CHE/ME 109 Heat Transfer in Electronics

1. CHE/ME 109 Heat Transfer in Electronics REVIEW FOR SECOND MID-TERM EXAM

2. ONE DIMENSIONAL NUMERICAL MODELS

3. NUMERICAL METHOD FUNDAMENTALS • NUMERICAL METHODS PROVIDE AN ALTERNATIVE TO ANALYTICAL MODELS • ANALYTICAL MODELS PROVIDE THE EXACT SOLUTION AND REPRESENT A LIMIT • ANALYTICAL MODELS ARE LIMITED TO SIMPLE SYSTEMS. • CYLINDERS, SPHERES, PLANE WALLS • CONSTANT PROPERTIES THROUGH THE SYSTEM • NUMERICAL MODELS PROVIDE APPROXIMATIONS • APPROXIMATIONS MAY BE ALL THAT IS AVAILABLE FOR COMPLEX SYSTEMS • COMPUTERS FACILITATE THE USE OF NUMERICAL MODELS; SOMETIMES TO THE POINT OF REPLACING ANALYTICAL SOLUTIONS

4. FORMULATION OF NUMERICAL MODELS • DIRECT AND ITERATIVE OPTIONS EXIST FOR NUMERICAL MODELS • DIRECT MODELS SET UP A MATRIX OF n LINEAR EQUATIONS AND n UNKNOWS • FOR HEAT TRANSFER, THE EQUATIONS ARE TYPICALLY HEAT BALANCES • ROOTS OF THESE ARE OBTAINED BY SOME REGRESSION TECHNIQUE

5. ITERATIVE MODELS • SET UP A SERIES OF RELATED EQUATIONS • INITIAL VALUES ARE ESTABLISHED AND THEN THE EQUATIONS ARE ITERATED UNTIL THEY REACH A STABLE “RELAXED” SOLUTION • THIS METHOD CAN BE APPLIED TO EITHER STEADY-STATE OR TRANSIENT SYSTEMS. • BASIC APPROACH IS TO DIVIDE THE SYSTEM INTO A SERIES OF SUBSYSTEMS. • SYSTEMS ARE SMALL ENOUGH TO ALLOW USE OF LINEAR RELATIONSHIPS • SUBSYSTEMS ARE REFERRED TO AS NODES

6. ONE DIMENSIONAL STEADY STATE MODELS • THE GENERAL FORM FOR THE HEAT TRANSFER MODEL FOR A SYSTEM IS: • FOR STEADY STATE, THE LAST TERM GOES TO ZERO • SIMPLIFYING FURTHER TO ONE-DIMENSION, WITH CONSTANT k, AND A PLANE SYSTEM, THE EQUATION FOR THE TEMPERATURE GRADIENT BECOMES (g’ = ė in text):

7. ONE DIMENSIONAL STEADY STATE • SYSTEM IS THEN DIVIDED INTO NODES. WHICH SEPARATE THE SYSTEM INTO A MESH IN THE DIRECTION OF HEAT TRANSFER. • THE NUMBER OF NODES IS ARBITRARY • THE MORE NODES USED, THE CLOSER THE RESULT TO THE ANALYTICAL “EXACT SOLUTION” • THE NUMERICAL METHOD WILL CALCULATE THE TEMPERATURE IN THE CENTER OF EACH SECTION • THE SECTIONS AT BOUNDARIES ARE ONE-HALF OF THE THICKNESS OF THOSE IN THE INTERIOR OF THE SYSTEM

8. ONE DIMENSIONAL STEADY STATE • NUMERICAL METHOD REPRESENTS THE FIRST TEMPERATURE DERIVATIVE AS: WHERE THE TEMPERATURES ARE IN THE CENTER OF THE ADJACENT NODAL SECTIONS • SIMILARLY, THE SECOND DERIVATIVE IS REPRESENTED AS SHOWN IN EQUATION (5-9) • SUBSTITUTING THESE EXPRESSIONS INTO THE HEAT BALANCE FOR AN INTERNAL NODE AT STEADY STATE AS PER EQUATION (5-11):

