1. CHE/ME 109 Heat Transfer in Electronics LECTURE 6 – ONE DIMENSIONAL CONDUTION SOLUTIONS

2. ONE-DIMENSIONAL HEAT CONDUCTION SOLUTIONS GENERAL METHOD FORMULATE THE DIFFERENTIAL EQUATION DEVELOP THE GENERAL SOLUTION USE THE BOUNDARY CONDITIONS TO OBTAIN THE INTEGRATION CONSTANTS

3. RECTANGULAR SYSTEM AT STEADY-STATE WITH NO GENERATION THE FORM OF THE MODEL IS THE FIRST INTEGRATION YIELDS THE GRADIENTTHE FIRST INTEGRATION YIELDS THE GRADIENT THE SECOND INTEGRATION YIELDS THE TEMPERATURE FUNCTIONTHE SECOND INTEGRATION YIELDS THE TEMPERATURE FUNCTION

4. EQUATION PARAMETERS EVALUATION OF EQUATION PARAMETERS BOUNDARY CONDITIONS ARE USED FOR EVALUATION OF C 1 AND C 2. AT x = 0, SUBSTITUTION IN TO THE FUNCTION YIELDS T(0) = C 2 FOR A CONSTANT VALUE OF k THE VALUE FOR C 1 BECOMES: WHERE L IS THE THICKNESS OF THE SECTION SUBSTITUTING THESE VALUES BACK INTO THE TEMPERATURE FUNCTION YIELDS A LINEAR RELATIONSHIP

5. FLUX & TEMPERATURE FUNCTIONS THE FLUX CAN THEN BE CALCULATED FROM THE FOURIER EQUATION: WHICH SHOWS HOW THE SLOPE DEPENDS ON THE RELATIVE VALUES OF FLUX AND CONDUCTIVITYTHIS RESULT CAN ALSO BE INSERTED INTO THE TEMPERATURE FUNCTION. WHICH SHOWS HOW THE SLOPE DEPENDS ON THE RELATIVE VALUES OF FLUX AND CONDUCTIVITY

6. 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 YIELDSTHE FIRST INTEGRATION YIELDS

7. CYLINDRICAL SYSTEM THE SECOND INTEGRATION YIELDS : NOTE THAT THIS EQUATION CANNOT BE SOLVED FOR r = 0, SO THIS EXPRESSION IS LIMITED TO PIPE-TYPE STRUCTURESNOTE 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(r.GENERAL VALUES FOR THE INTEGRATION CONSTANTS ARE BASED ON BOUNDARY CONDITIONS T(r1) AT r1 AND T(r2) AT r2:

8. CYLINDRICAL SYSTEM THE GENERAL TEMPERATURE EQUATION IS THEN TYPICAL TEMPERATURE PROFILETYPICAL TEMPERATURE PROFILE

9. CYLINDRICAL SYSTEM THE TOTAL HEAT EQUATION IS THE FLUX DEPENDS ON THE VALUE OF r AND WILL VARY OVER THE PIPE WALL THICKNESS

10. SPHERICAL SHELL SYSTEM REFER TO THE DEVELOPMENT IN EXAMPLE 2-16 TEMPERATURE PROFILE FOR r > 0 TOTAL HEAT FLOW

11. HEAT GENERATION

12. 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

13. HEAT GENERATION EXAMPLE OF A CYLINDER - AT A SPECIFIED VALUE OF r IN THE SOLID CYLINDER, THE HEAT BALANCE YIELDS:

14. HEAT GENERATION INTEGRATION WITH RESPECT TO r AND USING THE SURFACE TEMPERATURE, T s AND THE OUTSIDE RADIUS r o, AS BOUNDARY CONDITIONS YIELDS:

15. HEAT GENERATION RESULTING PROFILE FOR EXAMPLE 2-17 SIMILAR DEVELOPMENTS CAN BE USED FOR OTHER GEOMETRIC SHAPES

16. 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

17. VARIABLE THERMAL CONDUCTIVITY THE VALUE USED CAN BE AN ARITHMETIC AVERAGE, WHICH ASSUMES A LINEAR RELATION OF THE FORM AND HAS THE FORMAND HAS THE FORM ALTERNATELY THE LOG MEAN VALUE OF k FROM THE TABLE CAN BE USED, WHERE kALTERNATELY THE LOG MEAN VALUE OF k FROM THE TABLE CAN BE USED, WHERE k 2 IS THE CONDUCTIVITY AT T 2 AND k 1 IS THE CONDUCTIVITY AT T 1 :