# Thermal Analysis of Air-Core Power Reactors

Authors: Zhao Yuan; Jun-jia He; Yuan Pan; Xiao-gen Yin; Can Ding; Shao-fei Ning; Hong-lei Li
Journal: ISRN Mechanical Engineering
 (2013)

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## Abstract

A fluid-thermal coupled analysis based on FEM is conducted. The inner structure of the coils is built with consideration of both the structural details and the simplicity; thus, the detailed heat conduction process is coupled with the computational fluid dynamics in the thermal computation of air-core reactors. According to the simulation results, 2D temperature distribution results are given and proved by the thermal test results of a prototype. Then the temperature results are used to calculate the heat flux to predict the detailed heat transfer process in the packages of the reactors. The study in this paper may be useful in the design optimization in air-core reactors.


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## Body

## 1. Introduction

With the development of power system, the usage amount of power reactors continues to rise more and more large capacity air-core reactors appear. Air-core reactors with large capacities have more packages and greater heights, which may cause uneven distribution of temperature rise. Thus, the accurate temperature field computational methods are in need.Numerical simulation has been used to analyze the thermal condition of power equipment for a long time. Two categories of numerical approaches were developed as “Network Modeling” [1–6] and finite element method (FEM) [7–10]. The former can describe the heat transfer process clearly, but can only be used when critical temperatures such as hot spots are required and detailed local flow/temperature information is not [5]. The latter is adopted more and more due to its pinpoint accuracy and the detailed information it can provide.Air-core reactors have complex structures with paralleled cylinder packages and vertical air ducts. The heat transfer in them is a hybrid process with heat conduction, convection, and radiation. Thus FEM is obviously better suited than the other computational methods in the thermal computation of air-core reactors. In [11, 12], 2D and 3D fluid-thermal field coupled FEM models are built to compute the temperature field of power transformers and reactors. But the inner structures were not considered and the details of temperature distribution and heat transfer process are not analyzed.In this paper, a fluid-thermal coupled analysis based on FEM is conducted. The inner structure of the coils is built with consideration of both the structural details and the simplicity; thus, the detailed heat conduction process is coupled with the computational fluid dynamics in the thermal computation of air-core reactors. After the analysis, 2D temperature distribution results are given and proved by the test results of a prototype. Then the temperature results are used to calculate the heat flux to predict the detailed heat transfer process in the packages of the reactors. The study in this paper may be useful in the design optimization in air-core reactors.


## 2. Materials and Methods

### 2.1. Structure and Parameters of an Air-Core Reactor Prototype

The typical air-core power reactor contains some cylindrical packages connected in parallel and the vertical air ducts between them, as it is shown in Figure1. The packages comprise metal conductors and the insulating material. The air ducts between the packages are used as heat dissipation channels of the packages.Figure 1
Structure sketch of air-core reactor.In this paper, a prototype of air-core power reactor with five packages is taken as an example, the parameters of which are given in Table1.Table 1
The parameter of an air-core power  reactor prototype.
Package number
Diameter
Current
Ampere density
Height
(m)
(A)
(A/m2)
(m)
1
0.619
36.662
1.329 × 106
0.325
2
0.684
31.081
1.329 × 106
0.325
3
0.747
39.476
1.329 × 106
0.325
4
0.811
44.049
1.329 × 106
0.325
5
0.880
58.732
1.329 × 106
0.325


### 2.2. Mathematical Modal

#### 2.2.1. Computation of Heat Load

The heat sources in the packages include the resistive current loss and extra eddy loss [13–15]. According to [16, 17] the total loss in a package can be simplified as kc times of resistive current loss; thus, the loss of the ith coil can be expressed as
(1)Pi=πkc,iJi2SiWiDiκ,
where Ji is the ampere density of ith coil, Wi is the number of turns of ith coil, Di is the diameter of ith coil, Si is the total sectional area of the metallic conductor in the ith package, and κ is the conductivity of metal conductor.


