe
the
value
value
of
K
interpolated
at
Re
=
0.68
from
the
Kn
=
0.0576
data
is
approximately
three
times
that
given
by
the
simulationforforKn
Kn==0:0233
:233atatthe
thesame
sameReRe. .The
Thesimulation
simulationresults
resultsindicate
indicatethat
thatasasKn
Knincreases,
increases,KKdecreases.
decreases.
simulation
simulation
for
Kn
=
0.233
at
the
same
Re.
The
simulation
results
indicate
that
as
Kn
increases,
K
decreases.
Therefore,ififwewewere
weretotorun
runmore
moresimulations
simulationsforforKn
Kn!!0,0,the
theresulting
resultingvalues
valuesofofKKwill
willlielieabove
abovethe
thecurrent
current
Therefore,
Therefore,
if we
were toapproaching
run more simulations
for curve
Kn —>•or0,possibly
the resulting
values
of K will lie
above the current
datapoints,
points,
perhaps
theempirical
empirical
providing
aconsistent
consistent
continuation
the
data
perhaps
approaching
the
curve
or
possibly
providing
a
continuation
ofofthe
data
points, perhaps
approaching
theregion.
empirical curve or possibly providing a consistent continuation of the
experimental
data
into
the
lower
Re
experimental
data
into the
lower
Re region.
experimental
data into
the in
lower
Re
Canweweinclude
include
afactor
factor
Eq.(2)
(2)toregion.
toaccount
accountforforthis
thisreduction
reductionininKKasasKn
Knincreases?
increases?IfIfwewebase
basethis
thisfactor
factor
Can
a
in
Eq.
Can
we
include
a
factor
in
Eq.
(2)
to account for this reduction in K as Kn increases? If we base this factor
Knand
andinclude
includeititasas
f fononKn
/ on Kn and include it as
482
ρp (kg/m
(kg/m3))
2
3
P (n/m )
U
U (m/s)
(m/s)
25
25
113000
112000
- 1.2
1.2 _
20
20
111000
110000
109000
15
15
108000
- 1.15
1.15
107000
ρ
U
P
106000
105000
10
10
104000
1.1
5
103000
102000
101000
-20
-20
-15
-15
-10
-10
-5
-5
00
X,
µm
X,|am
5
10
15
20
0
FIGURE
FIGURE4.
4. Centerline
Centerline pressure,
pressure, velocity,
velocity, and
and density
density for
for the
the baseline
baseline case.
case.
K = f ;
2
0-2
t
33.5
:5 + 3
d
(10
=Re +
:22) ;
(lO/fle
+ 00.22),
(4)
(4)
@f
df < 0:
@Kn
dKn
(5)
(5)
wewemust
mustrequire
requiref/ to
tosatisfy
satisfy
ff(Kn
(Kn == 0)0)== 11,;
and
and
InInother
otherwords,
words,the
thescaling
scalinglaw
lawshould
should return
return the
the continuum
continuum limit
limit as
as Kn
Kn !
—»•0,0,and
andf/ decreases
decreases as
asKn
Kn increases
increases
from
fromKn
Kn ==0.0.AAsimple
simpleexpression
expression that
that satises
satisfies both
both of
ofthese
these requirements
requirements and
and has
has only
only one
one free
free parameter
parameter is
is
a
f/ ==
Kn
aa++Kn
(6)
(6)
Thisform
form has
has obvious
obviousproblems,
problems, such
such as
as predicting
predicting aa zero
zero pressure
pressure drop
drop for
for free-molecular
free-molecular ((Kn
—»•1
oo)
flow.
This
Kn !
) ow.
However,we
wedo
donot
not intend
intend to
to push
push the
the approximation
approximation that
that far.
far. Our
Our goal
goal is
is to
to provide
provide aa rough
However,
rough rule-of-thumb
rule-of-thumb
forlters
filtersthat
that can
canisolate
isolatecurrent
current biological
biological agents,
agents, which
which pins
pins us
us to
to aa range
range around
around the
the 11 micron
for
micron scale
scale under
under
atmosphericconditions.
conditions. Additional
Additionaltheoretical
theoreticalanalysis,
analysis, physical
physical reasoning,
reasoning, and
and comparisons
comparisons with
with experimental
experimental
atmospheric
andcomputational
computationalresults
results are
arerequired
required to
to develop
develop aa form
form of
of f/ with
with aa wider
wider range
range of
of applicability,
applicability, and
and will
will be
be
and
the focus
focus ofoffuture
future research.
research.
the
By choosing
choosing one
one data
data point
point at
at each
each of
of the
the four
four Knudsen
Knudsen numbers
numbers tested,
tested, the
the constant
constant in
By
in Eq.
