
Spreading Resistance of Circular Sources on
Compound and Isotropic Disks

Summary
This application calculates thermal spreading resistance
for a circular source on a
circular disk, semiinfinite circular cylinder,
or halfspace.
Solutions are available for a compound medium with
two layers having different thicknesses
and thermal conductivities,
and an isotropic medium with constant properties throughout.
For the finite disk cases, two boundary conditions
for the lower surface are considered:
a
uniform film coefficient, corresponding to convective or
contact cooling,
and a uniform
temperature boundary condition.
The heat flux over the circular source is either uniform, parabolic, or
distributed such that the resulting source is equivalent to an isothermal
boundary condition.
Background
Thermal spreading resistance occurs whenever heat leaves a source of finite
dimensions and enters a larger region. The particular case modeled by this
calculator involves a planar circular heat source of radius a in
perfect contact with the top surface of a
circular disk or semiinfinite circular cylinder of radius b.
For the finite disk, the overall thickness of the disk is t, and the
disk is cooled over its entire bottom surface either with a uniform
convective or contact conductance h, or an isothermal boundary.
The lateral and nonsource top surface boundaries of the disk or cylinder
are adiabatic.
The total system thermal resistance R_{total}
is defined by:
R_{total} = ( T_{source}  T_{sink} ) / Q
where: 
T_{source} = 
areamean source temperature
( ^{o}C) 

T_{sink} = 
mean heat sink temperature ( ^{o}C)

 Q = 
heat flow rate through the heat flux channel
(W) 
R_{total} = R_{s} + R_{1D}
where R_{s} is the thermal spreading resistance of the
system and R_{1D} is the onedimesional thermal resistance,
defined as:
R_{1D} =
( t_{1} / k_{1} +
t_{2} /
k_{2} +
1 / h ) /
b^{2}
For the general case of a rectangular source area on a finite, twolayer
rectangular heat flux channel, the spreading resistance will depend on
several geometric and the heat flux distribution:
R_{s} = f ( a, b,
t_{1}, t_{2},
k_{1}, k_{2},
h
)
Three flux distributions are included in this applications:
The results for total, onedimensional, and spreading resistance are
presented in both dimensional and dimensionless forms, where the resistance
is nondimensionalized by:
R^{*} = 4 k_{1} a R
All calculations are based on methods described in
M.M. Yovanovich, J.R. Culham and P. Teertstra,
"Modeling Thermal Resistance of Diamond Spreader on Copper Heat Sink
Systems," presented at the IEPS Electronics Packaging Conference,
Sept. 29  Oct. 1, Austin, TX, 1996, and
M.M. Yovanovich, C.H. Tien and G.E. Schneider, "General Solution of
Constriction Resistance Within a Compound Disk,"
Heat Transfer, Thermal Control and Heat Pipes, AIAA Progress in
Astronautics and Aeronautics, Vol. 70, 1980
Instructions
 Click on the image below that best describes your problem
 When the required tables are loaded, enter
all input values in the table on the left
 Browser will calculate when the Calculate button is clicked
 Depending on the speed of your machine and
the number of terms, the solution may take
a while to compute
Copyright © 2006 Microelectronics Heat Transfer Laboratory