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MODELING OF COLD WALL CHEMICAL VAPOR DEPOSITION REACTORS (FOR SEMICONDUCTOR
FABRICATION)
M. Pons, R. Klein, C. Arena, S. Mariaux
To cite this version:
M. Pons, R. Klein, C. Arena, S. Mariaux. MODELING OF COLD WALL CHEMICAL VAPOR
DEPOSITION REACTORS (FOR SEMICONDUCTOR FABRICATION). Journal de Physique Col-
loques, 1989, 50 (C5), pp.C5-57-C5-65. �10.1051/jphyscol:1989510�. �jpa-00229533�
JOURNAL DE PHYSIQUE
Colloque C5, suppl6ment au n05, Tome 50, mai 1989
MODELING OF COLD WALL CHEMICAL VAPOR DEPOSITION REACTORS (FOR SEMICONDUCTOR FABRICATION)
M. PONS"", R. KLEIN, C. ARENA and S. MARIAUX"
D . L E T I / I R D I , C E A , C E N / G , 85X, F - 3 8 0 4 1 G r e n o b l e , P r a n c e
* ~ ~ 2 1 , c h e m i n des
~ r s l e s ,
F - 3 8 2 4 0 M e y l a n , F r a n c e* * SZMC/CNRS UA-413, ENSEEG, I N P G , B P . 7 5 , F - 3 8 4 0 2
st
M a r t i n d ' ~ $ r e s , F r a n c eRESUME
-
Les phCnomines survenant a u cours du d6pbt chimiquei
partir d'une phase vapeur resultent du couplage des transferts d e chaleur e t d e m a t i i r e et d e la rdaction chimique donnant lieu a u dCpBt. La rCsolution du systhme d'dquations differentielles requiert I'emploi d e techniques numeriques. Nous avons au D-Leti utilise Flux-Expert, un logiciel faisant appel aux techniques Clkments finis pour simuler l e comportement des rbacteurs d e D.C.P.V. C e logiciel peut aisCrnent Cvoluer lorsque d e nouvelles Cquations sont ajout6es ; il e s t d e plus t r i s convivial. A I ' h e u r e a c t u e l l e , n o u s n e prCsentons q u e d e s r i s u l t a t s o b t e n u s pour d e s gCom6tries axisymCtriques ; des simulations tridimensionnelles sont e n cours d e dCveloppe- ment. L e but du travail present6 e s t d e montrer que cet outil peut p e r m e t t r e l a conception, 1'Cvaluation et l a prCdiction d e s r6acteurs.ABSTRACT - Basic chemical vapor deposition concepts involve fluid mechanics, heat and mass transfers associated with both g a s phase and surface chemical reactions. Powerful numerical techniques a r e necessary t o solve t h e relevant equations. Flux-Expert i s a general finite element program which is commercially available ; t h e adaptation of this general purpose software package t o t h e modeling of CVD reactors has been developed a t t h e D-Leti.
The original software c a n easily b e upgraded with additional new equations ; i t i s a fully interactive, menu driven program with built in graphics.
A t t h e present time, our software i s designed t o simulate only t w o dimensionnal axisymmetric p r o b l e m s ( c y l i n d r i c a l c o o r d i n a t e s y s t e m s ) . A t h r e e d i m e n s i o n a l v e r s i o n i s i n progress.
The purpose of this work is t o explore t h e performance of CVD reactors, optimize t h e possibilities of existing reactors, provide users with a helpful tool f o r t h e development of new reactor configurations and t h e prediction of related process parameters.
1
-
INTRODUCTIONThe design of CVD reactors involves cross-disciplinary sciences which will allow a complete description of t h e phenomena 111. In f a c t , t h e performances of CVD a r e depending on a wide range of variables. The geometric configuration of t h e reactor, t h e opCrating conditions a r e intimately related t o t h e kinetics of t h e film growth 11-91. I t i s important t o note that, for t h e most part, t h e design of t h e industrial equipment h a s been made empirically and cannot guarantee optimum conditions 1'61. For these reasons, t h e c u r r e n t trend is t o develop models t o describe t h e deposition process and t h e interdependence of t h e basic physical phenomena. The predictions which come from these models a r e limited (perhaps except for t h e Si deposition) by a n incomplete knowledge of t h e chemical phenomena (chemical reactions e i t h e r in t h e gas phase or on t h e substrate) 111. In order t o reach some insight, like in others studies 11-161, w e have adapted t h e general f i n i t e element code Flux-Expert 1171 t o t h e h e a t and mass transfers CVD problems. We have chosen a general code because CVD problems a r e numerous; t h e code must b e easily modified and upgraded. A t t h e present time, our software i s designed t o simulate only t w o dimensional problems (cylindrical coordinate systems). The t h r e e dimensional adaptation i s in progress, especially for t h e design of tubular reactors used for t h e low temperature deposition of silicon oxide and for t h e modeling of injector geometries.
