Abstract
Trench laying cables are often used at inlet and outlet regions of a power distribution cabinet. In order to improve the heat transfer performance and extend service life of a trench laying cable, the heat transfer and cable ampacity of the trench laying cable with a ceramic plate were numerically studied in the present paper and the results were compared with those of a traditional trench laying cable. The variations of conductor loss and eddy current loss of different loop cables were discussed in the trench with a ceramic plate, and the effects of ceramic plate parameters on heat transfer performance of the trench laying cable were optimized using the Taguchi method. It is found that for the trench with ceramic plates, although the ceramic plate restrains the natural convection in the trench, the total heat transfer for natural convection and thermal radiation are enhanced for the cables and the cable ampacity can be improved. The difference of electromagnetic loss between the upper- and lower-layer cables in the trench with ceramic plate is quite small. When the cable core current (I) increases from 700 A to 1100 A, the maximum difference of averaged electromagnetic loss between the upper- and lower-layer cables is 1.22%. With the Taguchi method, an optimum parameter combination is obtained. When the length, thickness, and surface emissivity of the ceramic plate are equal to 0.48 m, 0.0734 m, and 0.8, respectively, at I = 900 A, the cable maximum temperature in the trench is the lowest.
1. Introduction
Owing to the development of the power system and increasing demand of power supply in the city, underground power cables tend to be laid densely. The trench laying cable is often used at the inlet and outlet regions of the power distribution cabinet [1, 2]. Compared with other underground cable laying methods, the trench laying method can accommodate multiloop cables and costs less [3]. The cable ampacity is the maximum cable core current as cable core temperature reaches at 363.15 K (90 °C). When cable core temperature exceeds 363.15 K, the cable service life will be greatly shortened [4]. Therefore, it is necessary to enhance the heat transfer of the trench laying cable to ensure its safe and efficient operations.
The traditional underground cable ampacity is calculated according to the IEC 60287 standard [5, 6], which is based on the Kennelly hypothesis. However, the heat transfer process for the trench laying cable is complicated, and its real temperature distribution is different from that based on the Kennelly hypothesis. Therefore, the ampacity calculated according to the IEC 60287 standard would be questionable for the trench laying cable [7]. With the development of computer technology, numerical simulations are widely used to analyze the characteristics of underground power cables. With the given conditions of the cable laying mode, arrangement, and current load, the transport process for the whole trench laying cable domain can be solved with numerical simulations, and the results should be more reliable for the real situations [8]. For the trench laying cable, the heat transfer process inside is complex and the related studies are few. The related studies are usually focused on the effects of cable layouts, trench dimensions or forced ventilations, etc. Liu et al. [9] studied the variations of local Nusselt numbers near the cables laid on the trench bottom with different Rayleigh numbers. They found that the effect of soil thermal conductivity on the Nusselt number is remarkable, while the effect of the Rayleigh number was relatively small. The heat generated by the cables was mainly transferred through heat conduction at the trench bottom. Fu et al. [10] studied the variations of cable core temperatures with different currents when cables were regularly or irregularly arranged in the trench. It was found that when cables were irregularly arranged in the trench, the cable core temperature was a little higher than that of regular arranged cables. Xiong et al. [11] studied the effect of cable layouts on the cable ampacity. It was found that when cables were irregularly arranged on the trench bottom, the heat conduction between the cables and trench bottom was enhanced and the cable ampacity increased from 481 A to 542 A. However, in their work, only cable locations in the trench were studied and the effects of trench structure on the cable temperature and ampacity were not discussed. Hanna et al. [12] studied the effect of the trench width on the heat transfer performance of the cable. They found that the heat loss of cables increased as trench width increased. Klimenta et al. [13] studied the effects of outer surface emissivity of trench on the cable ampacity. It was found that the cable ampacity would increase when the heat released to the environment from the trench is higher than that absorbed from the environment. However, in their work, the effect of radiation heat transfer inside the trench was not concerned. Boukrouche et al. [14] studied the forced convection heat transfer for the trench laying cable and the effect of the distance between the cable and trench wall was analyzed. The results showed that when the distance between the cable and trench wall decreased, the convection transfer in the trench was weakened, while the radiation heat transfer was enhanced. Yang et al. [15] studied the effects of trench depth and cable layer distance on the cable ampacity. It was showed that the effect of the cable layer distance on the cable ampacity was more remarkable than that of trench depth. Yang et al. [1] also recommended to use air blowers and induced fans to form forced convection in the trench. They found that the cable ampacity was directly proportional to the air inlet velocity and inversely proportional to the trench length and air temperature in the trench. These studies showed that the cable ampacity would be improved by enhancing heat transfer in the trench. The cable ampacity will be greatly improved when air blowers or other ventilation equipment are used to form forced convection in the trench. However, as compared with the natural convection, the cost of forming forced convection in the trench using ventilation equipment should be much higher. If only changing cable layouts or trench dimensions, the heat transfer enhancement in the trench would not be so obvious.
