Proceedings of the 2008 International Conference on Electrical Machines Paper ID 787

978-1-4244-1736-0/08/$25.00 ©2008 IEEE 1

Numerical Modelling of the Coolant Flow in a High-Speed Electrical Machine

Zlatko Kolondzovski

Helsinki University of Technology - Finland Department of Electrical Engineering E-mail: zlatko.kolondzovski@tkk.fi

Abstract-In the paper, a numerical modelling of the air-flow in

a high-speed permanent-magnet electrical machine is presented. The method couples computational fluid dynamics and heat-transfer equations in order to estimate the turbulent and thermal properties of the flow. The machine under consideration is a high-speed PM motor that is designed for speed n = 31500 rpm and power P = 130 kW. The geometry of the fluid domain in this method is considered to be 2D axi-symmetric. At the end, results for the turbulent properties as well as for the temperature rises of the cooling fluid are presented.

I. INTRODUCTION

The thermal design of a high-speed machine is a very demanding task because a well designed high-speed machine operates at high temperatures that are close to the critical temperatures of its components. The reason for this is in the very high loss density in a high-speed machine because its dimensions are much smaller than the dimensions of a conventional machine of the same power. The problem is more serious for the rotor because rotor cooling is more difficult than for stator since it can be performed only through the air gap. This mostly refers to a permanent-magnet (PM) high-speed machine because it has a more complex rotor construction and thermally it is more sensitive in comparison with a high-speed induction machine. The most sensitive components in the rotor of a high-speed PM machine are the permanent magnets as an overheating can cause their demagnetization. An additional problem arises because the carbon-fibre sleeve which retains the magnets against the centrifugal forces is a very bad thermal conductor and makes the cooling of the magnets very difficult. The carbon fibre sleeve is also a thermally sensitive component on the rotor. Because of the very high surface speed, the outer sleeve surface generates a significant amount of air-friction losses in the air gap that can increase the temperature and cause overheating of the sleeve. If the stator is considered, a thermally sensitive part in it is the stator winding according to its class of insulation. The aforementioned constraints lead to the conclusion that an extensive and reliable thermal analysis for a high-speed PM machine should be performed. This kind of analysis must be performed during the design process of the machine since one should be sure that the critical temperatures in all machine parts will never be exceeded in order to achieve a long lasting life of the machine.

There are many references in which thermal analyses of electrical machines have been elaborated but they are mainly based on traditional analytical and empirical methods. An extensive description of a thermal-network model that is intended for the analysis of conventional induction motors is given in [1]. Thermal-network models for high-speed induction motors are elaborated in [2] and [3]. These models are implemented for determination of the temperature rises in two types of high-speed induction machines and they are successfully validated with experimental results. In [4], a thermal analysis of a high-speed PM machine is reported. Like in the previous methods this analysis also uses a thermal- network model based on traditional analytical and empirical approaches. Although these thermal network-methods are quite reliable and give good agreement with measured results, they have some disadvantages. The coarser is the thermal network the rougher is the thermal view of the machine. It means that these methods give only the mean temperatures of the machine parts but not a detailed temperature distribution in the whole domain. The determination of the coefficients of thermal convection over a cooling surface in these methods is usually based on empirical equations that give only the mean values of the coefficients but not their exact local values.

The developed numerical techniques today give the opportunity for advanced thermal designs of high-speed PM electrical machines. Instead of using a rough empirical estimation of the mean convection, using the computational fluid dynamics (CFD), a more realistic modelling of the turbulent flow in the machine can be done, which leads to an advanced estimation of the heat transfer by convection from the machine parts to the cooling fluid. There is a lack of references in which the numerical estimation of temperature distribution in an electrical machine is coupled with numerical estimation of the turbulent flow of the cooling fluid. In [5], a numerical thermal analysis of a conventional induction machine is done but the coefficients of thermal convection are calculated using empirical equations. In [6] and [7], an extensive numerical CFD approach for estimation of the turbulent-fluid properties in the air gap of a high-speed induction machine is reported. The temperature rise of the fluid and the local coefficient of thermal convection in the air gap are determined and experimentally validated, but the temperatures of stator and rotor outer surfaces are accepted as

Proceedings of the 2008 International Conference on Electrical Machines

constant values without estimation of the temperature distribution in the solid domain of the machine.

