Rarefied gas flows induced by temperature fields and their applications (Mathematical Analysis in Fluid and Gas Dynamics)

Rarefied

gas

flows induced

by temperature

fields and their

applications

京大・工・航空宇宙 杉元 宏 (HiroshiSugimoto)

Departmentof Aeronautics andAstronautics,

Graduate School ofengineering, Kyoto University

Thispaper consistsoftwo parts. Inthefirst$P^{aI}t$wepresent theresult of the numerical simulation

onthe steadygasflowinapump driven by thermal edgeflow, whichi8proposed in_RarefiedGas Dynmics,AIP,NewYork, _138-141 (2002). Anewfindingof thisanalysisisthepossibility ofthe

alternativedesign of the flow channelofthepump. According to thenewdesign ofthechannel, a rarefied gasflow is inducedthroughapair of parallel wire meshes with differenttemperatures. In

the secondpartofthispaper,weconflrmthuis phenomena by simpleexperiments.

1

Introduction

In

a

rarefied

gas,

where the

mean

ftee path of the

gas

molecule is not negligible

com-paredwith the scale ofthe system, the temperature field ofthe

gas

is deeply relatedto the

gas

motion. The flow of therarefied

gas

is induced by the temperature flelds ofthe

gas

even

ifthereis

no

extemal force. The thermal transpiration flow1,2) _whichis

a

_flow

inducedin

a

pipewith

a

temperature gradient alongit, isawell known example, and

now

it is known that various flows with different properties

are

induced by the temperature

fields.3)_{Thethermaltranspiration flowin}

_a

_pipesuggests

_a

_{possibility of}

_a

_Pumpwithout moving parts. That is, if

we

connect two tanks of different temperatures by

a

thin pipe of

a

uniforn

cross

section, the thernaltranspiration flowis inducedin thepipe and thus

we

can

maintain

a pressure

difference between these tanks. However the difference of

the pressure between the tanks is proportional to the temperature difference. Thus this

simple idea isnot practical since the variations of temperature of material is rather

lim-ited than that of

pressure

of the

gas.

In 1910, Knudsen camied out

an

experiment by

a

single pipewith periodic variations of diameter and$temperatu\infty$ (Fig. 1), and succeeded

to obtain the

pressuoe

ratio 10, _{which is fairly larger} than the temperature ratio in the system.4) _{Recently this type of}

_pumps

_{driven by temperature field attracts researchers}

again, since it does not require moving parts and has

a

possibility of the application

as

microdevices.5-14)

Theauthor andSone havebeen camied out twoexperimentsofthe

pumps

driven by the flows induced by the temperature field. Thefirst

one

is the

pump

driven by thethermal

transpiration flow.10) _{(This type of}

_pump

_{is called “Knudsen compressor’‘ now.) The} experiment showedthatit

can

reduce the

pressure

of

a

tank of8 literto

a

half in300 $sec$,

butits

energy

efficiencyislower thanstandard commercial

pump

byfar. This

pump

is

very

primitiveand

many

improvements

are

required. Oneof the

reason

of the low efficiency of

the Knudsen

compressor

isthe temperaturegradientofthepipe wall whichis essentialto

the thermal transpiration flow. Thatis,

a

heatfluxthrough

a

pipewall,whichis theloss of

the

energy,

isrequired tomaintainthe temperature gradient. Inordertoreducethis loss of

channelofthe

pump

isrequired. As for the Knudsen

compressor,

there

are some

amounts

of works. The corresponding analysis for the thermal edge

compressor

is

recently

published inJapanese13). _{Inthe presentPaper, themain resultsofthe Paperandthe}

new

results

on

the altemative design of the thermal edge

compressor

are

presented in Sec. 3.

The result

on

thealtemativedesignis uniquesinceitshowsthat

a new

typeofrarefied

gas

flow is induced through

a

pairof parallel wire meshes with differenttemperature. Inthe

next partof this

paper,

Sec. 4, theresultsof twopreliminaryexperiments

on

theexistence

ofthe flow through

a

pair ofwiremeshes

are

reported.

2

Thermal Edge Compressor

Thethermal edge

compressor

consistsof

a

number ofdrivingunitsconnectedinseries,

as

well

as

the Knudsen

compressor.

Figure 3shows

an

$2D$model of the

unit

of the thermal

edge

compressor

devised in Ref. 12. The channel is equipped with

a

pair oftwo

arrays

of plates,

one

isheated (temperature $T_{h}$) and the otherisunheated(temperature$T_{c}$). The

size ofthe unit in$X_{1}$ direction$D_{w}$ is largerthan the totalwidthofthe setof

arrays

ofthe

plates, thus

some

space

is left around the set of

arrays

ofthe plates. The mechanism of

thisunitis

as

follows. In theoverlapping regionofthetwo arraysof plates,

a

temperature

gradient in $X_{1}$ direction is induced. This temperature gradient induces the thermal edge

flow in $X_{1}$ direction. In the

gas

region around other edges of each plate, the temperature

ofthegas isroughly uniform since there arelots of plates of identicaltemperature in$X_{2}$

direction in

a space

left aroundthe

array.

Therefore the thermal edgeflow induced there

is weaker than those induced in the overlapping region. In the

narrow

regions between

theplates of each

array,

the flow isnotinducedsincethe temperatureof the

gas

isroughly

Flg.1: Theexperimental apparatus by Knudsen(Ref. 4). The diameter of the thin partis0.$4mm$

and that of thicker part is $10mm$, and the temperature difference between the heated part and unheatedpartisabout$500K$

.

