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

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 VISCOSITY DIVIDED 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:

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

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

51. FLOW REGIMES HEAT TRANSFER COEFFICIENTS
LOCAL - CONSTANT SURFACE TEMPERATURE
LOCAL - CONSTANT HEAT FLUX
AVERAGE - CONSTANT SURFACE TEMPERATURE OR CONSTANT HEAT RATE:

52. TURBULENT CORRELATIONS
5x105 < Re < 107
FRICTION FACTORS
LOCAL
AVERAGE
HEAT TRANSFER COEFFICIENTS

53. EXTERNAL CONVECTION IN SPECIFIC SYSTEMS

54. FLOW PARALLEL TO THE CYLINDER AXIS
MOMENTUM AND HEAT TRANSFER IS MODELED USING THE FLAT PLATE CORRELATIONS
FOR SPHERES THE SAME EFFECTS ARE PRESENT IN THREE DIMENSIONS
PRESSURE DROP CORRELATIONS ARE SHOWN IN FIGURE 7-17

55. HEAT TRANSFER COEFFICIENTS
HEAT TRANSFER COEFFICIENTS FOR CYLINDERS AND SPHERES ARE OF THE FORM:
EXAMPLES ARE (7-35) AND (7-36)
PROPERTIES ARE EVALUATED AT FILM TEMPERATURES, EXCEPT FOR THE WALL VISCOSITY
THESE CORRELATIONS INCLUDE A LAMINAR AND A TURBULENT PORTION

56. FLOW ACROSS A RANGE OF EXTERNAL FORMS
A MORE GENERAL FORM IS Nu = CRemPrn
VALUES FOR FLOW ACROSS A RANGE OF EXTERNAL FORMS ARE SHOWN IN TABLE 7-1
ALL FLUID PROPERTIES ARE BASED ON THE FILM TEMPERATURE
A VARIATION OF THIS EXPRESSION IS:
FOR THIS VERSION ALL PROPERTIES EXCEPT THE PrSurf ARE EVALUATED AT THE MEAN STREAM TEMPERATURE

57. LIMITATIONS FOR CORRELATIONS
THESE CORRELATIONS ARE ALL BASED ON:
A SPECIFIC FLUID
SPECIFIC FLOW REGIMES
SPECIFIC SURFACE ROUGHNESS
SPECIFIC RANGES OF Pr AND Re
EXPECTED ACCURACY IS + 20%

58. INTERNAL FORCED CONVECTION FUNDAMENTALS

59. CONVECTION HEAT TRANSFER CORRELATIONS
BASED ON MOMENTUM TRANSFER MODELS
ERRORS FOR CORRELATIONS + 20%
MINOR FACTORS SUCH AS VISCOUS HEATING MAY END UP IN THE NOISE FOR THESE CALCULATIONS, SO ARE IGNORED IN MANY SYSTEMS

60. MEAN VELOCITY AND MEAN TEMPERATURE
FLOW REGIMES
LAMINAR FLOW IS DEFINED BY Re < 2300
THE VELOCITY PROFILE IS TYPICALLY PARABOLIC FOR DEVELOPED LAMINAR FLOW
SEE DEVELOPMENT IN SECTION 8-2

61. MEAN VELOCITY
THE VELOCITY IS ZERO- VALUED AT EACH WALL AND GOES TO A MAXIMUM IN THE CENTER
THE MEAN VELOCITY IS OBTAINED FROM
NOTE THE MEAN VELOCITY WILL NOT BE AT THE CENTER OF THE FLOW

62. MEAN (MIXING CUP) TEMPERATURE
IS CALCULATED AS THE AVERAGE TEMPERATURE IN A DUCT CROSS SECTION
THE EQUATION FOR CALCULATION IS:

63. TURBULENT FLOW DEFINED BY Re>10000
AVERAGE VELOCITY AND MEAN TEMPERATURES ARE CALCULATED THE SAME AS FOR LAMINAR SYSTEMS
THE TURBULENT PROFILE IS TYPICALLY UNIFORM EXCEPT AT THE SURFACES

