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Research Papers

# Motor Cooling Modeling: An Inverse Method for the Identification of Convection CoefficientsOPEN ACCESS

[+] Author and Article Information
Tanguy Davin

Université de Lille Nord de France,
LAMIH UMR CNRS 8201,
Université de Valenciennes
et du Hainaut-Cambrésis,
le Mont Houy,
Valenciennes Cedex 9 59313, France;
Renault,
Direction de la Recherche,
1 avenue du golf,
Guyancourt 78280, France
e-mail: tanguy.davin@gmail.com

Université de Lille Nord de France,
LAMIH UMR CNRS 8201,
Université de Valenciennes
et du Hainaut-Cambrésis,
le Mont Houy,
Valenciennes Cedex 9 59313, France

Robert Yu

Renault,
Direction de la Recherche,
1 avenue du golf,
Guyancourt 78280, France

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF THERMAL SCIENCE AND ENGINEERING APPLICATIONS. Manuscript received October 5, 2016; final manuscript received February 20, 2017; published online April 25, 2017. Assoc. Editor: Hongbin Ma.

J. Thermal Sci. Eng. Appl 9(4), 041009 (Apr 25, 2017) (13 pages) Paper No: TSEA-16-1284; doi: 10.1115/1.4036303 History: Received October 05, 2016; Revised February 20, 2017

## Abstract

The present study focuses on oil cooling for electric motors. A 40 kW test machine in which oil was introduced at each side of the machine to directly cool the stator coil end-windings was previously implemented. The lumped system analysis is used to model the thermal behavior of this test electric machine. An inverse method is applied to interpret the data obtained by the experimental setup. The inverse method leads to interior convection coefficients that help describe the heat transfer mechanisms.

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## Introduction

An electrical motor is composed of elements that can be affected by rising temperatures. High electrical currents can generate significant heat loss in the windings. Cooling systems and materials should be optimized to protect these elements. Windings cannot endure a higher temperature than the limit determined by the motor insulation class—commonly, 180 °C for this type of automotive application—and the global efficiency can be affected by the Joule effect. Numerous cooling systems can be found among industry motors. For simplicity, indirect cooling, where air or water permits to cool external carters, is most used. Direct cooling is studied in the present work.

Convection on coil end-windings has been shown to have a great influence on the temperature level, as noted in several previous publications [1,2]. Unfortunately, water cannot be put in contact with the windings due to its electrical conductivity. Some authors propose solutions to improve the air convection around the end-windings [3,4]. Because our application is focused on an automotive powertrain (motors that are a few dozen centimeters long and wide and generate a few dozen kilowatts), the motors are very compact. The range of the air convection coefficients might therefore not reach a sufficient level.

Due to the presence of lubricating oil in the vicinity of the motor and the heat transfer enhancement that such a liquid provides, oil impingement has been considered. Practical issues must also be studied as the oil and the motor parts must be compatible and windage losses might reach critical values at high rotation speeds. The oil level should not reach the air gap. In this study, both impinging jets and the spray were investigated.

The literature provides abundant data for simplified geometries. Laminar jets impinging hot plates have been studied both locally [5,6] and globally [7]. High exchange rate can be obtained around the impinging point, but the Nusselt number profile sharply decreases. Viscosity play a key role in heat transfer, as was shown by a study [8] testing liquids with high Prandtl numbers like oil. Some authors have also shown a great influence of roughnesses on the impingement surface on heat transfer. The heat flux can be multiplied by four with microgrooves [9] or by 1.8 for special fins [10]. In these studies, the impinging zone is kept flat to preserve the high heat transfer in this zone, and the Reynolds numbers tested are high.

Cooling sprays were the subject of many studies gathered in a review [11]. As the temperature must not reach very high values in our application, only the single phase is of interest in the present investigation. Heat transfer increases with the flow rate, which is similar to jet impingement. The effect of the surface roughness, in particular straight fins, has been studied in Ref. [12]. The higher is the roughness, the greater is the heat flux. It is explained by the fact that wetted surface is increased and the favorable incidence of the liquid intercepted by the top of the fin. Conduction within the fin and liquid draining might counterbalance these effects. The heat flux might be improved (no quantification for single-phase), particularly for some combination of the nozzle inclination and the fins direction.

A comparison between a jet and a spray array using water has been performed on a flat surface [13,14]. It appeared that a spray could provide similar global heat transfer at a substantially lower liquid flow rate. Nevertheless, real oil flow is quite different from that described in these studies, and only a few studies have shown interest in oil cooling characterized by high Prandtl numbers and low Reynolds numbers. Locally, the roughnesses of the impinged surfaces are peculiar to the winding geometry. Globally, gravity and rotation effects disrupt the symmetries. The tested global flow rates related to the total surface are also much less important than those reported in the literature.

The thermal modeling of electric machines permits the prediction of the temperatures of the elements. The assessment of the temperature is essential to avoid deterioration of components. It is also interesting for multiphysical design. The finite elements method (which might be coupled to computational fluid dynamics) and lumped system analysis are the most commonly used analyses for thermal modeling. The assembly of electric parts forms quite complex geometries. Lumped system analysis allows an account of all of the heat transfer types for a reduced resolution time. This is appreciated for automotive application, for which many parameters are involved.

The principle of the method is to consider a system as a finite assembly of isothermal elements, namely, cells. Each cell is attributed a thermal capacitance, Cth, and an internal thermal power (mechanical or electromagnetic losses). Between two cells, heat transfer is regulated through a thermal resistance Rth. Thermal resistance can represent conduction as well as convection, radiation or transfer by fluid mass transport. The majority of authors have thermally represented a machine with a few cells [3,1517].

## Methodology

###### Experimental Setup and Procedure.

All of the equipment is briefly described here, but one may refer to a previous article for a detailed description [18]. A middle-range power (40 kW) radial electromagnetic flux machine was used in our experiments. The stator is composed of twelve concentrated windings, and the rotor is composed of eight salient poles covered with end-plates at each side of facing flanges. Thus, the main part of the injected oil must remain in the two side chambers. The rotor is entrained by a separate motor with a belt pulley system, up to 4600 rpm. A current source was installed to generate constant losses in the windings by the Joule effect. The winding power is the only source power except friction.

The heat transfer between the flanges and the oil–air mixture as well as between the flanges and the coil end-windings is studied. As a consequence, the flanges have been designed to be thermally separated from the stator as much as possible. Insulating rings were located between these two parts, and the stator was insulated from the ambient air. Cold-water ducts were added inside the flanges to maximize and to quantify the convective heat transfer between the flanges and the oil–air mixture. The test machine is represented in the left part of Fig. 1.

