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# Thermal contact conductance

In physics, thermal contact conductance is the study of heat conduction between solid bodies in contact. The thermal contact conductance coefficient, hc, is a property indicating the thermal conductivity, or ability to conduct heat, between two bodies in contact. The inverse of this property is termed thermal contact resistance.

## Definition

When two solid bodies come in contact, such as A and B in Figure 1, heat flows from the hotter body to the colder body. From experience, the temperature profile along the two bodies varies, approximately, as shown in the figure. A temperature drop is observed at the interface between the two surfaces in contact. This phenomenon is said to be a result of a thermal contact resistance existing between the contacting surfaces. Thermal contact resistance is defined as the ratio between this temperature drop and the average heat flow across the interface.[1]

According to Fourier's law, the heat flow between the bodies is found by the relation: $q=-kA\frac{dT}{dx}$ (1)
where q is the heat flow, k is the thermal conductivity, A is the cross sectional area and dT / dx is the temperature gradient in the direction of flow.

From considerations of energy conservation, the heat flow between the two bodies in contact, bodies A and B, is found as: $q=\frac{T_1 - T_3}{\Delta x_A/k_A A+1/h_c A + \Delta x_B/k_B A}$ (2)
One may observe that the heat flow is directly related to the thermal conductivities of the bodies in contact, kA and kB, the contact area, A and the thermal contact resistance, 1 / hc, which, as previously noted, is the inverse of the thermal conductance coefficient, hc. The thermal contact resistance may not be applied for the sandwich kind of materials since they are manufactured by rolling under high temperatures so that the decrease in thermal conductivity is negligible.

## Importance

Thermal contact conductance is an important factor in a variety of applications, largely because many physical systems contain a mechanical combination of two materials. Some of the fields where contact conductance is of importance are:[2][3][4]

• Electronics
• Electronic packaging
• Heat sinks
• Brackets
• Industry
• Nuclear reactor cooling
• Gas turbine cooling
• Internal combustion engines
• Heat exchangers
• Thermal insulation
• Flight
• Hypersonic flight vehicles
• Thermal supervision for space vehicles

## Factors influencing contact conductance

Thermal contact conductance is a complicated phenomenon, influenced by many factors. Experience shows that the most important ones are as follows:

### Contact pressure

The contact pressure is the factor of most influence on contact conductance. As contact pressure grows, contact conductance grows (And consequentially, contact resistance becomes smaller). This is attributed to the fact that the contact surface between the bodies grows as the contact pressure grows.

Since the contact pressure is the most important factor, most studies, correlations and mathematical models for measurement of contact conductance are done as a function of this factor.

### Interstitial materials

No truly smooth surfaces really exist, and surface imperfections are visible under a microscope. As a result, when two bodies are pressed together, contact is only performed in a finite number of points, separated by rather large gaps, as can be shown in Fig. 2. Since that actual contact area is reduced, another resistance for heat flow exists. The gasses/fluids filling these gaps may largely influence the total heat flow across the interface. Air is the most common interstitial material. The thermal conductivity of the interstitial material and its pressure are the two properties governing its influence on contact conductance.

In the absence of interstitial materials, such as the bodies are in vacuum, the contact resistance will be much larger, since flow through the intimate contact points is dominant.

### Surface roughness, waviness and flatness

One can characterise a surface that has undergone certain finish operations by three properties: Roughness, waviness and flatness. Among these, roughness is of most importance, and is usually indicated by an rms value, σ.

### Surface deformations

When the two bodies come in contact, surface deformation may occur on both bodies. This deformation may either be plastic or elastic, depending on the material properties and the contact pressure. When a surface undergoes plastic deformation, contact resistance is lowered, since the deformation causes the actual contact area to increase[5][6]

### Surface cleanliness

The presence of dust particles, acids, etc., can also influence the contact conductance.

## Measurement of thermal contact conductance

Going back to Formula 2, calculation of the thermal contact conductance may prove difficult, even impossible, due to the difficulty in measuring the contact area, A (A product of surface characteristics, as explained earlier). Because of this, contact conductance/resistance is usually found experimentally, by using a standard apparatus.

The results of such experiments are usually published in Engineering literature, on magazines such as Journal of Heat Transfer, International Journal of Heat and Mass Transfer, etc. Unfortunately, a centralized database of contact conductance coefficients does not exist, a situation which sometimes causes companies to use outdated, irrelevant data, or not taking contact conductance as a consideration at all.

CoCoE (Contact Conductance Estimator), a project founded to solve this problem and create a centralized database of contact conductance data and a computer program that uses it, was started in 2006.

## References

1. ^ Hollman, J. P. (1997). Heat Transfer, 8th Edition. McGraw-Hill.
2. ^ Fletcher, L. S. (November 1988). "Recent Developments in Contact Conductance Heat Transfer". Journal of Heat Transfer.
3. ^ Madhusudana, C. V.; Ling, F. F. (1995). Thermal Contact Conductance. Springer.
4. ^ Lambert, M. A.; Fletcher, L. S. (November 1997). "Thermal Contact Conductance of Spherical Rough Metals". Journal of Heat Transfer.
5. ^ Williamson, M.; Majumdar, A. (November 1992). "Effect of Surface Deformations on Contact Conductance". Journal of Heat Transfer.
6. ^ Heat Transfer Division (November 1970). "Conduction in Solids - Steady State, Imperfect Metal-to-Metal Surface Contact". General Electric Inc..