International Journal of Thermal Sciences

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1 International Journal of Thermal Sciences 49 (00) 73e74 Contents lists available at ScienceDirect International Journal of Thermal Sciences journal homepage: Experimental and CFD investigation of a lumped parameter thermal model of a single-sided, slotted axial flux generator C.H. Lim a, *, G. Airoldi a, J.R. Bumby a, R.G. Dominy a, G.I. Ingram a, K. Mahkamov a, N.L. Brown b, A. Mebarki b, M. Shanel b a Durham University School of Engineering and Computing Sciences, South Road, Durham DH 3LE, UK b Cummins Generator Technologies, Barnack Road, Stamford, Lincolnshire PE9 NB, UK article info abstract Article history: Received 0 July 009 Received in revised form 30 March 00 Accepted 30 March 00 Available online 6 May 00 Keywords: Axial flux generator Computational fluid dynamics (CFDs) Conduction thermal modelling Convection thermal modelling Lumped parameter model (LPM) Thermal network Thermal modelling A two dimensional lumped parameter model (LPM) which provides the steady state solution of temperatures within axisymmetric single-sided, slotted axial flux generators is presented in this paper. The two dimensional model refers to the heat modelling in the radial and axial directions. The heat flow in the circumferential direction is neglected. In this modelling method, the solid components and the internal air flow domain of the axial flux machine are split into a number of interacting control volumes. Subsequently, each of these control volumes is represented by thermal resistances and capacitances to form a two dimensional axisymmetric LPM thermal circuit. Both conductive and convective heat transfers are taken into consideration in the LPM thermal circuit by using annular conductive and convective thermal circuits respectively. In addition, the thermal circuit is formulated out of purely dimensional information and constant thermal coefficients. Thus, it can be easily adapted to a range of machine sizes. CFD thermal modelling and experimental testing are conducted to validate the temperatures predicted from the LPM thermal circuit. It is shown that the LPM thermal circuit is capable of predicting the surface temperature accurately and potentially replacing the CFD modelling in the axial flux machine rapid design process. Ó 00 Elsevier Masson SAS. All rights reserved.. Introduction There are several general purpose advanced computational fluid dynamic (CFD) codes commercially available in the market, e.g. ANSYS CFX and FLUENT. These CFD packages use the most modern solution technology and extremely efficient parallelization algorithms to perform - and 3-dimensional mass transfer and thermal modelling of internal and external flow systems. CFD modelling methods have been used extensively in the past decade, especially by electrical machine manufacturers to perform the thermal analysis of electrical machines [], cooling and air ventilation modelling [], and the thermal management of AC electrical motors [3]. However, sophisticated CFD modelling involves complicated and time consuming processes, including the geometrical meshing and the iterative calculation processes. Depending on the application and required accuracy, some of the complex models consume up to months of simulation time to obtain accurate numerical solutions. This makes it difficult to use advanced CFD techniques for machine rapid design, optimisation and what-if analyses. * Corresponding author. Tel.: þ44 (0) address: c.h.lim@durham.ac.uk (C.H. Lim). A feasible alternative to CFD modelling for the thermal modelling of electrical machines is the application of advanced lumped parameter modelling (LPM) methods. Instead of solving the heat conduction (Fourier) and convective heat transfer (Newton) equations analytically to simulate the temperature distribution inside the generator [4], in the LPM approach described in [5,6], the electrical machine is split into a number of lumped components (or control volumes), which are connected to each other in the calculation scheme through thermal impedances to form a thermal circuit. Several thermal circuits of electrical machines, such as induction motors [7], radial flux [8] and stationary axial flux generators [9] have been proposed. These papers demonstrate that the results of LPM modelling are in good agreement with experimental data. Similar research was conducted by using a commercially available LPM thermal modelling tool [0e], namely Motor-CAD [3]. One of the shortcomings of the LPM methods that were employed in previous works [4e,4] is that the air temperature variations in the electrical machines were neglected. The air temperature variations are crucial for thermal modelling of electrical machines, especially for axial flux permanent magnet (AFPM) machines, which typically have narrow and long flow passages and relatively high velocities of the cooling air as shown Fig.. Thevariationinthe temperature of the air flow affects the thermal state of the solid /$ e see front matter Ó 00 Elsevier Masson SAS. All rights reserved. doi:0.06/j.ijthermalsci