#### 2.2.2. Heat Transfer Equations

In most cases, heat transfer process contains heat conduction, convection, and radiation.Partial differential equation of heat conduction is given as(2)∂2t∂x2+∂2t∂y2+ϕλ=0,
where t is the temperature, ϕ is loss function, and λ is the thermal conductivity of the considered material.Heat convection process is closely related to the fluid states, which can be described by the continuity equation, the momentum equation, and the energy equation as shown in (3) the following equation:
(3)∂u∂x+∂v∂y=0,ρ(u∂u∂x+v∂u∂y)=Fx-∂p∂x+η(∂2u∂x2+∂2u∂y2),ρ(u∂v∂x+v∂v∂y)=Fy-∂p∂y+η(∂2v∂x2+∂2v∂y2),u∂t∂x+v∂t∂y=λρcp(∂2t∂x2+∂2t∂y2),
where u and v are velocities, ρ is density, Fx and Fy are x and y component of volume force, p is pressure, η is dynamic viscosity of fluid, and cp is specific heat at constant pressure.The heat radiation can be described by(4)q=εσ(TR4-TA4),
where q is the heat flux density, ε is the emissivity, σ is the Stefan-Boltzmann number, TR is the temperature of the radiating surface, and TA is the temperature of a receiving surface.In air-core reactors, most of the packages’ surfaces face the surface of another package, on which the temperature is nearly the same. Thus, for air-core reactors, the radiating heat transfer can be neglected according to (4).


### 2.3. Fluid-Thermal Coupled Analysis

#### 2.3.1. Finite Element Model

According to the mathematical modal mentioned above, a fluid-thermal coupled analysis should be adopted. A 2D axial symmetry finite element model in Flotran is built to conduct the analysis.The simulation model is comprised of five packages and four vertical air ducts. Rectangles are chosen to replace the real cross sections of metal conductors and insulating materials to simplify the modeling. But, in order to study the detailed distribution in the packages, the metal conductor is divided into three parts in the radial direction and 15 parts in the axial direction, as shown in Figure2(a).(a) Metallic conductors. (b) Insulating material.
(a)
(b)In the rectangles that indicate the metal conductors, equivalent surface heat densities computed according to (5) are loaded:
(5)Qf,i=kc,iJeq,i2κ,Jeq,i=JiSiWi(45ab),
where Jeq,i is the equivalent ampere density, a is the length of the rectangle in the radial direction, and b is the length of the rectangle in the axial direction.The insulating materials are also plotted as a combination of rectangles as shown in Figure2(b). The thermal conductivity of the insulating materials in the finite element model λm is defined according to
(6)λmd=λdm,
where d is the real length of insulating materials along the heat conduction direction, and dm is the corresponding equivalent length in the finite element model.


### 2.4. Boundary Conditions

The boundary conditions of the analysis are set as follows.(a)
Rotational symmetry conditions are defined on the left boundary line, the radial velocity of which is defined as 0.(b)
The radial and axial velocities of all the fluid-solid interface in the analysis model are set 0. The roughness coefficient of the packages’ surfaces is set 0.2.(c)
In the right boundary, the temperature is set 20°C, the relative pressure is set 1, and the velocities in all directions are set 0.(d)
In the lower boundary, the temperature is set 20°C, the velocities in all directions are set 0, and the pressure is defined variable.(e)
In the lower boundary, the temperature, the pressure, and the velocities in all directions are defined variable.(f)
The turbulence model is open and the SST turbulence model in Flotran is chosen to solve the problem.


## 2.1. Structure and Parameters of an Air-Core Reactor Prototype

The typical air-core power reactor contains some cylindrical packages connected in parallel and the vertical air ducts between them, as it is shown in Figure1. The packages comprise metal conductors and the insulating material. The air ducts between the packages are used as heat dissipation channels of the packages.Figure 1
Structure sketch of air-core reactor.In this paper, a prototype of air-core power reactor with five packages is taken as an example, the parameters of which are given in Table1.Table 1
The parameter of an air-core power  reactor prototype.
Package number
Diameter
Current
Ampere density
Height
(m)
(A)
(A/m2)
(m)
1
0.619
36.662
1.329 × 106
0.325
2
0.684
31.081
1.329 × 106
0.325
3
0.747
39.476
1.329 × 106
0.325
4
0.811
44.049
1.329 × 106
0.325
5
0.880
58.732
1.329 × 106
0.325


