Применение ЭВМ Для Решения Задач Теплопроводности_ЕгоровВИ_2006

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2006

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COMSOL Multiphysics, (PDE) . Heat Transfer [], Diffusion [] , , Fluid Dynamics [].

.

20 2005 . - 17 2006 .

: . -

, ., ... .. -

, , ... ..

- , 140402 140400

: .., 2006.

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, 2006.

COMSOL Multiphysics.. 41. COMSOL Multiphysics....

1.1. .. 1.2. .. 1.3. , 1.4. . 1.5. .. 1.6. 1.7. 1.8. ..

2. Heat Transfer()..... 2.1. 1D. .. 2.2. 1D axial symmetry ( ) 2.3. 1D axial symmetry ( ) 2.4. 2D. ...2.5. 3D. .. 2.6. Conduction and Convection..

3. Convection and Diffusion 3.1. 1D. ..

4. .. 4.1. . 4.2. ... 4.3. 4.4. .

.

558

12

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151820242727293337385156575959

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68

72

746

. 77

3

COMSOL Multiphysics

COMSOL Multiphysics - (PDE) . ( ) . . , PDE , : , , , .. . . (GUI), COMSOL Script MATLAB. .

. :

,

, (Weak form), PDE ,

.

, , :

PDE, COMSOL Multiphysics (FEM). . PDE, : , , , ,

4

, , , , , , .

, (coupling variables) .

.

1. , Model Navigator [ ] ( ) .

2. 3. ,

4. 5. 6. 7. . 8. 9.

5

1. COMSOL Multiphysics

1.1.

, , . , . , CAD , FEMLAB, , . Model Navigator [ ] (. 1.1) New Space Dimension [ ], 1D, 2D 3D Axial Symmetry (1D) (2D) .

, Heat Transfer [], Diffusion [] , , Fluid Dynamics []. Fluid Dynamics - . Heat Transfer , Conduction [], Convection and Conduction [ ], , , . Diffusion : Diffusion [] Convection and Diffusion [ ]. General Heat Transfer [ ] , , , Thin Conductive Layer , Bioheat equation , Non-Isotermal Flow .

6

1.1. .

, , : Multiphisics, Add geometry [ ], . , Add [] (Heat TransferConvection and Conduction), (Fluid DynamicsIncompressible Navier-Stokes). .

Steady-state analysis [ ] Transient analysis [ ], , .

New Model Navigator , Lagrange-Quadratic [-], , , .

7

New, Model Navigator, . Model Library [ ], , . User Models [ ], , . Settings [] ( , ). . COMSOL 3.2 , . Open User Models .

OK Model Navigator , . , .

File , , , CAD .

Edit , .

Options Axes/Grid settings, Constants, Expression, Function, Coupling Variables .

Draw , .

Physics Subdomain, Boundary, ( ) Periodic Condition Equation system.

Mesh . Solve

, , , .

Postprocessing , : , .

8

Multiphisics Model Navigator, .

Help .

1.2. , (. 1.2):

1D, . point [], line [], mirror ( ), move ( ) scale ( ). ; Mesh, Solve Postprocessing .

1.2. .

9

2D , , Array [] , . Rotate [] . , , , , .

Options>Axes/Grid settings [> /]. , . Axis [], Axis equal , , Y. , . , - .

x-y limits , 0 1.1 . Grid [] Auto . , (, ), , .

- , , , - . Visible .

SNAP, . x-y grid x y spacing . Extra x Extra y .

: Grid Mesh . Grid

10

. Mesh .

-. CAD MATLAB ( File>Import),

. , . Rectangle/Square [/] Rectangle/Square (Centered) [/ ()], , . Ctrl . Shift , . : Draw>Specify objects.

Size Width [] Height []. Rotation angle .

Position . Base x y. Corner , ( , ). Center .

Style, Solid Curve - . : , , Coerce to solid .

11

3D Curve Face . Name .

, Specify objects , , . . (, , , , ) . , Union [], Difference [], Intersection [] Draw>Create Composite Object , . Fillet/Chamfer Draw . Array, Mirror, Scale. Rotate Move . Draw>Modify.

3D, 2D, - . , - 1x0.5 .

Draw> Extrude

Extrude [] , ( , 2D 3D). Distance . , , , . Scale x y . , ( ) . Displacement

12

. Twist .

Draw>Embed . z=0. Draw>work plane settings

Draw>Revolve, . . .

, Scene Light , Zoom extents .

- , Draw>Draw Mode Draw Mode .

1.3. ,

, , . , (: ).

Comsol (Femlab) . Options. .

Constants [] , . , _air=30, _air. , . . Description [], .

Expression [] Scalar expression [ ], Subdomain, Boundary, Edge(

13

) Point expression. t; x, y, z; s ( 0 1 ) . Q_ist=100*exp(t). . , alfa. Boundary expression [ ] , alfa. alfa, .

Coupling Variables [ ] . .

Functions [] , . Interpolation function, , . , .

Coordinate systems [ ] , .

Material/Coefficients Library [ ] , ( ).

Visualization/Selection settings [ ] , .

Suppress [] . , , , , .

1.4.

, . Physics>Subdomain Settings F8: (.

14

1.3). , . . Heat Transfer by Conduction . Equation, . Subdomain selection, , . , . Ctrl+A.

1.3. Subdomain Settings.

Physics ,

, . Time-scaling coefficient [ ]. 1 . , . 1/60, 1/3600 ..