Four oil injection systems were tested. In all of the configurations, both sides supply oil in the same manner. The total oil flow rate was varied from 40 to 360 L/h and its input temperature was 50–75 °C. First, the mist effect was studied using two spray nozzles of different shape (full cone M8 and flat jet HVV). These nozzles only comprise an oil entry that is dispersed into droplets when forced into a shaped hole; no compressed air is added. The second type of injection is dripping. This injector is composed of five injection points, which are located above the five upper end-windings, to supply all end-windings under gravity. Eventually, the last injector includes three orifices in front of each end-winding, producing jets that impinge on different surfaces of the windings: one facing the flange, one facing the carter, and one facing the stator tooth.

All of the tests were conducted at a thermal steady state. Prior to any measurement, all of the operating parameters were set to the desired configurations in terms of the rotor speed, oil temperature, and flow rate. The windings were supplied with current until reaching a general steady state at a desired mean winding temperature.

Figure 1 right shows a sketch of the dissipation of the winding power. The principal powers for average conditions are represented. The heat fluxes exchanged with internal oil/air mixture are a little different from those listed in Table 1 as only the whole stator windings, stator stack side and carter arrow is shown. Power balances can be verified, in particular

Display Formula

(1)$Pwinding=Pstator→oil+Pstator→rotor+Pstator→flange+Pstator→airext$
Display Formula
(2)$Pstator→oil+Protor→oil=Poil→flange+Poil→oil bath$

Thanks to the insulation and the water temperature close to the ambient, the external convection influence is highly reduced. For a hot stator, approximately 30 W are dissipated by ambient air and 20 W for each side by the water through the insulating rings. The power supplied to the windings (Pwinding) thus mainly reflects the heat transfer efficiency on the side stator parts as it is insulated from the flanges. The winding power is mostly dissipated in the oil (around 550 W in total), but approximately half of this power is transmitted to the water before exiting. As bearings are in direct contact with the water-cooled flanges, the friction losses are easily dissipated into water, and the power drained from the shaft is relatively stable. It would not be the case for a standard motor for which the bearing temperature can be a problem.

###### Nodal Modeling of the Motor.

The experimental machine was modeled based on the lumped system analysis. A code was implemented using Matlab/Simulink® (MathWorks, Natick, MA). The model considers constant thermophysical properties for the solids and temperature-dependent for the fluids: only water and air, since the oil–air mixture heat transfer is expressed by the identified convection coefficients. The structure is quite similar to those found in the literature [3,1517], with a few improvements. Particularly, it involves cell parameters to refine zones with high temperature gradients. The model was experimentally validated during previous studies, and the mesh was optimized. This model is composed of 3900 cells to reach a maximum 3 °C difference, with a very detailed mesh of 100,000 cells. The stator mesh was fine, especially for the windings. A mesh grid representation is given in Fig. 2. Only a small portion of the motor is modeled, considering the symmetry of its geometry. One-half of a stator tooth and one rotor half-tooth are then modeled. Those cells are linked together by two thermal paths: the air gap and the assembly of the carter/flange/bearing on each side. Eventually, two cells were implemented to represent the fluid circulating on each side chambers. This fluid is either air only or a mixture of air and oil when oil is introduced. This implementation means that we assumed that the inside fluid temperature is homogeneous. Those assumptions are very important for the results provided. Indeed, the convection coefficients that are identified here might not be useful with different assumptions. The inaccuracies of the method are discussed in Sec. 4.

###### Inverse Method Introduction.

The aim of this paper is to quantify heat transfer due to internal convection. The convection coefficients on each internal surface of the side chambers are of interest. The distribution of this coefficient should help our understanding of the mechanisms of heat transfer in a way that is complementary to the direct analysis of the winding dissipation power that was previously achieved [18]. The thermal problem is represented in Fig. 3 (left).

From the nodal thermal model, we can calculate the temperature field. However, this calculation is only possible provided that all of the parameters are correctly implemented. In particular, the oil convection coefficients are tricky to assess in terms of the literature results. Even if it is impossible to directly measure those coefficients, it is possible to measure the temperature resulting from a given set of oil flow conditions. This is an inverse problem that must be solved according to mathematically approaching coefficients that generate a temperature field that is as close as possible to the measurements. The resolution algorithm applied to our inverse problem is shown in Fig. 4. First, some values for the coefficients are assumed, and then, the calculation loop begins. From the values of the coefficients, the temperatures are calculated by the model. The calculated temperatures are compared to the measurements. As long as the convergence criterion is not satisfied, the coefficients are modified to tend to a smaller temperature difference between the calculation and the measurements. The readjustment step is detailed in the following paragraphs. After successive iterations, a converged set of convection coefficients ${hj}$ is obtained.

The definition of the convergence criterion is crucial to evaluate the quality of the identification of the coefficients. Intuitively, we define a criterion function as the quadratic mean of the temperature difference between the calculation and measurements, at different measurement points. In Eq. (3), NT is the number of measurement points. The smaller the criterion function, the better the identification. Display Formula

(3)$crit=∑iNT(Tmeas−Tcalc)i2NT$

Although this algorithm is relatively simple, the readjustment step is complicated, as it can be performed in multiple ways. Our resolution can be seen as a parameter optimization as we try to minimize the criterion function. One may refer to book on parameter identification [19] that describes numerous optimization methods. A method can be based more on evolution or on exploration. It may involve one or multiple parameters. For the former, only one vector is readjusted for each iteration. For the latter (population methods), several vectors move in the parameter domain, hence, reinforced exploration. For the population methods, a stochastic approach can be selected. Methods based on exploration are especially helpful in coping with problems related to a criterion function with local minima.

Two methods were applied to our inverse problem. The first one involved a population of vectors, similarly to Beck's [20]. The second method was based on the evolution of only one vector. With the population algorithm, initialization values randomly comprised within the whole h physical range [5 W/m2 K; 2500 W/m2 K] were preliminary tested and both methods lead to very similar results on different data series (air only and oil injection). No local minimum on the criterion function was detected. As the time consumed by the population method was significantly higher, the second method was preferred and is the only method described below.

###### Mathematical Application of the Inverse Method to the Test Machine.

To obtain an ensemble of parameters hcalc which generates an approached temperature field as close as possible to the real temperature field, we proceeded in the simplest way, by trying to minimize the criterion function. This is actually similar to a previous application [21]. In the previous study, the entire thermal flux was reconstituted from the temperature field. The solution was obtained by minimizing the temperature difference between the calculation and measurements, and a spatial regularization condition was also added. This was performed by minimizing the flux gradients within the whole field. In contrast, in our study, the measurement points are restricted and the number of convection surfaces Ns (e.g., the number of coefficients) must be even smaller. Moreover, the fluid flow on the internal surfaces may be unequal (whether oil is present or not or due to the local flow speed). Regularization is used to smooth the identified parameter, and it may lead to difficult choices for the conditions of the relaxation. For the present problem, only one condition was considered: the minimization of the criterion function. This can be mathematically written as Display Formula

(4)$d critdhj=0 ∀j∈[1,Ns]$

This equation must be solved at each iteration of the algorithm. The parameters are now subscripted according to the current iteration (k), the following iteration (k + 1), or the measurements (meas). To derive iteration (k + 1) from iteration (k), the previous equation is written as the iteration (k + 1), replacing crit by its definition. Display Formula

