### LECTURE 05 - FORMULATION OF CONDUCTION EQUATIONS

```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
(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
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

L
T x =0
T YP IC A L B O U N D A R Y C O N D IT IO N
C O N FIG U R AT IO N S
T y =M
S T EA D Y-S T A TE
T E M PE R AT U R E
G R A D IE N T
T x =L
L
T x =0
M
T y =0
O N E D IM EN SIO N A L
H EA T F L O W
O
T W O D IM E N S IO N A L
H EA T F L O W
T x =L
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
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,
DEPEND ON THE RELATIVE VALUES OF k.
EXAMPLE
A
CHIP CARRIER IS BONDED TO A
TEMPERATURES AT THE SURFACE
OF THE CHIP AND THE PINS OF THE
CONDUCTION, DETERMINE THE
TEMPERATURE AT THE INTERFACE
BETWEEN THE BONDING LAYER
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

INTERFACE, CONSTANT FLUX
REQUIRES:


THE TEMPERATURE IS CONSTANT AND
THE HEAT FLUX DEPENDS ON THE
PROPERTIES OF THE MEDIA
 EXAMPLE
- A HEAT GENERATING
DEVICE HAS A SPECIFIED SURFACE
TEMPERATURE. DETERMINE THE
QUANTITY OF HEAT THAT IS
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
q  k
T
x
INSULATED SURFACE
- IDEAL SYSTEM WITH
NO HEAT TRANSFER
IN A SPECIFIED
T (0, t )
DIRECTION
k
0
x
SPECIFIED TEMPERATURE
 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
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.
```