Fig. 2: Thernal edge flow. The points A and $B$ are about

one mean

free path away from the

edgeof the plate. Near the edge, the isothermal linesaresharplycurved,_{and thusthe temperature}

is non-uniform along the plate. The molecules impinging

on

the edge region from theleft side

areroughlyrepresented bythosefrom A and the molecules from the right side bythosefrom$B$,

wherethegas ishotter than at A when the plateisheated. The situationis similartothat

over

a non-unifornly heatedplate, where the thernalcreepflow1)_{is induced. Thus,aflowis inducedin}

the directionofthearrowsaround theedgeofaheated plate.

Fig.3: Aunit ofthermaledgecompressor.

uniformthere. In

summary,

theflowismainly inducedinthe overlapping regionbetween

theheated

array

and unheated

array.

Thusthis unitinduces

a gas

flowin$X_{1}$ direction.

3

Numerical

simulation

Wecarryout two

cases

ofnumerical simulation

on

theflows in the thermal edge

com-pressor,

following thenumerical analysis

on

the

gas

flowsin theKnudsen

compressor

in

Refs.

6

and

7.

Problem-I: Investigatethe maximumflow rateobtained by thethermal edge

compressor.

The maximum flow rate here is the flow rate obtained when the

pressures

of the

gas

at

thebothendsof the system

are

closed by the walls.

The shape of the flow channel of thermal edge

compressor

israther complicated, thus

there

are

many

parameters

on

it. The

purpose

of this

paper

isnotthe optimization of these parameters, buttoshow the basic properties ofthethermal edge

compressor.

Furthermore the resultsgivenin Sec.3.$3B$will showthat the design of theflow channelis not restricted

tothe

one

shownin Fig.3. Therefore

we

firstrestrictourselvestothechannelshape shown

in Fig. 3, and omit the explanation ofdetailed shape

on

it. We

use

only the following

parameters ofthe

compressor

unit in this

paper.

$T_{h},$ $T_{c}$

:

the temperatuoe of two

arrays

of

plates (the temperature of side wall is also $T_{c}$)$;D_{h}$

:

thedistance berweenthe plates in

a

array;

$n$

:

the number of plates with temperature$T_{h}$ (orthe number of small flowchannels

of height$D_{h}$),$D_{w}$; thelengthof

a

unitin$X_{1}$ direction.

3.1

Basic Equation

The analysis is carried out

on

the basis of the Boltzmann equation for hard sphere molecules:17)

$\xi_{1}\frac{\partial f}{\partial X_{1}}+\xi_{2}\frac{\partial f}{\partial X_{2}}=J(f.f)$, (1)

$J(f,f)= \frac{d_{m}^{2}}{2m}\int_{\kappa.|<\infty.\psi|--1}[f(f_{*})g(\xi’)-f(\xi.)g(\xi)]|V\cdot a|d\Omega(a)oe_{*\prime}$ (2) $\xi’=\xi+(V\cdot a)a$, $\xi_{*}’=\xi$

.

$-(V\cdot a)a$

.

$V=\xi$

.

$-\xi$

.

(3)

The notation is

as

follows. $d_{m}$

:

the diameter of

a

molecule; $m$; the

mass

of

a

molecule;

$\xi_{i}$

:

the molecular velocity; $f$

:

velocity distribution function of

gas

molecules; $dg$

.

$=$ $d\xi_{1}d\xi_{2}d\xi_{3};d\Omega(a)$

:

solid angle elementinthe direction ofunit vector$a$

.

Theboundarycondition

on

solid walls isthediffusereflection:

$f( \xi\cdot n>0)=\frac{\sigma_{w}}{(2\pi RT_{w})^{3/2}}\exp(-\frac{|\xi|^{2}}{2RT_{w}})$, (4)

where$T_{w}$and $n$are,respectively, the temperatureandtheunit normalvectortothe

bound-$a\iota\gamma$, pointed to the

gas.

$R$ is the specific

gas

constant ($R=k_{B}/m,$ $k_{\beta}$

:

the Boltzmann

constant). Inthe numerical simulationthe specular reflectioncondition

$f(\xi\cdot n>0)=f(\xi-2(\xi\cdot n)n)$, (6)

is alsousedto represent

a

symmetric surface.

The macroscopic quantities

are

defined by the moments of the velocity distribution

function$f$

as

follows:

$\rho=\int fdg$, $\nu=\frac{1}{\rho}\int\xi foe$, $T= \frac{1}{3w}\int g_{-\nu|^{2}foe}$

.

$p=R\rho T$, (7)

where$\rho,$ $\nu,$ $T$, and$p$

are

the density, flowvelocity, temperatureand

pressure

ofthe

gas,

respecuvely. The

integrations

are

camiedout

over

thewhole

space

of$\xi$

.

An important length of scalein the Boltzmann equation is the

mean

free path ofthe

gas

molecule. Forhard spheremolecules, its sizeat equilibriumstate at rest$de\mu nds$

on

$d_{m},$ $m$, and density of the

gas.

Thus

we

deflne the reference

mean

free path $\ell_{a\nu}$ by using

the

average

density

over

a

period (Problem-l,

or average

density

over

the

pump

system (Pmblem-Il).Thatis,

$\ell_{av}=m/(\sqrt{2}\pi d_{m}^{2}\rho_{av})$

.

(8)

TheKnudsennumberKnisdefined by

$Kn=t_{a\nu}/D_{h}$

.

(9)

For theconvenienceof thedescription

we

introduce reference

pressure

$Po$by

$p_{0}=R\rho_{a\nu}T_{c}$

.

