Engineering task and calculation objective
This module calculates forced-convection heat transfer in pipe flow according to the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019, section G1). Pipe flow is by far the most common application of convective heat transfer calculation: shell-and-tube heat exchangers, cooling and heating coils, trace-heated piping, or heat loss calculations — everywhere, the internal heat transfer coefficient of a pipe carrying flow is needed.
Besides the circular tube, the module also covers non-circular cross sections such as rectangular ducts and oval tubes via the hydraulic diameter concept. Depending on the Reynolds number, laminar pipe flow with thermal or hydrodynamic entrance effects, the transition region, or fully turbulent flow according to the Gnielinski correlation is calculated. As the thermal boundary condition, you can choose between constant wall temperature and constant heat flux; on request, the module also calculates for comparison according to the relations of the 11th edition of the VDI Heat Atlas.
If you want to calculate the Nusselt number and heat transfer coefficient of a pipe flow, this gives you the central input quantity for the design and rating of heat exchangers in process equipment engineering.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Select cross section and boundary condition: First, the pipe cross section (circular, rectangular, oval) and the thermal boundary condition are defined — constant wall temperature or constant heat flux. For non-circular cross sections, the hydraulic diameter is formed as the characteristic length.
- Form the Reynolds number and determine the flow regime: The Reynolds number follows from the mean flow velocity, the hydraulic diameter, and the kinematic viscosity. Below Re = 2,300 the flow is laminar; above Re ≈ 10⁴ it is fully turbulent; in between, the transition region is handled by interpolating between the two solutions.
- Laminar flow: account for entrance effects: In the laminar regime, the mean Nusselt number is composed of the limiting values for fully developed flow and the entrance terms with the parameter Re·Pr·d/L. Short pipes therefore have significantly higher Nusselt numbers than the asymptotic limit (Nu = 3.66 for constant wall temperature).
- Turbulent flow: evaluate the Gnielinski correlation: In the turbulent regime, the friction factor ξ is first calculated from the Reynolds number using the Konakov/Filonenko relation. With this, the Gnielinski correlation yields the Nusselt number; the factor 1 + (d/L)²ᐟ³ accounts for the influence of the finite pipe length.
- Property and wall corrections, heat transfer coefficient: If there is a significant temperature difference between wall and fluid, the Nusselt number is corrected with the Prandtl number or temperature ratio. From Nu, the thermal conductivity, and the hydraulic diameter, the heat transfer coefficient follows; with the heat transfer area and temperature difference, the transferred duty is obtained.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Tube length | l | m |
| Tube inside diameter | di | m |
| Dynamic viscosity | η | mPa·s |
| Density | ρ | kg/m³ |
| Prandtl number | Pr | - |
| Prandtl number at wall temperature | PrW | - |
| Thermal conductivity | λ | W/(m·K) |
| Flow velocity | w | m/s |
| Total mass flow | mtot | kg/s |
| Number of tubes with parallel flow | Z | - |
| Mass flow per tube | m | kg/s |
| Wall temperature | ϑW | °C |
| Inlet temperature | ϑe | °C |
| Outlet temperature | ϑa | °C |
| Reynolds number | Re | - |
| Nusselt number | Nu | - |
| Heat transfer coefficient | α | W/(m²·K) |
| Heat duty | Q = mtot ∙ cp ∙ (ϑa - ϑe ) Q | W |
| Hydraulic diameter | dh | m |
| Cross sectional area of the tube | f | m² |
| Perimeter of the tube | u | m |
| Specific heat capacity | cp | J/(kg·K) |
| Tube circular / non-circular? | Rohrquerschnitts | – |
| Fluid liquid / gaseous? | gasförmig | – |
Calculation options
Tube circular / non-circular?
Circular tubes · Non-circular tubes
Fluid liquid / gaseous?
liquid · 1
Is the gas CO2 or H2O?
No · 1 · 2
Condition
Constant wall temperature · Constant heat flux
Worked example
Water at a mean temperature of 50 °C flows at w = 1.0 m/s through a pipe with an inside diameter d = 20 mm and a heated length L = 3.0 m. Find the Reynolds number, the Nusselt number according to Gnielinski, and the internal heat transfer coefficient (wall correction neglected) — a worked example of how to calculate pipe-flow heat transfer to the VDI Heat Atlas.
Given values
| Pipe inside diameter d | 20 mm |
| Heated pipe length L | 3.0 m |
| Flow velocity w | 1.0 m/s |
| Kinematic viscosity of water (50 °C) ν | 0.553 · 10⁻⁶ m²/s |
| Thermal conductivity λ | 0.644 W/(m·K) |
| Prandtl number Pr | 3.55 |
Solution
Reynolds number and flow regime
Re = w · d / ν = 1.0 · 0.02 / (0.553 · 10⁻⁶) ≈ 36,166
Re > 10⁴: The flow is fully turbulent, so the Gnielinski correlation applies.
Friction factor
ξ = (1.8 · lg Re − 1.5)−2 = (1.8 · lg 36,166 − 1.5)−2 ≈ 0.02224
Nusselt number according to Gnielinski
Nu = (ξ/8) · (Re − 1000) · Pr / [1 + 12.7 · (ξ/8)1/2 · (Pr2/3 − 1)] · [1 + (d/L)2/3]
Nu = (0.02224/8) · 35,166 · 3.55 / [1 + 12.7 · 0.0527 · (3.552/3 − 1)] · [1 + (0.02/3.0)2/3] ≈ 190.3
Heat transfer coefficient
α = Nu · λ / d = 190.3 · 0.644 / 0.02 ≈ 6,130 W/(m²·K)
Result
| Reynolds number Re | 36,166 |
| Friction factor ξ | 0.0222 |
| Nusselt number Nu | 190.3 |
| Heat transfer coefficient α | ≈ 6,130 W/(m²·K) |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
When must the hydraulic diameter be used?
For all non-circular cross sections (rectangular duct, annulus, oval tube), the hydraulic diameter d_h = 4·A/U — formed from the flow cross-sectional area and the wetted perimeter — is used as the characteristic length. For turbulent flow, this transfer is well established; in the laminar regime, however, the limiting Nusselt values depend on the cross-sectional shape, which is why the VDI Heat Atlas provides shape-specific values there.
What difference do the boundary conditions of constant wall temperature and constant heat flux make?
For laminar, fully developed pipe flow, the limiting values differ significantly: Nu = 3.66 for constant wall temperature and Nu = 4.36 for constant heat flux. For turbulent flow, the influence of the boundary condition is practically negligible, because turbulent exchange dominates the heat transport. Evaporators and condensers on the shell side approximately correspond to constant wall temperature; electrically heated pipes correspond to constant heat flux.
How is the transition region 2,300 < Re < 10⁴ treated?
In this range the flow is intermittent and neither of the two limiting solutions is valid on its own. The VDI Heat Atlas interpolates linearly with an intermittency factor γ between the laminar Nusselt number at Re = 2,300 and the turbulent one at Re = 10⁴. Designs should avoid this range where possible, since small operating fluctuations can cause large changes in heat transfer.
Why does the Gnielinski correlation contain the term (d/L)^(2/3)?
It accounts for the thermal entrance region at the pipe inlet: there, the thermal boundary layer is still thin and the local heat transfer is significantly higher than in the fully developed state. For short pipes (large d/L), the term noticeably increases the mean Nusselt number; for long pipes it tends to zero. The heated or cooled length must be used for the pipe length L.