2 C.H. Lim et al. / International Journal of Thermal Sciences 49 (00) 73e Nomenclature A cross-sectional area, m A v surface area of convective heat transfer, m C thermal capacitance, J/K c p specific heat capacity, J/kg K h convective heat transfer coefficient, W/m K h f average heat transfer coefficient of the rotor surface of free rotating disk, W/m K h p average heat transfer coefficient of the rotor peripheral edge, W/m K h rs average heat transfer coefficient of the flow passage between rotor and stator, W/m K HF CFD&LPM heat fluxes for CFD and LPM models, W/m HF electrical input heat fluxes due to electrical heater mat, W/m HF HF3 heat flux measured from heat flux sensor HF3, W/m k thermal conductivity, W/m K k a axial thermal conductivity, W/m K k r radial thermal conductivity, W/m K L length, m m mass, kg _m mass flow rate, kg/s q conv convection heat transfer, W Q volumetric flow rate, m 3 /s r r annulus ring outer radius, m annulus ring inner radius, m R electrical resistance, Ohm R a, R a and R a3 axial network resistances, K/W R r, R r and R r3 radial network resistances, K/W R c convection resistance, K/W R d conduction resistance, K/W R t thermal resistance, K/W T surf surface temperature, K T in inlet temperature, K T out outlet temperature, K T m annulus mean temperature, K V volume, m 3 ΔV voltage difference, V v average air velocity, m/s r density, kg/m 3 Dimensionless group Nu f Nusselt number on rotor surface of free rotating disk Nu p Nusselt number on the rotor peripheral edge Nu rs Nusselt number in the flow passage between rotor and stator Re rotor surface Reynolds number Re D rotor peripheral edge Reynolds number components of these generators. Lim et al. [5] proposed a new technique for constructing a D equivalent thermal circuit for AFPM generators which takes into account the temperature change in the air flow. In this approach the variation in the air flow temperature is taken into account by introducing additional thermal circuits to model the heat transfer between solid components and the air flow. However, accurate convective heat transfer coefficients are required to complete the new convective thermal circuit. Although the heat transfer in complex flow regimes has been investigated [6e], no suitable correlation was found. All of the correlations examined used the ambient temperature as the reference temperature to evaluate the heat transfer coefficient, but in order to be applicable to the new convective thermal circuit, correlations of heat transfer coefficient that are based on local bulk air temperature are needed. This paper describes the construction of a D equivalent thermal circuit of the single-sided slotted generator, as depicted in Fig.. A CFD model of the single-sided slotted generator is constructed to provide heat transfer coefficients for the thermal circuit. Experiments were conducted to validate the temperature predicted using the proposed D thermal circuit.. Theory: D lumped parameter thermal circuit The lumped parameter model works by splitting an electrical machine into a number of lumped components, and representing these lumped components by collections of thermal impedances and capacitances. Subsequently, these collections of thermal impedances and capacitances are interconnected together to form the thermal equivalent circuit by considering the heat flow paths in the electrical machines. Both steady and transient temperatures of the generator components and the local air temperatures can be obtained by considering these equivalent thermal circuits. The equivalent thermal circuit is an analogy of the electrical circuit. The heat, q (W), temperature difference, ΔT (K) and thermal resistance, R t (K/W) in the thermal equivalent circuit corresponds to the current, I (Amp), voltage difference, ΔV (V) and electrical resistance, R (Ohm) in the electrical circuit, respectively. However, the thermal resistances in the equivalent thermal circuit are defined differently from the electrical resistance in the electrical circuit. For conduction, the thermal resistance depends on the thermal conductivity of the material, k, the length, L, the cross-sectional area, A, of the heat flow path and is expressed as: R d ¼ L=Ak: () For convection, the thermal resistance is defined as: R c ¼ =A v h; () where A v is the surface area of convective heat transfer between two domains and h is the convective heat transfer coefficient. Since the temperature differences between the surfaces inside the generators are small, the heat transfer by radiation was ignored in the LPM. For transient analysis, in order to account for the variation in the internal energy of the machine components with time, thermal capacitances are introduced: C ¼ rvc p ¼ mc p ; (3) where c p is the specific heat capacity of the component material, r is the density and V and m are the part s volume and mass, respectively. Heat sources in the generator are represented as current sources in the thermal equivalent circuit. The temperature dependent resistive generator winding is taken into account during the simulation. Therefore, the magnitudes of the heat sources increase as the generator winding temperature increases. The proposed thermal equivalent thermal circuit is a combination of conductive and convective thermal circuits. Both conduction and convection heat transfers in the machine are modelled in the thermal equivalent thermal circuit. Both circuits are interconnected. The energy balance equations of these circuits are solved interactively and iteratively, to estimate temperatures in all solid and air control volumes for both steady and transient conditions.

