CHE/ME 109 Heat Transfer in

Electronics

LECTURE 5 – GENERAL HEAT CONDUCTION EQUATION

PRIMARY THREE DIMENSIONAL HEAT TRANSFER MODELS

DEVELOPED AS PARTIAL DIFFERENTIAL EQUATIONS, IN RECTANGULAR COORDINATES, OF THE FORM:

FORMS FOR SPECIFIC CONDITIONS

FOR UNIFORM k, THE POISSON EQUATION (STEADY-STATE) APPLIES (EQN. 2-40):

FORMS FOR SPECIFIC CONDITIONS

THE DIFFUSION EQUATION APPLIES FOR TRANSIENT HEAT TRANSFER WITH NO GENERATION (EQN. 2-41):

FORMS FOR SPECIFIC CONDITIONS

THE LAPLACE EQUATION APPLIES FOR STEADY-STATE WITH NO GENERATION: (EQN. 2-42)

FORMS FOR SPECIFIC CONDITIONS

SIMILAR EQUATIONS ARE DEVELOPED FOR CYLINDRICAL AND SPHERICAL COORDINATE SYSTEMS

CYLINDRICAL (EQN. 2-43) .SPHERICAL (EQN. 2-44)

SOLUTIONS TO DIFFERENTIAL EQUATIONS

EMPLOY EITHER BOUNDARY AND/OR INITIAL CONDITIONS

BOUNDARY CONDITIONS ARE TYPICALLY SPECIFIED AT AN INTERFACE IN THE SYSTEM

TWO BOUNDARY CONDITIONS MUST BE SPECIFIED IN EACH DIRECTION OF HEAT TRANSFER

SOLUTIONS TO DIFFERENTIAL EQUATIONS

O

ONE DIMENSIONALHEAT FLOW

Tx=0

Tx=L

L

STEADY-STATETEMPERATUREGRADIENT

Tx=0

Ty=0

Tx=L

Ty=M

L

M

TWO DIMENSIONALHEAT FLOW

TYPICAL BOUNDARY CONDITIONCONFIGURATIONS

BOUNDARY CONDITIONS TYPICALLY TAKE THE FORM OF FIXED

TEMPERATURES AT INTERFACES EXAMPLE - IF THE TEMPERATURES ARE

SPECIFIED AT THE BOUNDARIES OF A LARGE THIN PLATE, CALCULATE THE TEMPERATURE GRADIENT AND HEAT FLUX THROUGH THE PLATE

FOR MULTIDIMENSIONAL SYSTEMS, THE ANALYTICAL SOLUTIONS ARE MAY TAKE THE FORM OF EIGENFUNCTIONS AND NUMERICAL SOLUTIONS MAY BE USED

SPECIFIED HEAT FLUX HEAT FLUX BOUNDARIES CONDUCTION TO CONDUCTION (SERIES OF

SOLIDS) AT THE INTERFACE BETWEEN THE TWO

SOLIDS, THE EQUALITY OF HEAT FLUX REQUIRES:

THE TEMPERATURES WILL BE EQUIVALENT, BUT THE TEMPERATURE GRADIENTS WILL DEPEND ON THE RELATIVE VALUES OF k.

EXAMPLE A CHIP CARRIER IS BONDED TO A

LEAD FRAME USING A CONDUCTIVE ADHESIVE. FOR SPECIFIED TEMPERATURES AT THE SURFACE OF THE CHIP AND THE PINS OF THE LEADFRAME, CONSIDERING ONLY CONDUCTION, DETERMINE THE TEMPERATURE AT THE INTERFACE BETWEEN THE BONDING LAYER AND THE LEADFRAME SURFACE.

CONVECTION TO CONDUCTION FLUID TO SOLID

AT THE CONVECTION/CONDUCTION INTERFACE, EQUALITY OF HEAT FLUX REQUIRES THAT THE CONVECTED HEAT EQUAL THE CONDUCTED HEAT

THE TEMPERATURES WILL BE THE SAME AT THE INTERFACE AND THE FLUX WILL DEPEND ON THE RESISTANCE THROUGH THE TWO MEDIA

EXAMPLE - CONVECTION COOLING FOR A HEAT GENERATING DEVICE. FOR A SPECIFIED MAXIMUM TEMPERATURE AT THE INTERFACE AT A CONSTANT FLUX, DETERMINE THE NECESSARY BULK TEMPERATURE FOR THE COOLING FLUID FOR A SPECIFIED VALUE OF h

RADIATION TO CONDUCTION RADIATION SOURCE TO SOLID AT THE RADIATION/CONDUCTION

INTERFACE, CONSTANT FLUX REQUIRES:

THE TEMPERATURE IS CONSTANT AND THE HEAT FLUX DEPENDS ON THE RADIATIVE AND CONDUCTIVE PROPERTIES OF THE MEDIA

RADIATION TO CONDUCTION

EXAMPLE - A HEAT GENERATING DEVICE HAS A SPECIFIED SURFACE TEMPERATURE. DETERMINE THE QUANTITY OF HEAT THAT IS DISSIPATED BY RADIATION IF THE TEMPERATURE OF THE SURROUNDINGS ARE AT A SPECIFIED VALUE FOR A MATERIAL WITH A SPECIFIED EMISSIVITY

HEAT FLUX BOUNDARY CONDITIONS

FOURIER’S LAW APPLIES FOR A SPECIFIED HEAT FLUX IN ONE DIRECTION

INSULATED SURFACE - IDEAL SYSTEM WITH NO HEAT TRANSFER IN A SPECIFIED DIRECTION

x

Tkq

0),0(

x

tTk

SPECIFIED TEMPERATURE GRADIENTS

FOR STEADY STATE PROCESSES THE FLUX THROUGH A MEDIA IS

MODELED BASED ON THE TEMPERATURES THAT BOUND IT

FOR CONDUCTION (EQN. 2-46)

SIMILAR EQUATIONS CAN BE DEVELOPED FOR AN OVERALL ΔT FOR CONVECTION OR RADIATION

COMBINED MECHANISMS FOR SOME SYSTEMS THERE MAY BE PARALLEL

HEAT TRANSFER THROUGH THE SAME OR DIFFERENT MECHANISMS

HEAT MAY BE TRANSFERRED FROM A HEAT GENERATING UNIT BY

CONDUCTION - WHERE IT IS CONNECTED TO A SOLID HEAT SINK

CONVECTION - WHEN FLUID IS USED TO CONVECT AWAY HEAR

RADIATION - WHEN OTHER OPAQUE MEDIA AT DIFFERENT TEMPERATURES ARE ON A LINE OF SIGHT WITH THE HEAT TRANSFER SURFACE

INITIAL CONDITIONS INITIAL TEMPERATURE VALUES ARE SPECIFIED

AT ALL POINTS IN THE SYSTEM AT TIME ZERO THE SOLUTION REQUIRES SPECIFICATION OF

INITIAL TEMPERATURES AT ALL LOCATIONS IN THE SYSTEM AT TIMES > ZERO.

FOR EXAMPLE - A MOTHERBOARD IS INITIALLY UNIFORMLY AT AMBIENT TEMPERATURE, PRIOR TO INITIATING POWER. DETERMINE THE TEMPERATURES AT LOCATIONS ON THE MOTHERBOARD DURING OPERATION.

THIS TYPE OF PROBLEM WOULD NEED TO HAVE BOUNDARY CONDITIONS SPECIFIED AT TIME t > 0 TO DEVELOP A TRANSIENT SOLUTION.