Temperature Rising Analysis of Dry-type Reactor Based on Multi-physical Coupled-field Method

To study the temperature-rising characteristics of the windings of the dry-type reactor, a three-dimensional (3D) finite element (FE) simulation model of electromagnetic-fluid-temperature coupling of the reactor is constructed by using the method of the multi-physical field coupled modeling. Firstly, based on the magnetic-circuit coupled theory, the electromagnetic analytical model of the reactor is created and analyzed, and the heat density per unit volume of the reactor is obtained. Then, based on the heat transfer principle of fluid-temperature coupling, taking the heat density per unit volume of the reactor as the heat source, the 3D model of the reactor with fluid-temperature coupling is analyzed by the finite volume method, and the distribution of the internal temperature of the reactor is obtained. From the simulation results of the reactor, the temperature in the upper region is higher than that in the lower region, and the temperature of both sides is lower than that of the middle region. The change regulations of the temperatures in the different encapsulations are basically the same. The temperature gradually increases from the bottom to the top, and the maximum temperature locates at approximately 90% of the axial height of the reactor. The research results provide reference for structure modification, temperature-rising monitoring and overheating fault diagnosis of the reactor.


Introduction
Dry-type reactor, which has been widely used in long-distance transmission engineering, has the great advantages of easy maintenance, simple structural configuration, and low cost, and plays a significant role in voltage stabilization, filtering, current limiting, and reactive power compensation in the operation of the power grid [1,2].However, due to the external environment and its own structure characteristics, the reactor often produces local high temperature and overheating during the long-term operation process, which may lead to local burnout of the reactor [3].As the volume of the dry-type reactor tends to be larger, the number of encapsulation layers of the reactor increases accordingly, and the whole structure also becomes more and more complex.The structure with multiple encapsulations makes the temperature of the hot spots very difficult to monitor, which brings new challenges to the operation and maintenance of the equipment.Therefore, the study for temperature distribution characteristics of the reactor has great practical significance for the effective temperature rise monitoring and the prevention of overheating fault of the reactor.
At present, the methods such as the analytic method [4,5], experimental method [6,7] and finite element method [8][9][10] are often used for the research on characteristics of temperature distribution of the reactor.The distribution of the average temperature for the reactor windings can be quickly obtained using calculation formula on the average temperature rise [5].The characteristics of the temperature distribution of the reactor is acquired by solving the partial differential equations by the analytic method in the reference [5].Experimentally, the increase value of the reactor temperature is obtained using fiberoptic temperature sensors [6,7].In terms of numerical simulation, the finite element method for steadystate thermodynamic analysis is employed to achieve the distribution of the internal temperature and the temperature rise of the hot spots of the reactor [8].In recent years, with the great development of modern computer technology and the multi-physical coupling analysis technology, the finite element method based on fluid-temperature coupling is widely used in the temperature-field simulation of the reactor.The characteristics of the temperature distribution for the reactor under fluid-temperature coupling are obtained by the finite element simulation [9,10].However, the finite element simulation models in these literatures are simplified too much, and the temperature difference is ignored as each encapsulation is constructed without considering the number of the encapsulation layers.In this case, the model cannot accurately reflect the temperature distribution of the reactor.Therefore, a 3D FE model of the reactor with the encapsulations constructed in layer can describe the temperature distribution characteristics of the internal windings of the reactor encapsulation in detail.
In his paper, the BKGKL-2000/35 dry-type reactor is taken as the research object, and a 3D model with electromagnetic-fluid-temperature coupling is constructed for the simulation of temperature rise of the dry-type reactor by the multi-physical coupled-field method.The winding loss obtained by electromagnetic-circuit coupling using the 3D model is taken as the heat source.The temperature distribution of each layer of the reactor encapsulation are acquired by solving the fluid-temperature coupling 3D model of the reactor using the finite volume method.Base on the above, the axial and radial temperature changes of the reactor are analyzed.

