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CHE/ME 109 Heat Transfer in Electronics

CHE/ME 109 Heat Transfer in Electronics. LECTURE 10 – SPECIFIC TRANSIENT CONDUCTION MODELS. SEMI-INFINITE SOLID SOLUTIONS. SEMI-INFINITE SOLIDS HAVE ONE PLANE SURFACE ON AN INFINITE VOLUME

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CHE/ME 109 Heat Transfer in Electronics

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  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. SUPERPOSITION METHODS • FOR TOTAL HEAT TRANSFERRED, THE DIMENSIONLESS HEAT TERMS ARE USED: • FOR TWO DIMENSIONAL GEOMETRIES • FOR THREE DIMENSIONAL GEOMETRIES

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