The purpose of this work is t o provide users (engineers and even students) with (i) a helpfull tool for t h e development of new reactor configurations, (ii) t h e prediction of optimal operating conditions and (iii) general knowledge of CVD phenomena and problems. With built in graphics and fully interaction, this software may easily b e used by a broad range of users.
After t h e description of t h e used code and of the numerical procedures, we present i t s potential uses f o r t h e design of axisymmetrical reactors. This type of single wafer reactors is very a t t r a c t i v e for numerous microelectronic applications/l8/.
Many powerful and large scale, general purpose finite element or finite difference codes have been Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989510
C5-58 JOURNAL DE PHYSIQUE
developed and a r e commercially available. In t h e modeling of solids and s t r u c t u r e t h e interactive graphic accelerates popular use of finite element methods because t h e pre- and post- processing give users e a s y access 1191. For fluid mechanics, h e a t and mass transfer coupled problems, t o our knowledge, only few software packages a r e easily available. The Fidap finite element code (USA) allows solution of almost all incompressible fluid mechanics problems even coupled with h e a t transfer. Phoenics finite difference code (G.B.) allows solution of almost all h e a t and mass transfer problems and we think t h a t i t could solve CVD problems. In France, Modulef and Flux-Expert, general purpose finite element codes, allow solution of some chemical engineering problems.
However, t h e equations t o be solved must b e transformed t o a finite element formulation. We have chosen Flux-Expert. In this software package, some chemical engineering equations which a r e compatible with t h e finite element formulation c a n be solved. Consequently, i t may b e used (i) obviously f o r educational purpose, (ii) t o design r e a c t o r s for known chemical phenomena (iii) t o reach general insight about new phenomena.
3
-
THE EQUATIONS AND THE NUMERICAL PROCEDUREThe CVD process involves momentum, mass, energy and sqecies balance /1,20/. We only consider t h e stationary case-pecause t h e film growth r a t e ( I pm.s- ) is generally slow compared t o t h e g a s 1 m.s ) 1211. As in t h e most cases t h e r e a c t a n t s a r e diluted in t h e c a r r i e r g a s (HZ, is possible t o remove t h e dependence of t h e flow on t h e species balance; t h e field of t h e reactive species involves a negligible e f f e c t on t h e physical properties which c a n b e assumed t o be those of t h e pure c a r r i e r gas. The momentum, energy and mass balance a r e f i r s t solved. We obtain t h e velocities, pressure and temperature fields which become a n input t o species balance equations. The consumption of t h e reacting gas (i.e. t h e deposition kinetic) i s a boundary condition on t h e reacting areas. Generally, for CVD experiments t h e flow is laminar 11-161. Table 1 displays t h e general equations t o b e solved. Because of t h e assumption of dilute r e a c t a n t s t h e h e a t of t h e reaction will b e negligible except in some systems with oxidation (i.e.
SiO deposition). The consumption of t h e reacting g a s by reactions in t h e homogeneous phase is n o t takzn into account in t h e equation (4) but should be for special applications. The deposition c a l c u t a t i o n s w e r e p e r f o r m e d w i t h a n a x i s y m m e t r i c a l g e o m e t r y . T a b l e 2 s h o w s t h e b o u n d a r y conditions. In this example, the reactor walls and t h e wafer susceptor a r e w a t e r cooled. This configuration is desirable to avoid wall deposition. Configurations with insulated walls and/or with inductive heating will not b e presented in this paper for b e t t e r clarity. The wafer
(R
= 5 c m ) i s a t t a c h e d t o t h e heater. The distance between t h e wafer and t h e inlet will b e calledZ
t h e o u t l e t radius 1 and t h e inlet radius L.The physical and chemical properties of t h e mixture results from experimental d a t a o r from statistical calculations /20,22-231. They c a n generally be expressed a s a function of t e m p e r a t u r e and pressure /see for example 211.