Therefore, in the present paper, in order to improve heat transfer of the trench laying cable, ceramic plates were installed on the vertical walls of trench and then cables were laid on the ceramic plates. When the cables are laid on the ceramic plates, the ceramic plates can act as fins installed on the cable surface and the heat transfer areas for the cables should be significantly enlarged. Then, the overall heat transfer, including natural convection and radiation heat transfer, in the trench would be changed and the cable ampacity would also be affected. According to the authors’ knowledge, almost no such research was performed yet, which would be beneficial to enhance heat transfer and improve cable ampacity for trench laying cable. In the present work, the air flow, heat transfer, and cable ampacity of the trench laying cable with the ceramic plate were numerically studied based on the finite element method (FEM) and the performances were compared with those of the traditional trench laying cable. Meanwhile, the variations of conductor loss and eddy current loss of different loop cables were discussed in the trench with a ceramic plate. Finally, the effects of ceramic plate parameters on the heat transfer performance of trench laying cable were optimized with the Taguchi method. The present study would be meaningful for heat transfer enhancement and optimum design of the trench laying cable.
2. Physical Model and Computational Method
2.1. Physical Model and Geometric Parameters
In the present study, since the trench length is much larger than its cross-sectional dimension, the trench laying cable is simplified as a two-dimensional model for the computations [16]. The physical model for the trench laying cable and cable structure are presented in Figure 1, including the trench laying cable model without the ceramic plate (Figure 1(a)) and the trench laying cable model with the ceramic plate (Figure 1(b)). It is assumed that the soil temperature far away from the trench bottom is fixed at constant [17]. The dimension of the computational domain is set as 7 m (width) × 5 m (height). The dimension of the trench is set as 1.6 m (width) × 1.8 m (height), and the thickness of the concrete layer is 0.2 m. The inside and outside areas of the trench are filled with air and soil, respectively. Four loop cables are symmetrically laid in the trench, and three phase cables of the same loop are arranged horizontally. The cable core distance between two adjacent cables in the same loop is 0.09 m. The cable core distance between the upper and lower cables is 0.3 m. The distance between the lower cable core and trench bottom is 0.37 m. For the trench laying cable model with the ceramic plate (Figure 1(b)), the length (L1) and thickness (δ1) of the ceramic plate are 0.36 m and 0.033 m, respectively. The cable structure is presented in Figure 1(c). It shows that the cable is mainly composed of the cable core, insulation layer, sheath layer, and external layer. Typical geometric and physical parameters for the trench laying cable are listed in Tables 1 and 2.