(10)

3.2

Method of solution

The system ofequation(1)$-(6)$isanalyzed numerically by theDirectSimulationMonte

Carlo (DSMC) method18) _{where the gas molecules}

_are

_{replaced by}

_a

_{numbers of}

simu-lation particles, and the behaviorof the particles

over a

small time step is simulated by

two separate

processes,

the

process

ofmotionof the particles withoutcollisions andthe

process

ofchanging their velocities bycollisionsbetween particles. Thus the method is

$ume$-dependent. We obtained the steady solution of Pmblem-I

or

Problem-IIby chasing

a

long time behavior of the results. The DSMC method is widely used in the studies of rarefied

gas

flows, and the numerical procedureused here is just the

same

as

the

one

describedin Ref.

6.

What is specialin this

paper

is only thatthe shape of the flow field

is rather complicated by

many

pieces of plates. The DSMC method is convenient for

this type ofthe problems since

we

can

divide the flow fieldinto

many

small rectangular domainsbyintroducingvirtualboundariesin thegasregion. We

carry

outthenumerical simulation in each rectangular domain by using additional boundary conditions

on

the

(a) (b)

Fig. 4: One way flow in thethermal edge compressorI (Problem I). $T_{h}/T_{c}=3.n=10$, and

Kn $=1$

.

$(a)$Temperature fieldand (b): flow velocity field. The temperature field in(a)isshown

bytheshadeof thedarkness, and itsscaleisshown intheright end ofthepanel. The

arrows

in(b)

indicate the flow velocity attheir starting points,whose scaleisshown

on

the rightshoulderof the

panel.

from the virtual boundary ofcorresponding adjacent domain. This methodis also effec-tive for parallel computers since

we

can

carry

out the simulationforeach domainalmost

independentlyexcept small amount of information

on

the molecules thatpass thevirtual

boundaries ineachtimestep ofnumerical simulation.

3.3

Results

A One

way

flowfor basic channel

Here

we

explainthe result forProblem-I, themaximumflow rateobtainedby the ther-mal edge

compressor.

Figure4showsthe temperature and flow velocity fields for the

case

of$T_{h}/T_{c}=3$, Kn $=1$, and$n=10$

.

Onlythepart$X_{2}>0$ is analyzed sincethe system is

symmetricwithrespect to$X_{2}=0$, andthecyclic boundary condition isappliedto$X_{1}=0$

and $5D_{h}$

.

The temperature gradiem is large in the gas at the overlappingregion oftwo

arrays

of plates $(D_{h}\leq X_{1}\leq 1.5D_{h})$

.

On theotherhand, the temperature gradient issmall

around the other ends ofplates($X_{1}\sim 0,2.5D_{h}$

or

$5D_{h}$). As theresult, the

one-way

flow

is induced in$X_{1}$ direction. The flow spoed$v_{1}$ decreases

near

the side wall at$X_{2}=5D_{h}$

.

This isbecause ofthe temperature betweenthe sidewall and the nearest plate (theplate

at$X_{2}=4.5D_{h}$) isroughly symmetric withrespectto$X_{1}$ direction, andonly

a

small size of

netflow$v_{1}$ is inducedthere.

Figure 5 shows theone-way flow for the

case

of$T_{h}/T_{c}=3$, Kn $=1$, and$n=W$

.

The

effect of side wall at$X_{2}=nD_{h}/2$ is confined in several small channels (or several $D_{h}$in

$X_{2}$ direction)

near

the side wall, and

a

periodic structure oftheflow fieldin $X_{2}$ direction

(a) (b)

Fig.5: One way flow in the thermal edge compressorII (Problem$I$). $T_{h}/T_{c}=3,n=40$, and

Kn$=1$

.

$(a)$Temperaturefield and(b): flow velocity field. (Seethecaptionof Fig.4).

fig. 6: A simpler model ofa unit of the thermal edge compressor. $X_{2}=0$ and $D_{h}/2$ are the

symmetric surface. The part-Crepresentstheunheated plate, and pan-H does the heated wall.

This resultleads

a

simplified model ofthethermal edge

compressor

depicted in Fig.6.

Consider

a

rectangular domain$0<X_{1}<D_{w}$and$0<X_{2}<D_{h}/2$

.

The boundary condition

at

a

part of the lower boundary $X_{2}=0$ (say, part-C in Fig. 6) is the diffuse reflection (4) attemperature $T_{c}$, and

a

part ofthe

upper

boundary$X_{2}=D_{h}/2$ (part-Hin Fig. 6) is

the diffuse reflection at temperature $T_{h}$

.

The boundary condition at other partof

upper

and lower boundary is the specular reflection (6). The condition for boundaries normal

to$X_{1}$ direction will bedefined depending

on

the

purpose

of the problems; In this section

(P’vblem-b),

we

apply the cyclic boundary condition. This modelis denoted by$narrow\infty$

fortheconvenience ofexpression.

Flg. 7: Nondimensional

mass

flow rate per

cross

sectional

area

$m_{f}/\rho_{av}(2RT_{c})^{1/2}$

vs

Knudsen

numberKnof the thermaledgePump (Problem-l). $T_{h}/T_{c}=3$

.

$x:n=10,2:n=20,$ $\theta:n=40$,

$\bullet;narrow\infty$

.

flux

per

unittimeand

per

unit

area

ofthe

cross

sectionofthe

compressor

unit

$m_{f}=\{\begin{array}{ll}\frac{2}{nD_{h}}\int_{0}^{D_{h}/2}\rho v_{1}dX_{2} (n<\infty),\frac{2}{D_{h}}\int_{0}^{D_{h}/2}\rho v_{1}dX_{2}1 (narrow\infty),\end{array}$ (11)

are

plotted forKnudsennumberKn $=0.01,0.5,0.2,0.5,1,5$, and 10for$n=10$and $\infty$,

and Kn $=1$ for $n=20$ and

40.

