The Chemical Engineering Journal, 54 (1994) 137-145 137

The stability of a compressible fluid with variable transport properties in a horizontal chemical vapour deposition reactor

Steven Bradshaw* and Henk Viljoen? Department of Chemical Engineering, University of Stelknbosch, Stelknbosch 7600 (South Ama)

Cohn Wright Centre for Nonlinear Studies and Department of Computational and Applied Mathematics, University of the Witwate-rsrand, Wits 2050 (South Africa)

(Received August 2, 1993; in final form January 4, 1994)

Abstract

A linear stability analysis was performed to determine the point of onset of longitudinal rolls in a horizontal duct of rectangular cross-section with a heated bottom surface, approximating a typical horizontal chemical vapour deposition reactor. The fluid was considered to be compressible with temp~rat~e-dependent transport properties. It was found that nitrogen is a less stable carrier gas than helium. The effect of the temperature-dependent transport properties was to render less stable the flow in the duct with a bottom surface temperature of 700 K in comparison with the duct with the bottom surface at 1000 K. This interesting result emphasizes the importance of taking into account the temperature dependence of fluid properties. Comparison with the Bo~s~esq-fluid coopt-prope~es model showed that that model can be considerably in error, depending on the reference conditions for property evaluation and on the particular carrier gas.

1. Introduction

Chemical vapour deposition (0) reactors are used to produce a wide variety of thin uniform 6lms for electronic or optical devices, and the literature is rich with studies on CVD. In recent reviews Hess et al. [ I] and Jensen 12 1 provide overviews of the micro-reaction engineering involved in CVD. The need for uniformity of the deposited film has led to an interest in the flow profiles in CVD reactors, with emphasis being given to the effects of secondary flows on the uniformity of the deposit. The horizontal CVD reactor, while now little used for produ~ion purposes, offers a convenient configuration for fim- dame&al studies in CVD processes, and particularly for flow studies. In such a reactor, both natural convection and shear mechanisms can destabilize the laminar flow and cause secondary flows, which can adversely affect the uniformity of the deposit

*Author to whom correspondence should be addressed. tPresent address: Depment of Chemical Engineering, Uni-

versity of Nebraska-Lincoln, PO Box 880126, Lincoln, NE 68588- 0126, USA.

[ 1, 3, 41, and it is clear that the determination of the point of onset of secondary flows in horizontal CVD reactors is of some engineering importance.

In atmospheric pressure CVD reactors, the con- centration of reactive species is often very low, which means that concentration effects can be ne- glected [ 1, 5-71. This allows for a major simplifi- cation in the analysis of the flow patterns in the horizontal CVD reactor, allowing attention to be restricted to the study of flow instabilities in a horizontal duct with a heated bottom surface.

A first step in such an analysis would be to consider the fluid dynamics in the entrance region of the reactor. On entering the heated susceptor region, the fluid starts to heat up, and in the absence of natural convection it will develop a new linear temperature profile within the thermal entry length. It can be shown that horizontal temperature gra- dients will always be present in the entrance region, a s~cient condition for the existence of secondary flows. Indeed, two-dimensional results for the epi- taxial hot-wall reactor showed that rather small temperature differences between successive iso- thermal zones can cause backflows [8]. Depending

0923-0467/94/507.00 0 1994 Elsevier Science S.A. AU rig&s reserved SSDI 0923-0467(94)00205-C

138 S. Bradshaw et al. / Stability of a compressible &id

on the kind of reactor, these secondary flows will be confined to the entrance region or exist through- out the reactor [9, lo]. For discussions on entrance effects in rectangular channels and CVD reactors the interested reader is referred to inter alia refs. 11-14.

In the thermally developed region of the CVD reactor, the first possibility for secondary flows is the development of longitudinal rolls. This has been confirmed experimentally by Giling [ 151 and nu- merically by Moffat and Jensen [5] and Houtman et al. [7]. In the analysis of longitudinal rolls, only the span-wise and vertical perturbations come into effect and the resulting eigenvalue problem is iden- tical with the Rayleigh-Benard problem [ 161. For the case of incompressible flow, this problem has been extensively studied by several authors, for example Luijkx and Platten [ 171. LuQkx and Platten showed that channels of smaller aspect ratio were less susceptible to secondary flows. For an aspect ratio of unity they found Ra,=2585, while for an aspect ratio of 5, Ra, was found to be 1779. It appeared from their results that Ra, was asymptoting rapidly to the infinite parallel plate value of 1708.