3 734 C.H. Lim et al. / International Journal of Thermal Sciences 49 (00) 73e74 Fig.. The axial flux permanent magnet generator... Conductive thermal circuit: annular conductive circuit The solid parts of the single-sided slotted axial flux machine are split into a number of annuli control volumes. The heat transfers in the annulus ring are represented by two separate three-terminal circuits, one for the axial thermal network and the other for the radial thermal network. In each network, two of the terminals indicate the surface temperatures of the annulus components whilst the third represents the mean temperature of the annulus component. For example in Fig. 3, T and T represent the front and back surface temperatures of the annulus ring; T 3 and T 4 represent its inner and outer peripheral surface temperatures and T m is the mean temperature. The central nodes of the axial and radial thermal circuits are shown in Fig. 3 and the internal heat generation and thermal storage are introduced to the circuit at the mean temperature node. Fig.. Single-sided slotted generator.

4 C.H. Lim et al. / International Journal of Thermal Sciences 49 (00) 73e Fig. 3. D conductive thermal circuit for annulus. (a) Annulus ring. (b) Thermal network of annulus ring. The individual thermal resistances shown in Fig. 3(b) are calculated in terms of the physical dimensions and the thermal properties of the materials as described in equations (4)e(9). These equations are all derived from Fourier s heat conduction equations and the details of their derivation can be found in [5] which were based on the following assumptions: The radial and axial heat flows are independent of each other. The lumped component mean temperatures predicted from the radial and axial thermal circuits are assumed to be equal to each other. Conductive heat fluxes in the machine are assumed to be in the axial and radial directions only. There is no heat flow in the circumferential direction. The thermal capacities and heat generation are uniformly distributed in the volume. When superimposing the internal heat generation term at the mean temperature node, T m, of the axial and radial thermal circuits, it lowers the temperatures at the central node of those thermal circuits. This is reflected in the network by the negative values of the interconnecting resistances, R a3 and R r3, see equations (6) and (9). L R a ¼ pk a r r ; (4) L R a ¼ pk a r r ; (5) L R a3 ¼ 6pk a r r ; (6) " R r ¼ r ln r # r 4pk r L r r ; (7) " r r R r ¼ ln # r 4pk r L r r ; (8) 4r R r3 ¼ "r 8p r r kr L þ r r ln r # r r r : (9) The single-sided slotted generator can be split into a number of annulus parts with different radii and thickness, including the generator windings and magnets. The D conductive thermal network of the single-sided slotted generator can then be constructed by connecting these annulus conduction circuits together in the same manner as they physically interact inside the machine... Convective thermal circuit The convective thermal circuit is used to take into account the convective heat transfer, to determine the fluid temperature variation in different locations and to relate the total convective heat transfer to the change in the fluid temperature. For most air-cooled axial flux machines, the convective heat transfer mechanism dominates within the generator. Hence it is important to develop a circuit which compliments the conductive circuit to provide more accurate temperature prediction in a rotating electrical machine. Firstly, the convective thermal circuit splits the air domain inside the single-sided slotted generator into a number of fluid control volumes and each fluid control volume is represented as a separate circuit. Subsequently, the air temperature at the outlet of each control volume is predicted by solving the energy conservation equations. Fig. 4 shows a rectangular control volume with fluid passing through at a constant mass flow rate, _m. In this control volume, the fluid is heated from the bottom wall which is maintained at a constant temperature, T surf,andthefluid s temperature rises from T in to T out. Assuming that the upper wall is fully insulated, the energy conservation equation for this control volume can be expressed as: haðt surf T air bulk Þ¼q conv ¼ _mc p ðt out T in Þ: (0) Assuming that the change in the air temperature across the control volume is small, then T air-bulk can be simplified as T in, hence equation (0) can be rewritten as: haðt surf T in Þ¼q conv ¼ _mc p ðt out T in Þ: () mc p T in Fluid Inlet Insulated Wall (Heat Flux =0) mc p Tout Fluid Outlet T surf q conv Heated Wall Fig. 4. Fluid flow through a control volume.

