Temperature gradient in an iron core

Iron losses (hysteresis and eddy) are produced throughout the volume of the core. This heat must flow through the iron to the cooling surfaces and ducts, so the centre of each core packet is hotter than its surface. The resulting variation of temperature inside the core is the temperature gradient.

Anisotropy of a laminated core

• Along the laminations, heat flows through continuous iron, so the thermal resistivity $\rho_x$ is low.
• Across the laminations, heat must cross the insulation (varnish or oxide) and the air films between sheets, so the thermal resistivity $\rho_y$ is many times higher (roughly 20–50 times).

       cooling surface / duct
   +----------------------------+  ^
   | ========================== |  |
   | ======== hot spot ======== |  | l_y (across)
   | ========================== |  |
   +----------------------------+  v
   <----------- l_x ------------>  (along)

One-direction flow

For loss $p$ per unit volume flowing across a length $l$ between two cooled faces, consider a plane at $x$ from the centre. Heat crossing it per unit area is $p x$, so $\dfrac{d\theta}{dx} = -\rho p x$ and

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

The temperature is parabolic, with the hot spot at the centre.

Two-direction flow (actual core)

Let the loss divide into $p_x$ (along) and $p_y$ (across). Both paths share the same hot spot and surface temperature:

$$\theta_h - \theta_s = \frac{p_x\rho_x l_x^2}{8} = \frac{p_y\rho_y l_y^2}{8}, \qquad p_x + p_y = p$$ $$\theta_h - \theta_s = \frac{p}{8\left[\dfrac{1}{\rho_x l_x^2} + \dfrac{1}{\rho_y l_y^2}\right]}$$

Conclusions

• Most heat flows along the laminations because $\rho_x$ is small.
• The hot-spot rise varies as the square of the path length. Ventilating ducts that divide the core into packets of 40–80 mm therefore reduce the gradient sharply.
• Iron loss density $p$ should be kept moderate (a suitable $B_m$) in large cores.