Modelling of thermal field of electromotor's collector in service.

INTRODUCTION

Modelling of thermal state of node points of electrical machine in service is one of the most important tasks defining its work reliability. Collector is the most complicated node point, because it defines commutation conditions. Limit values of heating temperature is determined by temperature class of insulation, and exceeding them can make electromotor inoperative.

Methods:

For modelling collector's thermal state we use the finite elements method (FEM), which is very popular in these latter days for solving problems of this class [1, 2, 3, 4].

As collector is an axisymmetric structure and its active part is composed of homomorphic commutator segments, let's take a collector's sector having one commutator segment and one insulating pad as a design contour (fig. 1).

We included several details into design contour, so let's admit an assumption that their interfaces have tight thermal contact. This is close to the reality, because, to provide integrity, so-called "arch action" force is in effect in collector, and that produces a pressure in all interfaces. We also admitted an assumption about absence of heat flow going between adjoining commutator segments on lateral faces through insulating pads, that is main heat flow goes along vertical axis through commutator segment. Accepted design contour allows to investigate different thermal state of individual segments, because during motor's work, as a rule, not the whole collector runs hot, but individual segments or groups of segments "burn up".

Initial data for calculation of temperature in the specified contour are following: structure of details, thermophysical properties of materials and heat flows affecting collector. Details structure is presented by coordinates of node points of finite elements composing design contour, thermophysical properties of materials are expressed by heat conduction coefficients, and heat flows impact is taken into account in boundary conditions [5, 6]. According to heat flows affecting collector, the chosen design contour includes boundary conditions of the 2nd type (density of heat flow going through surface) and boundary conditions of the 3rd type (presence of convective heat exchange).

The boundary condition of the 2nd type works for:

1. Collector's contact surface, through which heat flow (arising due to brushes' frictions on collector and voltage drop) enters (fig. 3.1, pos. 2);

2. Thermally insulated surfaces, that is surfaces with heat flow rate equaling to zero (fig. 1, pos. 4). As thermally insulated surfaces we accepted: lateral surfaces of commutator segments, contacting with insulating pads; surfaces of a front pressure-exerting cone and a commutator lug, insulated from armature winding. This assumption was made because of low thermal conductivity coefficient of insulation materials--two orders lower that copper thermal conductivity coefficient.

Strong interrelation of heat generation sources eliminates the possibility of considering collector in isolation from other node points of electrical machine. Therefore, when defining collector's temperature we should take into consideration impact of three main factors [7]:

1. Environmental temperature, that is temperature of air inside the machine, which depends on value of motor's total loss.

2. Value of intrinsic losses on collector due to voltage drop in brushes contacts and brushes frictions on collector's outer surface.

3. Collector's thermal coupling with armature through armature winding sections contacts with commutator segments lugs.

Main part:

Temperature of air flowing around collector in case of ventilated motors can be accepted as such which is equal to temperature of incoming cooling air. When considering self cooled enclosed motors, with some approximation we can assume temperature of air around collector to be an average temperature of air inside motor:

[DELTA][[THETA].sub.B] = [P.sub.[SIGMA]]/([[alpha].sub.B] [S.sub.motor]), (1)

where [delta][[theta].sub.B] is an average exceeding of temperature of air inside machine over outer cooling air temperature, [S.sub.motor] is a conditional cooling surface of machine, [[alpha].sub.B]--coefficient of convective heat exchange of outer cooling surface, [P.sub.[SIGMA]] is total losses.

When defining motor's total losses we should consider all arising losses: fundamental and added ones.

Specific heat flow from collector's losses, which is relegated to collector's cooling surface, can be calculated by the following formula:

[p.sub.coll] = ([p.sub.c.b.] + [p.sub.f.b.])/[S.sub.coll], (2)

where [p.sub.c.b.] is electric losses in brushes' contacts; [p.sub.f.b.] is losses from brushes' frictions on collector; [S.sub.coll] is conditional cooling surface of collector.

Electric losses in sliding contact should be expressed by the formula:

[p.sub.c.b.] = [DELTA][U.sub.b] x [I.sub.2], (3)

where [delta] [U.sub.b] is voltage drop in brushes' contacts; [I.sub.2] is armature current.