5 736 C.H. Lim et al. / International Journal of Thermal Sciences 49 (00) 73e74 In order to calculate T out from equation (), it is necessary to find the heat transferred from the wall, q conv. On the other hand, q conv can be calculated by the Newton convection heat transfer equation, as shown in the first two terms of equation (). By rearranging the Newton convection heat transfer equation, q conv can be represented by equation (), q conv ¼ ðt surf T in Þ : () ha Furthermore, equation () can be represented by an electrical circuit as shown in Fig. 5, where q conv is analogous to the current; (T surf T in ) is analogous to the voltage difference and /ha is analogous to electrical resistance. By connecting the T surf node to the solid temperature sources, or to the temperature nodes from other conduction thermal circuits described in Section., q conv can be calculated by measuring the heat across the heat meter in the convective thermal circuit shown in Fig. 5. To obtain the temperature at the outlet of the control volume, T out, equation () is rearranged so that the fluid temperature at the outlet, T out, is written in terms of q conv and the known boundary conditions at the inlet, such as T in and c p. With q conv calculated from the convective thermal circuit in Fig. 5, T out can be predicted as: T out ¼ q conv _mc p þ T in : (3) Once computed, the predicted T out of the air control volume is passed to the neighboring air control volume. Similarly, the outlet temperature in the second air control volume is predicted using its corresponding equivalent convective thermal circuit and equation (3). By using this process, all the temperatures from the system inlet to the last air control volume at the outlet are determined. The calculation is run iteratively until the solution reaches convergence. The accuracy of the convective equivalent thermal circuit depends on the assumption made to derive equation (), i.e. that the change in temperature between the inlet and outlet is small. To improve the accuracy of the convective thermal circuit, it is necessary to discretise the air domain into finer control volumes to reduce the temperature change across it. Model discretisation studies of lumped parameter models are described in reference [5], and the authors conclude that splitting their model into three control volumes in the radial direction in their axial flux machine is sufficient to provide good results. Therefore, the air domain in the axial flux machine is discretised into three control volumes in radial direction..3. The construction of the single-sided slotted generator D lumped parameter thermal circuit The LPM thermal circuit of the simplified single-sided slotted generator is constructed as shown in Fig. 6. The simplified singlesided slotted generator consists of a rotor disk (on the left hand side) and a stator disk (on the right hand side); each of them is split in to four and three annular control volumes respectively. The annular control volumes are represented by the annular conductive circuit mentioned and they are connected at the axial and radial temperature terminals in the same way that they are physically connected in the real machine. The thermal resistances of the conductive thermal circuit are calculated by using equations (4)e(9), based on the geometry and material properties of each annular control volume. Single and double-sided slotted axial flux machines are often designed with very thin magnets protruding from the rotor disk surface. Typically, the magnet grooves range between and 4 mm. Furthermore, some commercial axial flux machine designs have magnets flush with the rotor disk for magnetic flux optimisation. Hence, in this analysis, the magnets on the rotor disk are omitted. The air domain inside the generator is split into four control volumes as shown in Fig. 6(a). The convective thermal circuit is connected to the conduction circuit to allow heat transfer from the air to the solid or vice versa. The temperature dependent Joule heating loss in the winding on the slotted stator is modelled by an independent transient heat source in the LPM thermal circuit. The accuracy of the temperature prediction of the LPM is closely related to the convective heat transfer coefficients used in the model. Nevertheless, accurately determining convection heat transfer coefficients is difficult due to the complexity of the flow regimes and it involves extensive theoretical