CHE/ME 109 Heat Transfer in

ElectronicsLECTURE 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.

• FOR SYSTEMS WHERE THE CONDUCTION RESISTANCE IS NEGLIGIBLE, THE TEMPERATURE OF A BODY CAN BE ASSUMED TO BE UNIFORM

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 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

• “RATE OF HEAT TRANSFER TO THE BODY IS EQUAL TO THE CHANGE IN INTERNAL ENERGY IN THE SYSTEM”

TRANSIENT HEAT BALANCE

• INTEGRATING FROM t = 0 TO t AND T(0) TO T(t) YIELDS

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

• Ω = OHMS, Ce = FARADS, E = VOLTS:

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

• GENERAL SOLUTION FOR THIS EQUATION ASSUMES THE FORM:

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 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 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• 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)

HEISLER CHARTS

HEISLER CHARTS