Numerical Analysis and Experimental Verification of Optical Fiber Composite Overhead Ground Wire (OPGW) Direct Current (DC) Ice Melting Dynamic Process Considering Gap Convection Heat Transfer

Abstract

An accurate analysis of the dynamic process of ice melting in an optical fiber composite overhead ground wire (OPGW) is of great reference significance for the selection of an ice melting current and the formulation of an ice melting strategy. Existing analytical models for the dynamic process of DC ice melting in an OPGW ignore the gap convective heat transfer after the formation of the air gap between the ground wire and the ice layer, and lack the study of the dynamic process of the phase transition of the ice layer. To this end, a finite element model of the DC ice melting process of OPGW was established by introducing the mushy zone constant to consider the influence of the convective heat transfer in the gap, and at the same time, the apparent heat capacity method was used to simulate the changes of the physical property parameters of the melted ice layer. The dynamic process of the ice layer phase transition and OPGW temperature rise during ice melting are calculated, and the effects of the half-width of phase transition interval dT and the mushy zone constant A_m on the DC ice melting process are summarized and analyzed. The accuracy of the OPGW DC ice melting model is verified by conducting DC ice melting experiments. The results show that during the ice melting process, the gap convection heat transfer mainly affects the temperature distribution of the air gap between the ice layer and the OPGW as well as the location of the phase transition interface, and the width of the air gap at the same height below the OPGW increases by about 3 mm after considering the gap convection; the half-width of phase transition interval, dT, mainly affects the location of the phase transition interface and the temperature rise of the modeled heat source, OPGW, while the mushy zone constant, A_m, mainly affects the temperature distribution in the mushy zone, the air gap region. The elliptical phase transition cross-section formed by the OPGW DC ice melting experiment is consistent with the shape of the ice melting simulation model results, and the measured temperature rise curves of the OPGW during DC ice melting are in good agreement with the simulation results, with a maximum difference of about 3.5 K in temperature and 10 min in ice melting time, but the overall trend is consistent, all showing as increasing first and then decreasing.

1. Introduction

Ground wire ice covering refers to the phenomenon that the supercooled water in the air condenses on the overhead ground wire when it gets cold, so that the ground wire is covered by ice over a large area. Ground wire ice cover is prone to cause serious defects, such as broken strands, broken wires, and damage to transmission line fittings and towers, which are a great threat to the safe and stable operation of the power system [1,2]. Existing thermal ice melting research mainly focuses on transmission conductors, and there are relatively few studies on ground wire ice melting methods and techniques. Optical fiber composite overhead ground wire (OPGW) is currently widely used in engineering; under the influence of ice melting working temperature, the difference in the elastic modulus of the optical fiber and the coating will cause the optical fiber to be deformed in many micro deformations, which will result in larger optical loss, and the long-term accumulation will inevitably lead to the damage of the optical fiber due to thermal aging and the reduction of its strength, life, and reliability [3]. So, it is particularly important to select the appropriate melting current and melting time, reasonably control the melting temperature, and reduce the impact on the OPGW as much as possible while guaranteeing the melting effect when melting the ice on the OPGW. The accurate simulation and analysis of the dynamic process of OPGW ice melting has an important reference role in the selection of ice melting current and the formulation of ice melting strategy. By constructing the numerical simulation model of the dynamic process of OPGW ice melting, the physical process of ice melting is finely simulated to obtain the rule of change, and then provides technical support for the melting of overhead ground wire, which is of great significance to theoretical research and engineering application value.