Electromagnetic-circuit coupling equation
In electromagnetic analysis, the reactor can be simplified and equivalent to an axisymmetric structure without considering the influence of non-ferromagnetic materials such as encapsulation braces, interturn insulation, and rain caps.There is alternating voltage on both sides of each layer winding of the reactor.The electromagnetic-circuit coupling method follows Maxwell equation, and the basic equation for solving the magnetic field in the domain [11] can be written as: Where   is the circumferential component of the magnetic vector potential, and 、z、r are the angle, axial coordinate, and radius in a cylindrical coordinate system.is the magnetic permeability of the aluminum wire; and the current densityflowing through each layer of the winding can be expressed as Where I is the winding current, and n is the number of the winding turns, andis the area of the windings, and i is the number of the windings layer.The current density outside the winding is 0, and the equation of the magnetic field is written as The circuit equation satisfied by the i-th layer winding is Whereis the winding voltage, andis the resistance of the winding, andis the winding current, and  is the magnetic linkage of the winding, andcan be expressed as

Heat source
The heat source of the reactor is mainly composed of the resistance loss Pr(i) and eddy current loss Pe(i) of each layer of the winding.The cross-section of the aluminum wire is circular.The expressions for the loss of the i-th winding layer [12] are as follows.
Where Ri is the resistance of the i-th winding layer.is the conductivity of the aluminum wire.Di is the winding diameter of the aluminum wire of the i-th winding layer.Ji is the current density flowing through the i-th winding layer.di is the diameter of the aluminum wire of the i-th winding layer.is the angular frequency of the current excitation.Br (i) and Bh (i), which are located at the center of aluminum conductor of the i-th winding layer, are respectively the radial component and axial component of magnetic induction intensity.
The heat density per unit volume of the i-th layer winding [2] is expressed as Where Vi indicates the volume of the i-th winding layer.

Analysis of fluid temperature Coupled field
During the working process, the reactor mainly transfers heat with the surrounding air through natural convection, radiation heat dissipation, heat conduction.When the coupling method of fluid temperature field is used for analysis, in addition to meeting the equation of heat conduction, the fluid governing equations are also followed.So the conservation equations for the mass, the momentum and the energy should be satisfied.
Based on the heat transfer theory [13], the steady-state heat conduction equation of the 3D model of the reactor is where  = /, and  = /.T indicates the surface temperature of the windings.k indicates the coefficient of the thermal conductivity.F indicates the intensity of the heat source.c indicates the capacity of the specific heat. indicates the density of the material.Usually, the air is regarded as incompressibility fluid, and it follows the Conservation of mass in the flow process Where  indicates the density of the air.u, v and w are respectively the components of the velocity field vector of the fluid on the x, y, and z axes.The flow of the fluid satisfies the conservation of momentum Where  indicates the kinematic viscosity of the fluid, and p is the pressure of the fluid.The energy inside the reactor is transmitted in the form of thermal energy which meets the conservation of energy, and its expression can be expressed as Where u is the velocity vector of the fluid.Tf is the temperature of the fluid. is the thermal conductivity of the fluid.S indicates the viscous and dissipation of the fluid.

Reactor structure
In this paper, the dry-type reactor studied is shown in Figure 1.The reactor has a rated voltage of 35 kV and a rated current of 989.74A.The inner diameter of the reactor is 2000 mm.The outer diameter of the reactor is 2700 mm.The height of the reactor is 2105mm.Table 1 gives the specific parameters of the reactor.The reactor chiefly consists of 5 layers of coaxial encapsulations, a rain cover, several support bars and star shaped brackets.Each encapsulation is wrapped in glass fiber impregnated with epoxy resin adhesive, and a pull rod is taken as the support between the adjacent encapsulations, which builds a good channel for heat dissipation.The encapsulation is compacted and fixed by two star shaped brackets, and each encapsulation is consist of 20 layers of wires, with the head and tail of each layer of wires welded to the star shaped bracket.

Model simplification and assumptions
The 3D electromagnetic-circuit coupling model is constructed for the dry-type reactor by Maxwell software.The following assumptions are taken into considertion for the model.
(1) The simulation model for electromagnetic field is based on each layer of the winding without considering the influence of other factors such as the braces, rain shields, and star frames on the electromagnetic field.
(2) The current is a mapping excitation, and the current of the windings is uniformly distributed.
(3) The material of the reactor encapsulation is considered as isotropic.