Moffat and Jensen [ 51 performed a numerical study of a horizontal CVD reactor. They showed the importance of the boundary conditions at the lateral walls and observed longitudinal rolls at su- percritical values of the Rayleigh number Ra. These critical values are associated with the classical Ray- leigh-Benard problem and for adiabatic walls the value of Ra results from an eigenvalue problem. In the case of conducting lateral walls, buoyancy-driven flows exist for all non-zero Rayleigh numbers and the associated boundary value problem is often studied numerically. Evans and Greif [4] studied travelling transverse wave instabilities numerically in a horizontal CVD reactor, and in a later paper [6] made a numerical study of unsteady three- dimensional mixed convection in a horizontal CVD reactor. The horizontal reactor configuration was also analysed by Kleijn and Hoogendoorn [ 111, who obtained numerical solutions to two- and three- dimensional models. The two-dimensional model was found to give reasonable results for reactors of large aspect ratio operated at sub-critical Rayleigh numbers. For other situations, the three-dimensional model with careful attention to the side-wall de- scription was necessary in order to obtain agreement with experimental results.

Under conditions typically found in CVD reactors, the Boussinesq approximation is not normally valid and the transport properties of the reaction mixture vary significantly because of the large temperature gradients [4-6, 181. Most of the existing analyses

of fluid flow phenomena in CVD reactors have invoked the Boussinesq approximation and assumed the transport properties to be constant. Among the exceptions are the recent numerical studies [4-6, 111. A number of related studies have also included the effects of variable transport properties, for ex- ample refs. 19-23. However, little attention appears to have been given to determination of the onset of secondary flows in CVD reactors when the fluid transport properties are considered to be temper- ature dependent.

To fill this gap in the literature, in the present study a duct of rectangular cross-section with a heated bottom surface (representing the susceptor of a CVD reactor, and hereafter referred to as the susceptor) is considered, and a linear stability anal- ysis is used to calculate conditions for the onset of longitudinal rolls for a compressible fluid with temperature-dependent properties. The limiting case of inllnite parallel plates is considered lirst, which can usefully provide conservative order of magnitude estimates of design parameters. This is particularly pertinent as many CVD reactors typically have aspect ratios of about 5 [ 41. This analysis is followed by the analysis of a duct of finite aspect ratio. For both cases, neutral stability curves are presented for two different gases, helium and nitrogen, and two different susceptor temperatures, T, = 1000 K and T, = 700 K. These results are compared with the incompressible Boussinesq model. The neutral stability curves can be used as guidelines for the design of horizontal CVD reactors, and also provide insight into the effect of temperature-dependent transport properties.

2. Model formulation

Consider a horizontal duct of rectangular cross- section with a heated lower surface (the susceptor), and room temperature on the top surface (see Fig. 1). To make the analysis more tractable the following assumptions are made.

(i) The flow entering the reactor is at room temperature T, and hydrodynamically and thermally fully developed.

(ii) The reactive species are present in a very low concentration so that heat effects due to the chemical reaction and concentration-driven natural convection can be safely neglected [ 5, 71.

(iii) Viscous dissipation effects are neglected. (iv) Thermal diffusion effects are neglected. (v) The susceptor is horizontal. (vi) The susceptor is kept at a constant temper-

ature T,.

S. Bmdshuw et al. / Stability of a compressible Juid 139

Z=B

isothermal top wall

Z=-B

y=-A //

Gas flow

y=A

Fig. 1. Schematic diagram of the system.

(vii) The reversible rate of energy change due to expansion or compression is negligible.

(viii) Axial pressure gradients are decoupled from the transverse gradients. The pressure field may be represented as the sum of a term driving the axial flow only and a term describing the variation of pressure about the mean [5].