In this paper, a coupled CFD heat-transfer analysis of the coolant properties in a high-speed PM machine is presented. The analysis is performed using the COMSOL Multiphysics® commercial software. The CFD and heat-transfer modelling is performed simultaneously using a 2D axi-symmetric model of the whole machine domain but the results presented in this paper give an emphasis on the fluid domain of the machine. The machine under consideration is a high-speed PM motor that is designed for speed n = 31500 rpm and power P = 130 kW intended for a compressor application. The permanent magnets are mounted on an aluminium cage that shields the magnets from eddy-current heat generation. A carbon-fibre composite material retains the magnets and the aluminium cage against the huge centrifugal forces that arise during the high-speed operation.

II. METHOD

A multiphysics coupled CFD heat-transfer model is created for a simultaneous estimation of the thermal and turbulent properties of the fluid in the machine. The model is 2D axi-symmetric one. The air flow in the high-speed PM machine is always turbulent. This is desired since in the case of turbulent flow the heat extraction due to convection is much more effective than in laminar flow [8]. The turbulent model of the flow is performed using COMSOL Multiphysics® and the complete CFD analysis of this method is elaborated in [9]. The flow of an incompressible fluid is described by the Reynolds Averaged Navier-Stokes Equations (RANS)

( )ρ η ρ ρ ' '

0

Pt

∂ − ∇ ⋅∇ + ⋅∇ + ∇ + ∇ ⊗ =∂

∇ ⋅ =

U U U U u u F

U

(1)

Here η denotes the dynamic viscosity, U is the averaged velocity field, u is the velocity vector, ρ is the density of the fluid, P is the pressure, and F is the volumetric force vector. The last term on the left-hand side in the first equation represents fluctuations around a mean flow and it is called the Reynolds stress tensor. The derivation of the governing equation for the Reynolds stress tensor in a general 3D case will introduce six additional equations and six additional unknowns. This will not solve the flow problems since the resulting equations will contain additional unknowns, emanating from higher order statistics. This could be solved by involving assumptions about the flow that is often called the closure of a turbulence model. One of the most used turbulence models is the κ-ε model and it is used for the CFD analysis in this paper. This model gives a closure to the system and results in the following equations for the conservation of momentum and continuity

( )( )2

μκρ η ρ ρε

0

TC Pt

⎡ ⎤⎛ ⎞∂ − ∇⋅ + ⋅ ∇ + ∇ + ⋅∇ + ∇ =⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦∇⋅ =

U U U U U F

U

(2)

The two new variables in this equation are the turbulent kinetic energy κ and the dissipation rate of the turbulent energy ε. Two extra equations for κ and ε are solved for these two introduced variables

( )( )

( )( )

2μ

κ

2 2

μ

2μ

ε

22

ε1 ε2

κ κρ η ρ κ ρ κσ ε

κρ ρε2ε

ε κρ η ρ ε ρ εσ ε

κ ερ ρ2 κ

T

T

Ct

C

Ct

C C

⎡ ⎤⎛ ⎞∂ − ∇ ⋅ + ∇ + ⋅ ∇ =⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦

∇ + ∇ −

⎡ ⎤⎛ ⎞∂ − ∇ ⋅ + ∇ + ⋅ ∇ =⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦

∇ + ∇ −

U

U U

U

U U

(3)

The values of the model constants are: Cμ = 0.09, Cε1 = 1.44, Cε2 = 1.92, σκ = 0.9 and σε = 1.3. They are determined from experimental data. The κ-ε turbulence model gives an isotropic turbulence, which is turbulence constant in all directions. However, close to solid walls the fluctuations in the turbulence vary greatly in magnitude and direction so in these places the turbulence cannot be considered as an isotropic one. The most convenient approach for modelling the properties of the thin boundary layer near the solid wall is by using an empirical relation between the values of velocity and wall friction. This approach is implemented and it has the best accuracy for high Reynolds numbers and situations in which pressure variations along the wall are not very large. If we assume a flow parallel to a solid wall, there is one velocity component U parallel to the wall, which is a function of y, the coordinate perpendicular to the wall. Another assumption in this approach is that the flow close to the wall is not influenced by the flow in a region far from the wall. The complete set of boundary conditions for the solid wall is elaborated in [9]. The following boundary conditions for κ and ε are obtained

2 3τ τ

μ

κ ; εa

u uk yC

= = (4)

where uτ is the friction velocity and ka ≈ 0.42 is the Kármán’s constant.

