CHE/ME 109 Heat Transfer in Electronics REVIEW FOR SECOND
MID-TERM EXAM
Slide 2
ONE DIMENSIONAL NUMERICAL MODELS
Slide 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
Slide 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
Slide 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
Slide 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 = in text):
Slide 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
Slide 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):
Slide 9
ONE DIMENSIONAL STEADY STATE FOR THE BOUNDARY NODES AT
SURFACES, WHICH ARE 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
Slide 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)
Slide 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
Slide 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.
Slide 13
MULTI- DIMENSIONAL NUMERICAL MODELS
Slide 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
Slide 15
TWO DIMENSIONAL STEADY STATE CONDUCTION THE SIZE OF THE NODE IS
DEFINED BY x AND y 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
Slide 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
Slide 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
Slide 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
Slide 19
CONVECTION FUNDAMENTALS
Slide 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 NEWTONS LAW OF COOLING APPLIES
Slide 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:
Slide 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
Slide 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
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
Slide 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
Slide 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
Slide 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
Slide 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/m 2
Slide 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:
Slide 31
CHARACTERIZATION OF FLOW REGIMES REYNOLDS NUMBER
(DIMENSIONLESS) IS USED TO CHARACTERIZE THE FLOW REGIME: THE
CHANGES IN FLOW REGIME ARE CORRELATED WITH THE Re NUMBER
Slide 32
REYNOLDS 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
Slide 33
CONVECTION HEAT AND MOMENTUM ANALOGIES
Slide 34
TURBULENT FLOW HEAT TRANSFER REYNOLDS 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
Slide 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
Slide 36
FLAT PLATE SOLUTIONS NONDIMENSIONAL EQUATIONS DIMENSIONLESS
VARIABLES ARE DEVELOPED TO ALLOW CORRELATIONS THAT CAN BE USED OVER
A RANGE OF CONDITIONS THE REYNOLDS 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
Slide 37
FLAT PLATE SOLUTIONS A SIMILAR DEVELOPMENT LEADS TO THE
CALCULATION OF LOCAL FRICTION COEFFICIENTS ON THE PLATE
(6-54):
Slide 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):
Slide 39
HEAT TRANSFER COEFFICIENT HEAT TRANSFER COEFFICIENT
CORRELATIONS FOR THE HEAT TRANSFER COEFFICIENT FOR LAMINAR FLOW
OVER PLATES ARE OF THE FORM: http://electronics-
cooling.com/articles/2002/2 002_february_calccorner.ph p
Slide 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:
Slide 41
MOMENTUM AND HEAT TRANSFER ANALOGIES REYNOLDS ANALOGY APPLIES
WHEN Pr = 1 (6-79): USING THE STANTON NUMBER DEFINITION: THE
REYNOLDS ANALOGY IS EXPRESSED (6-80):.
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
Slide 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:
Slide 46
DRAG CORRELATIONS VISCOUS DRAG IS CORRELATED USING THE REYNOLDS
NUMBER WHERE THE LENGTH TERM IS IN THE DIRECTION OF FLOW FORM DRAG
IS CORRELATED WITH THE REYNOLDS 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/photog allery/data/514/VW.jpg
Slide 47
RELATIONSHIP BETWEEN DRAG AND HEAT TRANSFER THE REYNOLDS
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
Slide 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
Slide 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
Slide 50
FLOW REGIMES TYPICAL VALUES FOR THE TRANSITION FROM LAMINAR TO
TURBULENT ARE AT Re VALUES OF ABOUT 5 X 10 5 LAMINAR CORRELATIONS
Re < 5x10 5 FRICTION FACTORS LOCAL AVERAGE
Slide 51
FLOW REGIMES HEAT TRANSFER COEFFICIENTS LOCAL - CONSTANT
SURFACE TEMPERATURE LOCAL - CONSTANT HEAT FLUX AVERAGE - CONSTANT
SURFACE TEMPERATURE OR CONSTANT HEAT RATE:
Slide 52
TURBULENT CORRELATIONS 5x10 5 < Re < 10 7 FRICTION
FACTORS LOCAL AVERAGE HEAT TRANSFER COEFFICIENTS LOCAL AVERAGE
Slide 53
EXTERNAL CONVECTION IN SPECIFIC SYSTEMS
Slide 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
Slide 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
Slide 56
FLOW ACROSS A RANGE OF EXTERNAL FORMS A MORE GENERAL FORM IS Nu
