Irt. J. oJApplieclMechanicsand Engineering,200l, vol.9, No.3,pp.47I_4gl
THERMOPHORESISFREE CONVECTIONFROM A VERTICAL CYLINDER EMBEDDEDIN A POROUSMEDIUM A.J. CHAMKHA ManufacturingEngineeringDepartment The PublicAuthority fbr AppliedEducationandTraining P. O. Box 42325,Shuweikh,i0654, KUWAIT
M. JARADATandI. popFacuityof Mathematics, Universityof Cluj R-3400Clui. Cp 253,ROMANIA e-mail:
[email protected]
This paperdealswith the stcadyfree convectionover rrnisothernralrcrtjcal circulrr crlinder enrbedilcd in a fluid-saturated porousrnediumin the presenceof the therrnophoresi' plrrric).-dr-nosiriuneff'ect.'l'he governine partialdifferentialequatiorrsare tritnsformedinto a sct ol'non-sirnilarequations.rvhich are sol'cd nurnericall. using an implicit finite-difference melhod.Comparisonswith the previouslypublishedwork are perfbrmeclancl the resultsare found to be in excellent agrecment.Many resultsare obtained and a representative set of these results is displayedgraphically to illustratc the intluencc of the various physical paramcters on the wall thermophoretic depositionvelocityand concentration profiles. Key words: free convection,thermophoresis, verticalcylincler.boundarylayer,porousmedium.
1. Introduction Convectiveheat transferin fluicl-saturated porousmedia has beenstudiedquite extensivelydulng -'e last few decadesThis has been motivatedby its importancein many naturaland industrialpioblems. 'i:ominentamong theseare the utilizationof geothermaL energy,chemicalengineering,thermalinsulation " slems'nuclearwastemanagement,erain storage.fruits and vegetable,migrition of moisturethroughair - ntainedin fibrous insulation,food processingand storageand conraminanirransportin groundrvater and : :ny others.A detailedreview of the subjectof convectiveflorv in porousmedia,includingan exhaustive 't of references, was recently0,
Q+0
as
r-)m.
The thermophoresisand buoyancyparametersN, and N are definedas
N,=
T* Tru-T*
N = Fc(c*-c*)l\r\* -r*).
(2.15)
Thermophoresisfree convection from a vertical cr-linder ...
473
;v t -_ - K ; 1,_., v 0 7
(2.s)
tor
subjectto the boundaryconditions i=0,
T=Tr,
u-+0,
C=Cnr,
i=rg,
T-->T*, C_+C*,
(2.6)
F_+*.
In orderto transform theseequations, we introduce thefbllowingnon-dimensional variables r = X a 1 l(2ff , r ) ,
x = I f 1 6,
v,=pr-1t2 (r,lu,), 'ihereRa and u
"
u = u fU , ,
I , =n a t , ,(zr ff L , ) ,
o= (z_r*)l(r*_L) O=(c _ c*)l@,,,_c_)
(2.7)
arethe Rayleighnumberand the characteristic velocity,respectively, which are definedas
R a= g K B T( r * _ r * ) r o f u * v ,
( J "= S K F r ( f * _ f * ) 1 v .
(2.8)
Thus,Eqs.(2.1)-(2.5) canbe wrirtenas
i
a, r d, .o x \ r - n l + ; - ( r v ) = ( / ,
(2.e)
dr'
a=0+N0,
(2.t0)
ao v _ao= _rafas) _t r_ I
U_+
r0rl'dr)'
(2.11)
/ f a [,'aq) ,,$*,?o+a(u'o)=
(2.12)
dx
Ex
dr
0r
dr
Ler0rl'ar),
pr a0
, r-ot 't--KN
(2.r3)
: theboundaryconditions(2.6) become v=0, u--s0,
0=1, 0)0,
0=1, Q-+0
on
r=1, as
r_)6.