Loads and boundary conditions
The boundary conditions for the simulation model of the reactor are as follows.
(1) Reynolds number [14] of the innermost winding of the reactor is defined as Re=2Vr/v.The average flow velocity of the air V is equal to 0.87m/s, the minimum coil radius of the winding r is equal to 1m.  is the kinematic viscosity of the air, and it is 16.96×10 -6 m2/s.Therefore, the turbulent model is determined for calculation as Reynolds number Re>2320.
(2) The heat source of the reactor is mainly composed of the resistance loss and eddy current loss of each layer of the winding, and the heat density per unit volume is taken as the heat source applied to the temperature field.
(3) According to the actual situation, the ambient temperature is suppose to be 27 ℃, namely 300.15K.
(4) The fluid domain at the bottom of the reactor is the inlet boundary with an axial air flow velocity V of 0.87m/s and a radial flow velocity of zero.The top fluid domain is the pressure-outlet boundary with the relative pressure zero.

Modeling for Multi-physics coupling simulation
The temperature rise model of the dry-type reactor is established with the method of Multi-physics coupling, and the modeling and calculation process is as follows.
(1) The 3D Geometric model of the reactor is constructed by means of CAD software UG NX.
(2) The Geometric model is imported into the electromagnetic module Maxwell 3D in the software Workbench.The eddy current field analysis is performed as the analysis type, and the electromagnetic simulation model, as shown in Figure 2, is established to solve the eddy current loss.

Figure 2. Electromagnetic simulation model of reactor
(3) When the eddy current loss is obtained, the resistance loss is solved by the analytic method, and the heat density per unit volume for the reactor is obtained through the above formula (9).
(4) In Fluent, the simulation model is established to realize the coupling of fluid and temperature.The heat density per unit volume is taken as the heat source which is added to the model with fluid temperature coupled.The distribution of the temperature of the reactor is acquired by the steps of setting the boundary conditions, solving the equations and doing the post-processing.
The modeling and analysis process for the reactor simulation model of the electromagnetic-thermalfluid coupling on ANSYS Workbench platform is shown in Figure 3.

Simulation of electro-magnetic circuit coupling model
The electro-magnetic circuit coupling simulation of the reactor is perforemed, and then the distribution diagram of the magnetic induction intensity is obtained, as shown in Figure 4.It can be seen that the reactor magnetic field is basically symmetrical about the central axis line of the encapsulation.The maximum value of the magnetic induction intensity is located at the central height position of the first layer of the encapsulation, and the direction of the entire magnetic field complies with the ampere principle and superposition theorem.   5.It can be seen from figure 5 that the magnetic induction intensity of each layer of the winding has very little difference in radial direction.The change trend of the magnetic induction intensity is that both ends decrease towards the central height position, and it is close to zero at the central height position.From Figure 6, the change for the magnetic induction intensity of the first to the third layers increases first and then decreases in the axial direction.The change for the fourth and fifth layers is opposite.The magnetic induction intensity decreases first and then increases.It is obvious that from the fourth layer, the axial component of the magnetic field has a reversal of 180 °.The magnetic induction intensity is the superposition of the magnetic fields generated by each layer of the winding currents in space.When the coil radius of the reactor is greater than a certain distance, the direction of the magnetic field changes.

