Rectangular pancake geometry creates routes with different lengths and connections around long sides, short sides and corners. Their flows depend on pressure differences and resistance, not simply on nominal duct height.
Build a network from the actual inlet to the outlet. Include corner connections and possible bypasses. Identify which branches pass the regions with significant loss and which mainly provide an easier route around them.
Use an appropriate branch model
For a fully developed laminar flow between wide parallel plates, a simplified resistance is R = 12μL/(bh³), with Δp = RQ. Here viscosity μ is in pascal-seconds, length L, width b and gap h in metres, volume flow Q in cubic metres per second and pressure drop Δp in pascals.
This expression assumes a narrow uniform gap and neglects entrance, turning and other local effects. It demonstrates the strong sensitivity to gap height; it does not qualify a particular coil channel.
| Route | Geometry to record | Thermal connection |
|---|---|---|
| Long side | Length, width and local opening | Losses adjacent to that route |
| Short side | Its distinct path and connections | Different available flow and heated area |
| Corner link | Turns and restrictions | Mixing or bypass between sides |
| Spacer interruption | Footprint and remaining passage | Local redistribution |
| Outlet connection | Shared pressure boundary | Interaction between parallel branches |
Solve mass conservation at the network nodes and pressure-flow relationships in the branches. Assigning equal flow to parallel sides without solving their resistance relationship can conceal a weakly cooled region.
Couple flow with local heat input
A low-flow branch is not necessarily the hottest if it also receives little heat. Conversely, a high-loss region can remain demanding despite a larger flow. Pair each route with the local loss estimate and heat-transfer assumptions. Record rated quantities and test conditions on the general power-transformer basis of IEC 60076-1 [1].
Liquid temperature changes viscosity and other properties, so a coupled or iterative calculation may be needed. State whether the model assumes forced circulation or natural circulation and how its driving pressure is obtained.
Check sensitivity to assembled gap variation. A small local restriction can matter substantially under the slit assumptions, while a corner bypass can change the pressure available to an entire side. Use actual tolerance bounds rather than a nominal-only network.
Validate the branch picture
Compare representative flow or temperature observations with the model where approved access permits. Temperature alone does not uniquely identify flow: loss and heat-transfer errors can produce similar readings.
If a side runs hotter than predicted, inspect its geometry and pressure connections before increasing overall pump flow. The added flow may preferentially take an existing bypass. A local network explains where cooling is delivered, which is the decision a total circulation rate cannot make by itself.
Quantify how a restricted duct loses flow
Assume fully developed laminar flow at one fixed oil viscosity, with μ = 0.010 Pa·s, L = 1.0 m, b = 0.10 m, h = 2.0 mm and Δp = 100 Pa. The slit model gives R = 1.50×10⁸ Pa·s/m³ and Q = 0.667 mL/s. Reducing h to 1.8 mm at the same pressure gives 0.486 mL/s, a 27.1% decrease. If local heat input is unchanged, mass-flow heat removal needs reassessment. Prioritize the restricted passage and bypass network before increasing total circulation; the assumed slit relation omits the actual corner and entry losses. Assess the relevant measured quantities with a defined measurement model and propagated uncertainty [2].
References
- IEC, IEC 60076-1:2011, Power transformers – Part 1: General.
- JCGM, JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement, 2008.