64. TURBULENT/TRANSITION FLOW
THE VALUES FOR AVERAGE VELOCITY AND MEAN TEMPERATURES ARE VERY CLOSE TO THE CENTERLINE VALUES FOR TURBULENT FLOW
TRANSITION FLOW IS 2300 < Re < 10000
THERE ARE NO CORRELATIONS FOR THE TRANSITION REGION

65. NON-CIRCULAR DUCTS ADAPTING THESE CORRELATIONS TO NON- CIRCULAR DUCTS
ACCOMPLISHED USING THE HYDRAULIC DIAMETER IN THE SAME EQUATIONS.
SAME LIMITS FOR FLOW REGIMES ARE NORMALLY APPLIED TO NON-CIRCULAR DUCTS

66. LIMITING SYSTEMS
IDEAL SYSTEM MODELS ARE BASED ON EITHER CONSTANT SURFACE TEMPERATURE OR CONSTANT SURFACE FLUX
FOR CONSTANT SURFACE HEATING, THE VALUE OF ΔT = Ts - Tm STAYS CONSTANT
Ts INCREASES AS Tm INCREASES

67. LIMITING SYSTEMS
FOR CONSTANT VALUES OF Cp AND As THE RATE OF INCREASE CAN BE EVALUATED AS:
THIS RELATIONSHIP DOES NOT APPLY IN THE ENTRY LENGTH

68. LIMITING SYSTEMS FOR CONSTANT SURFACE TEMPERATURE
THE VALUE OF ΔT IS ALWAYS CHANGING
EVENTUALLY THE BULK TEMPERATURE WILL MATCH THE WALL TEMPERATURE
THE DIMENSIONLESS TEMPERATURE CAN BE EXPRESSED AS AN EXPONENTIAL DECAY FUNCTION:

69. CONSTANT SURFACE TEMPERATURE
TOTAL HEAT TRANSFER OVER THE DUCT
USE AN AVERAGE ΔT FOR THE CALCULATIONS
MATH AVERAGE ΔT:
LOG-MEAN AVERAGE ΔT

70. FLOW IN TUBES

71. LAMINAR FLOW - MEAN VELOCITY
MEAN VELOCITY FROM THE INTEGRATED AVERAGE OVER THE RADIUS:
IN TERMS OF THE MEAN VELOCITY

72. HEAT TRANSFER TO LAMINAR FLUID FLOWS IN TUBES
ENERGY BALANCE ON A CYLINDRICAL VOLUME IN
LAMINAR FLOW YIELDS:
SOLUTION TO THIS EQUATION USES BOUNDARY CONDITIONS BASED ON EITHER CONSTANT HEAT FLUX OR CONSTANT SURFACE TEMPERATURE

73. CONSTANT HEAT FLUX SOLUTIONS
BOUNDARY CONDITIONS:
AT THE WALL T = r = R
AT THE CENTERLINE FROM SYMMETRY:

74. CONSTANT WALL TEMPERATURE
SUBSTITUTING THE VELOCITY PROFILE INTO THIS EQUATION YIELDS AN EQUATION IN THE FORM OF AN INFINITE SERIES
RESULTING VALUES SHOW: Nu = 3.657

75. HEAT TRANSFER IN NON-CIRCULAR TUBES
USES THE SAME APPROACH AS DESCRIBED FOR CIRCULAR TUBES
CORRELATIONS USE Re AND Nu BASED ON THE HYDRAULIC DIAMETER:
SEE TABLE 8-1 FOR LIMITING VALUES FOR f AND Nu BASED ON SYSTEM GEOMETRY AND THERMAL CONFIGURATION

76. TURBULENT FLOW IN TUBES
FRICTION FACTORS ARE BASED ON CORRELATIONS FOR VARIOUS SURFACE FINISHES (SEE PREVIOUS FIGURE FOR f VS. Re)
FOR SMOOTH TUBES:

77. TURBULENT FLOW
FOR VARIOUS ROUGHNESS VALUES (MEASURED BY PRESSURE DROP):
TYPICAL ROUGHNESS VALUES ARE IN TABLES 8.2 AND 8.3