. . - . , MATLAB, , , Options>Expression.

15

, , Artificial Diffusion [ ] . . . . - . Solve>Restart [>] . .

Init , . Element .

COMSOL 3.2 Groups Color. Group . , , Color.

1.5.

1.4. Boundary Settings.

,

, . Point Mode, Edge

16

Mode, Boundary Mode Subdomain Mode Physics>Selection Mode>

Physics> Boundary Settings F7. (. 1.4) Boundary selection. , Interior boundaries . Coefficients . . Equation .

Subdomain Settings Boundary Settings - OMSOL 3.2 Groups Color . General Heat Transfer, , , . , ( ) . Member of Group(s) Boundary Condition.

( ) . Fluid Flow>Incompressible Navier-Stokes .

, , . . Physics>Periodic Condition. Boundary Point Edge. .

, (, ) Physics>Identity Conditions.

COMSOL 3.2 . (PDE). Physics>Equation system.

17

PDE, , .

.[1] 1.1:

fuuuuctud

tuc aa =++++

+

)(2

2

1.1

. , ( Heat Transfer>Conduction) :

QTktTC p =

)( 1.2

ca, , , a . C Heat Transfer>Convection and Conduction =*Cp*u ( u ) :

QTuCTktTC pp =+

)( 1.3

,

, ca [3], .

Physics>Equation system>Subdomain settings Ctrl+F8, (Subdomain) PDE . , . Physics>Subdomain settings. Physics>Equation system>Subdomain settings, , Physics>Subdomain settings. Reset Equation.

Physics>Equation system>Boundary settings Ctrl+F7. . Physics>ODE settings COMSOL 3.2 .

18

1.6.

. , Mesh>Initialize Mesh ( ) Mesh>Refine mesh . Mesh Mode, .

Mesh>Mesh Mode. , , : , - , Refine selection . . , . , .

1.5. Mesh parameters.

19

, COMSOL , . Mesh>Mesh parameters F9. , Global ( 1.5) . Predefined mash sizes Extremely fine [ ] Extremely coarse [ ], . . Maximum element size . 1/15 , . Maximum element size scaling factor , ( 0.5, 1/30 , 0.1 1/150). Element growth rate [ ] , , . Mesh curvature factor Mesh curvature cut off , : . Resolution of narrow regions , .

Mesh geometry to level , . Subdomain. Refinement method Regular Longest Refine mesh. Regular 2D , 3D. Longest, . Subdomain, Boundary, Edge Point . Advanced .

Remesh . ,

Mesh>Map mesh Ctrl+F9.

Draw>Extrude Draw>Revolve. , . 2D ( ),

20

Mesh>Extrude Mesh Mesh>Revolve Mesh . , . , , .

1.7.

, . , . , , , Athlon64 3000+ 1Gb , .

Solve Solve>Solve problem . Restart Solve>Restart ( ) . , . . , , . , .

F11 Solve>Solver parameters . ( 1.6).

Auto select solver, , .

, . , Time Dependent.

21

.1.6. Solver parameters.

, , , Eigenvalue.

, . , : , , ; , .

=>. (, ) () , . , Ctrans ( -) . , Incompressible Navier-Stokes .

, , . , - , . Solver, .

22

Time Dependent General Time stepping [ ] Times 0:10:60 . , 0 60 10 . , Physics>Subdomain Settings Time scaling coefficient. 1 1/60, 1 . tolerance . Allow complex number , PDE . Time Stepping . Times to store in output . Specified Times, General. Time steps from solver. , , , Times General. (, ) Time steps taken by solver Strict( ) Intermediate( General) Free. , Manual Tuning of step size.

Eigenvalue Desired number of eigenvalues Search for eigenvalues around.

, Name of parameter List of parameter values. , 0:10:100 linspace(1,100,25), 1 100.

Nonlinear , ( 25). Highly nonlinear problem .

Time Dependent Adaptive Mesh Refinement, . , , . .

23

, Matrix symmetry Symmetric . , . . , -.

, Linear system solver. Direct(UMFPACK) . , Direct (SPOOLES) , . GMRES. 1 Direct Cholesky (TAUCS) Conjugate Gradients. , , .

, . Geometric multigrid. - , , . . (UMFPAK) , , , .

Solve Solve>Solve Problem. , ( ) . , Solve>Update model, , . Restart, , Init , . . 1 [symmetric, positive-definite systems] [2,4]. .

24

: ( ), . , , .

1.8. Postprocessing mode [ ] .

Postprocessing> Postprocessing mode. , , - , - - . , : . , . , . , . Postprocessing>Plot parameters F12. Plot parameters ( 1D).

General Plot type , Line [], Max/min marker [ ] Geometry edges [ ] . Solution to use [ ] ( , ) Solution at time [ ]. Interpolated, Time .

25

.1.7. . , , .

Line Line plot General, Height data , . Heat transfer Predefined quantities [ ] , : Heat flux [ ], Temperature gradient [ ], PDE (, .. ) (, ). Expression [] . , : gradT_ht. , , - .

Line color Uniform [] , , Line color expression .

Max/Min . Animation (time dependent)

26

(parametric) . .

. . 2D Surface Plot (, ) . 3D Surface Plot . Contour plot ( - ). Arrow plot ( , ) . Streamline plot . Line Plot . Animate . Postprocessing>Plot Parameters F12, . General , . , , .