Ill C O H M U N EOUATKM FOR THE MASS
IZ1
coNTwm
EQUATION FOR 'I-& MOMPCrUA4 4 4 4
p . (
-
v . g m d v ) = div(i)+
p . g+ +
t 4 + 47 = p . ( g r a d v
+
gradv)-
( W 3 . p . d i v v+
P ) . I 131 C O N T W Y EWATDN FOR TI€ BERGYI41 COMlMICrY E o U A m FOR TI€ COMPONEHT A,
div ( p
. +
V . w ~ ) PP . r v c
%(
&dwAi+ K + . I ~ . & ~ L L ~ T) ) )-Fickian hC-
-
Thermaldiffusion diffusion Table I : General equations
+- +
eq. (R) :
R
= pDAB (grad W A+
DT W A grad LnT)4
-
v
= 0SUBSTRATE
Table 2 : Boundary conditions for a stagnation point flow reactor.
The modeling equations, t h e boundary conditions and t h e variable physical properties form a complex s e t of partial differential equations. The Galerkin finite e l e m e n t method was chosen t o solve this problem. Each equation is projected on polynomial basis functions. A mixed order polynomial approximation was used ; for t h e components of velocity, temperature and concentra- tion t h e order 2 was chosen, for pressure t h e order 1. The stability of t h e method depends on t h e element dimensions and t h e characteristic number which is t h e r a t i o of advection t o diffusion. The limit is R e = 2 on each element. This limit is, a t the present time, acceptable for t h e most part of CVD experiments. The complete s e t form of nonlinear algeabric equations c a n be found in Ref.24.
The finite element method is a powerful means of solving complex equations ; no simplifying assumptions a r e necessary, but extensive computation time is required and many simulations a r e necessary t o reach a general insight 151.
Table 3 summarizes t h e resolution scheme for cold wall reactors.
i
- (a)
IENERGY
E E > I Ryj"b-T (
SPECIES
I
RESULTS(a)
Table 3 : Resolution scheme for cold wall reactor. I
This sofware allows t o calculate deposition r a t e s from deposited flux under t h g following conditions :
*
stationary c a s ; laminar flow*
isothermal/non isothermal reactors*
concentration and thermal gradient driven diffusion*
surface reaction o r diffusion controlled deposition*
any inlet flow profile*
variable physical properties ; axisymmetrical geometryJOURNAL DE PHYSIQUE
4
-
SIMULATION RESULTSOur a i m is not t o show results for a special problem b u t t o demonstrate t h a t our s o f t w a r e can solve a wide range of problems, a t t h e present time, for axisymmetrical cylindrical reactors ;'so, t h e results, a s in some recent studies 11-151, will be displayed with dimensionless characteristic numbers.
Although t h e goal of t h e simulation i s t o predict t h e deposition r a t e and t h e thickness homogeneity of t h e film, we have separated t h e thermal gas flow study from t h e deposition r a t e study. Indeed, t e m p e r a t u r e gradient driven recirculations arising from operating conditions must generally be avoided because they limit t h e deposition uniformity and they c a n bring t h e by-products back into c o n t a c t with t h e wafer. We have numerically explored t h e geometric and operating conditions t h a t would lead t o a flow field f r e e of recir'culations. From t h e operating conditions previously defined, w e have calculated t h e deposition rate.