Over the years, scholars have proposed ice melting dynamic models, elliptical ice melting models, etc. for the simulation calculation of the thermal ice melting method [4,5,6,7]. Jiang Xingliang [8] from Chongqing University relied on the long-term observation of the ice-cover melting process of the line by the artificial climate laboratory and the natural ice covering test station, and through the analysis of the thermal equilibrium process of ice melting, they established the physical-mathematical model of the critical ice melting conditions of the DC, put forward the method of calculating the ice surface temperature and the critical ice melting current, and analyzed the factors affecting the ice surface temperature and the critical ice melting current. Meng Zhigao [9] established a dynamic numerical calculation model for DC ice melting of OPGW, considered the distribution of ice melting current inside the OPGW and the dynamic change in heat exchange between the ice layer and the environment in the process of ice melting, and analyzed the growth process of the air gap, and obtained the change rule of the DC ice melting time of OPGW and the temperature of the optical fiber inside the OPGW in the process of DC ice melting. Yang Guolin [10] analyzed and compared the DC ice melting time for single conductor circular and wing shaped ice cover based on the fact that the shape of the ice cover varies across the conductor. Zhou Yushen [11,12] from Changsha University of Science and Technology established a high-frequency excitation ice melting model for transmission conductors by considering local convective heat transfer, and obtained the effects of local convective heat transfer on the temperature field of the high-frequency ice melting model and its windward and leeward sides. He Zhu [13] constructed a mathematical model of the melting ice vibration characteristics based on the temperature distribution of the ground wire melting process and the characteristic matrix of the ground wire structure. Hu Qin [14] established a dynamic numerical calculation model for DC ice melting, and analyzed the influence of the temperature change in the optical fiber in terms of ambient temperature, wind speed, and ice melting current for the temperature distribution of the OPGW during ice melting and after complete ice melting. Nidhal Ben Khedher [15,16] offered in-depth modeling and experimental evaluation methods for heat transfer improvement with curved fins in latent heat systems and presented comprehensive 3D numerical simulations of melting processes, which combine fin design and heat propagation to benchmark phase change and heat transfer models for this research, while closely aligning these approaches with the thermal modeling of melting phenomena in OPGW systems.

As there are both heat transfer and ice phase change in the ice melting process, the current study has less simulation for the ice phase change process and does not consider the convective heat transfer in the air gap generated in the ice melting process, which affects the accuracy of the dynamic simulation model of ice melting and leads to a discrepancy in the calculation results of the ice melting current and ice melting time with the actual situation. Therefore, considering the gap convective heat transfer, the finite element analysis model of OPGW DC ice melting dynamic process is established in this paper. The phase change problem of the ice layer is solved by using the apparent heat capacity method, and the influence of the half-width of phase transition interval dT and the mushy zone constant A_m on the phase change simulation of ice layer is analyzed. Finally, the accuracy of the ice melting model and calculation results in this paper are verified by OPGW DC ice melting experiments.

The structure of this paper is as follows: Section 2 explains the three stages of the physical process of OPGW DC ice melting. Section 3 gives the boundary conditions and control equations of the dynamic numerical simulation of DC ice melting and establishes the geometric model of OPGW DC ice melting. Section 4 analyzes whether to consider the influence of gap convection, half-width of phase transition interval dT, and mushy zone constant A_m on the simulation of OPGW DC ice melting. In the Section 5, the OPGW DC ice melting experiment was carried out and compared with the simulation results.

2. Physical Process of OPGW DC Ice Melting

When DC ice melting starts, the surface temperature of the OPGW increases under the effect of Joule heat. When the surface temperature of the OPGW reaches 0 °C, the ice in contact with the OPGW starts to melt. The ice melting process can be divided into three stages as follows:

① After the current flows through the OPGW, the Joule heat heats the OPGW and the ice, and the heat is transferred to the outer surface of the ice to exchange heat with the surroundings by radiation and convection. The surface temperature of the OPGW does not reach 0 °C at this stage, and the overlying ice does not undergo phase change.

② When the surface temperature reaches 0 °C, the ice-covered ground wire is in the critical ice melting state, and the ice in the grooves on the surface of OPGW absorbs heat and melts. After the ice begins to melt, the melted water is lost through the ice gap infiltration, and an air gap is formed between the OPGW and the ice, and the ice gradually moves down by gravity, and the upper surface of the OPGW is in contact with the ice, and the distance between the lower surface and the ice gap increases, so the melting speed of the inner surface of the ice layer is different at each place, and the air gap is in the shape of an ellipse.

③ As the ice moves down, when the thickness of the ice cover on the upper surface of the OPGW reaches a certain level, the upper ice cover cannot withstand its own weight and is cut off by the OPGW, and the OPGW DC ice melting process ends.

3. Numerical Simulation of OPGW DC Ice Melting Process

3.1. Numerical Calculation Modeling of OPGW Ice Melting

3.1.1. Control Equations and Boundary Conditions for Electrical and Thermal Fields

The Joule heat generated by the OPGW flux during the DC ice melting process is partly used to heat the OPGW and the ice and provide heat for the latent heat of phase change; the other part is transferred to the outer surface of the ice by heat conduction through the OPGW and the ice, and is finally lost in the external environment through convective and radiative heat exchanges. Therefore, the energy equation during ice melting can be expressed as [17]:

$$I^{2}Rt={\displaystyle \int h(T_{\mathrm{i}}-T_{\mathrm{a}})dl·\Delta t}+{\displaystyle \iint {\rho }_{\mathsf{\theta }}{C}_{\mathsf{\theta }}dSdT}+{\rho }_{\mathrm{ice}}{L}_{\mathrm{f}}\Delta S$$