(ix) Inertial terms in the momentum equation are neglected.

Only longitudinal rolls are considered, restricting the analysis to the span-wise and vertical coordi- nates. The equations are made dimensionless with the following scales: length H/2 =B, velocity v,/B, time B ‘/v,, density P,,,M/RT,, pressure P,MvzI BRT,. The dimensionless temperature was deflned 8 = (T - T,)/(T, - TJ. Under the assumptions given above, the following set of dimensionless equations describes the system (see Appendix A):

continuity equation

$ + v*pu=o

momentum balance

- = -VP-- V-puu- V-r+pGa at

where

(2)

energy balance

ae at +u.ve= 1

- 1 V-(~(~)WI imm

(3)

(4)

equation of state

p= 3 T (5)

Conditions of no-flow and no-slip were assumed on all bounding walls, while the upper and lower walls were assumed to be isothermal, as discussed above. The side walls for the duct of finite aspect ratio were assumed to be adiabatic. Boundary con- ditions for the cases of infinite and finite aspect ratio are given explicitly in Sections 3 and 4 re- spectively. In eqns. (3) and (4),f(f3), g(e) and h(0) are second-order polynomials representing the tem- perature dependence of the transport properties Y, C, and k respectively. Second-order polynomials were fitted to v, C, and k data, which had been normalized by v,, C, and k,, for the temperature range 200-1000 K [24], i.e.

Y = urf(e) OW

c,= cd(e) C6b) k=k&(O) (6~)

The subscript r refers to the reference temperature which was chosen as the temperature of the upper wall of the channel (300 K). Note that other con- venient relationships could have been chosen to represent the temperature dependence of the trans- port properties, for example power law relationships [ 2 1 ] or Sutherland formulae [ 20 1. F’itting of the temperature dependence of transport properties with second-order polynomials was also done by Kleijn and Hoogendoorn [ 111. Values of v,, which are useful for converting the results of Sections 3 and 4 from Gr, to B, the duct half-height, are for nitrogen v,= 1.5686~ 10e5 m2 s-r, and for helium v,= 12.263~ lo-’ m2 s-l.

The basic state (PO, 0, = (1 - 2)/2, v. = 0, w. = 0) is perturbed as follows. It is assumed that 6 is small enough to allow a linear theory to be for- mulated:

P=P,+6P,+***** (7a)

e=80+6e,+a..ms Ub) v=vo+&l, + * *. . . (7c)

w=wo+&n* +. * * * * U’d)

Gr=Gr,+6Gr,+...*. (7e)

Inserting eqns. (7a)-(7e) into eqns. (l)-(4) and retaining terms of order 6 yields the first-order perturbation equations:

continuity equation

2ae, k, +0 --PO% +Poay +w1 .& - +po2 =o (8)

140 S. Bradshaw et al. / Stability of a compressible &id

v equation

(9)

w equation

(10) 8 equation

30, at+w’d&z 1

prpog(eo)

where

h(e) =a + b(e, + 6e1) + c(e, + selJ2

=a+beo+ce~+2@el+2ceoel)

= h(B,) + 6h(8,) (12) An analogous equation also holds for g(0).

In order to solve the model, the Galerkin method [25] was applied in the horizontal dimension. This reduced the system to an eigenvalue problem in the vertical dimension x, and the eigenvalue problem was solved by a Chebyshev pseudo-spectral method [26] to determine the conditions for the onset of longitudinal rolls.

3. Onset of longitudinal rolls for infinite parallel plates

In this section, critical parameter values were calculated for the onset of secondary flows when the fluid was considered to be compressible. The calculations were performed for two different carrier gases, helium and nitrogen, and for two different

susceptor temperatures, T, = 1000 K and T,= 700 K.

