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CHE/ME 109 Heat Transfer in
Electronics
LECTURE 6 – ONE DIMENSIONAL CONDUTION
SOLUTIONS
ONE-DIMENSIONAL HEAT CONDUCTION SOLUTIONS
• GENERAL METHOD• FORMULATE THE DIFFERENTIAL
EQUATION• DEVELOP THE GENERAL SOLUTION• USE THE BOUNDARY CONDITIONS TO
OBTAIN THE INTEGRATION CONSTANTS
RECTANGULAR SYSTEM• AT STEADY-STATE WITH NO
GENERATION THE FORM OF THE MODEL IS
• THE FIRST INTEGRATION YIELDS THE GRADIENT
• THE SECOND INTEGRATION YIELDS THE TEMPERATURE FUNCTION
EVALUATION OF EQUATION PARAMETERS
• BOUNDARY CONDITIONS ARE USED FOR EVALUATION OF C1 AND C2.
• AT x = 0, SUBSTITUTION IN TO THE FUNCTION YIELDS T(0) = C2
• FOR A CONSTANT VALUE OF kTHE VALUE FOR C1 BECOMES:WHERE L IS THE THICKNESSOF THE SECTION
• SUBSTITUTING THESE VALUES BACK INTO THETEMPERATURE FUNCTIONYIELDS A LINEAR RELATIONSHIP
FLUX & TEMPERATURE FUNCTIONS
• THE FLUX CAN THEN BE CALCULATED FROM THE FOURIER EQUATION:
• THIS RESULT CAN ALSO BE INSERTED INTO THE TEMPERATURE FUNCTION . WHICH SHOWS HOW THE SLOPE DEPENDS ON THE RELATIVE VALUES OF FLUX AND CONDUCTIVITY
L
LTTkkC
dx
dTk
A
)()0(1
CYLINDRICAL SYSTEM
• AT STEADY-STATE WITH NO GENERATION
• SINCE r IS A CHANGING VALUE IN A CYLINDRICAL SYSTEM THE FORM OF THE PRIMARY EQUATION IS
• THE FIRST INTEGRATION YIELDS
CYLINDRICAL SYSTEM• THE SECOND INTEGRATION YIELDS :
• NOTE THAT THIS EQUATION CANNOT BE SOLVED FOR r = 0, SO THIS EXPRESSION IS LIMITED TO PIPE-TYPE STRUCTURES
• .GENERAL VALUES FOR THE INTEGRATION CONSTANTS ARE BASED ON BOUNDARY CONDITIONS T(r1) AT r1 AND T(r2) AT r2:
CYLINDRICAL SYSTEM
• THE GENERAL TEMPERATURE EQUATION IS THEN
• TYPICAL TEMPERATURE PROFILE
CYLINDRICAL SYSTEM
• THE TOTAL HEAT EQUATION IS
• THE FLUX DEPENDS ON THE VALUE OF r AND WILL VARY OVER THE PIPE WALL THICKNESS
SPHERICAL SHELL SYSTEM
• REFER TO THE DEVELOPMENT IN EXAMPLE 2-16
• TEMPERATURE PROFILE FOR r > 0
• TOTAL HEAT FLOW
HEAT GENERATION• HEAT BALANCE EQUATION FOR STEADY-STATE
CONDITIONS: HEAT TRANSFERRED = HEAT GENERATED• TEMPERATURE DISTRIBUTION IN GENERATING SOLID• ASSUME THE GENERATION OCCURS UNIFORMLY IN A
SOLID, SO THE MAXIMUM TEMPERATURE IS AT THE CENTER OF THE SOLID
• SURFACE TEMPERATURE DEPENDS ON THE RATE HEAT IS TRANSFERRED FROM THE SURFACE. ASSUMING ONLY CONVECTION TRANSFER
HEAT GENERATION
• THE INTERNAL TEMPERATURE IS EVALUATED BY TAKING A SHELL BALANCE IN THE SOLID
• THE FLUX WILL CHANGE WITH POSITION AS THE TOTAL HEAT GENERATED IS BASED ON THE ENCLOSED VOLUME
HEAT GENERATION
• EXAMPLE OF A CYLINDER - AT A SPECIFIED VALUE OF r IN THE SOLID CYLINDER, THE HEAT BALANCE YIELDS:
HEAT GENERATION
• INTEGRATION WITH RESPECT TO r AND USING THE SURFACE TEMPERATURE, Ts AND THE OUTSIDE RADIUS ro, AS BOUNDARY CONDITIONS YIELDS:
HEAT GENERATION
• RESULTING PROFILE FOR EXAMPLE 2-17
• SIMILAR DEVELOPMENTS CAN BE USED FOR OTHER GEOMETRIC SHAPES
VARIABLE THERMAL CONDUCTIVITY
• WHEN k CHANGES WITH TEMPERATURE, A k(T) FUNCTION NEEDS TO REPLACE k IN THE HEAT BALANCE EQUATION
• IT IS UNDER THE INTEGRAL FOR CALCULATION PURPOSES (SEE EXAMPLE 2-20)
• AN ALTERNATE IS TO USE AN AVERAGE VALUE FOR k OVER THE RANGE OF TEMPERATURE AND TAKE k OUTSIDE OF THE INTEGRAL (SEE EXAMPLE 2-21):
• THIS METHOD IS USEFUL WHEN k DATA IS PROVIDED IN TABULAR INSTEAD OF FUNCTION FORM AND A SMALL TEMPERATURE RANGE IS BEING CONSIDERED
VARIABLE THERMAL CONDUCTIVITY
• THE VALUE USED CAN BE AN ARITHMETIC AVERAGE, WHICH ASSUMES A LINEAR RELATION OF THE FORM
• AND HAS THE FORM
• ALTERNATELY THE LOG MEAN VALUE OF k FROM THE TABLE CAN BE USED, WHERE k2 IS THE CONDUCTIVITY AT T2 AND k1 IS THE CONDUCTIVITY AT T1:
)(
)(ln
)()(.
1
2
12
Tk
Tk
TkTkk avg