CHE/ME 109 Heat Transfer in

ElectronicsLECTURE 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 Ti

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

2

12

x

t

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