(5)$d ∑iNT(Tmeas−Tk+1)i2dhjk+1=0 ∀j∈[1,Ns]$
Display Formula
(6)$∑iNTd (Tmeas−Tk+1)i2dhjk+1=0 ∀j∈[1,Ns]$
Display Formula
(7)$−2∑iNTdTik+1dhjk+1⋅ (Tmeas−Tk+1)i =0 ∀j∈[1,Ns]$

This equation system can be expressed using a matrix, introducing the matrix of the sensitivities of the temperature to the convection coefficients Display Formula

(8)$J⋅[Tmeas−Tk+1]i=0$
$where J=[dT1k+1dh1k+1⋯dTNik+1dh1k+1⋮⋱⋮dT1k+1dhNjk+1⋯dTNjk+1dhNjk+1]$

The temperature at the new iteration (k + 1) is then expressed as a function of the previous iteration (k) using Taylor's approximation (a first order limited development). A relationship containing $Δh$ is then obtained, which permits the readjustment of the coefficients and thus leads to the next iteration. This allows the appearance of the J matrix transposed. Display Formula

(9)$Tik+1=Tik+dTik+1dTjk+1⋅(hjk+1−hjk) ∀j∈[1,Ns]or written as [Tik+1]=[Tik]+Jt⋅[hjk+1−hjk]=[Tik]+Jt⋅Δh$

Melding equations (8) and (9) generates the following relationship: Display Formula

(10)$J⋅[Tmeas−([Tik]+Jt⋅Δh)]i=0$

The direct temperature difference between the measurements and calculation at iteration (k) is defined as $ΔT$Display Formula

(11)$ΔT=[Tmeas=Tk]i$

It can then be written as Display Formula

(12)$J⋅[ΔT+Jt⋅Δh]i=0$

And eventually Display Formula

(13)$Δh=−inv(JtJ)⋅JΔT$

In this last relationship, the J matrix is described for iteration (k + 1). To express $Δh$, it is assumed that this matrix remains the same between two iterations. The variations in coefficients hj must be restricted. This assumption permits the consideration of the temperature T = f(h) as a locally linear function. In the algorithm, the sensitivities matrix is calculated by introducing a small variation $ε$ (1% in our method application) in the coefficients Display Formula

(14)$J(i,j)=∂Tik∂Tjk=Tik(hjk(1+ε))−Tik(hjk)εhjk$

A relaxation coefficient ξ is introduced for convergence stability Display Formula

(15)$hk+1=hk+ξΔh$

As shown in Fig. 4, the last given relationships are implemented in the algorithm: once the J matrix is calculated by varying each hj parameter (14), all parameters were recalculated using Eqs. (13) and (15). If the temperatures are barely influenced by a change in one coefficient, the sensitivity values considered are very small. This might affect the accuracy of the parameter identification. A preliminary study was carried out to find the relevant measurement points during the bench design. This led us to add thermocouples (TC) to the core of the stator and the water channels in the flanges. Without those cold water channels, the heat flux between the oil and the flanges would be very small and uncontrollable. It would then be impossible to identify the corresponding convection coefficient. The measurement points are shown in Fig. 5. Three thermocouples (TC) at three different windings were inserted at the core of the machine. Other thermocouples were stuck to the walls: flange (two TC), one end-winding (three TC), yoke side and carter. Eventually, two thermocouples were located in the fluid to assess the internal air temperature.

###### Choices for Parameters.

To obtain a correct distribution of convection coefficients (small criterion function, physical sense, algorithm stability and rapidity), attention is focused on some parameters involved in the method.

• convergence parameters

• convection surface grid (number Ns of identified h)

• considered measurement points

###### Convergence parameters.

The calculation progress is mainly driven by the relaxation coefficient $ξ$. As the function $T=f(h)$ is considered to be locally linear, the variation between the two steps should not be too large. To avoid a very quick divergence at the beginning of the calculation, the relaxation coefficient is first set to be small (0.1–0.3). To accelerate the convergence, this $ξ$ coefficient is then adapted in regard to the criterion evolution. Thanks to this adaptation, the algorithm is robust and convergence is quick. The initial values for the hj parameters were set to 20 and 100 W/m K, respectively, for air only and oil configurations. As soon as convergence is assured (number of identified parameters Ns and relaxation coefficient $ξ$ low enough), the initialization values for the coefficients hj do not change the solution nor significantly affect the calculation time. An example of the calculation progress for a test with air only is seen in Fig. 6. For Ns = 3, the convection coefficients and the temperature difference (components of the vector $ΔT$) at the measurement points are shown. The criterion (crit = 2.3 at the end of the calculation) is also plotted in this last graph. It is important to note that some of the measured temperatures are quite far from the calculation (in this example, up to 7 °C); these inaccuracies are discussed in Sec. 4.

###### Convection surface grid.

The indentation of the convection surface is very important for the study. It corresponds to the number Ns of h parameters that are identified, but it also involves the consideration of surfaces as a group of convection areas. First, no measurement has been performed concerning the rotating parts. The convection coefficients are approximated, assuming an air flow on a rotating disk or cylinder, respectively, for the rotor side and the shaft. This assumption can be discussed, but the convection coefficients barely influence the temperature. Indeed the rotor was made from a highly insulated material; the convection surface on the shaft is very small; and the involved powers (almost entirely friction) are not high. The resulting error made on the convection coefficients is thus reasonably low.

The identification is thus focused on the windings, the stator side, the carter, and the flanges. The choice for the indentation is first driven by a physical sense. The surfaces that have a similar flow could be combined. Another standard is mathematical coherence. Power passing through each surface must be of similar value to keep a uniform precision of identification. There should not be too much discrepancy of value in the sensitivities matrix J.

First, the “carter” (outer cylindrical surface of convection) part is composed of part of the carter and the protruding part of the flange. Considering the carter (both of the surfaces in purple in Fig. 3 left) as one convection surface leads to a particular effect: the oil cools the carter extension but heats the flange extension, as the oil is at an intermediate temperature. It numerically enables the modification of the distribution of the thermal flux from the stator between water and oil. This can significantly disturb the transverse convection power and the identified coefficients on the other surfaces. The outer cylinder is then considered to be a convection surface with the flange disk (upper/right parts in black–blue in Fig. 3 right). The algorithm becomes more stable, and the resulting coefficients have a better physical meaning.

Considering the stator stack, there are both the side of the tooth and the side of the yoke. The power flux is quite low due to the small axial conductivity of the stack and the presence of an insulating sheet covering the stack side. The entire stator stack side (red in Fig. 3 right) is thus considered as another convection surface. The winding surface is taken apart because oil was directly impinged on it, generating a very different flow. Eventually, a few configurations were considered for the refinement of the convection surface of the windings (referring to the windings surface numbers in Fig. 3 right). In all of the configurations, the “stator side” and the “flange + carter” are selected as the other two convection surfaces.