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(b)
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2.2. Governing Equations and the Computational Method
In the present study, the electromagnetic effect, heat transfer, and air flow are coupled with each other [18]. The computational domain is divided into a solid domain and fluid domain for the coupling computations. For the solid domain, the governing equations for the heat transfer are as follows:where λ is the solid thermal conductivity, W/(m·K); T is the temperature, K; and is the electromagnetic heat loss, W/m3, which is defined as follows:where qrh and qml are the electrical loss density and magnetic loss density, respectively, W/m3; Re is the real part of imaginary number; is the current density, A/m2; j is the unit of complex number; ω is the angular frequency, rad/s; is the magnetic induction intensity, T; is the magnetic vector potential, Wb/m; is the electric field intensity, V/m; is magnetic field intensity, A/m; and are the conjugate complex numbers of and , respectively; and σ1 is the electronic conductivity, S/m, which is defined as follows:where Tref is the reference temperature, 293.15 K; ρref is the reference electrical resistivity, Ω·m; and α is the temperature coefficient, K−1.
For the fluid domain, the air flow in the trench is two-dimensional turbulent natural convection, and the governing equations are as follows:where is the velocity vector, m/s; T0 is the air reference temperature, K; ρ and ρ0 are the air density at T and T0 respectively, kg/m3; and p, , and β are the pressure, gravitational acceleration, and volumetric expansion coefficient, respectively. μf and μt are the air dynamic viscosity and turbulence viscosity, respectively, (kg/(m·s)); λf is the thermal conductivity of air, W/(m·K); cp is the specific heat at the constant pressure of air, J/(kg·K); and σt is the turbulence Prandtl number in the energy equation.
For the turbulent flow, the standard model is adopted, which is defined as follows:where k is the turbulent kinetic energy, m2/s2; ε is the turbulent dissipation rate, m2/s3; Pk is the shear production of turbulence, kg/(m·s3); σk and σε are the turbulence Prandtl numbers in k and ε equations, respectively; and Cε1 and Cε2 are the turbulent model parameters in k and ε equations, respectively.
The boundary conditions are set as follows:where T1 is the deep soil temperature, K; h1 is the equivalent heat transfer coefficient on the ground surface, which is set as 12.5 W/(m2·K); Tsur is air temperature on the ground, 298 K. λi (i = 2, 3, 4) represents the thermal conductivity of the external layer, ceramic plate, and concrete layer, respectively, W/(m·K); and hi (i = 2, 3, 4) are the convective heat transfer coefficient on the cable surface, ceramic plate surface, and concrete surface, respectively, W/(m2·K). εi (i = 2, 3, 4) are the radiant emissivity on the cable surface, ceramic plate surface, and concrete surface, respectively. Ji (i = 2, 3, 4) are the effective radiation on the cable surface, ceramic plate surface, and concrete surface, respectively, W/m2. Tf is the air temperature in the trench, K; and σ is the blackbody radiation constant.
In the present study, the governing equations are solved with the commercial code Comsol Multiphysics 5.2, and the Pardiso solver is employed for the computations. The current frequency is set as 50 Hz. The conservative interface flux conditions for mass, momentum, and heat transfer are adopted at the solid–solid and solid–fluid interfaces. For convergence criteria, all residuals of the calculations are less than 10−3.
3. Grid Independence Test and Model Validations
3.1. Grid Independence Test
First, the grid independence test is performed. The computational grid for the trench laying cable with the ceramic plate is presented in Figure 2, where the cable core current is fixed at 850 A, and current frequency is 50 Hz. The unstructured grid is constructed for the computations, and the grid is intensified near the surfaces of the cable and ceramic plate. Four sets of grids are adopted for the tests. The total element numbers are 8783, 21361, 48516, and 89812, and the detailed mesh settings are listed in Table 3. The variations of cable maximum temperature with different grids are presented in Figure 3. It shows that as the total element number changes from 48516 (Grid 3) to 89812 (Grid 4), the difference of cable maximum temperature is 0.37%. Therefore, Grid 3 with a total element number of 48516 should be good enough for the test. For Grid 3, the minimum lengths of the grid element in the air zone (Zone 1), ceramic zone (Zone 2), and cable zone (Zone 3) are 2.6 mm, 2.5 mm, and 1.1 mm, respectively. Finally, similar grid settings to the test grid of Grid 3 were used for the following simulations.
