Contact Thermal Resistance

I had this problem some years ago for heat flowing through a stiffening ring on a vessel

I developed an expression (using conduction only) dependent on the material properties and contact pressure which is:

ΔT = q/K * 2√3*RMS*Ln(CP/YS) - in any compatible unit system

where

ΔT = Temperature drop across the boundary

q = heat flow per unit area

K = Thermal conductivity

RMS = Root Mean Square surface finish

CP = Contact Pressure

YS = yield strength of the materials

A numerical example for typical values

q=1kW/ft^2: RMS=128micro in; CP=20ksi; YS=120ksi; K=18BTU/hr ft °F

gives

ΔT = 12.6°F

This correlated well for my application

Thank you for the information.

How did you come to the relationship? why Ln(CP/YS) and why 2*3^0.5 ? I would want to understand the physics behind the relation.

I choosed to deveop a formula which considers the contact according to the Abbott curve and the fact that usually roughness is filled with a low conductive gas (air for instance).

How did you come to the relationship? why Ln(CP/YS)

I chose a triangular profile, initially starting "point to point". Since there is contact pressure the points will flatten generating two trapezoids with touching faces. The ratio CP/YS is a measure of how much of the faces are in contact (and therefore conduct) .The expression occurs in a log form as an integration of q/(K.A) is performed wrt x. A is a linear function of x. Hence the Log.

and why 2*3^0.5

The model is symmetrical about the contact plane. I found the temperature drop for one side and then of course doubled it for the total temperature drop.

The √3 comes from converting the triangle to an RMS value to correlate it to roughness

Hi,

The fact that your approximation was convergent with results you obtained confirmed my approach and I am very thankful for it.

As mentioned I started from the Abbott curve as in following figure approximated with a sinus function. Of course it is not always a symmetrical curve but the advantage of the sinus is that is considers the peaks as well hill side as valley side.

I made a comparison of the 2 results for same conditions i.e. same conductivity and same thermal flow (W/m² i am "metric") and the differences are well accorded to the form of the area versus depth accepted approximation, linear for you and sinusoidal for me.

As parameter I used the ratio p/Re and I have the feeling that for low pressures the sinus model could lie nearer to the actual result. In the middle range both are near to each other and show that it is useless to stress the contact over 30...40% of Re.

If you have by any chance some of measurements I would appreciate if it could be possible to check against the second approach. It is correct to get for lower ratios a lower temp drop with the sinus model since the surface crushes more.

As an answer to Rhabe: I asked for values because I have not the possibility, now, to make measurements not having all instrumentation at my disposal. In fact I know how to do them but thanks for the hint.

Hi,

better you do a simple measurement.

If you take two plane parts of known thermal conductance, bring these into contact with known contact force/pressure and measure temperature rise (on the outside surfaces) and heat flow.

This is best done by placing an electrical plane heater on one of the outside surfaces and cooling the other surface with water.

Then insulating other side of heater and periphery of assembly so that the heat (simply measured by voltage x amperes of electrical heater) is only flowing perpendicular to plates and contact zone into the constant temperature water (or oil).

Thus you have delta-temperature/power or K/W that is thermal resistance.

This needs correction as being the sum of the 2 thermal resistances of the 2 plates plus the wanted contact resistance.

This to be optimised by choosing enough power to get a good temperature difference and establishing the time to wait until steady state temperature have established.

If you are not certain about being correct, then make another measurement with thinner or thicker plates to check if the same result i s there.

RHABE