Three Dimensional Numerical Modeling of Convective Heat and Mass Transport under Realistic Microgravity Environment

4th International Conference on Computational Heat and Mass Transfer

Proceedings of 4th ICCHMT

May 1720, 2005, Paris-Cachan, FRANCE

Paper number____EDUCATION & TUTORIALS IN MODELING IN COMPUTATION HEAT TRANSFER: FROM ELEMENTARY PROCESSES ON THE BASIS OF COMPUTER LABORATORY TO THE INDUSTRIAL COMPLEXES

A.I. Fedyushkin*, V.I. Polezhaev, and V.P.Yaremchuk The Institute for Problems in Mechanics, Russian Academy of Sciences, Prospect Vernadskogo 101, building 1, Moscow, 119526, Russia

*Correspondence author: Phone: +7 095 4343283; Fax: +7 095 9382048; E-mail: fai@ipmnet.ruABSTRACT

A goal of the paper is to point on the steps of training the students from using of the computer laboratory for initial stage of the education of the convective heat/mass transfer on the basis of Navier-Stokes equation till usage of professional industrial CFD codes. Presented examples are based on the experience of students research works and serve as a link between initial stage and studies on the basis of complicated professional industrial commercial codes for preliminary stage of the modeling.

NOMENCLATURE

Hlength scale

Ttemperature scale

tension factor on a free surface

g

gravity acceleration

(

density

thermal conductivity

a

heat transfer coefficient

(

kinematics viscosity

thermal diffusivity

( thermal expansion coefficient

GrGrashof number, ( T gH3/(2

MnMarangoninumber, (d/dT) (TH)/()

PrPrandtl number, (/a

1. INTRODUCTION

Computational Fluid Dynamics (CFD) is important instrument for investigation transfer phenomena of fluid and has a huge range of the applications. However a problem how to find an efficient way to the access to this instrument is actual because of elementary flows and heat/mass transfer processes in realistic industrial applications are nonlinear, unsteady, multi parametric and multi scale. A number of the books and textbooks devoted to the fundamentals of the fluid mechanics and applications in heat and mass transfer discuss the problem (see, for instance, [1-5]). Last decades efficient professional tools for modelling on the basis of Navier-Stokes equations and the industrial complexes of programs are developed. They have a number of the tutorials, which mainly oriented for the users guide, following complicated instructions. Development of the numerical heat transfer needs intermediate systems, which help to access to the scientific nature of the problems for education process with the use of the modern tools of the modelling for new generation of researchers. One can find one of the first steps on the basis of one-dimensional and boundary layer type approach oriented on the engineering education, using Mathcad [5]. Education and tutorials in modelling of the elementary flows on the basis of Navier-Stokes equations in ground-based and microgravity environments using computer system COMGA (COnvection in MicroGravity and Applications) and ASTRA (Automated System for Theoretical Research and Analysis) are developed and based on the previous experience of the modelling [6,7]. Computer laboratory as intellectual shell of these systems includes microgravity and ground-based applications [6]. Basic tutorials of fluid dynamics and heat/mass transfer such as hydrostatic equilibrium, stability of the steady flow, study of the elementary of the buoyancy-driven, surface-tension gradients driven and forced convective flows and multi-parametric analysis and some applied tutorials for microgravity fluid mechanics and technological fluid dynamics presented in [8, 9, 12].

In this paper a plan of the education and tutorials for convective heat/mass transfer processes on the basis of Navier-Stokes equation from preliminary stage of the modelling with a link to the comprehensive commercial codes similar [10] is presented.

2. COMPUTER EDUCATION USING MATHCAD SYSTEM The understanding of the general engineering problem is very important point for initial stage of the education. Engineering-physical projects executed in Mathcad include problem-setting and physical model formulation, designing on the basis of one-dimension or boundary layer-type of models [5] (www.thermal ru). The initial step shows on real examples how to use effectively Mathcad at all development stage of an engineering project (analytical preprocessing the mathematical description (during normalization, research of the special points, identical transformations, etc.), analytical decisions where it the opportunities of symbolical Mathcad-processor allows, the numerical decision when analytical decisions are impossible or inefficient, results presentation and visualization.

3. EDUCATION & TUTORIAL IN MODELING ON THE BASIS OF NAVIER-STOKES EQUATIONS

Next step of the education may be study and tutorials using relatively small computer systems as COMGA, ASTRA [6-7]. The COMGA system (see www.ipmnet.ru/~ermakov) supports modeling problems of free and forced convection on the basis of the Navier-Stokes equations (Boussinesq approach). A basic version of the system and computer laboratory includes classical problem of convective heat and mass transfer with different types of the boundary conditions, external forces and fluid (gas) properties. In Fig.1 general classification of the elementary convective processes of the gravity- and non gravity types which contains such kind of computer laboratory is shown.