## 2.2. Mathematical Modal

### 2.2.1. Computation of Heat Load

The heat sources in the packages include the resistive current loss and extra eddy loss [13–15]. According to [16, 17] the total loss in a package can be simplified as kc times of resistive current loss; thus, the loss of the ith coil can be expressed as
(1)Pi=πkc,iJi2SiWiDiκ,
where Ji is the ampere density of ith coil, Wi is the number of turns of ith coil, Di is the diameter of ith coil, Si is the total sectional area of the metallic conductor in the ith package, and κ is the conductivity of metal conductor.


### 2.2.2. Heat Transfer Equations

In most cases, heat transfer process contains heat conduction, convection, and radiation.Partial differential equation of heat conduction is given as(2)∂2t∂x2+∂2t∂y2+ϕλ=0,
where t is the temperature, ϕ is loss function, and λ is the thermal conductivity of the considered material.Heat convection process is closely related to the fluid states, which can be described by the continuity equation, the momentum equation, and the energy equation as shown in (3) the following equation:
(3)∂u∂x+∂v∂y=0,ρ(u∂u∂x+v∂u∂y)=Fx-∂p∂x+η(∂2u∂x2+∂2u∂y2),ρ(u∂v∂x+v∂v∂y)=Fy-∂p∂y+η(∂2v∂x2+∂2v∂y2),u∂t∂x+v∂t∂y=λρcp(∂2t∂x2+∂2t∂y2),
where u and v are velocities, ρ is density, Fx and Fy are x and y component of volume force, p is pressure, η is dynamic viscosity of fluid, and cp is specific heat at constant pressure.The heat radiation can be described by(4)q=εσ(TR4-TA4),
where q is the heat flux density, ε is the emissivity, σ is the Stefan-Boltzmann number, TR is the temperature of the radiating surface, and TA is the temperature of a receiving surface.In air-core reactors, most of the packages’ surfaces face the surface of another package, on which the temperature is nearly the same. Thus, for air-core reactors, the radiating heat transfer can be neglected according to (4).


## 2.2.1. Computation of Heat Load

The heat sources in the packages include the resistive current loss and extra eddy loss [13–15]. According to [16, 17] the total loss in a package can be simplified as kc times of resistive current loss; thus, the loss of the ith coil can be expressed as
(1)Pi=πkc,iJi2SiWiDiκ,
where Ji is the ampere density of ith coil, Wi is the number of turns of ith coil, Di is the diameter of ith coil, Si is the total sectional area of the metallic conductor in the ith package, and κ is the conductivity of metal conductor.


## 2.2.2. Heat Transfer Equations

In most cases, heat transfer process contains heat conduction, convection, and radiation.Partial differential equation of heat conduction is given as(2)∂2t∂x2+∂2t∂y2+ϕλ=0,
where t is the temperature, ϕ is loss function, and λ is the thermal conductivity of the considered material.Heat convection process is closely related to the fluid states, which can be described by the continuity equation, the momentum equation, and the energy equation as shown in (3) the following equation:
(3)∂u∂x+∂v∂y=0,ρ(u∂u∂x+v∂u∂y)=Fx-∂p∂x+η(∂2u∂x2+∂2u∂y2),ρ(u∂v∂x+v∂v∂y)=Fy-∂p∂y+η(∂2v∂x2+∂2v∂y2),u∂t∂x+v∂t∂y=λρcp(∂2t∂x2+∂2t∂y2),
where u and v are velocities, ρ is density, Fx and Fy are x and y component of volume force, p is pressure, η is dynamic viscosity of fluid, and cp is specific heat at constant pressure.The heat radiation can be described by(4)q=εσ(TR4-TA4),
where q is the heat flux density, ε is the emissivity, σ is the Stefan-Boltzmann number, TR is the temperature of the radiating surface, and TA is the temperature of a receiving surface.In air-core reactors, most of the packages’ surfaces face the surface of another package, on which the temperature is nearly the same. Thus, for air-core reactors, the radiating heat transfer can be neglected according to (4).


