Engineering task and calculation objective
The module calculates natural convection heat transfer in vertical and inclined channels as well as in the open vertical annular gap according to the VDI Heat Atlas (12th edition, 2019). Unlike the enclosed fluid layer, the fluid here flows through a channel that is open at both ends: the heated walls generate buoyancy, which drives a through-flow from bottom to top (chimney effect). In addition, superimposed forced convection can be taken into account when an imposed flow overlays the buoyancy.
Calculating natural convection in vertical channels is required for the passive cooling of electronics and switch cabinets, for cooling fin arrays and radiator convection shafts, for ventilated facades and solar chimneys, or for the emergency cooling of vessels via open annular gaps. The decisive geometric quantities are the channel width or pipe diameter, the channel height and the channel length; from these, the characteristic length of the correlations is formed.
The special feature compared with the free plate: in narrow channels, the boundary layers of the opposing walls interfere with each other, while in wide channels the walls behave like individual plates. The correlations of the VDI Heat Atlas bridge both limiting cases and thus also provide the basis for optimizing the wall spacing.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation scope
- Vertical channels
- Inclined channels
- Open vertical annuli
- Superimposed forced convection
Calculation workflow
- Define the channel geometry: The inputs are channel width or pipe diameter, channel height or pipe height, and channel length, plus the inclination angle for inclined channels. From these the module determines the characteristic length, the heated surface and the flow cross-section.
- Evaluate temperatures and fluid properties: The driving temperature difference is formed from the wall temperature and the inlet temperature of the fluid. The fluid properties — density, heat capacity, dynamic and kinematic viscosity, thermal conductivity, thermal diffusivity and expansion coefficient — are evaluated at the governing mean temperature.
- Form the dimensionless numbers: Using gravitational acceleration, expansion coefficient, temperature difference and characteristic length, the Grashof and Rayleigh numbers are formed; the Prandtl number describes the fluid behaviour. For channels, the Rayleigh number is additionally scaled with the ratio of channel width to channel height, which controls the transition between narrow-gap and single-plate behaviour.
- Calculate the Nusselt number and heat transfer: The channel correlation delivers the mean Nusselt number between the limiting case of the fully developed narrow gap and that of independent single plates. From this follow the heat transfer coefficient and, via the heated surface, the heat flow removed.
- Check for superimposed forced convection: If an imposed through-flow is also present (e.g. a weak fan or chimney draught from other sources), mixed convection is taken into account: co-current flow assists the buoyancy, counter-current flow can weaken the circulation and reduce the heat transfer considerably.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Textfeld | Textfeld | – |
| Channel width | d | m |
| Tube height | h | m |
| Channel length | b | m |
| Angle of inclination to the vertical line | Θ ≤ 45° | ° |
| Outside radius | ra | m |
| Inside radius | ri | m |
| Ratio of the radii | ra/ri | – |
| Hydraulic diameter | Dh | m |
| Flow velocity | w | m/s |
| Characteristic length | s | m |
| Heat transfer area | A | m² |
| Cross-section | f | m² |
| Acceleration due to gravity | g | m/s² |
| Wall temperature | ϑW | °C |
| Inlet temperature | ϑE | °C |
| Temperature difference | (ϑW-ϑE) | K (diff) |
| Mean temperature | 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) |
| Thermal diffusivity | a | m²/s |
Frequently asked questions
What distinguishes the open channel from the enclosed fluid layer?
In the enclosed gap, the same fluid circulates between the heated and the cooled surface; the heat flow passes across the layer. In the open channel, fresh fluid continuously enters from below, heats up at the walls and exits at the top — the heat is carried away convectively with the mass flow. The correlations and the governing temperature difference (wall versus inlet temperature) differ accordingly; the two cases must not be confused.
Is there an optimum wall spacing for cooling fins and channels?
Yes. If the channel is too narrow, friction and the merging boundary layers throttle the throughput; if it is too wide, surface area goes unused because the walls act as single plates and the space between them no longer contributes. For a given stack of parallel heated plates there is therefore an optimum spacing, which depends on the Rayleigh number (i.e. on the excess temperature and channel height) — exactly this relationship can be found with the channel correlations.
Why does the heat transfer not increase indefinitely with channel height?
With increasing height the chimney effect and thus the throughput grow, but at the same time the fluid heats up along the channel and the local temperature difference to the wall decreases towards the top. In the limiting case the fluid leaves the channel almost at wall temperature — additional height then yields no extra performance, only additional friction resistance. The design must balance both effects against each other.
When must superimposed forced convection be taken into account?
As soon as the imposed flow velocity approaches the order of magnitude of the buoyancy velocity — recognizable by the ratio of Grashof number to the square of the Reynolds number being close to unity. Counter-current flow is particularly critical: if the forced flow opposes the buoyancy, the channel flow can partially collapse and the heat transfer drops below both individual values. In such cases, simply taking the larger of the two Nusselt numbers is never permissible.