Thermal characteristic length (JCA / JCAL)

A length that characterises the pore surface available for heat exchange between the gas and the fibres. It sets where the bulk modulus of the gas in the pores changes from isothermal (low frequency) to adiabatic (high frequency). It is an input of the JCA and JCAL models only; a blank field takes twice the viscous length.

Last updated 2026-11-11

Theory

Champoux and Allard (Champoux and Allard, 1991) defined $\Lambda'$ as twice the ratio of pore volume to pore surface. At low frequency the gas in the pores exchanges heat with the fibres and is compressed isothermally (bulk modulus $P_0/\phi$ ); at high frequency it is compressed adiabatically ( $\gamma P_0/\phi$ ). The transition is controlled by the thermal length and, in JCAL, by the static thermal permeability, equation (1). In JCA, $k'_0$ is not an input but follows from $\Lambda'$ , equation (2).

Equations

K_{eq} = \frac{\gamma P_0/\phi}{\gamma - (\gamma - 1)\left[1 - i\,\dfrac{\phi\,\eta/\mathrm{Pr}}{k'_0\rho_0\omega}\sqrt{1 + i\,\dfrac{4 k_0'^2\rho_0\omega}{(\eta/\mathrm{Pr})\Lambda'^2\phi^2}}\right]^{-1}}

(1)

k'_0 = \frac{\phi\Lambda'^2}{8}\ \ \text{(JCA; JCAL when its permeability is blank)}, \qquad \Lambda' = 2\Lambda\ \ \text{(when blank)}

(2)

where	meaning	unit
$\Lambda'$	Thermal characteristic length (this setting)	m
$\Lambda$	Viscous characteristic length	m
$k'_0$	Static thermal permeability	m²
$K_{eq}$	Equivalent bulk modulus	Pa
$\gamma, P_0, \eta, \mathrm{Pr}, \rho_0$	Gas: ratio of specific heats, absolute pressure, viscosity, Prandtl number, density	SI
$\phi$	Porosity	-
$\omega$	Angular frequency	rad/s

Default and rule

Default: blank = 2 × viscous lengthComputed by the program
Rule: Blank: $\Lambda' = 2\Lambda$ , the usual ratio for fibrous materials.
Source: The equations are cross-checked against Atalla and Panneton (2000); the thermal length is defined by Champoux and Allard (1991). The ratio 2 is a common estimate for fibrous materials; it is not taken from a cited source.

Valid range > 0; ≥ viscous length

Values

Above 0; entered in µm.

Shorter than the viscous length

The QA gate warns when $\Lambda' < \Lambda$ .

Verified

Checked in a full acoustic run of a packed resonator (2026-11-11): every quality check passes; with estimated parameters the TL agrees with Miki within about 1 dB below 1 kHz and falls 7-9 dB below it at 2 kHz.

Theory

Equations

Default and rule

Valid range > 0; ≥ viscous length

Related 4

References 2