Engineering task and calculation objective
This module calculates the heat transfer by natural convection at bodies in external flow, as well as mixed convection – the superposition of natural and forced convection – using the component approach of the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019). Covered are vertical and inclined plane surfaces, horizontal plates, vertical and horizontal cylinders, spheres, cubes, and finned tubes. The results are the Nusselt number, the heat transfer coefficient, and the convective heat flow of the respective arrangement.
Engineers need to calculate natural convection whenever heat is released or absorbed solely by buoyancy-driven flow, without a fan or pump: heat losses from vessels, equipment walls, and control cabinets, passive cooling of components, or the cool-down of idle plants. As soon as a weak external flow is additionally present – for example a draft in a hall – mixed convection occurs, where neither of the two mechanisms alone describes the heat transfer.
The basis is the set of dimensionless numbers of buoyancy-driven flow: from the Grashof or Rayleigh number and the Prandtl number, the geometry-specific correlations yield the Nusselt number; in the component approach, the Nusselt numbers from natural and forced convection are superimposed according to the interpolation rule of the VDI Heat Atlas.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation scope
- Combined natural and forced convection: Component balance
- Horizontally curved surfaces: Cubes
- Horizontally curved surfaces: Spheres
- Horizontally curved surfaces: Fin tubes
- Horizontally curved surfaces: Cylinder
- Horizontal plane surfaces
- Inclined flat surfaces
- Vertical cylinders
- Vertical flat surfaces
- Fundamentals
Calculation workflow
- Define the geometry and characteristic length: First, the body shape is chosen (plate, cylinder, sphere, cube, finned tube) and the characteristic length is determined – for vertical plates the height, for horizontal plates the ratio of projected area to perimeter of the projected area, for the horizontal cylinder the overflowed arc.
- Evaluate temperatures and material properties: The temperature difference is formed from the temperature at the surface and the temperature of the fluid outside the boundary layer; the fluid properties (density, heat capacity, viscosity, thermal conductivity, coefficient of expansion, Prandtl number) are evaluated at the mean temperature of the boundary layer. For ideal gases, the coefficient of expansion is the reciprocal of the absolute fluid temperature.
- Form the dimensionless numbers: The Grashof number is formed from the acceleration due to gravity, coefficient of expansion, temperature difference, characteristic length, and kinematic viscosity; multiplied by the Prandtl number, it gives the Rayleigh number. For density-driven flows without a purely thermal cause, the generalized Grashof number with the relative change of density can be used.
- Evaluate the Nusselt correlation: For the chosen geometry, the corresponding VDI Heat Atlas correlation is evaluated, which describes the laminar and turbulent regimes continuously and accounts for the Prandtl number via a correction function. The result is the mean Nusselt number of the arrangement.
- Superimpose mixed convection: If a forced flow is additionally present, its Nusselt number is calculated separately and superimposed on the natural convection according to the component approach; depending on whether buoyancy and flow are aligned, opposed, or crosswise, the contributions reinforce or weaken each other.
- Determine the heat transfer and heat flow: From the Nusselt number, the heat transfer coefficient follows via the characteristic length and the thermal conductivity, and the convective heat flow via the surface of the body in the flow and the temperature difference.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Fluid volume | V | m³ |
| Exchange surface | A | m² |
| Perimeter of the projection surface | U | m |
| Height of the area | h | m |
| Outside diameter | D | m |
| Width of the area | b | m |
| Edge length | a | m |
| Tube diameter | d | m |
| Fin height | h | m |
| Fin thickness | s | m |
| Fin spacing | b | m |
| Angle of inclination to vertical line | γ | ° |
| Length of the cylinder | L | m |
| Characteristic length | l | m |
| Acceleration due to gravity | g | m/s² |
| Temperature on the surface | ϑ0 | °C |
| Temperature difference | (ϑ0 - ϑ∞) Δϑ | K (diff) |
| Mean temperature | (ϑ0 + ϑ∞) / 2 ϑm | °C |
| Density | ρ | kg/m³ |
| Specific heat capacity | cp | J/(kg·K) |
| Dynamic viscosity | η | mPa·s |
| Kinematic viscosity | ν | m²/s |
| Thermal conductivity | λ | W/(m·K) |
| Coefficient of thermal expansion | β | 1/K |
Frequently asked questions
When must I apply mixed convection instead of pure natural or forced convection?
The governing quantity is the ratio of the Grashof number to the square of the Reynolds number (Richardson number). If it is very large, buoyancy dominates (natural convection); if it is very small, the external flow dominates (forced convection). If it is of order one – typical for a weak draft over warm surfaces – the component approach must be used, because otherwise the heat transfer is significantly over- or underestimated depending on the case.
At which temperature do I evaluate the fluid properties?
According to the VDI Heat Atlas, at the mean temperature between the surface and the fluid outside the boundary layer. Since the properties – above all viscosity and density – are strongly temperature-dependent, evaluating them at the wrong reference temperature leads to noticeable errors in the Rayleigh and Nusselt numbers. For large temperature differences, it should additionally be checked whether the Boussinesq approximation (small relative change of density) still holds.
Why does the orientation of the surface have such a large influence?
Buoyancy acts vertically. On a heated vertical plate, the boundary layer flow can rise freely; on a heated horizontal plate facing upward, it detaches unstably (good heat transfer), while on the heated underside the warm layer remains trapped (poor heat transfer). Separate correlations therefore apply to each orientation, and inclined surfaces are treated via the component of the acceleration due to gravity parallel to the surface.
Is radiation included in the calculated heat flow?
No, the module delivers the convective heat flow. On free surfaces in air, however, the radiation contribution at usual emissivities is of the same order of magnitude as the natural convection and must be calculated separately and added to obtain the total heat release – for example via a radiation exchange calculation with the surroundings.