The nondimensional

mass

flux $m_{f}/\rho_{0}(2RT_{c})^{1\prime 2}$ takes its

maximum value around Kn – 0.5, and it decreases

as

$Knarrow 0$

or

$\infty$

.

The effect ofthe

side wall is clearly

seen

in the

cases

for $n=10.20,40$, and $\infty$ at Kn $=1$

.

The

mass

flux approaches to that for$narrow\infty$

as

$n$ increases and the deviation of$m_{f}$ fromthe

case

of$narrow\infty$ is roughly proportional to $1/n$

.

This supports the previous discussion that the

effectof sidewall isconfined in

a

smallregion

near

the sidewall.

B Alternativedesignofflow channel

The preceding result shows that the velocity ofthe

one-way

flow (which _is roughly representedby $m_{f}/\rho_{av}$) in the thermal edge

compressor

takes its maximumvalue when

the scaleof thesmall flow channel$(D_{h})$iscomparableto the

mean

freepath$\ell_{a\nu}$ of the

gas

molecules. The $p_{a\nu}$ athigh

pressure

is, however, quite small [cf. Eq. (8). $\ell_{av}\sim 0.07\mu m$ under the atnospheric

pressure

and the standardtemperature], and it

can

be difficultto

realize

a

complicated flow channels by the usualengineering process. Here

we

investigate the possibility of altemative design of flow channel of the thermal edge

compressor.

In the following numerical analysis,

we

arrange

objects ofvarious $sha\mu$ in the flow

channel, and the

gas

region is not represenoed only by $X_{1}=$ Const

or

$X_{2}=$ Const. In

this

paper

such shapes of

gas

region is also approximately represented by the union of

increase of computational time, since, in

some

cases,

we

have to

prepare

large number of rectangularregionsto represent the shape of the boundary. Therefore

we

carryout the

numerical simulation onlyfor thelimiting

case

formany smallflowchannels ($narrow\infty$ in Sec.A).

Some of the results

are

shown in Fig.

8.

Inthe figures, the temperature fieldis shown by the shade of the darkness and flow velocity is represented by

arrows.

In the

case

of

Fig.8(a), two

squares

with

one

side$\ell_{av}$ in length

are

put in thechannel. Thetemperature

of square at smaller$X_{1}$ is $T_{c}$, and that of larger $X_{1}$ is $T_{h}$

.

In this

case

the temperature

field around each

square

is slightlyasymmetricwith respect to$X_{1}$ direction,and

one-way

flowis inducedin$X_{1}$ direction. The

mass

flux_{$m_{f}$}

per

unit

area

ofthe

cross

sectionof the

compressor unitl

is$m_{f}/\rho_{av}(2RT_{c})^{1/2}=0.\omega 32$

.

The

mass

flux_{$m_{f}$} increases if

we

round

off the edges of thesesquares. In the

case

ofFig. 8(b), the radius ofthe curvatureis$l_{av}/4$ and$m_{f}/\rho_{0}(2RT_{c})^{1/2}=0.\alpha)43$

.

In Fig. 8(c), the objectis cylinder with radius$\ell_{a\nu}/2$, and

$m_{f}/\rho_{0}(2RT_{c})^{1/2}=0.\infty 53$

.

In Fig. 8(c), the objects put in the channel

are

the circular cylinders, and thus there is

no

“sharp edge” of

a

solid body which is usuallyrequiredto induce the thermal edge

flow. Of

course

thereis

no

thermaledgeflow around

a

circular cylinder put isolatedfrom

otherboundaries. Inthe

case

ofFig. 8(c),

we

are

considering thebehaviorofthegas in

a

compressor

unit whichconsistsoftwoparallelmeshes withdifferent$tem_{K^{ratu\infty S}}$, since

the boundaries$X_{2}=0$ and$D_{h}/2$

are

the symmetric surface. Thesetwo meshesinduce

a

temperature gradient ofthegas in$X_{1}$ direction between these meshes. This temperature

gradient induces

a

one

way

flow similar to those

seen

in the thermal edge

compressor,

because the size of the wires and distancebetween the meshes

are

ofthe order of the

mean

free pathofthe

gas

molecules$\ell_{av}$ ($=D_{h}/2$inFig. 8).

Asis discussed inSec.$A$,thewidth of the smallchannelofthethermaledge

compressor

would be of the order of the

mean

freepath of the

gas

molecules$\ell_{av}$

.

Fromthe resultin

this section, it is possible to infer that

one-way

flows

are

induced by

any

object with

a

radius ofcurvature of the order of$\ell_{av}$

.

These two informations

means

that

one can use

various $\mu rous$ materials with pole size of$t_{av}$ to constructthe thermal edge

compressor

unit. From theviewpoint ofengineering, it is importantresult whichenablesthethermal

edge

compressors

withmicrochannels worksathigher

pressure

of

gases.

CPump effect

Next

we

considerProblem-lI, the steady

pressure

distribution in

a pump

system

con-sisting of

a

number of

compressor

units connected in series with their two ends ofthe

Pumpsystem beingclosedbythe walls. The number of the

compressor

units$N$islimited

in the DSMC simulation. The accuracy and computational speed of DSMC dcpend

on

thenumberdensity of simulation particles, whichshow unbalanced distribution

as

$N$

in-creases

inthis problem. Thisresultsinthe low

accuracy

atlowdensityregions andalow

1Amodiflcationofthedomainof integrationinthedefinition of$m_{f}$in(11)isrequiredsincesomepart

of the domain of integrationcanbeinsidethe solidbody. Theintegrationhereiscarriedoutonlyinthegas region.

(c)

Fig.8: Oneway flow inthe thermal edgecompressor$b$ (Problem 7). $T_{h}/T_{c}=3$

.

