1. CHE/ME 109 Heat Transfer in Electronics LECTURE 10 – SPECIFIC TRANSIENT CONDUCTION MODELS

2. SEMI-INFINITE SOLID SOLUTIONS  SEMI-INFINITE SOLIDS HAVE ONE PLANE SURFACE ON AN INFINITE VOLUME  THIS MODEL APPLIES TO SYSTEMS THAT CAN BE TREATED AS VERY THICK SLABS, SUCH AS THE SURFACE OF THE EARTH.  THE HEAT TRANSFER IS MODELED IN ONE DIMENSION, NORMAL TO THE SURFACE  PRIMARY MODEL EQUATION IS:

3. SEMI-INFINITE TRANSIENT CONDUCTION MODELS  CRITERIA FOR SOLUTIONS ARE  INITIAL TEMPERATURE IS UNIFORM IN THE SOLID  A UNIFORM HEAT FLOW IS INTRODUCED AT THE PLANE SURFACE AT t = 0, SO THE SURFACE TEMPERATURE BECOMES T  THE CONVECTION HEAT TRANSFER COEFFICIENT AT THE SURFACE, h, IS UNIFORM AND CONSTANT FOR t >0.

4. SEMI-INFINITE TRANSIENT CONDUCTION MODELS  VARIATIONS ON SOLUTIONS FOR INFINITE h VALUE (NO THERMAL RESISTANCE AT THE SURFACE)  USING THE GAUSSIAN ERROR FUNCTION:  OR USING THE COMPLEMENTARY ERROR FUNCTION :

5. TRANSIENT CONDUCTION EXAMPLE  AN EXAMPLE OF THIS CALCULATION IS SHOWN FOR A TEMPERATURE CHANGE IN A CONCRETE SLAB. TIME IS IN ½ HOUR INCREMENTS AND DEPTH IS IN 5 cm INCREMENTS

6. TRANSIENT CONDUCTION EXAMPLE

7. 3 DIMENSIONAL OUTPUT

8. TRANSIENT CONDUCTION EXAMPLE  THE SURFACE GRADIENT CAN BE CALCULATED AS:  THE TOTAL HEAT CHANGE OVER TIME IS THEN:

9. TRANSIENT CONDUCTION EXAMPLE  SOLUTION FOR A FINITE VALUE OF THE CONVECTION COEFFICIENT, USING THE GAUSSIAN ERROR FUNCTION:  USING THE COMPLEMENTARY ERROR FUNCTION:

10. SUPERPOSITION METHODS  FOR SOLID TRANSIENT SYSTEMS  THE PRODUCTS OF ONE DIMENSIONAL SOLUTIONS ARE USED TO OBTAIN THE TEMPERATURE GRADIENTS IN TWO DIMENSIONAL SYSTEMS  FOR TEMPERATURE PROFILES  ONE DIMENSIONAL SOLUTIONS USED INCLUDE:  PLANE WALL  INFINITE CYLINDER  SEMI-INFINITE SOLID  APPLICATION WILL RESULT IN THE TEMPERATURE WITHIN THE SOLID AT A SPECIFIC LOCATION AND TIME  TABLE 4-5 PROVIDES A SUMMARY FOR VARIOUS SYSTEMS

11. SUPERPOSITION METHODS

12.  FOR TOTAL HEAT TRANSFERRED, THE DIMENSIONLESS HEAT TERMS ARE USED:  FOR TWO DIMENSIONAL GEOMETRIES  FOR THREE DIMENSIONAL GEOMETRIES