Effective axial thermal conductivity in packed beds

In continuation of my recent essay on the local thermal equilibrium model, I wanted to discuss the mechanisms underpinning axial conduction in packed beds. Recall the LTE equation:

$$(\epsilon_b \rho_f C_{p,f} + (1 - \epsilon_b) \rho_s C_{p,s}) \frac{\partial{T}}{\partial{t}} = \frac{\partial}{\partial{z}}(k_{e,z} \frac{\partial{T_f}}{\partial{z}}) - \rho_f C_{p,f} u \frac{\partial{T}}{\partial{z}} - \frac{4h_w}{d_i}(T_f - T_w) + S_e$$

The effective axial thermal conductivity of the bed is $k_{e,z}$. We call this the effective thermal conductivity as it is composed of several mechanisms. In the original work of Yagi and Kunii, they differentiate the heat transfer in packed beds into seven distinct processes. They can generally be grouped into three main contributions:

• Conduction (gas conduction + solid conduction)
• Convection (gas phase)
• Radiation

The insight provided by the authors is to consider each contribution to be independent and acting in parallel. Thus, their contributions are additive:

$$k_{e,z} = (k_{e,z})_{cond} + (k_{e,z})_{conv} + (k_{e,z})_{rad}$$

Conduction

The conductive term is not simply the thermal conductivity of the solid or gas - it is much more complex. Solids conduct heat through the porous matrix via contact points with other solids, which will depend on the solid shape, packing density, and contact area. The gas phase will similarly conduct heat in the space between the solid matrix. Therefore, the conduction term is defined as the conduction through a stagnant bed, i.e. no flow.

Convection

Under non-stagnant conditions, heat is dispersed through the same mechanisms as in mass dispersion. This term, therefore, depends on the Reynolds number and Prandtl number.

Radiation

In low-temperature applications, the radiative contribution can be neglected.

In 1957, Yagi and Kunii proposed the following equation for calculating the effective thermal conductivity in a packed bed:

$$\frac{k_{e}}{k_g} = \frac{k_{e}^0}{k_g} + \delta Pr Re$$

$k_{e}^0$ is the effective thermal conductivity of the stagnant bed.

Although originally proposed to describe the radial effective thermal conductivity, the same empirical equation was shown to apply to the axial thermal conductivity by Yagi, Kunii and Wakao in 1960. The author also demonstrated that the stationary effective thermal conductivity was effectively equal in the axial and radial directions, which makes sense for a randomly packed, homogeneous bed.

In their seminal paper, values for $\delta$ ranged from 0.7 to 0.8, and values for $\frac{k_{e,z}^0}{k_g}$ ranged from 7 to 13.

Ultimately, the actual thermal conductivity of the bed depends on the material properties, including shape, size, packing structure, and the ratio of particle diameter to bed diameter, among others. Correlations used in the literature should be employed only when experimental data for the system of interest are unavailable. Coefficients to the general equation can be fitted for a more reliable prediction during scale-up.