Calculation of Temperature Rise in Dry-type Air-core Reactors Using Strong Coupling of Fluid-Temperature Field

The ventilation system design of dry-type air-core reactor is a complex task that must determine the thermal loads to achieve the maximum insulation material exploitation. In this study, the temperature rise in reactor is due to Joule’s losses and heat dissipation by air convection, convection and radiation. The Joule’s losses calculated by coupled magnetic field-circuit analysis are used as the input for the thermal field by finite-element analysis, which is directly coupled with fluid analysis. Finally, the temperature distributions of reactor can be calculated. Therefore, the thermal performance analysis of air-core reactor could be conducted in the early design stage to guarantee the insulation material requirements.


INTRODUCTION
In power system, dry-type air-core reactor is employed to limit current, steady voltage and compensate reactive power.In practical applications, air-core reactor during operation is prone to faults, including partial discharge, overheating and burnout (Liu et al., 2003).In the case of dry-type reactors with fully encapsulated windings, the insulation between turns is usually a solid dielectric material.In selecting appropriate materials to be utilized as turns' insulation on the conductors, it is necessary to evaluate dielectric properties, mechanical properties and aging characteristics under operating conditions.Performance at temperature is one of the main evaluating criteria for materials, so the temperature rise is one of the main aging mechanisms that have to be considered, especially for the overall system (IEEE Std., 1996).
Accurate calculation of the temperature distribution is pivotal to the optimal design, which is a multiphysical coupled process that involves electromagnetic losses as well as fluid dynamic and thermal behavior.In traditional research, the average temperature rise of reactor was calculated by empirical formula (Wu et al., 1997).However, even if average temperature rise satisfies design requirement, the hottest-spot temperature in reactor might exceed the maximum temperature limits of insulation materials.Moreover, thermal field was simulated by finite-element method (FEM) after calculating the heat transfer coefficient by using Nusselt number (Liu, 1991;Wu et al., 2002;Ho et al., 2006Ho et al., , 2007)).Nevertheless, the fluid dynamic behavior cannot be accurately described not by simulating fluid field.The calculation results were very different with the measurement results, so the heat transfer coefficient must be repeatedly modified.Therefore, a coupled analysis of fluid field and thermal field is mandatory to compute the temperature rise in the design stage (Zhang et al., 2012a).
In this study, we calculated temperature distributions by using the coupled magnetic field-circuit method and coupled fluid-dynamical and thermal finite-element analysis in a type of dry-type air-core reactor, as shown in Fig. 1.Through the big ventilation duct, cooling air taken in from the exterior to the interior of reactor by fan is sent to exothermic parts such as the windings.On the top of reactor, there are two layers of rainproof shield for protecting the reactor against rain.A 2-D computational model can be established to analyze the multi-physics simulation due to the axial symmetry.The Joule's losses calculated by the coupled field-circuit analysis are used as the input for the thermal field analysis, which is deeply dependent on accurate air fluid field analysis.

THEORY AND FORMULATION
Coupled magnetic field-circuit calculation: The drytype air-core reactor contains parallel several encapsulated windings and each of them contains parallel several layers of small-diameter aluminum conductors.The ac voltage, whose frequency is 50 Hz, is applied to the conductor in each layer.In cylindrical coordinate system, the magnetic field equation can be given as Liu et al. (2003): The terminal voltage of the conductors in the i th layer is given as: where, Ψ i = The flux linkage of conductors in the i th layer N i = The number of turns From Eq. (1-3), the voltage equation can be described with magnetic vector potential (A).These equations can be solved by FEM.After obtaining the value of A, the current can be calculated.Then, Joule's losses can be calculated by Eq. ( 4).For the encapsulated windings, the main heat dissipation mode is heat conduction.For the surface between encapsulated windings and surrounding air, the main heat dissipation mode is heat forced convection and thermal radiation.Finally, our concern is the temperature distributions of each encapsulated winding.
• Heat conduction: For the encapsulated windings, the steady state heat conduction equation for solid is given as Zhang et al. (2012b): where, k = The coefficient of heat conductivity Q = The heat generation of unit volume in aluminum conductors • Heat convection: The forced convection of air satisfies the Navier-Stokes Equations, which consist of three groups of equations.For two-dimensional incompressible steady fluid, the Navier-Stokes Equations in cylindrical coordinate system can be simplified as following (Wu et al., 2002): Continuity equation: ( ) Momentum conservation equations: ( ) Energy equation is given as: where, ρ = The density of air μ = The viscosity coefficient p = Pressure c = The specific heat T = The fluid temperature and υ r and υ z are the velocity in the r-and z-directions, respectively Because of the large Reynolds number (>2300), the fluid gets turbulent.The standard k-ε turbulence model was used in the turbulence calculation (Zhang et al., 2012a).
• Radiation: For a system of two surfaces (surface i and j) radiating to each other, the heat transfer rate between surfaces i and j is expressed as Wu et al. (2002):  The thermal field and fluid field are coupled directly according to Navier-Stokes equations.Firstly, coupled magnetic field-circuit calculation is carried out, then, the Joule heat is obtained.Secondly, the Joule heat is coupled into the thermal field as heat generation rate, in addition to some other proper fluid and thermal boundary conditions, the thermal field and fluid field is simultaneously calculated.Proper boundary conditions should be set as followings.
• Set non-slip boundary condition (υ x = 0, υ y = 0) on the surface of reactor • Set surrounding temperature (here 20°C in accordance with normal temperature) Here all the emissivity is set to 0.9.

CALCULATION RESULTS AND ANALYSIS
After coupled magnetic field-circuit analysis, the Joule's losses can be calculated.Table 2 gives the losses of every aluminum conductor in each encapsulated winding.
In forced air cooling reactor, the cooling air blows from bottom to top and flows through the air passages between two encapsulated windings.From Fig. 4, the temperature distribution in one encapsulated winding is non uniform because the heat conductivity of insulation material is much smaller than that of aluminum conductor.
From Fig. 5, we can find that the flow velocity of air in passage between encapsulated windings, which are near the outside and inside wall, is faster.Therefore, the hottest point is on the encapsulated winding in middle of model, as shown in Fig. 6.The rainproof shield has an important influence on the air velocity among the encapsulated windings.Thus, the air cooling should be considered in the design of rainproof shield.The uniform temperature rise distribution might be achieved by improvement of the geometrical structure of rainproof shield.

CONCLUSION
In this study, the temperature rise of a forced air cooling dry-type air-core reactor is calculated by finite element method for coupled magnetic-thermal-fluid field.The current distribution among the encapsulated windings is analyzed by using coupled magnetic fieldcircuit method.The temperature rise distribution of reactor is tightly related with current Joule's losses in each encapsulated windings, the thermal parameters of insulation materials and the structures of diverse components in reactor, such as rainproof shield which may influence on the air flow among the encapsulated windings.

Fig. 1 :
Fig. 1: This type of reactors in substation of aluminum J = Current density Strong coupling of fluid-temperature field calculation: For the system of the reactor shown in Fig. 1, there are three heat dissipation modes, involving heat conduction, heat convection and thermal radiation.
Fig. 2: The calculation model of reactor

Fig. 4 :
Fig. 4: The hottest temperature of the encapsulated winding

Table 1 :
Main geometry and material properties of the model

Table 2 :
The heat generation rate of every aluminum conductor in each encapsulated winding (W/m 3 ) Layer of aluminum conductor