. . SNAP . Draw Point for Cross-Section Point Plot. , . Draw Line for Cross-Section Line Plot . Postprocessing>Cross-Section Plot Parameters . General ( ) , . Point , , . Line , . .

Domain Plot parameters .

Slice Plot [ ],

27

. Isosurface Plot . Subdomain Plot . Boundary Plot . 2D. Postprocessing>Plot Parameters F12

.

, . Postprocessing>Subdomain/ Boundary/Edge Parameters : , . , (, ) 1.

, . File>Export>Postprocessing Data.

2. Heat Transfer ()

2.1. 1D.

. , , . Axial symmetry (1D).

I III c . 10000 /3.

1. Model Navigator. 2. Space Dimension 1D. 3. Application Mode Heat

Transfer>Conduction>Steady-state analysis. 4. OK. .

1. Draw>Specify Objects>Line.

28

2. Coordinates. 3. 5 15 . 0 0.05 4. OK. , I1. 5. Draw>Specify Objects>Line 6. 0.05 0.2 .

, Line Shift

7. OK. 8. Zoom Extents .

. 1. Physics>Subdomain Settings ( F8). 2. Subdomain selection . . 3. 1-1

k [/*]

Q [/3]

1 5 400 10 000 000 2 15 0.11 0 4. Density () Heat capacity () . .

. 1. Physics>Boundary Settings (F7). 2. Boundary selection . .

2 , . , Interior boundaries.

3. 1-2 1 - Temperature T0=373 2 - Continuity 3 - Heat flux h=10 /(2*) Tinf=293

29

h ( + ), : Const - 5,67*10-8 /(2*4) Tamb . , , ( ).

, .

1. , Mesh>Initialize Mesh . .

[Mesh>Refine Mesh]. .

1. Solve[] Solver Parameters[ ].

2. Solver[] Stationary linear[ ].

3. OK.

4. Solve[]. [Solve>Solve Problem]. .

2.2. 1D axial symmetry ( )

. , 1000. , .. . 300K.

R1, R2. 0.98.

30

R1 0.05 R2 0.15 Tliq 1000 Tamb 300 emissivity 0.98 sigma 5.67

/*4 -

1. Model Navigator Axial symmetry (1D) Space dimension.

2. , COMSOL Multiphysics>Heat Transfer Conduction.

3. Steady-state analysis. 4. OK.

1. Draw, Specify Objects Line.

2. Line, 0.05 0.15 r Coordinates.

3. OK. 4. Zoom Extents.

1. Options Constants. 2.

-: NAME EXPRESSION emissivity 0.98 sigma 5.67e-8

1. Physics Subdomain Settings. 2. Subdomain Settings 1

: SUBDOMAIN 1 k (isotropic) 55.563 7850 Cp 460

31

, . , :

1. Init. 2. 1000 Temperature. 3. OK.

1. Physics Boundary Settings. 2. Boundary Settings 1. 3. Boundary condition Temperature. 4. 1000 Temperature. 5. 2. 6. Boundary condition Heat flux. 7. emissivity*sigma Problem-dependent constant. 8. 300 Ambient temperature [

]. 9. OK. ,

. ( ), -, . , , . .

Initialize Mesh ( ). 15 .


2. Solver[] Stationary nonlinear[ ].

3. OK. 4. Solve[].

32

, ( 2.1).

2.1. .

- , . Postprocessing Point Evaluation, OK . 1 1000 K. 2 15 T=895.401929 . T=895.401901 .

, . , . .

33

2.3. 1D axial symmetry ( )

, . :


2. , COMSOL Multiphysics>Heat Transfer Conduction.

3. Transient analysis. 4. OK.

.


2. Solver[] Time dependent [ ].

3. General Times 0:0.1:1 0:20:3000. , 2 20 . 3000 .

4. OK. 5. Solve[].

, 3000 . , , .

1. Postprocessing>Plot Parameters. 2. General Solution at time.

.

3. 300. 4. Apply.

2 , . .. .

34

. .

5. Line Line Color Uniform Color [ ] Use expression to color lines[ ]. .

6. Animate[]. , .

7. , . Ctrl+A

8. Start Animation. , . AVI MOV .

9. OK. 10. Postprocessing>Cross-Section Plot Parameters. 11. General , . 12. Point. 13. Coordinates r: 0.15

14. OK.

:

2.3. .

35

Zoom window . , . 15. Point Expression 895.402-T 16. OK.

17. Zoom window , 1 .

2.4. .

18. , 1 1800 (30 ) ,

.

19. Postprocessing>Domain Plot Parameters. 20. Plot type Line/Extrusion Plot.

36

21. Solutions to use [ ] Select via: Interpolated times [ ].

22. Times : 100 200 500 900 1300 2000

23. Line/Extrusion Plot type Line Plot.

24. Expression T. Subdomain Selection .

, , , , - .

25. Line settings. 26. Line color Color.

Line style Cycle. Line marker Cycle. 27. Color . 28. Legend. 29. OK Line settings. 30. OK Domain Plot Parameters.

( 2.5).

2.5. .

37

2.4. 2D.

, , [3]. T0=293 K, L=0,02 , H=0.03 . , ( ). :

=2800 /3

=2000 /(*) k=1.7 /(2*)

, : )]exp(1[10)(10 tTTtT += )exp(*)sin(22)(20 ttTTtT += ,

1=393 2=373 , 1=1-0=100 , 2=50, =0,01 -1, =0,05 -1, =0,003 -1, 1=60000 /(2*) 2=40000 /(2*). .