$(a)$ Thecase

where theplates arereplaced with squareboxes with temperature $T_{c}$ (shownby

a

darkbox) and

$T_{h}$ (shown by blight box); (b) Thecase with square boxes with the round off(theradius of the

curvature is $D_{h}/4$); and (c) The

case

with a cylinder with diameter $D_{h}/2$

.

(See the

capuon

of

Fig.4.) Thesizeof the reference

mean

freepath$t_{av}$isshownonthe leftshoulderofeach panel.

computational speed athigh densityregions. In Ref. 6, itis shown that thecompression

ratio ofa

compressor

unit is

a

functionof the local Knudsen number$Kn_{L}$ defined bythe

average

density of the

gas

in each unit in the system, and theirrelation is determined

as

follows: (i) Carry out the numerical simulation with $N=10$ for Knudsen number Kn defined by the density$\rho_{av}$ of the

gas

averaged

over

wholePump system;(ii) Obtain

a

part

oftherelation between thecompression ratio and $Kn_{L}$ from the result of(i); (iii)Repeat

(i) and(ii)forvarious Kn. In thisPaper,

we

follow this method. We also consider

a

pump

system of 10

compressor

units of basictype shown inFigs. 3

or

6, and theboth ends of thePump system

are

closed by the walls with temperature$T_{c}$

.

Wefirst define the

pressure

averaged

over

the

cross

sectionofthe

pump

$p(X_{1})$by

$K_{1}\Phi_{i}$

(a) (b)

Fig.9: Thedistribution of theaverage poessure$\overline{p}(X_{1})$ and local Knudsen number$Kn_{L}(X_{1})$in the

steady state of

a

closed system of10 compressorunits(Problem II).$T_{h}/T_{c}=3$

.

$\cdots\cdots\cdots$ :$n=10$,

–$\cdot$ –: $n=20$,–: _{$narrow\infty$}

.

$(a)Kn=0.1,$$(b)Kn=1$

.

(a) (b)

Fig.10: Thecompression ratioofacompressorunit$P(X_{1})$vsthe local Knudsen number$Kn_{L}(X_{1})$

(Problem II. $T_{h}/T_{c}=3$

.

$(a)n=10,$ $O$: Kn $=$ 0.1,0.4, 2, 5; $\bullet$: 0.2, 1,3.5. (b) _{$narrow\infty,$} $O$:

Kn $=0.05,0.1$,0.3,0.75, 1.5, 2.75, 4; $\bullet$: 0.075,0.2, 0.4, 1, 2,3.5. Theranges oflocal Knudsen

number $Kn_{L}(X_{1})$ for each value of Kn are shown in the bottom of the figure. The white line

representstheapproximationcurveEq. (15).

Thecompressionratio$P(X_{1})$of

a

compressor

unitbetween$X_{1}$ and$X_{1}+D_{w}$isdefinedby

$P(X_{1})=\overline{p}(X_{1}+D_{w})/\overline{p}(X_{1})$, (13)

and the local Knudsennumber$Kn_{L}(X_{1})$by

$Kn_{L}(X_{1})=\frac{m}{\sqrt{2}\pi d_{m}^{2}\overline{\rho}(X_{1})D_{h}}=\frac{\rho_{av}}{\overline{\rho}(X_{1})}Kn$, (14)

where$\overline{\rho}(X_{1})$ isthe

average

densitybetween_{$X_{1}$} and_{$X_{1}+D_{w}$}

.

The examples of the distribution ofthe

average

pressure

$\overline{p}(X_{1})$ and thelocal Knudsen

Flg.11: Thecompressionratio$\Pi_{N}$obtained by_{$Ncomp\infty ssor$}units (Problem II)._{$T_{h}/T_{c}=3.narrow$}

$\infty,$$Kn_{L(1)}=5.7$

.

The marks $\bullet$ show the points$(i,\Pi_{i})$where$Kn_{I4^{i)}}\sim 0.05$,0.1,0.2,0.5, 1, 2,3.5.

pressure

$\overline{p}(X_{1})$ increases showing

some

vibrations

as

$X_{1}$ increases. The profile of local

Knudsen number $Kn_{L}(X_{1})$ is smoother than that of$\overline{p}(X_{1})$

.

Itis because $Kn_{L}$ is defined

by the

average

density

over

a

unit length $D_{w}$ and $\overline{p}(X_{1})$ by the values at

a

cross

section

at$X_{1}$

.

The profile of$Kn_{L}$ shows local variations atboth endsof the

pump

system. This

corresponds totheeffect of end walls of the

pump

systemwhichis also

seen

inRef. 6.

The sets of$(Kn_{L}(X_{1}),P(X_{1}))$ for various Knudsen numbers Kn

are

plotted in $Kn_{L}-P$ plane for the

case

$T_{h}/T_{c}=3$ and $n=10$ and $\infty$ in Fig. 10. Thepoints $(Kn_{L},P)$ form

a

curve

$P’(Kn_{L})$ in the $Kn_{L}-P$plane. The data for$X_{1}$

near

the ends of the Pump system

deviate from the above

curve

$P’(Kn_{L})$due tothe effect of the end walls ofthe system. An $aPproXimanon$of$P(Kn_{L})$

:

$P’(x)\sim$

exp

$[C_{0}+C_{1}\ln x+C_{2}(\ln x)^{2}]+1$, $C_{i}$ : Constant, (15)

is also shown in the figure, where $C_{i}$

are

determined by the least

square

method from

the data $(Kn_{L},P)$ except those close to the ends of the _Pump system. The values of$C_{l}$

are

$(C_{0}, C_{1},C_{2})=(-2.24, -0.276, - 0.244)$ for $n=10$ and $(-2.11, -0.221. - 0.238)$ for $narrow\infty$

.