Simulation and Analysis of Fluid Temperature Field
For the purpose of realizing the coupling of the fluid temperature field, the volume mesh is generated by defining the polyhedron mesh and Boundary layer mesh, and then the mesh is generated with the number of nodes 354207.As shown in Figure 7, the fluid temperature simulation model is established by the finite element method.The heat density per unit volume of each layer of the winding is applied to the simulation mdoel as the excitation of the heat source.The distribution for the temperature of the reactor is obtained through iterative calculation for the fluid temperature simulation model, as shown in Figure 8.
From Figure 8, the distribution of the temperature for the reactor shows the trend that the temperature of the upper region is higher than that of the lower region.The temperature of the middle region for the encapsulation is higher than that of the two sides.The change regularity of the inner encapsulation can be better reflected in the unit of layer.The hot spots are mainly concentrated in the upper area of the third and fourth encapsulations.The maximum temperature is 359.51Klocated at 1.98m above the top of the third encapsulation, and the temperature has increased by 59.51K.The temperature of the reactor encapsulations decreases from top to bottom along the axis direction.The encapsulations of the two side directly exchanges heat with the air outside, which ensures sufficient convection and conduction.Therefore, the temperature on both sides of the encapsulations is lower than that of the middle region.9, is obtained by the simulation for multiphysical fields of the reactor.From Figure 9, the maximum flow velocity of the air in the solution domain is 1.096m/s, and it is located in the entrance area at the bottom of the first layer of the encapsulation.As the sectional area of the entrance between the two layers of the encapsulation decreases, the flow velocity of the air increases.After absorbing heat from the bottom of the encapsulation, the air accelerates to rise under the influence of natural convection and gets into the air duct at high speed.The effect of the air convection and heat dissipation become weaker after the air flowing into the duct.As the air flow velocity is accelerated to a certain value, the speed of the air tends to be stable.When the hot air flows to the top outlet, it is gradually farther from the heat source and the flow velocity decreases accordingly.11 shows that the radial distribution of the reactor temperature is similar to the square distribution.The peak of the square wave represents the temperature value of the encapsulation, and the trough of the square wave represents the air passage temperature between the two adjacent encapsulations.In the encapsulation, the loss of the winding is the heat source, and it gives rise to a relatively high temperature.The temperature of the air passage is relatively low between every two encapsulations due to the influence of heat dissipation.There is a little difference in temperature in different layers of the encapsulation in the radial direction.The temperature of each layer within the same encapsulation are relatively close, and the temperature of the middle layer of the same encapsulation is slightly higher than that of the both sides.Because the heat dissipation performance is slightly poor in the middle layer of the reactor, which leads to an increase in temperature.12 shows that there are differences in temperature among different encapsulations.However, the changes for the temperature are basically the same along the axial direction of the encapsulation.The temperature shows an upward trend from the bottom to the top.The maximum temperature is located at approximately 90% of the axial height of the reactor.The air outlet is at the top of the encapsulation, and the air temperature decreases as air flows out and comes into direct contact with the external air.
From the analysis results, it is known that the heat is dissipated through the air duct to keep the temperature within the specified range in the working condition of the reactor.However, the top region of the reactor is the area with significant temperature rise, and the local high temperature may arise easily in this area.Therefore, more attentions should be paid to the top region of the reactor.If the reactor operates at high temperature for a long time, fire accidents may occur and result in the equipment damage and shutdown.(2) The temperature distributions of the radial and axial directions for the reactor are obtained.The temperature in the middle region of the encapsulation is higher than that on both sides in the radial direction.The encapsulation temperature gradually increases from bottom to top in the axial direction, and the maximum temperature is located at approximately 90% of the axial height in the encapsulation.
(3) The temperature distribution of the reactor with layer as the unit can better reflect the difference caused by the location of the heat source, which provides reference for the online monitoring, operation and maintenance of the reactor.

Figure 1 .
Figure 1.Structural composition of dry-type reactor

Figure 3 .
Figure 3. Process of Multi-physics coupling simulation of reactor by ANSYS Workbench

Figure 4 .
Figure 4. Distribution diagram for magnetic induction intensity of reactorThe radial distribution of the magnetic induction intensity of the reactor is shown in Figure5.It can be seen from figure5that the magnetic induction intensity of each layer of the winding has very little difference in radial direction.The change trend of the magnetic induction intensity is that both ends decrease towards the central height position, and it is close to zero at the central height position.From Figure6, the change for the magnetic induction intensity of the first to the third layers increases first and then decreases in the axial direction.The change for the fourth and fifth layers is opposite.The magnetic induction intensity decreases first and then increases.It is obvious that from the fourth layer, the axial component of the magnetic field has a reversal of 180 °.The magnetic induction intensity is the superposition of the magnetic fields generated by each layer of the winding currents in space.When the coil radius of the reactor is greater than a certain distance, the direction of the magnetic field changes.