9. ONE DIMENSIONAL STEADY STATE • FOR THE BOUNDARY NODES AT SURFACES, WHICH ARE ½ THE THICKNESS OF THE INTERNAL NODES AND INCLUDE THE BOUNDARY CONDITIONS, THE TYPES OF BALANCES INCLUDE: • SPECIFIED TEMPERATURE - DOES NOT REQUIRE A HEAT BALANCE SINCE THE VALUE IS GIVEN • SPECIFIED HEAT FLUX • AN INSULATED SURFACE, q` = 0, SO

10. ONE DIMENSIONAL STEADY STATE • OTHER HEAT BALANCES ARE USED FOR: • CONVECTION BOUNDARY CONDITION WHERE: • RADIATION BOUNDARY WHERE • COMBINATIONS (SEE EQUATIONS 5-26 THROUGH 5-28) • INTERFACES WITH OTHER SOLIDS (5-29)

11. ONE DIMENSIONAL STEADY STATE • WHEN ALL THE NODAL HEAT BALANCES ARE DEVELOPED, THEN THE SYSTEM CAN BE REGRESSED (DIRECTLY SOLVED) TO OBTAIN THE STEADY-STATE TEMPERATURES AT EACH NODE. • SYMMETRY CAN BE USED TO SIMPLIFY THE SYSTEM • THE RESULTING ADIABATIC SYSTEMS ARE TREATED AS INSULATED SURFACES

12. ITERATION TECHNIQUE • THE ALTERNATE METHOD OF SOLUTION IS TO ESTIMATE THE VALUES AT EACH POINT AND THEN ITERATE UNTIL THE VALUES REACH STABLE VALUES. • WHEN THERE IS NO HEAT GENERATION, THE EQUATIONS FOR THE INTERNAL NODES SIMPLIFY TO: • ITERATIVE CALCULATIONS CAN BE COMPLETED ON SPREADSHEETS OR BY WRITING CUSTOM PROGRAMS.

13. MULTI- DIMENSIONAL NUMERICAL MODELS

14. TWO DIMENSIONAL STEADY STATE CONDUCTION • BOUNDARY CONDITIONS • THE BASIC APPROACH USED FOR ONEDIMENSIONAL • NUMERICAL MODELING IS APPLIED IN TWO DIMENSIONAL MODELING • A TWO DIMENSIONAL MESH IS CONSTRUCTED OVER THE SURFACE OF THE AREA • TYPICALLY THE NODES ARE SUBSCRIPTED TO IDENTIFY THOSE IN THE x AND y DIRECTIONS, WITH A UNIT DEPTH IN THE z DIRECTION

15. TWO DIMENSIONAL STEADY STATE CONDUCTION • THE SIZE OF THE NODE IS DEFINED BY Δx AND Δy AND THESE ARE DEFINED AS 1 FOR A SQUARE UNIFORM MESH. • THE BASIC HEAT BALANCE EQUATION OVER AN INTERNAL NODE HAS THE FORM: • CRITERIA FOR THIS SIMPLIFIED MODEL INCLUDE CONSTANT k AND STEADY-STATE • WHEN THERE IS NO GENERATION, THIS • SIMPLIFIES TO

16. NODES AT BOUNDARIES • HEAT BALANCES FOR BOUNDARIES ARE MODELED USING PARTIAL SIZE ELEMENTS (REFER TO FIGURE 5-27) • ALONG A STRAIGHT SIDE THE HEAT BALANCE IS BASED ON TWO LONG AND TWO SHORT SIDE FACES. • THE EQUATION IS

17. TWO DIMENSIONAL STEADY STATE CONDUCTION • SIMILAR HEAT BALANCES ARE CONSTRUCTED • FOR OTHER SECTIONS (SEE EXAMPLE 5-3); • OUTSIDE CORNERS • INSIDE CORNERS • CONVECTION INTERFACES • INSULATED INTERFACES • RADIATION INTERFACES • CONDUCTION INTERFACES TO OTHERSOLIDS