The compression ratio $\Pi_{N}$ of

a

Pump system consists of$N$units

can

be estimatedby

usingEq. (15). Thatis,

we

estimatethe local Knudsen number$Kn_{I40}$ ofi-thunit and the total compressionratioof thesystem $\Pi_{N}$ from

a

initiallocal Knudsen numberati-thunit

by

$Kn_{L\langle i+1)}=Kn_{L(i)}/P(Kn_{L\langle l)})$, $\Pi_{N}=\prod_{i\overline{-}1}^{N}P(Kn_{L(l)})$

.

(16)

An example of the resultis shown in Fig. 11,_{for the}

_case

of$T_{h}/T_{c}=3$ and$n\cdotarrow\infty$ with

(a) (b) (c)

Fig. 12: The device that forns a flow channel ofthe scale of lmm. (a) Unheated part. It is a

rectangular copperplate (thickness lmm) of200mm in height and 200mm in width. There isa rectangularhole of80mm in widthand 100mmin height. A large numberofcopper wires with diameter lmmisattached in thehorizontal directiontothe hole with the solder. (b)Heatedpart.

An aluminum frame forms a square flow channel of side 100mmin length. A large number of Kanthal heater wires with diameter lmm invertical direction areattached by alumina adhesive

on

onesideof theframe. (c)_{The assembled device. The wires aroundtheheatedpartsupply the} electric current totheheaterwire.

4

Experiment

In the

course

ofthenumericalsimulationin Sec.3,

_{a one-way}

_flowunder thecondition ofperiodic flow along the

pump

system (Pmblem-I) is induced by the newly designed

unit,whichconsistsof

a

pair of parallelwiremeshes with differenttemperatures. It

means

that there will be

a

rarefied

gas

flow through

a

pair of parallel wire meshes withdifferent temperatures. The flow is, however,notfoundinliterature. Therefore,

we

win

carry

out

a

preliminaryexperiment inthissectiontoobserve this phenomenain

a

rarefied

gas.

4.1

Experinent

_on

_{the channel width of}

_lmm

A Experimentalapparatus

Here

we

presentthe setup for theexperimentof the channel width of lmm. The device consistsoftwoparts:

one

isthe unheated part, and the other is the heated part. Thefomer

is

a

rectangularcoPperplate (thickness lmm) of$200m\iota n$inheightand 190mm in width.

This plate has

a

hole of 100

mm

in height and 80mm in width [Fig. 1$2(a)$], and

many

copper

wires with diameter lmm in horizontal direction

are

arranged in the hole. The latteris

a

aluminum frame[Fig. $12(b)$]whichforms

a

square

flow channel of side $1\infty\iota nm$

in length, and

many

Kanthal heater wires with diameter lmm in vertical direction

are

attached by alumina adhesive. The

gap

between eachof thesewires,$cop\mu r$andKhantal,

is about lmm.

The aluminum frameishung

on

the

copper

plate by several Nylon

suppon

parts

so

that the side ofthe mesh of heater wire faceto the wire mesh

on

the

copper

plate. The

dis-Fig. 13: The overviewofthe experimental apparatus. The deviceis setinside the glass belljar. Themotionof the filmorwindmillisobserved by thecamerapositioned appropriately.

(a) (b)

Flg. 14: (a) _{Wlndmill to detect} the vertical gas flow. The

vane

is made of mica. The

vane

is supported byabearing oftinyglasscupandasteelneedle. (b)_{The aluminum duct and}ahole for

thewindmmill, attached to the unheated side ofthe device. Thewindmilldetects upward flow ifa

gasflowsthrough the device from unheated side to heated side.

tance between thesewiremeshes is kept

as

close

as

possible, while thereis

a

small

space

between them. (We

can

detect thecontactbetween them by measuningthe electric

resis-tance.) Then thewhole systemisputin

a

glassbelljar, whoseinner

pressure

iscontrolled

fromthe atnospheric

pressure

($1.01\cross 10^{5}$Pa)down to about0.5 Pa byextemal oil-sealed

vacuum pump

(Fig. 13). The coPperplate ofthe unheated

pan

is connected to the steel

base of the

vacuum

chamber to keepthetemperatureclose to the

room

temperature.

We supply the electric current to the heater wire with the

gas pressure

$p$ in the bell

jar being kept at

some

constant value. Thenthe

gas

flow through thispair ofwires

are

detected by two separate experiments: (A) A thin aluminum film (thickness $4\mu m$) of

width

60mm

andheight80mm ishung

on

the flowchannel inthe side of theheatedwire.

We observe the motion ofthe film by

a

camera

(Fig. 13); (B) Prepare

a

windmill which

detects

a

flow in the vertical direction[Fig. $14(a)$]. The windmill is set in

an

aluminum ductattachedtothe hole ofunheated

copper

platein the opposite sideofthe heatedmesh

(a) (b) (c)

Fig. 15: The movement ofthe film in experiment (A) _{for the} case ofchannel width of lmm. I: The results forconstantpressure$p=1Pa$ and variousenergysupply $E$to the heater. (a) $E=OW$,

(b)5.$2W,$ $(c)12W$,and(d)$21W$

.

after the heaterisput

on

in ordertowaitthe steadytemperature of the device. B Res\omega 籾

Two series of experiments

are

camied out for experiment (A). In the first series

we

observe the film forvarious

energy

to the heater $E=0,5.2,12$, and 21W with keeping

constant

gas pressure

$p=$ 1Pa. The result is shown in Fig.

15.