1. Model Navigator 2D Space dimension. 2. COMSOL

Multiphysics [FEMLAB]>Heat Transfer Conduction. 3. Transient analysis. 4. OK.

1. Draw Specify Objects Rectangle. 2. Rectangle, Size 0.03 Width

[], 0.02 Height [] 3. OK. 4. Draw Specify Objects Point. 5. 0.015, y 0.02.

. 6. OK. 7. Zoom Extents.

1. Options Constants.

38

2. :

NAME EXPRESSION T1 393 T2 373 deltaT1 100 deltaT2 50 sigma 0.01 omega 0.05 eta 0.003 T0 293

3. OK. 4. Options>Expressions>

Boundary Expression [ ].

5. 1, Name Tliq, Expression T0+deltaT1*(1-exp(-sigma*t)).

6. 4, Expression

T2-deltaT2*sin(omega*t)*exp(-eta*t). Tliq, .

7. OK.

2.6. .

.

1. Physics Subdomain Settings. SUBDOMAIN 1 k (isotropic) 1.7 2800

2. Subdomain Settings 1 :

Cp 2000 .

4. Init.

39

5. T0 Temperature. 6. OK.

1. Physics Boundary Settings. 2. Boundary Settings 1. 3. Boundary condition Heat flux. 4. 60000 h - Heat transfer coefficient. 5. Tliq Tinf - External Temperature 6. 4. 7. 40000 h - Heat transfer coefficient. 8. Tliq Tinf - External Temperature 9. OK.

: , Tliq, , Boundary Expressions.

Initialize Mesh ( ). 629 .

1. Solve[] Solver Parameters. 2. Solver[] Time dependent. 3. General Times 0:0.1:1

0:10:3000. 4. OK. 5. Solve[].

3000 . , Animation . Contour plot . Arrow plot , Streamline plot . - , , .

40

2.5. 3D.

, . . . , , , .

2.7. .

. , , .

16 . 3, 4. 4 , 10 . , , , , , .

0)( = Tk 2.1

41

k - [*-1-1], T [K]. :

)()( inf0 TThqTkn += 2.2

n - , h - [ -2-1], Tinf - [K].

h 15 -2-1, , .

h . , . :

4141 )()( Grf

LkhGrfNu ==>= 2.3

k - (=0,06 -1K-1), L - , (44 ), f()

[] F()

[] F()

0 0.48 130 0.38 90 0.46 140 0.35 100 0.45 150 0.28 110 0.435 160 0.22

120 0.42 180 0.15 Gr , [1]

2

3

=TLgGr 2.4

- (1/), 1/T , g - (9.81 /2), (18x10-6 2/)

=> , .

42

Model Navigator New . Space dimension 2D, OK.

1. Options Constants. , OK: NAME EXPRESSIONNAME EXPRESSION D 4.4e-2 Hh 15 K 0.06 visc 18e-6 Tinf 298 beta 1/Tinf Tinner 363 grav 9.81

2. Options Interpolation Functions [ ].

3. New. Function name graph, Interpolation metod Linear. OK.

4. Interpolation Functions, x f(x); , OK: X F(X) X F(X) X F(X) X F(X) 0 0.48 110 0.435 140 0.35 180 0.15 90 0.46 120 0.42 150 0.28 100 0.45 130 0.38 160 0.22

5. Options Axes/Grid Settings. 6. Axis equal, : x min -1.4 y min 0.2 x max 2.6 y max 2.8

7. Grid, Auto, , OK: x spacing 0.1 y spacing 0.1

:

43

R1 :

1. Shift, Rectangle/Square . , OK:

Width 0.8 x-position 0 Height 0.4 y-position 0.8 Base Corner

CO1 : 1. 2nd Degree Bezier Curve . 2. (0, 1.2), (0.2, 1.2), (0.2, 1.4). 3. Line, (0.6, 1.4). 4. 2nd Degree Bezier Curve. 5. (0.6, 1.4),

(0.6, 1.2), (0.8, 1.2). 6. .

, .

CO2 : 1. Line ,

(0.2, 1.4), (0.2, 2.0), (0.4, 1.9), (0.6, 2.0), and (0.6, 1.4). 2. , .

CO4: 1. 2nd Degree Bezier Curve. 2. (0.2, 2.0), (0.2, 2.2), (0.4, 2.2), (0.6, 2.2), and

(0.6, 2.0). 3. Line, (0.4, 1.9)

(0.2, 2.0). 4. . CO4,

. 5. Shift Rectangle/Square

(Centered).

44

6. , OK: Width 0.1 45 Base Center x-position 0.4 y-position 2.05

7. Line, , (0.2, 2.1), .

8. Line, , (0.6, 2.1), .

9. Line, , (0.4, 2.2), .

10. Line, , (0.4, 1.9), .

11. , CO3, SQ1, B1, B2, B3, B4, Coerce to Solid . ( , Shift )

: 1. R1, CO1, CO2, CO4, Array

.( Shift )

2. ; OK: displacement - x 0.8 displacement - y 0 array size - x 3 array size - y 1

CO10: 1. Line. 2. R1,

(-1.2, 0.8), (-1.2, 1.1), (-0.8, 1.1). 3. 3rd Degree Bezier Curve[

]. 4. (-0.4, 1.1), (-0.4, 1.2), (0, 1.2). 5. CO10.