(1)

where I represents the ice melting current, R represents the OPGW DC resistance, h represents the heat transfer coefficient between the outer surface of the ice layer and the external environment, T_i and T_a represent the temperature of the outer surface of the ice layer and the temperature of the external environment, respectively, ρ_θ represents the density of the microelement of the area of the solution area, C_θ represents the specific heat capacity of the microelement of the area of the solution area, ρ_ice represents the density of the ice layer, L_f represents the enthalpy of the phase change of the ice, and ΔS represents the area of ice melting in the Δt area of the melted cross-section.

The OPGW through-current region during DC ice melting is the aluminum layer, steel core, and stainless-steel tube, and the current distribution in the OPGW is assumed to be I for the ice melting current:

$$\left\{\begin{array}{l}{I}_{\mathrm{al}}={\displaystyle \frac{{\sigma }_{\mathrm{al}}{S}_{\mathrm{al}}}{{\sigma }_{\mathrm{al}}{S}_{\mathrm{al}}+{\sigma }_{\mathrm{st}}{S}_{\mathrm{st}}}}I\hfill \\ I_{\mathrm{st}}={\displaystyle \frac{{\sigma }_{\mathrm{st}}{S}_{\mathrm{st}}}{{\sigma }_{\mathrm{al}}{S}_{\mathrm{al}}+{\sigma }_{\mathrm{st}}{S}_{\mathrm{st}}}}I-I_{\mathrm{st}}^{\prime }\hfill \\ I_{\mathrm{st}}^{\prime }={\displaystyle \frac{S_{\mathrm{st}}^{\prime }}{S_{\mathrm{st}}}}{I}_{\mathrm{st}}\hfill \end{array}\right.$$

(2)

where I_al, I_st, I′_st represent the current in the region of the aluminum layer, steel core, and stainless steel tube, respectively, σ_al, σ_st represent the conductivity of aluminum and steel, respectively, and S_al, S_st, S′_st represent the cross-sectional area of the aluminum layer, steel core, and stainless steel tube, respectively.

The inner region of the ice layer is mainly for heat conduction and satisfies the following differential equation:

$$\rho c{\displaystyle \frac{\partial T}{\partial t}}=\nabla ·(\lambda \nabla T)+Q$$

(3)

where ρ represents the density of the material, c represents the specific heat capacity of the material, λ represents the heat transfer coefficient of the material, and Q represents the intensity of the internal heat source, which can be expressed as follows:

$$Q=\left\{\begin{array}{c}{\displaystyle \frac{{I_{\mathrm{al}}}^2{R}_{\mathrm{al}}}{S_{\mathrm{al}}}}\\ {\displaystyle \frac{{I_{\mathrm{st}}}^2{R}_{\mathrm{st}}}{S_{\mathrm{st}}}}\end{array}\right.$$

(4)

There are convective and radiative heat exchanges between the outer surface of the ice layer and the external environment, so the third type of boundary conditions should be satisfied [18]:

$$-\lambda {\displaystyle \frac{\partial T}{\partial n}}=h(T_{\mathrm{i}}-T_{\mathrm{a}})$$

(5)

where h represents the sum of the convective heat transfer coefficient and the radiative heat transfer coefficient; the convective heat transfer coefficient includes the natural convective heat transfer coefficient and the forced convective heat transfer coefficient, expressed as follows:

$$\left\{\begin{array}{l}h=h_{\mathrm{D}}+h_{\mathrm{R}}\hfill \\ h_{\mathrm{D}}=h_{\mathrm{N}}+h_{\mathrm{F}}\hfill \\ h_{\mathrm{N}}={\displaystyle \frac{{\lambda }_{\mathrm{air}}B{(GrPr)}^b}{D}}\hfill \\ h_{\mathrm{F}}={\displaystyle \frac{{\lambda }_{\mathrm{air}}C{(Re)}^n{(Pr)}^{1/3}}{D}}\hfill \end{array}\right.$$