Fig 1. Classification of the elementary 2D convection and heat/mass transfer processes

Classification deals with the convection problems in an enclosure with rigid (solid lines) or free boundaries (dashed line) in the field of gravity (g), vibration and slow rotation for binary mixture (heat and mass transfer). Solid arrows show direction of the heat flux, dashed arrows are direction of the mass flux. First four items correspond to the case, when mechanical equilibrium exists in uniform (two first) or binary (third and forth) systems. Two last items correspond to the case of advection convection which induced by the horizontal temperature or concentration gradient which are gown in the same or in opposite directions. Four intermediate items correspond to the coupling between stratified and advective flows. It should be taken into account multidimensional nature of the convection and possibility to provide for students an experience in simplest parametrical analysis of the searched characteristics of heat/mass transfer.

Computation code ASTRA (www.ipmnet.ru/~burago) similar elementary items also contains (Fig2). Using adaptive Eulerian-Lagrangian unstructured grids and the finite element method, ASTRA provides numerical solution of the one, two and three Fig. 2. ASTRA computer system as efficient tool for education of the elementary heat mass transfer processes

dimensional unsteady nonlinear problems in continuum (fluid and solid) mechanics. ASTRAs distinguishing features are as on-line preprocessors for generation of unstructured finite element grids in two and three dimensional regions; on-line editor for preparation of the geometrical and physical starting data. Free format rules (NAMELIST style), online help for keywords and online diagnostics of errors facilitate this part of job; on-line symbolic debugger for providing detail analysis and management of data, breakpoints, dumping, restart etc; an inbuilt postprocessor for on-line visualization of the results in form of tables, graphs, contour lines, colour zones. With ASTRA it is easy to produce movies and to perform hard copies of pictures in PCL, HPGL or PostScript formats; possibility of modification and replacement of the code component. Source code of problem orientated part of ASTRA and high level software for pre- and post- processing are opened for interested users.

Basic education and tutorials

Initial stage of education with the use of the COMGA, ASTRA can be start from the definition of the physical properties of the typical liquids and gases, hydrostatic equilibrium, driving forces, induced by the buoyancy and gradient of the surface tension (Marangoni) as wall as general reasons for onset of convection and convective instability. It is necessary to note, that a first step of preparation can be acquaintance of learning colleges and universities with classical experimental results and with the elementary analytical exact solutions of the equations of conduction and hydrodynamics (Poiseuille flow in a pipe, boundary layer Blaziuse flow on a plate, slow convection flows in layers etc.)

Elements of the modeling theory as well as the basic knowledge of numerical methods should be done for understanding the options of the FDM, FEM methods. For instance, some versions of the implicit scheme for viscous terms, second-order approximation applied on staggered mesh, FFT method for the pressure correction step, significance of the uniform (non uniform) meshes with approximation of the Navier-Stokes equations should be explained.

Starting from the elements of the convection tutorials, classical problems of the gravity-driven convection with side heating, Rayleigh-Benard convection with rigid and free melt surfaces, and similar types of the Marangoni convection will be studied.

Fig. 3. Example of the tutorial for the onset of Marangoni convection in a layer with free surface and bottom heating. Menu of the basic part of the computer laboratory.A fragment of problem solution on thermo-capillary convection in horizontal layer during bottom heating (Marangoni instability), presented in Fig. 3, illustrates onset of Marangoni convection after the loss of hydrostatic equilibrium stability and formation of the steady state regime for the classical Pearson problem [11]. Stream functions illustrate roll structure on the top and isotherms of thermo-capillary convection - below. On the right side in middle a temporal evaluation of the maximum of stream function is shown. A fragment of Menu of computer system COMGA_W is shown on the left side. One can see here a title of the problem and items inside a "Problem category", which only a given problem characterizes: parameters Mn, Pr, region L/H, body force (g=0 in this case) and type of initial conditions. The menu makes it possible also to change mesh size, visualization type, or to use information from a file. Following the classification in Fig.1 most of the elementary convective problems can be studied and illustrate in real time calculation processes by the students during the tutorial process.

Therefore the basic tutorial can make support of the number textbooks on heat transfer and fluid mechanics and convective stability [1,4]. A number of examples which discussed in a monograph [5] can be solved again and illustrate in details. Note that complex experimental and computer tutorial [8] is also important step on this way as a link with concrete physical nature and validation of the computer modeling results.

