CHE/ME 109 Heat Transfer in Electronics LECTURE 12 – 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

THREE DIMENSIONAL STEADY- STATE SOLUTIONS USE THE SAME METHODS AS FOR TWO DIMENSIONAL MODELS THE SYSTEM IS DIVIDED INTO THREE DIMENSIONAL SHAPES, MOST CONVENIENTLY CUBES

THREE DIMENSIONAL STEADY- STATE SOLUTIONS THE HEAT BALANCE FOR AN INTERIOR CUBE HAS THE FORM AT STEADY STATE MODELS FOR BOUNDARY NODES ARE DEVELOPED IN A SIMILAR FASHION TO THAT USED FOR TWO DIMENSIONAL SYSTEMS. SOLUTIONS FOR THE TEMPERATURE DISTRIBUTION CAN BE EITHER MATRIX OR BY ITERATION NOTE THAT FOR SPREADSHEET ITERATION, THE METHOD USES A SERIES OF TWO DIMENSIONAL SYSTEMS ON A SERIES OF LINKED SHEETS

TRANSIENT HEAT CONDUCTION THE GENERAL MODEL FOR TRANSIENT HEAT CONDUCTION RETAINS THE A SIMILAR CONFIGURATION AS THE STEADY-STATE MODEL THE PRIMARY DIFFERENCE IS ADDING THE CAPACITANCE TERM TO ALLOW FOR CHANGES IN THE HEAT CONTENT OF THE CONTROL VOLUME THE METHOD OF ESTABLISHING NODES FOR THE ANALYSIS IS THE SAME THE SOLUTIONS ARE TYPICALLY CARRIED OUT IN SUCCESSIVE TIME STEPS, SO THIS IS A FINITE DIFFERENCE SOLUTION IN TIME AND SPACE

TRANSIENT HEAT CONDUCTION MODEL FOR ONE-DIMENSIONAL TRANSIENT HEAT BALANCE ON AN INTERIOR NODE FOR A TIME INCREMENT: THIS MODEL CAN BE SOLVED USING TWO ITERATIVE METHODS EXPLICIT - WHICH ASSUMES THE TEMPERATURE OF THE CONTROL VOLUME IN TIME INCREMENT i IS BASED ON THE TEMPERATURE VALUES IN ADJACENT NODES AT THE PREVIOUS TIME INCREMENT i-1 IMPLICIT - WHICH ASSUMES THE TEMPERATURE OF THE CONTROL VOLUME IN TIME INCREMENT i IS BASED ON THE TEMPERATURE VALUES IN ADJACENT NODES AT THE SAME TIME INCREMENT

TRANSIENT SOLUTION METHODS IMPLICIT METHOD IS INHERENTLY STABLE AND WILL CONVERGE THROUGH ITERATION REGARDLESS OF THE TIME INCREMENT SELECTED (SEE EQUATION 5-49) EXPLICIT METHOD HAS A STABILITY CRITERION THAT MUST BE SATISFIED TO OBTAIN A CONVERGENT SOLUTION (5-52) EXPLICIT EQUATION CAN BE RESOLVED FOR THE NODE TEMPERATURE (5-47)

TRANSIENT SOLUTION METHODS THE STABILITY CRITERION REQUIRES THAT THE COEFFICIENT FOR T i m REMAIN POSITIVE OR τ ≤ 1/2 SINCE THE NODE SIZE IS NORMALLY SPECIFIED, THEN THE MAXIMUM TIME INCREMENT IS CALCULATED FROM THE STABILITY CRITERION

TWO DIMENSIONAL TRANSIENT CONDUCTION SOLUTIONS THE HEAT BALANCE FOR AN INTERIOR NODE WITH TWO-DIMENSIONAL TRANSIENT HEAT CONDUCTION HAS THE FORM: THE STABILITY CRITERION FOR THIS SYSTEM REQUIRES THAT τ ≤ 1/4

TWO DIMENSIONAL TRANSIENT CONDUCTION SOLUTIONS BOUNDARY NODES ARE MODELED BASED ON THE GEOMETRY AND HEAT CONDITION (SEE EXAMPLE 5-7) BALANCES FOR STRAIGHT SIDES, INSIDE CORNERS, OUTSIDE CORNERS BALANCES FOR CONVECTION, CONDUCTION AND RADIANT HEAT FLUXES

THREE DIMENSIONAL TRANSIENT SOLUTIONS THE FORM OF THE INTERIOR NODE HEAT BALANCE IS THE STABILITY CRITERION IS τ ≤ 1/6 BOUNDARY NODES ARE MODELED BASED ON THE GEOMETRY AND HEAT CONDITION BALANCES FOR STRAIGHT SIDES, INSIDE CORNERS, OUTSIDE CORNERS BALANCES FOR CONVECTION, CONDUCTION AND RADIANT HEAT FLUXES