Engineering task and calculation objective
This module calculates forced-convection heat transfer in helically coiled tubes according to the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019, section G3). Coiled tubes are ubiquitous in process equipment: as heating and cooling coils in stirred vessels, in helical-coil heat exchangers, in continuous-flow heaters, or as half-pipe coils on vessel shells. Compared with a straight pipe, the tube curvature produces a secondary flow (Dean vortices) that enhances heat transfer and stabilizes the flow.
The calculation accounts for these curvature effects consistently: the critical Reynolds number of the laminar-turbulent transition rises with the ratio of tube diameter to coil diameter well above the straight-pipe value of 2,300, and the Nusselt correlations for laminar and turbulent flow contain the diameter ratio d/D as an additional parameter. Depending on the selected coil type — helical tube coil or half-pipe coil — the geometric parameters are formed accordingly.
If you want to calculate the heat transfer of a coiled tube, you obtain the Nusselt number and heat transfer coefficient on the tube inside as the basis for the design of helical-coil heat exchangers and vessel heating systems.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define coil type and geometry: First, the coil type is selected (helical coil with circular cross section or half-pipe coil). The effective mean curvature diameter of the coil is calculated from the tube inside diameter, coil diameter, and pitch; for half-pipe coils, the hydraulic diameter takes the place of the tube diameter.
- Determine the Reynolds number and the critical Reynolds number: The Reynolds number follows from velocity, characteristic diameter, and viscosity. The critical Reynolds number of the transition is calculated as a function of the diameter ratio d/D — for curved tubes it lies well above 2,300, because the secondary flow stabilizes the laminar flow.
- Laminar regime: Nusselt number with curvature effect: Below the critical Reynolds number, the Nusselt number is calculated with a correlation that captures the curvature influence via d/D. The Dean vortices improve cross-mixing, so the Nusselt number lies well above that of the straight pipe and — unlike there — depends on the Reynolds number even for fully developed laminar flow.
- Turbulent regime: modified pipe correlation: Above the critical Reynolds number, the turbulent Nusselt number is calculated with a friction factor increased as a function of curvature. In the transition region between laminar and turbulent flow, the two solutions are interpolated.
- Evaluate the heat transfer coefficient: From the Nusselt number, the thermal conductivity of the fluid, and the (hydraulic) tube diameter, the internal heat transfer coefficient of the coil follows; for large temperature differences, the property correction between wall and fluid temperature is additionally applied.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Pitch | h | m |
| Tube length | l | m |
| Number of turns | n | - |
| Dynamic viscosity | η | mPa·s |
| Density | ρ | kg/m³ |
| Spec. heat capacity | cp | J/(kg·K) |
| Thermal conductivity | λ | W/(m·K) |
| Volume flow | V | m³/s |
| Inlet temperature | ϑi | °C |
| Outlet temperature | ϑa | °C |
| Prandtl number | Pr | - |
| Coil diameter | DW | m |
| Prandtl number at wall temperature | PrW | - |
| Tube outside diameter | da | m |
| Type of coil: | Wendelschlange=0/Halbrohre=1 | - |
| Outside vessel diameter | dBa | m |
| Flow velocity | w | m/s |
| Tube inside diameter | di | m |
| Mass flow | m | kg/s |
| Mean temperature | ϑm | °C |
| Kinematic viscosity | ν | m²/s |
| Tube wall thickness | s | m |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Thermal diameter (dth = di) | dth | m |
| Reynolds number | Re | - |
| Critical Reynolds number | Recrit | - |
| Nusselt number | Nu | - |
| Nusselt number | Nu | - |
| Nusselt number | Nu | - |
| Heat duty | Q | W |
| Heat transfer coefficient | α | W/(m²·K) |
Calculation options
Type of coil:
Helical coil · Helically welded half pipe
Frequently asked questions
Why is heat transfer in a helical coil better than in a straight pipe?
Due to the centrifugal force, a secondary flow of two counter-rotating Dean vortices forms in the curved tube. It continuously transports fluid from the core to the wall and back, improving cross-mixing. In the laminar regime, the Nusselt number can thereby reach a multiple of the straight-pipe value; in the turbulent regime the gain is smaller but still present — at the cost of a higher pressure drop.
Why does the flow in the coil become turbulent only at higher Reynolds numbers?
The secondary flow damps disturbances in the main flow and thus stabilizes the laminar flow regime. The critical Reynolds number therefore increases with the ratio d/D; for tight coils (large d/D) it can reach three to four times the straight-pipe value of 2,300. If you calculate with the straight-pipe transition limit, the flow is wrongly classified as turbulent and the heat transfer in the transition region is overestimated.
Which diameter should be used as the curvature diameter for a coil with pitch?
The governing quantity is not the winding diameter of the coil alone, but the mean curvature diameter corrected for the pitch: for a significant pitch h, the effective radius of curvature increases, captured in the VDI Heat Atlas by D = D_W·[1 + (h/(π·D_W))²]. For typical tightly wound heating coils the difference is small, but for strongly stretched coils it is not negligible.
Do the relations also apply to half-pipe coils on vessels?
Yes, the module supports half-pipe coils as a separate coil type. Since the cross section is not a full circle, the hydraulic diameter of the semicircular channel is used. Note that for half-pipe coils only part of the perimeter transfers heat to the vessel wall — for the design of the vessel heating, the actual heat-transferring surface must therefore be used.