45

, , , - :

1. Ctrl-A.

2. Scale . 3. ,

OK: Scale factor - x 1e-2 Scale factor - y 1e-2 Scale base point - x 0 Scale base point - y 0

4. Zoom Extents. .

1. Mesh Map Mesh. .

2. Boundary, :

, , Ctrl. 3. Constrained edge element distribution (

1 Number of edge elements). 4. :

46

5. Constrained edge element distribution, Number of edge elements 2.

6. Remesh, OK. 7. Mesh Revolve Mesh [ ]. 8. , 2

180. 9. Angle from x-axis, 0. 10. OK.

, . . , . , Draw>Revolve , . , . , - , . , , . . , , .

1. Multiphysics Model Navigator. 2. Multiphysics Geom2 (3D). 3. Heat Transfer

Module>General Heat Transfer>Steady-state analysis. , FEMLAB>Heat Transfer>Conduction>Steady-state analysis

4. Add, OK.

47

1. Physics Subdomain Settings. 2. . 3. ,

. Load. , Materials Library1>Silica Glass. OK. :

K CP 1.38 2203 703

4. Init, Temperature 323.

5. OK.

1. Physics Boundary Settings. 2. Ctrl+A. 3. Boundary condition Heat Flux,

: h Hc Tinf Tinf

4. Apply. 5. , .. 4

. , , . , . , ( ):

1,2,5,8,12,15,21,22,24,27,33,34,37,40,43,46,48,52,61,62,72,75,81,82,84, 87,93,94,97,100,103,106,108,112,121,122,132,135,141,142,144,147,153, 154,157,160,163,166,168,172,181,182,191,192

48

6. Boundary condition Thermal insulation

[ ], Apply. 7. , ,

: 4,6,16,18,76,78,136,138

8. Boundary conditions Heat flux,

: h Hh Tinf Tinner

9. OK. 10. Options Expressions>Boundary Expressions. 11. , ,

.

49

Physics>Boundary Settings Select by group. , 3. . Boundary Expressions.

12. ,

OK:

EXPRESSION VALUE Hc K/D*graph(angle)*Gr^0.25 angle 90+180*atan(y/z)/pi Gr grav*beta*(T-Tinf)/visc^2*D^3

1. Solve Solver Parameters. 2. Solver Stationary nonlinear.

. , .

3. Linear system solver GMRES. .

4. Preconditioner Algebraic multigrid. 5. Matrix symmetry Symmetric. (

, Symmetric Matrices). . .

50

. .

6. OK. 7. Solve . Athlon

3000 c 1 . ( ).

, :

1. Postprocessing Plot Parameters ( Comsol 3.2 F12).

2. General. Slice[], Boundary.

3. Boundary. Predefined quantities[ ] Temperature.

4. Apply. , Slice . Expression . , T-273.15 , . , . , 363 40 , 15 . , . , . .

Boundary Expression Gr. . : 1. Postprocessing>Plot Parameters,

Boundary, Predefined quantities Heat transfer coefficient. OK. FEMLAB, Expression h_ht h , ht

51

( Multiphysics)

2. , , Boundary Integration Postprocessing. ,

3. , Boundary Settings Select by group.

4. 3 . .

5. Boundary Integration, Predefined quantities Normal total heat flux.( Normal heat flux) OK.

6. . , , .

1.6 .

2.6. Conduction and Convection

Heat Transfer . , Heat Transfer>Conduction, , . (1.3) .

. , , . , 0 2 /. ( 2 ., 20 ) , , , . 2 . . 2 . , .

52

:

=7805 /3

=475 /(*) k=44,5 /(2*)

. , , , .


2. COMSOL Multiphysics [FEMLAB]>Heat Transfer Convection and Conduction.

3. Steady-State analysis. 4. OK.

1. Draw Specify Objects . 2. Rectangle, Size 0.01 (

) Width [], 0.2 Height []

3. OK. 4. Draw Specify Objects Rectangle. 5. Rectangle, Size 0.02 Width [],

0.2 Height [], Position r 0.01.

6. OK. 7. Zoom Extents.

1. Options Expressions>Global Expressions. 2. Name S, Expression -(0.01-(r))*50,

S .

3. OK.

53

4. Options Integration Coupling Variables>Boundary Variables [ > ]

5. Source 6 ( )

6. Name Tint, a Expressions T*2*pi*r/(0.0008*pi) OK.

, , 2*pi*r. , , S=pi*(R^2-r^2)=pi*(0.03^2-0.01^2)=0.0008*pi ^2. 2D T .

4-6 . . , , . , . , Postprocessing>Boundary Integration , . 7. Options>Integration Coupling Variables>Subdomain

Variables [ > ]

8. 1, Name Vsterj, Expression 2*pi*r, .

, 2*pi*r. , 1. 9. OK.

1. Physics Subdomain Settings. 2. Subdomain Settings

1 2: SUBDOMAIN 1 2 k (isotropic) 44.5 0.61 7850 995.6 p 475 4178 Q 2000/Vsterj 0

54

u 0 0 v 0 6*Vmean*S*(1-S)

. 3. Init. 4. 300 Temperature

. 5. OK.

