FLOW CHARACTERISTIC AND HEAT
TRANSFER FOR SINGLE PHASE GASEOUS IN SLIP文明与我同行
Received date :2007-08-05;revision received date :2007-10-15 E -mail :jhsun @nuaa .edu
REGIME OF MICROCHANNEL FLOW
Sun J ianhong ,Cai Zhoux in
(Co lleg e of A ero space Engineer ing ,N U A A ,27Y udao Street ,N anjing ,P.R.China)
Abstract :Bad o n analyzing so me sim ulation m odels of single pha gaous flo w in micr ochannels (0.001<K n <0.1),a numer ical simula tio n of N -S equatio ns w ith t he slip model is pr ent ed.In the simulatio n,the collo-cated g r id and the SIM PL E scheme a re ud .Result s show t hat the pressur e in t he inlet is changed w it h K nudn num ber .T he slip speed a nd the t emper ature cr eep ar e incr ead with the aug ment o f Knudn number .T he dr ag for ce decr eas and the resistance of t he heat trensfer has a lit tle incr ea.Key words :micr o-channel;Knudn number ;N -S equatio n
CLC number :V 211.1+7; Document code :A Article ID :1005-1120(2007)04-0288-06
INTRODUCTION
The developm ent of techno logies in m icro-electro-m echanical systems (M EM S)has promo t-ed the study of fluid flow s in microchannels.In-vestigation in micro flow characteristic and heat transfer can optimize the desig ns of m icrofluidic devices ,such as m icro -pum p ,micro -valve and micro-co oler,etc
挑衅英文
[1-2]
.As w e know n,most o f
the dev ices are compod by microchannels.A microchannel cooling structur e is propod as a microelectronic coo ling [3]
.The micro channels hav e a rectangular cross-ctio n w ith an optimum channel w idth of a =50L m ,channel wall thick-ness o f t =50L m,and a channel aspect ratio o f b /a =8.T he exper im ent on the str ucture co n-structed by a silicon w afer show s that a heat o f 790W/cm 2
m ay be rem oved using water with a pr essure drop of about 2atm and 71℃ri abov e the fluid.
As w e kno wn,the ratio o f surface to volum e increas with channel size decreasing ,and the time o f heat conduction decreas in the cond order of magnitude as the channel size decreasing.If using classical theory ,the conclusions w ould
promo te heat transm ission.For the microscale g as flow ,Knudn num ber (K n )is g enerally ud to identify the effect o f rarefaction or micro scale.If the Knudn num ber is betw een 0.001and 0.1,the flow can be appro ached to the slip flo w g overned by the Navier-Stokes (N-S )equations w ith slip co nditions on the w alls,in w hich,K n =K /D ,and K is the molecular mean free path K =
L Q
P 2RT =L Q P Q
2P
(1)w her e L is fluid density ,P the pressur e ,and R
the universal gas constant.
In this paper ,the g aous flow inside tw o parallel planes is sim ulated w ith an infinite aspect ratio
and a m icroscale height.M eanw hile,the m icroscale flow characteristics on different K n num bers are analyzed .
1 THEORETICAL ANALYSIS
For the 2-D micr oflow on parallel plates,the g overning equations can be w ritten as
5(Q u )5x +5(Q v )5y
=0
(2a)5(Q u j u i )5x j =-5p 5x i +55x j L
5u i
5x j
(2b)
5(Q u j T) 5x j=
5
re是什么意思
5x j
k
c p
5T
5x j(2c)
and the state equation is
p=Q RT(3) In order to study the micr oflow w ith velocity slip and temper ature jump under the boundary co ndition,in recent decades,there are som e dif-ferent models to be dev elo ped in slip r eg ion,such as the slip mo del[4],Knudn layer m odel[5-7],sur-face ro ug hness viscosity mo del[8],and m icro-polar fluid model[9].The mo dels ar e all bad on N-S equations.On the o ther hand,lattice Bo ltzmann metho d(LBM),M onte-Calor direct simulatio n (DSM C),and IP m ethod[10]are also developed to carr y o ut the num er ical simulation in the m i-cro flo w.In the m ethods,the slip m odel is w idely ud for its simplicity and practicality,and tho results w ith the slip model ar e in agreement w ith ex perimental data.T her efore,as show n in Eqs.(4,5),the slip model is
ud to clo N-S and energ y equations w ith the v elo city slip and the temperature jump boundary
u s=B v K n d u
d y
solanum+3
R T
8P
0.5K
T
5T
5x wall(4)
T s-T w=B(B v K n)d T
d y
+
1
4R
u2
w all
(5)
w here u is the stream w i velocity,u s the fluid slip velocity(stream wi fluid velocity on the w all),y the norm al dir ection of w all,x the flow direction,T s the fluid temperature o n the w all, and T w the wall temper ature.B v,B are the non-di-mensional parameters including inform ation o f slip flow on the boundaries as follo w s
B v=2-R v
R v(6)
B=2-R T
R T
2C
C+1
1
P r
B v(7)
w here R v,R T are the tang ential m omentum and therm al acco mmo datio n coefficients,respectiv e-ly.C is the ratio of specific heats,and Pr the Prandtl number.