The film is gradually

inclined

as

the

energy

$E$ to the heater increases, and keeps at

some

angle

as

long

as

$E$

and$p$

are

kept atconstant value. The temperatures of the frame ofthe heated part and

that ofthe

copper

plate of the unheatedpart

are

measuredin

a

separateexperiment. The

temperature difference between themis 0.OK at $E=OW,$ $6.6K$ at2.$0W,$ $11.4K$ at4.$0W$,

and

19.

$8K$ at

8.

$5W$

.

In the second series the

energy

to the heater $E$ is fixedto $21W$, and theobservationiscamiedoutforvarious $p$inthe

range

from 1Pato$2\ovalbox{\tt\small REJECT} Pa$

.

Some ofthe

results

are

shown inFig.

16.

Themovementof thefilmtakes maximumvalue ataround

$5Pa$,where the

mean

free pathoftheairisabout lmm. Themovementofthefilm vanishes

as

pressure

increases. In this preliminary experiment it is difficult toconclude that this shows that the flow vanishes

as

Knudsen number decreases; We will discuss about the

problem laterin Sec.4.3.

The experiment (B) _{is camied out for the}

case

_with $(E,p)=(0W, 5Pa),$ $(2W, 5Pa)$,

$(18W, 5Pa),$ $(18W, 20Pa)$, and($18W,$ lmpa). There is

no

rotationof thewindmill for the

case

of(OW,$5Pa$)

or

$(18W. 100Pa)$

.

_{For other}cases,_thewindmill rotates andthedirection

of therotationshows that the

gas

flowisupward. Fromthepositionof the windmill shown

inFig. 14(b), it

means

thatthe

gas

flows through thepair ofwire meshes from unheated wiremeshtoheatedwiremesh. Therotationspeed of the windmillis 95rpmat$(2W, 5Pa)$_,

667rpmat$(18W, 5Pa)$, and 176rpmat $(18W, 20Pa)$

.

4.2

Experiment

on

the

channel width

of

$100\mu m$

A Experimentalapparatus

In order to confirmthe applicability ofthe flows induced through thewire meshes to

(a) $\zeta b$)

Mg. 17: Experimental setup for the exporiment of flow channel of $1\infty\mu m$ in width. (a) The

stainless mesh. The scale of 100pm is shown at theright bottom

comer

of the figure. (b) The schematic figure (not to scale) _{of the device for the} experiment. This figure shows the

cross

sectionalongtheflow channel to explain the layered sbuctureof thedevice.

mesh size is about $10\mu m$ [Fig. $17(a)$_]. Inthis

case we

have to_suPpontwo wire meshes

with differenttemperaturesin

a

smalldistance. For this

purpose we

constructed alayered

structure

as

shown inFig. 17(b). Thebaseis

a

copperplate A (thickness$20mn$) _with

a

circular hole whichis

one

end oftheflow channel [Fig. $17(b)$]. _{The diameter of}thehole

is 10mm

on

one

side andit is enlargedto20mm

on

theother side. This plateisconnected to

a

bigger

copper

basewhichisconnectedtothesteelbaseof

a

vacuum

chamberto$k\infty p$

the$tem\mu rature$ closeto the

room

temprature[the coPperbase is omitted in Fig. $17(b)$_]. In order to support two wire meshes, _{the following material} is layered

on

this

copper

plate: (i) A stainless wire mesh B. (ii) _{Several mica plates} $C$ (thickness _{$20\mu m$}). which

forms

a space

around the hole ofthe

copper

plate A. (iii) Another stainless wire mesh $D$, to whichtwoof electricalleads, shown by

a

dotted lines inFig. 17(b),

are

attachedto

suPply

an

electric currentto the mesh. (iv) A mica plate $E$ (thickness $2\ovalbox{\tt\small REJECT}\mu m$). $(v)$ A

aluminum plate $F$(thickness $10mm$) with

a

hole whose size is similarto that

on

_{$\bm{m}p\mu r$}

plate A. The main

purpose

ofmicaplates$C$ is to keep

a space

between theheated wire mesh$B$ andtheunheatedwire meshD. Another_{$pu\iota pose$} ofmicaplates $C$ istomaintain

(a) (b) (c)

(d) (e) (f)

Fig. 18: The movement of the film forthe caseof the channel width of $100\mu m$

.

Theresults for

constantenergy supply$E=1.5W$ _{and various}pressure$p$ in the bell jar. (a) $p=5Pa,$ $(b)20Pa,$ $(c)$

$40Pa,$ $(d)100Pa,$$(e)600Pa$,and$(D1000Pa$

.

the electrical insulation between them. The electrical insulationisrequiredtopreventthe

electriccurrent through wiremesh$D$ fromheating the wire mesh B. The mica plate $E$is

alsoinserted tokeep theelectrical isolation betweenmesh$D$andplate F.

Theselayers

are

fastened tightly by severalbolts, and the electrical isolation (including

the space between meshes $B$ and D) is confirmed by measuring the electrical resistance

between mesh $B$ and other metallic materials. Then the hole systemis put in the

same

vacuum

chamberas in section 4.1. In this system, the flow is detectedbythe movement

ofathinfilm ofmicaoraluminum hung in thehole of thealuminum plateF. B Results

We supply theelectriccurrent to

a

wiremesh(mesh$D$in Fig. 17)andobservethetilting ofthefilm. Twoseries ofexperiments

are

camiedout;Thefirst

one

isthe

case

forvarious

energy

supply $E=0,0.06,0.23$, and 0.$53W$for$p=10Pa$, and the second is the

case

for various

pressure

$p=5$,_{10,20,40,100,600, and 1000Pafor}

_a

_fixed

_energy

_supply$E=1.5W$

.

In

any case

the film is constant if $E$ and $p$

are

kept constant. Some of the results for

of thethin fllm of windmill put channel of the flow shows that

a

flow is inducedfromthe colder sideto hotter side, and these phenomenais

seen

only in the rarefied

gas

regime. This flow is attributed to the flow induced by the

temperaturedifference oftwowiremeshes. These experiments,however,

are

preliminary

ones, and there should be several discussions

on

the results.