Basic tutorial with technology applications

Special versions of the systems COMGA and ASTRA were developed for the cases of technology geometries and microgravity environment [6, 12]. Possibility of computer laboratory for study fluid flow and heat transfer in technology applications may be explain on hydrodynamics (or so-called idealized) Czochralski model, which represents a nonlinear multi parametrical hydrodynamic system with wide range values of governing parameters. Its control function (for instance, amplitude of temperature oscillation, A) seems as follows:

A = f (r, z, , Res, Rec, Gr, Mn, Pr, Rs/Rc, H/Rc, 1, 2)

(*)

Here Res, Rec are two dynamical criteria: Reynolds numbers related to the seed and crucible rotation (forced convection). Gr, Mn are two natural convection criteria Grashof and Marangoni numbers, Pr number - is a criteria of the physical properties, H/Rc, Rs/Rc are two geometrical parameters - non dimensional height of the melt and width of the seed, and 1, 2 are thermal boundary and initial conditions. One part of the tutorial (see more information in [12]) consists with a number of examples deals with typical crystal growth configurations for silicon, GaAs, oxides melts and other kinds of the crystal growth on the basis of the axis-symmetric model following the paper [13].

Fig. 4. Simulation by system INTEX regimes of coupling convection and crucible rotation visualized by temperature (left) and stream function (right) fields. Res=0, Mn=0, Gr=7.8107, Pr=0.07, H=Rc=Rs/Rc=0.578, a)Rec=1.6104 , b) Rec=1.9103, c) Rec=103

Formulation (*) shows principal possibility for control of fluid flow/temperature fields in the melt using thermal (power input, including boundary condition), dynamical (steady state crystal/crucible rotation or angular acceleration, artificial vibration etc.) geometrical (dimensionless melt level or width of the melt surface) and gravity level control. Fig. 4 illustrates possibilities of the parametric analysis of the coupling between gravity driven convection and fluid flow induced by the crucible rotation. Crucible rotation induces the temperature oscillation in the melt for typical GaAs parameters [13] which may be reduced by the lower rotation rates.

4.A LINK BETWEEN COMGA, ASTRA and FLUENT TUTORIALS

Above mentioned tutorials make it possible to understand some elementary convective processes deals with more complicated process. Next step on the way of training of the students can be mastering by a package FLOWLAB [10], giving an opportunity in a uniform package will familiarize with all chain of process of modeling of tasks CFD. During work with this package, the user FLOWLAB, will familiarize. First, with elementary principles CAD system of construction of geometrical bodies (which will be useful to any engineer going to work with professional CAD by the programs, (for example, such as Pro/ENGINEER, I-DEAS, CATJAV4, UNIGRAFICS and etc.) secondly, with application of the generator of grids, thirdly, with solver, and, fourthly, about a post-processing and elements of the analysis of numerical results.

An example of the FLUENT tutorial which deals with modeling solidifications (following tutorial No20 solid [10]) illustrates how to set up and solve a problem involving solidification. In this tutorial, you will learn how to define a solidification problem, define pull velocities for simulation of continuous casting, define a surface tension gradient for Marangoni convection and solve a solidification problem. This tutorial assumes that users are familiar with the menu structure in FLUENT, and that you have solved previous tutorials. Some steps in the setup and solution procedure will not be shown explicitly. It demonstrates the setup and solution procedure for a fluid flow and heat transfer problem involving solidification, namely the Czochralski growth process. The geometry considered is a 2D axisymmetric bowl, containing a liquid metal, free surface, mushy zone and crystal in Fig5). The bottom and sides of the bowl are heated above the liquidus temperature, as is the free surface of the liquid. The liquid is solidified by heat loss from the crystal and the solid is pulled out of the domain at a rate of 0.001 m/s and a temperature of 500 K. There is a steady injection of liquid at the bottom of the bowl with a velocity of 1.01x10-3 and a temperature of 1300 K. Material properties are corresponded to semiconductor. Starting with an existing 2D mesh, the details regarding the setup and solution procedure for the solidification problem are presented.

Fig. 5. Simulation flow in Czohralski Crystal Growth by using FLUENT. Contours of temperature (left) and stream function (right) in moment time t=22 sec.

Then, the fluid flow is turned on to investigate the effect of natural and Marangoni convection in an unsteady fashion with able moving solid-liquid interface. The results of calculation of flow and heat transfer in Czochralski Crystal Growth carried out by authors using FLUENT is showed for moment t=22sec in Fig. 5. Dynamical changing of the structure of flow and relocation of interface user also can to show using animation options. If it needs user can to change of values of parameters, properties of the problem, tested of different physical and numerical models on one tutorial, but for correct using the models he must has introductory knowledge about that and to be able rightly to apply these models. For that preliminary training computer laboratory similar COMGA, ASTRA FLOWLAB program can be efficient tools.