3.2. Model Validations
Subsequently, the computational model and methods are validated. Since the numerical study on multiphysics coupling simulation for the trench laying cable is few, the numerical simulations for the pipe laying cable [18] are restudied. The physical model for the pipe laying cable [18] is presented in Figure 4. Three phase cables are laid in a polyvinyl chloride (PVC) pipe. Both natural convection and radiation heat transfer were considered on the cable surfaces and pipe inner surfaces. The cable core distance between two adjacent cables is 151.95 mm. The outer diameter of the PVC pipe is 620 mm, and the thickness of the pipe is 10 mm. The equivalent heat transfer coefficient on the outside surface of the PVC pipe (hr) is 5 W/(m2·K), and the environment temperature outside the PVC pipe (Tr) is 293.15 K. Typical geometric and physical parameters for the pipe laying cable [18] are listed in Table 4. The computational results for cable maximum temperature, cable core loss, and sheath loss are presented in Figure 5. It shows that the present computational results can agree well with those as reported in Ref. [18]. The maximum differences for cable maximum temperature, cable core loss, and sheath loss are 1.49%, 0.29%, and 0.82%, respectively.
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(b)
In addition, since the air flow inside the PVC pipe is laminar flow in Ref. [18], the turbulence natural convection heat transfer in a two-dimensional square cavity [19] is restudied. The physical model for the two-dimensional square cavity [19] is presented in Figure 6. It shows that the top and bottom walls of the cavity are adiabatic. Two vertical walls are high-temperature and low-temperature walls, where the wall temperature is kept constant and the temperature difference is 40 K. The computational results for the average Nusselt number (Nuave) are presented in Table 5. It shows that the average Nusselt numbers (Nuave) obtained from present simulations can agree well with those in Ref. [19], and the maximum difference is 10.06%.
4. Results and Discussion
4.1. Temperature and Velocity Distributions in the Cable Trench
First, the flow and heat transfer performances and cable ampacity variations were analyzed for different trench models. For the present case, the cable core current is fixed at 900 A, and the imbalance of the three-phase current is not considered [20]. The velocity and temperature distributions in the trench with and without the ceramic plate are presented in Figure 7. In Figures 7(a) and 7(b), it shows that the velocity distributions in the trench with and without the ceramic plate are quite different. When ceramic plates were installed on trench vertical walls, the air natural convection was restrained and the air maximum velocity in the trench decreases from 0.23 m/s to 0.09 m/s. Therefore, when ceramic plates were installed in the trench, the natural convection should be weakened. In Figures 7(c) and 7(d), it shows that the cable temperature in the trench with ceramic plate is lower than that in the trench without ceramic plate. The maximum temperatures (Tmax) of upper- and lower-layer cables in the trench without the ceramic plate are 346.55 K and 350.22 K, respectively, while those in the trench with a ceramic plate are 341.33 K and 343.93 K, respectively. For different trench models, the temperature of the middle-phase cable is the highest for both the upper- or lower-layer cables. For the trench with ceramic plates, all the cables are laid on the ceramic plates. These ceramic plates are similar to the fins installed on the cable surface, and the heat transfer areas for the cables are significantly enlarged. Although the ceramic plate restrains the natural convection in the trench, the total heat transfer for natural convection and thermal radiation are enhanced for the cables. Therefore, the cable temperatures in the trench with ceramic plates decrease.
