CHE/ME 109 Heat Transfer in Electronics LECTURE 8 – SPECIFIC CONDUCTION 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 SOLUTION TO THIS SECOND-ORDER LINEAR DIFFERENTIAL EQUATION IS: THE GENERAL SOLUTION TO THIS SECOND-ORDER LINEAR DIFFERENTIAL EQUATION IS: AT THE BOUNDARY CONDITION REPRESENTED BY THE BASED CONNECTION TO THE PLATE: T = T AT THE BOUNDARY CONDITION REPRESENTED BY THE BASED CONNECTION TO THE 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 HEAT TRANSFER CAN BE EVALUATED WITH THIS PROFILE, THE TOTAL HEAT TRANSFER CAN BE EVALUATED CONSIDERING THE CONDUCTION THROUGH THE BASE AS EQUAL TO THE TOTAL CONVECTION CONSIDERING THE CONDUCTION THROUGH THE BASE AS EQUAL TO THE TOTAL CONVECTION

SOLUTIONS FOR FINS TAKING THE DERIVATIVE OF (3-60) AND SUBSTITUTING AT x = 0, YIELDS THE SAME RESULT COMES FROM A CALCULATION OF THE TOTAL CONVECTED HEAT THE SAME RESULT COMES FROM A CALCULATION OF THE TOTAL CONVECTED HEAT

SOLUTIONS FOR FINS FINITE LENGTH WITH INSULATED TIP OR INSIGNIFICANT SO THE TEMPERATURE GRADIENT AT x = L WILL BE VALUES CALCULATED FOR C VALUES CALCULATED FOR C 1 AND C 2 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, h L, 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 TOTAL HEAT TRANSFER INCREASES RELATIVE TO THE NON-FINNED SURFACE FIN EFFECTIVENESS INDICATES HOW MUCH THE TOTAL HEAT TRANSFER INCREASES RELATIVE TO THE NON-FINNED SURFACE IT IS A FUNCTION OF IT IS A FUNCTION OF RELATIVE HEAT TRANSFER AREA RELATIVE HEAT TRANSFER AREA TEMPERATURE DISTRIBUTION TEMPERATURE DISTRIBUTION CAN BE RELATED TO EFFICIENCY CAN BE RELATED TO EFFICIENCY

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

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

TEMPERATURE PROFILES THIS SKETCH SHOWS THE CORNER WITH ISOTHERMAL WALLS AT TEMPERATURES T 1 AND T 2 TAKEN FROM Kreith, F., Principles of Heat Transfer, 3rd Edition, Harper & Row, 1973

TEMPERATURE PROFILES THIS SKETCH SHOWS THE CORNER WITH ISOTHERMAL WALLS AT TEMPERATURES T 1 AND T 2 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 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, SO THE FLUX EQUATION BECOMES:. THE SHAPE FACTOR IS DEFINED AS S = m/n, SO THE FLUX EQUATION BECOMES:. GENERATION OF THE MESH IS THE CRITICAL COMPONENT IN THIS TYPE OF CALCULATION GENERATION OF THE MESH IS THE CRITICAL COMPONENT IN THIS TYPE OF CALCULATION TABLE 3-5 SUMMARIZES THE VALUES FOR EQUATIONS FOR VARIOUS SHAPE FACTORS.TABLE 3-5 SUMMARIZES THE VALUES FOR EQUATIONS FOR VARIOUS SHAPE FACTORS