• grid 3 (Ns = 3): the entire winding surface {1 + 2 + 3 + 4}

• grid 4 (Ns = 4): the winding surface divided in {1 + 2} and {3 + 4}

• grid 5 (Ns = 5): the winding surface divided in {1}, {2} and {3 + 4}

The grids were chosen to highlight the influence of the rotation, which was expected to be more important for surfaces 3 and 4. The influence of the convection surface grid was studied considering data from the tests using the full cone mist (M8 nozzle) with 50 °C oil (29 tests).

The three grids were tested, but some irrelevant results were detected for the finest grid. A physical condition was set during the calculation progress. The coefficients were kept inside the physical limits [5 W/m2 K; 2500 W/m2 K]. Without these limits, some of the identified coefficient values were even negative. For the three grid configurations, the identified coefficients are plotted as a function of the oil flow rate in Fig. 7.

For all 29 tests, the mean convergence criterion is, respectively, 2.28, 2.26, and 2.11 for grids 3, 4, and 5. When refining the grid, the identified coefficients better approximate the measured temperatures. This is logical, as the degrees of freedom grow with the number of identified parameters Ns. In theory, the obtained convergence criterion might be null if there were as many coefficients as the measured temperatures (Ns = NT). Therefore, the model does not perfectly represent the thermal behavior of the machine. The geometry is simplified, particularly the winding cross section, which is not strictly rectangular. This may explain the greater scatter as the grid is refined. For the finest grid (grid 5), the algorithm is less stable and the coefficients become physically inconsistent. A less accurate but interpretable solution is preferred. Also, for models dedicated to engineering applications, the consideration of a mean convection coefficient might be sufficient. Grid 3 (the winding surfaces altogether) was then chosen for the resolution of this issue.

###### Considered measurement points.

For the tests with oil, according to the definition of a convection coefficient, the input oil temperature Toil_input was chosen as the reference for the oil/air mixture in the side chamber. This temperature was applied as a measurement point: the model calculates the temperature of the internal cell but approaches the measurement at the convergence. It must be noted that the temperature rise between input and output oil very rarely reaches more than 10 °C (only for a low flow rate and small heat transfer with flanges). Considerations for the fluid temperature are quite different for air only; this is discussed in Sec. 2.3.4.

In most tests with oil, there were ten measurement points (NT = 10) considered for the identification (see location of the thermocouples in Fig. 5): two at the flange wall, one at the yoke, one at the carter, one for the internal oil/air mixture (a PT100 probe immersed into the oil channel before injection), and two for the water outputs (PT100 probes immersed into small baths after each flange). The previous paper [18] highlighted some oil flow and temperature disparities along angular position. From the nine thermocouples inserted at the winding core, three measurement points (the mean temperature on the three angular positions for each position on the winding section) were then considered. For the tests with air only, the disparities were small. The three thermocouples stuck at one end-winding side surface were only taken into account for tests without oil.

Some of the measurements were not taken for the identification for a few test series. One thermocouple at the core of the winding was faulty for the tests with multijet injectors. In addition, some of the thermocouples that were stuck to walls generated inconsistent temperatures. Measurements on the yoke side were not considered for multijet tests as well as on the flange for multijets and dripping tests. This inconsistence is assumed to be due to the local oil flow on thermocouples.

###### Reference Configuration: Tests With Air Only.

The definition of a convection coefficient is crucial and depends on two temperatures. In theory, the fluid temperature that is taken into account is the mean temperature at the core of the fluid. The solid temperature must be the one at the wall. Some special cells were added to the model to better assess the temperatures at the walls. The wall temperatures are slightly different from those at the closest points from the surface.

For the internal convection coefficients, the temperature of the internal cell (air only or oil/air mixture) was considered. For convection with air only, the machine is completely closed and the internal temperature is not fixed at all. For the consideration of the fluid reference temperature, there are two possibilities:

• Not taking the measured internal temperatures into account, NT = 12 (the calculation of the temperature at the internal cell is free).

• Adding a measurement point, NT = 13 (the temperature of the internal cell is calculated but is very close to the measurement by the application of the inverse method).

During tests with air only, important temperature differences inside the side chamber were measured. One thermocouple was positioned between an end-winding and the carter and another closer to the rotor. Those two thermocouples are represented in Fig. 5. The temperature measured close to the rotor was significantly lower than that next to the winding (an average of 27 °C lower for a 90 °C difference between the windings and water). The literature [22,23] provides descriptions of the flow in the case of a shrouded discoid rotor–stator system. Air is tangentially entrained by the rotating disk due to its viscosity. As a result, air is driven centrifugally at the rotor side and centripetally close to the flange. This explains the temperature difference: the air expelled from the rotor is heated by the end-windings (hot thermocouple), then flows back to the axis and is cooled in the vicinity of the flange (cold thermocouple).

Identification was made for four assumptions concerning the measurement of the internal air. The first assumption does not take into account any measurement. The other three assumptions take the thermocouples measurements into account for the temperature of the internal air cell: hot measurement, cold measurement, and the average of both. The criteria obtained were 3.36, 3.03, 3.18, and 3.05 °C, respectively, for the hypotheses without any air measurement and with the hot, cold, and average measurement. The results are shown in Fig. 8.

For this first assumption, the calculated internal air temperature can be very far from the thermocouple values. The coefficients are then hard to interpret. At the stator, they are very high and are not monotonic functions of the rotation speed. The measurements should thus be taken into account, which leads to better physical results.

As the rotation speed increases, the air velocity in the side chambers increases and the convection coefficients grow. The coefficients are higher on the windings. This is logical because the end-windings are located directly downstream of the centrifugal, flat jets emanating from the rotor. The augmentation of the coefficient on the windings (slope) seems to decrease with the rotation speed, while the slope of the coefficients on the flange is relatively constant. It is noteworthy that the power passing through the stack side is very low; therefore, the identification of the resulting coefficient is not accurate.

Convection effects in the side-chamber of motors have been studied using in particular computational fluid dynamics analysis, but regarding the present approach, a comparison with the literature can be made only under the assumption of internal air at a homogeneous temperature. Staton et al. [2] carried out tests with two motors and lumped systems to analyse the data. The authors used a thermal model to identify only one internal convection coefficient on all of the surfaces of the side-chamber (end-windings and flange). For the purpose of this comparison only, a simple grid (equivalent to grid 1, Ns = 1) was adopted in our inverse method, with a mean 3.25 °C criterion. It can be noted that the values of this unique coefficient are more similar to the flange coefficient than the end-windings coefficient. In Fig. 9, hinternal_surfaces is represented for the present data and for the two motors in the previous study [2]. For our data, the air average measured temperature was considered. The peripheral rotor velocity is taken as a reference, as in the representation of Staton. The coefficients report to the tests without any blade on the rotor (simple rotating disk).