18. TWO DIMENSIONAL STEADY STATE CONDUCTION • SOLUTIONS FOR THESE SYSTEMS ARE NORMALLY OBTAINED USING ITERATIVE TECHNIQUES OR USING • MATRIX INVERSION FOR n EQUATIONS/n UNKNOWNS • SIMPLIFICATION IS POSSIBLE USING SYMMETRY • IRREGULAR BOUNDARIES MAY BE APPROXIMATED BY A FINE RECTANGULAR MESH • MAY ALSO BE REPRESENTED BY A SERIES OF TRAPEZOIDS

19. CONVECTION FUNDAMENTALS

20. MECHANISM FOR CONVECTION • CONVECTION IS ENHANCED CONDUCTION • FLOW RESULTS IN MOVEMENT OF MOLECULES THAT WILL EFFECTIVELY INCREASE THE VALUE OF THE DRIVING FORCE (dT/dX) FOR CONDUCTION • CONVECTION OCCURS AT A SURFACE • NEWTON’S LAW OF COOLING APPLIES

21. MECHANISM FOR CONVECTION • HEAT FLUX AT THE SURFACE IS BASED ON THE TEMPERATURE PROFILE AT THE SURFACE (WHERE A ZERO VELOCITY FOR THE FLUID IS ASSUMED: • THE RESULTING DEFINITION OF h IS:

22. NUSSELT NUMBER • PROVIDES A RELATIVE MEASURE OF HEAT TRANSFER BY CONDUCTION VERSUS HEAT TRANSFER BY CONVECTION • THE VALUE OF THE L TERM IS ADJUSTED ACCORDING TO THE SYSTEM GEOMETRY

23. TYPES OF FLOWS • THERE ARE A WIDE RANGE OF FLUID FLOW TYPES • VALUES OF h ARE BASED ON CORRELATIONS • CORRELATIONS ARE BASED ON FLUID FLOW REGIME, GEOMETRY, AND FLUID CHARACTERISTICS

24. VISCOUS/INVISCID (FRICTIONLESS) INTERNAL/EXTERNAL COMPRESSIBLE/NON-COMPRESSIBLE LAMINAR/TURBULENT/TRANSITION NATURAL/FORCED CONVECTION STEADY/UNSTEADY ONE-TWO-THREE DIMENSIONAL FLOWS TYPES OF FLOWS

25. VELOCITY BOUNDARY LAYER • THERE IS A VELOCITY GRADIENT FROM THE HEAT TRANSFER SURFACE INTO THE FLOW REGIME. • AS THE FLOW INTERACTS WITH THE SURFACE, MOMENTUM IS TRANSFERRED INTO VELOCITY GRADIENTS NORMAL TO THE SURFACE

26. BOUNDARY LAYER • DEFINED AS THE REGION OVER WHICH THERE IS A CHANGE IN VELOCITY FROM THE SURFACE VALUE TO THE BULK VALUE • THE TYPE OF FLOW ADJACENT TO THE SURFACE IS CHARACTERIZED AS • LAMINAR – TURBULENT OR TRANSITION

27. BOUNDARY LAYER FLOWS • LAMINAR - SMOOTH FLOW WITH MINIMAL VELOCITY NORMAL TO THE SURFACE • TURBULENT - FLOW WITH SIGNIFICANT VELOCITY NORMAL TO THE SURFACE • THE TURBULENT LAYER MAY BE FURTHER SUBDIVIDED INTO THE LAMINAR SUBLAYER, THE TURBULENT LAYER, AND THE BUFFER LAYER • THE BREAKS OCCURS AT VALUES RELATIVE TO THE CHANGES IN VELOCITY WITH RESPECT TO DISTANCE • TRANSITION - THE REGION BETWEEN LAMINAR AND TURBULENT

28. VISCOSITY • DYNAMIC VISCOSITY - IS A MEASUREMENT OF THE CHANGE IN VELOCITY WITH RESPECT TO DISTANCE UNDER A SPECIFIED SHEAR STRESS • KINEMATIC VISCOSITY IS THE DYNAMIC VISCOSITYDIVIDED BY THE DENSITY AND HAS THE SAME UNITS AS THERMAL DIFFUSIVITY