(2.14)
The thermophoresisand buoyancyparametersN, and N are definedas
r,' =J=, N =Fc(c,-c*)l;r(r,-r*). Tr, -T*
(2.1s)
ltililH A..1.Chantkha,M. Jaradat tntd L t' rt1'
474 of the form We now look for a solutionto Eqs.(2'9)-(2'12)
l (r'-,/e, \=2xtt2. T=
v = G l z ) r \ , n ) , 0 = o ( E , q ) ,o=o(E,n)
( 2 .1 6
way as where ry is the streamfunction, which is definedin the r,rsuai
l l
=
"'r
1ov
,I
r dr
r ox
- -
dllt "Y
( 2 . 1 1'
10)-(2.I 2) become Thus,Eqs.(2. Af
( 2 . 1 8t
";' =0+NO' dn
.i,#=j.[## ##) *[,'.En)#] .i' #.,#{roho#-' E,)#] **[,'-'
(2.19
(2.20
aoao o fae)rJ t,(a1aa a/aal a2o*rnun+ou4 ana€-aqdnl
^,,.etrr,l ]=ttl
subjectto the boundaryconditions
f =0,
0=/,
U-g,
0n
0=1
ol.l
o-+0, Q-+o
I = 0,
(2.21
n-+-.
Nu and Sh' whici The physical parametersof interestare the local Nusseltand Sherwoodnumbers, aregivenby
=-[+'l Nu/na{' \d
I I,r=,r
sh/Ra112 it ^dl Q l
t1 1.
\ d 1l /,1=o
number whereRa..is thelocalRa1'lei-eh 3. Results and discussion
an initial-valueproblemwith ( playingthe role of time ar Equations(2.18)-(2.2Urihich represenr solution'Therefbre,the non-linear,coupled,partial differentialequations.uhich possessno closed-fonn The implicit,iteratir given by Eq.(Z-21). must be solvednumericallysubjectto the boundarl'conditions of this ty'p merhoddiicusseaby Blottner(1970)hasprovento be adequatetor the solution finite-difference (2'20) ar (2'19) and Equations work. present in the of equations.For this reason,this methodis employ'ed with respectto ( beir discretizedusing three-pointcentraldifferencequotientsivith the first derivatives
:'()p
(r)
discretizedusing two-point backwarddifferencequotients.This convertsthesedifferentialequationsinto linearsetsof algebraicequationsat eachline of constent(. which can be readily solvedby the well-known Thomasalgorithm(Blottner, 1970).On the other hand,Eq.(2.18)is discretizedand solved subjectto the appropriateboundaryconditionby the trapezoidalruie. The computationaldomain (q,n) was made up of 196 non-unilbrrngrid pointsin the q directionand 101 uniformgrid pointsin the ( direction.It is expected that mostchangesin the dependentvariablesoccurin the regioncloseto the cylindersurfacewhereviscous effects dominate.However, small changesin the dependentvariablesare expectedfar away from the cylindersurface.For thesereasons,variablestepsizesin the '4 directionare employed.The initial stepsize Aq, and the growth factor K* employedsuchthat Ar1,-, - K'AI,
nl
9)
_1))
475
Thennophoresis free convectionfromd vertical cyliruler
(wherethe subscripti indicatesthe grid
location) were 10--i and 1.03, respectively.These vahresu'ere found (by perfbrming many numerical experimentations) to give accurateand grid-independent solutions.Hovuel'er. constautstepsizesof 0.01 were usedin the ( directionsincethe changesin the dependentvariablesiu this directionare not expectedto be great.The problernwassolvedline by line startingwith (=0 and rnalchinslolrrn'd in the ( directionuntil reduceto a setof ordinarydifferentialequations the desired( valueis reached. At (=0, Eqs.