The identification performed with our motor is relatively close to Staton's in terms of magnitude. Therefore, the slope of the increase of the internal coefficients with the rotation speed is higher. Of note, considering motors with blades, the slope is much more important than for motors without blades [2,3]. Although interesting, one should critically examine this comparison, as many parameters are involved. The motor geometries are quite varied, the motors tested by Staton were significantly smaller, and the model used was composed of only a few dozen cells. It appears to be difficult to clearly validate the results with air only, but the comparison with the literature gives the same order of coefficient values and the influence of the velocity is consistent. Also, the criterion was reasonably low.

## Results From the Identification for Oil Injection

In this section, the coefficients are presented that result from the identification of oil tests. Visualizations of the oil flow and thermal measurements help our understanding of these data. In a previous paper [18], it was highlighted that the heat transfer was always enhanced by raising the oil flow rate, while the effect of the rotation was quite uncertain.

For most of the convection coefficients results, a general trend is observed. On the stator, the activation of rotation improves the coefficients. When the rate of rotation is increased, these coefficients decrease. The coefficient on the flange is increased when the rotation is increased. Characteristic coefficient curves are displayed in Fig. 10 for the M8 nozzle at the maximum flow rate (102 L/h).

In the following paragraphs, each coefficient is analysed. The coefficients are presented as a function of the rotor speed for different flow rates. The results are compared for the four injectors at 75 °C oil. The air-only coefficient appears on each group in the graph. Some curves fluctuate, which shows the inaccuracy of the method, but clear trends can be observed.

###### Coefficient on End-Windings.

In Fig. 11, data are presented for four injectors under similar experimental conditions. For all of the tests, an increase in the oil flow rate improves the coefficient. The effect of the rotation is however clearer. Rotation seems harmful beyond 900 rpm. This decrease is particularly visible for mist generation (33% for M8 nozzle, augmenting rotation from 900 to 4600 rpm). For the dripping injector, the decrease is smaller. For multijet injection, the coefficient seems almost independent of the rotation. Rotation only has an influence on h for a high flow rate (see the purple curve in the bottom-right graph in Fig. 11). These results can be explained by flow visualizations: The break-up of the jets streaming from upper end-windings forms droplets. A part of these droplets is driven toward the flange due to rotational effects. The amount of oil impinging on the lower end-windings is thus reduced at high rotation speed. This flow phenomenon must be important because the convection coefficient on windings is significantly affected. The mean convection might also be affected by the disparities generated by rotation.

A comparison between injectors is performed in Fig. 12. The scattering of points shows the influence of the rotor speed and the oil temperatures on the windings coefficient. The coefficient for air only (identified with the averaged internal air temperature) is also given. The coefficient with air only is increased with the rotation speed, contrary to the oil injection configuration. However, the presence of oil is clearly beneficial, as the coefficient is noticeably superior. The coefficient values are quite comparable for all injection types. At a fixed flow rate, the multijet injection seems slightly more efficient on windings than dripping.

The windings coefficient varies analogously to the relative global dissipation efficiency, previously defined as the relative winding power in direct analysis [18]. This is logical, as the winding power is mainly dissipated through the end-windings surface.

###### Coefficient on the Flange.

The convective coefficient on the flange and carter is shown in Fig. 13. Interestingly, the h profile is similar to the one identified for air only at the lowest flow rate. With oil, the coefficient is slightly higher. This must be due to heat transfer with the oil at the oil exit point (bottom of the side chamber) or at the injection point (residual power through an insulating part fastening the injectors).

The coefficient on the flange is substantially improved, with an increase in the flow rate. This is in agreement with the visualizations: there are fewer droplets for low flow rates. An important amount of oil in the domain was observed at a high flow rate, especially for the mist tests. For a high rotation speed, the droplets are particularly concentrated and impinge on the flange to then form a wall film. This phenomenon seems to be intensified by rotation. This is also in agreement with the calculated coefficient on the flange: it improves with an increase in rotation speed. The coefficient for the last two injectors is similar, except at the highest flow rate for which the dripping injector is more efficient. This must be due to the number of droplets, which is more important for the dripping case. The concentration of droplets is indeed directly linked to the amount of oil which is brought to the upper end-windings. The fact that more oil impacts on the upper windings seems to significantly intensify the amount of droplets.

###### Coefficients on the Stack Side.

It is difficult to analyse the results for this coefficient (Fig. 14). The influence of the flow rate is not obvious, and the coefficients obtained in counter-rotation are different (the profiles are asymmetrical). These coefficients are relatively high for the first three injectors and are very low for the multijets. These observations might be explained by the fact that the concentration of droplets seems to be lower in the latter configuration. The identification on the stack side must be taken carefully, as the heat power through this surface is poor.

###### Influence of the Oil Temperature.

In a previous study [18], it was shown that the global dissipation power was enhanced when the oil input temperature was decreased at a fixed mean winding temperature and a fixed oil flow rate. Indeed, the temperature difference between the stator and the oil becomes higher. However, it is important to observe the relative impact of the oil temperature. The oil viscosity is 2.5 times higher at 50 °C than at 75 °C (e.g., 30 mm2/s at 50 °C and 12 mm2/s at 75 °C). We are reminded that the mean winding temperature was 110 °C for all tests, except for mist nozzles with oil at 50 °C, for which the winding temperature was 90 °C.

All coefficients (windings and flange) are improved by the increase in oil temperature. Table 2 shows the increase in these coefficients by raising the temperature from 50 to 75 °C for the maximum flow rates relative to each injector. Due to the difference of the mean winding temperature condition, the two nozzles and the other two injectors must be analysed separately. They can nevertheless be compared in terms of magnitude.

In the case of a simple laminar jet on a flat plate, considering Ma's correlation [5], the increase in the temperature from 50 to 75 °C generates an increase of 18% in the convection coefficient. Applying Perreira's correlation for multiple sprays [24], the temperature rise causes an increase of 16% in the coefficient. For the identified coefficients on windings at a maximum flow rate, this increase reaches comparable or slightly higher values. Concerning the coefficient on the flange, the increase is enormous. This is attributed to the fact that the oil is not impinging directly on the flange. The viscosity clearly modifies the oil flow downstream of the first impact with a wall.

For both mist nozzles, the effects of the oil temperature are similar. Passing from 50 to 75 °C, the sheets emanating from the nozzles do not seem to change a lot, but the mean diameter of the droplets must be reduced and the velocity increased at constant flow rate. Both effects must contribute to the global improvement of the convection.

Trends are quite different for the two injectors with higher flow rates. For the multijet injection, the improvement on the windings is higher than for dripping. On the flange, the influence of the oil temperature is similar. Furthermore, Fig. 15 shows that the coefficient on the flange is almost independent of the oil flow rate for tests with 50 °C oil. This profile is then close to the profile identified with air only. For the dripping injector, the influence of the oil injection is only visible for the maximum flow rate. However, with 75 °C oil, the coefficient becomes substantially higher. For both injection types that do not generate mist, the oil in contact with the flange is mainly driven from the upper end-windings by the rotor. These results show that the effects of the rotation (i.e., jet breakups generating droplets driven toward the flange) are amplified when the oil temperature is increased.