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The cable ampacity is the cable core current when the cable maximum temperature reaches 363.15 K (90°C). The variations of cable maximum temperature (Tmax) with cable core current for different trench models are presented in Figure 8. According to the temperature distributions in Figures 7(c) and 7(d), it shows that with the same cable core current, the temperature of the lower-layer middle-phase cable is the highest in the trench without the ceramic plate. As shown in Figure 8, when the maximum temperature (Tmax) of the lower-layer middle-phase cable in the trench without the ceramic plate reaches 363.15 K (90°C), its cable core current (I) is 1005 A, while the maximum temperature (Tmax) for the upper-layer middle-phase cable is 359.21 (86.06°C) at I = 1005 A. Furthermore, with the same cable core current (I = 1005 A), the maximum temperature (Tmax) of the lower- and upper-layer middle-phase cables in the trench with the ceramic plate are 356.16 K (83.01°C) and 352.89 (79.74°C), respectively, which are 7.77% and 7.34% lower than those in the trench without ceramic plates. In addition, for the trench with ceramic plates, when the maximum temperature (Tmax) of the lower-layer middle-phase cable reaches 363.15 K (90°C), its cable core current is 1059 A, which is 5.37% higher than that in the trench without the ceramic plate. Therefore, using ceramic plates in the trench, the heat transfer for the cables would be enhanced and the cable ampacities would also be improved.
4.2. Electromagnetic Loss for the Trench Laying Cable with the Ceramic Plate
The electromagnetic loss distributions for the trench laying cable with the ceramic plate are presented in Figure 9. The electromagnetic loss in the cable core is conductor loss, and the electromagnetic loss in the sheath layer is eddy current loss. The conductor loss in the cable core increases along the radial direction, while the eddy current loss in the sheath layer decreases along the radial direction. When the cable core is connected with three-phase current, the current will concentrate near the cable core surface because of the skin effect, which will make actual current cross section area in the cable core decrease, and make electrical resistance increase. Therefore, the conductor loss in the cable core increases along the radial direction. In the sheath layer, as it is closer to the cable core, the magnetic field changes more dramatically and the induced current is higher. The induced current will form a closed loop in the sheath layer and the eddy current loss decreases along the radial direction.
The variations of conductor loss and eddy current loss with cable core current for the trench laying cable with the ceramic plate are presented in Figure 10. It shows that both the conductor loss and eddy current loss increase as core current increases and the eddy current loss is one order of magnitude smaller than the conductor loss. The difference of electromagnetic loss between the upper- and lower-layer cables in the trench is quite small. When the cable core current increases from 700 A to 1100 A, the maximum difference of averaged conductor loss between the upper- and lower-layer cables is 1.09%, and the maximum difference of averaged eddy current loss is 1.22%.
4.3. Parameter Optimization for the Trench Laying Cable with the Ceramic Plate
In this part, the effects of ceramic plate parameters on heat transfer performance of trench laying cable were analyzed and optimized using the finite element method (FEM) combined with the Taguchi method [21]. In the present study, the cable core current is fixed at 900 A. For practical applications, the lower the cable maximum temperature (Tmax), the better for the cable safe operations. Therefore, the Tmax is selected as the objective parameter for cable optimizations, and its value should be as small as possible. This is consistent with the small-the-better characteristics of the Taguchi method.The optimization parameters for the Taguchi method are presented in Table 6. The length (L1), thickness (δ1), and surface emissivity (ε3) of the ceramic plate are selected as optimization parameters, and each parameter has five levels. If all parameter combinations are considered, it is necessary to study 125 cases. For the Taguchi method, an orthogonal array table is formed for the study with selected representative parameters. With this method, the studied cases can be greatly reduced and the effects on the cable maximum temperature (Tmax) will be obtained for different optimization parameters. As shown in Table 7, an orthogonal array table of L25 (53) is formed and only 25 cases are required for the present study.
The signal-noise ratio (SN) of cable maximum temperature (Tmax) in the trench is calculated as follows [21]:
The optimization parameters and corresponding signal-noise ratios (SN) for different cases are presented in Table 8. According to the small-the-better characteristics of the Taguchi method, the corresponding SN should be as large as possible. Table 8 shows that the SN of Case 25 is the largest and the corresponding Tmax is the lowest (Tmax = 341.22 K). For this case (Case 25), the optimum parameters combination are L1 = 0.48 m, δ1 = 0.0734 m, and ε3 = 0.7. To further analyze the contribution ratio (CR) of each optimization parameter on Tmax, it is necessary to carry out statistical analysis for SN in Table 8. Each level of the single optimization parameter has a performance statistics value (Pi,j), where i and j are the parameter index and level index, respectively. The detailed calculation method for Pi,j is formulated with the following example. As for level 1 of parameter A, it includes case 1, case 2, case 3, case 4, and case 5 in Table 8. PA,1 is calculated as follows [22]:
Then, the contribution ratio (CR) of each optimization parameter is calculated as follows [22]:where i and j are the parameter index and level index, respectively.