and experimental explorations. In this LPM model, the convective heat transfers are evaluated by a number of existing correlations [] based on the flow characteristic in the axial flux machine..3.. Free rotating dics The average heat transfer coefficient on the left hand side of the rotor surface is developed using the formula developed for a combination of laminar and turbulent flow of a free rotating plate [3], which is, h f ¼ k r Nu f ; (4) where Nu f ¼ 0:05Re 4=5 u 00 r c ¼ In equation (5) rc r : (5) :5 0 5 v ; (6) u Fig. 5. Convective thermal circuit for control volume. where r c is the radius at which the transition occurs from laminar flow to turbulent flow, m; v is the fluid kinematic viscosity, m /s; u is the rotational speed, rad/s; r is the disk outer radius, m; Re u is the rotational Reynolds number, which is defined as Re u ¼ ur =v; k is the air thermal conductivity, W/m K.

6 C.H. Lim et al. / International Journal of Thermal Sciences 49 (00) 73e Fig. 6. (a) Simplified single-sided slotted axial flux generator and (b) the corresponding lumped parameter thermal circuit. Considering the single slotted axial flux generator described in Fig. 6(a), with an outer radius, r c, of 0.5 m, rotational speed, u, of 495 rpm (or 56.5 rad/s) and air kinematic viscosity and thermal conductivity of m /s and 0.07 W/m K respectively, the convection heat transfer coefficient on the rotor side surface, h f, calculated from equations (4)e(6) is 6.83 W/m K..3.. Rotor peripheral edge The heat transfer coefficients for the radial periphery of the rotor disk are similar to the rotating cylinder in air. Hence the average heat transfer coefficient is given as []: h p ¼ k D Nu p; (7) where Nu p ¼ 0:33Re =3 D Pr=3 : (8) In equation (8) Re D ¼ ud v ; (9) where D is the rotor disk diameter, m, Pr is the air Prandtl s Number. Subsequently, when D ¼ 0.30 m, Pr ¼ 0.7 and u ¼ 56.5 rad/s, the average convection heat transfer coefficient at the peripheral edge of the rotor disk, h p, is found to be 94.7 W/m K Flow passage between the rotor and the stator Owen [] provided an approximation solution for the flow between a rotating and a stationary disk, which relates the statorside average Nusselt number to the volumetric flow rate, Q (m 3 /s) by the following equation: h rs ¼ k r Nu rs; (0) where Nu rs ¼ 0:333Q : () pyr Since no mass flow correlation has been developed for the single slotted axial flux generator, the mass flow was determined from the experiments and found to be equal to 3.6 g/s. This is used to calculate an average stator-side heat transfer coefficient, h rs of.63 W/m K was obtained from equations (0) and ().Also,Wang [] suggests that the convection heat transfer coefficient on the rotating disk can be assumed to be the same as on the stator-side. In this semi-empirical lumped parameter thermal circuit, experimental results are required to evaluate the mass flow rates and heat transfer coefficients in the electrical machines. In the future, sophisticated parametric variation studies of convective heat transfer coefficient and mass flow rate will be performed to develop the empirical correlations that relate the heat transfer coefficient and mass flow rate with different flow conditions and geometrical parameters. Hence, the LPM will be able to provide accurate temperature values inside the AFPM generators independently from the numerical models or CFD model. 3. The single-sided slotted generator CFD model An axisymmetric CFD model of the simplified single-sided slotted generator was constructed and simulated in FLUENT package. In this model, only the heat and air flows in the axial and radial directions are simulated. The heat flux and air flow rate in circumferential direction is assumed to be constant. The CFD model mesh grid shown in Fig. 7(b) consists of 40,000 nodes. The additional air volumes at the inlet and outlet are modelled to eliminate the boundary interference in the CFD model. On a modestly powered desktop computer (.773 GHz Core Duo Intel processor, GB RAM machine), the meshing process and iterative calculation of the CFD modelling took up to 9 h of computational time. The input data and the boundary conditions applied in the CFD model are as follows: 0.3 m outer diameter and 0.0 m thick rotor disk. 0.3 m outer diameter, 0.07 m inner diameter and m thick stator disk. 3 mm rotorestator clearance. 5 C ambient temperature. Boundary layers are used in the near wall region.

