Engineering task and calculation objective
This module calculates the effective thermal conductivity of packed beds – both permeable and non-permeated – according to the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019). A bed of particles – spheres, cylinders, or other packing material – with gas-filled or liquid-filled voids conducts heat quite differently from the pure solid: the effective thermal conductivity results from the interplay of particle conductivity, fluid conductivity, porosity, particle shape, and the contact conditions between the particles.
Engineers who need to calculate the thermal conductivity of packed beds meet this task in fixed-bed reactors and adsorbers, in regenerators and thermal storage systems, in drying processes, in heated or cooled bulk-material silos, and when evaluating porous insulating and packing materials. The effective conductivity of the stagnant bed is the basic quantity on which wall heat transfer and through-flow models are built.
The calculation is based on the cell model described in the VDI Heat Atlas (after Zehner, Bauer, and Schlünder), which is adapted to the respective particle geometry and problem via the model selection and can also account for effects such as radiation in the pore space and the pressure dependence of the gas conductivity (Smoluchowski effect).
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Select the model and particle geometry: The model selection defines the type of bed – for example a bed of spheres, a bed of cylinders, or other packing material – and thereby the shape factor of the cell model, which describes the heat conduction paths between the particles.
- Specify material and structural data: Required inputs are the thermal conductivity of the particle material, the thermal conductivity of the fluid in the pore space (gas or liquid), the porosity of the bed, and the particle dimensions. For gases, pressure and temperature may additionally enter, since the gas conductivity in narrow pores decreases with pressure.
- Calculate the core conductivity of the unit cell: The cell model divides the bed into a representative unit cell of particle and fluid fractions and calculates from it the conductivity of the core region. The deformation parameter accounts for the particle shape; secondary parameters capture flattening and contact points between the particles.
- Superimpose additional effects: Depending on the application, thermal radiation in the pore space (relevant at high temperatures and for coarse particles) and the pressure dependence of the gas conductivity (Smoluchowski effect at reduced pressure or in very fine pores) are included in the effective conductivity.
- Evaluate the effective thermal conductivity: As a result, the module delivers the effective thermal conductivity of the stagnant bed. It can subsequently be used as the property of the homogenized layer in overall heat transfer, storage, or reactor models.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Fluid: 0=liquid 1=gas | 1=Gas | - |
| Temperature | T | K |
| Pressure | P | Pa |
| Thermal conductivity gas | λG | W/(m·K) |
| Heat capacity gas | cpGas | J/(kg·K) |
| Mol weight gas | mGas | kg/kmol |
| Mean free path length | lf | m |
| Thermal conductivity particle | λP | W/(m·K) |
| Thermal conductivity fluid | λF | W/(m·K) |
| Number of particle fractions | Npoly | - |
| Reference thermal conductivity particle | kp | - |
| Effective particle diameter | d | m |
| Thickness particle coating | s | m |
| Thermal conductivity coating | λS | W/(m·K) |
| Particle shape 0..3 | pShape | - |
| Particle material 0..3 | pMaterial | - |
| Inside diameter Raschig ring | di,RR | - |
| Outside diameter Raschig ring | da,RR | - |
| Shape factor | cf | - |
| Deformation parameter | φ | - |
| Porosity | ψ | - |
| Emission coefficient | ε | - |
| Deformation parameter | b | - |
| Auxiliary variable | n | - |
Calculation options
Model
Zehner/Bauer/Schlünder model (with secondary influences) · Zehner/Bauer/Schlünder model (w/o secondary influences) · Maxwell model · Krischer model
Frequently asked questions
Why is the effective conductivity of a bed so far below that of the solid?
The heat flow must overcome the poorly conducting fluid-filled interstices between the particles; the particles touch each other only at points. In a gas-filled bed with metal or ceramic particles, the bottleneck at the contact points therefore dominates: the effective conductivity is typically only one to ten times the gas conductivity, even if the solid conducts orders of magnitude better.
Does the calculated conductivity also apply to beds with through-flow?
The calculated value applies to the stagnant, non-permeated bed. With through-flow, a dispersive transport contribution is superimposed that grows with the flow velocity (Péclet number); in the VDI Heat Atlas it is modeled as an addition to the stagnant conductivity. The stagnant effective conductivity, however, remains the fundamental building block of these models.
When must radiation in the pore space be taken into account?
The radiation contribution grows with the third power of the absolute temperature and with the particle diameter. At room temperature and for fine beds it is usually negligible; at temperatures from a few hundred degrees Celsius upward and for coarse particles – for example in regenerators, lime shaft kilns, or pebble-bed applications – it can dominate the effective conductivity and must not be omitted.
What role does the gas pressure in the bed play?
If the gas pressure drops far enough that the mean free path of the gas molecules approaches the order of the pore width, the effective gas conductivity decreases (Smoluchowski effect). This is relevant for vacuum insulation, fine-pored powder beds, or evacuated storage systems. At atmospheric pressure and with particles of about one millimeter or larger, the effect is generally negligible.