CHE/ME 109 CHE/ME 109 Heat Transfer in Heat Transfer in

ElectronicsElectronics

LECTURE 8 – SPECIFIC LECTURE 8 – SPECIFIC CONDUCTION MODELSCONDUCTION MODELS

SOLUTIONS FOR EXTENDED (FINNED) SURFACES

FINS ARE ADDED TO A SURFACE TO PROVIDE ADDITIONAL HEAT TRANSFER AREA

THE TEMPERATURE OF THE FIN RANGES FROM THE HIGH VALUE AT THE BASE TO A GRADUALLY LOWER VALUE AS THE DISTANCE INCREASES FROM THE BASE

SOLUTIONS FOR FINS

BASIC HEAT BALANCE OVER AN ELEMENT OF THE FIN INCLUDES CONDUCTION FROM THE BASE, CONDUCTION TO THE TIP, AND CONVECTION TO THE SURROUNDINGS WHICH MATHEMATICALLY IS

SOLUTIONS FOR FINS FOR UNIFORM VALUES

OF k AND h, THIS EQUATION CAN BE WRITTEN AS:

THE GENERAL THE GENERAL SOLUTION TO THIS SOLUTION TO THIS SECOND-ORDER LINEAR SECOND-ORDER LINEAR DIFFERENTIAL DIFFERENTIAL EQUATION IS: EQUATION IS:

AT THE BOUNDARY AT THE BOUNDARY CONDITION CONDITION REPRESENTED BY THE REPRESENTED BY THE BASED CONNECTION TO BASED CONNECTION TO THE PLATE: T = TTHE PLATE: T = To AT x = 0, THE SOLUTION BECOMES:

SOLUTIONS FOR FINS

NEED ONE MORE BOUNDARY CONDITION TO SOLVE FOR THE ACTUAL VALUES

THERE ARE 3 CONDITIONS THAT PROVIDE ALTERNATE SOLUTIONS

INFINITELY LONG FIN SO THE TIP TEMPERATURE APPROACHES T∞:

SOLUTIONS FOR FINS SO THE FINAL FORM OF THIS MODEL

IS AN EXPONENTIALLY DECREASING PROFILE

WITH THIS PROFILE, THE TOTAL WITH THIS PROFILE, THE TOTAL HEAT TRANSFER CAN BE EVALUATEDHEAT TRANSFER CAN BE EVALUATED

CONSIDERING THE CONDUCTION CONSIDERING THE CONDUCTION THROUGH THE BASE AS EQUAL TO THROUGH THE BASE AS EQUAL TO THE TOTAL CONVECTION THE TOTAL CONVECTION

SOLUTIONS FOR FINS

TAKING THE DERIVATIVE OF (3-60) AND SUBSTITUTING AT x = 0, YIELDS

THE SAME RESULT COMES FROM A THE SAME RESULT COMES FROM A CALCULATION OF THE TOTAL CALCULATION OF THE TOTAL CONVECTED HEAT CONVECTED HEAT

SOLUTIONS FOR FINS

FINITE LENGTH WITH INSULATED TIP OR INSIGNIFICANT SO THE TEMPERATURE GRADIENT AT x = L WILL BE

VALUES CALCULATED FOR CVALUES CALCULATED FOR C1 AND C2 USING THIS BOUNDARY CONDITION ARE

SOLUTIONS FOR FINS THIS LEADS TO A TEMPERATURE PROFILE OF THE

FORM:

THE TOTAL HEAT FROM THIS SYSTEM CAN BE EVALUATED USING THE TEMPERATURE GRADIENT AT THE BASE TO YIELD

SOLUTIONS FOR FINS ALLOWING FOR CONVECTION AT THE

TIP THE CORRECTED LENGTH (3-66)

APPROACH CAN BE USED WITH EQUATIONS (3-64 AND 3-65)

ALTERNATELY, ALLOWING FOR A DIFFERENT FORM FOR THE CONVECTION COEFFICIENT AT THE TIP, hL, THEN THE HEAT BALANCE AT THE TIP IS

SOLUTIONS FOR FINS

THE RESULTING TEMPERATURE PROFILE IS

AND THE TOTAL HEAT TRANSFER BECOMES

FIN EFFICIENCY THE RATIO OF ACTUAL HEAT TRANSFER TO

IDEAL HEAT TRANSFER WITH A FIN IDEAL TRANSFER ASSUMES THE ROOT

TEMPERATURE EXTENDS OUT THE LENGTH OF THE FIN

REAL TRANSFER IS BASED ON THE ACTUAL TEMPERATURE PROFILE

FOR THE LONG FIN

FIN EFFICIENCY SIMILARLY, FOR A FIN WITH AN INSULATED TIP:

FIN EFFECTIVENESS INDICATES HOW MUCH THE FIN EFFECTIVENESS INDICATES HOW MUCH THE TOTAL HEAT TRANSFER INCREASES RELATIVE TO TOTAL HEAT TRANSFER INCREASES RELATIVE TO THE NON-FINNED SURFACETHE NON-FINNED SURFACE

IT IS A FUNCTION OF IT IS A FUNCTION OF RELATIVE HEAT TRANSFER AREARELATIVE HEAT TRANSFER AREA TEMPERATURE DISTRIBUTIONTEMPERATURE DISTRIBUTION CAN BE RELATED TO EFFICIENCYCAN BE RELATED TO EFFICIENCY

HEAT SINKS HH

EXTENDED AREA DEVICES TYPICAL DESIGNS ARE SHOWN IN TABLE 3-6 TYPICAL LEVELS OF LOADING

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OTHER COMMON SYSTEM MODELS

USE OF CONDUCTION SHAPE FACTORS TO CALCULATE HEAT TRANSFER

FOR TRANSFER BETWEEN SURFACES MAINTAINED AT CONSTANT TEMPERATURE, THROUGH A CONDUCTING MEDIA

FOR TWO DIMENSIONAL TRANSFER THE SHAPE FACTOR, S, RESULTS IN

AN EQUATION OF THE FORM Q`= SkdT

SHAPE FACTORSSHAPE FACTORS THE METHOD OF SHAPE FACTORS COMES FROM A

GRAPHICAL METHOD WHICH ATTEMPTS TO DETERMINE THE ISOTHERMS AND ADIABATIC LINES FOR A HEAT TRANSFER SYSTEM

AN EXAMPLE IS FOR HEAT TRANSFER FROM AN AN EXAMPLE IS FOR HEAT TRANSFER FROM AN INSIDE TO AN OUTSIDE CORNER, WHICH INSIDE TO AN OUTSIDE CORNER, WHICH REPRESENTS A SYMMETRIC QUARTER SECTION OF REPRESENTS A SYMMETRIC QUARTER SECTION OF A SYSTEM WITH THE CROSS-SECTION AS SHOWN IN A SYSTEM WITH THE CROSS-SECTION AS SHOWN IN THIS SKETCHTHIS SKETCH

TEMPERATURE PROFILESTEMPERATURE PROFILES THIS SKETCH SHOWS THE CORNER WITH

ISOTHERMAL WALLS AT TEMPERATURES T1 AND T2

TAKEN FROM Kreith, F., Principles of Heat Transfer, 3rd Edition, Harper & Row, 1973

TEMPERATURE PROFILESTEMPERATURE PROFILES THIS SKETCH SHOWS THE CORNER WITH

ISOTHERMAL WALLS AT TEMPERATURES T1 AND T2

THE CONSTRUCTION IS CAN BE MANUAL OR AUTOMATED

n LINES ARE CONSTRUCTED MORE OR LESS PARALLEL TO THE SURFACES THAT REPRESENT ISOTHERMS

A SECOND SET OF m LINES ARE CONSTRUCTED NORMAL TO THE ISOTHERMS AS ADIABATS (LINES OF NO HEAT TRANSFER) AND THE NUMBER IS ARBITRARY

TEMPERATURE PROFILESTEMPERATURE PROFILES THE TOTAL HEAT FLUX FROM SURFACE 1 TO

SURFACE 2, THROUGH m ADIABATIC CHANNELS AND OVER n TEMPERATURE INTERVALS IS:

THE SHAPE FACTOR IS DEFINED AS S = m/n, THE SHAPE FACTOR IS DEFINED AS S = m/n, SO THE FLUX EQUATION BECOMES: .SO THE FLUX EQUATION BECOMES: .

GENERATION OF THE MESH IS THE CRITICAL GENERATION OF THE MESH IS THE CRITICAL COMPONENT IN THIS TYPE OF CALCULATIONCOMPONENT IN THIS TYPE OF CALCULATION

.TABLE 3-5 SUMMARIZES THE VALUES FOR TABLE 3-5 SUMMARIZES THE VALUES FOR EQUATIONS FOR VARIOUS SHAPE FACTORSEQUATIONS FOR VARIOUS SHAPE FACTORS