## 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 ~ A^{1/2} 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## Nomenclature

- a
actual contact area (πδ

^{2})- a
_{o} area when local yielding starts

- A
nominal contact area (πr

_{o}^{2})- A
_{m} Area between elements

- C
_{j} coefficient Bessel expansion of T

- D
_{i, j} coefficient of generalized expansion

- F
force

- F
_{o} force when local yielding starts

- g2m
Taylor expansion coefficients of T’

- i, j, m
indices

- (i, j)
i

^{ th }, j^{ th elemen}**J**_{0}**J**_{1}Bessel functions

- k
thermal conductance

- L
_{m} distance between elements

- n
number of contacts

- n
_{0} n when local yielding starts

- Q
heat flow

- Q
_{j} Q at z = z

_{j}- r
radius

- r
_{i} = i Δr

- r
_{o} radius of cylinder

- Δr
width of toroidal element

- T
temperature

- T
_{i, j} temperature of element (i, j)

- T
_{m} temperature of element (m)

- T
_{o} temperature at a perfect contact

- T
_{x} excess temperature

- T
^{✓} axial temperature gradient (∂T/∂z)

- T
^{✓}∞ T

^{✓}far from contact- ΔT
excess temperature

- ΔT
_{n} excess temperature for n contacts

- ΔT
_{1} excess temperature for 1 contact

- ΔT
mean approx. excess temperature

- ΔT
_{axial} approx. on axis excess temperature

- ΔT
_{Y} Yovanovich’s approximation

- x
= λ

_{j}δ- Δx
change in x

- z
axial distance from contact

- z
_{j} = j Δz

- z
_{max} maximum z

- Δz
height of toroidal element

- δ
radius of contact

- λ
_{j} constant = j

^{ th }zero of**J**_{1}- ν
variance

- σ
_{y} yield stress

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## References

- 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.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.Hildebrand, F.B., Advanced Calculus for Applications. Prentice-Hall, New Jersey (1962)Google Scholar