Transport Correlations

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 (its dispersion and heat-transfer models) 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.

Constant¶

\[ D_{ax} = \text{constant} \]

Input: Dispersion coefficient [m² s⁻¹].

Ruthven (1984)¶

Combines molecular diffusion and mechanical (eddy) dispersion:

\[ D_{ax} = 0.7\thinspace D_m + 0.5\thinspace \frac{|u|}{\epsilon_b}\thinspace d_p \]

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.

Constant¶

\[ \lambda = \text{constant} \]

Ruthven (1984)¶

The thermal analogue of the mechanical dispersion term:

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

Yagi and Kunii (1957)¶

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

\[ \lambda = \lambda_{z}^{0} + 0.75\thinspace k_g\thinspace \mathrm{Pr}\thinspace \mathrm{Re} \]

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

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.

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,

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

then the effective radial conductivity,

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

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. It is set with the wall settings.

Constant¶

\[ h_o = \text{constant} \]

Holman (2008)¶

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

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

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:

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

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