In Eqs.(4,5),the cond item in the rig ht hand can be neg lected since it is the smaller order one than o thers.Eqs.(4,5)may be simplified as
u s=B
T s-T w=B(B v K n)
d T
d y wall
(9) w her e B v K n prov ides a measure of the rarefaction or the slip flo w.It is assum ed to v ary fro m the continuum regime w here the non-slip flow occurs (B v K n=0)to the accepted upper lim it fo r slip (B v K n=0.1).B is a pro perty related to the gas-surface interactio n,and v ar ies in the range fr om0 to10.T her efore,the no n-slip flo w and the slip flow could be com pared in different cas of B v K n=0and other values.
On the o ther hand,the concept of fully de-veloped flow is accepted in the numerical solution for the slip flow.M eanw hile,the length of the flow inlet ctio n w ill is confirm ed by R e number and the hydraulic diam eter.
2 NUMERICAL METHODS新东方在线论坛
T he collocated g rid and SIM PLE schem e will be ud to simulate the micro flo w in this paper. As w e know n,SIM PLE ries(SIMPLER,SIM-PLEC,etc.)are go od numerical methods and have a w ide application on numerical simulatio n. And SIMPLEC is w orth to increa the conver-gence precisio n[11].In order to sim ulate the slip flow w ith N-S equatio ns,the boundar y co nditions are m odified according to the disperd formats of Eqs.(8,9),r espectively.
T he sample in this paper is a two-dimension-al steady micro flo w on parallel planes w ith 1.2L m hig h and6L m long,in w hich the upper plate is adiabatic and the boundary co ndition on the butto n plate is the co nstant heat flux,and the auxiliary Neum ann condition is ud at the o utlet for the velocity and the temperature.T he g as is nitr ogen,in w hich heat conductivity coefficient J=0.0259W/(m・k)and v isco sity coefficient L= 1.85×10-5(N・s/m2).And K n is v ar ied w ith the chang e of the inlet pressure fo r the con-stant inlet tem perature(T in=298K)and the fixed R e num ber(Eq.1).Ho wever,Re and M a can be also changed w ith the variation of B v K n at the in-let w hen the tem peratur e is constant.
In iteration processing,w hen the velocity and the pressure distributio n are obtained by the
N -S equations ,the tem peratur e distributio n can be also got by the energ y equation w ith the velo
ci-ty profile,and then the property coefficients co u-pled w ith the temperature ar e m odified .If the ab-solute max im um value of the r em ained m ass in each contro l volume is no t small enough,then re-turn to the SIM PLEC cy cle so lver,as far as the iteration reaches a convergence .M eanw hile ,co mpar ed w ith different grid numbers,the mesh w ith 100×20g rids is suitable and ud in this pa-per .
3 RESULTS AND DISCUSSION
Fig s .1-2show the pressure distributio n o n different values of B v K n .T he pressure is de-cread with the increasing of B v K n .Results show that the rarefactio n w ill be enlar ged in the ca of hig her value of B v K n .And it w ill create the nonlinear flo w near the
outlet.
F ig.1 Pr essur e distr ibutio n in B v K n =0.001,B =0.2,
Pe =
5
F ig .2 Pr essur e distr ibutio n in B v K n =0.1,B =0.2,
Pe =5
Fig.3show s the aver ag ed pressure along streamw i on different K n number s.It is very clear that the ca of K n >0.01,the variation o f pr essure profile is visible com pared w ith that in the ca of K n <0.01.T he pressure decreas w ith the K n number increasing since the gas rar-
efication decreas the effect of the v isco sity and increas g as speed.With the viscosity decreasing
and Re num ber increasing caud by gas rarefica-tion,the pr essure w ill decrea along the stream -w i (Fig.4).Ther efore,tho cas o f K n >0.01will be consider ed in the fo llow ing .
