Engineering task and calculation objective
This module calculates forced-convection heat transfer in cross flow through single tube rows and tube bundles according to the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019, section G7). The tube bundle in cross flow is the core geometry of the shell-and-tube heat exchanger: on the shell side, the medium — guided by baffles — flows predominantly across the tubes, just as in air coolers, economizers, and flue-gas heat exchangers. Besides circular tubes, oval tube cross sections are also supported.
The calculation builds on the correlation for the single cylinder in cross flow and extends it by the bundle effects: the Reynolds number is formed with the velocity in the void space between the tubes (void fraction ψ from the transverse pitch ratio), and an arrangement factor captures whether the tubes are in-line or staggered. Staggered arrangements direct the fluid onto the following tube rows and thereby achieve higher heat transfer coefficients than in-line arrangements. For bundles with fewer than ten tube rows, a tube-row factor is additionally applied.
If you want to calculate the shell-side heat transfer of a tube bundle, this module delivers the Nusselt number and heat transfer coefficient of the bundle — the basis of every thermal design of shell-and-tube equipment according to the VDI Heat Atlas.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define tube cross section and bundle geometry: The tube cross section (circular or oval) and the tube arrangement (in-line or staggered) are selected, with the transverse pitch ratio a = s₁/d and the longitudinal pitch ratio b = s₂/d formed from the tube pitches and the tube diameter.
- Form the void fraction and the effective Reynolds number: The transverse pitch ratio yields the void fraction ψ of the bundle (for b ≥ 1: ψ = 1 − π/(4a)). The Reynolds number is formed with the approach velocity, the streamed length l = (π/2)·d of the single tube, and the velocity in the bundle increased by 1/ψ.
- Calculate the Nusselt number of the single tube: With the effective Reynolds number and the Prandtl number, the mean Nusselt number of the single cylinder in cross flow is calculated according to Gnielinski — as the quadratic superposition of the laminar and the turbulent boundary layer contributions.
- Apply the bundle arrangement factor: The tube arrangement factor f_A raises the single-tube Nusselt number to the bundle value. For staggered arrangements it depends essentially on the longitudinal pitch ratio; for in-line arrangements, on both pitch ratios.
- Tube-row correction and heat transfer coefficient: For bundles with fewer than ten tube rows, the Nusselt number is reduced by a factor for the number of tube rows, since the turbulent approach flow profile only develops after a few rows. From the bundle Nusselt number, the thermal conductivity, and the streamed length, the mean heat transfer coefficient of the bundle follows.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Number of tube rows | n | - |
| Outside diameter of the tubes | da → l = π/2·da | m |
| Crosswise pitch | s1 → a = s1/da | m |
| Longitudinal pitch | s2 → b = s2/da | m |
| Staggering pitch | s3 → c = s3/da | m |
| Free flow cross-section | F | m² |
| Angle (channel/tube axis) | Θ | ° |
| Outside diameter of the tubes | da → l = π/2·da | m |
| Crosswise pitch | s1 → a = s1/da | - |
| Longitudinal pitch | s2 → b = s2/da | - |
| Staggering pitch | s3 → c = s3/da | - |
| Mean pressure | 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 | - |
| Fluid liquid / gaseous? | <2> | – |
Calculation options
Fluid liquid / gaseous?
liquid · 2
Tube cross section
Circular tube · Oval tube
Geometry
Single tube rows · Tube bundle
Worked example
A staggered tube bundle (tube outside diameter d = 25 mm, transverse pitch ratio a = 2.0, longitudinal pitch ratio b = 1.5, more than ten tube rows) is exposed to a cross flow of air at 20 °C with an approach velocity w = 2 m/s. Find the mean heat transfer coefficient of the bundle (wall correction neglected) — a worked example of how to calculate tube-bundle heat transfer to the VDI Heat Atlas.
Given values
| Tube outside diameter d | 25 mm |
| Transverse pitch ratio a | 2.0 |
| Longitudinal pitch ratio b | 1.5 (staggered) |
| Approach velocity w | 2 m/s |
| Kinematic viscosity of air (20 °C) ν | 15.32 · 10⁻⁶ m²/s |
| Thermal conductivity λ | 0.0259 W/(m·K) |
| Prandtl number Pr | 0.71 |
Solution
Void fraction and effective Reynolds number
ψ = 1 − π/(4a) = 1 − π/8 ≈ 0.607 (valid for b ≥ 1)
Streamed length l = (π/2) · d = 0.0393 m
Reψ = w · l / (ψ · ν) = 2 · 0.0393 / (0.607 · 15.32 · 10⁻⁶) ≈ 8,440
Nusselt number of the single tube
Nulam = 0.664 · Reψ1/2 · Pr1/3 ≈ 54.4
Nuturb = 0.037 · Reψ0.8 · Pr / [1 + 2.443 · Reψ−0.1 · (Pr2/3 − 1)] ≈ 45.6
Nul,0 = 0.3 + (54.4² + 45.6²)1/2 ≈ 71.3
Arrangement factor for the staggered arrangement
fA = 1 + 2/(3b) = 1 + 2/(3 · 1.5) ≈ 1.444
Nubundle = fA · Nul,0 = 1.444 · 71.3 ≈ 103.0
Tube-row factor = 1 (more than ten rows).
Heat transfer coefficient
α = Nubundle · λ / l = 103.0 · 0.0259 / 0.0393 ≈ 67.9 W/(m²·K)
Result
| Void fraction ψ | 0.607 |
| Reynolds number Re_ψ | 8,440 |
| Nusselt number, single tube | 71.3 |
| Arrangement factor f_A | 1.444 |
| Nusselt number, bundle | 103.0 |
| Heat transfer coefficient α | ≈ 67.9 W/(m²·K) |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
Why does the staggered tube arrangement give a better heat transfer than the in-line arrangement?
In an in-line arrangement, the rear tubes lie partly in the wake of the front ones, where the velocity is small. In a staggered arrangement, by contrast, the flow is directed onto the stagnation points of the next row, and each tube receives the full approach flow. The higher heat transfer comes at the cost of a higher pressure drop — the choice of arrangement is always a trade-off between transfer performance and pump or fan power.
With which velocity is the Reynolds number formed?
The basis is the approach velocity referred to the free cross section upstream of the bundle (superficial velocity). It is divided by the void fraction ψ, giving the mean velocity in the void space between the tubes. A common mistake is to use the velocity in the narrowest cross section instead — the VDI correlation is explicitly calibrated to the void-space velocity w/ψ.
When must the tube-row factor be taken into account?
The standard correlation applies to bundles with ten or more tube rows in the flow direction, in which a bundle-typical turbulence level has developed. With fewer rows, the mean heat transfer is lower because the first rows still see the low-turbulence approach flow; the correction factor weights the first row with the single-tube value to account for this. With only one or two rows (e.g. tube registers), the effect is largest.
Can the shell side of a shell-and-tube heat exchanger be calculated directly with this module?
The module delivers the heat transfer of the ideal bundle in cross flow. In a real shell space with baffles, window longitudinal flow, leakage streams through the clearances at the baffles, and bypass between bundle and shell are added, all of which reduce the heat transfer. These effects are captured in the cell model of the VDI Heat Atlas (section G8) through correction factors based on the bundle Nusselt number calculated here.