European Congress on Computational Methods in Applied Sciences and Engineering
ECCOMAS 2000
Barcelona, 11-14 September 2000
c
ECCOMAS
A FINITE-VOLUME SOLUTION PROCEDURE FOR HEAT
AND MASS TRANSFER PROBLEMS COUPLED WITH
HETEROGENEOUS CHEMICAL REACTIONS
M. Selder , L. Kadinski
Institute of Fluid Mechanics, University of Erlangen–Nuremberg
Cauerstr. 4, D–91058 Erlangen, Germany
phone: +49-9131-761244, fax: +49-9131-761275
e-mail: [email protected]
Key words: Computational Fluid Dynamics, Non-linear Partial Differential Equations,
Finite-Volume Method, Multigrid-Technique, Heat and Mass Transfer, Heterogeneous
Chemical Reactions, Crystal Growth.
Abstract.
A modeling approach for the simulation of fluid flow problems including
heat transfer, mass transfer and heterogeneous chemical reactions is presented. The fluid
flow is described mathematically by the conservation equations for mass (continuity equation), momentum (Navier–Stokes equations), energy and chemical species resulting in a
coupled set of partial differential equations. Heat transfer by radiation and species generation/consumption by heterogeneous chemical reactions is accounted for by an appropriate
formulation of the boundary conditions.
The governing partial differential equations are discretized on non-orthogonal blockstructured grids. The equations are solved sequentially by a Finite-Volume algorithm with
a colocated arrangement of the variables. Non-linearities and coupling of variables is
accounted for by an iterative solution procedure. A multi-grid algorithm is used to speed
up convergence.
The model is used to simulate growth processes in inductively heated growth reactors.
Results of the physical vapour transport (PVT) growth of silicon carbide (SiC) are given
to demonstrate the capabilities of the method and to analyze its efficiency. Physical phenomena involved in the growth process are investigated and discussed.
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M. Selder, L. Kadinski
1
Introduction
The numerical modeling of crystal growth processes is an important tool in the development of growth conditions designed to produce crystals matching the high industrial
demands on crystal quality and substrate prices. The development of reliable mathematical models is often an enormous task as the physical and chemical processes involved in
crystal growth are highly complex and mostly coupled. Physical phenomena which are
typically involved in crystal growth are generation of heat sources, radiative heat transfer,
heat conduction, convection, diffusion, and chemical reactions.
As the mentioned processes are usually coupled, a numerical solution procedure very
often requires a large amount of calculation time. For that reason, the development
of effective solution algorithms is a prerequisite for the wide employment of simulation
techniques. In this paper, a modeling approach for the simulation of typical crystal growth
problems is introduced, and methods for the improvement of its efficiency are discussed.
2
Modeling Approach
A reactor configuration consisting of solid and gas parts is considered. In the solid
parts, heat transfer by conduction and heat generation by external sources are taken into
account. In the gas phase, heat transfer by convection, conduction and radiation and
mass transfer by diffusion and convection are modeled. The mass transfer problem is
coupled with the generation/consumption of chemical species by heterogeneous chemical
reactions. The gas flow is assumed to be sufficiently slow to use the model of low Mach
number flows, i.e. the gas velocity is small compared to the speed of sound [1].
The mathematical model of the flow used in this work consists of the stationary conservation equations for mass, momentum, energy and chemical species [2]:
=0
∇ · (ρ V)
V)
= −∇p + ρg + 2 ∇ · (µ Ṡ) − 2 ∇ (µ ∇ · V)
∇ · (ρ V
3
T) = ∇ · (λ ∇T)
cp ∇ · (ρ V
ρ=
N
P0 xi Mi ,
R T i=0
ωi ) = ∇ · (ρ Di ∇ωi ), i = 1 . . . N ,
∇ · (ρ V
(1)
(2)
(3)
(4)
(5)
velocity vector, T: temperature, p: dynamic pressure, ρ: density, µ: dynamic
where V:
viscosity, cp : specific heat at constant pressure, λ: thermal conductivity, g: gravitation
acceleration vector, Ṡ: deformation rate tensor, P0 : hydrostatic pressure, xi : molar fraction, ωi : mass fraction, Mi : molar mass, Di : diffusion coefficient of component i in the
gas mixture, N: number of chemical species diluted in the carrier gas. It should be noted
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M. Selder, L. Kadinski
that the density is calculated by summing up the contributions of the diluted species and
the carrier gas.