The performance statistics and contribution ratios of different optimization parameters are presented in Table 9. It shows that the effects of different optimization parameters on Tmax are different. The CR of surface emissivity (ε3) of the ceramic plate is the highest, which is 51.99%. The CRs of length (L1) and thickness (δ1) of the ceramic plate are similar, which are 27.14% and 20.87%, respectively. The performance statistics of different optimization parameters for the cable maximum temperature (Tmax) are presented in Figure 11. It shows that as optimization parameters (L1, δ1 and ε3) increases, the performance statistics values for Tmax increase monotonically. When the levels of optimization parameters (L1, δ1 and ε3) are all equal to 5, the Tmax should be the lowest and the optimum parameters combination are L1 = 0.48 m, δ1 = 0.0734 m, and ε3 = 0.8. The cable maximum temperature (Tmax) is calculated for Case 26 with these optimum parameters combination (L1 = 0.48 m, δ1 = 0.0734 m and ε3 = 0.8). The temperature distributions in the trench with the ceramic plate for Case 25 and Case 26 are presented and compared in Figure 12. It shows that the temperature distributions for Case 25 and Case 26 are similar. The cable maximum temperature (Tmax) for Case 26 is 340.77 K, which is lower than that of Case 25 (341.22 K), and it means the Tmax for Case 26 is the lowest for all 26 cases studied in the present paper.
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5. Conclusion
In the present paper, the air flow, heat transfer, and cable ampacity of trench laying cable with ceramic plate were numerically studied based on the FEM, and the performances were compared with those of the traditional trench laying cable. Meanwhile, the variations of conductor loss and eddy current loss of different loop cables were discussed in the trench with the ceramic plate. The effects of ceramic plate parameters on the heat transfer performance of the trench laying cable were optimized with the Taguchi method. The main findings are as follows:(1)The effect of the ceramic plate on the air flow, heat transfer, and cable ampacity in the trench are remarkable. For the trench with the ceramic plate, all the cables are laid on the ceramic plates. Although the ceramic plate restrains the natural convection in the trench, the total heat transfer for natural convection, and thermal radiation are enhanced for the cables. With the same cable core current (I = 1005 A), the maximum temperature (Tmax) of the lower-layer middle-phase cable in the trench with ceramic plate is 7.77% lower than that in the trench without the ceramic plate. With the same maximum temperature (Tmax = 363.15 K), the cable ampacity of the lower-layer middle-phase cable in the trench with the ceramic plate is 5.37% higher than that in the trench without the ceramic plate.(2)There exists conductor loss in the cable core and eddy current loss in the sheath layer of the trench laying cable with the ceramic plate. The conductor loss in the cable core increases along the radial direction, while the eddy current loss in the sheath layer decreases along the radial direction. The eddy current loss is one order of magnitude smaller than the conductor loss. The difference of electromagnetic loss between the upper- and lower-layer cables in the trench is quite small. When the cable core current increases from 700 A to 1100 A, the maximum difference of averaged conductor loss between the upper- and lower-layer cables is 1.09%, and the maximum difference of averaged eddy current loss is 1.22%.(3)The effects of length (L1), thickness (δ1), and surface emissivity (ε3) of the ceramic plate on the cable maximum temperature (Tmax) in the trench are different. The CR of surface emissivity (ε3) of the ceramic plate is the highest, which is 51.99%. The CRs of length (L1) and thickness (δ1) of the ceramic plate are similar, which are 27.14% and 20.87%, respectively. With optimum parameters combination (L1 = 0.48 m, δ1 = 0.0734 m and ε3 = 0.8) at I = 900 A, the value of Tmax is 340.77 K, which is the lowest for all the cases studied in the present paper.