29. FRICTION FACTOR • IS A VALUE RELATED TO THE SHEAR STRESS AS A FUNCTION OF VELOCITY AND VISCOSITY FOR A SYSTEM: • IT IS RELATED TO THE VELOCITY BOUNDARY LAYER AND HAS UNITS N/m2

30. THERMAL BOUNDARY LAYER • GENERAL CHARACTERIZATION IS THE SAME AS FOR THE VELOCITY BOUNDARY LAYER • THE PRANDTL NUMBER (DIMENSIONLESS RATIO) IS USED TO RELATE THE THERMAL AND VELOCITY BOUNDARY LAYERS:

31. CHARACTERIZATION OF FLOW REGIMES • REYNOLD’S NUMBER (DIMENSIONLESS) IS USED TO CHARACTERIZE THE FLOW REGIME: • THE CHANGES IN FLOW REGIME ARE CORRELATED WITH THE Re NUMBER

32. REYNOLD’S NUMBER PARAMETERS • THE VALUE FOR THE LENGTH TERM, D, CHANGES ACCORDING TO SYSTEM GEOMETRY • D IS THE LENGTH DOWN A FLAT PLATE • D IS THE DIAMETER OF A PIPE FOR INTERNAL OR EXTERNAL FLOWS • D IS THE DIAMETER OF A SPHERE OR THE EQUIVALENT DIAMETER OF A NON-SPHERICAL SHAPE

33. CONVECTION HEAT AND MOMENTUM ANALOGIES

34. TURBULENT FLOW HEAT TRANSFER • REYNOLD’S NUMBER (DIMENSIONLESS) IS USED TO CHARACTERIZE FLOW REGIMES • FOR FLAT PLATES (USING THE LENGTH FROM THE ENTRY FOR X) THE TRANSITION FROM LAMINAR TO TURBULENT FLOW IS APPROXIMATELY Re = 5 x 105 • FOR FLOW IN PIPES THE TRANSITION OCCURS AT ABOUT Re = 2100

35. TURBULENT FLOW • CHARACTERIZED BY FORMATION OF VORTICES OF FLUID PACKETS - CALLED EDDIES • EDDIES ADD TO THE EFFECTIVE DIFFUSION OF HEAT AND MOMENTUM, USING TIME AVERAGED VELOCITIES AND TEMPERATURES

36. FLAT PLATE SOLUTIONS • NONDIMENSIONAL EQUATIONS • DIMENSIONLESS VARIABLES ARE DEVELOPED TO ALLOW CORRELATIONS THAT CAN BE USED OVER A RANGE OF CONDITIONS • THE REYNOLD’S NUMBER IS THE PRIMARY TERM FOR MOMENTUM TRANSFER • USING STREAM FUNCTIONS AND BLASIUS DIMENSIONLESS SIMILARITY VARIABLE FOR VELOCITY, THE BOUNDARY LAYER THICKNESS CAN BE DETERMINED: • WHERE BY DEFINITION u = 0.99 u∞

37. FLAT PLATE SOLUTIONS • A SIMILAR DEVELOPMENT LEADS TO THE CALCULATION OF LOCAL FRICTION COEFFICIENTS ON THE PLATE (6-54):

38. HEAT TRANSFER EQUATIONS • BASED ON CONSERVATION OF ENERGY • DIMENSIONLESS CORRELATIONS BASED ON THE PRANDTL AND NUSSELT NUMBERS • A DIMENSIONLESS TEMPERATURE IS INCLUDED WITH THE DIMENSIONLESS VELOCITY EXPRESSIONS: • WHICH CAN BE USED TO DETERMINE THE THERMAL BOUNDARY LAYER THICKNESS FOR LAMINAR FLOW OVER PLATES (6-63):

39. HEAT TRANSFER COEFFICIENT • CORRELATIONS FOR THE HEAT TRANSFER COEFFICIENT FOR LAMINAR FLOW OVER PLATES ARE OF THE FORM: http://electronics-cooling.com/articles/2002/2002_february_calccorner.php

40. COEFFICIENTS OF FRICTION AND CONVECTION • THE GENERAL FUNCTIONS FOR PLATES ARE BASED ON THE AVERAGED VALUES OF FRICTION AND HEAT TRANSFER COEFFICIENTS OVER A DISTANCE ON A PLATE • FOR FRICTION COEFFICIENTS: • FOR HEAT TRANSFER COEFFICIENTS:

41. MOMENTUM AND HEAT TRANSFER ANALOGIES • REYNOLD’S ANALOGY APPLIES WHEN Pr = 1 (6-79): • USING THE STANTON NUMBER DEFINITION: • THE REYNOLD’S ANALOGY IS EXPRESSED (6-80): .