7 738 C.H. Lim et al. / International Journal of Thermal Sciences 49 (00) 73e74 Fig. 7. (a) The schematic plan of the simplified single-sided slotted axial flux generator and (b) corresponding CFD mesh model. The realisable k-epsilon turbulent model with Enhanced Wall Treatment (EWT) is used to model the turbulence in the flow. EWT is a near wall modelling method that is used in turbulent modelling. The details of EWT can be found in [4]. Zero total pressure and zero static pressure conditions are specified at the inlet and outlet of the simplified AFPM generator respectively. The rotor parts are identified to have a rotational speed of 495 rpm with the use of a rotating reference plane. 553 W/m heat flux input is specified at the back of the stator to model the winding Joule losses. The heat flux specified in CFD model is obtained from the experimental results so that the temperatures simulated can be compared with the temperatures measured from the test rig. 4. The experimental simplified single-sided slotted axial flux generator The purpose of commencing the experiments was to provide sufficient heat transfer data to validate the temperatures predicted by the LPM thermal circuit and to give confidence in the use of the lumped parameter modelling method to develop better cooling arrangement for other axial flux machines, e.g. double-sided slotted axial flux machines. An experimental rig was constructed as shown in Fig. 8. The experimental rig was configured to measure the air flow rate, the local heat transfer coefficients, and the temperatures and heat fluxes on the stator surface. The test rig replicates the slotted windings of the single-sided axial flux machine by using six, 00 mm diameter, 8 W heater mats and the rotor disk by a 0 mm thick Perspex disk. The magnets on the single-sided axial flux machine were omitted in this test rig. In addition, a cylindrical duct of length 300 mm was welded onto the upstream side of the stator, allowing air mass flow measurements to be made by using the hot wire anemometer. The geometrical dimensions of the experimental rig are shown in Fig. 8(c). Since the main purpose of conducting the experiment was to validate the application of the convective thermal circuit in the LPM thermal circuit, the thermal circuit and CFD model were constructed so that they matched the experimental rig dimensions shown in Fig. 8(c). A hot wire anemometer was used to measure the axial flow velocity passing through the simplified single-sided slotted generator. The inlet duct cross-sectional average velocity, v, was measured by taking 0 measurements at different positions at the inlet. All the flow readings were taken by placing the hot wire perpendicular to the rotational axis, to ensure that the axial velocity measurement was not influenced by the swirl component of the flows. Furthermore, the cylindrical stator duct inlet was extended to 300 mm length, to minimise the swirl component at the inlet. With the duct cross-section area, A, and the air density, r, known, the air mass flow rate can be calculated as, _m ¼ v A r: () The temperatures and local heat fluxes on the stator surface were measured by using fast response T-type surface thermocouples and thin film heat flux sensors (HFS). Four surface thermocouples (which are TC, TC, TC3 and TC4) and three heat flux sensors (HF, HF and HF3) were attached to the surfaces on the stator. HF and HF were attached on the aluminum stator front surface, where as HF3 was attached at the back side of one of the heater mats. The positions of these thermal sensors are illustrated in Fig. 9(a). With the induction motor regulated at 495 rpm, the time dependent temperatures and local heat fluxes were recorded in an Excel spreadsheet via PICO TC-08 USB data loggers. The experiment was stopped after the experimental rig had reached thermal equilibrium. During the experiment, the back of stator was thermally insulated using fibre glass matting. However, a significant portion the heat produced from the heater mats escapes through the back of the stator. So the total heat input to the front of the stator disk, P front, is determined as the difference between the electrical heat input and the heat lost through the back and is given by equation (3): P front ¼ P electrical input P back : (3) Since aluminum has high thermal conductivity, the heat flux across the stator front surface can be assumed to be uniform. As a result, the uniform heat flux boundary conditions can be applied on the stator disk in both CFD and lumped parameter models. The uniform heat fluxes specified in the two models are calculated by dividing the total heat input to the front of stator, P front, by the total area of the stator front surface. The uniform heat flux boundary condition assumption is verified by measuring the heat fluxes at two different positions on the stator front surface. Heat flux sensors, HF and HF are used and the position of these sensors is shown in Fig. 9(a). The measured surface heat fluxes from these two heat flux sensors are shown and compared in Section Comparison of CFD and LPM predictions with measured data Both CFD and experimental measurements are used to verify the temperatures predicted by the lumped parameter model of the single-sided slotted axial flux machine. Fig. 0(a) and (b) shows the temperature contours and velocity vectors, respectively, of the simplified single-sided slotted axial flux machine obtained from the CFD model. Table summarises the mass flow rate measured from the experiments and predicted from the numerical models (CFD and LPM). The mass flow rate predicted from the CFD model shows