1. Physics Boundary Settings. 2. Boundary Settings 1. 3. Boundary condition Axial symmetry. 4. 5. 5. Temperature 300. 6. 6 Convective flux. 7. Thermal

insulation. 8. OK.

, 300 .

Mesh>Mesh parameters Predefined mesh sizes Global, Finer Remesh. OK. 416 .

1. Solve[] Solver Parameters. 2. Solver[] Parametric linear. 3. General Name of parameter Vmean,

List of parameter values [ ] 0:0.1:2.

4. OK. 5. Solve[].

2 /. Animation , , . Postprocessing>Plot parameters (F12) General Parameter value, .

1. Postprocessing>Cross-Section Plot Parameters.

55

2. General , . 3. Point.

- OK, . , , 300 . Artificial Diffusion [ ] Subdomain Settings. , . 4. Expression Tint. 5. OK.

:

56

3. Convection and Diffusion

, 1.2 1.3 . ,

RcDtc

ts =+

)( , 3.1

D 2/, R , , . ts

)()( 0 cckNcDn bc += 3.2 N0 - , kc - , cb - . 3.1 :

RcucDtc

ts =++

)( 3.2

, :

0)( NcucDn =+ 3.3 [3]: L = 2, , , NaOH. 1, 0. . , , [0; 0.2 L/c0]. . NaOH D=10-9 2/. 1=0,01 /3, 0= 0 /3, 0 =1 /. , /3 /3. NaOH M=30 /. 1 = 1/3 /3, 0,2L/c0 400 .

57

3.1. 1D ( )

1. Model Navigator 1D Space dimension. 2. , COMSOL

Multiphysics>Convection and Diffusion Convection and Diffusion.

3. Transient analysis. 4. OK.

1. Draw>Specify Objects>Line. 2. Coordinates. 3. 0 1 4. OK. 5. Draw>Specify Objects>Line 6. 1 2 . 7. OK. 8. Zoom Extents

.

1. Physics>Subdomain Settings . 2. D 1e-9, u 1e-3 3. Artificial diffusion, Isotropic

diffusion, tuning parameter 0.1 4. OK 5. Init. 1 1/3,

2 0 6. OK.

1. Physics>Boundary Settings (F7). 2. 1 Concentration c0=1/3 /3

2 3 Concentration c0=0 /3

3. OK

58

1. Mesh>Mesh parameters. 2. Global Maximum element size 0.05 3. Boundary 2 4. Maximum element size 0.005

, .


2. Solver[] Time dependent [ ].

3. General Times 0:0.1:1 0:10:400.

4. OK. 5. Solve[].

, . . , , [3], , , .

Postprocessing>Plot Parameters. 1. General Solution at time.

.

2. 0. 3. Apply.

. , Artificial diffusion, .

4. Animate[]

. 5. Postprocessing>Cross-Section Plot Parameters. 6. General , . 7. Point.

59

8. Coordinates r: 1.1 9. OK. 10. 1.1

.

4. . , , , , . , , . , , . COMSOL Multyphysics . . -.

4.1. , ,

. 4.1: , ( ).

. 4.1. . A ,

A-B .

60

, h. . , . h, .

, . : General Heat Transfer Non-Isothermal Flow. , .4.1.

.4.2. (a), (b).

. , ( A .4.1). . 4.2 . :

: ( ) 0.13 , 0.002

: 0.02 , 0.002

0.010

61

, 2 (b) , .

: .

(The Non-Isothermal Flow), -, , u, , p, :

0)()(]))(3/2())(([)( 0

=+++=

ugIukuupuu T 4.1

- , , , , 0. (-0)g. , .

.

TTuCQTk p = )( 4.2

k - ; Cp - Q - 1,25 /3 (1 /) . 2/3 , , .

.

()

(FR4)

[/3] 2330 1900 Cp [/( )] 703 1369 k [/( )] 163 0.30

62

:

)/()( 0 RTMp w= , 4.3

p0 = 101.3 , Mw = 0.0288 /, R = 8.314 /*. , Cp=1100 log k = (-3,723+0,865 log(T)) 4.4

T86 100.4100.6 += 4.5

(normal flow) . , . , ( ), uy, umax, 1 /. :

uy=s(1-s)4umax 4.6 s - , COMSOL Multiphysics 0 1 . normal flow/zero pressure. , no-slip . 300 K ( ). . , y. , .

63

.4.3. .

( 4.3) ,

, . 30 K, 90 K. , ( 1 ) . , , . .

- . ( 4.4) , .

. 4.4.

.

64

,

. 4.5 . , . , .

.4.5. (a) (b).

65

, , . 4.6, , 50 60 K.

. 4.6.

.

. 4.7, , .

. 4.7. .

66

4.2.

: Heat_Transfer_Module/Electronics_and_Power_Systems/circuit_board_nat_2D

1. COMSOL Multiphysics, Model Navigator New.

2. Space dimension 2D. 3. Multiphysics.

Heat Transfer Module>General Heat Transfer, Add.

4. Heat Transfer Module>Non-Isothermal Flow, Add, OK. Femlab, Heat Transfer.

Femlab>Heat Transfer>Conduction and Convection> Steady-state analysis, Add. Fluid dynamics> Incompressible Navier-Stocks>Steady-state analysis, Add, OK.