By considering the problem of infinite parallel plates, it is possible to assume purely periodic trial functions in the horizontal direction for vl, wl, O1 and PI. These trial functions are

e1 = %4 SW&Y) exp(A0

w1 = w(x) sir&~) exp(At)

v1 = V(z) cos(k,y) exp(ht)

Pl =P(@ r(9) exp(At)

Wa)

(13b)

(13c)

(13d)

Note that it is not necessary to specify the form of the 3 dependence of the pressure perturbation, as this term cancels in the analysis. Substitution of trial functions of this form into the perturbation equations and application of the Gale&in method in the horizontal direction followed by manipulation gives two ordinary differential equations in terms of W(x) and a(z) and the parameters Gr, Pr and E, where E = (T, - T,.)/T,.

The boundary conditions to be satisfied on the horizontal surfaces are

flfl)=O

W(i-l)= ; [W(&-l)]=O

(1 da)

(14b)

Note that boundary conditions for only two trial functions need be specified. The two equations in W(z) and 19(z) constitute the eigenvalue problem, which on application of a Chebyshev pseudo-spectral method yields the (2N + 2) X (Uv f 2) non-linear gen- eralized eigenvalue problem:

CX= ABX+ A’GX (15)

The AZ term in eqn. (15) arises because the term +,/at was retained in the continuity equation.

Two different carries gases, He and Na, were considered, as examples of typical light and heavy carrier gases, and the critical Grashof numbers were calculated for the onset of secondary flows. The calculations were performed for TS = 1000 K and T,= 700 K. The results of these calculations are shown in Pig. 2. It can be seen that Gr, occurs at Icy= 1.5 for both helium and nitrogen. Of interest is the fact that for both helium and nitrogen, the duct with T, = 1000 K shows the onset of secondary flows at a higher value of Gr than the duct with T, = 700 K. The increased stability with higher sus- ceptor temperature, which is not seen with the incompressible constant-properties model, is due to the increase in gas viscosity with increasing temperature. This result, which has been commented

S. Bradshaw et al. / Stability of a compressible fluid 141

30 000

Gr

20 000

10 000

0 0 1 2 3 4 5

kY Fig. 2. Neutral stability curves for the channel of infinite aspect ratio: - helium T, = 1000 K, - - - nitrogen 7’. = 1000 K, - - helium T, = 700 K, . . . . . nitrogen T. = 700 K.

TABLE 1. Comparison of minimum reactor half-heights for the onset of secondary flows for different carrier gases for the infkite parallel plate model, T.= 1000 K (t-=2.3333)

Carrier gas Value of B at which secondary flow appears

(ml

Compressible temperature dependent

Boussinesq fluid

Reference Reference susceptor top plate

Helium 0.0184 0.0123 0.00322 Nitrogen 0.00440 0.00548 0.00230

on by Giling [ 151 and Moffat and Jensen [5], high- lights the importance of taking into account the temperature dependence of transport properties.

Of interest is the fact that for a given T,, the ratio of the Grashof numbers for the two carrier gases was constant over the range of wave numbers considered.

The minimum reactor half-heights for the onset of secondary flows are shown in Tables 1 and 2. This alternative presentation of the results provides easier comparison of stability, for engineering pur-

TABLE 2. Comparison of minimum reactor half-heights for the onset of secondary flows for different carrier gases for the infinite parallel plate model, T,= 700 K (E= 1.3333)

Carrier gas Value of B at which secondary flow appears

(m)

Compressible temperature dependent

Boussinesq fluid

Reference Reference susceptor top plate

Helium 0.0163 0.00969 0.005 14 Nitrogen 0.00389 0.0105 0.00246

poses, between different carrier gases than the graph- ical presentation of Fig. 2. Note that v, appears in the definition of Gr, thus preventing direct com- parison on the basis of Gr alone. Also shown in the tables are the results for the Boussinesq in- compressible model. For the Boussinesq model the usual dehnition of Ra is

Ra = gPG”s - TJW13 (161

KV

In order to calculate B for a Boussinesq fluid, the right-hand side of eqn. (16) was equated to 1708. Upper and lower bounding values of B were then obtained by choosing the reference temperature for the evaluation of fluid properties as either the sus- ceptor or upper plate temperature. These values are reported in the tables. (Note that other choices of reference temperature could be made, to give greater or lesser weight to the hot and cold wall temperatures.)