Besides the turbulent fluid flow, the application presented in this paper involves a thermal interaction of the flow and solid objects. In fact, when cooling the machine, heat is transferred from the hotter rotor and stator surfaces to the cooler turbulent fluid. In this case, a multiphysics coupling between the κ-ε turbulence equations and heat-transfer equations is performed. The thermal conductivity of the fluid is automatically corrected to take into account the effect of mixing due to eddies. The turbulence results in an effective thermal conductivity, keff, according to the equations

;eff c T T p Tk k k k C η= + = (5) Here kc is the physical thermal conductivity of the fluid, kT is the turbulent conductivity, ηT denotes the turbulent cinematic viscosity and Cp is the heat capacity. On the boundaries between the fluid and solid domains, the correct amount of heat flux across the boundary should be calculated. An vV

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Proceedings of the 2008 International Conference on Electrical Machines

algebraic relationship describes the momentum transfer at the solid-fluid interface. This means that the modelled fluid domain ends at the top of the laminar boundary layer where the fluid experiences a significant wall-tangential velocity. Similar to the fluid velocity, the temperature is not modelled in the laminar sub-layer. According to [9], instead of assuming a continuity of the temperature across the layer, there will be a jump in the temperature from the solid surface to the fluid due to the omitted laminar sub-layer. The velocity, pressure and temperature fields are simultaneously solved in the presented multiphysics model.

A disadvantage of the κ-ε turbulence model is that it shows poor results in the description of rotating flows [9]. This disadvantage can influence the accuracy of the CFD modelling of the turbulent flow in the air gap because the rotor accelerates the fluid to a tangential movement. A good circumstance in a high-speed machine is that the rotational movement of the fluid is undeveloped [6]. Using the equations in [3] and [6] the rotating speed of the fluid can be calculated according to the surface speed of the rotor. Further, taking into account that rotating speed together with the axial speed of the fluid and the axial length of the machine, it is easy to calculate that the fluid makes less than one rotation when passing in the machine. This means that the fluid does not have a developed rotational movement in the air gap and the κ-ε turbulence model can be implemented also in this part of the machine. In the 2D axy-symmetric model, the total speed of the fluid that represents a superposition of its axial and tangential components is taken into account. The finite-element mesh of the 2D geometry is presented in Fig. 1. The mesh is refined on the boundaries between the solid and fluid domains. It is also refined close to the sharp corners.

III. RESULTS

The results that are shown here are related to the steady state performance of the high-speed PM machine when its rotational speed is 31500 rpm and the mechanical power on the shaft is 130 kW. The machine operates as an electric motor intended to drive a high-speed compressor and the whole air for supplying the compressor previously passes as a coolant in the electrical machine. When it flows through the machine it is distributed into two parts. One small part of the coolant flows in the air gap extracting the heat which is generated in the rotor and some amount of heat generated in the stator. This flow is estimated to q1 = 0.05 m3/s. The main part of the coolant flows between the stator yoke and the frame of the machine and its amount is q2 = 2 m3/s.

The velocity field of the coolant in the whole fluid domain of the electrical machine is presented in Fig. 2. The inlet side of the machine is down and the outlet side is up. More precise presentation of the fluid velocity along the air gap is given in Fig. 3. The results of the fluid velocity refer to the middle line of the air gap. One can notice that when the fluid is entering the narrow air gap, its velocity is dropping at the beginning and after that it has the tendency to rise in the axial direction. The

velocity drop at the beginning of the air gap can be explained due to the interaction between the fluid and the sharp corners of the rotor and stator. The moderate velocity rise in axial direction is due to the temperature rise and decreasing of the air density [6] that leads to increased volume of the fluid that should be blown through the air gap. The velocity profile of the fluid in the air gap is presented in Fig. 4. The figure represents the values of the fluid velocity versus the radial length of the air gap in a position equal to half of the axial length of the air gap. The fluid velocity is lower close to the solid boundaries of the rotor (0 p.u. of radial length) and stator (1 p.u. of the radial length) due to the friction between the fluid and the solid surfaces. However, the velocity of the fluid on the boundary is not close to zero because the thin laminar sub-layer is omitted in the analysis.

Fig. 1. Finite element mesh of the 2D axi-symmetric geometry of the high-speed PM machine.

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Proceedings of the 2008 International Conference on Electrical Machines

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Besides the velocity field of the coolant, another important parameter that is obtained with this method is the coefficient of thermal convection on each surface between the solid and fluid domains. The local value of this coefficient is calculated in the postprocessor using the equation

"local

w f

qhT T

=−

(6)

Here q” is the local heat flux, Tw is the local wall temperature of the solid domain and Tf is the local average temperature of the fluid next to the wall. The local heat-transfer coefficient is presented as a function of a position along the flow of the fluid. As an example, the local coefficient of thermal convection for the rotor surface in the air gap is presented in Fig. 5. In this way, all local heat-transfer coefficients of convection over all the outer surfaces of the solid domain in the machine can be calculated.

