CHE/ME 109 Heat Transfer in Electronics LECTURE 5 – GENERAL HEAT CONDUCTION EQUATION
Dec 23, 2015
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.