From Tables 1 and 2 it can be seen that nitrogen is a considerably less stable carrier gas than helium, although this result is not immediately obvious from Fig. 2. This result is expected in view of the lower kinematic viscosity of nitrogen. This con6rm.s the experimental results of inter alia Curtis and Dis- mukes [27], van de Ven et al. [3] and Giling [15], who used interference holography to show that horizontal reactors with Ar or N2 were less stable than reactors with Hz or He.

From Tables 1 and 2 it is also apparent that the Boussinesq incompressible model gives a conserv- ative result for helium, in that the reactor half- height B with transport properties evaluated at either the susceptor or the top plate is always less than the value calculated from the compressible fluid model with temperature-dependent properties. This is not true for nitrogen, for which the values of B with transport properties evaluated at the susceptor and the top plate spanned the value calculated for the compressible fluid model. This means that the

142 S. Bradsh.aw et al / Stability of a campressibk J&&i

Boussinesq model for nitrogen is conservative if the properties are evaluated at the temperature of the top plate (i.e. ambient temperature), but if they are evaluated at the susceptor temperature, the value of B is not conservative. These observations once again highlight the importance of taking into account the compressibility of the carrier gas, and the temperature dependence of the transport prop- erties, when designing CVD reactors. The results of the analysis presented in this section can be used to provide conservative estimates for design parameters for horizontal CVD reactors and, as many reactors of this type typically have aspect ratios of 5, provide reasonable estimates of maximum safe duct height.

4. Onset of longitudinal rolls for a duct of finite aspect ratio

For a duct of finite aspect ratio, conditions of no flow and no slip are applied to all four walls, while the horizontal surfaces are again maintained at isothermal conditions and the vertical walls are assumed to be adiabatic. The boundary conditions are as follows:

vertical walls

v,(*y, z)=O (no flow) (I 7a)

w1 ( -t y, z) = 0 (no slip) (I 7b)

z ( + y, Z) = 0 (adiabatic walls) (I 7c)

horizontal walls

vl(y, + 1) =0 (no slip) (I8a)

w,(y, * 1) = 0 (no flow) (I 8b)

8(y, 1) = 0 (ambient temperature) (18~)

Q, - 1) = 0 (susceptor) (I8d)

The basic solution satisfies these conditions. The first-order term in the temperature perturbation (eqn. (11)) on the boundaries is

&(*Y,Z)=&(y, *1>=0 (1%

Consideration of the first-order perturbation form of the continuity equation (eqn. (8)) shows that the following condition must be satisfied:

M&r,z)= $ [v1(+r,z)l=O (20)

A suitable trial function for v1 satisfying this con- dition is given by Chandrasekhar [ 161. It is also

known that when vi reaches a maximum, w1 must be 0, and vice versu. This condition, together with the no-slip condition, can be satisfied by choosing the function representing the y-variation of w1 to be the derivative with respect to y of the function representing the y-variation of vi. A further condi- tion that should be applied is that 19~ should be in phase with wl. This can be satisfied by choosing the same trial function to represent both w1 and &. Suitable trial functions are

for an odd number of rolls

e1 = fl4 sinhmy .

cash my + sm exp(ht)

cos my 1 (2Ia)

VI =V(z) (

cash my

cash my - sY

1 exp(At) (21b)

w,-,qx) exp00 WC)

where m is the root of the transcendental equation

tanh myftan my=0 (22)

for an even number of rolls

(23a)

- - G=)

(23~)

where n is the root of the transcendental equation

coth ny-cot ny=O (24)

As for the case of the infinite aspect ratio, note that it is not necessary to specify the form of the pressure perturbation. The number of rolls in the duct is determined by the value of m or n.

When the trial functions (eqns. (21)-(24)) are inserted into eqns. (8)-( 11) and the Galerkin method applied in the span-wise direction, an eigenvahre problem in the z-direction results, as described in Section 3. The eigenvalue problem was solved by the Chebyshev pseudo-spectral method.

