Engineering task and calculation objective
The module calculates natural convection heat transfer on bodies in external flow according to the VDI Heat Atlas (12th edition, 2019): on plane vertical and inclined plates, on horizontal plates with heat release upward or downward, on the horizontal cylinder in cross-flow, and on spheres. The geometry selection fixes the respective configuration with its correlation and characteristic length; the results are the Nusselt number, heat transfer coefficient and heat flow.
Calculating natural convection heat transfer is required wherever surfaces exchange heat with their surroundings without forced flow: vessel and equipment walls, piping in still air, enclosures and electronic components, heating surfaces, or components cooling down after shutdown. Natural convection then determines the heat losses and surface temperatures together with radiation.
The basis are the continuous correlations of the VDI Heat Atlas (after Churchill and Chu for plate and cylinder), which describe the mean Nusselt number from the Rayleigh number and a Prandtl number function all the way from the laminar to the turbulent regime — with no case distinction at the transition boundary.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation scope
- Flat plates: Plane vertical or inclined plates
- Flat plates: Horizontal plates
- Cylinder in horizontal crossflow
- Spheres
Calculation workflow
- Select the geometry: The geometry selection fixes the configuration: vertical or inclined plate, horizontal plate (heat release upward or downward), horizontal cylinder or sphere. This determines the correlation and the characteristic length — the height for the vertical plate, the ratio of area to perimeter for the horizontal plate, and the overflow length formed from the diameter for the cylinder and sphere.
- Determine temperature difference and fluid properties: The driving temperature difference is formed from the surface temperature and the fluid temperature outside the boundary layer; the fluid properties (kinematic viscosity, thermal conductivity, Prandtl number, expansion coefficient) are evaluated at the mean boundary layer temperature. For ideal gases the expansion coefficient is the reciprocal of the absolute temperature.
- Form the Rayleigh number: From gravitational acceleration, expansion coefficient, temperature difference, characteristic length and viscosity, the Grashof number is formed and multiplied by the Prandtl number to give the Rayleigh number. For inclined plates, the effective component of the gravitational acceleration is used.
- Evaluate the Nusselt correlation: The correlation of the selected geometry, together with the Prandtl number function, delivers the mean Nusselt number over the entire Rayleigh number range. For horizontal plates, a distinction is made between the unstable orientation (heat release upward) and the stable one (heat release downward), which yield markedly different Nusselt numbers.
- Calculate heat transfer coefficient and heat flow: From the Nusselt number, the heat transfer coefficient follows via the thermal conductivity and the characteristic length; multiplied by area and temperature difference, this gives the convective heat flow released. For the total heat release, the radiation share must be added separately.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Wall temperature | ϑw | °C |
| Fluid temperature | ϑ∞ | °C |
| Mean temperature | ϑm | °C |
| Difference of temperature | Δϑ | K (diff) |
| Thermal conductivity of fluid | λ | W/(m·K) |
| Density of fluid | ρ | kg/m³ |
| Viscosity of fluid | η | mPa·s |
| Coefficient of volumetric expansion | β | 1/K |
| Specific heat capacity | cp | J/(kg·K) |
| Prandtl number | Pr | - |
| Acceleration due to gravity | g | m/s² |
| Grashof number | Gr | - |
| Reynolds number | Re | - |
| Rayleigh number | Ra | - |
| Mean velocity of the fluid | u | m/s |
| Inclination to the vertical | γ | ° |
| Flow length over plate | l | m |
| Plate width | b | m |
| Diameter of the sphere | D | m |
| Tube length | LR | m |
| Cocurrent flow? | J | – |
| Geometry | Bauform | – |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Characteristic length (Flow length over plate) | L | m |
| Critical Rayleigh number | Rac | - |
| Nusselt number of mixed convection | Numix | - |
| Nusselt number of free convection | Nufree | - |
| Nusselt number of forced convection | Nuforced | - |
| Nusselt number of forced convection (turbulent) | Nuforced,turb | - |
| Nusselt number of forced convection (laminar) | Nuforced,lam | - |
| Heat transfer area | A | m² |
| Heat duty | Q | W |
| Heat flux | q̇ | W/m² |
| Heat transfer coefficient | α | W/(m²·K) |
Calculation options
Cocurrent flow?