(6)

where h_D, h_R, h_N, h_F represent convective heat transfer coefficient, radiative heat transfer coefficient, natural convective heat transfer coefficient, and forced convective heat transfer coefficient, respectively, λ_air represents the air heat transfer rate, D represents the geometrical diameter of the ice-covered area, and Re, Gr, Pr represent the Reynolds number, the Prandlt number, and the Grashof number, respectively:

$$\left\{\begin{array}{l}Re={\displaystyle \frac{V_{\mathrm{a}}D{\rho }_{\mathrm{air}}}{{\mu }_{\mathrm{air}}}}\hfill \\ Gr={\displaystyle \frac{2g(T-T_{\mathrm{a}})D^3}{{\upsilon }_{\mathrm{air}}^2(T+T_{\mathrm{a}})}}\hfill \\ Pr={\displaystyle \frac{{\mu }_{\mathrm{air}}{C}_{\mathrm{air}}}{{\lambda }_{\mathrm{air}}}}\hfill \end{array}\right.$$

(7)

where g represents the gravitational constant, ν represents the air kinematic viscosity, μ represents the coefficient of kinetic viscosity of the air, C_air represents the specific heat capacity of the air, V_a represents the external ambient wind speed, ρ_air represents the air density, and B, b, C, and n represent the coefficients determined by the Grashof number and Reynolds number, which can be obtained by looking up the table [18].

The radiative heat transfer coefficient h_R is expressed as follows:

$$h_{\mathrm{R}}=\epsilon {\sigma }_{\mathrm{r}}(T^2+{T_a}^2)(T+T_a)$$

(8)

where ε represents the emissivity of the outer surface of the ice layer, ε = 0.9. σ_r represents the radiation constant, σ_r = 5.67 × 10⁻⁸ W/(m²·K⁴).

3.1.2. Characterization of Ice Layer Material Properties by Apparent Heat Capacity Method

During the physical process of ice melting, there are three types of regions in the phase change distribution of the ice: solid phase, mushy zone, and liquid phase. The paste region is a semi-solid region that exists as an interface between the melted and unmelted regions of the phase change material.

Based on the relative magnitudes of the actual temperature T in the solution region with respect to the melting point T_m of the ice layer and the width of the phase transition interval ΔT, the phase transition process of the ice layer during the DC ice melting process of the OPGW is solved by using the apparent heat capacity method, and the three types of regions can be characterized by the segmented function liquid phase fraction φ(T) [19]:

$$\phi (T)=\left\{\begin{array}{ll}0\hfill & T<T_{\mathrm{m}}-{\displaystyle \frac{\Delta T}{2}}\hfill \\ {\displaystyle \frac{T-(T_{\mathrm{m}}-\frac{\Delta T}{2})}{\Delta T}}\hfill & T_{\mathrm{m}}-{\displaystyle \frac{\Delta T}{2}}\le T\le T_{\mathrm{m}}+{\displaystyle \frac{\Delta T}{2}}\hfill \\ 1\hfill & T>T_{\mathrm{m}}+{\displaystyle \frac{\Delta T}{2}}\hfill \end{array}\right.$$

(9)

where T_m represents the melting point of the ice layer and ΔT represents the width of the phase transition interval.

The material properties during the phase transition of the ice layer in each of the three regions: solid phase, mushy zone, and liquid phase can be expressed as [20]:

① Equivalent heat capacity C_e(T)

$$C_{\mathrm{e}}(T)=\left\{\begin{array}{ll}{C}_{\mathrm{fs}}(T)\hfill & T<T_{\mathrm{m}}-{\displaystyle \frac{\Delta T}{2}}\hfill \\ C_{\mathrm{fs}}(T)+[C_{\mathrm{fg}}(T)-C_{\mathrm{fs}}(T)]\phi (T)+L_{\mathrm{f}}\cdot D(T)\hfill & T_{\mathrm{m}}-{\displaystyle \frac{\Delta T}{2}}\le T\le T_{\mathrm{m}}+{\displaystyle \frac{\Delta T}{2}}\hfill \\ C_{\mathrm{fg}}(T)\hfill & T>T_{\mathrm{m}}+{\displaystyle \frac{\Delta T}{2}}\hfill \end{array}\right.$$

(10)

where C_fs and C_fg represent the specific heat capacity of ice and air, respectively, and L_f represents the latent heat of phase transition; the contribution in the equivalent specific heat capacity is represented by a Gaussian function D(T) centered on T_m [21]:

$$D(T)={\displaystyle \frac{e^{-\frac{{(T-T_{\mathrm{m}})}^2}{{(\Delta T/4)}^2}}}{\sqrt{\pi {(\Delta T/4)}^2}}}$$

(11)