Eq. (6)
(6) was
was
determined. Using
UsingEq.
Eq.(2)
(2)and
and the
the DSMC
DSMC result
result for
for each
each case,
case, aa target
target value
determined.
value for
for /f isis obtained
obtained for
for each
each Kn.
Kn.
Then,the
the constant
constant ccin
in Eq.
Eq. (6)
(6) isisdetermined
determined by
by minimizing
minimizing the
the error
error in
in matching
matching these
these four
four ((Kn,f)
data
Then,
Kn; f ) data
points. This
This procedure
procedure gives
gives cc == 00.05765.
close-up of
of the
the DSMC
DSMC results
results are
are shown
shown in
in Figure
Figure 6,
6, along
points.
:05765. AA close-up
along
withEq.
Eq.(2)
(2)and
and the
the scaled
scaled version
version Eq.
Eq. (4)
(4) evaluated
evaluated at
at the
the four
four values
values of
of Kn
Kn that
that correspond
correspond to
to the
the DSMC
with
DSMC
data.The
Thescaling
scalingprovides
providesreasonable
reasonableagreement
agreement considering
considering that
that we
weare
are trying
trying to
to match
match data
data at
at four
four dierent
different
data.
Knudsennumbers
numberswith
withone
onefree
free parameter
parameter and
and aacrude
crude form
form for
for f/.. The
The scaled
scaled curves
curves at
at the
the highest
highest and
Knudsen
and lowest
lowest
483
K β / [3.5(t/d) + 3]
102
co
10 1
2
u
10°
10
0
•
•
•
10
-1
10 -1
j_i
DSMC,
DSMC, Kn
Kn== 0.0576
0.0576
DSMC,
DSMC, 0.0309
0.0309 << Kn
Kn<<0.233
0.233
Equation
Equation of
of Yang
Yang et
et al.
al.
Experimental
Experimental Results
Results of
ofYang
Yang et
etal.
al.
i i i i 111___i_i
0
10
i i i i 111___i_i
1
10
i i i i 111
10 2
Re
Re
FIGURE 5.
Nondimensional pressure
pressure drop
FIGURE 5. Nondimensional
drop for
for various
various cases,
cases, including
including DSMC
DSMC calculations,
calculations, experimental
experimental results,
results, and
and
Yang
et
al.'s
empirical
t
given
in
Eq.
(2).
Yang et al.'s empirical fit given in Eq. (2).
Kn values
values miss
miss their
their respective
respective DSMC
Kn
DSMC data
data points
points by
by around
around 15%.
15%. The
The middle
middle values
values do
do better,
better, especially
especially
the
Kn
=
0
:
0576
cases.
Only
one
Kn
=
0
:
0576
data
point
was
used
in
determining
c
,
so
the
the Kn = 0.0576 cases. Only one Kn = 0.0576 data point was used in determining c, so the good
good agreement
agreement
is not
not due
due to
to this
this data
data being
being weighted
weighted improperly
is
improperly when
when cc was
was determined.
determined. This
This error
error of
of 15%
15%isissubstantial,
substantial,
but
it
is
a
considerable
improvement
over
the
factor
of
2
to
5
dierence
seen
between
the
unscaled
but it is a considerable improvement over the factor of 2 to 5 difference seen between the unsealed empirical
empirical
formula and
and the
the DSMC
DSMC results.
results.
formula
IV
IV CONCLUSIONS
CONCLUSIONS
As microfilters
microlters are
fabricated with
As
are fabricated
with smaller
smaller and
and smaller
smaller holes,
holes, noncontinuum
noncontinuum eects
effects will
will become
become more
more imporimportant. The
The scales
scales required
required for
for filtering
ltering bio-agents
tant.
