Static thermal permeability (JCAL)

Lafarge's static thermal permeability: the low-frequency limit of the thermal exchange between gas and fibres. It is the additional parameter of the JCAL model (shown only for JCAL) and corrects the low-frequency bulk modulus that JCA derives from the thermal length alone.

Last updated 2026-11-11

Theory

Lafarge et al. (Lafarge et al., 1997) introduced $k'_0$ , the thermal analogue of the static viscous permeability $\eta/R$ , to describe the bulk modulus correctly at low frequency. In the bulk modulus of equation (1) it appears in the first term (low-frequency behaviour) and in the square root (the transition). With $k'_0 = \phi\Lambda'^2/8$ , JCAL reduces exactly to JCA; the program's tests confirm this.

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)

0.2 < \frac{k'_0}{\phi\Lambda'^2/8} < 5 \quad \text{(otherwise a QA warning)}

(2)

where	meaning	unit
$k'_0$	Static thermal permeability (this setting)	m²
$\Lambda'$	Thermal characteristic length	m
$\phi$	Porosity	-
$\gamma, P_0, \eta, \mathrm{Pr}, \rho_0$	Gas: ratio of specific heats, absolute pressure, viscosity, Prandtl number, density	SI

Default and rule

Default: blank = φ Λ'² / 8 (the JCA value)Computed by the program
Rule: Blank: $\phi\Lambda'^2/8$ , so that JCAL behaves as JCA.
Source: The JCAL model is that of Lafarge et al. (1997). Equation (1) follows an open reference description of the model and is implemented together with the JCA equations, which are cross-checked against Atalla and Panneton (2000). The 0.2-5 consistency band is the program's own check: a factor that catches a unit error, such as mm² entered for m².

Valid range > 0; warning outside 0.2-5 × φ Λ'² / 8

Values

Above 0, in m² (fibrous materials: of the order of $10^{-9}$ m²).

Consistency

Outside 0.2-5 times the JCA value the QA gate warns: check the unit.

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; warning outside 0.2-5 × φ Λ'² / 8

Related 4

References 2