CHE/ME 109 Heat Transfer in Electronics

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CHE/ME 109 Heat Transfer in Electronics

• LECTURE 10 SPECIFIC TRANSIENT CONDUCTION MODELS

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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

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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 gt0.

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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

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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

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TRANSIENT CONDUCTION EXAMPLE
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3 DIMENSIONAL OUTPUT
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TRANSIENT CONDUCTION EXAMPLE

• THE SURFACE GRADIENT CAN BE CALCULATED AS
• THE TOTAL HEAT CHANGE OVER TIME IS THEN

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TRANSIENT CONDUCTION EXAMPLE

• SOLUTION FOR A FINITE VALUE OF THE CONVECTION
  COEFFICIENT, USING THE GAUSSIAN ERROR FUNCTION
• USING THE COMPLEMENTARY ERROR FUNCTION

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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

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SUPERPOSITION METHODS
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SUPERPOSITION METHODS

• FOR TOTAL HEAT TRANSFERRED, THE DIMENSIONLESS
  HEAT TERMS ARE USED
• FOR TWO DIMENSIONAL GEOMETRIES
• FOR THREE DIMENSIONAL GEOMETRIES