## 2.3. Fluid-Thermal Coupled Analysis

### 2.3.1. Finite Element Model

According to the mathematical modal mentioned above, a fluid-thermal coupled analysis should be adopted. A 2D axial symmetry finite element model in Flotran is built to conduct the analysis.The simulation model is comprised of five packages and four vertical air ducts. Rectangles are chosen to replace the real cross sections of metal conductors and insulating materials to simplify the modeling. But, in order to study the detailed distribution in the packages, the metal conductor is divided into three parts in the radial direction and 15 parts in the axial direction, as shown in Figure2(a).(a) Metallic conductors. (b) Insulating material.
(a)
(b)In the rectangles that indicate the metal conductors, equivalent surface heat densities computed according to (5) are loaded:
(5)Qf,i=kc,iJeq,i2κ,Jeq,i=JiSiWi(45ab),
where Jeq,i is the equivalent ampere density, a is the length of the rectangle in the radial direction, and b is the length of the rectangle in the axial direction.The insulating materials are also plotted as a combination of rectangles as shown in Figure2(b). The thermal conductivity of the insulating materials in the finite element model λm is defined according to
(6)λmd=λdm,
where d is the real length of insulating materials along the heat conduction direction, and dm is the corresponding equivalent length in the finite element model.


## 2.3.1. Finite Element Model

According to the mathematical modal mentioned above, a fluid-thermal coupled analysis should be adopted. A 2D axial symmetry finite element model in Flotran is built to conduct the analysis.The simulation model is comprised of five packages and four vertical air ducts. Rectangles are chosen to replace the real cross sections of metal conductors and insulating materials to simplify the modeling. But, in order to study the detailed distribution in the packages, the metal conductor is divided into three parts in the radial direction and 15 parts in the axial direction, as shown in Figure2(a).(a) Metallic conductors. (b) Insulating material.
(a)
(b)In the rectangles that indicate the metal conductors, equivalent surface heat densities computed according to (5) are loaded:
(5)Qf,i=kc,iJeq,i2κ,Jeq,i=JiSiWi(45ab),
where Jeq,i is the equivalent ampere density, a is the length of the rectangle in the radial direction, and b is the length of the rectangle in the axial direction.The insulating materials are also plotted as a combination of rectangles as shown in Figure2(b). The thermal conductivity of the insulating materials in the finite element model λm is defined according to
(6)λmd=λdm,
where d is the real length of insulating materials along the heat conduction direction, and dm is the corresponding equivalent length in the finite element model.


## 2.4. Boundary Conditions

The boundary conditions of the analysis are set as follows.(a)
Rotational symmetry conditions are defined on the left boundary line, the radial velocity of which is defined as 0.(b)
The radial and axial velocities of all the fluid-solid interface in the analysis model are set 0. The roughness coefficient of the packages’ surfaces is set 0.2.(c)
In the right boundary, the temperature is set 20°C, the relative pressure is set 1, and the velocities in all directions are set 0.(d)
In the lower boundary, the temperature is set 20°C, the velocities in all directions are set 0, and the pressure is defined variable.(e)
In the lower boundary, the temperature, the pressure, and the velocities in all directions are defined variable.(f)
The turbulence model is open and the SST turbulence model in Flotran is chosen to solve the problem.


## 3. Results and Discussion

After the analysis, results of temperature field, fluid velocity and the pressure distribution, and so on are given. According to the abovementioned results the temperature distribution regulatory and the heat transfer processes of air-core power reactors are studied.


