CHE/ME 109 Heat Transfer in Electronics

1. CHE/ME 109 Heat Transfer in Electronics LECTURE 4 – HEAT TRANSFER MODELS

2. HEAT TRANSFER MODEL PARAMETERS • MODELS ARE BASED ON FOUR SETS OF PARAMETERS • TIME VARIABLES • GEOMETRY • SYSTEM PROPERTIES • HEAT GENERATION

3. TIME VARIABLES • STEADY-STATE - WHERE CONDITIONS STAY CONSTANT WITH TIME • TRANSIENT - WHERE CONDITIONS ARE CHANGING IN TIME http://ccrma-www.stanford.edu/~jos/fp/img609.png

4. GEOMETRY • THE COORDINATE SYSTEM FOR THE MODELS IS NORMALLY SELECTED BASED ON THE SHAPE OF THE SYSTEM. • PRIMARY MODELS ARE RECTANGULAR, CYLINDRICAL AND SPHERICAL- BUT THESE CAN BE USED TOGETHER FOR SOME SYSTEMS • HEAT TRANSFER DIMENSIONS • HEAT TRANSFER IS A THREE DIMENSIONAL PROCESS • SOME CONDITIONS ALLOW SIMPLIFICATION TO ONE AND TWO DIMENSIONAL SYSTEMS

5. SYSTEM PROPERTIES • ISOTROPIC SYSTEMS HAVE UNIFORM PROPERTIES IN ALL DIMENSIONS • ANISOTROPIC MATERIALS MAY HAVE VARIATION IN PROPERTIES WHICH ENHANCE OR DIMINISH HEAT TRANSFER IN A SPECIFIC DIRECTION http://www.feppd.org/ICB-Dent/campus/biomechanics_in_dentistry/ldv_data/img/bone_17.jpg

6. HEAT GENERATION • GENERATION OF HEAT IN A SYSTEM RESULTS IN AN “INTERNAL” SOURCE WHICH MUST BE CONSIDERED IN THE MODEL • GENERATION CAN BE A POINT OR UNIFORM VOLUMETRIC PHENOMENON • TYPICAL EXAMPLES INCLUDE: • RESISTANCE HEATING WHICH OCCURS IN POWER CABLES AND HEATERS • REACTION SYSTEMS, CHEMICAL AND NUCLEAR • IN SOME CASES, THE SYSTEM MAY ALSO CONSUME HEAT, SUCH AS IN AN ENDOTHERMIC REACTION IN A COLD PACK

7. SPECIFIC MODELS • RECTANGULAR MODELS CAN BE DEVELOPED AS SHOWN IN THE FOLLOWING FIGURES • THE HEAT TRANSFER ENTERS AND EXITS IN x, y, AND z PLANES THROUGH THE CONTROL VOLUME • DIMENSIONS OF THE VOLUME ARE Δx, Δy AND Δz • THE OVERALL MODEL FOR THE SYSTEM INCLUDES GENERATION TERMS AND ALLOWS FOR CHANGES IN THE CONTROL VOLUME WITH TIME

8. DIFFERENTIAL MODEL • THIS SYSTEM CAN BE REDUCED TO DIFFERENTIAL DISTANCE AND TIME, USING THE EXPRESSIONS FOR CONDUCTION HEAT TRANSFER AND HEAT CAPACITY TO YIELD:

9. DIFFERENTIAL MODEL FOR SPECIFIC SYSTEMS • STEADY STATE: • STEADY STATE WITH NO GENERATION: • TRANSIENT WITH NO GENERATION: • TWO DIMENSIONAL HEAT TRANSFER (TWO OPPOSITE SIDES ARE INSULATED). • .ONE DIMENSIONAL HEAT TRANSFER (FOUR SIDES ARE INSULATED- OPPOSITE PAIRS)

10. OTHER VARIATIONS ON THE EQUATION FOR SPECIFIC CONDITIONS • SIMILAR MODIFICATIONS CAN BE APPLIED TO THE ONE AND TWO DIMENSIONAL EQUATIONS FOR: • STEADY STATE • AND NO-GENERATION CONDITIONS http://www.emeraldinsight.com/fig/1340120602047.png

11. OTHER GEOMETRIES • CYLINDRICAL USE A CONTROL VOLUME BASED ON ONE DIMENSIONAL (RADIAL) HEAT TRANSFER FOR THE CONDITIONS: • THE ENDS ARE INSULATED OR THE AREA AT THE ENDS IS NOT SIGNIFICANT RELATIVE TO THE SIDES OF THE CYLINDER • THE HEAT TRANSFER IS UNIFORM IN ALL DIRECTIONS AROUND THE AXIS. • THE CONTROL VOLUME FOR THE ANALYSIS IS A CYLINDRICAL PIPE AS SHOWN IN FIGURE 2-15 • RESULTING DIFFERENTIAL FORMS OF THE MODEL EQUATIONS ARE SHOWN AS (2-25) THROUGH (2-28)

12. SPHERICAL SYSTEMS • MODELED USING A VOLUME ELEMENT BASED ON A HOLLOW BALL OF WALL THICKNESS Δr (SEE FIGURE 2-17) • FOR UNIFORM COMPONENT PROPERTIES, THE MODEL BECOMES ONE DIMENSIONAL FOR RADIAL HEAT TRANSFER. • THE RESULTING EQUATIONS ARE (2-30) - (2-34) IN THE TEXT

13. GENERALIZED EQUATION • GENERAL ONE-DIMENSIONAL HEAT TRANSFER EQUATION IS • WHERE THE VALUE OF n IS • 0 FOR RECTANGULAR COORDINATES • 1 FOR CYLINDRICAL COORDINATES • 2 FOR SPHERICAL COORDINATES

14. GENERAL RESISTANCE METHOD • CONSIDER A COMPOSITE SYSTEM • CONVECTION ON INSIDE AND OUTSIDE SURFACES • STEADY-STATE CONDITIONS • EQUATION FOR Q http://www.owlnet.rice.edu/~chbe402/ed1projects/proj99/dsmith/index.htm

15. COMPOSITE TRANSFER EQUATION

16. OVERALL RESISTANCE VERSION