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
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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 (225) 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
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COMPOSITE TRANSFER EQUATION
T
q



Overall

i


T
Overall
T3  T0
i

Res is tance term s:
Internal Convection:
Ri
1
2    r0  L  hi
External Convection:
Ri
1
2    r3  L  ho
Acros s Annual s ections:
R1
 r1 
ln
 r0 
 
2    k1  L
R2
 r2 
ln
 r1 
 
2    k2  L
T
Subs tituting : q
R3
 r3 
ln
 r2 
 
2    k3  L
Overall
Ri  R1  R2  R3  Ro
OVERALL RESISTANCE VERSION
In terms of Overall Heat Transfer Coefficient;
T
q
Overall
Rtotal
U  A T
If U is defined in terms of inner Area, A
0:
U0
1
ro
 r1  ro  r2  ro  r3  r0 i
i

 ln

 ln

 ln








hi
k1  r0  k2  r1  k3  r2  ho ho
U0  A0
U1  A1
U2  A2
U3  A3

i
1
Ri