Engineering task and calculation objective
This module calculates the pressure drop for cross-flow over tube bundles with plain circular or oval tubes according to Section L1.4 of the VDI-Wärmeatlas (VDI Heat Atlas, 12th edition, 2019). Anyone who wants to calculate the gas-side or liquid-side pressure drop across a tube bundle in cross-flow – for example in the shell side of a heat exchanger, in air coolers or economizers – needs exactly these drag coefficients, because the pressure drop directly determines the required fan or pump power.
The calculation is based on the Gaddis and Gnielinski correlation documented in the VDI Heat Atlas: the drag coefficient is composed of a laminar and a turbulent contribution and adapted via geometry factors to the in-line or staggered tube arrangement with its transverse and longitudinal pitch ratios. The method therefore covers a wide Reynolds number range from creeping flow to fully turbulent conditions.
In practice this calculation is an integral part of every thermal-hydraulic design of shell and tube heat exchangers: heat transfer increases with the approach velocity, but the pressure drop grows disproportionately – the module provides the data basis for this trade-off.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define the bundle geometry: Tube outside diameter, tube arrangement (in-line or staggered) as well as the transverse pitch ratio a and the longitudinal pitch ratio b are entered. From these follow the void fraction of the bundle and the velocity reference that governs the Reynolds number.
- Determine the reference velocity in the narrowest cross-section: From the volume or mass flow rate and the narrowest free flow cross-section between the tubes, the governing velocity is calculated; the Reynolds number and the dynamic pressure both refer to this velocity.
- Determine Reynolds number and drag coefficient: The Reynolds number is formed with the fluid properties at the mean temperature. Following Gaddis/Gnielinski, the drag coefficient is composed of a laminar contribution (proportional to 1/Re) and a turbulent contribution, each with geometry factors for the chosen tube arrangement.
- Apply corrections: Influences such as the temperature-dependent viscosity change at the tube wall (heating or cooling) and the number of tube rows – with only a few rows the flow is not yet fully developed – are accounted for by correction factors.
- Calculate the pressure drop: The pressure drop follows from the drag coefficient, the number of main flow resistances (tube rows in the flow direction) and the dynamic pressure of the reference velocity.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Crosswise pitch | s1 | m |
| Longitudinal pitch | s2 | m |
| Tube diameter | d | m |
| Velocity in the free cross-section | wf | m/s |
| Number of tube rows | NR | - |
| Density of the fluid | ρ | kg/m³ |
| Dynamic viscosity | η | mPa·s |
| Frictional pressure drop | Δp | Pa |
| Density at the outlet | ρA | kg/m³ |
| Density at the inlet | ρE | kg/m³ |
| Velocity in the free cross-section at the outlet | wf,A | m/s |
| Velocty in the free cross-section at the inlet | wf,E | m/s |
| Additional pressure difference | Δpi | Pa |
| Geodesic height difference | h | m |
| Geodesic pressure difference | Δph | Pa |
| Total pressure drop | Δp + Δpi + Δph | Pa |
| Longitudinal pitch ratio | b | - |
| if | b ≥ = 0.5 ∙ √(2 a + 1) | - |
| Pressure drop factor | ξ | - |
| Laminar part of the pressure drop factor | ξl | - |
| Turbulent part of the pressure drop factor | ξt | - |
| Arrangement factor (laminar - in-line) | fa.l.f | - |
| Arrangement factor (turbulent - in-line) | fa.t.f | - |
| Arrangement factor (laminar - staggered) | fa.l.v | - |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Reynolds number | Re | - |
| Crosswise pitch ratio | a | - |
| Longitudinal pitch ratio | b | - |
| Diagonal pitch ratio | c | - |
| Number of main resistances in direction of flow | NW | - |
| Velocity in the narrowest cross-section | we | m/s |
Calculation options
Bauform
In-line tube arrangement (circular bare tubes) · Staggered tube arrangement (circular bare tubes) · Staggered oval tubes according to Brauer · Staggered oval tubes according to Merker and Hanke · Staggered lens tubes according to Ruth
Frequently asked questions
How does the in-line arrangement differ from the staggered arrangement with respect to pressure drop?
With a staggered arrangement the flow is deflected more strongly and the narrowest cross-section may lie diagonally between the tubes; for the same pitch, the drag coefficient is usually higher than for the in-line arrangement. In return, heat transfer is also better. The geometry factors of the correlation distinguish both cases explicitly, as does the definition of the number of main flow resistances.
Which velocity does the drag coefficient refer to?
To the velocity in the narrowest cross-section of the bundle, not to the approach velocity upstream of the bundle. Confusing these two velocities is a common source of error – with a tight pitch, the velocity in the narrowest gap is considerably higher than the superficial velocity.
Does the method also apply to bundles with only a few tube rows?
The correlation was derived for bundles with many rows. With fewer than about ten tube rows in the flow direction, the flow is not yet periodically developed; the VDI Heat Atlas provides correction factors for this case. With only one or two rows the result should be assessed critically.
Does the module also cover the shell side of a shell and tube heat exchanger with baffles?
No. Section L1.4 deals with the ideally cross-flowed bundle. For the shell side with baffles, bypass and leakage streams, the method of Section L1.5 (available as a separate module in the program) must be used; it builds on the cross-flow correlation and extends it with window and leakage resistances.