8 C.H. Lim et al. / International Journal of Thermal Sciences 49 (00) 73e Fig. 8. The schematic (a), snapshot (b) and geometrical dimensions of the simplified experimental rig. reasonable agreement with the experimental result, being about % higher than the mass flow rate measured by the hot wire anemometer, see Table. Fig. shows the measured and predicted temperature at four different radii along the stator surface from the LPM thermal circuits, CFD model and experimental rig respectively. It can be observed that the temperatures measured (from the experiments) and predicted (from the CFD models) are low as compared to the winding temperatures in commercial electrical machines. Normal commercial electrical machines usually operate at stator surface Fig. 9. Thermocouples and heat flux sensors positioning on the stator front (a) and back (b) surface.

9 740 C.H. Lim et al. / International Journal of Thermal Sciences 49 (00) 73e74 Fig. 0. Temperature contour and velocity vectors plot of the single-sided slotted axial machine. temperatures of 80e0 C, but the surface temperature measured or predicted from the experiments and CFD model are of the order of 30e35 C. This is because the power of the heater mats used in the experiment is low. The rated power of each of the heat mat is 8 W and six heater mats were used in total. Hence, the total heat input is 48 W. Compared with the winding losses of commercial electrical machines, which range in between 90 and 00 W (for.5 kw axial flux machines), the temperatures measured from the experiments and predicted from the CFD are correspondingly low. Table shows the heat fluxes measured from the heat flux sensors attached on the front and back sides of the stator disk. The results show that the local heat fluxes measured on the stator surface are similar. Therefore, a uniform heat flux boundary condition on the stator front surface can be applied with confidence in the CFD and lumped parameter models for the thermal modelling of the test rig. By specifying the convection heat transfer coefficients derived by the correlations suggested from [,] in the LPM model, the temperatures predicted show a big discrepancy with the results obtained from the measurements. The highest absolute discrepancy occurs on the stator disk at position TC4 (see Fig. ), which is 0.6% (5.5 C difference in temperature rise). This confirms that the convective heat transfer coefficients derived from the correlations are not sufficient to providing an acceptable temperature approximation for the LPM model. Hence, to improve the LPM predictions, the existing correlations [,] were replaced by local convection heat transfer coefficients extracted from the CFD model. These local convection heat transfer coefficients are evaluated based on the local bulk working fluid (or air) temperatures. Like the LPM model, the working fluid in the CFD model is split into four control volumes as shown in Fig. 6(a). The local bulk air temperatures used for the evaluation of the CFD local convection heat transfer coefficients are calculated by taking the volumetric average air temperature for these fluid control volumes. With the new local heat transfer coefficients obtained from CFD input to the D LPM, the absolute discrepancies of the temperature predictions have reduced significantly as shown in Fig.. For example, the absolute discrepancy at TC4 has halved to 0.4% (3 C difference in temperature rise). The discrepancies between the two LPMs are due solely to the heat transfer coefficients used in the circuits and they demonstrate that LPMs are very sensitive to convective resistances as opposed to conductive resistances. For most air-cooled axial flux machines, the magnitude of the convective resistance is about two orders of magnitude above the conductive resistance. For example the convective resistance at T4 is 3.35 K/W while the radial conductive resistance at T4 position is 0.0 K/W. This highlights the necessity of developing a more sophisticated parametric variation study of convective heat transfer coefficients and mass flow rates for axial flux machines to complete the LPM model. Unlike the CFD modelling technique, the determination of the local heat transfer coefficients experimentally is difficult, because it is almost impossible to determine local air bulk temperature accurately by using thermocouples in the stator-rotor gap. Therefore, only global heat transfer coefficients are measured from the Table Mass flow rates comparison. Experiments CFD model Mass flow rate (g/s) Fig.. The temperatures measured and predicted from experimental rig and numerical models (CFD and LPM) respectively.

