Engineering task and calculation objective
This module calculates forced-convection heat transfer on moving surfaces with parallel flow according to the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019, section G5). Unlike the stationary plate, here the surface itself moves relative to the fluid — or surface and fluid move at different velocities in the same or opposite directions. Typical applications are continuously drawn wires, filaments, and films during cooling, moving belts in dryers and coating lines, and the cooling of extruded or rolled strand material.
The boundary layer on a moving surface develops fundamentally differently from that on a stationary plate in parallel flow: when the surface is drawn out of a nozzle or die into a quiescent fluid, the boundary layer grows with the running length from the exit point, and the Nusselt numbers differ from the Blasius solution. The module delivers the mean Nusselt numbers for the governing cases — a moving plane wall and a longitudinally moving cylinder (wire, filament) — for both laminar and turbulent boundary layers.
If you want to calculate the cooling of moving strips or wires, you obtain the Nusselt number and from it the heat transfer coefficient as an input for the thermal balance of the cooling line.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define the motion case: First, the case is selected: a plane moving surface (strip, film) or a longitudinally moving cylinder (wire, filament), each either in a quiescent fluid or with a superimposed parallel flow in the same or opposite direction.
- Determine the governing relative velocity and running length: The governing velocity is formed from the surface velocity and the fluid velocity; the characteristic length is the running length of the surface from the exit point (nozzle, die, roll gap), where the boundary layer starts to grow.
- Form the Reynolds number and check the boundary layer state: The Reynolds number is calculated with the velocity, running length, and kinematic viscosity, and the boundary layer state (laminar, turbulent, transition region) is determined.
- Calculate the Nusselt number for the respective case: For the selected motion case, the Nusselt numbers of the laminar and turbulent boundary layers are evaluated according to the correlations of section G5. The prefactors differ from the stationary-plate case, since the velocity profile in the boundary layer at the moving wall is different; for a thin wire, the curvature effect of the cylindrical boundary layer is added.
- Evaluate the heat transfer coefficient: From the Nusselt number, the thermal conductivity of the fluid, and the characteristic length, the mean heat transfer coefficient follows, with which the cooling curve of the moving product along the cooling line can be balanced.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Density | ρ | kg/m³ |
| Kinematic viscosity | ν | m²/s |
| Thermal conductivity | λW | W/(m·K) |
| Cylinder coordinate | (axial) x | m |
| Cylinder coordinate | (radial) r | m |
| Cylinder length | L | m |
| Cylinder diameter | d | m |
| Cylinder radius | rW | m |
| Fluid velocity | uδ | m/s |
| Cylinder velocity | uw | m/s |
| Velocity difference | ur | m/s |
| Parameter of curvature (at x = L) | K | – |
| Reynolds number at point x | Rex | – |
| Reynolds number at point L | ReL | – |
| Prandtl number | Pr | – |
| Nusselt number | Nud | – |
| Nusselt number | Nux | – |
| Nusselt number | Nudm | – |
| Nusselt number | Nuxm | – |
| Nusselt number | Nud | – |
| Nusselt number | Nux | – |
| Nusselt number | Nudm | – |
| Nusselt number | Nuxm | – |
| Prandtl-number for Table 3 | Pr | – |
Frequently asked questions
Why can a moving surface not simply be calculated as a stationary plate in parallel flow?
At the stationary plate, the fluid is at rest at the wall and flows at full velocity outside; for a surface drawn from a nozzle it is the other way around — the wall moves and the fluid is at rest far away. The velocity profile in the boundary layer is therefore fundamentally different, and the Nusselt numbers deviate from the Blasius/Pohlhausen solution even though the relative velocity is the same. For accurate cooling calculations on strips and wires, the special correlations of section G5 must therefore be used.
What role does the diameter play for moving wires and filaments?
For thin wires, the boundary layer thickness quickly reaches the order of the wire diameter or beyond. The boundary layer is then cylindrically curved, which increases the heat transfer compared with the plane treatment — the curvature parameter formed from running length, diameter, and Reynolds number enters the correlation. The thinner the wire and the longer the cooling line, the larger the deviation from the plane solution.
What applies for co-current and counter-current motion of fluid and surface?
If surface and fluid move in the same direction, the relative velocity and thus the heat transfer are smaller than with a quiescent fluid; with counter-current motion, the heat transfer increases. The correlations contain the velocity ratio of the two motions for this. The limiting case of a stationary surface with flowing fluid reduces to the classical plate solution.
Does this fully describe the cooling of the product?
No, the module delivers the convective heat transfer coefficient at the surface. For the cooling curve of the product, the transient heat conduction within the product (Biot number), the radiative contribution at high temperatures, and possibly evaporation effects must additionally be balanced. For fast-running thin films and filaments, heat conduction across the cross section is usually uncritical; for thick strand material, however, it governs the design.