Engineering task and calculation objective
The module calculates heat transfer to non-Newtonian liquids in flow through pipes and parallel-plate gaps according to chapter M4 of the VDI Wärmeatlas (VDI Heat Atlas, 12th edition 2019). Non-Newtonian media — such as polymer solutions and melts, suspensions, pastes, paints or many food products — have no constant viscosity; their apparent viscosity depends on the shear rate. From the flow law of the medium and the chosen geometry, the module determines the heat transfer coefficient between wall and fluid.
This calculation is needed wherever shear-thinning media are pumped under temperature control: tubular heat exchangers and heaters in the food and cosmetics industries, cooling and heating sections for polymer solutions, gap-type heat exchangers. To calculate heat transfer of non-Newtonian liquids, standard Newtonian correlations cannot be applied unchanged, because the velocity profile in the pipe is deformed by the flow behavior and the near-wall shear rate differs from that of a Newtonian fluid.
For this purpose, the VDI Wärmeatlas provides correction approaches based on the power law (Ostwald–de Waele), with which the proven relationships for laminar and turbulent pipe and gap flow are transferred to shear-thinning media.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Select the geometry: Pipe or parallel-plate gap is selected as the flow geometry; from this follow the characteristic length (inside diameter or hydraulic diameter of the gap) and the geometry-dependent constants of the correlations.
- Describe the flow behavior of the medium: The rheological behavior is captured by the power law with consistency factor and flow index; a flow index below one indicates shear-thinning, above one dilatant behavior. The parameters must have been measured in the relevant shear rate and temperature range.
- Form the representative viscosity and dimensionless numbers: From the flow law and the near-wall shear rate, a representative apparent viscosity is calculated. With it, generalized Reynolds and Prandtl numbers are formed, which define the flow regime (laminar or turbulent).
- Evaluate the heat transfer correlation: In the laminar regime, the relationship for the thermal entrance region is evaluated with a correction of the velocity profile depending on the flow index; in the turbulent regime, the Newtonian correlations with the representative viscosity are used approximately. The temperature influence on viscosity is corrected via the ratio of viscosities at bulk and wall temperature.
- Determine the heat transfer coefficient and design quantities: From the Nusselt number follows the heat transfer coefficient, and from it — with the surface area and temperature difference — the transfer duty; in parallel, the flow law provides the pressure drop, which for viscous media often dominates the design.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Inlet temperature | ϑE | °C |
| Wall temperature | ϑW | °C |
| Thermal conductivity | λ | W/(m·K) |
| Thermal diffusivity | a | m²/s |
| Temperature coefficient | β | 1/K |
| Flow index | m | - |
| Gap width | d | m |
| Gap length | l | m |
| Mean axial flow velocity | w | m/s |
| Graetz number | Gz | - |
| Nusselt number (temperature-independent viscosity) | Nuo | - |
| Temperature weighting factor | θ | - |
| Regression coefficient | A | - |
| Regression coefficient | A' | - |
| Relative pressure drop | φ | - |
| Relative rate of shear at wall | ΓA | - |
| Nusselt number (temperature-dependent viscosity) | Nuθ | - |
| Heat transfer coefficient | αθ | W/(m²·K) |
| Outlet temperature | ϑA | °C |
| Density | ρ | kg/m³ |
| Specific heat capacity | cp | J/(kg·K) |
| Fluidity at inlet temperature | ΦE | (Pa^-m)/s |
| Fluidity at wall temperature | ΦW | (Pa^-m)/s |
| Pressure drop (temperature-independent viscosity) | Δp0 | Pa |
Calculation options
Geometry
Pipe with circular cross-section · Flat gap
Frequently asked questions
Why can Newtonian Nusselt correlations not be used directly with a viscosity measured at an arbitrary point?
In shear-thinning media, the apparent viscosity varies over the pipe cross-section by orders of magnitude: at the wall the shear rate is high and the viscosity low, at the pipe center the reverse. As a result, the velocity profile becomes flatter than the Newtonian parabolic profile, which changes the near-wall temperature gradient and hence the heat transfer. Only the combination of a representative viscosity and a flow-index-dependent profile correction makes the Newtonian correlations transferable.
In which flow regime do apparatus for non-Newtonian media typically operate?
Because of the high consistency of most shear-thinning media, almost always in the laminar regime, often with a pronounced thermal entrance region over the entire apparatus length. Turbulence is achievable only with thin liquids or very high velocities. Entrance-region correlations and the question of whether the temperature profile is developed are therefore more important here than for Newtonian liquids.
What must be observed with the rheological property data?
Consistency factor and flow index are valid only in the shear rate and temperature window in which they were measured; extrapolating the power law to very small or very large shear rates is inadmissible, since real media exhibit limiting viscosities there. The strong temperature dependence of the consistency must additionally be captured, because the medium changes rheologically along the pipe during heating or cooling.
Does the approach also apply to media with a yield stress or viscoelastic behavior?
Only to a limited extent. Media with a pronounced yield stress (Bingham behavior) form an unsheared plug in the pipe core, which requires separate approaches; viscoelastic effects such as die swell or secondary flows are not captured by the power law. In such cases, the correlations provide only indicative values that must be verified by experiments.