4-1 Thermal e f f e c t s
For hydrogen c a r r i e r gas, Reynolds numbers (see table 4 for t h e definition) from 1 t o 10 have been chosen f o r t h e c a l c ~ a ~ o n s s ; they r e f l e c t realistic CVD conditions. The different geometries presented a r e not always common b u t they a r e technologically possible. Figure 1 shows a n example of recirculation arising from thermal gradients for a n atmospheric pressure reactor. With these conditions, t h e flow is dominated by a buoyancy driven recirculation cell above t h e substrate. The dimensionless number which may c h a r a c t e r i z e t h e magnitude of this e f f e c t i s t h e ratio Gr/Re2 (see table 4) ; it can b e used a s a diagnostic factor/2/. I t represents the r a t i o of natural convection velocity squared t o t h e forced convection velocity squared. The dimensionless number G (see table 4) defined by Jensen and al. 1211 c a n characterize, for a given R e number, t h e magnitude of the thermal e f f e c t s in relation t o t h e inlet-plate distance Z (note t h a t if
Z=L,
~ r = G . ( A p / p ) . According t o Wahl 121, t h e influence of gravity on t h e gas flow c a n b e neglected if Gr/Re2<Co, where Co is a complex function of t h e previously selected geometric parameters ; nevertheless Co seems difficult t o calculate. I t is t h e reason why we have chosen for t h e stagnation point reactor t o present t h e transition between t h e different flow features with t h e number G ; i t may give some insight more easily.The intent i s t o lower o r avoid thermal recirculation. The first solution would b e t o increase t h e flow r a t e (i.e. t o decrease Gr/Re2) ; at !jmospheric pressure, even with hydrogen carrier gas, high flow r a t e a r e required (more than 5 1.mn STP). The second solution is t o diminish t h e wafer-inlet length (see t h e cubic dependence of C on t h e p l a t e separation 2). We have shown t h a t i t is necessary t o lower t h e a s p e c t r a t i o down t o 0.3 t o avoid recirculation. Figure 2 is an example of existing reactors whose aspect r a t i o have been decreased. The third solution is t o decrease t h e density of t h e g a s and eventually t h e a s p e c t ratio. As hydrogen carrier g a s was chosen, t h e only way t o decrease t h e density is t o reduce t h e pressure (figure 3).
Table 4 : Dimensionless numbers
Fig. 1 ZITypical velocitiy field and i ~ o t h e r m s for H2 flow in a vertical reactor ( I 1.mn a t the mass flow) ; P = 10 Pa
(a) Re = 2 ; A = 1.5 ; IA = 1.1 ; PA = 0.25 ; G = 187000 (b) R e = 2 ; A = 0.65 ; IA = 0.8 ; PA = 0.25 ; G = 12000
Fig. 2 Solution of t h ~ thermal effects : Z is lowered (i.e. G ) 1 1.mn STP ; P = I0 Pa
R e = 2 ; A = 0.3 ; IA = 0.8 ; PA = 0.25 ; G = 1500
JOURNAL
DE
PHYSIQUEFig 3
-
Solution of t h e Y e r m a l e f f e c t s : pressure is reduced (i.e. G and Gr) 1 l.mn-' STP : P = 5.10 P aA diagnostic can b e made using t h e G ratio. For a n inlet flow r a t e of I 1.mn- STP, if t h e r a t i o is 1 lower than 2000 t h e flow is forced convection dominated ; if G i s higher than 2000 natural convection appears. For higher inlet flow r a t e s this critical value is higher. These results a r e in good agreement with t h e literature 11-21.
We a r e able t o predict, for realistic CVD conditions, a flow f r e e of thermal gradient driven recirculation. In general, for low pressure reactors, with low molecular c a r r i e r gas, if t h e aspect ratio i s lower than 1, no recirculation is found. For atmospheric pressure reactors, without using prohibitive flow rates, t h e a s p e c t ratio must be decreased t o avoid t h e thermal effects. It is important t o note t h a t for nitrogen o r argon c a r r i e r gas, i t is much more difficult t o solve this p r o b l e m 1211. T h e t h e r m a l g r a d i e n t i s s t e e p e r a n d t h e i n f l u e n c e of g r a v i t y i s i n c r e a s e d . If t h e substrate i s heated by induction instead of by a heater, thermal e f f e c t s a r e also found in t h e e x i t area. We have shown t h a t by narrowing t h e outlet region (i.e. decreasing t h e magnitude of PA down t o 0.2) recirculation c a n b e lowered o r eliminated. Jensen and al. 121,251 have also shown t h a t t h e thermal e f f e c t s a r e lowered by inverting t h e reactor.
4-2 Film homogeneity
Our a i m i s t o predict t h e condition leading t o a uniform deposition (homogeneity less than 5%) over the wafer. I t - i s obvious t h a t t h e desigi of t h e r e a c t o r - w i l l be all tThe m o r e difficult a s the substrate i s large. Although i t is possible t o obtain a uniform deposition in a mixed flow regime (natural and forced convection) by a proper selection of G o r Gr/Re2, i t is a much simpler task t o specify pure forced convection. The mixing action of the vortices is t o be avoided since mixing will bring t h e by-products of t h e reaction back into c o n t a c t with t h e wafer, r a t h e r than sweeping them out of t h e reactor 161.