10 C.H. Lim et al. / International Journal of Thermal Sciences 49 (00) 73e74 74 Table Local heat fluxes measured on the stator front and back surfaces. HF (Front) HF (Front) HF3 (Back) Heat flux (W/m K) experiments, by using the inlet air temperature as the reference temperature. The results are shown in Fig.. A new set of heat transfer coefficients was derived from CFD, by changing the reference temperature from air bulk temperatures to the inlet air temperature. The results are then directly comparable with the experimental results, as well as the convection heat transfer coefficients derived from the correlations shown in [,], as also shown in Fig.. It observes that the CFD better predicts the global convective heat transfer coefficient than the correlations. The temperatures predicted from the lumped parameter model are slightly higher than the experimental ones. It should be noted that this lumped parameter model did not take into account the extra heat transfer from the stator pipe (Fig. 8) to simplify the thermal circuit. Hence in the lumped parameter model, the heat generated from the heater mats is all transferred through the stator surface. In the experiments, a certain portion of heat is transferred from the stator pipe surface to the surroundings, see Fig. 8(c). Therefore, the local temperatures, as well as the local heat transfer coefficients for the stator surface, predicted from the numerical models, are higher than the experiments ones, as shown in Figs. and. From Fig., it can be noted that the outlet air temperature predicted by the LPM is 0.9 C when the convection heat transfer coefficients predicted by the correlations [,] are used. However, when the local convection heat transfer coefficients obtained from CFD model are used, which are generally higher than the coefficients predicted by the correlations, the outlet air temperature increased to 3. C showing that the outlet air temperature from the LPM when the convection heat transfer coefficient increases. This is because with higher convection heat transfer coefficients, more heat is removed from the stator surface, by the moving fluid. Therefore, the solid temperatures are reduced, and subsequently the outlet temperature is increased. This is illustrated in Fig.. In general, the local temperatures and heat transfer coefficients predicted from the LPM thermal circuit show good agreements with the CFD models. This demonstrates that both the annular conductive circuit and the convective thermal circuit used in the Fig.. The global heat transfer coefficients measured and predicted from experimental rig and numerical models (CFD and LPM) respectively. LPM work well in predicting the conduction and convection heat transfers in the simplified single-sided slotted generator. 6. Conclusion A fast LPM thermal circuit of single-sided slotted generators has been investigated. Temperatures predicted from the LPM thermal circuit show a reasonable correspondence with by both CFD and experimental results. The research confirms that the LPM method provides a fast reliable design tool for axial flux generators. References [] C.M. Liao, C.L. Chen, Thermal analysis of high performance motors, in: The Sixth Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, 998, ITHERM 98, May 998, pp. 44e433. [] S.J. Pickering, D. Lampard, M. Shanel, Modelling ventilation and cooling of the rotors of salient pole machines, in: IEEE International Electric Machines and Drives Conference (IEMDC) 00, June 00, pp. 806e808. [3] C.M. Liao, C.L. Chen, T. 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