Apointin questionoftheexperimentsgiven in thissectionis thatthe

energy

supply to

the heater$E$ is kept constant in the experiments fordifferent

pressure

of the

gas

$p$

.

This

will give

some

effect

on

the

pressure

dependence of the flow obtained by these

experi-ments. The temperature of heated wire isdetermined by the given

energy

supply $E$and theloss of the

energy.

The lossconsistsof the radiation and heat flow through thesupport parts of the heated wire mesh and the loss of the

energy

camied by the

gas

molecules. The later

energy

lossis roughly proportionalto$p$and the temperaturedifference between

thewiremeshes. Therefore,in theseexperiments,thetemperaturedifferencebetween the

wire meshes decreases

as

$p$ increas

es.

(Atthe

same

timethe temperature of theunheated

wire mesh increases because the heat conduction between the unheated wire mesh and extemal environmentis rather limitedin

a

vacuum

chamber.) Thus inthese experiments, the flow through the pair of wire meshes

may

vanish in higher

pressures

by following two

reasons:

(1)the

gas

flow induced by rarefied effect vanishessincethe

mean

free path

becomessmaller; (2)the temperaturedifferencebetween twowire meshesvanishes. Thus

thedependence of the strength of theflow

on

pressure

$p$isnotclearbythese experiments.

The difficulty will be solved by the experiments with the temperature difference between

two wire meshesbeing kept constant. It ispossibleforthe system givenin Sec. 4.1, and theresult will bereported inthe futurework. In the systemgiven in Sec.4.2,it isdifficult

to

measure

the temperature ofwire meshes since the wires

are

thin and total size ofthe heated

area

israthersmall. A

more

sophisUcatedversionof the device

may

berequiredto obtain thepressuredependence ofthegasflow.

Inthe results for lower

pressure,

themotionsofthefilmshow that theeffectofthe flow decreases

as

the

pressure

decreases. This isbecause the

gas

molecule thatamivesto the

surfaceofwireexperienceslastmolecularcollisioninthe

gas

farfrom thepairofmeshes,

wherethetemperatureof the

gas

isclosetothe

room

temperature.

Wehaveto examinewhat

we

observe with the tilting of the filmin these experiments.

The weight of the aluminumfilm

per

unit

area

used in Sec. 4.1 is about$0.01kym^{2}$, and

bythemomentumtransfer through theflowchannel

per

unit time. The momentum

trans-fer(perunit

area

andperunittime) through

a

channel

cross

sectionconsists of two parts,

the contribution of themomentumcamiedby the

gas

motion$(\rho\#_{1})$and thecontribution of

the stress. In the experiments shown in this section, the density of the

gas

is

very

small ($\rho\sim 10^{-5}kym^{3}$ at_$p=1Pa$). Therefore, it is impossible to support the film only by the

contribution of$\rho v_{1}^{2}$, since the flow speed

may

not be

so

large by the temperature

differ-ence

given in these experiment. The contribution of the stress

or pressure

is important.

That is, the motion ofthe gas in the flow channel is blocked bythe film, and it induces

pressure

differencebetween both sides of the film. Thereis

no

analytical information

on

the

pressure

difference obtained by blocking the

gas

flow induced through wire meshes

withdifferenttemperature. Instead,

_we

estimatethe

pressure

gradientobtained by

block-ing the thermal transpiration flow betweentwoparallel plates with

a

temperaturegradient

$dT/dX$

.

Accordingto the result of linearized Boltzmannequation,the

mass

flux through

the twoparallelplates isProportionalto

$( \frac{1}{p_{0}}\frac{dp}{dX})M_{P}(Kn)+(\frac{1}{T_{0}}\frac{dT}{dX})M_{T}(Kn)$,

where $p_{0},$ $T_{0}$

are

the reference

pressure

and temperature, respectively.17) Then

we can

obtain the relation between the

pressure

and temperature differences$\Delta p$and$\Delta T$ in

some

range

of$X$whenthere is

no mass

flux,byusingthenumerical values for$M_{P}$ and$M_{T}$

.

For

example, $\Delta p\sim 0.3p_{0}\Delta T/T_{0}$ (Kn $=1$₎ for hardsphere molecules, whichis, forexample,

0.1Pawhen$p_{0}=5Pa$and$\Delta T/T_{0}=0.07$

.

5

Concluding Remarks

In this

paper, we

camied out the numerical simulation ofthe rarefied

gas

flows inthe thermal edge

compressor,

and clarified the maximum

mass

flow (Problem-I in Sec. 3)

and

pressure

ratio (Problem-II_{in Sec.} 3). In the

course

ofthe analysis

we

also tried to

develop the altemative design of the thermal edge

compressor

adequateto high

pressure

range

or

compressor for micro channels, and it is shown that there is

a

wide variety of

the design of the compressor. The result

on

the altemative design of the unit is also

important

as

the fundamental studyof the rarefied

gas

flows induced by the temperature

fields, _{since it shows the}

_presence

_of

_a

_rarefied

_gas

_{flow through}

_a

_{pair ofwire meshes}

with differenttemperatures. The flow,

a

new

type ofthe flow which is not pointed out

before, is demonstratedby simple experiments in Sec.

4.

The experiment succeeded in

showing that the flow is actuallyinduced, _{but its dependence}

_on

_the

_gas

_pressure

_or

_the Knudsen numberisnotstillclear bythese preliminary experiments.

Acknowledgment

The author express his thanks to Mr. T. Yamada for his help in the experiments in Sec.4.

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