Values of Gr, for the onset of secondary flows were calculated for two different carrier gases, He and Nz, for two different susceptor temperatures T,= 1000 K and T,= 700 K, as was described in Section 3. The calculations were performed for 2.5 < ye 8, where y is the aspect ratio of the duct. Gr, for each y was obtained by varying m or n, i.e. the number of convection cells, and selecting the result with the smallest Gr,. The results of these

S. Brad&au, et al. / Stability of a compressible fluid 143

calculations are shown in Figs. 3 and 4. From these figures it can be seen that Gr, decreases with increasing aspect ratio as expected. Calculation of

20 000

17 500

Gr

n=2

i \ m=3 \ \

15 000

12 500

10000

_

I -

2

‘t, n=2 \ \ n=4 \ \

-\ \

m=.5 \ ‘1, m=3

\ --\ y,,n=4

--_\ ,_ m=5 ----

/ I I / /

3 4 5 6 7 8 Y

Fig. 3. Neutral stability curves for the rectangular duct, T, = 1000 K: - helium, --- nitrogen, m and n are the numbers of convection cells.

8000 1

‘\ n=2

7000

Gr

6000

5000 \ \

‘-_\ \ n- -4 ‘\.__. ._ m=5 --__

4000 ~ , , I I I

2 3 4 5 6 7 8 Y

Fig. 4. Neutral stability curves for the rectangular duct, T, = 700 K: - helium, - -- nitrogen, m and n are the numbers of convection cells.

B, the half-height of the duct at which natural convection first appears, indicates that nitrogen is a less stable carrier gas than helium, confirming the result discussed in Section 3. For the purposes of engineering calculations, B is a more convenient parameter than Gr, and can be simply obtained from the latter by using appropriate values for the reference kinematic viscosity v,, given in Section 2, and the dimensionless temperature difference E.

Comparison of the results shown in Figs. 3 and 4 with the results of Luijkx and Platten [ 171, which were obtained for a Boussinesq fluid with constant properties, showed the same trends as discussed in Section 3. For helium, the reactor half-height B at the onset of natural convection is always predicted to be lower with the Boussinesq model, irrespective of whether the transport properties are evaluated at the susceptor temperature or at the top wall temperature. Thus, the Boussinesq constant-prop- erties model gives conservative results, although its results can be considerably in error. As an example, for T,=lOOO K (~=2.3333) and r=3, the model in this study predicts that natural convection will commence for B = 0.0226 m. For the Boussinesq model with properties evaluated at the susceptor temperature, the critical value of B for these con- ditions is B = 0.0127 m, while if the properties were evaluated at the top temperature the result would be B= 0.00330 m. Indeed, such a large range of values prevents the presentation of this comparison graphically.

For nitrogen, the value of B obtained from the Boussinesq model with properties evaluated at the susceptor temperature is larger than for the com- pressible fluid model of this study, while B is smaller if the properties are evaluated at the top wall temperature. Using as a comparison once more the case T, = 1000 K, y = 3 one finds that the model of this study predicts B=0.0054 m, while the Bous- sinesq model predicts B = 0.0057 m for properties evaluated at the susceptor and 0.0024 m when properties are evaluated at the top wall temperature. Thus one can seethat the Boussinesq model agrees reasonably well with the compressible model for property evaluation at T,, while agreement is less good for the other case. However, one can see that the Boussinesq model does predict an unsafe reactor half-height for property evaluation at T,. This com- parison again emphasizes that great care should be taken when designing CVD reactors if the fluid is modelled as Boussinesq-incompressible with con- stant transport properties.

Comparison of Figs. 3 and 4 with Fig. 2 shows that the neutral stability curve for the finite aspect ratio duct approaches the infinite parallel plate

144 S. Bradshaw et al. / Stability of a compressible j&id

asymptote rather slowly. This is possibly due to the fact that only a single-term trial function was used, rather than an infinite series. This was done to make the problem more tractable, as the eigenvalue prob- lem even for the single-term trial function is al- gebraically very complicated. In view of the possible inaccuracy introduced in this way, the results from the infinite parallel plate problem of Section 3 could be used to give conservative design parameters for ducts of finite aspect ratio. As an example, for a duct of aspect ratio y= 8, using He with T, = 700 K, the maximum half-height of the duct is B = 0.0 184 m, which can be compared with the result for the iniinite parallel plate problem for the same carrier gas and susceptor temperature, B= 0.0163 m. Typ- ical horizontal CVD reactors have aspect ratios of approximately 5, and for this value the critical parameter value for the onset of longitudinal rolls is close to the value for infinite parallel plates. This indicates that the value of B calculated for the parallel plates could reasonably be applied to a typical CVD reactor, and would give a conservative estimate of the allowable reactor half-height.