The viscosity reduction generally leads to an enhancement of heat transfer. However, this reduction also seems to improve the oil spread into the domain, particularly toward the flange. The coefficient for the flange is thus significantly increased (1.5–3 times). This represents a substantial amount of oil streaming on the flange, rather than on lower end-windings. However, the decrease in convection due to this lack of flow on windings appears to be lower than the improvement of the convection due to viscosity. In all configurations, the temperature rise increases the coefficient for the windings.

We are reminded that the convection coefficients are relative to the temperature difference between the oil and the winding wall. This does not mean that the global cooling is improved for an oil temperature rise. For a fixed mean winding temperature, a more important dissipation power (Pwinding) was actually obtained at 50 °C than at 75 °C.

## Validity of the Method

The identification method generates important uncertainties. These errors are due to the representativeness of the thermal model, to the measurements or to the method application.

###### Representativeness of the Thermal Model.

Some hypotheses implemented in the model generate errors in the calculated temperature. These are mostly caused by the assumption of a homogenous temperature within the internal fluid (only one cell representing the air or the oil/air mixture) as well as the symmetries (only a half-tooth modeled).

The assumption of a homogenous temperature is particularly biased for tests with air only. The considered value is thus approximate, but the mean temperature tends to an intermediate value that must be more representative of the core of the internal air. For tests with oil, the air mass percentage and the temperature rise of the oil remain reasonably low. The choice of the input oil temperature seems to be more justified.

The second main weakness of the model is the use of symmetries. Orthoradial heat transfer is not permitted. The previous study highlighted asymmetries because of rotational and gravity effects when injecting oil. In the algorithm, the winding temperature measurements considered for the identification are the temperatures of each position in the slot were averaged for all three radial positions (every 120 deg). This is not strictly representative but better renders the general trend of the temperatures. The coefficients obtained reflect a mean value of the convection level. Both the positive and negative rotation directions were tested. The symmetry problem can then be observed on graphs of coefficient functions of the rotation speed (mainly visible in Fig. 14). While the profiles of the coefficients on windings and on the flange are rather symmetrical, it is clearly not the case on the stack side.

###### Measurement Errors.

The measurement errors include all of the direct uncertainties already described in the previous article. Here, the position problem of the thermocouples is described. All of the thermocouples, except those at the core of the machine, are stuck with a resin on the internal surfaces of the side chamber (example in Fig. 5). They do not guarantee a strict measurement of the wall temperature, as accounted for in the inverse method. The resin can add resistance to thermal conduction between the oil/air mixture and the thermocouple welding or between the welding and the wall. A local oil stream might also directly impinge on a thermocouple, implying a temperature closer to the oil temperature. The thermocouples on the flange at a radius of 75 and 90 mm were stuck in a diametrically opposed position. High temperature differences were detected, which is not consistent with the model. This was found even though specific attention was paid to the modeling of the water channels inside the flange. Local oil fluxes are supposed to occur, particularly due to rotation.

Actually, the problem of the temperature errors involving thermocouples is linked to the model assumptions (symmetries and homogeneity of the internal fluid) because the measurement is local and not averaged on all of the radial positions.

###### Inaccuracies of the Method Application.

The criterion function crit is a good indicator to describe the temperature difference between measurement and the converged solution. The criterion values are given in Table 3 averaged for each dataset.

The coefficients are inherently identified according to knowledge of two values: the power crossing the considered surface and the temperature difference between the wall and the fluid. The accuracy of the identification depends on the accuracy of those two values. It is thus important to maximize the power and temperature difference. The temperature differences were intended to be high when designing the bench by adding cold water channels inside the flanges (allowing higher accuracy on the flange coefficient). The transverse power is directly dependent on the quality of the convection (that is assessed) and the temperature difference. The mean winding temperature was then chosen to be as high as possible within a thermal safety limit (no measurement over 140 °C). The target value was 110 °C. This allows reasonably high winding dissipation power to be obtained by the use of oil injection. For air only, the mean winding power is 190 W, for a minimum of 400 W for the oil tests (mean 650 W). Errors calculated for tests with air only must then be significantly more important. However, the results remain coherent. This involves a greater trust in the results for oil injection, although the symmetry hypothesis leads to other inaccuracies.

The following paragraph analyses heat transfer for average conditions for tests with oil injection. These conditions are listed in Table 1 in the top part, based on the identified coefficients averaged on all of the tests. The power balance is also given, considering both oil/air mixtures (one for each side). The power is positive if heat is brought to the mixture. It can be noted that a residual part of the winding power is directed toward the carter, then dissipated through the oil or the insulating parts. That is why the 650 W cannot be found in the oil/air mixture power balance.

The values for the power exchanged on the three identification surfaces are quite different. Particularly, transfer through the stator sides is low due to the poor axial conductivity. The convection resistance is not predominant compared to the assembly of the conduction resistances between the heat source and the wall. The sensitivity of this convection coefficient is low, and the resulting errors are substantially higher than for the other coefficients.

The relatively unstable identification of this coefficient might disturb the identification on the other surfaces. In particular, the winding power is divided into two major fluxes, the convective flux on the stack sides and that on the end-windings. Knowing the value of the power exchanged from the whole stator to the oil, over-assessing the flux on stack side would under-assess the flux on windings. Fortunately, even for a very high coefficient on the stack side, the flux is not highly augmented. Identification on the windings is thus relatively undisturbed. It can be observed that despite clearly dissymmetrical profiles of the stack (Fig. 14), the symmetry of the profiles on the windings (Fig. 11) are maintained. The identification on the flange should not be impacted as the power considered for the identification is dissipated into water.

The identification method generates errors, and the results must be critically analysed. These uncertainties are especially high for tests with air only and for the stack side coefficient. It would be complicated to give uncertainties for every coefficient because the uncertainty due to the model assumptions is not known. Attention was focused on the criterion function and the physical coherence of the results.

## Conclusions

In this study, an inverse method is applied to identify the internal convection coefficients. The results are based on experimental data. The application parameters and inaccuracies for this method are presented and discussed. Coefficients for air only and for different oil injection types are given.

The results of identified convection coefficients show the same general trend than the power dissipation studied in the previous direct measurement analysis. In particular, the winding coefficient responds similarly as the total winding power dissipation in direct analysis. However, as several exchange surfaces were considered, this identification method permits a better understanding of some localized effects. The analysis of the coefficients highlights the oil spread. In particular, the rotation effects (jet breakups generating droplets driven toward the flange) are found to be important phenomena. A strong influence of the oil temperature on the convection coefficients is also observed. This could not be clearly shown with direct measurement. The method, however, contains inaccuracies. The identification on the stack side is quite uncertain, and the results do not take into account the disparities along the angular position.