(2.18)-(2.20) which can be easilysolvedby the Thomasalgorithm.This solutionis then usedas the initial solutionto set off the marching process.The convergencecriterion employedin the presentwork was based on the differencebetweenthe valuesof the dependent variablesat the currentand the previousiterations.When this differencereached10-5, the solutionwas assumedconvergedandthe iterationprocesswas terminatecl. The resultsare gir,enfor severalvaluesof the parameters k, Pr, Le, N and lV,. However.to checkthe presentnumericalresuits,we have calculatedtlre valuesof the reclucedheat transfer,-O'(O,O) und mass t r a n s f e r-,0 ( 0 , 0 ) , f o r ( = 0 ( f l a t p l a t e )w i t h k = 0 . N = 0 , 1 a n d L e = 1 . T h u s ,f o r 1 V= 0 w e o b t a i n e d -O'(O,O)=0.44325, whiie the valuefbunclby Minkowyczand Cheng(19'/6)is -0'(0,0) =0.414. Also, for k = 0 a n d N = L e = 1 , w e g e t - g ' ( 0 , 0 ) = - Q ' ( . 0 . 0 ) = 0 . 6 2 7 8 3w,h i l e B e j a n a n d K h a i r ( 1 9 8 5 )o b t a i n e d -O'(O,O)=-q'(0,0)=0.628. A comparisonof the present results for the local Nusselt number. -a0(E,O)lan with thosereportedby Minkowycz andCheng(1916)is also given in Tab.1for N = 0 and
ll)
;lr
different values of the curvatureparalneter(. It can thus be corrcludedthrt the presentresultsare in :xcelientagreement with thoseof Bejanand Khair (1985),and thoseof Minkorvy'cz and Chengi1976)and ,,\'eare,therefore,confidentthat the presentnumericalresultsare ver\raccurate. l'able 1. Valuesof the local Nusseltnumber.- a 0(i. o )/0 r1. for ,V = 0 anclsomer aluesof the pararneter( .
t: _t t )
0.25
0.s0
rey rVc '
?v
,llo ,,'b
0.75 t.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 t0.00
Minkowyczand Cheng(19"76)
Presentresults
0.4855 0.5272
0.490341
0.5664 0.6049 0.7517
rB 0.5784 0.620125 r1 0.7763
0 . 8 951 1.024 t.t51 t.283
r23 0.922
r.4t3
r.44646I 1.570056 1.69r356
1.514 1.678 t.815
0.s35r89
1.0s9965 1.t91509
r.320432
I . B1 3 8 7 5
4',76
A.J.Chamkha, M.Jaradat and I.Pop
0.0040
0.0035
0.0030
G o.oozs 0.0020
0.0015
0.0010 0.00
0.25
0.50 F
0.15
1.00
Fig.l. Effects of N on local wall thermophoreticdeposition velocity.
N = - 1 0, , l , 3 , 5 , l 0
? o.so
n Fig.2.Effectsof A'on concentration profiles. Typicalconcentration profile>. o(: 11). rr all thelmophoretic deposition velocity,V,,,(\) and wall mass t r a n s f e r-,a 0 ( 8 , O ) l a n , a r e s h o * n i n F r s s . l - E i o rP r = 0 . 7 2 a n d , s o m e v a l u e st o h fe g o v e r n i n g p a r a m e t e r s k . Le, N and N,. These figures shori hori the concentrationboundarylayer and the wall thermophoretic depositionvelocity reactto changesin the sorernins pararneters. Thus, the concentrationprofiles, (r(E.n), indicatethe characteristic shapeof a non-dimensional concentration with a rapid developmentcloseto the plate.Thus,the concentration profilesdecreaseu ith an increaseof the buoyancyparameterN and also with an increaseof the Lewis numberLe (Figs.2and 7). rvhichis on line with the resultsreportedby Bejan and
*
Pnp
. ttermophnresis.free convectionJrom a vertical cylinder ...