② Equivalent heat transfer coefficient k_e(T)

$$k_{\mathrm{e}}(T)=\left\{\begin{array}{ll}{k}_{\mathrm{fs}}(T)\hfill & T<T_{\mathrm{m}}-{\displaystyle \frac{\Delta T}{2}}\hfill \\ k_{\mathrm{fs}}(T)+[k_{\mathrm{fg}}(T)-k_{\mathrm{fs}}(T)]\phi (T)\hfill & T_{\mathrm{m}}-{\displaystyle \frac{\Delta T}{2}}\le T\le T_{\mathrm{m}}+{\displaystyle \frac{\Delta T}{2}}\hfill \\ k_{\mathrm{fg}}(T)\hfill & T>T_{\mathrm{m}}+\Delta T/2\hfill \end{array}\right.$$

(12)

where k_fs and k_fg represent the heat transfer coefficients for ice and air, respectively.

③ Equivalent density ρ_e(T)

$${\rho }_{\mathrm{e}}(T)=\left\{\begin{array}{ll}{\rho }_{\mathrm{fs}}(T)\hfill & T<T_{\mathrm{m}}-{\displaystyle \frac{\Delta T}{2}}\hfill \\ {\rho }_{\mathrm{f}}(T)\hfill & T_{\mathrm{m}}-{\displaystyle \frac{\Delta T}{2}}\le T\le T_{\mathrm{m}}+{\displaystyle \frac{\Delta T}{2}}\hfill \\ {\rho }_{\mathrm{fg}}(T)\hfill & T>T_{\mathrm{m}}+{\displaystyle \frac{\Delta T}{2}}\hfill \end{array}\right.$$

(13)

where ρ_fs and ρ_fg represent the densities of ice and air, respectively.

ρ_f(T) is related to the solid phase rate φ_fs, which is 1 in the solid phase and 0 in the liquid phase for phase change materials, and linearly correlates in the paste region between the solid phase rate and the temperature distribution in the paste region, expressed as follows:

$$\left\{\begin{array}{l}{\rho }_{\mathrm{f}}(T)={\varphi }_{\mathrm{fg}}{\rho }_{\mathrm{fg}}(T)+{\varphi }_{\mathrm{fs}}{\rho }_{\mathrm{fs}}(T)\hfill \\ {\varphi }_{\mathrm{fs}}={\displaystyle \frac{(T_{\mathrm{m}}+\Delta T/2)-T}{\Delta T}}\hfill \\ {\varphi }_{\mathrm{fs}}+{\varphi }_{\mathrm{fg}}=1\hfill \end{array}\right.$$

(14)

where φ_fs represents the solid phase rate and φ_fg represents the liquid phase rate.

3.1.3. Convective Flow in Fiber Optic Paste, Paste Region, Air Gap

The fiber optic fiber paste as well as the air gap solution region from the ice melting is an incompressible flow, which can be represented by the Navier–Stokes equations [22,23]:

$$\left\{\begin{array}{l}\rho \nabla ·u=0\hfill \\ \rho ({\displaystyle \frac{\partial u}{\partial t}}+u·\nabla u)=\nabla ·[-pI+\mu (\nabla u+{(\nabla u)}^T)]-S(T)·u\hfill \end{array}\right.$$

(15)

where ρ represents the fluid density, u represents the fluid velocity vector, p represents the fluid pressure, I represents the unit matrix, μ represents the hydrodynamic viscosity, and S(T) represents the damping term.

In order to describe the convective flow after a phase change in the ice layer, while preventing the solid from generating flow, a Darcy’s law damping term S(T) is added to the momentum equation, incorporating the Carman–Kozeny relation, which has a large value when the liquid phase fraction tends to 0, forcing the solid-phase velocity field to be 0, and eliminating the term when the liquid-phase fraction region is 1. It can be expressed as follows:

$$S(T)=A_{\mathrm{m}}{\displaystyle \frac{{(1-\phi (T))}^2}{\phi {(T)}^3+\epsilon }}$$

(16)

where A_m denotes the mushy zone constant, for incompressible flow, A_m ≥ 1 × 10⁴ kg/(m³·s), and the constant ε, ε = 0.001, is added to avoid de-zeroing in the case of φ(T) = 0.

