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

ONE DIMENSIONAL NUMERICAL MODELS

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

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

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

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):

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

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):

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

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)

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

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.

MULTI- DIMENSIONAL NUMERICAL MODELS

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

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

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

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

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

CONVECTION FUNDAMENTALS

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

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:

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

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

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

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

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

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

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

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

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:

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

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

CONVECTION HEAT AND MOMENTUM ANALOGIES

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

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

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∞

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

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):

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

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:

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): .

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

EXTERNAL CONVECTION FUNDAMENTALS

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

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:

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

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

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

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

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

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

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

EXTERNAL CONVECTION IN SPECIFIC SYSTEMS

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

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

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

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%

INTERNAL FORCED CONVECTION FUNDAMENTALS

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

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

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

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

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

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

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

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

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

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:

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

FLOW IN TUBES

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

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

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

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

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

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

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

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)

NATURAL CONVECTION FUNDAMENTALS

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

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

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

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):

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

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

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∞

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

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:

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

SPECIFIC NATURAL CONVECTION MODELS

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

EXTENDED SURFACES FOR CONSTANT HEAT FLUX:

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

OPTIMUM VERTICAL FIN SPACING ISOTHERMAL FINS: OPTIMUM NUSSELT: Nu = 1.307 = 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

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:

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

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

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

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