CHE/ME 109 Heat Transfer in Electronics

LECTURE 4 – HEAT TRANSFER MODELS

HEAT TRANSFER MODEL PARAMETERS

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MODELS ARE BASED ON FOUR SETS OF PARAMETERS

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TIME VARIABLES GEOMETRY SYSTEM PROPERTIES HEAT GENERATION

TIME VARIABLES

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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

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

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

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HEAT TRANSFER DIMENSIONS

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HEAT TRANSFER IS A THREE DIMENSIONAL PROCESS

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SOME CONDITIONS ALLOW SIMPLIFICATION TO ONE AND TWO DIMENSIONAL SYSTEMS

SYSTEM PROPERTIES

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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 http://www.feppd.or

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HEAT GENERATION

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

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GENERATION CAN BE A POINT OR UNIFORM VOLUMETRIC PHENOMENON

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TYPICAL EXAMPLES INCLUDE:

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RESISTANCE HEATING WHICH OCCURS IN POWER CABLES AND HEATERS

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REACTION SYSTEMS, CHEMICAL AND NUCLEAR

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

SPECIFIC MODELS

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RECTANGULAR MODELS CAN BE DEVELOPED AS SHOWN IN THE FOLLOWING FIGURES

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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

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

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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)

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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

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

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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

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GENERAL ONE-DIMENSIONAL HEAT TRANSFER EQUATION IS

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WHERE THE VALUE OF n IS

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0 FOR RECTANGULAR COORDINATES

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1 FOR CYLINDRICAL COORDINATES 2 FOR SPHERICAL COORDINATES

GENERAL RESISTANCE METHOD

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CONSIDER A COMPOSITE SYSTEM CONVECTION ON INSIDE AND OUTSIDE SURFACES STEADY-STATE CONDITIONS EQUATION FOR Q http://www.owlnet.rice.

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COMPOSITE TRANSFER EQUATION

q  T Overall   i i  T Overall Resistance terms: Internal Convection: Ri 2   1  r0  L  hi External Convection: Ri 2   1  r3  L  ho Across Annual sections: R1 ln   2   r1 r0    k1  L R2 ln   2   r2 r1    k2  L R3 ln   2   r3 r2    k3  L Substituting : q  T Overall  R2  R3  Ro

OVERALL RESISTANCE VERSION

In terms of Overall Heat Transfer Coefficient; q  T Overall Rtotal   T If U is defined in terms of inner Area, A 0 : U0 i hi  ro k1  ln   r1 r0    ro k2  ln   1 r2 r1    ro k3  ln   r3 r2    r0 ho  i ho  i 1 Ri