Figure 5 .Figure 6 .
Figure 5. Radial distribution for magnetic induction intensity of reactor

Figure 7 .Figure 8 .
Figure 7. Simulation model of fluid temperature field of reactor

Figure 9 .
Figure 9. Flow velocity distribution of the air after simulation for multiphysical fields5.3.Temperature distribution of encapsulationFigure10shows the cloud diagrams of the temperature distributions for the first, third and fifth layers of the encapsulations respectively.The temperature gradient of the encapsulation can be seen from the diagrams in Figure10.The radial and axial temperature distributions for the different encapsulations can be simulated by the software ANSYS.It is obvious that the encapsulation in the middle layer has higher temperature than the encapsulation on both sides.

Figure 10 .
Figure 10.Temperature distribution of each encapsulation of the reactorTo measure the distribution of the reactor temperature, 100 measurement points are set radially from the 1st layer encapsulation in the inner side to the 5th layer encapsulation in the outer side.Figure11shows that the radial distribution of the reactor temperature is similar to the square distribution.The peak of the square wave represents the temperature value of the encapsulation, and the trough of the square wave represents the air passage temperature between the two adjacent encapsulations.In the encapsulation, the loss of the winding is the heat source, and it gives rise to a relatively high temperature.The temperature of the air passage is relatively low between every two encapsulations due to the influence of heat dissipation.There is a little difference in temperature in different layers of the encapsulation in the radial direction.The temperature of each layer within the same encapsulation are relatively close, and the temperature of the middle layer of the same encapsulation is slightly higher than that of the both sides.Because the heat dissipation performance is slightly poor in the middle layer of the reactor, which leads to an increase in temperature.
Figure 10.Temperature distribution of each encapsulation of the reactorTo measure the distribution of the reactor temperature, 100 measurement points are set radially from the 1st layer encapsulation in the inner side to the 5th layer encapsulation in the outer side.Figure11shows that the radial distribution of the reactor temperature is similar to the square distribution.The peak of the square wave represents the temperature value of the encapsulation, and the trough of the square wave represents the air passage temperature between the two adjacent encapsulations.In the encapsulation, the loss of the winding is the heat source, and it gives rise to a relatively high temperature.The temperature of the air passage is relatively low between every two encapsulations due to the influence of heat dissipation.There is a little difference in temperature in different layers of the encapsulation in the radial direction.The temperature of each layer within the same encapsulation are relatively close, and the temperature of the middle layer of the same encapsulation is slightly higher than that of the both sides.Because the heat dissipation performance is slightly poor in the middle layer of the reactor, which leads to an increase in temperature.

Figure 11 .
Figure 11.Radial temperature distribution of the encapsulation Figure12shows that there are differences in temperature among different encapsulations.However, the changes for the temperature are basically the same along the axial direction of the encapsulation.The temperature shows an upward trend from the bottom to the top.The maximum temperature is located at approximately 90% of the axial height of the reactor.The air outlet is at the top of the encapsulation, and the air temperature decreases as air flows out and comes into direct contact with the external air.From the analysis results, it is known that the heat is dissipated through the air duct to keep the temperature within the specified range in the working condition of the reactor.However, the top region of the reactor is the area with significant temperature rise, and the local high temperature may arise easily in this area.Therefore, more attentions should be paid to the top region of the reactor.If the reactor operates at high temperature for a long time, fire accidents may occur and result in the equipment damage and shutdown.

Figure 12 .
Figure 12.Axial temperature distribution of the encapsulation 6. Summary (1) The established 3D FE model of electromagnetic-fluid-temperature coupling can accurately predict the results of the temperature rise of the reactor.The maximum temperature is 359.51Klocated at 1.98m above the top of the third encapsulation, and the temperature has increased by 59.51K.The obtained results are consistent with engineering practice, which validates the effectiveness of the method.(2)The temperature distributions of the radial and axial directions for the reactor are obtained.The temperature in the middle region of the encapsulation is higher than that on both sides in the radial direction.The encapsulation temperature gradually increases from bottom to top in the axial direction, and the maximum temperature is located at approximately 90% of the axial height in the encapsulation.(3)The temperature distribution of the reactor with layer as the unit can better reflect the difference caused by the location of the heat source, which provides reference for the online monitoring, operation and maintenance of the reactor.