Thanks to the method, a database of coefficients is obtained. This database can help improve the thermal modeling of electrical machines using this type of cooling, provided that the assumptions are consistent with those used for the method presented here. Especially, the model must include only one cell that represents the air or the oil/air mixture.

## Nomenclature

• crit =

criterion function (°C)

• h =

convection coefficient (W/m2 K)

• J =

sensitivity of the temperature to the convection coefficient (m2 K2/W)

• Ns =

number of convection surfaces

• NT =

number of measurement points

• P =

power (W)

• T =

temperature (°C)

• $Twinding¯$ =

mean winding temperature (°C)

• X, Y, Z =

axes (m)

• $ε$ =

variation parameter for the sensitivity matrix

• $ξ$ =

relaxation coefficient

Subscripts and Superscripts
• calc =

calculation

• i =

relative to the temperature

• j =

relative to the surface

• k =

relative to the current iteration

• meas =

measurement

Abbreviations
• Aext =

external ambient air

• O =

oil

• TC =

thermocouple

• W =

water

## References

Bertin, Y. , 2006, Refroidissement des Machines Tournantes. Études Paramétriques, Techniques de l'ingénieur, D3462 v1, pp. 1–17.
Staton, D. , Boglietti, A. , and Cavagnino, A. , 2005, “ Solving the More Difficult Aspects of Electric Motor Thermal Analysis in Small and Medium Size Industrial Induction Motors,” IEEE Trans. Energy Convers., 20(3), pp. 620–628.
Staton, D.-A. , Popescu, M. , Hawkins, D. , and Boglietti, A. , 2010, “ Influence of Different End Region Cooling Arrangements on End-Winding Heat Transfer Coefficients in Electrical Machines,” Energy Conversion Congress and Exposition (ECCE), Atlanta, GA, Sept. 12–16, pp. 1298–1305.
Hay, N. , Lampard, D. , Pickering, S. , and Roylance, T.-F. , 1995, “ Heat Transfer From the Stator End-Windings of a Low-Voltage Concentric-Wound Electric Motor,” Seventh International Conference on Electrical Machines and Drives, Durham, UK, Sept. 11–13, Paper No. 412.
Ma, C. , and Zheng, Q. , 1997, “ Local Heat Transfer and Recovery Factor With Impinging Free-Surface Circular Jets of Transformer Oil,” Int. J. Heat Mass Transfer, 40(18), pp. 4295–4308.
Zhou, D. , and Ma, C. , 2006, “ Radial Heat Transfer Behavior of Impinging Submerged Circular Jets,” Int. J. Heat Mass Transfer, 49, pp. 1719–1722.
Pais, M. , and Chow, L. , 1995, “ Jet Impingement Cooling in VSCF Generators,” SAE Paper No. 951451.
Sun, H. , Ma, C. , and Chen, Y. , 1998 “ Prandtl Number Dependence of Impingement Heat Transfer With Circular Free-Surface Liquid Jets,” Int. J. Heat Mass Transfer, 41(10), pp. 1360–1363.
Wadsworth, D. , and Mudawar, I. , 1992, “ Enhancement of Single-Phase Heat Transfer and Critical Heat Flux From an Ultra-High-Flux Simulated Microelectronic Heat Source to a Rectangular Impinging Jet of Dielectric Liquid,” ASME J. Heat Transfer, 114(3), pp. 764–768.
Womac, D. , Aharoni, G. , and Incropera, F. , 1993, “ Correlating Equations for Impingement Cooling of Small Heat Sources With Single Circular Liquid Jets,” ASME J. Heat Transfer, 115(1), pp. 106–115.
Kim, J. , 2007, “ Spray Cooling Heat Transfer: The State of the Art,” Int. J. Heat Fluid Flow, 28(4), pp. 753–767.
Silk, E. A. , Kim, J. , and Kiger, K. , 2006, “ Spray Cooling of Enhanced Surfaces: Impact of Structured Surface Geometry and Spray Axis Inclination,” Int. J. Heat Mass Transfer, 49(25–26), pp. 4910–4920.
Oliphant, K. , Webb, B. , and McQuay, M. , 1998, “ An Experimental Comparison of Liquid Jet Array and Spray Impingement Cooling in the Non-Boiling Regime,” Exp. Therm. Fluid Sci., 18(1), pp. 1–10.
Karwa, N. , Kale, S. , and Subbarao, P. , 2007, “ Experimental Study of Non-Boiling Heat Transfer From a Horizontal Surface by Water Sprays,” Exp. Therm. Fluid Sci., 32(2), pp. 571–579.
Okoro, O. , 2003, “ Thermal Analysis of an Induction Motor Using MATLAB,” Greenwich J. Sci. Technol., 4(1), pp. 5–15.
Lindström, J. , 1999, “ Thermal Model of a Permanent-Magnet for a Hybrid Electric Vehicle,” Licentiate thesis, Chalmers University of Technology, Göteborg, Sweden.
Fan, J. , Zhang, C. , Wang, Z. , and Dong, Y. , 2010, “ Thermal Analysis of Permanent Magnet Motor for the Electric Vehicle Application Considering Driving Duty Cycle,” IEEE Trans. Magn., 46(6), pp. 2493–2496.
Davin, T. , Pellé, J. , Harmand, S. , and Yu, R. , 2015, “ Experimental Study of Oil Cooling Systems for Electric Motors,” Appl. Therm. Eng., 75, pp. 1–13.
Luke, S. , 2013, Essentials of Metaheuristics, 2nd ed., Lulu, Morrisville, NC.
Beck, T. , Bieler, A. , and Thomas, N. , 2012, “ Numerical Thermal Mathematical Model Correlation to Thermal Balance Test Using Adaptive Particle Swarm Optimization (APSO),” Appl. Therm. Eng., 38, pp. 168–174.
Bornschlegell, A. , 2012, “ Etude Aérothermique d'une Machine Synchrone Lente,” Ph.D. thesis, University of Lille, Villeneuve-d'Ascq, France.
Kreith, F., 1969, “ Convection Heat Transfer in Rotating Systems,” Adv. Heat Transfer, 5, pp. 129–251.
Owen, J. , and Rogers, M. , 1989, “ Flow and Heat Transfer in Rotating Disk Systems,” Rotor-Stator Systems, Vol. 1.
Pereira, R. H. , Braga, S. L. , and Parise, J. A. R. , 2012, “ Single Phase Cooling of Large Surfaces With Square Arrays of Impinging Water Sprays,” Appl. Therm. Eng., 36, pp. 161–170.
View article in PDF format.