78. TURBULENT FLOW HEAT TRANSFER IN TUBES
FOR FULLY DEVELOPED FLOW DITTUS-BOELTER EQUATION:
OTHER EQUATIONS ARE INCLUDED AS (8-69) & (8-70)
SPECIAL CORRELATIONS ARE FOR LOW Pr NUMBERS (LIQUID METALS) (8-71) AND (8-72)

79. NATURAL CONVECTION FUNDAMENTALS

80. NATURAL CONVECTION MECHANISMS
NATURAL CONVECTION IS THE RESULT OF LOCALIZED DENSITY DIFFERENCES
THESE CAN BE DUE TO DIFFERENCES IN COMPOSITIONS
FOR HEAT TRANSFER THEY ARE GENERALLY RELATED TO TEMPERATURE DIFFERENCES
CONCENTRATION BASED CONVECTION INCLUDES CLOUD FORMATIONS

81. DENSITY DIFFERENCES DEFINED IN TERMS OF VOLUME EXPANSION COEFFICIENT
DERIVATION OF CHANGES IN DENSITY FOR FLUIDS:
VOLUME EXPANSIVITY:
ISOTHERMAL COMPRESSIBILITY:

82. DENSITY DIFFERENCES FOR IDEAL GASES: SO AROUND AMBIENT TEMPERATURE
β = 3.3x10-3 K-1 = 1.8x10-3 R-1
FOR LIQUIDS THE VALUES ARE ON THE ORDER OF β = 3x10-4 K

83. GRASHOF NUMBER
FLUID MOTION OCCURS DUE TO BOUYANCY EFFECTS AS PER (FIGURE 9-6)
ONCE THE FLUID IS IN MOTION, THEN VISCOUS EFFECTS OCCUR
COMPLETING A MOMENTUM BALANCE FOR A NATURAL CONVECTION FLOW WITH VELOCITIES IN THE x AND y DIRECTION (u AND v RESPECTIVELY) CONSIDERED YIELDS (9-13):

84. GRASHOF NUMBER
GRASHOF NUMBER IS THE RATIO OF THE BOUYANCY FORCES TO THE VISCOUS FORCES
VALUE OF THE GRASHOF NUMBER CAN BE LINKED TO FLOW REGIMES FOR NATURAL CONVECTION

85. NATURAL CONVECTION OVER SURFACES
FOR NATURAL CONVECTION HEAT TRANSFER PROCESSES
THE CORRELATIONS FOR HEAT TRANSFER COEFFICIENTS ARE BASED ON THE RAYLEIGH NUMBER: Ra = GrPr
Ra IS THE NATURAL CONVECTION EQUIVALENT OF THE PECLET NUMBER, Pe = RePr FOR FORCED CONVECTION

86. NATURAL CONVECTION OVER SPECIFIC SHAPES
VERTICAL FLAT PLATES
BOUNDARY LAYER STAYS AGAINST THE SURFACE AND THE FLOW REGIME CHANGES WITH DISTANCE.
TRANSITION TO TURBULENCE IS GENERALLY DEFINED IN TERMS OF THE Ra NUMBER AT Ra > 109.
EQUATIONS ARE DEVELOPED
FOR CONSTANT TEMPERATURE OR CONSTANT HEAT RATE
BASED ON FILM TEMPERATURE EQUAL TO (Ts - T )/2
APPLY EQUALLY TO HOT OR COLD WALLS, RELATIVE TO T∞

87. NATURAL CONVECTION OVER SPECIFIC SHAPES
VERTICAL CYLINDERS CAN BE ANALYZED WITH THE VERTICAL PLATE EQUATIONS AS LONG AS THE DIAMETER IS LARGE ENOUGH

88. HORIZONTAL CYLINDERS
THE BOUNDARY LAYER FORMS AROUND THE RADIUS AS SHOWN IN FIGURE 9-12
SINGLE CORRELATION IS PROVIDED (9-25) APPLIES TO
LAMINAR CONDITIONS Ra < 1012
FOR TURBULENT FLOW Ra > 109:

89. OTHER CORRELATIONS
FOR CONSTANT SURFACE TEMPERATURE , VALUES ARE BASED ON THE GENERAL FORMULATION:
SPHERES ARE MODELED USING (9-26)
FROM IRVINE & HARTNETT (Eds), ADVANCES IN HEAT TRANSFER, Vol 11, 1975, Pp