The r a t e flow and temperature files acccording t o table 3 scheme, were injected in t h e species b a l a n c e e q u a t i o n s . T h e r e s u l t s h a v e l e d t o o b t a i n c o n c e n t r a t i o n p r o f i l e s a c c o r d i n g t o t h e consumption mode of t h e reacting species. As previous studies /1,10,13/ have shown t h a t thermal diffusion cannot b e neglected in modeling CVD, w e consider only t h e results where thermal diffusion was included. The major e f f e c t of thermal diffusion i s generally t o lower t h e deposition r a t e (i.e. t h e deposited flux on t h e reacting areas).
Numerous kinetic schemes c a n be used (i.e. R i s a function of t h e species concentration; the simpliest form is R = kwA
,.
k is t h e r a t e constant). For diffusion limited deposition, t h e r a t e constant has been chosen s u f f l c ~ e n t l y large ; according t o Moffat and Jensen 1251, i t i s a b e t t e r approach than setting t h e partial pressure of t h e reacting gas t o z e r o a t t h e wafer surface.When surface kinetics control t h e reaction (generally a t low t e m p e r a t u r e and /or pressure) a n uniform distribution ( 5%) was found. When complex kinetics, involving g a s phase reactions, control t h e deposition rate, homogeneity may b e altered. T h e e f f e c t s of unsaturated species a r e n o t described here. With t h e previous results concerning natural convection, several design guidelines o r rules of thumb may b e developed. For example, a contribution t o t h e optimization of t h e selectivity in a tungsten CVD reactor 161 c a n proceed from these trends ; t h e use of (i) a cold wall reactor, (ii) H c a r r i e r gas, (iii) low pressure, (iiii) high g a s velocities leads t o a r e a c t o r with no wall growth, wi4h no recirculation, thus f r e e of by-products in t h e reacting area. I t must b e noted t h a t f o r t h e most p a r t these rules have already been incorporated in commercial r e a c t o r design. In t h e c a s e of tungsten, t h e s e rules a r e required but they a r e n o t sufficient 1261.
In t h e mass transfer limited regime (generally a t higher t e m p e r a t u r e and/or pressure) non linear interactions exist between buoyancy, viscous and inertia t e r m s ; multiple flow fields leading t o uniform films could b e predicted. Figure 4 shows t h e reduced growth r a t e for typical configura- tions. Reactor (a) leads t o a n increase of t h e thickness near t h e o u t e r edge since (i) buoyancy dominates t h e tranport and (ii) t h e diffusion layer i s thinned at t h e h e a t e r edge by a large flow.
For lower values of R e ( R e 0.1) an opposite profile has been found because t h e reactive gas is depleted along t h e surface. R e a c t o r (b), due t o i t s small a s p e c t r a t i o is forced convection dominated ; t h e film thickness decreases away from t h e c e n t e r of t h e wafer, e x c e p t near t h e edge due t o t h e bending of t h e flow. R e a c t o r (c), a low pressure reactor, obviously leads t o t h e more uniform profile since t h e diffusion coeficient is increased and t h e deposition r a t e lowered.
I t i s important t o note t h a t t h e most p a r t of a c t u a l vertical CVD r e a c t o r s rarely have t h e wide inlets (IA=0.8 t o I ) used in t h e above calculations. Generally, a thin t u b e a s inlet is used and t h e a s p e c t ratio i s o f t e n g r e a t e r than 1. With t h e s e configurations, homogeneity is never achieved /I/.
We have shown, with selected examples, some relationships existing between t h e performances (i.e.
uniformity) and t h e operating conditions (temperature, flow patterns, r a t e limiting process
...I
; o t h e r s technological problems, a s for example t h e t e m p e r a t u r e homogeneity of t h e wafer w e r e n o t taken into a c c o u n t 1261.Fig 4
-
Homogeneity of t h e film 5-
CONCLUSIONHOMOGENEITY
1
The presented calculations depict t h e influence of t h e design on t h e technological performances of axisymmetrical reactors. Numerical models may provide a complete description of t h e transport processes. But, even with t h e best numerical models, t h e r e is t o o much lack of d a t a in growth chemistry and in t h e values of physical constants t o r e a c h t h e absolute gro6tth r a t e s a priori.
Therefore, they allow t o r e a c h a general insight i n t o t h e physical processes and t o predict t h e uniformity of t h e film. This c o d e obviously cannot solve all t h e problems found in CVD but i t may b e a starting point for f u t u r e development.