Mechanical losses from brushes' frictions on collector should be expressed by the formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

where k is coefficient of brushes' frictions on collector; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is total area of brushes' contact; [V.sub.coll] is collector's periphery speed.

Expression (2) is used during designing of electrical machines to find average exceeding of collector's temperature. Thereat one should make an assumption that all heat losses in brushes and collector contact are firstly absorbed by collector and then transferred into air inside machine via collector's surface.

For analysis of distribution of temperature inside collector let us transform the expression (2) taking into account part of heat flow passed by convection into the air flowing around the collector:

[p.sub.coll] = ([p.sub.c.b.] + [p.sub.f.b.] - [k.sub.o] [[alpha].sub.coll] [S.sub.coll] [DELTA][[THETA].sub.coll])/ [S.sub.coll], (5)

where [k.sub.o] is coefficient considering ratio of heat flow effect and convection; [[alpha].sub.K] is coefficient of convective heat exchange of collector's outer surface; [delta][[theta].sub.K] is exceeding of collector's outer surface temperature over ambient temperature.

Exceeding of collector's outer surface temperature over ambient temperature should be taken for established thermobalance.

Heat flow between armature and collector arises as a result of difference in temperatures of active elements --armature winding sections and commutator segments, having strong interrelation because of a high thermal conductivity coefficient of the material (copper). Thermal coupling is conducted with winding sections along the armature winding through commutator segments lugs joint surfaces (splined groove).

In this regard let us assume collector's thermal coupling with armature to be a one-dimensional field, and for defining direction of motion of heat flow between armature and collector and also for estimating its rate, let us use values of average temperatures of collector copper and armature winding:

Q = ([DELTA][t'.sub.2] - [DELTA][t'.sub.coll]) [lambda]S/[delta], (6)

where [delta] [t'.sub.2] is average exceeding of collector copper temperature over temperature of air inside the machine; [delta] [t'.sub.coll] is average exceeding of armature winding temperature over temperature of air inside the machine; [lambda] is thermal conductivity coefficient of armature winding material (copper); S is winding sections cross-section area; [[delta]] is distance between the commutator segment mid-point and the mid-point of armature section in axial direction.

Sign of the expression (6) characterizes heat flow direction. Sign "+" means heat movement from armature to collector, sign "-"--from collector to armature. This formula defines thermobalance between armature and collector, allowing to use initial data--heat flow rates of armature and collector, via values of their average temperatures.

Actually heat flow between armature and collector is not uniform. Part of heat is passed by the air flowing around armature and collector, also there is a heat flow from the details which are more heated to those which are less heated, that is from copper segments to steel cones in collector and from winding to body in armature. Presence of these two types is indirectly considered in definition of average temperatures of copper of armature and collector, in expression of which heat flows passed into air and contacting details are included. Average exceeding of collector copper temperature over temperature of air inside the machine:

[DELTA][t'.sub.coll] = [p.sub.coll]/[[alpha].sub.coll], (7)

where [[alpha].sub.coll] is coefficient of convective heat exchange of collector's outer cooling surface.

During estimation of collector's thermal state, several expressions ((1), (5), (7)) include coefficient of convective heat exchange. Accuracy of the whole thermal calculation depends to a substantial degree on accuracy of its calculation. Coefficient of heat transmission defines speed of the process and other main characteristics [8, 9].

To estimate coefficient of heat transmission let us use criterial equations [10]. For the case of natural convection (heat transmission from the electromotor body outer surface)

Nu = C[([Pr.sub.0] x [Gr.sub.0]).sup.n], (8)

for the case of forced convection (heat exchange of rotating collector with the air inside the machine)

Nu = C x [Re.sub.0.sup.m] x [Pr.sub.0.sup.n] x [Gr.sub.0.sup.p] ([Pr.sub.0]/[Pr.sub.s]), (9)

Nu = [alpha] x l/[lambda], (10)

where Nu is a dimensionless group called Nusselt criterion; C, n, m, p are coefficients; [Pr.sub.0], [Pr.sub.s] are Prandtl numbers correspondingly for cooling medium and cooling surface; [Gr.sub.0] is Grashof number for cooling medium; [Re.sub.0] is Reynolds criterion for cooling medium; [alpha] is value of heat exchange coefficient which is average for the exposed surface W/[m.sup.2][degrees]C; l is characteristic dimension, m; [lambda] is medium's coefficient of heat conductivity W/m[degrees]C.

