# Mathematical expression

The mathematical definition of the static thermal permeability $k'_0$ given by Lafarge, Lemarinier, Allard and Tarnow is:

$k'_0 = \lim_{\omega \rightarrow 0} k'(\omega)$

where the dynamic thermal permeability $k'(\omega)$ is defined as:

$\phi\tau = \frac{k'(\omega)}{\kappa}\frac{\partial p}{\partial t}$

In this last expression, $\phi$ is the open porosity of the material. $\tau$ is the temperature excess inside the porous medium subjected to a variation of pressure with respect to time $\partial p/\partial t$. $\kappa$ is the thermal conductivity of air.

# Physical description

In 1997, Lafarge & et al. [LLAT97] use a thermal equivalent to the dynamic Darcy's law to describe thermal effects in a porous medium. This work leads to the introduction of a new parameter: the static thermal permeability i.e. the low frequency limit of the dynamic thermal permeability.

The static thermal permeability plays, in the description of the thermal exchanges between frame and saturating fluid, a role similar to the viscous permeability in the description of the viscous forces.

This static thermal permeability characterizes, partly, the thermal effects at low frequencies (when the thermal boundary layer is of the order of magnitude of the characteritic size of the pores).

The static thermal permeability $k'_0$ is a geometrical parameter and does not depend on the fluid properties. It is equivalent to the inverse of the trapping constant of the solid frame used by some authors.

For acoustical materials, the range of values for the static thermal permeability is approximately [$10^{-10}$ $10^{-8}$] m$^{2}$. In addition, it can be shown $k'_0$ should always be greater than or equal to the static [viscous] permeability $k_0$ which is defined as the ratio of the dynamic viscosity of air $\eta$ over the static air-flow resistivity $\sigma$:

$k'_0 \ge k_0 \Big(\equiv \displaystyle\frac{\eta}{\sigma}\Big)$

The static thermal permeability can be estimated from impedance tube or ultrasonic measurements.