It is assumed that the chemical species do not undergo any homogeneous chemical
reactions. This condition is fulfilled in typical PVT-growth processes of silicon carbide
where species are desorbing from a porous source and are transported to the material sink
without any further reactions [3].
The modeling of a typical SiC PVT-growth process requires a special approach to
the chemical boundary conditions [4]. It is usually assumed that the gaseous species
Si, Si2 C and SiC2 are diluted in a carrier gas (mostly argon) and that these species
undergo heterogeneous chemical reactions with a material source (porous SiC powder)
and a material sink (seed crystal). At the surface of the material source, for example, the
following heterogeneous reactions are considered:
SiC2 (g) + Si(g)
SiC2 (g)
SiC2 (g) + 3 Si(g)
2 SiC(s)
2 C(s) + Si(g)
2 Si2 C(g) ,
(6)
The mass action law equations relate the partial molar concentrations of the species with
each other:
xSiC2 · xSi
xSiC2
xSiC2 · (xSi )3
=
=
=
K1 (T)
K2 (T) · xSi
K3 (T) · (xSi2 C )2
(7)
For each reaction, the reaction constant Ki (T) is calculated as function of the chemical
potentials of solid and gaseous species taking part in the reaction. In general, a set of
equations of the form
N
νij
i=1
xi = Kj (T) ,
(8)
where N is the number of gaseous chemical species and νij the stoichiometric coefficient
of component i in reaction j, has to be solved. To determine the boundary concentrations
of the gaseous species at the interfaces gas/source and gas/sink, equation system (8) has
to be completed by flux balance equations, if necessary.
Resulting from the volume change on evaporation/crystallization, the gas velocity component normal to the respective wall is different from zero [5]. It is given by
vSte = −
N
i=1
Di ∇ωi
,
1− N
j=1 ωj
(9)
where the carrier gas is excluded from the summations.
Equation (3) includes heat transfer by convection and conduction. Heat transfer by
radiation is accounted for by an appropriate formulation of the temperature boundary
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M. Selder, L. Kadinski
conditions. All solid materials are considered to be opaque media, the gas mixture is
assumed to be transparent. The radiation model consists of the computation of greydiffuse viewfactor-based heat exchange between all internal surface elements of the reactor,
and the balancing of the heat flux contributions (radiation, convection, conduction) at
the internal solid/gas boundaries [2].
The solution of equations (1–5) taking account of the boundary conditions allows to
calculate the actual growth rate distribution of the crystal. Assuming that this distribution is valid for a fixed time step (e.g. 30 min.), the shape of the growing crystal after that
time step can be determined. After updating the investigated geometry, the solution procedure is repeated. Following this strategy, a complete growth process can be analyzed.
This quasi-stationary approach is based on the assumption that the time-scale characteristic for the macroscopic growth process is large compared to the time-scale characteristic
for the physical and chemical processes.
3
Numerical Method
The coupled partial differential equations (1–3) and (5) are discretized on non-orthogonal block-structured grids. The block-structuring serves as base for the distinction
between gas and solid reactor parts, and allows for the definition of material-dependent
properties in the solid parts (e.g. heat conductivity). The solution procedure is based on a
Finite-Volume approach using a colocated arrangement of the variables. Convective and
diffusive fluxes at the controlvolume faces are approximated by central differences using
the deffered correction approach. Sources are calculated using the variable values at each
controlvolume center as mean values of the whole cell.
Using an iterative solution procedure, non-linearities and the coupling of variables is
accounted for through outer iterations. At the beginning of each outer iteration, the
physical properties (viscosity, diffusion coefficients, etc.) are updated, and the density
is calculated by equation (4). After solving the momentum equations, the SIMPLEalgorithm [6] is used which couples pressure and mass fluxes to ensure that continuity
equation and momentum equations are simultaneously fulfilled. Finally, the energy equation and the species equations are solved. In each of these steps, the actual equation is
linearized using the respective variable values resulting from the preceding steps/iteration.
All equations are solved by applying the Strongly Implicit Procedure (SIP) of Stone [7].
The iteration number of the equation solver (inner iterations) can be chosen for each
variable independently.
In each outer iteration, the boundary values of temperature and species concentrations
at solid/gas interfaces are updated. The interface temperatures are calculated before
solving the energy equation by balancing the heat flux contributions of convection, conduction and radiation at both sides of each interface segment. As the radiative heat flux
depends on the fourth power of temperature, the balance equation is linearized by a Newton algorithm. At interfaces between different solid materials a similar approach is used,
but as there are no radiative contributions no linearization procedure is necessary. The
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M. Selder, L. Kadinski
viewfactors used for the calculation of the radiative heat fluxes are calculated by numerical
integration using a shadowing algorithm to take account of complex geometries.