1. Options Constants, , OK: NAME EXPRESSION q_source 2/3*1/(2e-3*20e-3^2) T0 300 rho0_air 1.013e5*28.8e-3/(8.314*T0)

2. Options Expressions>Scalar Expressions, , OK: NAME EXPRESSION k_air 10^(-3.723+0.865*log10(abs(T))) rho_air 1.013e5*28.8e-3/(8.314*T) Cp_air 1.1e3 eta_air 6e-6+4e-8*T

1. , Draw, Specify Object>Rectangle, , OK:

OBJECT WIDTH HEIGHT BASE X Y

67

R1 0.002 0.13 Corner 0 0 R2 0.01 0.13 Corner 0.002 0 R3 0.002 0.02 Corner 0.002 0.01

2. Zoom Extents. 3. R3,

Array . , Displacement y 0.03; Array size y 4. OK.

1. Multiphysics Non-Isothermal Flow ( Incompressible Navier-Stocks).

2. Physics Subdomain Settings. 3. 1 36. Active in this

domain, Non-Isothermal Flow(Incompressible Navier-Stocks) .

4. 2 : PARAMETER EXPRESSION rho_air eta_air Fy 9.81*(rho0_air-rho_air)

5. OK Subdomain Settings. 6. Multiphysics General Heat

Transfer (Convection and Conduction). 7. Physics Subdomain Settings

, Init T0 Temperature. 8. Conduction (Physics)

. SUBDOMAIN 1 2 k (isotropic) 0.3 k_air 1900 rho_air Cp 1369 Cp_air

9. 2 , Convection, Enable convective heat transfer. ( Convection and Conduction ). x-velocity u, y-velocity v.

10. Conduction 3-6. Load, Silicon, OK.

11. Q q_source. 12. OK Subdomain Settings.

68

1. Physics Boundary Settings. 2. OK (

Radiation type ): BOUNDARY 5 22 Boundary condition Temperature Convective flux T0 T0 Radiation type None

3. Multiphysics Non-Isothermal Flow(Incompressible Navier-Stocks).

4. Physics Boundary Settings. 5. No slip. 6. 5 22, Normal flow>Pressure

(pressure) 0. 7. OK. 8. Physics>Periodic Conditions Periodic

Boundary Conditions. 9. Source [] 1,

Expression [] T, Enter. 10. Destination [],

27, Expression T. 11. Source Vertices Vertex selection,

1 2, >>. 12. Destination Vertices.

21 22. OK.

1. Mesh Mesh Parameters. Global.

Predefined mesh sizes Normal. 2. Subdomain 2, Maximum

element size 1.5e-3. OK.

3. Mesh Initialize Mesh.

Solve .

1. 2-3 Plot Parameters Postprocessing.

69

2. Surface Temperature Predefined quantities Surface data. OK.

3. 2-5 (a) Surface data Velocity field OK.

4.3.

: Heat_Transfer_Module/Electronics_and_Power_Systems/circuit_board_nat_3D : Space dimension 3D.

1. . , Draw Block. OBJECT LENGTH X LENGTH Y LENGTH Z BASE X BASE Y BASE Z BLK1 0.015 0.002 0.13 0 0 0 BLK2 0.01 0.002 0.02 0 -0.002 0.01 BLK3 0.015 0.01 0.13 0 -0.01 0

2. Zoom Extents . 3. BLK2 Array

. Displacement, z 0.03; Array size z 4. OK.

1. Physics Subdomain Settings. 2. 2-6. Active in this

subdomain Non-Isothermal Flow .

3. 1 :

PARAMETER EXPRESSION rho_air eta_air Fz 9.81*(rho0_air-rho_air)

4. Artificial Diffusion, Streamline diffusion, OK.

70

5. OK Subdomain Settings. 6. Multiphysics General Heat

Transfer. 7. Physics Subdomain Settings.

. Init, Temperature T0.

8. Conduction : SUBDOMAIN 1 6 k (isotropic) k_air 0.3 rho_air 1900 Cp Cp_air 1369

9. 1, Convection. Enable convective heat transfer, x-, y- z-velocity u, v, w, .

10. Artificial Diffusion. Streamline diffusion OK.

11. Conduction, 2-5. Load, Silicon, OK.

12. Q q_source. 13. OK Subdomain Settings.

1. Physics Boundary Settings. 2. ;

OK. BOUNDARY 3 4 Boundary condition Temperature Convective flux T0 T0 Radiation type None

3. Multiphysics Non-Isothermal Flow. 4. Physics Boundary Settings. 5. Ctrl+A,

No slip. 6. 3 4,

Normal flow/Pressure pressure of 0. 7. 1 34, Slip/Symmetry. 8. OK. 9. Physics Periodic Conditions>Periodic Boundary

Conditions ( ). 10. Source 2. Expression T,

Enter.

71

11. Destination, 29. Expression T.

12. Source Vertices ( >>), , 1, 2, 39, 40.

13. Destination Vertices 21, 22, 43, 44, . OK.

1. Mesh Mesh Parameters. Global Predefined mesh sizes Finer.

2. Boundary 2. Maximum element size ( ) 3e-3.

3. Advanced. x-direction scale factor 0.5. OK.


1. Solve Solver Parameters. 2. Solver Stationary nonlinear, Linear

system solver Direct (UMFPACK). 3. Solve .

, . , : . 500 MB . COMSOL Multiphysics 15 1.5-GHz.

1. 4 Postprocessing Plot Parameters.

2. General , Boundary.

3. Boundary. Surface data Predefined quantities Temperature. Apply.

4. 5 (b), General, Slice Boundary.