No · Yes
Geometry
Plane vertical or inclined layers · Plane horizontal plate · Cylinder in horizontal cross-flow ( Flow cocurrent ) · Sphere (Flow cocurrent)
Worked example
A vertical equipment wall 1.0 m high has a surface temperature of 40 °C and stands in still room air at 20 °C. Find the mean natural convection heat transfer coefficient and the area-specific heat release (excluding radiation) — a worked example of how to calculate natural convection to the VDI Heat Atlas. Air properties at the reference temperature of 30 °C: ν = 16.0·10⁻⁶ m²/s, λ = 0.0264 W/(m·K), Pr = 0.71; expansion coefficient β = 1/303.15 K⁻¹.
Given values
| Wall height H | 1.0 m |
| Surface temperature ϑW | 40 °C |
| Air temperature ϑ∞ | 20 °C |
| Kinematic viscosity ν (30 °C) | 16.0·10⁻⁶ m²/s |
| Thermal conductivity λ (30 °C) | 0.0264 W/(m·K) |
| Prandtl number Pr | 0.71 |
Solution
Form the Rayleigh number
Gr = g · β · Δϑ · H³ / ν² = 9.81 · (1/303.15) · 20 · 1.0³ / (16.0·10⁻⁶)² = 2.53·10⁹
Ra = Gr · Pr = 2.53·10⁹ · 0.71 = 1.80·10⁹
Prandtl number function
f1(Pr) = [1 + (0.492/Pr)9/16]−16/9 = [1 + (0.492/0.71)9/16]−16/9 = 0.347
Nusselt number after Churchill and Chu
Nu = {0.825 + 0.387 · [Ra · f1(Pr)]1/6}²
Ra · f1 = 1.80·10⁹ · 0.347 = 6.23·10⁸
Nu = (0.825 + 0.387 · 6.23·10⁸1/6)² ≈ 147
Heat transfer coefficient and heat flux
α = Nu · λ / H = 147 · 0.0264 / 1.0 ≈ 3.9 W/(m²·K)
q = α · Δϑ = 3.9 · 20 ≈ 78 W/m²
For the total heat release of the wall, the radiation share must be added, which at typical emissivities is of a similar order of magnitude.
Result
| Rayleigh number Ra | 1.80·10⁹ |
| Nusselt number Nu | ≈ 147 |
| Heat transfer coefficient α | ≈ 3.9 W/(m²·K) |
| Heat flux q | ≈ 78 W/m² |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
At what point does natural convection on a vertical plate become turbulent?
Transition begins at Rayleigh numbers around 10⁹. For a wall 1 m high in room air, that is reached at only about 20 K excess temperature — real equipment walls therefore frequently lie in the transitional or turbulent regime. The Churchill and Chu correlation covers the laminar and turbulent regimes in a single equation, so no error-prone case distinction is needed.
Why do the top and bottom faces of a horizontal plate differ so strongly?
When a heated plate releases heat upward, the warmed air rises unhindered — the stratification is unstable and the heat transfer is good. On the heated underside, the warm, light air clings to the surface and can only drain away sideways — the stratification is stable and the Nusselt number is considerably smaller. For cooled plates the situation is exactly reversed. The orientation must therefore always be chosen correctly.
Which characteristic length applies to the horizontal plate?
According to the VDI Heat Atlas, the ratio of plate area to the perimeter of the projected area (for a circular disc, one quarter of the diameter). This definition makes the correlation shape-independent for rectangles, circles and other footprints. Anyone who simply inserts the edge length instead obtains systematically deviating Rayleigh and Nusselt numbers.
Is natural convection sufficient to calculate the total heat loss of a surface?
No. On surfaces with typical emissivities (paint, plastic, oxidized steel: 0.8 to 0.95), the radiation share at small and moderate excess temperatures is of the same order of magnitude as natural convection. The radiative exchange with the surroundings must therefore be calculated separately and added; only for bare metal surfaces is it small.