Engineering task and calculation objective
This module calculates forced-convection heat transfer in the concentric annulus and the plane gap according to the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019, section G2). The annulus is the basic geometry of the double-pipe heat exchanger and also occurs in the shell spaces of immersion tubes, in tube-in-tube coolers, and in gap flows between parallel plates. The characteristic length is the hydraulic diameter — for the annulus, the difference between the inside diameter of the outer tube and the outside diameter of the inner tube.
Unlike in a circular tube, the Nusselt number in the annulus depends on which wall is heated: only the inner tube, only the outer tube, or both walls. The VDI Heat Atlas captures these cases through correction functions with the diameter ratio di/da, which build on the Nusselt number of the circular tube; laminar, turbulent, and transitional flow are treated separately. The plane gap results as the limiting case of the annulus with a diameter ratio approaching one.
If you want to calculate heat transfer in an annulus, you obtain the Reynolds number, the Nusselt number, the heat transfer coefficient, and — together with heat transfer area and temperature difference — the transferred duty of the double-pipe apparatus from the geometry, flow velocity, and fluid properties.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Determine geometry and hydraulic diameter: The hydraulic diameter d_h = d_a − d_i is formed from the inside diameter of the outer tube and the outside diameter of the inner tube; together with the tube length, it defines the geometry of the annulus. In addition, it is specified whether the inner wall, the outer wall, or both walls transfer heat.
- Form the Reynolds number and determine the flow regime: The Reynolds number is calculated with the flow velocity in the gap cross section and the hydraulic diameter. As for the circular tube, the flow is laminar below Re = 2,300 and fully turbulent above Re ≈ 10⁴; in between, interpolation is used.
- Calculate the Nusselt number with the wall-case correction: The Nusselt number is calculated starting from the circular-tube relations (laminar with entrance terms, turbulent according to Gnielinski) and corrected with the factor for the selected wall case, which depends on the diameter ratio d_i/d_a. For heat transfer at the inner tube, the Nusselt number lies above the circular-tube value; at the outer tube, below it.
- Evaluate the heat transfer coefficient and area: From the Nusselt number, the thermal conductivity of the fluid, and the hydraulic diameter, the heat transfer coefficient follows. The heat transfer area results from the heated wall (inner or outer tube surface) and the tube length.
- Determine the duty: With the temperature difference between inlet and outlet or the driving temperature difference to the wall, the module finally delivers the transferred duty of the gap — the check quantity for the design of the double-pipe heat exchanger.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Outside diameter of the internal tube | di | m |
| Inside diameter of the external tube | da | m |
| Hydraulic diameter | dh | m |
| Length of gap | l | m |
| Pressure (abs.) | p | Pa |
| Inlet temperature | ϑe | °C |
| Outlet temperature | ϑa | °C |
| Mean temperature (ϑe+ϑa)/2 | ϑm | °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 | - |
| Mass flow | m | kg/s |
| Temperature difference (ϑa-ϑe) | Δϑ | K (diff) |
| Duty | Q | W |
| Velocity | w | m/s |
| Reynolds number | Re | - |
| Nusselt number | Nu | - |
| Heat transfer coefficient | α | W/(m²·K) |
| Heat transfer area | A | m² |
Calculation options
Medium liquid / gaseous?
liquid · 1
Is the gas CO2 or H2O?
No · 1 · 2
Frequently asked questions
Why is it not enough to simply calculate the annulus as a pipe with the hydraulic diameter?
The hydraulic diameter transfers the Reynolds number and pressure drop to the annulus well, but the heat transfer additionally depends on which wall is heated. At the convexly curved inner tube, the Nusselt number is higher; at the concave outer tube, lower than in a circular tube of the same hydraulic diameter. The VDI Heat Atlas therefore corrects the circular-tube Nusselt number with functions of the diameter ratio d_i/d_a for the respective wall case.
What does the case of the wall insulated on one side mean in practice?
In the classic double-pipe heat exchanger, only the inner tube transfers heat while the outer tube is insulated to the outside — this is the most important wall case. If, on the other hand, an annulus is heated from both sides (e.g. a gap between two process spaces), different corrections apply. Choosing the wrong wall case is a typical source of error and can distort the heat transfer coefficient by double-digit percentages.
How is the plane gap between parallel plates treated?
The plane gap is the limiting case of the concentric annulus for a diameter ratio d_i/d_a approaching one. Its hydraulic diameter is twice the gap width. The laminar limiting values of the Nusselt number differ from the circular tube; different values apply to the gap heated on both sides and the gap heated on one side, which the code provides.
Does the calculation also apply to eccentric annuli?
No, the correlations assume a concentric annulus. With eccentricity, the fluid preferentially flows through the wide part of the gap; in the narrow part, velocity and local heat transfer drop significantly. Even moderate eccentricity can noticeably reduce the mean heat transfer — for horizontal double pipes with long unsupported inner tubes, centering should be ensured by design.