CONCLUSIONS

Presented plan of the education and tutorials is oriented mainly on the convective processes and heat transfer and applications and suppose before the study industrial packages tutorials (see for instance [10]) preliminary stage of education as follows:

1. Basic Heat Mass Transfer Educations, Fluid Mechanics and Engineering education textbooks [1 -5].

2. Computer laboratories [6, 8, 12] for the elementary convective flows which supports of the above mentioned fluid mechanics and heat/mass transfer textbooks.

3.Complex tutorials [10] of the professional industrial code, for example, FLUENT.

Note, that computation methods and solvers techniques extremely rise during last years, but most of them are outside of the computer laboratories now. For instance, efficient codes for DNS problems, including direct simulation of the stability and turbulence, three dimensional stability [13-14], a hierarchy of compressible fluid flow models including near critical ones [15] and a number of the new applied problems will be included in the education process in the near future.

ACKNOWLEDGEMENTS

This work are supported partly by RFBR grants 03 01- 0682, RFBR - NSFC 04-01-39021, a well as project Integratziya by the Ministry of Education of Russian Federation by the guidance of Rostov State University No. 74. The authors express their gratitude to the Prof. A.P. Solodov for the information related engineering education, Prof. N.G.Bourago, Dr. M.K.Ermakov and S.A.Nikitin for the access to the ASTRA, COMGA and INTEX systems and to the company Processflow Inc. for the opportunity of the use the program FLUENT.

REFERENCES

1. Incropera F., De Witt D., Fundamentals of Heat and Mass Transfer, Wiley, 1996, 886p.

2. Shih T.M. Numerical heat transfer, Hemisphere, 1984

3. Solodov A., Ochkov V. Differential models. An Introduction with Mathcad, Springer, p.232, 2005

4. Landau L.D., Lifshitz E.M. Mechanics of continuous media, M., Nauka, 1986 (in Russian).

5. V.I. Polezhaev, M.S. Bello., N.A.Veresub et al. Convective processes in microgravity, (Nauka, Moscow, Russia, Moscow), 240 p. (in Russian).

6. M.K. Ermakov, S.A. Nikitin and V.I. Polezhaev, A system and computer laboratory for modelling of the convective heat and mass transfer processes, Fluid Dynamics 32, 3, 1997, 338.

7. Bourago N.G. Numerical methods for non-linear processes in elastic plastic media. In: Lectures of FEM-94 Seminar, Chalmers Univ. Techn., Struct.Mech.Dept., Publ:1994.1, Gothenberg, pp. 1-15.

8. V.P.Yaremchuk, M.K. Ermakov, S.A. Nikitin, V.I. Polezhaev. Education & Tutorial on Fluid Mechanics on the basis of Computer Laboratory, 21st International Congress of Theoretical and Applied Mechanics, August 15-21, 2004, Warsaw, Poland. In: ICTAM04 Abstract Book and CD-ROM Proceedings, IPPT PAN, Warszawa, p 418

9. Zyuzgin A.V., Polezhaev V.I., Putin G.F. The new concept of the convective tutorial Proceedings of 4th ICCHMT, 2005

10. Manuals of FLUENT6.1 and FLOWLAB1.1. User's Guides, Fluent.Inc 1864p. and 262p., 2003, www.fluent.com11. Pearson J.K.A. On convection cells induced by surface tension, J. Fluid Mech., 4, 1958, 489

12. Ermakov M.K., Nikitin S.A., Polezhaev V.I. and Yaremchuk V.P. Education and Tutorial in Modeling of Elementary Flows, Heat and Mass Transfer During Crystal Growth in Ground-based and Microgravity Environment, J. of Crystal Growth. 2004. V. 266. P. 388-395.

13. Polezhaev V.I.,. Bessonov O.A, Nikitin N.V., Nikitin S.A., S.A. Convective inerraction and instabilities in GaAs Czochralsky model, J. of Crystal Growth, 230, 2001, 40-47

14. Nikitin N.V., Polezhaev V.I., Three dimensional features of transitional and turbulent convective flows in Czochralski model Izv. RAS, MZG, 1999 (English translation in Fluid Dynamics Vol.34, No 6)

15. Polezhaev V.I., Gorbunov A.A., Soboleva E.B. Unsteady near critical flows in microgravity environment. In: Transport phenomena in microgravity, Annals of the New York Academy of Sciences. V.1027, 2004, p. 286-302.