Engineering task and calculation objective
This module calculates the heat transfer coefficient for forced convection along plane walls in longitudinal flow according to the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019, section G4). The flat plate in parallel flow is the fundamental case of convective heat transfer: it describes heated or cooled vessel walls, sheets in dryers, duct walls, solar absorbers, or electronic assemblies over which a fluid flows parallel to the surface.
At the core of the calculation are the Nusselt correlations for the laminar and turbulent boundary layers. From the heated plate length, the flow velocity, and the fluid properties, the Reynolds number is formed first. The Nusselt numbers of the laminar and the turbulent boundary layer are then calculated and combined into an averaged Nusselt number across the transition region — so a single calculation covers the entire technically relevant Reynolds range from purely laminar to fully turbulent flow.
If you want to calculate heat transfer on a flat plate in parallel flow, you obtain not only the Reynolds and Nusselt numbers but also the Nusselt number with wall correction for temperature-dependent fluid properties, and the mean heat transfer coefficient as a direct input for the overall heat transfer coefficient of a heat exchanger.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define geometry and approach flow: The inputs are the heated plate length in the flow direction as the characteristic length and the undisturbed flow velocity of the fluid. The fluid properties (viscosity, thermal conductivity, Prandtl number) are evaluated at the mean fluid temperature.
- Form the Reynolds number and check the flow regime: The Reynolds number follows from velocity, plate length, and kinematic viscosity. It determines the boundary layer state: below about Re = 10⁵ the boundary layer is laminar; above that, laminar-turbulent transition sets in.
- Calculate the laminar Nusselt number: For the laminar boundary layer, the mean Nusselt number is calculated from the Reynolds and Prandtl numbers using the Pohlhausen solution (proportional to Re¹ᐟ² and Pr¹ᐟ³).
- Calculate the turbulent Nusselt number: For the turbulent boundary layer, the mean Nusselt number is calculated from Re⁰ᐧ⁸ and the Prandtl number using the Petukhov/Gnielinski correlation; the denominator of the relation accounts for the Prandtl number dependence of turbulent transport.
- Superimpose and apply the wall correction: The averaged Nusselt number is obtained as the quadratic superposition of the laminar and turbulent contributions and is thus valid across the entire transition region. If there is a significant difference between wall and fluid temperature, the Nusselt number is corrected with a factor for temperature-dependent properties (Prandtl number ratio or temperature ratio).
- Evaluate the heat transfer coefficient: From the corrected Nusselt number, the thermal conductivity of the fluid, and the plate length, the mean heat transfer coefficient of the plate follows.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Heated plate length | l | m |
| Flow velocity | w | m/s |
| Mean pressure | p | Pa |
| Mean temperature | ϑ | °C |
| Density | ρ | kg/m³ |
| Specific heat capacity | cp | J/(kg·K) |
| Thermal conductivity | λ | W/(m·K) |
| Dynamic viscosity | η | mPa·s |
| Kinematic viscosity | ν | m²/s |
| Prandtl number | Pr | - |
| Mean wall temperature | ϑW | °C |
| Prandtl number at wall temperature | PrW | - |
| Fluid liquid /gaseous? | <2> | – |
| Exponent for liquids | nF | – |
| Exponent for gases | nG | – |
| Reynolds number | Re | – |
| Nusselt number laminar | Nulam (1) | – |
| Nusselt number turbulent | Nuturb (2) | – |
| Nusselt number average | Nul,0 (5) | – |
| Nusselt number with wall correction | Nu (6) | – |
| Heat transfer coefficient | α | W/(m²·K) |
Calculation options
Fluid liquid /gaseous?
Liquids · Gases
Worked example
A plane heated wall with a heated length L = 1.0 m is exposed to a longitudinal air flow at 20 °C with a velocity w = 5 m/s. Find the Reynolds number, the mean Nusselt number, and the mean heat transfer coefficient (wall correction neglected) — a worked example of how to calculate heat transfer on a flat plate to the VDI Heat Atlas.
Given values
| Heated plate length L | 1.0 m |
| Flow velocity w | 5 m/s |
| Kinematic viscosity of air (20 °C) ν | 15.32 · 10⁻⁶ m²/s |
| Thermal conductivity of air λ | 0.0259 W/(m·K) |
| Prandtl number Pr | 0.71 |
Solution
Reynolds number
Re = w · L / ν = 5 · 1.0 / (15.32 · 10⁻⁶) ≈ 3.26 · 10⁵
The Reynolds number lies in the transition region, so the laminar and turbulent contributions are superimposed.
Nusselt number of the laminar boundary layer
Nulam = 0.664 · Re1/2 · Pr1/3 = 0.664 · (3.26 · 10⁵)1/2 · 0.711/3 ≈ 338.4
Nusselt number of the turbulent boundary layer
Nuturb = 0.037 · Re0.8 · Pr / [1 + 2.443 · Re−0.1 · (Pr2/3 − 1)]
Nuturb = 0.037 · (3.26 · 10⁵)0.8 · 0.71 / [1 + 2.443 · (3.26 · 10⁵)−0.1 · (0.712/3 − 1)] ≈ 787.0
Averaged Nusselt number and heat transfer coefficient
Nu = (Nulam² + Nuturb²)1/2 = (338.4² + 787.0²)1/2 ≈ 856.7
α = Nu · λ / L = 856.7 · 0.0259 / 1.0 ≈ 22.2 W/(m²·K)
Result
| Reynolds number Re | 3.26 · 10⁵ |
| Nusselt number, laminar | 338.4 |
| Nusselt number, turbulent | 787.0 |
| Nusselt number, averaged | 856.7 |
| Heat transfer coefficient α | ≈ 22.2 W/(m²·K) |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
At what Reynolds number does the boundary layer on the plate become turbulent?
Depending on the free-stream turbulence and surface roughness, laminar-turbulent transition on a flat plate in parallel flow sets in at Reynolds numbers between about 3·10⁵ and 5·10⁵; Re ≈ 5·10⁵ is usually quoted as the critical reference value. The VDI Heat Atlas avoids fixing the transition point exactly by superimposing the laminar and turbulent contributions quadratically — the correlation is thus valid continuously up to Re ≈ 10⁷.
Why does the superimposed Nusselt number give useful values even at low Reynolds numbers?
At low Reynolds numbers, the turbulent contribution Nu_turb is negligibly small compared with the laminar contribution, so the square root of the sum of squares practically reduces to the laminar solution. Conversely, at large Reynolds numbers the turbulent contribution dominates. The superposition is therefore not an approximation with range limits, but a continuous interpolation across the transition region.
At which temperature must the fluid properties be evaluated?
The fluid properties are evaluated at the mean fluid temperature (arithmetic mean of the approach flow and near-wall conditions, or the mean temperature of the external flow). The influence of temperature-dependent properties between wall and fluid is then captured by the wall correction — via the Prandtl number ratio Pr/Pr_W for liquids and via the ratio of absolute temperatures for gases.
Is the correlation also valid for short, unheated starting lengths?
The standard correlation assumes that the velocity and thermal boundary layers start together at the leading edge of the plate. If heating only begins after an unheated starting length, the thermal boundary layer is thinner than the velocity boundary layer and the local heat transfer at the start of heating is higher; the VDI Heat Atlas provides separate corrections for this. The heated plate length must always be used as the characteristic length.