Thermal contact conductance

Fig. 1: Heat flow between two solids in contact and the temperature distribution.

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:

${\displaystyle q=-kA{\frac {dT}{dx}}}$

(1)

where ${\displaystyle q}$ is the heat flow, ${\displaystyle k}$ is the thermal conductivity, ${\displaystyle A}$ is the cross sectional area and ${\displaystyle dT/dx}$ 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:

${\displaystyle q={\frac {T_{1}-T_{3}}{\Delta x_{A}/(k_{A}A)+1/(h_{c}A)+\Delta x_{B}/(k_{B}A)}}}$

(2)

One may observe that the heat flow is directly related to the thermal conductivities of the bodies in contact, ${\displaystyle k_{A}}$ and ${\displaystyle k_{B}}$, the contact area ${\displaystyle A}$, and the thermal contact resistance, ${\displaystyle 1/h_{c}}$, which, as previously noted, is the inverse of the thermal conductance coefficient, ${\displaystyle h_{c}}$.

Most experimentally determined values of the thermal contact resistance fall between 0.000005 and 0.0005 m² K/W (the corresponding range of thermal contact conductance is 200,000 to 2000 W/m² K). To know whether the thermal contact resistance is significant or not, magnitudes of the thermal resistances of the layers are compared with typical values of thermal contact resistance. Thermal contact resistance is significant and may dominate for good heat conductors such as metals but can be neglected for poor heat conductors such as insulators.[2] 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:[3][4][5]

• 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
• Residential/Building Science
  • Performance of building envelops

Fig. 2: An enlargement of the interface between two contacting surfaces. The finish quality is exaggerated for the sake of the argument.

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

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, ${\displaystyle A}$ (A product of surface characteristics, as explained earlier). Because of this, contact conductance/resistance is usually found experimentally, by using a standard apparatus.[9]

The results of such experiments are usually published in Engineering literature, on journals 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.

While a finite thermal contact conductance is due to voids at the interface, surface waviness, and surface roughness, etc., a finite conductance exists even at near ideal interfaces as well. This conductance, known as thermal boundary conductance, is due to the differences in electronic and vibrational properties between the contacting materials. This conductance is generally much higher than thermal contact conductance, but becomes important in nanoscale material systems.

• Heat transfer

• Project CoCoE - Free software to estimate TCC