Nomenclature
| : | Magnetic vector potential (Wb/m) |
| : | Magnetic induction intensity (T) |
| cp: | Specific heat at the constant pressure of air (J/(kg·K)) |
| Ci: | Contribution value |
| Cε1, Cε2: | Turbulent model parameters |
| CRi: | Contribution rate |
| d1: | Cable core diameter (mm) |
| d2: | Insulation layer diameter (mm) |
| d3: | Sheath layer diameter (mm) |
| d4: | External layer diameter (mm) |
| : | Electric field intensity (V/m) |
| : | Conjugate complex numbers of |
| : | Gravitational acceleration (m/s2) |
| h1: | Equivalent heat transfer coefficient on the ground surface (W/(m2·K)) |
| h2: | Convective heat transfer coefficient on the cable surface (W/(m2·K)) |
| h3: | Convective heat transfer coefficient on the ceramic plate surface (W/(m2·K)) |
| h4: | Convective heat transfer coefficient on the concrete surface (W/(m2·K)) |
| hr: | Equivalent heat transfer coefficient on PVC pipe outer surface (W/(m2·K)) |
| H: | Computational domain height (m) |
| : | Magnetic field intensity (A/m) |
| : | Conjugate complex numbers of |
| I: | Cable core current (A) |
| j: | Unit of complex number |
| : | Current density (A/m2) |
| J2: | Effective radiation on the cable surface (W/m2) |
| J3: | Effective radiation on the ceramic plate surface (W/m2) |
| J4: | Effective radiation on the concrete surface (W/m2) |
| k: | Turbulent kinetic energy (m2/s2) |
| L1: | Ceramic plate length (m) |
| Nuave: | Average Nusselt number |
| p: | Pressure (Pa) |
| Pi,j: | Performance statistics value |
| Pk: | Shear production of turbulence (kg/(m·s3)) |
| qml: | Magnetic loss density (W/m3) |
| qrh: | Electrical loss density (W/m3) |
| : | Electromagnetic heat loss (W/m3) |
| Re: | Real part of imaginary number |
| SN: | Signal-noise ratio |
| T: | Temperature (K) |
| T0: | Air reference temperature (K) |
| T1: | Deep soil temperature (K) |
| Tc: | Temperature on the low-temperature wall (K) |
| Tf: | Air temperature in the trench (K) |
| Th: | Temperature on the high-temperature wall (K) |
| Tmax: | Maximum temperature (K) |
| Tr: | Environment temperature outside PVC pipe (K) |
| Tref: | Reference temperature (K) |
| Tsur: | Air temperature on the ground (K) |
| : | Velocity vector (m/s) |
| : | Computational domain width (m). |
| α: | Temperature coefficient (K−1) |
| β: | Volumetric expansion coefficient of air (K−1) |
| δ1: | Ceramic plate thickness (m) |
| ε: | Turbulent dissipation rate (m2/s3) |
| ε2: | Radiant emissivity on the cable surface |
| ε3: | Radiant emissivity on the ceramic plate surface |
| ε4: | Radiant emissivity on the concrete surface |
| λ: | Solid thermal conductivity (W/(m·K)) |
| λf: | Thermal conductivity of air (W/(m·K)) |
| μf: | Dynamic viscosity of air (kg/(m·s)) |
| μt: | Turbulent viscosity of air (kg/(m·s)) |
| ρ: | Air density (kg/m3) |
| ρ0: | Air density at T0 temperature (kg/m3) |
| ρref: | Reference electrical resistivity (Ω·m) |
| σ: | Blackbody radiation constant |
| σ1: | Electric conductivity (S/m) |
| σk, σt, σε: | Prandtl number |
| ω: | Angular frequency (rad/s). |
| FEM: | Finite element method |
| PVC: | Polyvinyl chloride. |
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by Science and Technology Project No. 52094019006Z of State Grid Shanghai Electric Power Company.