Engineering task and calculation objective
The module calculates heat transfer from a heating surface to stationary or mechanically agitated packed beds according to chapter M6 of the VDI Wärmeatlas (VDI Heat Atlas, 12th edition 2019). The basis is the model of wall-contact heat transfer: between the heating surface and the adjacent particles acts a contact resistance, behind it the penetration resistance of the bulk. From these, the module determines the time-dependent or effective heat transfer coefficient, optionally for the stationary bed or for mechanically agitated bulk material.
This calculation is needed for contact dryers (paddle, conical and tumble dryers), heated screw conveyors, rotary kilns, vacuum dryers and sterilizers — wherever granular or powdery goods are heated or cooled via a heated wall and no gas stream carries in the heat. To calculate heat transfer to a packed bed, the surrounding gas must be considered in addition to the particle properties, since its thermal conductivity and pressure largely determine the contact resistance; under vacuum, the heat transfer drops significantly.
With mechanical mixing, the bulk at the wall is renewed periodically; the contact time between two mixing events thus becomes the central design quantity.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define bulk and gas data: Inputs are the particle diameter, thermal conductivity, density and heat capacity of the particle material, the porosity of the bed, and the type and pressure of the surrounding gas; via the options, a distinction is made between a stationary bed, an agitated bed and, if applicable, vacuum operation.
- Calculate the wall-contact heat transfer: The contact coefficient between the heating surface and the first particle layer is determined from gas conduction in the wedge-shaped gap between particle and wall, including the Smoluchowski effect: at low pressures or with fine particles, the mean free path of the gas molecules limits the gap conduction and lowers the contact coefficient.
- Capture the penetration behavior of the bed: The bulk behind the first particle layer is treated as a semi-infinite body with effective thermal conductivity and heat capacity. The transient penetration coefficient decreases with the square root of the contact time, because the near-wall layer increasingly warms up to wall temperature.
- Apply the contact time or mixing behavior: For the stationary bed, the heat transfer follows directly from the process time. For mechanically agitated beds, the turnover frequency of the mixing element determines the mean contact time of a particle packet at the wall; from this, a time-averaged effective heat transfer coefficient is formed.
- Assemble the total heat transfer: Contact and penetration resistance are connected in series; at high temperatures, the radiative component is added. The result is the effective wall-to-bed heat transfer coefficient as the basis for the required heating surface and for heat-up and drying times.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Density | ρb | kg/m³ |
| Thermal conductivity | λb | W/(m·K) |
| Residence time | t | s |
| Mixing time | tR | s |
| Time constant of mixer | tmix | s |
| Mixing efficiency | Nmix | - |
| Rotation speed | n | 1/s |
| Apparatus diameter | D | m |
| Surface coverage | φ | - |
| Thermal conductivity | λG | W/(m·K) |
| Particle diameter | d | m |
| Surface roughness | δ | m |
| Emissivity (bed) | εb | - |
| Pressure | p | Pa |
| Specific heat capacity | cpG | J/(kg·K) |
| Molar mass | MW | kg/kmol |
| Size C for the calculation of the accommodation coefficient Air | C He H2O | - |
| Into the bed | tc | W/m² |
| Wall temperature | ϑW | °C |
| Temperature of the bed at wall | ϑ0 | °C |
| Mean temperature (bed) | ϑb | °C |
| Specific heat capacity | cp,b | J/(kg·K) |
| Emissivity (wall) | εW | - |
| Disc dryers | C = 25 x=0.2 C | - |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| bed | αbed,s αWS αS | W/(m²·K) |
| Schüttung | αbed,s αWS αS | W/(m²·K) |
| Heat transfer coefficient wall-to-solids | αWS | W/(m²·K) |
| tc | tc | s |
| bed | αbed αWs α | W/(m²·K) |
| Mixing time | tR | s |
| Heat transfer coefficient wall-to-particle | αWP | W/(m²·K) |
| Heat transfer coefficient radiation | αrad | W/(m²·K) |
| Modified mean free path | l | m |
| Accommodation coefficient | γ | - |
| Into the bed | tc | W/m² |
| Temperature of the bed at wall | ϑ0 | °C |
| Schüttung | tR | W/m² |
| Schüttung | αbed αWs α | W/(m²·K) |
| Schüttung) | αbed,w αWS αwet | W/(m²·K) |
| Schüttung | αbed,w αWS αwet | W/(m²·K) |
| Heating element | q0 | W/m² |
| For vaporation | qlat | W/m² |
| Trocknungsgeschwindigkeit | m ΔX Δϑbed K | kg/(m²·s) |
| Feuchtegehalts | m ΔX Δϑbed K | - |
| Schüttung | m ΔX Δϑbed K | K |
| Dimensionless thermal conductivity | kG | - |
| Deformation parameter | B | - |
| Parameter | N | - |
Calculation options
Options
Heat transfer from a heated surface to stirred beds without heat sinks · Heat transfer from a heated surface to stirred beds with heat sinks in an atmosphere of pure vapour · Heat transfer from a heated surface to stirred beds with heat sinks in an atmosphere of inert gas
Frequently asked questions
Why does mechanical mixing improve heat transfer so strongly?
In a stationary bed, the heated zone grows into the bulk and the heat flow decays with the square root of time. Mixing periodically replaces the near-wall, already heated particle layer with cold material from the core; each renewal restarts the efficient initial state. The shorter the contact time between two mixing events, the closer the effective heat transfer comes to the contact coefficient of the first particle layer as the upper limit.
What role does the gas pressure play, for example in vacuum dryers?
The heat from the wall into the first particle layer flows predominantly through the gas in the wedge between particle and wall. If the pressure drops so far that the mean free path of the gas molecules approaches the size of the gap (Smoluchowski effect), this gap conduction collapses — with fine powders already at moderate vacuum. Vacuum contact dryers must therefore be designed with significantly reduced heat transfer coefficients, even though the vacuum is advantageous for evaporation.
Why is the particle diameter so decisive?
The contact coefficient of the first particle layer is approximately inversely proportional to the particle diameter: fine particles form many contact points with thin, well-conducting gas gaps, coarse particles few with large gap widths. Fine-grained goods therefore achieve very high transfer coefficients at the wall — as long as they do not cake, or the Smoluchowski effect does not become limiting under vacuum.
How does the thermal conductivity of the particle material enter?
Only in attenuated form: what governs penetration is the effective thermal conductivity of the bed, which is determined by porosity, gas conductivity and contact points and typically lies far below that of the solid material. A bed of metal particles therefore conducts only slightly better than one of ceramic — the limiting path always leads through the gas wedges between the particles.