C5-64 JOURNAL
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PHYSIQUEREFERENCES
/ I / Jenskn, K.F., Chem. Eng. Sci.,
frZ
(1987) 923 /2/ Wahl, G., Thin Solid Films,g
(1977) 1 3/3/ Moffat, H.K., Jensen, K.F., J. Electrochem. Soc.,
135
(1988) 459 /4/ Kisker, D.W., Mc Kenna, D.A., Jensen, K.F., Mat. Lett.,6
(1988) 123 /5/ Field, R.J., Scholz, F., J. Crystal Growth,88
(1988) 371/6/ Raupp, G.B., in Wells, V.A. (ed.) Tungsten and O t h e r Refractory (Metals for VLSI Applications 111, Material Research Society, Pittsburg, (1988) 1 5
/7/ Vinante, C , Bertrand, J., Couderc, J.P., in R. P o r a t (ed.) Proc. 6 t h European Conference on Chemical Vapor Deposition, Jerusalem, Israel, 29 march-3 april 1987, (1987) 42
/8/ Coltrin, M., Kee, R.J., Miller, J.A., J. Electrochem. Soc.,
129
(1982) 1627 /9/ Rebenne, H., Pollard, R., J. Electrochem. Soc.,132(8)
(1985) 1932/ l o / Rosemberg, F.,in G.W. Cullen, J.H. Blocher (eds.), Proc. 10th International Conference on Chemical Vapor Deposition, The Electrochemical Society Pennington, NJ,
87-X
(1987) 11 /11/ Wahl, G., S c h m a d e r e r , F., H u b l e r , R., W e b e r , R., i d e m r e f . 10,87-8
(1987) 42 /12/ L e e P., Mc Kenna D., Kapur D., Jensen K.F., J. Crystal Growth,77
(1986) 1201131 Rebenne, H., Pollard, R., J.Am. Cer. Soc.,
70(12)
(1987) 9071141 Roenigk, K.F., Jensen, K.F., J. Electrochem. Soc.,
134(7)
(1987) 1777 1151 Wilke, T.E., Turner, K.A., Takoudis, C.G., Chem. Eng. Sci., L11(4) (1986) 643 /16/ He, Y., Sahai, Y., id ref 10,87-8(1987) 1931171 Flux-Expert, DT21, Meylan, France
/18/ Wahl, G., Schmaderer, F., Huber, R., in P o r a t (ed.) Proc. 6th European Conference on Chemical Vapor Deposition, Jerusalem, Israel, 29 march-3 april 1987, (1987) 50
1191 N. K i k u c h i , F i n i t e E l e m e n t M e t h o d in M e c h a n i c s , C a m b r i d g e U n i v e r s i t y P r e s s , 1987 1201 Bird, R.B., S t e w a r t , W.E., L i g h t f o o t , E.N., T r a n s p o r t P h e n o m e n a , Wiley, NY, 1 9 6 0 1211 H o u t m a n , C., G r a v e s , D.B., J e n s e n , K.F., J. E l e c t r o c h e m . Soc.,
133
(1986) 9 6 1 1221 Reid, R.C., Sherwood, T.K., The properties of Gases and Liquids, Mc Graw Hill, NY, 1966 1231 Encyclopedie des G a z (1'Air Liquide), Elsevier, 19761241 Mariaux, S., Rapport DT21, Meylan, France, January 1988 1251 Moffat, H., Jensen, K.F., J. Crystal Growth,
77
(1986) 108 1261 Papapietro, M., Arena, C., Noel, P., t o be publishedp : dynamic viscosity
U : kinematic viscosity
: thermal diffusivity : dilatability Variables and physical quantities
o (index) : properties a t 300 K, a t t h e inlet Ai (index) : r e a c t i v e dilute g a s
B (index) : leading g a s
R
: reactive variableT : t e m p e r a t u r e W ~ i : mass fraction of Ai
V : velocity vector
Vr : radial velocity vz : axial velocity
g : acceleration due t o gravity
k : thermal conductivity
P : pressure
C : h e a t capacity
"
AB : binary diffusion coefficient"T : thermal diffusion coefficient
M : molar mass
R : ideal gas law constant
Dimensionless numbers
Gr : Grashof
R e : Reynolds
Fr : Froude
G : Gravity number
Sh : Sherwood
A : a s p e c t ratio J A : inlet aspect ratio PA : pumping a s p e c t ratio R e a c t o r dimensions
Z : inlet-substrate length
R : substrate radius
L : inlet radius
I : radial length of the o u t l e t
