CHE/ME 109 Heat Transfer in Electronics

LECTURE 4 – HEAT TRANSFER MODELS

HEAT TRANSFER MODEL PARAMETERS

• MODELS ARE BASED ON FOUR SETS OF PARAMETERS– TIME VARIABLES– GEOMETRY– SYSTEM PROPERTIES– HEAT GENERATION

TIME VARIABLES

• STEADY-STATE - WHERE CONDITIONS STAY CONSTANT WITH TIME

• TRANSIENT - WHERE CONDITIONS ARE CHANGING IN TIME

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GEOMETRY

• THE COORDINATE SYSTEM FOR THE MODELS IS NORMALLY SELECTED BASED ON THE SHAPE OF THE SYSTEM.

• PRIMARY MODELS ARE RECTANGULAR, CYLINDRICAL AND SPHERICAL- BUT THESE CAN BE USED TOGETHER FOR SOME SYSTEMS

• HEAT TRANSFER DIMENSIONS

• HEAT TRANSFER IS A THREE DIMENSIONAL PROCESS

• SOME CONDITIONS ALLOW SIMPLIFICATION TO ONE AND TWO DIMENSIONAL SYSTEMS

SYSTEM PROPERTIES

• ISOTROPIC SYSTEMS HAVE UNIFORM PROPERTIES IN ALL DIMENSIONS

• ANISOTROPIC MATERIALS MAY HAVE VARIATION IN PROPERTIES WHICH ENHANCE OR DIMINISH HEAT TRANSFER IN A SPECIFIC DIRECTION

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HEAT GENERATION

• GENERATION OF HEAT IN A SYSTEM RESULTS IN AN “INTERNAL” SOURCE WHICH MUST BE CONSIDERED IN THE MODEL

• GENERATION CAN BE A POINT OR UNIFORM VOLUMETRIC PHENOMENON

• TYPICAL EXAMPLES INCLUDE:– RESISTANCE HEATING WHICH OCCURS IN

POWER CABLES AND HEATERS– REACTION SYSTEMS, CHEMICAL AND

NUCLEAR

• IN SOME CASES, THE SYSTEM MAY ALSO CONSUME HEAT, SUCH AS IN AN ENDOTHERMIC REACTION IN A COLD PACK

SPECIFIC MODELS

• RECTANGULAR MODELS CAN BE DEVELOPED AS SHOWN IN THE FOLLOWING FIGURES

• THE HEAT TRANSFER ENTERS AND EXITS IN x, y, AND z PLANES THROUGH THE CONTROL VOLUME

• DIMENSIONS OF THE VOLUME ARE Δx, Δy AND Δz• THE OVERALL MODEL FOR THE SYSTEM INCLUDES

GENERATION TERMS AND ALLOWS FOR CHANGES IN THE CONTROL VOLUME WITH TIME

DIFFERENTIAL MODEL

• THIS SYSTEM CAN BE REDUCED TO DIFFERENTIAL DISTANCE AND TIME, USING THE EXPRESSIONS FOR CONDUCTION HEAT TRANSFER AND HEAT CAPACITY TO YIELD:

DIFFERENTIAL MODEL FOR SPECIFIC SYSTEMS

• STEADY STATE:

• STEADY STATE WITH NO GENERATION:

• TRANSIENT WITH NO GENERATION:

• TWO DIMENSIONAL HEAT TRANSFER (TWO OPPOSITE SIDES ARE INSULATED).

• .ONE DIMENSIONAL HEAT TRANSFER (FOUR SIDES ARE INSULATED- OPPOSITE PAIRS)

OTHER VARIATIONS ON THE EQUATION FOR SPECIFIC

CONDITIONS• SIMILAR MODIFICATIONS CAN BE APPLIED TO THE

ONE AND TWO DIMENSIONAL EQUATIONS FOR:• STEADY STATE • AND NO-GENERATION CONDITIONS

http://www.emeraldinsight.com/fig/1340120602047.png

OTHER GEOMETRIES

• CYLINDRICAL USE A CONTROL VOLUME BASED ON ONE DIMENSIONAL (RADIAL) HEAT TRANSFER FOR THE CONDITIONS:

• THE ENDS ARE INSULATED OR THE AREA AT THE ENDS IS NOT SIGNIFICANT RELATIVE TO THE SIDES OF THE CYLINDER

• THE HEAT TRANSFER IS UNIFORM IN ALL DIRECTIONS AROUND THE AXIS.

• THE CONTROL VOLUME FOR THE ANALYSIS IS A CYLINDRICAL PIPE AS SHOWN IN FIGURE 2-15

• RESULTING DIFFERENTIAL FORMS OF THE MODEL EQUATIONS ARE SHOWN AS (2-25) THROUGH (2-28)

SPHERICAL SYSTEMS

• MODELED USING A VOLUME ELEMENT BASED ON A HOLLOW BALL OF WALL THICKNESS Δr (SEE FIGURE 2-17)

• FOR UNIFORM COMPONENT PROPERTIES, THE MODEL BECOMES ONE DIMENSIONAL FOR RADIAL HEAT TRANSFER.

• THE RESULTING EQUATIONS ARE (2-30) - (2-34) IN THE TEXT

GENERALIZED EQUATION

• GENERAL ONE-DIMENSIONAL HEAT TRANSFER EQUATION IS

• WHERE THE VALUE OF n IS– 0 FOR RECTANGULAR COORDINATES– 1 FOR CYLINDRICAL COORDINATES– 2 FOR SPHERICAL COORDINATES

GENERAL RESISTANCE METHOD

• CONSIDER A COMPOSITE SYSTEM

• CONVECTION ON INSIDE AND OUTSIDE SURFACES

• STEADY-STATE CONDITIONS

• EQUATION FOR Q

http://www.owlnet.rice.edu/~chbe402/ed1projects/proj99/dsmith/index.htm

COMPOSITE TRANSFER EQUATION

qT Overall

i

i

T Overall T3 T0

Resistance terms:

Internal Convection: Ri1

2 r0 L hi

External Convection: Ri1

2 r3 L ho

Across Annual sections:

R1

lnr1

r0

2 k1 LR2

lnr2

r1

2 k2 LR3

lnr3

r2

2 k3 L

Substituting : qT Overall

Ri R1 R2 R3 Ro

OVERALL RESISTANCE VERSION

In terms of Overall Heat Transfer Coefficient;

qT Overall

RtotalU A T

If U is defined in terms of inner Area, A0:

U01

i

hi

ro

k1ln

r1

r0

ro

k2ln

r2

r1

ro

k3ln

r3

r2

r0

ho

i

ho

U0 A0 U1 A1 U2 A2 U3 A3

i

1

Ri