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

Qq

)()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