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Cryocoolers 8 pp 755-764 | Cite as

Modeling Thermal Contact Resistance

  • Peter Kittel

Abstract

One difficulty in using cryocoolers is making good thermal contact between the cooler and the instrument being cooled. The connection is often made through a bolted joint. The temperature drop associated with this joint has been the subject of many experimental and theoretical studies. The low temperature behavior of dry joints has shown some anomalous dependence on the surface condition of the mating parts. There is also some doubt on how well one can extrapolate from test samples to predicting the performance of a real system.

Both finite element and analytic models of a simple contact system have been developed. The models show that in the limit of actual contact area ≪ the nominal area (a ≪ A), that the excess temperature drop due to a single point of contact scales as a−1/2. This disturbance only extends a distance ~ A1/2 into the bulk material. A group of identical contacts will result in an excess temperature drop that scales as n−1/2, where n is the number of contacts and n·a is constant. This implies that flat rough surfaces will have a lower excess temperature drop than flat smooth surfaces.

Keywords

Finite Element Model Excess Heat Local Yielding Lower Order Term Excess Temperature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

a

actual contact area (πδ2)

ao

area when local yielding starts

A

nominal contact area (πro 2)

Am

Area between elements

Cj

coefficient Bessel expansion of T

Di, j

coefficient of generalized expansion

F

force

Fo

force when local yielding starts

g2m

Taylor expansion coefficients of T’

i, j, m

indices

(i, j)

i th , j th elemen

J0J1

Bessel functions

k

thermal conductance

Lm

distance between elements

n

number of contacts

n0

n when local yielding starts

Q

heat flow

Qj

Q at z = zj

r

radius

ri

= i Δr

ro

radius of cylinder

Δr

width of toroidal element

T

temperature

Ti, j

temperature of element (i, j)

Tm

temperature of element (m)

To

temperature at a perfect contact

Tx

excess temperature

T

axial temperature gradient (∂T/∂z)

T

T far from contact

ΔT

excess temperature

ΔTn

excess temperature for n contacts

ΔT1

excess temperature for 1 contact

ΔT

mean approx. excess temperature

ΔTaxial

approx. on axis excess temperature

ΔTY

Yovanovich’s approximation

x

= λj δ

Δx

change in x

z

axial distance from contact

zj

= j Δz

zmax

maximum z

Δz

height of toroidal element

δ

radius of contact

λj

constant = j th zero of J 1

ν

variance

σy

yield stress

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References

  1. 1.
    Yovanovich, M.M. “General Expression for Circular Constriction Resistances for Arbitrary Flux Distributions,” Progress in Astronautics and Aeronautics. vol. 49 (1976) p. 381.Google Scholar
  2. 2.
    Salerno, L.J., Kittel, P., Brooks, W.F., Spivak, A.L., and Marks Jr., W.G., “Thermal Conductance of Pressed Brass Contacts at Liquid Helium Temperatures,” Cryogenics, vol. 26, (1986) p. 217.ADSGoogle Scholar
  3. 3.
    Hildebrand, F.B., Advanced Calculus for Applications. Prentice-Hall, New Jersey (1962)Google Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Peter Kittel
    • 1
  1. 1.NASA Ames Research CenterUSA

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