### 3.1. Analysis of Temperature Distribution

Temperature field of the analyzed area is given in Figure3.Figure 3
Temperature field of an air-core reactor prototype.The temperature field shown in Figure3 agrees with the thermal test results of the prototype with parameters shown in Table 1 and thus proves the computational accuracy of the coupled analysis.Data of the temperature along the radial center of the packages are read and plotted as the curves shown in Figure4.Figure 4
The axial temperature distribution along the midline of the packages.Data of the temperature along the radial center of the vertical air ducts are read and plotted as the curves shown in Figure5.Figure 5
The axial temperature distribution along the midline of the vertical air ducts.In order to study the temperature distribution and heat transfer in the radial direction, 3 horizontal paths are defined, as the horizontal lines shown in Figure3. Temperature data along the paths are read and plotted as the curves shown in Figure 6.Figure 6
Radial temperature distribution along the “Bottom,” “Middle,” and “Top” paths defined in Figure3.According to Figures3–6, the following can be seen.In the axial direction of the packages, temperature rises from the bottom up. Due to the end effects the maximum temperature appears near 80% the package height.Along the radial direction of the packages, the maximum temperature appears near the center of the package. Along the path from the center to the side surface, temperature declines slowly and nonlinearly, and the temperature gradient in the insulating material is bigger than that in the metal conductors.In the air ducts, temperature rises linearly from the bottom up. In the radial direction, along the path from the center of the ducts to the packages’ side surfaces, temperature rises nonlinearly. A remarkable temperature gradient appears in the lamina near the fluid-solid interface, which is defined as thermal boundary layer. From Figure6 it can be seen that the thicknesses of thermal boundary layer also rise from the bottom up, which indicates the change of the fluid flow state.


### 3.2. Study of the Heat Transfer Process

Heat flux reflects the heat flow in the air-core reactor, which is a good tool to study the heat transfer process. Due to the complex construction, the heat flux inside the packages is hard to acquire. Thus, the heat flux passes through the surfaces of the packages are considered instead.Using the temperature distribution results and conduction equation given in (2), the heat flux passes through the packages’ surfaces can be calculated. The heat fluxes go through the side surfaces of the 3rd and 5th packages’ are given as the curves in Figures 7(a) and 7(b), the axial heat flux through the bottom and top of the coils are calculated as the columns in Figure 8. Meanwhile, the average heat flux q-w,i conducted through the surfaces of the packages can be calculated according to (1) and (7):
(7)q-w,i=PiAi,
where Ai is the superficial area of the ith package.(a) The average heat flux and the axial heat flux through the inner and outer surfaces of the 3rd package. (b) The average heat flux and the axial heat flux through the inner and outer surfaces of the 5th package.
(a)
(b)Figure 8
Axial heat flux through the bottom and top of the packages.The results ofq-w,3, q-w,5 are shown as the full straight lines in Figures 7(a) and 7(b).From Figures7 and 8 the following can be seen.Most of the heat load generated inside the packages are dissipated through the side surfaces of the packages and are carried away by the flowing air. A small part of the heat load dissipated through the bottom and top of the packages. The heat flux through the bottom surfaces is larger than that through the top surfaces.For side surfaces that are facing the air ducts (inside surfaces), the envelope of the heat flux curves decline from the bottom up, which indicates that an axial heat flow goes from above to below inside the package. The mentioned axial heat flux is much smaller than the radial heat flux that goes through the side surfaces.For inner packages, both the side surfaces of which face the air ducts, the heat flux through the two side surfaces equals each other. For outer packages, the side surfaces of which face the large air space (outside surfaces) and the air ducts (inside surfaces) separately, the heat convection in the air duct is stronger than that in the large air space, which indicates a heat flux from the outer part of the outer packages into the inside surface.Near the entrance of the air ducts, the local temperature difference between the conductors and air is relatively small, but the local heat flux is considerably higher due to a stronger local heat convective effect.