3.1.4. Dynamic Mesh Following at the Ice Layer Phase Transition Interface

In the ice layer melting model, there are thermal and mass balances at the phase transition interface, from which the Stefan condition, which the interface velocity V_s should satisfy, can be derived [24]:

$$\left\{\begin{array}{c}{V}_{\mathrm{s}}={\displaystyle \frac{Q_{\mathrm{s}}}{{\rho }_{\mathrm{ice}}(T)L_{\mathrm{f}}}}\\ Q_{\mathrm{s}}={\lambda }_{\mathrm{ice}}{\displaystyle \frac{\partial T}{\partial n}}\end{array}\right.$$

(17)

where Q_s represents the sudden change in normal heat flux at the interface, ρ_ice is the density of ice, and λ_ice represents the heat transfer coefficient of ice.

With the OPGW DC ice melting process, the air gap between the ice layer and the OPGW gradually increases, and the ice layer will gradually move downward after melting due to the influence of gravity, and the process is simulated in the model by a moving mesh, and the mesh moving speed is the phase transition interface velocity V_s, which can be determined by Equation (17).

3.2. Geometric Modeling

Fiber optic composite overhead ground wire OPGW-24B1-98 [124.2; 51.8] was selected to establish its DC ice melting two-dimensional model, and the corresponding parameters are shown in

Table 1. OPGW technical parameters.

Ground Type	Calculated Cross-Sectional Area (mm2)		Outer Diameter (mm)	20 °C DC Resistance (Ω/km)
	Aluminum	Steel
OPGW-24B1-98 [124.2; 51.8]	26.069	79.377	13.20	0.884

and

Table 2. OPGW electric heating parameters.

Material	Density (kg·m−3)	Thermal Conductivity (W·(m·K)−1)	Specific Heat Capacity (J·(kg·K)−1)	Conductivity (S/m)
Steel	7850	45.00	460	7.69 × 106
Aluminum	2700	238	900	3.77 × 107

. The density, specific heat capacity, and heat transfer coefficient of the ice layer are determined according to the apparent heat capacity method in Section 3.1.2. In addition, the initial conditions of ice melting are ambient temperature T_a = 268.15 K, wind speed V_a = 5 m/s, ice melting current I = 500 A, and the thickness of ice cover d = 15 mm.

The ice cover OPGW simulation model established in this paper is glaze icing. Due to the complexity of the actual ice cover morphology and ice melting conditions of OPGW, the following assumptions and simplifications are made for the convenience of DC ice melting modeling and calculations while taking into account its accuracy [25,26,27]:

① Ice-covered cross-section is circular: Due to the small diameter of OPGW, the torsional stiffness is small; when the ice cover is piled up on the windward side, it is more likely to torsion under the action of the gravity of the ice layer to make the shape of the ice-covered cross-section close to circular.

② Neglecting the axial heat transfer of OPGW: When the OPGW is de-icing in the same period or the de-icing part of the wire is long, the axial heat transfer of OPGW can be neglected, so the two-dimensional field is used to analyze and solve the temperature distribution of OPGW.

③ Neglect de-icing shear: the process of ice melting on the ground wire is the process of expanding the air gap, and when the thickness of the ice layer on the upper surface of the conductor is zero, the ice layer is dislodged from the surface of the conductor.

④ Neglecting the effect of OPGW strand gap on heat transfer: the OPGW diameter is relatively small, and the strand gap is also small, which has less influence on the heat transfer process of ice melting. In order to improve the quality of the dissecting mesh and the convergence of the solution, the geometry of the optical fiber is taken into account, and the geometrical models of the corresponding aluminum and steel parts are established with the equivalent aluminum and steel cross-sectional areas as a whole, respectively, so that the overall geometrical model of the OPGW is finally formed, as shown in Figure 1.

3.3. Numerical Calculation of the Dynamic Process of Ice Melting in OPGW

By establishing a numerical analysis model of OPGW DC ice melting and using the finite element method, the laws of the phase change process and temperature distribution of the OPGW ice layer considering the interstitial convective heat transfer are calculated, and the calculation flow is shown in Figure 2.

In the simulation of the dynamic process of ice melting, the difficulty lies in the tracking of the dynamic mesh following the phase transition interface; therefore, the first step is to determine the location of the phase transition interface. After solving for the heat flow density inside the model, the heat and mass balances existing at the phase transition interface can be determined, and then the interface velocity, i.e., the rate of phase transition at the top of the inner surface of the ice layer, can be derived from Equation (17) and used as the following velocity of the moving mesh. Equation (9) is used as a judgement condition for whether or not to perform moving mesh tracking; when the liquid phase fraction is less than 1, the mesh does not move, i.e., no tracking is performed. When the liquid phase fraction is equal to 1, the mesh starts to move, and the mesh tracking is performed.