Fig.3 M ean pr essure distribut ions a lo ng st reamw i
星期四
on different K n number s
Fig.4 M ean pr essure distribut ions a lo ng st reamw i
on different R e number s
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Fig s.5,6show the v elo city and temperature profiles at the lo cation of x /L =1/2on different K n num bers (K n =0.001—0.1)w hen R e =1,R v =1,R T =1,and q =500kW/m.T he velocity near the w all becomes a little larg e and the max i-mum v alue w ill decrea w ith the increasing of K n num ber,thoug h they r em ain to be the parabolic curv es.It is interesting that the pro perty is re-mained that the v elo city in y /H =
12±36
equals the m ean velocity.How ev er ,the tempera-tur e that equals the mean o ne is about in y /H =0.
42,and decreas along the traver direction since the larg er slip velocity caus hig her coeffi-cient o f the heat convection .It is ver y clear that the flow s on higher K n number (K n >0.01)are different from the cas of lo wer K n num ber.
F ig .5 V elocity pr of iles at lo cat ion o f x /L =1/2on
differ ent K n numbers (K n =0.001—0.1)w hen Re =1,R v =1,R T =1and q =500kW /
m.
F ig.6 T em per ature pro files at lo ca tio n of x /L =1/2
o n differ ent K n num ber s (K n =0.001—0.1)w hen R e =1,R v =1,R T =1,and q =500kW /m .
Non-dimensional velocity u m (x )/u in and tem-peratur e T m (x )/T in alo ng the streamw i ar e show n in Figs .7,8.Since the slip v elo city and the pr essure dro p (Fig.3)w ill be enlar ged with the increasing o f K n number ,the local m ean velocity w ill increa along stream direction.Sam e as v e-locity ,the local mean temperature will incr ea w ith the increasing of K n num ber since the tem-peratur e jum p and heat transfer.The effect is very clear when K n number is larger than 0.01.
thaadFig .9show s the v ar iation of the drag for ce in a microflow .In the flo w developm ent regime near the channel inlet ,as show n in Figs .1,2,the drag fo rce is lar ger than that in flow developed regim e w here there is a little change along str eam wi for sm aller K n ,and it is decread w ith the incr ea of
K n since the effect of g as rarefication.奥林匹斯十二神歌词
atprentOn the other hand,the effect on heat trans-Fig .7 Distr ibutio ns o f mean velocity on different K n
num ber s along streamw i when Re =1and T in =298K
Fig.8 Distr ibutio ns of mean temper ature o n differ ent
K n number s along str eamw i w hen T in =298K
Fig.9 V ariatio ns o f drag fo rce o n differ ent K n num-ber s
fer is sho wn in Fig .10.T he cur ves show that the value of N u functio n is decreasing w ith the in-cr easing of K n number ,although the r ar efication is str ong er in the ca of hig h K n num ber and the slip velocity is larger.Then the convection heat transfer is po w erful and the value of N u function
increas .The reaso n is that the heat resistant is cr eated since the tem peratur e jum p is near the w all.The tem perature jump will sharply decr ea the gradient of the temperature and the heat co n-vection .M eanw hile the m olecular collision is w eakened by rarefication.Under the effect,the resistant w ill be mo re enlarg ed so that the advan-tag e of slip velocity can be ignored
.
F ig .10 Va ria tio ns of N u functio n o n different K n
number s
4 CONCLUSION
The flow pressure contours,flow resistance and N u num ber variety are obtained by simulating N-S equations and ener gy equatio ns under the slip flow condition.T wo param eters B v K n and B ar e ud to note the effect of rarefaction or micro scale and the temper ature jump condition.Results show that the pressure drops are nonlinear at the inlet and outlet ctions .There are positive pres-sure area appeared near to w alls at the inlet ,and ano ther positive pressure area appear ed at the outlet w hen B vK n is a larg er one.It is in ag ree-ment w ith that the velocity slip beco mes mor e acute as the gas is m ore rarefied.Meanw hile,the fr iction dr ops w ith B vK n incr easing.
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