The conservation equations of the chemical species (5) are coupled with the mass action
law equations (8) which specify the concentrations at the boundaries. Although for special
conditions an analytical solution of equation system (8) is possible, in general, a numerical
approach has to be applied. Writing the equations (8) in the form


f1 (x)


F(x) =  ...  = 0 ,


fM (x)
(10)
where x = (x1 , . . . , xM ), it is obvious that this task is equivalent to finding the solution
of a non-linear equation system. This is realized by a quasi-Newton algorithm which
minimizes the summed squares of the components fi of (10).
The convergence of the procedure is ensured by employing under-relaxation of all variables. For the energy and species equations, relaxation parameters between 0.8 and
0.95 were used. For the momentum equations, stronger under-relaxation (0.3–0.5) was
necessary. To speed up the calculations, a multi-grid algorithm based on the Full Approximation Scheme is used [8]: The solution is obtained first on the coarsest grid. This
solution is extrapolated to the next finer grid where it is used as initial guess for a V-cycle.
This procedure is repeated until the finest grid level is reached. It should be mentioned
that in the multigrid cycles the boundary species concentrations are only calculated on
the currently finest grid, and are restricted to the coarser grids by the usual restriction
procedure.
After the convergence of the solution procedure for the actual time step, the modification of the reactor geometry requires the updating of grid data, viewfactors, interpolation
factors etc. The results of the last time step are used as starting values. The equation
system is solved again by the multi-grid algorithm using V-cycles to speed up convergence.
4
Results
To demonstrate that the presented model is appropriate to analyze complex problems,
the growth of SiC bulk crystals in inductively heated reactors has been studied. This
problem includes inductive heating in solid blocks, heat transfer by convection, conduction
and radiation, heterogeneous chemical reactions at solid/gas interfaces and mass transfer
by convection and diffusion [4].
A model of an axisymmetric growth reactor is shown schematically in Fig. 1. By inductive heating (realized numerically by an additional source term in the energy equation)
a negative temperature gradient between source powder and seed crystal is established,
resulting in evaporation of species at the source and their nucleation at the seed. Transport of species through the enclosure (filled with carrier gas) results from diffusion and
convection. Details concerning the process conditions can be found in [9].
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M. Selder, L. Kadinski
The goal of the present study was the development of an efficient numerical solution
procedure for the prediction of flow, heat transfer and growth rate distribution for the
described reactor configuration. It has been checked recently that the modeling results
are in good agreement with experimental data [9].
To check the efficiency of the multigrid algorithm, the
growth rate distribution was calculated using 3 grid levels
with the finest grid consisting of 124.576 controlvolumes.
The calculation time on a SUN Ultra 10 workstation for
the three grid levels with and without using the multigrid
technique, respectively, are shown in Fig. 2. For each grid
Source
level, the calculations were stopped when the residual sums
were reduced by three orders of magnitude. This rather
slight criterion was only used to compare calculation times
and seems to be justified as the logarithm of the residual
sums is decreasing linearly. For the other calculations preCrystal
sented in this paper, a stronger reduction of the residual
sums has been used as convergence criterion. Using the
multigrid algorithm, the calculation time is reduced by the
factor 18 which shows the efficiency of the method.
Figure 1: Schematic Plot of a
The growth rate dependence on the radial coordinate is
PVT growth reactor.
shown in Fig. 3 for the two finest grid levels. It is seen that
at no position the difference between the two solutions is larger then 5% which indicates
that the discretization error is less then 1.7% [2]. It should be mentioned, however, that
10
350
MG
SG
Growth rate [µm/h]
Computing Time [h]
102
1
100
300
Grid level 2
Grid level 3
250
200
150
100
-20
10000
100000
Number of Controlvolumes
-10
0
10
Radial position [mm]
20
Figure 3: Growth rate dependence on the radial
position for the two finest grid levels.
Figure 2: Calculation time for three grid levels
for the SG and the MG approach.
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M. Selder, L. Kadinski
25
Crystal length [mm]
15
28
2498
10
5
0
-20
Crystal/gas interface
-10
0
10
Radial position [mm]
20
Figure 5: Calculated evolution of the crystal/gas
interface. The time difference between each two
interface lines is 5 hours.
