Engineering task and calculation objective
The WROK module calculates the heat transfer in cross flow through tube bundles with small longitudinal pitch. While the classic tube bundle correlations apply to the usual pitch ratios, closely staggered bundles behave differently: with a small longitudinal pitch, the lanes between the tube rows act like short channels, the approach flow to the subsequent rows changes, and the mean heat transfer coefficient must be determined with a method tailored to this case.
From the tube arrangement (in-line or staggered), the number of tube rows, the outside diameter of the tubes, and the crosswise, longitudinal and staggering pitch, the program determines the flow cross-section and the velocity in the narrowest cross-section. Building on this, the Reynolds number, the Nusselt number and finally the mean heat transfer coefficient of the bundle are calculated — the central quantity for sizing the outside heat transfer of shell and tube heat exchangers, economizers, air coolers and waste heat surfaces.
In practice, this calculation is needed whenever installation space is to be saved and the tube rows are packed more tightly than the standard correlations cover — for example in compact flue gas heat exchangers or closely finned boiler heating surfaces.
Calculation workflow
- Define the bundle geometry: The tube arrangement (in-line or staggered), the number of tube rows, the outside diameter of the tubes, and the crosswise, longitudinal and staggering pitch are specified; from these, the dimensionless pitch ratios of the bundle follow.
- Determine the narrowest flow cross-section: From the crosswise pitch — for staggered arrangements, if applicable, from the diagonal pitch — the narrowest free flow cross-section is determined, and from it the velocity in the narrowest cross-section, which serves as the reference velocity.
- Form the Reynolds number: With the reference velocity, the characteristic length and the fluid properties at mean temperature, the Reynolds number of the bundle flow is formed and the validity range of the correlation is checked.
- Nusselt number of the bundle flow: Using the correlation valid for small longitudinal pitches, the Nusselt number is determined from the Reynolds and Prandtl numbers; arrangement, pitch ratios and the number of tube rows enter as correction factors, since short bundles with few rows exhibit a lower mean heat transfer.
- Output the mean heat transfer coefficient: From the Nusselt number follows the mean heat transfer coefficient of the bundle, which can be adopted as the outside film coefficient in the overall heat transfer calculation of the apparatus (U-value).
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Number of tube rows | n | - |
| Outside diameter of the tubes | da → l = π/2·da | m |
| Crosswise pitch | s1 → tq = s1/da | m |
| Longitudinal pitch | s2 → tl = s2/da | m |
| Staggering pitch | s3 → tv = s3/da | m |
| Flow cross-section | F | m² |
| Outside diameter of the tubes | da → l = π/2·da | m |
| Crosswise pitch | s1 → tq = s1/da | - |
| Longitudinal pitch | s2 → tl = s2/da | - |
| Staggering pitch | s3 → tv = 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 (wall) | PrW | - |
| Medium liquid / gaseous? | (1/2) | - |
| Mass flow | m | kg/s |
Calculation options
Medium liquid / gaseous?
Liquid · 2
Tube arrangement
in-line · staggered
Frequently asked questions
What is special about tube bundles with small longitudinal pitch?
With a small longitudinal pitch, the tube rows stand so closely one behind the other that fully developed wake regions no longer form behind the tubes — the flow resembles a channel flow through the tube lanes rather than flow around individual tubes. The usual bundle correlations, which build on the flow around a single cylinder with arrangement factors, are outside their validity range here; that is why a dedicated calculation method exists for this case.
Why is the velocity in the narrowest cross-section used as the reference quantity?
The heat transfer is dominated by the highest local velocity, which occurs in the narrowest free cross-section between the tubes. For staggered arrangements, the narrowest cross-section may lie on the diagonal — the module checks this via the staggering pitch. Confusing the approach velocity with the gap velocity is one of the most common sources of error in hand calculations and leads to markedly wrong Reynolds numbers.
What is the influence of the number of tube rows?
The first tube rows are still hit by the undisturbed incoming flow and transfer less heat than the subsequent rows, which benefit from the bundle-induced turbulence. For bundles with fewer than about ten rows, the mean heat transfer coefficient is therefore reduced by a row-number factor; the module accounts for this via the entered number of tube rows.
Does the module also provide the pressure drop of the bundle?
WROK is focused on heat transfer and delivers the Reynolds number, the Nusselt number and the mean heat transfer coefficient. The bundle-side pressure drop is calculated with the dedicated pressure drop modules of the library; both calculations use the same geometry and property data and can be coupled within the project.