Engineering task and calculation objective
This module calculates forced-convection heat transfer in cross flow around single tubes, wires, and profiled cylinders according to the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019, section G6). The cylinder in cross flow is the basic building block of many pieces of equipment: individual pipes exposed to wind, immersion tubes and thermocouple protection tubes in ducts, heating rods, hot-wire anemometer wires, or profiled bars in air flows. The tube-bundle calculation also builds on the solution for the single cylinder.
The calculation uses the concept of the streamed length: the characteristic length is not the diameter, but half the circumferential length of the profile swept by the fluid. This allows circular cylinders, wires, and non-circular profiled cylinders to be treated with the same correlations. The mean Nusselt number is formed as the superposition of a laminar and a turbulent boundary layer contribution and is thus continuously valid over a wide Reynolds range from creeping to highly turbulent flow.
If you want to calculate heat transfer on a tube in cross flow — for example, for the heat losses of exposed pipes or the design of measuring probes — this module delivers the Nusselt number and heat transfer coefficient based on one of the best-validated correlations of the VDI Heat Atlas.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Select geometry and determine the streamed length: First, the geometry is defined: circular cylinder, wire, or profiled cylinder (e.g. square, elliptical). From this, the streamed length follows as the characteristic length — for the circular cylinder, half the circumference l = (π/2)·d.
- Evaluate fluid properties and form the Reynolds number: The fluid properties are evaluated at the mean temperature between the approach flow and the wall. The Reynolds number is formed with the approach velocity, the streamed length, and the kinematic viscosity.
- Calculate the laminar and turbulent Nusselt contributions: The laminar contribution follows the boundary layer solution with Re¹ᐟ² and Pr¹ᐟ³, the turbulent contribution the correlation with Re⁰ᐧ⁸. Both contributions are superimposed quadratically; the constant base value 0.3, which describes the heat transfer at very small Reynolds numbers, is added.
- Apply profile and wall corrections: For non-circular profiles and for the temperature influence on the fluid properties (Prandtl number ratio for liquids, temperature ratio for gases), the correction factors given in the code are applied.
- Evaluate the heat transfer coefficient: From the mean Nusselt number, the thermal conductivity of the fluid, and the streamed length, the mean heat transfer coefficient over the circumference of the cylinder follows.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Angle of flow | Θ | ° |
| Dynamic viscosity | η | mPa·s |
| Streamed length | l | m |
| Density | ρ | kg/m³ |
| Heat transfer area | A | m² |
| Flow velocity | w | m/s |
| Reynolds number | Rel | - |
| Prandtl number | Pr | - |
| Prandtl number at wall temperature | PrW | - |
| Mean temperature | ϑ | °C |
| Mean wall temperature | ϑW | °C |
| Specific thermal conductivity | λ | W/(m·K) |
| Heat transfer coefficient | α | W/(m²·K) |
| Temperature difference | Δϑ | K (diff) |
| Transfered heat rating | Q | W |
| Nusselt number | Nul | - |
| Fluid liquid /gaseous? | <2> | - |
| Mean pressure | p | Pa |
| Specific heat capacity | cp | J/(kg·K) |
| Kinematic viscosity | ν | m²/s |
| Heat flux | qPkt | W/m² |
| Flow velocity in the cross-section of the empty channel | w0 | m/s |
| Void fraction | ψ | - |
| Diameter of the cylinder | d | m |
Calculation options
Fluid liquid /gaseous?
liquid · 2
Geometry
cross-flow around individual tubes, wires, and profiled cylinders · single cylinder in a duct
Worked example
A single tube with an outside diameter d = 25 mm is exposed to a cross flow of air at 20 °C with w = 3 m/s. Find the Reynolds number, the mean Nusselt number, and the heat transfer coefficient (wall correction neglected) — a worked example of how to calculate heat transfer on a cylinder in cross flow to the VDI Heat Atlas.
Given values
| Tube outside diameter d | 25 mm |
| Approach velocity w | 3 m/s |
| Kinematic viscosity of air (20 °C) ν | 15.32 · 10⁻⁶ m²/s |
| Thermal conductivity λ | 0.0259 W/(m·K) |
| Prandtl number Pr | 0.71 |
Solution
Streamed length and Reynolds number
l = (π/2) · d = (π/2) · 0.025 ≈ 0.0393 m
Re = w · l / ν = 3 · 0.0393 / (15.32 · 10⁻⁶) ≈ 7,690
Laminar and turbulent Nusselt contributions
Nulam = 0.664 · Re1/2 · Pr1/3 = 0.664 · 7,6901/2 · 0.711/3 ≈ 51.9
Nuturb = 0.037 · Re0.8 · Pr / [1 + 2.443 · Re−0.1 · (Pr2/3 − 1)] ≈ 42.4
Mean Nusselt number
Nu = 0.3 + (Nulam² + Nuturb²)1/2 = 0.3 + (51.9² + 42.4²)1/2 ≈ 67.3
Heat transfer coefficient
α = Nu · λ / l = 67.3 · 0.0259 / 0.0393 ≈ 44.4 W/(m²·K)
Result
| Streamed length l | 39.3 mm |
| Reynolds number Re | 7,690 |
| Nusselt number Nu | 67.3 |
| Heat transfer coefficient α | ≈ 44.4 W/(m²·K) |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
Why is the streamed length used instead of the diameter?
The Gnielinski correlation transfers the flat-plate boundary layer solution to curved surfaces. The governing quantity is the path a fluid particle travels along the surface — for the circular cylinder, from the stagnation line to separation, half the circumference l = (π/2)·d. This concept makes the correlation transferable to wires and profiled cylinders as well. When comparing with literature values, note which length the Nusselt number refers to: Nu values based on the diameter must be converted, being smaller by the factor π/2.
In which Reynolds range is the correlation valid?
The superimposed correlation is approximately valid from Re ≈ 10 to 10⁷ and for Prandtl numbers from about 0.6 to 1,000, thus covering gases, water, and many organic liquids. In the critical range around Re ≈ 2·10⁵, where the boundary layer on the cylinder becomes turbulent and the drag coefficient drops, somewhat larger deviations must be expected than for the plate.
What must be considered for heat loss calculations on outdoor pipes?
The calculated coefficient applies to forced cross flow with a defined velocity. At low wind speeds, natural convection is superimposed, which increases the heat transfer compared with pure forced convection; in addition, radiation often contributes to the heat loss in the same order of magnitude. For reliable loss calculations, mixed convection and the radiative contribution must be considered separately.
Does the mean Nusselt number also apply locally around the circumference?
No. The local heat transfer varies strongly around the cylinder circumference: it is high at the stagnation point, falls toward the separation point, and rises again in the wake region. The correlation provides the value averaged over the circumference, which is decisive for design calculations (transferred duty, heat loss). For local questions such as thermal shock or material stress, separate distributions must be consulted.