Heat transfer methods

In general, there are three mechanisms to dissipate the heat energy due to losses in electrical machines; convection, conduction, and radiation, of which the last one is usually of least significance, especially in the case of motors with forced cooling, as the frame temperature remains low. With self-cooled motors, the frame temperature may reach values, at which the radiation cannot be neglected any longer. With the prototype motor, the effect of the radiation was emphasized, as the frame was matt black, having thus high emissivity.

4.2.1 Convection

By convection, heat is transferred from solid to either gas or liquid through the surface layer, always from the higher temperature to the lower. Convection can be either natural or forced convection, depending on whether there is an external device, such as a fan or a pump, which forces the gas or cooling liquid to flow. In natural convection, heat transfer is caused by the gravity acting on the density variations near the boundary section of the two medium. Dissipated power due to convection can be expressed

T A h

P_conv = _c ∆ , (4.15)

Where A is the area, T is the temperature, and the coefficient hc is the convection heat transfer coefficient (also known as the boundary film coefficient), which depends on many variables, such as the shape and dimensions of the surface, flow characteristics, temperature, and material

characteristics of the fluid. In electrical machines, heat transfer by convection occurs mainly in three regions

•Convective heat transfer between the frame and the ambient air

•Convective heat transfer between the end windings and the end-region air

•Convective heat transfer between the stator or the rotor and the air gap

Typically, the frame of electric machines is equipped with cooling fins, which makes the calculation of convective heat transfer from the frame to the ambient very difficult, and this coefficient is usually determined by experimental tests. The convective heat transfer coefficient from the frame to the ambient can be found by running the motor continuously so that the steady-state temperature is reached, and by measuring the frame outer surface temperature and the total machine loss. Similarly, the accurate calculation of the convection heat transfer from the end windings is practically impossible, as the geometry of the end winding is too complex. The convective heat transfer from the end winding can also be determined experimentally.

Also the heat transfer from the rotor to the stator across the air gap is extremely difficult to model, due to the slotting of the stator (and sometimes also the rotor) that cause fluid disturbances in the air gap, the accurate modelling of which is practically impossible. Originally, Taylor (1935) solved the convective heat transfer between two rotating concentric smooth cylinders, but due to slotting, these equations cannot be directly applied. For example, Gazley (1958) suggests that the slotting increases the heat flow by approximately 10 % in the air gap. However, in most of the research reports, convective heat transfer across the air gap is solved based on Taylor’s work, with some modifications depending on the author.

Convective heat transfer coefficient in the air gap is obtained from Nusselt’s number by first defining Taylor’s and Prandtl’s numbers for the air gap. A dimensionless Taylor’s number is obtained from Couette Reynold’s number (for enclosed cylinder), defined as

µ δ ρ _circ

δ

Re = v , (4.16)

where ρ is the density of the fluid, vcirc the circumferential speed of the rotor, δ the radial air gap length, and µ the dynamic viscosity of the fluid. Now Taylor’s number can be obtained from

Re r

Ta δ

δ2

= , (4.17)

where r is the radius of the rotor. Equation for Prandtl’s number can be expressed as

air th

λ µc

Pr= , (4.18) where cth is the specific thermal capacity and λ the thermal conductivity of the fluid. As the peripheral speed of the rotor is low below rated speed (below 10.5 m/s), and the air gap is very

short (0.3 mm), Taylor’s number becomes low. This means that the air flow in the air gap can be assumed laminar, and according to Boglietti (2002), Nusselt’s number is then Nu = 2.0. If the air gap flow would be turbulent, the following equation given by Boglietti could be used

^0.27

Finally, a convective heat transfer coefficient in the air gap, hδ, can be calculated from

δ λ_air

δ Nu

h = . (4.20) The value for the thermal resistance used in the calculations that describes the convective heat transfer across the surface, can be calculated as

R Ah1

conv

th, = (4.21)

4.2.2 Conduction

Conduction is the mechanism that transfers the heat inside the solid medium. In general case, conduction can be described with the diffusion equation based on Fourier’s law



where T is the temperature, t the time, ρ the density, c the specific thermal capacity, λ the thermal conductivity, and p the generated power density. Eq. (4.22), however, cannot be directly applied, due to complex boundary value problems arising due to complex and unknown geometries in different machine parts. In thermal circuit modelling, heat transfer is solved in one dimension at a time, and thus Eq. (4.22) can be re-written in the x-direction to yield the heat flux Φth

1

and by introducing the thermal resistance Rth due to conduction

A

Heat transfer can thus be modelled in a manner that is analogous to Ohm’s law. This is very convenient, since commercial circuit simulators can then be applied to model the heat transfer.

4.2.3 Radiation

Heat flux dissipated by radiation can be expressed with Stefan-Boltzmann equation

(

)

1 SB

th T T

Φ =εσ − , (4.25)

where ε is the emissivity, σSB Stefan-Boltzmann’s constant (5.67×10-8 J/s/m²/K⁴), and T the temperature. Emissivity ε varies from 0 of a “transparent” material to 1 of a black body. In practice, however, emissivity for a painted black surface is typically around 0.9. In this analysis, radiation only from the frame to ambient is taken into account, and radiation for example inside the machine is neglected. As the emissivities for typical machine materials, such as copper and steel are low, in addition to small surface areas inside the machine, this assumption is justified, as the convective and conduction heat transfer dominate inside the machine. Thermal resistance of the radiation from the motor frame to the ambient can be calculated

(

)

frame SB

amb frame

th A T T

T R T

−

= −

εσ . (4.26)