Transport Correlations

Beyond the isotherm and rate model, the bed model needs closures for axial dispersion, effective axial thermal conductivity, and the wall heat-transfer coefficients. Each can be set to a constant value or computed from a built-in empirical correlation.

These are configured per adsorbent layer in the Adsorbent Properties window (Dispersion and Heat Transfer tabs) and, for the external coefficient, alongside the wall settings. Choosing a correlation instead of a constant makes the coefficient respond automatically to local velocity, temperature and gas properties.

Axial Mass Dispersion

Sets the axial dispersion coefficient $D_{ax}$ in the component mass balance. Select on the Dispersion → Axial tab.

Constant

D_{ax} = \text{constant}

Input: Dispersion coefficient [m² s⁻¹].

Ruthven (1984)

Combines molecular diffusion and mechanical (eddy) dispersion:

D_{ax} = 0.7\, D_m + 0.5\, \frac{|u|}{\epsilon_b}\, d_p

Symbol	Description	Unit
$D_m$	Molecular diffusivity	m² s⁻¹
$u$	Superficial velocity	m s⁻¹
$\epsilon_b$	Bed void fraction	-
$d_p$	Particle diameter	m

Axial Thermal Conductivity

Sets the effective axial conductivity $\lambda$ in the energy balance. Select on the Dispersion → Axial Heat tab.

Constant

\lambda = \text{constant}

Ruthven (1984)

The thermal analogue of the mechanical dispersion term:

\lambda = 0.5\, \frac{|u|}{\epsilon_b}\, d_p\, \rho_g\, C_{p,g}

Yagi and Kunii (1957)

A stagnant-plus-flow correlation that accounts for the solid/gas conductivity ratio:

\lambda = \lambda_{z}^{0} + 0.75\, k_g\, \mathrm{Pr}\, \mathrm{Re}

\lambda_{z}^{0} = k_g \left(\frac{k_s}{k_g}\right)^{n}, \qquad n = 0.28 - 0.757 \log_{10}(\epsilon_b) - 0.057 \log_{10}\!\left(\frac{k_s}{k_g}\right)

Symbol	Description	Unit
$\lambda_z^0$	Stagnant (no-flow) conductivity	W m⁻¹ K⁻¹
$k_g, k_s$	Gas and solid thermal conductivity	W m⁻¹ K⁻¹
$\mathrm{Pr}, \mathrm{Re}$	Prandtl and Reynolds numbers	-

Internal Heat Transfer Coefficient

The gas–wall coefficient $h_i$ in the energy and wall energy balances. Select on the Heat Transfer → Internal tab.

Constant

h_i = \text{constant}

Input: Heat transfer coefficient [W m⁻² K⁻¹]. This is the most common choice for fitting experimental breakthrough data.

Dixon (1996)

An apparent wall coefficient derived from a two-dimensional pseudo-homogeneous model, folding the near-wall resistance and the effective radial bed conductivity into a single 1-D coefficient. First the wall film and stagnant contributions,

h_w = \frac{k_g}{d_p}\left[2\epsilon_b + \frac{1-\epsilon_b}{\tfrac{1}{3}(k_g/k_s) + \theta_w}\right] + \frac{k_g}{d_p}\left(0.0835\,\mathrm{Re}^{0.91}\right), \qquad \theta_w = 0.0024\left(\frac{D_c}{d_p}\right)^{1.58}

then the effective radial conductivity,

k_e = k_g\left[\epsilon_b + \frac{1-\epsilon_b}{\tfrac{2}{3}(k_g/k_s) + 0.22\,\epsilon_b^2}\right] + \frac{|u|\,\rho_g C_{p,g}\, d_p}{\mathrm{Pe}_H}, \qquad \mathrm{Pe}_H = 8.65\left[1 + 19.4\left(\frac{d_p}{D_c}\right)^2\right]

and finally the apparent internal coefficient via the wall Biot number $\mathrm{Bi} = h_w D_c / (2 k_e)$ ,

\frac{1}{h_i} = \frac{1}{h_w} + \frac{D_c}{6 k_e}\cdot\frac{\mathrm{Bi} + 3}{\mathrm{Bi} + 4}

where $D_c$ is the column (internal) diameter. The film term is valid for roughly $10 < \mathrm{Re} < 1200$ .

External Heat Transfer Coefficient

The wall–ambient coefficient $h_o$ in the wall energy balance. Select alongside the wall settings.

Constant

h_o = \text{constant}

Holman (2008)

Forced convection over the outside of the column from a cross-flow of air:

h_o = \frac{\mathrm{Nu}\, k_{air}}{d_e}, \qquad \mathrm{Nu} = c\,\mathrm{Re}_{air}^{\,m}\,\mathrm{Pr}_{air}^{\,n}, \qquad \mathrm{Re}_{air} = \frac{d_e\, v_{ext}\, \rho_{air}}{\mu_{air}}

Input: External air flow velocity $v_{ext}$ . The air properties ( $k_{air}$ , $\rho_{air}$ , $\mu_{air}$ , $c_{p,air}$ ) and the correlation constants $c$ , $m$ , $n$ are built in; $d_e$ is the external column diameter.

Particle Heat Transfer (LTNE only)

When the energy balance uses local thermal non-equilibrium (LTNE), a gas–particle coefficient is needed. The Constant model converts an entered coefficient into a volumetric value using the specific surface area of the packing:

h_p^{vol} = h_p\, a_v, \qquad a_v = \frac{6(1-\epsilon_b)}{d_p}

Input: Heat transfer coefficient $h_p$ [W m⁻² K⁻¹]. Under local thermal equilibrium (LTE, the default) this model is not used.