CHE/ME 109 Heat Transfer in

Electronics

LECTURE 7 – EXAMPLES OF CONDUCTION MODELS

ONE DIMENSIONAL CONDUCTION SOLUTIONS

• CONDUCTION SOLUTIONS FOR PLANE WALLS• TOTAL HEAT TRANSFER CAN BE EXPRESSED IN TERMS OF

OVERALL OR LOCAL HEAT TRANSFER• LOCAL APPLIES TO A SYSTEM WITH A SINGLE MECHANISM

• WHERE TEMPERATURES ARE SPECIFIED OVER A FIXED THICKNESS AND THERE IS CONSTANT k IS A TYPICAL LOCAL SYSTEM

• ALTERNATELY THIS CAN BE EXPRESSED IN TERMS OF THERMAL RESISTANCE

• COMPARING WITH THE PREVIOUS EQUATION THEN

ELECTRICAL ANALOGS• SHOWN IN FIGURE 3-3 FOR SINGLE

LAYER

• RESISTANCE TERM IS SPECIFIC FOR EACH MODE OF HEAT TRANSFER (EQNS. 3-5, 3.8 & 3.10)

• FIGURE 3.5 SHOWS MULTIPLE MODES – PARALLEL TRANSPORT

OVERALL NETWORK• SERIES OF CONVECTION AND

CONDUCTION MODES

MULTIPLE LAYERS• FOR A SERIES OF LAYERS WHERE

SYSTEM THE FLUX THROUGH EACH LAYER IS CONSTANT, SEE FIGURE 3-9

MULTIPLE LAYERS

• THE FLUX THROUGH EACH LAYER IS THE SAME SO:

• IN TERMS OF RESISTANCE THIS RELATIONSHIP BECOMES:

MULTIPLE LAYERS

• IN OVERALL TERMS, CONSIDER THE DRIVING FORCE TO BE T∞1 - T∞2 AND THEN EXPRESS THE OVERALL RESISTANCE AS

• SO THE OVERALL HEAT TRANSFER CAN THEN BE EXPRESSED AS

THERMAL CONTACT RESISTANCE

• CONTACT RESISTANCE IS CONSIDERED WHEN ONE LAYER OR COMPONENT IS ATTACHED TO ANOTHER

• THE QUALITY OF THE CONTACT DEPENDS ON HOW COMPLETELY THE SURFACES ARE MATED

• SURFACE ROUGHNESS CAN DECREASE THE CONTACT (REFER TO FIGURE 3-14 IN THE TEXT)

THERMAL CONTACT RESISTANCE

• THE QUALITY OF THE CONTACT TYPICALLY INCREASES WITH THE MOUNTING PRESSURE AND THE SURFACE ROUGHNESS

• SEE TABLE 3-2 FOR TYPICAL VALUES

THERMAL CONTACT RESISTANCE

THERMAL CONTACT RESISTANCE

• ALSO SEE RANGE OF VALUES IN THE FOLLOWING TABLE:

THERMAL CONTACT RESISTANCE

• FOR CRITICAL CONNECTIONS, THE RESISTANCE CAN ALSO BE REDUCED BY THE FOLLOWING METHODS:

• USING A SOFT METAL FOIL SHEET• USING A CONDUCTIVE ADHESIVE

(EPOXY)• USING A THERMAL GREASE (SILICON)• USING A GAS WITH A HIGH

CONDUCTIVITY IN THE REGION

THERMAL RESISTANCE NETWORKS

• THE GENERALIZED FORM FOR THE THERMAL RESISTANCE NETWORK IS BASED ON THE ELECTRICAL ANALOGY

• FOR PARALLEL PATHS, THE DRIVING FORCES ARE THE SAME FOR THE SAME TERMINAL TEMPERATURES, AS PER FIGURE (3-19)

THERMAL RESISTANCE NETWORKS

• TOTAL HEAT TRANSFER

• RESISTANCE THROUGH EACH LAYER

• OVERALL EQUATION

• OVERALL RESISTANCE FOR PARALLEL FLOWS:

FOR PARALLEL/SERIES PATHS

• CONSIDER A FLOW THROUGH A SYSTEM OF UNIT WIDTH, WITH FIXED SURFACE TEMPERATURES AND NO CONTACT RESISTANCE

• OVERALL RESISTANCE NETWORK

FOR PARALLEL/SERIES PATHS

• FOR THIS TYPE OF SYSTEM, THE OVERALL RESISTANCE CAN BE EXPRESSED AS A SERIES OF THREE RESISTANCE TERMS

• THE FIRST AND THIRD TERMS ARE BASED ON SINGLE PLANE TERMS

• THE SECOND TERM IS A PARALLEL RESISTANCE TERM OF THE FORM:

• THE REAL SITUATION WILL PROBABLY INCLUDE HEAT TRANSFER BETWEEN THE MIDDLE LAYERS AND A NON-UNIFORM TEMPERATURE AT THE INTERFACES WITH THE MIDDLE SECTION.

FOR PARALLEL/SERIES PATHS

• OTHER RESISTANCE TERMS THAT CAN BE INCLUDED IN THE NETWORK

• CONVECTION RESISTANCE • RADIATION RESISTANCE • .CONTACT RESISTANCE

CONDUCTION IN SPHERES AND CYLINDERS

• RESISTANCE NETWORKS CAN ALSO BE USED FOR CIRCULAR SYSTEMS

• THE PRIMARY CHANGE IS TO ALLOW FOR VARIATION IN THE SURFACE AREA WITH RADIUS, WHICH RESULTS IN A CHANGE IN THE FLUX

CONDUCTION IN SPHERES AND CYLINDERS

• FOR THE CYLINDRICAL SYSTEM FROM AN INNER RADIUS, r1 AND TEMPERATURE T1, TO AND OUTER

RADIUS r2 AND TEMPERATURE T2:

Lk

rr

ThenTT

QFor

rr

TTLkQorkdTdr

rL

Q

partsbygIntegratindr

dTk

A

Q

CYLCYL

r

r

T

T

2

ln

`

ln

2`2

`

`

1

2

21

1

2

212

1

1

2

CONDUCTION IN SPHERES AND CYLINDERS

• FOR A SERIES OF CYLINDRICAL SHELLS, THE SAME ANALYSIS IS USED

• FOR A SPHERICAL SYSTEM (HOLLOW BALL) THE SAME METHOD IS USED

• FOR MULTIPLE LAYERS, THE RESISTANCE FOR EACH LAYER IS INCLUDED IN THE OVERALL TOTAL RESISTANCE

CRITICAL RADIUS OF INSULATION (Section 3.5)

• A SPECIFIC APPLICATION OF THE RESISTANCE CONCEPT

• THIS IS A CALCULATION TO DETERMINE THE OPTIMUM RADIUS FOR AN INSULATING LAYER ON A CYLINDER, WITH CONVECTION ON THE OUTSIDE OF THE INSULATION (SEE FIGURE 3-30)

• AS THE THICKNESS OF THE INSULATION IS INCREASED, THE RESISTANCE OF THE INSULATION INCREASES (SEE FIGURE 3-31)

CRITICAL RADIUS OF INSULATION

• AS THE THICKNESS OF THE INSULATION IS INCREASED, THE EXTERNAL AREA INCREASES, WHICH REDUCES THE CONVECTION RESISTANCE AND INCREASES HEAT TRANSFER

• THE OPTIMUM OCCURS AT THE CRITICAL RADIUS

• AT THINNER RADII, THERE IS MORE HEAT LOSS

• AT THICKER RADII, THERE IS MORE HEAT LOSS