As the mesh moves following the phase transition interface, the mesh cells are stretched or squeezed, which may lead to non-convergence or inaccurate results, and there must also be an automatic updating of the mesh, which uses the minimum relative cell volume as the mesh redrawing condition, and the mesh is redrawn when it is less than a threshold value of 0.5. Otherwise, the mesh continues to move until it rejoins the outer boundary of the ice layer.

After the initial mesh dissection, in addition to the domain boundaries, the computational domain mesh is a free triangular mesh, the domain boundaries are surrounded by a quadrilateral boundary layer mesh, and corner refinement is used in the corner region, and the overall mesh delineation is shown in Figure 3, with the largest cell size of 1.61 mm and the number of divided meshes of 15,862. The outer boundary of the OPGW is a moving mesh boundary, which is used to simulate the downward movement of the ice layer in the process of ice-melting, and the movement speed is determined according to Equation (17).

3.4. Mesh Independence Verification

In order to minimize the influence of the mesh density on the accuracy of the numerical calculation results and to ensure the reliability of the results and the efficiency of the calculation. Three mesh scales with mesh numbers of 11,266, 15,862, and 21,240 were selected. By choosing the truncated path along the Y-axis as shown in Figure 1, the temperature profiles and phase transition profiles for different mesh densities were obtained, as shown in Figure 4.

It can be seen that as the mesh density increases, its impact on the accuracy of the calculation results decreases, and the number of meshes of 15,862, 21,240, the two temperature curves and phase transition curves are basically the same. Therefore, considering the calculation accuracy and cost, the mesh scale of 15,862 is selected for numerical simulation in the subsequent study.

4. Simulation Analysis of OPGW DC Ice Melting

4.1. Influence of Interstitial Convective Heat Transfer on the Temperature and Phase Change Distribution of the Dynamic Process of Ice Melting

Considering the interstitial convective heat transfer, the temperature and phase change distributions of the dynamic process of OPGW ice melting were obtained by applying the calculation method shown in Figure 2 to the established OPGW DC ice melting model. Figure 5a,b show the curves of the temperature and convective heat transfer coefficient of the outer surface of the top and bottom of the ice layer with time during ice melting, respectively.

Prior to the start of DC ice melting, there is no temperature difference between the outer surface of the ice layer and the outside environment, and there is no heat exchange. After the ice melting current is applied, the convective heat transfer coefficient increases as the temperature of the OPGW and the ice layer increases under the action of Joule heat, and there is a temperature difference between the ice layer and the external environment.

From Figure 6, it can be seen that in the initial stage of ice melting, the temperature rise of the outer surface of the ice layer is faster and then gradually tends to the melting point, so the convective heat transfer coefficient shows the same growth trend; but, as the ice layer continues to melt and move downward, the thermal resistance of the ice layer at the top decreases, and at the bottom, due to the continuous growth of air gaps hindering the transfer of heat, the temperature rise of the ice layer as well as the convective heat transfer coefficient grows more gently, so that in the process of melting, the OPGW temperature. Therefore, the maximum value of the OPGW temperature during the ice melting process occurs at the bottom outer surface.

The temperature distributions and phase transition distributions of the OPGW ice melting model with or without considering the gap convection were obtained by selecting three typical time points in the pre-melting, mid-melting, and deglazing processes, as shown in Figure 6a–d, respectively.

In Figure 6b,d, the phase transition distribution is characterized by Equation (9), where the dark blue region is 0, indicating that no phase transition has occurred, i.e., the ice layer, and the dark red region is 1, indicating that the ice layer has undergone a phase transition, i.e., the air gap, and the region between 0 and 1 indicates the phase transition interval, i.e., the mushy zone, of the ice layer.

Comparing and analyzing the temperature distribution at the same time point in Figure 6a,c, it can be found that there is a significant difference in the temperature distributions in the air gap region between the two, with the latter having a lower temperature in the air gap region, suggesting that it is necessary to consider interstitial convection in the numerical simulation of DC ice melting. In addition, near the lower part of the OPGW, the maximum difference in the width of the air gap at the same phase change interface location of the ice layer is about 3 mm after considering the gap convection, and the phase transition cross-section shape of the ice layer in Figure 6b,d is elliptical.

Selecting the paths along the X-axis intercept and along the Y-axis intercept as shown in Figure 1, the temperature and phase transition calculations for the corresponding locations at the moment of OPGW de-icing are extracted with or without consideration of interstitial convection, as shown in Figure 7.