Collector is a body of rotation, so we should take its diameter as a characteristic dimension.

Reynolds criterion characterizes speed of medium's movement in relation to solid body:

Re = w x l/v, (11)

where w is flow's speed, m/s; v is medium's kinematic coefficient of viscosity, [m.sup.2]/s.

Certain range of Reynolds criterion numbers defines the circumambient medium's movement mode--laminar or turbulent. In case of collector the circumambient medium's movement may be considered as turbulent one.

Prandtl number characterizes capability of heat to distribute in this medium.

Pr = v /[omega], (12)

where [omega] is medium's temperature conductivity.

Grashof number considers natural convection's impact inside the medium. In case of forced convection, arising during collector's rotating, exponent of power in a criterial equation for Grashof number is equal to zero, therefore, this number equals to one and it can be excluded from the equation (9). In case of natural convection Grashof number is estimated by the expression

Gr = [beta] [gl.sup.3]/[v.sup.2] ([[THETA].sub.s] - [[THETA].sub.0]), (13)

where [beta] is medium's coefficient of volumetric expansion, 1/[degrees]C; g is free fall acceleration, g = 9.81 m/[s.sup.2]; [theta]s, [theta]0 are temperatures of cooling surface and cooling medium correspondingly, [degrees]C.

In case of cross flow around cylindrical surfaces and turbulent nature of the flow the following exponents of power in the criterial equation (9) take place: C = 0.28; m = 0.6; n = 0.36. For conditions of natural convection--(8) C = 0.135; n = 0.33.

Thus, let's write expressions for estimation of heat-conduction coefficient during collector's rotating

[alpha] = 0,28 x [Re.sub.0.sup.0,6] [Pr.sub.0.sup.0,36] [([Pr.sub.0]/[Pr.sub.s]).sup.0.25] [lambda/l, (14)

in case of natural convection of electromotor body

[alpha] = 0,135 [([Gr.sub.0] x [Pr.sub.0]).sup.0,33] [lambda]/l. (15)

For computational experiment we used finite-element packet COSMOS/Works, integrated into CAD-system of solid modeling SolidWorks.

Computational experiment includes following main stages:

1. Creation of a solid model of design contour (fig. 1) via means of SolidWorks.

2. Description of structure materials' properties. For thermal calculation we should set heat conductivity coefficient and volumetric heat capacity of the material. Materials employed may be isotropic and orthotropic. Possibility of setting orthotropic properties is very important, because insulating material consists of resin-impregnated pressed mica, therefore its properties differ through the length and breadth of fibres.

3. Digitalization of computational domain, generation of finite elements grid. For solid bodies COSMOSWorks uses three-dimensional tetrahedral elements which can be of two types: linear (Draft quality mesh) and parabolical (High quality mesh), they are represented in fig. 2.

Linear elements have four node points in vertices. Such configuration provides linear variation of required parameter, in this case--temperature, within the elements volume. Parabolical elements, apart from node points in vertices, have node points in the middle of edges. The required parameter is described by quadratic polynomial. Edges and, consequently, planes can be curvilinear. This allows to reproduce a curvilinear geometrical pattern quite adequately. Taking into account these factors, let's use linear finite elements for development of design models: selection of correct boundary conditions, mode of interaction of details in the construction, parameters of grid seal. After estimating these factors, let's use higher order elements to obtain final results. This approach allows to save a lot of time for preparation of initial data [10].

Fig. 3 shows grid of finite elements of design model of the dirrect current excavating electromotor DPE-52 ([TEXT NOT REPRODUCIBLE IN ASCII]-52).

