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CHE/ME 109 Heat Transfer in Electronics. LECTURE 9 – GENERAL TRANSIENT CONDUCTION MODELS. GENERAL TRANSIENT MODEL. BASED ON THE CHANGE IN TEMPERATURE IN SYSTEMS WITH THE TRANSFER OF HEAT. LUMPED CAPACITANCE MODELS THE MODEL THAT APPLIES FOR TRANSIENT DEPENDS ON THE CONTROLLING RESISTANCE .

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che me 109 heat transfer in electronics

CHE/ME 109 Heat Transfer in Electronics

LECTURE 9 – GENERAL TRANSIENT CONDUCTION MODELS

general transient model
GENERAL TRANSIENT MODEL
  • BASED ON THE CHANGE IN TEMPERATURE IN SYSTEMS WITH THE TRANSFER OF HEAT.
  • LUMPED CAPACITANCE MODELS
  • THE MODEL THAT APPLIES FOR TRANSIENT DEPENDS ON THE CONTROLLING RESISTANCE.
  • FOR SYSTEMS WHERE THE CONDUCTION RESISTANCE IS NEGLIGIBLE, THE TEMPERATURE OF A BODY CAN BE ASSUMED TO BE UNIFORM
lumped capacitance
LUMPED CAPACITANCE
  • DETERMINATION OF THIS CONDITION IS OBTAINED BY TAKING THE RATIO OF THE RESISTANCE TERMS, THE BIOT NUMBER:
  • WHEN THE INTERNAL RESISTANCE (CONDUCTION) IS LESS THAT 10% OF THE EXTERNAL (CONVECTION) RESISTANCE,
lumped capacitance1
LUMPED CAPACITANCE
  • THE BIOT NUMBER IS THE VALUE THAT IS USED TO DETERMINE WHETHER THE INTERNAL RESISTANCE IS NEGLIGIBLE
  • THE VALUE USED FOR x IN THIS EQUATION IS BASED ON THE CHARACTERISTIC LENGTH
heat balance equation for transient states
HEAT BALANCE EQUATION FOR TRANSIENT STATES
  • “RATE OF HEAT TRANSFER TO THE BODY IS EQUAL TO THE CHANGE IN INTERNAL ENERGY IN THE SYSTEM”
transient heat balance
TRANSIENT HEAT BALANCE
  • INTEGRATING FROM t = 0 TO t AND T(0) TO T(t) YIELDS
heat balance equation
HEAT BALANCE EQUATION
  • THIS EQUATION HAS A DIMENSIONLESS TEMPERATURE FORM
  • FOR A CONSTANT Cp, THE FRACTION OF TOTAL CHANGE IN THE HEAT IN THE SYSTEM IS
  • THE MAXIMUM HEAT THAT CAN BE TRANSFERRED IS REPRESENTED BY
electrical analog
ELECTRICAL ANALOG
  • Ω = OHMS, Ce = FARADS, E = VOLTS:
transient system models
TRANSIENT SYSTEM MODELS
  • WITH TEMPERATURE GRADIENTS IN THE SYSTEM
  • GENERAL FORM OF THE ONE DIMENSIONAL TRANSIENT HEAT TRANSFER EQUATION,AS APPLIED TO A PLANE WALL AS SHOWN IN FIGURE 4-11(a)
transient system solutions
TRANSIENT SYSTEM SOLUTIONS
  • GENERAL SOLUTION FOR THIS EQUATION ASSUMES THE FORM:
transient system solutions1
TRANSIENT SYSTEM SOLUTIONS
  • THIS EQUATION IS SOLVED ANALYTICALLY BY SEPARATION OF VARIABLES:
  • THE SOLUTION TO THIS EQUATION IS TWO ODE”S OF THE FORM:
transient system solutions2
TRANSIENT SYSTEM SOLUTIONS
  • SPECIFIC SOLUTIONS TO THIS TRANSCENDENTAL EQUATION RESULT DEPEND ON THE GEOMETRY OF THE SYSTEM, THE BOUNDARY CONDITIONS AND THE INITIAL CONDITIONS
  • THE CHARACTERISTIC SOLUTIONS ARE SUMS OF EIGENFUNCTIONS
  • THE NUMBER OF SIGNIFICANT TERMS DEPENDS ON THE BIOT NUMBER
transient system solutions3
TRANSIENT SYSTEM SOLUTIONS
  • SOLUTIONS ARE CONVENIENTLY PRESENTED IN TABULAR OR GRAPHICAL FORM USING DIMENSIONLESS VARIABLES
  • TABLE 4-2 AND FIGURES 4-15 THROUGH 4-17 (HEISLER CHARTS) PROVIDE SOLUTIONS
  • PARAMETERS ARE
  • DIMENSIONLESS TEMPERATURE:
  • BIOT NUMBER:
  • FOURIER NUMBER (DIMENSIONLESS TIME):
heisler charts
HEISLER CHARTS
  • GRAPHICAL TRANSIENT SOLUTIONS ARE TYPICALLY PROVIDED IN FIGURES FOR SYSTEMS OF PLANES, CYLINDERS AND SPHERES
  • TEMPERATURE IN THE CENTER OF THE SYSTEM (a)
  • TEMPERATURE DISTRIBUTION AT DIFFERENT LOCATIONS (DISTANCE) (b)
  • TOTAL HEAT TRANSFERRED, NORMALIZED WITH THE TOTAL THAT CAN BE TRANSFERRED (c)