bio-agents of
of current
current interest
interest experience
experience strong
strong Knudsen
Knudsen number
number eects,
effects,
which requires
requires an
an expansion
expansion of
which
of the
the classic
classic scaling
scaling laws
laws used
used to
to predict
predict lter
filter performance.
performance. By
By including
including aa simple
simple
factor
based
on
the
Knudsen
number,
a
reasonable
adjustment
can
be
made
in
the
pressure
factor based on the Knudsen number, a reasonable adjustment can be made in the pressure drop
drop calculation
calculation
that isis within
within about
that
about 15%
15% of
of the
the solutions
solutions calculated
calculated in
in the
the current
current study
study using
using the
the DSMC
DSMC method.
method. Additional
Additional
theoretical
analysis,
physical
reasoning,
and
comparisons
with
experimental
and
computational
theoretical analysis, physical reasoning, and comparisons with experimental and computational results
results will
will be
be
pursued
in
order
to
develop
scaling
laws
that
accurately
predict
lter
performance
over
a
wider
range
of
pursued in order to develop scaling laws that accurately predict filter performance over a wider range of lter
filter
parameters and
and flow
ow conditions.
parameters
conditions.
ACKNOWLEDGMENTS
ACKNOWLEDGMENTS
This work
work isis funded
funded by
This
by DARPA
DARPA Design
Design for
for Mixed
Mixed Technology
Technology Integration
Integration Program.
Program. D.
D. Mott
Mott isis aa National
National
Research
Council-Naval
Research
Laboratory
Postdoctoral
Research
Associate.
The
authors
Research Council-Naval Research Laboratory Postdoctoral Research Associate. The authors would
would like
like to
to
thank
Professors
Chih-Ming
Ho
and
Yu-Chong
Tai
for
suggesting
this
problem
and,
along
with
Dr.
Joon
thank Professors Chih-Ming Ho and Yu-Chong Tai for suggesting this problem and, along with Dr. Joon Mo
Mo
Yang, discussing
discussing their
their results
results with
Yang,
with us.
us.
484
K β / [3.5(t/d) + 3]
2
10
102
DSMC,
DSMC, Kn
Kn==0.0309
0.0309
DSMC,
Kn
DSMC, Kn==0.0576
0.0576
DSMC,
DSMC, Kn
Kn==0.1166
0.1166
DSMC,
Kn
DSMC, Kn==0.2330
0.2330
Equation,
Equation,Yang
Yanget
et al.,
al., no
noscaling
scaling
Equation,
scaled
by
f(Kn
= 0.0309)
- - - - Equation, scaled by f(Kn=
0.0309)
Equation,
Equation,scaled
scaledby
byf(Kn
f(Kn==0.0576)
0.0576)
— — — - Equation,
Equation,scaled
scaledby
byf(Kn
f(Kn==0.1166)
0.1166)
Equation,
scaled
by
f(Kn
_.._.._.._ Equation, scaled by f(Kn==0.2330)
0.2330)
CO
u
2
si
1
10
101
0
10
1 -1
1010
10'1
10 0
10 1
Re
Re
FIGURE
FIGURE 6.
6. Comparison
Comparison of
of DSMC
DSMC results
results with
with the
the scaled
scaled empirical
empirical t
fit given
given in
in Eq.
Eq. (4).
(4).
REFERENCES
REFERENCES
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K.E.G. \On the Resistance of Screens." The Aeronautical Quarterly, Vol. IV, February 1953.
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2.2. Derbunovich,
G.I., Zemskaya, A.S., Repik, Ye.U., and Sosedko, Yu.P. \Hydraulic Drag of Perforated Plates." Fluid
Derbunovich, G.I., Zemskaya, A.S., Repik, Ye.U., and Sosedko, Yu.P. "Hydraulic Drag of Perforated Plates." Fluid
Mechanics
{
Soviet
Research, Vol. 13, No. 1, January{February 1984.
Mechanics - Soviet Research, Vol. 13, No. 1, January-February 1984.
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X., Yang, J.M., Tai, Y.-C., and Ho, C.-M., Sensors and Actuators A: Physical 73, No. 1-2, pp. 184-191, 1999.
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C.-M., and
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485