Engineering task and calculation objective
The module calculates heat transfer by natural convection in enclosed fluid layers according to the VDI Heat Atlas (12th edition, 2019): in horizontal, inclined and vertical gaps between a heated and a cooled surface, in horizontal and vertical annular gaps, and in porous layers. In addition, the radiative exchange between the bounding surfaces is captured via the radiation exchange coefficient, so that the total heat flow from conduction, convection and radiation is available.
Calculating heat transfer in enclosed layers is required for double walls and jackets of process equipment, for air gaps in insulating glazing and solar collectors, for annular gaps between a pipe and its containment pipe, or for gas-filled cavities in vessels. Unlike surfaces in open flow, an enclosed gap develops a circulation flow whose onset and intensity depend on the Rayleigh number, the inclination angle and the direction of heating.
The result is conveniently expressed as a Nusselt number referred to pure heat conduction across the layer: Nu = 1 means that no convection sets in and the gap acts like a stagnant conducting layer — the basis of every gap sizing aimed at minimum losses.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation scope
- General relations
- Layers (horizontal, inclined or vertical)
- Annuli (horizontal or vertical)
- Porous layers
Calculation workflow
- Define the configuration: The configuration is selected: horizontal, inclined or vertical layer, horizontal or vertical annular gap, or porous layer. Added to this are the layer thickness as the characteristic length, the heated surface and the orientation of the heating (from below, from above, from the side).
- Evaluate temperatures and fluid properties: The temperature difference is formed from the temperatures at the heated and the cooled surface; the fluid properties (density, heat capacity, viscosity, thermal conductivity, thermal diffusivity, expansion coefficient) are evaluated at the mean temperature of the layer.
- Form the dimensionless numbers: Using gravitational acceleration, expansion coefficient, temperature difference and layer thickness, the Grashof and Rayleigh numbers are formed, supplemented by the Prandtl number. The Rayleigh number decides whether the stratification remains stable or convection cells set in.
- Determine the Nusselt number of the layer: The Nusselt number is calculated with the correlation for the respective configuration — for horizontal layers heated from below it exceeds unity only above the critical Rayleigh number, for vertical layers it additionally depends on the height-to-thickness ratio. From this follow the heat transfer coefficient and the heat flow from conduction and convection.
- Superimpose radiation and form the total heat flow: In parallel, the radiative heat flow is calculated from the radiation exchange coefficient of the two surfaces and the difference of the fourth powers of the absolute temperatures. The sum gives the total heat flow; for comparison, the module also delivers the equivalent heat transfer coefficient of the layer.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Outside radius | ra | m |
| Inside radius | ri | m |
| Annulus length | h | m |
| Layer thickness | s | m |
| Layer height | h | m |
| Angle of inclination to the vertical line | Θ | ° |
| Angle of inclination to the horizontal line | Θ' | ° |
| Number of partitions | N | - |
| Heated area | A | m² |
| Emission rate of the heated surface (Ka) | ε1 | - |
| Emission rate of the cooled surface (Ka) | ε2 | - |
| Particle diameter | d | m |
| Total volume | Vg | m³ |
| Gap volume | Vz | m³ |
| Porosity | ψ (20) | - |
| Permeability | K (21) | m² |
| Acceleration due to gravity | g | m/s² |
| Temperature on the heated surface | ϑ1 | °C |
| Temperature on the cooled surface | ϑ2 | °C |
| Temperature difference (ϑ1 - ϑ2) | Δϑ | K (diff) |
| Temperature difference (T14 - T24) | ΔT | K^4 |
| Mean temperature | ϑm | °C |
| Density | ρ | kg/m³ |
| Specific heat capacity | cp | J/(kg·K) |
Frequently asked questions
When does a fluid layer remain purely conducting, without convection setting in?
In a horizontal layer heated from below, convection sets in only above the critical Rayleigh number of about 1708 (Rayleigh-Bénard instability); below it, Nu = 1. A horizontal layer heated from above is stably stratified and always remains purely conducting. In vertical gaps there is no sharp threshold, but at small Rayleigh numbers conduction likewise dominates. Thin gaps are therefore effective insulating layers.
Why is the optimum gap width limited — does a wider air gap not always help?
As the thickness grows, the conduction contribution initially drops (longer conduction path). At the same time, however, the Rayleigh number grows with the third power of the layer thickness, so beyond a certain thickness convection sets in and increases the heat transport again. There is therefore an optimum gap width — for air gaps under typical conditions on the order of one to two centimetres — above which a wider gap has little or even a negative effect.
How large is the radiation share in the gap, and how can it be reduced?
Between surfaces of typical emissivities (0.8 to 0.95), the radiation share in gas-filled gaps at moderate temperatures is often equal to or greater than the convection share. It can be reduced drastically by low-emissivity surfaces (bare metal foils, low-E coatings), which directly reduce the radiation exchange coefficient — the best-known example being coated insulating glazing.
What is different about porous layers compared with an open gap?
In a porous layer (e.g. a packed bed or fibrous insulation in the gap), the flow is retarded by the flow resistance of the matrix; the governing quantity is the modified Rayleigh number involving the permeability, and the base conductivity is the effective conductivity of the saturated material. Convection sets in much later than in the open gap — which is why fibrous insulation suppresses convection in insulation packages so effectively.