4'77
{hair (1985) for the case when the thermophoesis effectsare absent.This will give rise to a large wall , rncentration gradient,-a0(E, O)lAn, as the parameters 1/ and Le increase(Figs.3and 8). It also causesa :gh depositionof the velocityon the surface,which increases parameterk increases, as the thermophoretic .. can be seenliom Fig.4. It can be also seenin Figs.1,5 and 6 that the wall thermophoreticdeposition :iocity,V,,rl\), becomessensitiveto the variationof the parametersLe, N and N, . Thus, the deposition ,',ocity profile on the wall is reducedas N and Le are decreased(Figs.l and 6). However, the wall :rniophoretic depositionvelocity profiles decreaseas the valuesof the therrnophoreticparameterN, are -'reased. This is of particularbenefitin processes, which requireextremecleanliness of the surfaces. 5.0
k=0.5 Le=10 N,=100 Pr=O.12
4.0
3.-5 3.t) O !u,
^ _
0.25
0.50
0.75
\ Fig.3.Effectsof N on local wall concentration gradient. Le=10 N=3.0
0.0025 0.0020 lr.n a
0.0015 0.0010
,
. INASS
. -.:ers k, ' ,oretic
;,n/, . : ttt the _ ..r with : .:ll flfid
0.000-5 0.0000 0.00
0.25
0.50 F
0.75
Fig.4.Effectsof k on local wall thermophoretic depositionvelocity.
100
A.J.Chamklta,M.Jaradat and I'P'
478 0.028 0.026 0.024 0.022 0.020 0.018 0.016 !\t
0.014
t
0.012 0.010 0.008 0.006 0.004 0.002 0.000 0.00
0.50 F
0./)
velocity' Fig.5. Effects of N, on local wall thermophoreticdeposition 0.0045
0.0040
0.0035
tl o.ooro
t
Fig.6'EffectsofLeonlocalwailthermophoreticdepositionvelocity.
4. Concludingremarks steadyboundary layer free convection over a Numerical solutions for heat and mass transfer by mediumin the presenceof thermophoresisparticle heatedisothermalvertical cylinder embeddeJin a porous as proposedby Blottner (1970)' has beenusedto depositioneffect were reported.A finite-differencemethod, of purti"t.r, the concentrationprofiles and the calculate the effect of thermophoresison ihe d"position clearly show the effect of thermophoresison concentrationgradient at the surfaceof cylinder. calculations there are no experimental data on particle particle deposition. To the best of the authors' knowledge,
-
Pop
'[hermophoresis free convectionfroma yertical cylinder ..
479
depositionfrom naturally convected flows in porous media. It is hoped that the present treatment will facilitatefuture comparisonsof the naturalconvectiondepositiontheory with laboratorydata. 1.00
Le=0.1.1.0,10.100,1000 0.75
? o.so
n Fig.7.Effectsof Le on concentration profiles.
Le=1000
Le=500
l0
0 0.50
Fig.8. Effects of Le on local concentrationgradient. -'r 'le
:l to : rhe .. on : lcle
\,Jmenclature C - concentration D,,, - massdiffusivity / - reducedstreamfunction
k=0.5 N=3.0 N,=100 Pr=0.72
A.J.Chamkha,M'Jaradat and I.Po1t
480
k /\ Le N N,
-
Nu Pr r r6
-
q
gravitationalacceleration thermophoreticParameter permeabilityof the porousmedium Lewis number buoyancyparameter
thermoPhoreticParameter localNusseltnumber Prandtlnumber radial coordinate radiusof the cYlinder Rayleighnumber basedon r7,for a porousmedium Ra - local Rayleighnumberbasedon i Rat Sh - Sherwoodnumber - fluid temperature I - velocity componentsin the x- and r-directions U'l - thermoPhoretic velocitY vt x - axial coordinale d,n - equivalentthermaldiffusivity 0c - chemicalexpansioncoefficient gr 11 0 v E q \l,
- thermalexpansioncoefficient - pseudo-similaritY variable - dimensionlesstemperature - kinematicviscositY - stretchedstreamwisecoordinate - d i m e n s i o n l e sc so n c e n t r a t i o n - streamfunct'ion
Subscripts w - conditionat the wall @ - conditionin the ambientfluid
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,'Iyer rt ical
Received:December1. 100,i Revised: April l. 200-1