## 3.1. Analysis of Temperature Distribution

Temperature field of the analyzed area is given in Figure3.Figure 3
Temperature field of an air-core reactor prototype.The temperature field shown in Figure3 agrees with the thermal test results of the prototype with parameters shown in Table 1 and thus proves the computational accuracy of the coupled analysis.Data of the temperature along the radial center of the packages are read and plotted as the curves shown in Figure4.Figure 4
The axial temperature distribution along the midline of the packages.Data of the temperature along the radial center of the vertical air ducts are read and plotted as the curves shown in Figure5.Figure 5
The axial temperature distribution along the midline of the vertical air ducts.In order to study the temperature distribution and heat transfer in the radial direction, 3 horizontal paths are defined, as the horizontal lines shown in Figure3. Temperature data along the paths are read and plotted as the curves shown in Figure 6.Figure 6
Radial temperature distribution along the “Bottom,” “Middle,” and “Top” paths defined in Figure3.According to Figures3–6, the following can be seen.In the axial direction of the packages, temperature rises from the bottom up. Due to the end effects the maximum temperature appears near 80% the package height.Along the radial direction of the packages, the maximum temperature appears near the center of the package. Along the path from the center to the side surface, temperature declines slowly and nonlinearly, and the temperature gradient in the insulating material is bigger than that in the metal conductors.In the air ducts, temperature rises linearly from the bottom up. In the radial direction, along the path from the center of the ducts to the packages’ side surfaces, temperature rises nonlinearly. A remarkable temperature gradient appears in the lamina near the fluid-solid interface, which is defined as thermal boundary layer. From Figure6 it can be seen that the thicknesses of thermal boundary layer also rise from the bottom up, which indicates the change of the fluid flow state.


## 3.2. Study of the Heat Transfer Process

Heat flux reflects the heat flow in the air-core reactor, which is a good tool to study the heat transfer process. Due to the complex construction, the heat flux inside the packages is hard to acquire. Thus, the heat flux passes through the surfaces of the packages are considered instead.Using the temperature distribution results and conduction equation given in (2), the heat flux passes through the packages’ surfaces can be calculated. The heat fluxes go through the side surfaces of the 3rd and 5th packages’ are given as the curves in Figures 7(a) and 7(b), the axial heat flux through the bottom and top of the coils are calculated as the columns in Figure 8. Meanwhile, the average heat flux q-w,i conducted through the surfaces of the packages can be calculated according to (1) and (7):
(7)q-w,i=PiAi,
where Ai is the superficial area of the ith package.(a) The average heat flux and the axial heat flux through the inner and outer surfaces of the 3rd package. (b) The average heat flux and the axial heat flux through the inner and outer surfaces of the 5th package.
(a)
(b)Figure 8
Axial heat flux through the bottom and top of the packages.The results ofq-w,3, q-w,5 are shown as the full straight lines in Figures 7(a) and 7(b).From Figures7 and 8 the following can be seen.Most of the heat load generated inside the packages are dissipated through the side surfaces of the packages and are carried away by the flowing air. A small part of the heat load dissipated through the bottom and top of the packages. The heat flux through the bottom surfaces is larger than that through the top surfaces.For side surfaces that are facing the air ducts (inside surfaces), the envelope of the heat flux curves decline from the bottom up, which indicates that an axial heat flow goes from above to below inside the package. The mentioned axial heat flux is much smaller than the radial heat flux that goes through the side surfaces.For inner packages, both the side surfaces of which face the air ducts, the heat flux through the two side surfaces equals each other. For outer packages, the side surfaces of which face the large air space (outside surfaces) and the air ducts (inside surfaces) separately, the heat convection in the air duct is stronger than that in the large air space, which indicates a heat flux from the outer part of the outer packages into the inside surface.Near the entrance of the air ducts, the local temperature difference between the conductors and air is relatively small, but the local heat flux is considerably higher due to a stronger local heat convective effect.


## 4. Conclusion

The agreement of the thermal results between the analysis and the prototype testing proves the computational accuracy of the fluid-thermal coupled analysis.The fluid-thermal coupled analysis gives more details than the prototype tests. The computational results of temperature field, fluid velocity distribution, and so forth can be used to study the field distribution of a design and thus are useful to judge its reliability.By means of the data obtained in the fluid-thermal coupled analysis, the heat transfer process, including the heat conduction in the packages and the thermal conduction/convection process in the ducts and large air spaces, can be studied distinctly. As a result, the temperature distribution results could be explained. Furthermore, the study results of detailed heat transfer process in this paper could be used to optimize the heat load distribution in the packages, as well as the structural improvement of the packages and air ducts.



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*Source: 865015-2013-03-24.xml*