5. Conclusion

Linear stability analyses have been performed to determine the critical parameter values for the onset of longitudinal rolls between infinite parallel plates and in horizontal ducts of rectangular cross-section with heated lower surfaces. The fluid has been treated as compressible and with temperature-dependent properties. The practical implication of these results is to give an indication of the maximum allowable duct height below which natural convection will not occur.

Comparison with the results from the Boussinesq incompressible-fluid constant-properties model showed that the results from that model can be considerably in error and that account should be taken of the compressibility of the fluid and tem- perature dependence of the fluid properties. For helium, it was found that the Boussinesq-fluid con- stant-properties model gave conservative results for the maximum allowable duct height, irrespective of whether the transport properties were evaluated at the susceptor or top plate (ambient) temperature. For nitrogen, evaluation of transport properties at the top plate temperature gave conservative results, while the opposite was true when property evaluation was done at the susceptor temperature.

Of particular interest was the result that the effect of the temperature-dependent transport properties was to render less stable the flow in the duct with

a bottom surface temperature of 700 Kin comparison with the duct with the bottom surface at 1000 K. This interesting result emphasizes the importance of taking into account the temperature dependence of fluid properties.

Acknowledgment

Dr. J.J. Thiart provided helpful comments.

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Appendix A: Nomenclature

it B B c C

_%)

g g

G Ga Gr

coefficient of h(e) etc., eqn. (12) coefficient of h(e) etc., eqn. (12) half-height of reactor (m) matrix of coefficients, eqn. (15) coefficient of h(B) etc., eqn. (12) matrix of coefficients, eqn. (15) heat capacity (J kg-’ K-‘) function describing temperature depen- dence of V, eqn. (6a) acceleration due to gravity (m” s- ‘) gravitational acceleration vector (O.O,g) (m” s- ‘) function describing temperature depen- dence of C,, eqn. (6b) matrix of coefficients, eqn. (15) Galileo number, Ga =gB3/$ Grashof number, Gr = Eg@/yz

h(e)

H k

k

k, m iI4

Ii

PC4

P

Pill Pr

r(y)

R Ra t T

u, v, w

u

w4

W(z)

x7 Y, x X

function describing temperature depen- dence of k, eqn. (6~) height (m) thermal conductivity (W m-l K- ‘) unit vertical vector horizontal wave number wave number molecular weight (kg krnol- ‘) wave number number of collocation terms in pseudo- spectral method part of trial function for PI dependent on z dimensionless pressure mean pressure (Pa) Prandtl number, Pr=C, h q/k,

part of trial function for P, dependent on

Y universal gas constant (J kmol- ’ K- ‘) Rayleigh number, eqn. (16) dimensionless time temperature (K) axial, span-wise and vertical coordinates of velocity velocity vector part of trial function for v1 dependent on z part of trial function for w1 dependent on z axial, span-wise and vertical coordinates vector [IV, aIT, eqn. (15)

Greek letters

: aspect ratio y/z perturbation parameter

&j Kronecker delta E (T, - T=)lTr e dimensionless temperature, 8 = (T- TJ/

(T, - Tr) J%) part of trial function for el dependent on

z

; thermal diffusivity (m2 s- ‘) eigenvalue

Y kinematic viscosity (m2 s- ‘) P dimensionless density 7 stress tensor (kg se2 m-‘)

7ij component of stress tensor, eqn. (3) (kg sm2 m-‘)

Subscripts C critical value r reference state (300 K) S susceptor 0 basic solution 1 perturbation variable