Figure 4: Temperature isolines at the beginning
of the growth process. The temperature difference between two neighbouring isolines is 6 K.
the absolute value of the external heat sources has been slightly modified for the different
grid levels to simulate equivalent growth conditions.
The temperature distribution in the crystal and in the reactor enclosure at the beginning of the growth process is shown in Fig. 4. The temperature distribution gives rise
to non-uniform crystal growth. The evolution of the growing crystal is demonstrated in
Fig. 5 where the crystal/gas interface at different stages of the growth process is shown.
At the beginning of the growth process, nucleation is restricted widely to the center of the
seed crystal (compare Fig. 3). During the growth process, the growth rate distribution is
smoothed out more and more resulting in nearly uniform crystal growth. Possible origins
of this behaviour are discussed in [4].
The flow field in the reactor enclosure is shown in Fig. 6. The maximum value of
the gas velocity is about 5 cm/sec. The directions of the velocity vectors and their high
absolute values result from the advective Stefan flow [5]: Diffusion of the species at the
source/sink induces convection normal to the walls (equation 9). Neglecting this effect
would result in thermally induced convection rolls with significantly lower velocities.
The influence of this advective contribution on the growth process is demonstrated in
Fig. 7 where the growth rate at the center of the source at the beginning of the growth
process is shown as function of the hydrostatic pressure. The growth rate calculated
taking into account advection (indicated in the figure by vSte = 0) is compared to the
growth rate neglecting it. In contrast to the preceding calculations, helium was used as
carrier gas for these simulations. It can be seen that the Stefan flow is of major importance
especially at lower pressures where the mass fraction of the carrier gas is low.
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M. Selder, L. Kadinski
Growth rate [µm/h]
450
vSte≠ 0
vSte=0
400
350
300
250
200
150
20
100
Figure 7: Dependence of the growth rate on the
advective Stefan flow.
Figure 6: Flow field inside the reactor enclosure.
5
40
60
80
Pressure [mbar]
Conclusion
A mathematical model for the simulation of fluid flow problems including heat transfer
by convection, conduction and radiation, mass transfer by convection and diffusion and
heterogeneous chemical reactions was presented. The governing system of partial differential equations is solved by a Finite-Volume solution procedure, heterogeneous chemical
reactions and radiation phenomena are accounted for by an adequate formulation of the
boundary conditions. It was shown that the calculation time could be reduced significantly by using the multigrid-technique. The model was used to analyze the growth
process of silicon carbide in an axisymmetric PVT-growth reactor. Physical phenomena
involved in the growth process were discussed.
Acknowledgements
This work is supported by the Bavarian Science Foundation (contract 176/96) and the
Deutsche Forschungsgemeinschaft (contract Du 101/47). The contribution of Dr. D. Hofmann and his group (Department of Materials Science, University of Erlangen–Nuremberg) concerning the experimental verification of the model is gratefully acknowledged.
References
[1] Yu. N. Makarov, A. I. Zhmakin, Journal of Crystal Growth 94 (1989) 537.
[2] F. Durst, L. Kadinski, M. Schäfer, Journal of Crystal Growth 146 (1995) 202–208.
[3] A.S. Segal, A.N. Vorob’ev, S.Y. Karpov, Y.N. Makarov, E.N. Mokhov, M.G. Ramm,
M.S. Ramm, A.D.Roenkov, Y.A. Vodakov, A.I. Zhmakin, Materials Science and
Engineering B61–62 (1999) 40–43.
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M. Selder, L. Kadinski
[4] M. Selder, L. Kadinski, Yu. Makarov, F. Durst, P. Wellmann, T. Straubinger, D. Hofmann, S. Karpov, M. Ramm, Journal of Crystal Growth (accepted for publication).
[5] D.W. Greenwell, B.L. Markham, F. Rosenberger, Journal of Crystal Growth 51
(1981) 413–425.
[6] S.V. Patanker, D.B. Spalding, Int. Journal of Heat and Mass Transfer 15 (1972)
1787.
[7] H.L. Stone, SIAM Journal of Numerical Analysis 5 (1968) 530–558.
[8] F. Durst, L. Kadinski, M. Perić, M. Schäfer, Journal of Crystal Growth 125 (1992)
612–626.
[9] M. Selder, L. Kadinski, F. Durst, T. Straubinger, D. Hofmann, P. Wellmann, Proceedings of the ICSCRM’99 (accepted for publication).
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