90. SPECIFIC NATURAL CONVECTION MODELS

91. SPECIFIC NATURAL CONVECTION MODELS
EXTENDED SURFACES
THE NUSSELT NUMBER FOR FINNED SYSTEMS IS BASED ON THE SPACING BETWEEN FINS, S, AND THE FIN HEIGHT, L
FOR CONSTANT SURFACE TEMPERATURE

92. EXTENDED SURFACES
FOR CONSTANT HEAT FLUX:

93. VERTICAL FINS PARAMETERS FOR THESE EQUATIONS:
VERTICAL ISOTHERMAL FINS (EQN 9-31)
TRANSFER FROM BOTH SIDES: C1 = 576, C2 = 2.87
ONE SIDE ADIABATIC:
C1 = 144, C2 = 2.87
VERTICAL CONSTANT HEAT FLUX FIND (EQN 9-36)
TRANSFER FROM BOTH SIDES: C1 = 48, C2 = =2.51 ONE SIDE ADIABATIC: C1 = 24, C1 = 2.51

94. OPTIMUM VERTICAL FIN SPACING
ISOTHERMAL FINS:
OPTIMUM NUSSELT: Nu = = hSopt/K
TRANSFER FROM BOTH SIDES (EQN 9-32): Sopt = 2.714(S3L/Ras) 1/4
CONSTANT HEAT FLUX
TRANSFER FROM BOTH SIDES (EQN 9-37): Sopt = 2.12(S4L/Ra*s)1/5
PROPERTIES FOR THESE CORRELATIONS ARE ALL BASED ON AN AVERAGE VALUE FOR THE FILM TEMPERATURE

95. NATURAL CONVECTION INSIDE ENCLOSURES
THERE ARE MANY RESEARCH PROJECTS FOR THIS SYSTEM, SO THEREFORE MANY CORRELATIONS
HEAT FLUX ACROSS AN ENCLOSURE IS TYPICALLY EXPRESSED AS Q = hA(T1 - T2)
h DEPENDS STRONGLY ON THE ASPECT RATIO, H/L
THE Ra NUMBER FOR THIS SYSTEM IS DEFINED IN TERMS OF THE SPACING BETWEEN HEATED PLATES, L:

96. NATURAL CONVECTION INSIDE ENCLOSURES
FOR LOW RALEIGH NUMBERS, Ra < 1000, DUE TO CLOSE PLATE SPACING:
THERE IS MINIMAL BOUYANCY DRIVEN FLOW
THIS BECOMES A CONDUCTION SYSTEM

97. CONCENTRIC CYLINDERS
FOR VERTICAL SYSTEMS, THE VERTICAL RECTANGULAR CORRELATIONS MAY BE USED
FOR HORIZONTAL SYSTEMS
EQUATIONS USE A MODIFIED CONDUCTION MODEL:
kEff IS CALCULATED FROM:
L = Do - Di AND Lc = (Do - Di)/2
PROPERTIES ARE BASED ON AVERAGE TEMPERATURE

98. COMBINED NATURAL & FORCED CONVECTION
FACTOR APPLIED WHEN MODELING A SYSTEM WITH BOTH FORMS OF CONVECTION IS Gr/Re2
WHEN Gr/Re2 << 1, THEN NATURAL CONVECTION CAN BE IGNORED
WHEN Gr/Re2 >> 1, THEN FORCED CONVECTION CAN BE IGNORED

99. COMBINED NATURAL & FORCED CONVECTION
FOR CONDITIONS WHERE 0.1 < Gr/Re2 < 10, THEN BOTH MECHANISMS ARE SIGNIFICANT
THE NUSSELT FOR THIS COMBINED CONDITION IS TYPICALLY MODELED WITH
n = 3 FOR A WIDE RANGE OF SYSTEMS
n = 7/2 OR 4 APPEARS TO WORK BETTER FOR TRANSVERSE FLOWS OVER HORIZONTAL PLATES OR HORIZONTAL CYLINDERS