CHE/ME 109 Heat Transfer in Electronics LECTURE 11 – ONE DIMENSIONAL NUMERICAL MODELS

NUMERICAL METHOD FUNDAMENTALS NUMERICAL METHODS 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

EXAMPLE USING NUMERICAL METHODS NEWTON-RAPHSON PROVIDES AN EXAMPLE TO MODEL A COMPLEX SYSTEM NEWTON-RAPHSON EXAMPLE GIVEN: EQUATION OF THE FORM

NEWTON-RAPHSON EXAMPLE WANTED: FIND THE ROOTS OF THIS EQUATION BASIS: USE ANALYTICAL OR NUMERICAL METHODS SOLUTION: A PLOT OF THIS EQUATION HAS THE FORM

NEWTON-RAPHSON EXAMPLE SOLUTION: USING NEWTON-RAPHSON TO OBTAIN THE ROOTS, START BY EVALUATING THE FUNCTION AT x1= 1. THE VALUE OBTAINED IS y1 = A SECOND CALCULATION IS COMPLETED AT x2 = 1.05 AND FROM THIS THE RESULT IS y2 = USING THESE VALUES TO CALCULATE THE DERIVATIVE NUMERICALLY THE NEXT VALUE OF x CAN BE ESTIMATED: AND x3 = USING THIS VALUE, THE RESULT IS y3 = THIS VALUE IS STILL NOT ZERO, SO THE PROCESS IS REPEATED. RESULTS ARE SHOWN IN THE FOLLOWING TABLE.

NEWTON-RAPHSON EXAMPLE FUNCTION HAS A STEEP SLOPE AND IS SENSITIVE TO SMALL CHANGES IN x, BUT THE METHOD STILL WORKS. TAKING ADDITIONAL VALUES COULD REDUCE THE VALUE OF y TO A TARGET LEVEL.

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.