42. MODIFIED ANALOGIES • MODIFIED REYNOLD’S ANALOGY OR CHILTON-COLBURN ANALOGY (EQN, 6-83):

43. EXTERNAL CONVECTION FUNDAMENTALS

44. DRAG AND HEAT TRANSFER RELATIONSHIPS • TYPES OF DRAG FORCES • VISCOUS • DUE TO VISCOSITY OF FLUID ADHERING TO THE SURFACE • FORCES ARE PARALLEL TO THE SURFACE • SOMETIMES CALLED FRICTION DRAG • PRESSURE • DUE TO FLUID FLOW NORMAL TO THE SURFACE • FORCES ARE NORMAL TO THE SURFACE • SOMETIMES CALLED FORM DRAG

45. DRAG COEFFICIENTS • DRAG FORCES CAN MODELED USING DRAG COEFFICIENTS • FOR FORM DRAG, THE AREA IS NORMAL TO THE FLOW : • FOR VISCOUS DRAG, THE AREA IS PARALLEL TO THE FLOW:

46. DRAG CORRELATIONS • VISCOUS DRAG IS CORRELATED USING THE REYNOLD’S NUMBER WHERE THE LENGTH TERM IS IN THE DIRECTION OF FLOW • FORM DRAG IS CORRELATED WITH THE REYNOLD’S NUMBER WHERE THE LENGTH TERM IS A CHARACTERISTIC DIMENSION OF THE AREA NORMAL TO FLOW • REAL SYSTEMS TEND TO EXHIBIT BOTH FORMS OF DRAG • EXTREME CASE FOR FORM DRAG IS REPRESENTED BY THE DEVICE SHOWN IN THIS PHOTO • THERE IS SOME VISCOUS DRAG, BUT IT IS NOT SIGNIFICANT COMPARED TO THE FORM DRAG http://www.photoclub.eu/photogallery/data/514/VW.jpg

47. RELATIONSHIP BETWEEN DRAG AND HEAT TRANSFER • THE REYNOLD’S ANALOGY LINKS HEAT AND MOMENTUM TRANSFER USING DIMENSIONLESS NUMBERS: Nu = Nu (Re,Pr) • LOCAL AND OVERALL VALUES • LOCAL FRICTION FACTORS AND HEAT TRANSFER COEFFICIENTS CAN BE CALCULATED AT A SPECIFIC LOCATION USING LOCAL CORRELATIONS • AVERAGE OVERALL VALUES FOR COEFFICIENTS CAN BE OBTAINED FROM THE LOCAL VALUES BY INTEGRATING OVER THE FLOW LENGTH

48. HEAT TRANSFER FACTORS • FILM TEMPERATURES ARE USED TO CALCULATE BOUNDARY LAYER PROPERTIES • SYSTEMS CAN BE MODELED USING TWO LIMITING CONDITIONS • CONSTANT SURFACE TEMPERATURE • CONSTANT SURFACE HEAT RATE

49. FLOW OVER FLAT PLATES • FLOW REGIMES CHANGE AS FLOW MOVES DOWN A PLATE • THE ACTUAL TRANSITION BETWEEN REGIMES IS BASED ON THE ROUGHNESS FACTOR FOR THE MATERIAL • ROUGHNESS IS CALCULATED BY MEASURING PRESSURE DROP AND DOES NOT RELATE TO ACTUAL SURFACE DIMENSIONS

50. FLOW REGIMES • TYPICAL VALUES FOR THE TRANSITION FROM LAMINAR TO TURBULENT ARE AT Re VALUES OF ABOUT 5 X 105 • LAMINAR CORRELATIONS Re < 5x105 • FRICTION FACTORS • LOCAL • AVERAGE