CHE/ME 109 Heat Transfer in Electronics LECTURE 4 – HEAT TRANSFER MODELS.
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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
http://ccrma-www.stanford.edu/~jos/fp/img609.png
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
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