## References

Bertin, Y. , 2006, Refroidissement des Machines Tournantes. Études Paramétriques, Techniques de l'ingénieur, D3462 v1, pp. 1–17.
Staton, D. , Boglietti, A. , and Cavagnino, A. , 2005, “ Solving the More Difficult Aspects of Electric Motor Thermal Analysis in Small and Medium Size Industrial Induction Motors,” IEEE Trans. Energy Convers., 20(3), pp. 620–628.
Staton, D.-A. , Popescu, M. , Hawkins, D. , and Boglietti, A. , 2010, “ Influence of Different End Region Cooling Arrangements on End-Winding Heat Transfer Coefficients in Electrical Machines,” Energy Conversion Congress and Exposition (ECCE), Atlanta, GA, Sept. 12–16, pp. 1298–1305.
Hay, N. , Lampard, D. , Pickering, S. , and Roylance, T.-F. , 1995, “ Heat Transfer From the Stator End-Windings of a Low-Voltage Concentric-Wound Electric Motor,” Seventh International Conference on Electrical Machines and Drives, Durham, UK, Sept. 11–13, Paper No. 412.
Ma, C. , and Zheng, Q. , 1997, “ Local Heat Transfer and Recovery Factor With Impinging Free-Surface Circular Jets of Transformer Oil,” Int. J. Heat Mass Transfer, 40(18), pp. 4295–4308.
Zhou, D. , and Ma, C. , 2006, “ Radial Heat Transfer Behavior of Impinging Submerged Circular Jets,” Int. J. Heat Mass Transfer, 49, pp. 1719–1722.
Pais, M. , and Chow, L. , 1995, “ Jet Impingement Cooling in VSCF Generators,” SAE Paper No. 951451.
Sun, H. , Ma, C. , and Chen, Y. , 1998 “ Prandtl Number Dependence of Impingement Heat Transfer With Circular Free-Surface Liquid Jets,” Int. J. Heat Mass Transfer, 41(10), pp. 1360–1363.
Wadsworth, D. , and Mudawar, I. , 1992, “ Enhancement of Single-Phase Heat Transfer and Critical Heat Flux From an Ultra-High-Flux Simulated Microelectronic Heat Source to a Rectangular Impinging Jet of Dielectric Liquid,” ASME J. Heat Transfer, 114(3), pp. 764–768.
Womac, D. , Aharoni, G. , and Incropera, F. , 1993, “ Correlating Equations for Impingement Cooling of Small Heat Sources With Single Circular Liquid Jets,” ASME J. Heat Transfer, 115(1), pp. 106–115.
Kim, J. , 2007, “ Spray Cooling Heat Transfer: The State of the Art,” Int. J. Heat Fluid Flow, 28(4), pp. 753–767.
Silk, E. A. , Kim, J. , and Kiger, K. , 2006, “ Spray Cooling of Enhanced Surfaces: Impact of Structured Surface Geometry and Spray Axis Inclination,” Int. J. Heat Mass Transfer, 49(25–26), pp. 4910–4920.
Oliphant, K. , Webb, B. , and McQuay, M. , 1998, “ An Experimental Comparison of Liquid Jet Array and Spray Impingement Cooling in the Non-Boiling Regime,” Exp. Therm. Fluid Sci., 18(1), pp. 1–10.
Karwa, N. , Kale, S. , and Subbarao, P. , 2007, “ Experimental Study of Non-Boiling Heat Transfer From a Horizontal Surface by Water Sprays,” Exp. Therm. Fluid Sci., 32(2), pp. 571–579.
Okoro, O. , 2003, “ Thermal Analysis of an Induction Motor Using MATLAB,” Greenwich J. Sci. Technol., 4(1), pp. 5–15.
Lindström, J. , 1999, “ Thermal Model of a Permanent-Magnet for a Hybrid Electric Vehicle,” Licentiate thesis, Chalmers University of Technology, Göteborg, Sweden.
Fan, J. , Zhang, C. , Wang, Z. , and Dong, Y. , 2010, “ Thermal Analysis of Permanent Magnet Motor for the Electric Vehicle Application Considering Driving Duty Cycle,” IEEE Trans. Magn., 46(6), pp. 2493–2496.
Davin, T. , Pellé, J. , Harmand, S. , and Yu, R. , 2015, “ Experimental Study of Oil Cooling Systems for Electric Motors,” Appl. Therm. Eng., 75, pp. 1–13.
Luke, S. , 2013, Essentials of Metaheuristics, 2nd ed., Lulu, Morrisville, NC.
Beck, T. , Bieler, A. , and Thomas, N. , 2012, “ Numerical Thermal Mathematical Model Correlation to Thermal Balance Test Using Adaptive Particle Swarm Optimization (APSO),” Appl. Therm. Eng., 38, pp. 168–174.
Bornschlegell, A. , 2012, “ Etude Aérothermique d'une Machine Synchrone Lente,” Ph.D. thesis, University of Lille, Villeneuve-d'Ascq, France.
Kreith, F., 1969, “ Convection Heat Transfer in Rotating Systems,” Adv. Heat Transfer, 5, pp. 129–251.
Owen, J. , and Rogers, M. , 1989, “ Flow and Heat Transfer in Rotating Disk Systems,” Rotor-Stator Systems, Vol. 1.
Pereira, R. H. , Braga, S. L. , and Parise, J. A. R. , 2012, “ Single Phase Cooling of Large Surfaces With Square Arrays of Impinging Water Sprays,” Appl. Therm. Eng., 36, pp. 161–170.

## Figures

Fig. 1

Schematics of the test machine (left) and characteristic principal dissipation powers (right): from the winding to the oil (O), water (W) and external air (Aext), total values for both sides

Fig. 2

Mesh grid represented in the axial and radial sections

Fig. 3

Internal oil/air mixture convection on different surfaces of the machine (left); grid 3 used for identification (right)

Fig. 4

Applied algorithm of the inverse method (minimizing the criterion function)

Fig. 5

Sketch of the thermocouples in the test machine (left) and photograph of a thermocouple stuck on a flange (right)

Fig. 6

Evolution of the identified coefficients (left) and the temperature differences (right) during the algorithm progress

Fig. 7

Influence of the convection surface grid for M8 tests at a 50 °C oil

Fig. 8

Influence of the assumptions on the internal air measurements, tests with air only, Twinding¯ = 110 °C

Fig. 9

Comparison with the literature [2] of the convection coefficient on the internal surfaces of the side-chamber

Fig. 10

Characteristic identified coefficients for oil tests (here for M8 nozzle injection, 50 °C oil, 102 L/h)

Fig. 11

Coefficients identified on end-windings for tests with oil at 75 °C

Fig. 12

Comparison of the coefficient identified on end-windings for all datasets

Fig. 13

Coefficients identified on the flange+carter zone for tests with a 75 °C oil

Fig. 14

Coefficients identified on the stack side for tests with oil at 75 °C

Fig. 15

Influence of the oil temperature on the convection coefficient on the flange for the dripping (top) and multijet (bottom) injectors

## Tables

Table 1 Characteristic configuration: conditions (top) and power balance (bottom) on heat transfer with the oil/air mixture
Table 2 Increase in the convection coefficient between 50 and 75 °C at the maximum relative flow rate (averaged on the rotor speed range)
Table 3 Mean criterion function by dataset

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