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 = 22.348.

• A SECOND CALCULATION IS COMPLETED AT x2 = 1.05 AND FROM

• THIS THE RESULT IS y2 = 17.099. USING THESE VALUES TO CALCULATE THE DERIVATIVE NUMERICALLY THE NEXT VALUE OF x CAN BE ESTIMATED:

• AND x3 = 1.213. USING THIS VALUE, THE RESULT IS y3 = 4.716. 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.