4. Application of loads and boundary conditions. Let's use expressions (1)-(15).

5. Assignment of analysis type. COSMOSWorks provides stationary and nonstationary calculation of temperature. Stationary calculation should be used for estimation of collector's thermal field at the moment of thermobalance establishment. Nonstationary calculation is not less important. It allows to define collector's thermal field in transient conditions. Nonstationary thermal calculation does not consider changes of boundary conditions through time. If necessary, this limitation may be compensated via dividing of an assigned task into a range of subtasks connected with each other: thermal field, estimated in a previous task, is used as the initial datum for the following one. With the help of this stratagem one can imitate varying boundary conditions and loads, consequently changing them in different subtasks, at the same time within one subtask boundary conditions and loads are constant [11].

Computational experiment was carried out for a mining traction motor DPE-52. According to technical characteristics, heat flow parameter limits for a contact surface are shown in table 1. The electromotor under investigation can work both in blowing mode (with cooling) for a long time and in enclosed design (without cooling) up to 45 min. under nominal speed 1200/2200 rpm.

Conclusion:

Fig. 4 and 5 show results of estimation of collector's thermal field in the established mode with cooling under different nominal speeds.

Results show that commutator's segment has the highest temperature: 87[degrees]C and 108[degrees]C correspondingly under 1200 and 2200 rpm. The higher rotation frequency intensifies heat flow going through the contact surface, therefore the segment heats up at a more intensive rate. At the same time cooling rate also increases due to increase of the coefficient of convective heat exchange, therefore pressure-bearing cones temperature in different modes of functioning varies unissentially. All collector's details temperature is in permissible range from the viewpoint of temperature class of insulation.

Fig. 6, 7 show changes of collector's thermal field in transient condition during functioning without cooling under rotation frequency 1200 rpm: fig. 6 a--after 15, fig. 6 b--after 30, and fig.7--after 45 minutes of working.

Results show (fig. 7) that after 45 minutes of functioning collector's temperature is close to the nominal established one. The further exploitation of the motor without cooling leads to its overheat. As well as during functioning with cooling, commutator segment has the highest temperature. Pressure bearing cones' heating nature changes to some degree. Left cone is more heated, because it is heated by convection with the internal air, which absorbs all heat losses of the motor functioning without forced cooling.

Thermal fields for any stage and mode of the work may be created likewise, and that allows to rationalize construction solutions and technology concepts [12].

Resume:

The developed model of collector's thermal state allows to model collector's thermal field, both in established thermobalance and in transient condition, taking into account heat flows functioning while in service in normal work mode and in modes of different overloads.

ARTICLE INFO

Article history:

Received in revised form 8 May 2014

Accepted 20 July 2014

Available online 18 August 2014

REFERENCES

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Tatyana Aleksandrovna Duyun, Anna Vladimirovna Grinek, Roman Valentinovich Manzhos

Belgorod State Technological University named after V. G. Shukhov, Russia, 308012, Belgorod, Kostyukov str., 46

Corresponding Author: Tatyana Aleksandrovna Duyun, Belgorod State Technological University named after V. G. Shukhov, Russia, 308012, Belgorod, Kostyukov str., 46
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Table 1: Parameters of heat flow along the contact surface.

Title                   Mark                 Value

1    Electric losses in     [p.sub.[kappa].[??].]        375 W
contact

2   Mechanical losses in
contact:             [p.sub.T.[??].]
min under rotation                                  150 W
frequency 1200 rpm
max under rotation                                  450 W
frequency 2200 rpm

3   Part of total losses,
passed by the flowing
air
min under rotation                                  257 W
frequency 1200 rpm
max under rotation                                  352 W
frequency 2200 rpm

4       Total losses
min under rotation                                  268 W
frequency 1200 rpm
max under rotation                                  473 W
frequency 2200 rpm

5    Area of conditional       [S.sub.[kappa]]       0.07 [m.sup.2]
cooling surface

6    Total specific heat       [p.sub.[kappa]]
flow going through
contact surface:
min under rotation                             3829 W/[m.sup.2]
frequency 1200 rpm
max under rotation                             6757 W/[m.sup.2]
frequency 2200 rpm

7      Coefficient of       [[alpha].sub.[kappa]]
convective heat
exchange min under
rotation frequency
1200 rpm                                   46 W/[m.sup.2]
max under rotation                              63 W/[m.sup.2]
frequency 2200 rpm
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