5. Slice Slice data Predefined quantities - Velocity field.

6. x, y, z levels 1, 0, 8, . OK.

72

4.4.

: Heat_Transfer_Module/ Electronics_and_Power_Systems/circuit_board_forced_3D

. . , .

1. Draw mode Draw Main toolbar.

2. , Draw>Modify>Rotate. 3. Rotation -90. 4. Rotation axis direction vector, x, y, z 1,

0, 0, . OK. 90 , , .

1. Multiphysics General Heat Transfer. 2. Physics Boundary Settings. 3. ,

OK: BOUNDARY 5 29 Boundary condition Convective flux Temperature T0 T0 Radiation type None

4. Multiphysics Non-Isothermal Flow.

5. Physics Boundary Settings. 6. 29, Boundary condition

Inflow/Outflow velocity. 7. y-velocity

-4*(1e4)*(z)*(0.01-(z))*(z>0). OK.

73

1. Mesh Mesh Parameters. Global

Predefined mesh sizes Finer. 2. Resolution of narrow regions 1. 3. Boundary 2. Maximum element

size 3e-3; Element growth rate 2.

4. Advanced. x-direction scale factor 0.5. OK.


1. Solve Solver Manager. 2. Initial Value Initial value

Initial value expression. 3. Solve For Non-Isothermal

Flow. 4. Solve .

7 500 MB .

5. COMSOL Multiphysics , Init, Initial value Current Solution.

- , . : 6. Solve For . 7. Solve .

7 .

1. 6, Plot Parameters Postprocessing.

2. General Slice Boundary. 3. Slice Surface data Predefined

quantities Temperature. 4. x, y, z levels 1, 10, 0, .

Apply. 5. 7, Slice

Predefined quantities Velocity field. 6. x levels 0, OK.

74

1

. . 300 . 300 . 40 /2. 5 /2. 0.5. T, . A . B . . 1 , . [5].

, - - . , .

A, B, C, T,

1 10 3 5 350 2 15 3 4 350 3 15 3 10 350 4 25 2 5 400 5 20 2 4 400 6 10 2 5 400 7 5 0.5 1 400 8 50 3 10 400 9 40 5 5 450 10 25 2 10 450 11 25 1 5 450 12 80 3 6 450 13 25 2.5 15 450 14 35 2.5 20 373 15 45 3 8 373 16 8 0.4 2 373 17 25 2 5 373 18 20 2 5 373 19 10 1 4 500 20 5 0.2 1 500

75

2

, , . 1. h=20 /2. [5].

3

34 (2.3), . . . T1, T2, ,1/c ,1/c , 1/c 1,

/(2*) 2, /(2*)

1 393 373 0.01 0.06 0.001 60000 40000 2 393 373 0.02 0.05 0.002 70000 45000 3 393 373 0.03 0.04 0.003 65000 35000 4 393 373 0.04 0.03 0.004 55000 30000 5 393 373 0.05 0.07 0.005 70000 40000 6 373 393 0.01 0.08 0.001 45000 45000 7 373 393 0.02 0.06 0.002 50000 35000 8 373 393 0.03 0.05 0.003 60000 30000 9 373 393 0.04 0.04 0.004 70000 40000 10 373 393 0.05 0.03 0.005 65000 45000 11 400 380 0.01 0.07 0.001 55000 35000 12 400 380 0.02 0.08 0.002 70000 30000 13 400 380 0.03 0.06 0.003 45000 40000 14 400 380 0.04 0.05 0.004 50000 45000 15 400 380 0.05 0.04 0.005 50000 35000 16 450 400 0.04 0.03 0.001 60000 30000 17 450 400 0.03 0.07 0.002 80000 40000 18 450 400 0.02 0.08 0.003 60000 45000 19 400 450 0.01 0.05 0.004 70000 35000 20 400 450 0.025 0.04 0.005 65000 30000

76

1. .. .. .. : . .: , 1981. 416 . 2. .. .. .. : . . . . . .: . ., 1990. 207 . 3. .. . : . .: - , 2002. 576 . 4. . . .: , 1979. 392 . 5. : . . . .. , .. . .: , 1991.1232 . 6. www.matlab.exponenta.ru

77

COMSOL Multiphysics 1.2. , (. 1.2): 2D , x-y limits , - , : Grid Mesh -,

, NAMET1393T2373deltaT1100deltaT250sigma0.01omega0.05eta0.003T0293

4. 4.1.

4.3.

4.4.

1 . B . , A,B, C, T,1103535021534350315310350425254005202440061025400750.5140085031040094055450102521045011251545012803645013252.51545014352.5203731545383731680.423731725253731820253731910145002050.21500 2 T1,T2,,1/c,1/c, 1/c1, /(2*)2,13933730.010.060.001600004000023933730.020.050.002700004500033933730.030.040.003650003500043933730.040.030.004550003000053933730.050.070.005700004000063733930.010.080.001450004500073733930.020.060.002500003500083733930.030.050.003600003000093733930.040.040.0047000040000103733930.050.030.0056500045000114003800.010.070.0015500035000124003800.020.080.0027000030000134003800.030.060.0034500040000144003800.040.050.0045000045000154003800.050.040.0055000035000164504000.040.030.0016000030000174504000.030.070.0028000040000184504000.020.080.0036000045000194004500.010.050.0047000035000204004500.0250.040.0056500030000

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