Ventilating ducts across laminations

The statement is true. Iron loss is produced throughout the volume of a core, and the heat must flow through the iron to a cooling surface. The internal (hot-spot) temperature rise depends on the square of the heat path length. Ventilating ducts shorten this path and add cooling surface, so they reduce the temperature rise inside the material sharply.

Justification

For a core packet of length $l$ between two cooled faces, with loss $p$ per unit volume and thermal resistivity $\rho$, the hot spot is at the middle:

$$\theta_h - \theta_s = \frac{p\,\rho\,l^2}{8}$$

If the core is divided by $n$ radial ducts into $(n+1)$ packets, each packet has length $\dfrac{l}{n+1}$:

$$\theta_h' - \theta_s = \frac{p\,\rho}{8}\left(\frac{l}{n+1}\right)^2 = \frac{\theta_h - \theta_s}{(n+1)^2}$$

For example, one central duct ($n = 1$) reduces the internal rise to one-quarter. Three ducts reduce it to one-sixteenth.

 without duct            with one duct
 +-----------+          +----+  +----+
 |    hot    |          |    |  |    |
 +-----------+          +----+  +----+
   rise ~ l^2           rise ~ (l/2)^2 = 1/4
                               ^ air

Why the ducts cut across the laminations

• Heat flows easily along the laminations (through iron), but with difficulty across them, through the insulation between sheets, where the thermal resistivity is much higher.
• A duct "across the lamination" (between packets, at right angles to the core length) gives every sheet its own short path along its plane to cooling air, so heat does not have to cross many insulation layers.

Further benefits

1. Extra cooling surface: the duct faces dissipate heat, so $\theta_s$ (surface temperature) also falls.
2. Air is blown through the ducts by the rotor (radial ventilation), raising the heat transfer coefficient.
3. Lower core temperature protects the winding insulation and permits higher flux density or current density, giving more output from the same frame.

Typical practice: packets of 40–80 mm with ducts about 10 mm wide in the stator and rotor cores of large machines, and oil ducts in transformer cores.