Fig. 5. Local heat-transfer coefficient of convection h [W/(mm2·K)] of therotor surface in the air gap as a function of the axial length of the rotor. 5 5

2 s

j

j sjj

4 Fig. 4. Velocity profile of the fluid in the air gap of the electrical machine.The result is for the position which is half (0.5 p.u.) of the axial length of theair gap.

Fig. 3. Velocity v [m/s] of the fluid along the air gap of the electricalmachine.

Fig. 2. Velocity field v [m/s] of the coolant flow.

Proceedings of the 2008 International Conference on Electrical Machines

Finally, the temperature distribution in the fluid domain of the machine can be determined. The main sources of heat in the machine are the losses. The electromagnetic losses are calculated using the Finite Element Method (FEM). The procedure of their calculation is given in [10]. Another type of losses are the mechanical losses caused from the fluid flow and they are usually divided into air-friction and cooling losses. The procedure of their determination is described in [2]. The distribution of the temperature rise in the whole fluid domain of the machine is presented in Fig. 6. One can conclude that the temperature rise of the fluid cannot be neglected since it extracts a huge amount of heat due to the very high loss density which is typical for the high-speed electrical machines. In Fig. 7 is presented the temperature rise of the coolant flow when is passing between the rotor and stator parts of the machine. The temperature of the coolant part which is passing between the rotor and stator is increased for more than 20 K. This significant temperature rise of the fluid usually leads to higher temperatures of the solid parts of the machine that are close to the outlet side.

IV. CONCLUSION

The thermal design of a high-speed PM machine is a very demanding task because due to the decreased size of the machine the density of the losses is very high and some parts can be heated very close to their critical temperatures. The rotor is the most critical part of the machine because it has thermally sensitive components like the permanent magnets and the carbon-fibre sleeve and its cooling is more difficult than the cooling of the stator. Using the 2D coupled CFD heat-transfer method, the turbulent and thermal properties of the flow such as the velocity field, the coefficients of thermal convection and temperature rise of the fluid are estimated. The temperature rise of the coolant in the air gap is very significant and cannot be neglected in a high speed PM machine. It contributes to unequal temperature distribution in the machine, so the machine parts close to the outlet side of the machine are more heated than the parts located close to the inlet side.

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for electrical machines of TEFC design”, IEE Proceedings-B , vol. 138, No. 5, pp. 205-218, 1991.

[2] J. Saari, Thermal Modelling of High-Speed Induction Machines, Acta Polytechnica Scandinavica, Helsinki, 1995.

[3] J. Saari, Thermal Analysis of High-Speed Induction Machines, Acta Polytechnica Scandinavica, Espoo, 1998.

[4] O. Aglén, Å. Anderson, “Thermal analysis of a high-speed generator”, Conference record of the 38th IAS annual meeting, pp. 547-554, 2003.

[5] Y. Huai, R.V.N. Melnik, P.B. Thogersen, “Computational analysis of temperature rise phenomena in electric induction motors”, Journal of Applied Thermal Engineering, vol. 23, No.7, pp. 779-795, 2003.

[6] M. Kuosa, P. Sallinen, J. Larjola, “Numerical and experimental modelling of gas flow and heat transfer in the air gap of an electric machine”, Journal of Thermal Science, vol.13, No.3, pp. 264-278, 2004.

[7] M. Kuosa, P.Sallinen, A. Reunanen, J. Backman, J. Larjola, L. Koskelainen, “Numerical and experimental modelling of gas flow and heat transfer in the air gap of an electric machine. PartII: Grooved surfaces”, Journal of Thermal Science, Vol.14, No.1, pp. 48-55. 2005.

[8] F.P. Incropera, D.P. DeWitt, Fundamentals of Heat and Mass Transfer, 3rd ed., John Wiley and sons, USA, 1990.

[9] Comsol Multiphysics, Heat Transfer Module, User’s Guide, Version 3.3, 2006.

[10] A. Arkkio, Analysis of Induction Motors Based on the Numerical Solution of the Magnetic Field and Circuit Equations, Acta Polytechnica Scandinavica, Helsinki, 1987.

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f

5Fig. 7. Temperature rise ΔT [K] of the coolant flow in axial direction when passing between the rotor and stator parts of the machine.

Fig. 6. Distribution of the temperature rise ΔT [K] in the fluid domain of theelectrical machine.