Engineering task and calculation objective
The PILLOW module calculates heat transfer and pressure drop in cooling pockets — flat flow channels formed by spot or seam welding, used as pillow plates (dimple jackets) on vessel walls, agitated vessels and plate-type equipment for cooling or heating. The irregular, bulged flow cross-section of a cooling pocket is described as an equivalent channel via the hydraulic diameter.
From the properties of the heat transfer medium (density, dynamic viscosity at mean and at wall temperature, thermal conductivity, specific heat capacity), the flow area, the length of the flow path and the volume or mass flow, the module forms the flow velocity, the Reynolds number and the Prandtl number. Using a pipe flow correlation with pocket-specific adjustment factors, the heat transfer coefficient on the pocket side follows, and — with the friction factor — the pressure drop over the flow path.
In practice, this calculation is needed to design or rate cooling pockets on vessels: the pocket-side heat transfer coefficient enters the overall heat transfer of the vessel wall, while the pressure drop determines the required pump head of the cooling circuit.
Calculation workflow
- Describe the geometry of the cooling pocket: The flow area, the length of the flow path and the equivalent hydraulic diameter describe the equivalent channel; the adjustment factors capture the deviation of the bulged pocket geometry from a smooth pipe.
- Specify fluid properties and throughput: Density, dynamic viscosity (at mean temperature and at wall temperature), thermal conductivity and specific heat capacity of the heat transfer medium as well as the volume or mass flow are entered; from these follow the flow velocity, the mass flux and the Prandtl number.
- Determine the flow regime: With the velocity, the hydraulic diameter and the viscosity, the Reynolds number is formed and the laminar or turbulent flow regime is established.
- Calculate the heat transfer coefficient: From the Reynolds and Prandtl numbers, the Nusselt number is determined via the implemented correlation with the adjustment factors; the viscosity ratio between mean and wall temperature corrects for the influence of the temperature-dependent properties near the wall. The result is the heat transfer coefficient of the pocket side.
- Determine the pressure drop: With the friction factor, the length of the flow path, the hydraulic diameter and the dynamic pressure, the pressure drop over the flow path of the cooling pocket is calculated.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Density | ρ | kg/m³ |
| Viscosity at average temperature | μ | mPa·s |
| Viscosity of fluid at wall temperature | μw | mPa·s |
| Prandtl number | Pr | - |
| Reynolds number | Re | - |
| Specific heat | cp | J/(kg·K) |
| Volume flow | v | m³/s |
| Mass flow | M | kg/s |
| Mass flux | G | kg/(m²·s) |
| Flow area | A | m² |
| For Re < 2300 | f | - |
| Factor | F1 | - |
| Factor | F2 | - |
| Factor | F3 | - |
| Length of path of travel | L | m |
| Thermal conductivity | λ | W/(m·K) |
| Flow velocity | v | m/s |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Heat-transfer film coefficient | hi | W/(m²·K) |
| Equivalent hydraulic diameter | DH | m |
| Pressure drop | ΔP | Pa |
| Equivalent hydraulic diameter | (DH * F1) DH | m |
Frequently asked questions
Why is the hydraulic diameter used for cooling pockets?
The flow cross-section of a cooling pocket is not a circle but a flat, bulged gap, additionally constricted between the weld spots. The hydraulic diameter — four times the flow cross-section divided by the wetted perimeter — converts this geometry into an equivalent channel to which pipe flow correlations can be applied. The remaining deviation from a smooth pipe is captured by the module's adjustment factors.
Why are two viscosity values (mean and wall temperature) needed?
During heating or cooling, the temperature of the near-wall layer differs significantly from the bulk temperature. Since viscosity is strongly temperature-dependent, this alters the velocity and temperature profiles. Correlations correct for this via the ratio of the viscosities at mean and at wall temperature — when cooling viscous media, the heat transfer turns out noticeably lower than without this correction.
What role do the weld spots of the pillow plates play for heat transfer and pressure drop?
The spot welds act as turbulence promoters: they increase the heat transfer compared with a smooth channel, but at the same time raise the friction factor and thus the pressure drop. Both effects depend on the spot spacing and the pillow height and are accounted for in the module via the adjustment factors. In the design, one must always seek the compromise between high heat transfer and the allowable pressure drop of the cooling circuit.
Does the calculation also apply to evaporating or condensing media in the pocket?
No, the module covers single-phase flow with liquid or gaseous heat transfer media. If the medium evaporates in the pocket (e.g. direct refrigerant evaporation), two-phase correlations with completely different pressure drop and heat transfer behavior apply — that case must be calculated separately.