= CRe m Pr n 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 Pr Surf ARE EVALUATED AT THE MEAN STREAM
TEMPERATURE
Slide 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%
Slide 58
INTERNAL FORCED CONVECTION FUNDAMENTALS
Slide 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
Slide 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
Slide 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
Slide 62
MEAN (MIXING CUP) TEMPERATURE IS CALCULATED AS THE AVERAGE
TEMPERATURE IN A DUCT CROSS SECTION THE EQUATION FOR CALCULATION
IS:
Slide 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
Slide 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
Slide 65
NON-CIRCULAR DUCTS 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
Slide 66
LIMITING SYSTEMS IDEAL SYSTEM MODELS ARE BASED ON EITHER
CONSTANT SURFACE TEMPERATURE OR CONSTANT SURFACE FLUX FOR CONSTANT
SURFACE HEATING, THE VALUE OF T = T s - T m STAYS CONSTANT T s
INCREASES AS T m INCREASES
Slide 67
LIMITING SYSTEMS FOR CONSTANT VALUES OF C p AND A s THE RATE OF
INCREASE CAN BE EVALUATED AS: THIS RELATIONSHIP DOES NOT APPLY IN
THE ENTRY LENGTH
Slide 68
LIMITING SYSTEMS FOR CONSTANT SURFACE TEMPERATURE THE VALUE OF
T IS ALWAYS CHANGING EVENTUALLY THE BULK TEMPERATURE WILL MATCH THE
WALL TEMPERATURE THE DIMENSIONLESS TEMPERATURE CAN BE EXPRESSED AS
AN EXPONENTIAL DECAY FUNCTION:
Slide 69
CONSTANT SURFACE TEMPERATURE TOTAL HEAT TRANSFER OVER THE DUCT
USE AN AVERAGE T FOR THE CALCULATIONS MATH AVERAGE T : LOG-MEAN
AVERAGE T
Slide 70
FLOW IN TUBES
Slide 71
LAMINAR FLOW - MEAN VELOCITY MEAN VELOCITY FROM THE INTEGRATED
AVERAGE OVER THE RADIUS: IN TERMS OF THE MEAN VELOCITY
Slide 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
Slide 73
CONSTANT HEAT FLUX SOLUTIONS BOUNDARY CONDITIONS: AT THE WALL T
= Ts @ r = R AT THE CENTERLINE FROM SYMMETRY:
Slide 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
Slide 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
Slide 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:
Slide 77
TURBULENT FLOW FOR VARIOUS ROUGHNESS VALUES (MEASURED BY
PRESSURE DROP): TYPICAL ROUGHNESS VALUES ARE IN TABLES 8.2 AND
8.3
Slide 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)
Slide 79
NATURAL CONVECTION FUNDAMENTALS
Slide 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
http://blogs.sun.com/staso/resource/cumulonimbus-cloud-akbhhf-sw.jpg
Slide 81
DENSITY DIFFERENCES DEFINED IN TERMS OF VOLUME EXPANSION
COEFFICIENT DERIVATION OF CHANGES IN DENSITY FOR FLUIDS: VOLUME
EXPANSIVITY: ISOTHERMAL COMPRESSIBILITY:
Slide 82
DENSITY DIFFERENCES FOR IDEAL GASES: SO AROUND AMBIENT
TEMPERATURE = 3.3x10 -3 K -1 = 1.8x10 -3 R -1 FOR LIQUIDS THE
VALUES ARE ON THE ORDER OF = 3x10 -4 K
Slide 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):
Slide 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
Slide 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
Slide 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 > 10 9. 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
Slide 87
NATURAL CONVECTION OVER SPECIFIC SHAPES VERTICAL CYLINDERS CAN
BE ANALYZED WITH THE VERTICAL PLATE EQUATIONS AS LONG AS THE
DIAMETER IS LARGE ENOUGH
Slide 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 < 10 12 FOR TURBULENT FLOW Ra
> 10 9 :
Slide 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. 199- 264
Slide 90
SPECIFIC NATURAL CONVECTION MODELS
Slide 91
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
Slide 92
EXTENDED SURFACES FOR CONSTANT HEAT FLUX:
Slide 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
Slide 94
OPTIMUM VERTICAL FIN SPACING ISOTHERMAL FINS: OPTIMUM NUSSELT:
Nu = 1.307 = hS opt /K TRANSFER FROM BOTH SIDES (EQN 9-32): S opt =
2.714(S 3 L/Ra s ) 1/4 CONSTANT HEAT FLUX TRANSFER FROM BOTH SIDES
(EQN 9-37): S opt = 2.12(S 4 L/Ra* s )1/5 PROPERTIES FOR THESE
CORRELATIONS ARE ALL BASED ON AN AVERAGE VALUE FOR THE FILM
TEMPERATURE
Slide 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:
Slide 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
Slide 97
CONCENTRIC CYLINDERS FOR VERTICAL SYSTEMS, THE VERTICAL
RECTANGULAR CORRELATIONS MAY BE USED FOR HORIZONTAL SYSTEMS
EQUATIONS USE A MODIFIED CONDUCTION MODEL: k Eff IS CALCULATED
FROM: L = D o - D i AND L c = (D o - Di)/2 PROPERTIES ARE BASED ON
AVERAGE TEMPERATURE
Slide 98
COMBINED NATURAL & FORCED CONVECTION FACTOR APPLIED WHEN
MODELING A SYSTEM WITH BOTH FORMS OF CONVECTION IS Gr/Re 2 WHEN
Gr/Re 2 > 1, THEN FORCED CONVECTION CAN BE IGNORED
Slide 99
COMBINED NATURAL & FORCED CONVECTION FOR CONDITIONS WHERE
0.1 < Gr/Re 2 < 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