Horoshenkov Model

The Horoshenkov model consists of three parameters -- the porosity \((\phi)\), median pore size \((\bar s)\), and pore size distribution \((\sigma_{s})\).

From these three parameters, the rest of the JCAL parameters -- the static airflow resistivity \((\sigma)\), porosity \((\phi)\), tortuosity \((\tau)\), viscous characteristic length \((\Lambda)\), thermal characteristic length \((\Lambda^\prime)\), and thermal permeability \((k_{0}^\prime)\) -- can be calculated as seen below:

\[ \tau = \exp\Bigg(4[\sigma_{s}ln(2)]^2\Bigg) \]

\[ \sigma = \frac{8\eta\tau}{\phi{\bar s^2}}\exp\Bigg(6[\sigma_{s}ln(2)]^2\Bigg) \]

\[ \Lambda = \bar s \exp\Bigg(-\frac{5}{2}[\sigma_{s}ln(2)]^2\Bigg) \]

\[ \Lambda^\prime = \bar s \exp\Bigg(\frac{3}{2}[\sigma_{s}ln(2)]^2\Bigg) \]

\[ k_{0}^\prime = \frac{\phi{\bar s^2}}{8\tau}\exp\Bigg(-6[\sigma_{s}ln(2)]^2\Bigg) \]

The dynamic mass density and bulk modulus are then determined within the _calc_dynamics method using the calculated JCAL parameters, following the implemention found here: JCAL.

The Add_Horoshenkov_Layer method then converts The dynamic mass density and bulk modulus to the characteristic impedence \((Z_{c})\) and wavenumber \((k_{c})\) for use in the layer transfer matrix.

\[ Z_{c} = \sqrt{\tilde{\rho}\widetilde{K}} \]

\[ k_{c} = {\omega}\sqrt{\frac{\tilde{\rho}}{\widetilde{K}}} \]

Model Parameters:

Using the following nomenclature --- Symbol = [Units] (name)

\[ \phi = \Bigg[unitless\Bigg]\tag{porosity} \]

\[ \bar s = \Bigg[{\mu}m\Bigg]\tag{median pore size} \]

\[ \sigma_{s} = \Bigg[unitless\Bigg]\tag{pore size distribution} \]

Defining Other Symbols:

Using the following nomenclature --- Symbol = [Units] (name)

\[ \eta = \Bigg[Pa*s\Bigg]\tag{viscosity of air} \]