As an example, the range of intercept distances for the ice layer, mushy zone, air gap, and OPGW without considering gap convection is indicated as shown in Figure 7a,b. The location of the mushy zone for the case of whether or not to consider gap convection is shown in Figure 7c,d.

From Figure 7a,b, it can be seen that the temperature curve is almost fit in the range of the ice layer, mushy zone, and OPGW truncation distance, while in the range of air gap truncation distance. The temperature difference along the X-axis intercept is about 1.5 K, and the temperature difference along the Y-axis intercept is more than 3 K. From Figure 7c,d, it can be seen that whether the gap convection is considered, the width of the mushy zone is different and the phase transition distribution is also different, but the starting point of the mushy zone overlaps.

The above analysis illustrates that during the ice melting process, whether or not to consider interstitial convection mainly affects the temperature distribution in the air gap region and the location of the ice layer phase transition interface, and has no significant effect on the temperature change of the OPGW. This is due to the fact that after considering gap convection, there is no longer only a heat conduction process in the air gap region, but it also contains the convective heat transfer process, which makes the heat in the air gap transfer to the inner surface of the ice layer better and accelerates the melting of the ice layer, which results in the difference of the air gap width with a maximum difference of about 3 mm as shown in Figure 6b,d; the melting time of ice melting without considering the gap convection is 4776.87 s, i.e., about 80 min, while the melting time after considering the gap convection is about 4563.34 s, i.e., about 76 min. Therefore, it is necessary to consider the gap convection in the numerical simulation of DC ice melting.

4.2. Influence of the Half-Width of the Phase Transition Interval and Mushy Zone Constants on the Dynamic Process of Ice Melting

When performing the phase change analysis of the OPGW DC ice melting process, since the volume and shape of the air gap during the ice melting process directly affects the convective motion and convective strength of the air, which in turn, also affects the heat transfer process within the gap, and thus in turn, affects the volume and shape of the air gap, determining the location of the phase change interface and its changes, and simulating the volume and shape of the air gap that agrees with the actual situation have a significant impact on the ice melting dynamic physical process simulation results.

In this paper, the apparent heat capacity method is used to characterize the properties of phase change materials, and the half-width of the phase transition interval dT (dT = ΔT/2) and the constant A_m of mushy zone are used to determine and simulate the position of phase change interface.

From Equation (9), the half-width of phase transition interval dT is used to define the temperature interval in which the material undergoes a phase transition, i.e., the start temperature of the phase transition and the end temperature of the phase transition, which determines the size of the mushy zone. The mushy zone constant A_m is used to indicate the amount of change in fluid velocity as the ice melting; the larger the value, the faster the velocity decreases. It has been shown that the values of dT and A_m have an important influence on the accuracy of the simulation calculation of the phase transition process [28], and reasonable values of the parameters should be selected for different phase transition materials and different geometries [22].

According to the literature [20], three groups of typical half-widths of phase transition intervals and mushy zone constants were selected, respectively, to analyze their influence laws on the dynamic process of ice melting and to provide reference for the corresponding parameter selection in the numerical analysis of ice melting. The half-width dT values of the three groups of phase transition intervals are 0.25 K, 0.5 K, and 0.75 K, and the values of the three groups of mushy zone constants A_m are 10⁴, 10⁵, and 10⁶.

In this paper, the influence of the half width dT of the phase transition interval in Equation (9) and the constant A_m of the mushy zone in Equation (16) on the phase transition interface is compared and analyzed. The X-axis and Y-axis cross-section paths shown in Figure 1 are selected to extract the temperature and phase transition calculation results of the corresponding position at the time of OPGW de-icing, as shown in Figure 8 and Figure 9.

4.2.1. Simulation Analysis of the Dynamic Process of Ice Melting for Different Phase Transition Interval Half-Widths

Taking dT = 0.5 K as an example, the range of cut-off distance of the ice layer, mushy zone, air gap, and OPGW is indicated, as shown in Figure 8a,b. The position of the half-wide mushy zone of the three groups of phase transition intervals is shown in Figure 8c,d.

Figure 8. Temperature profiles and phase transition profiles during ice melting at different half-width of phase transition intervals. (a) Temperature profile along the X-axis intercept. (b) Temperature profile along the Y-axis intercept. (c) Phase transition curve along the X-axis intercept. (d) Phase transition curve along the Y-axis intercept.

Figure 8. Temperature profiles and phase transition profiles during ice melting at different half-width of phase transition intervals. (a) Temperature profile along the X-axis intercept. (b) Temperature profile along the Y-axis intercept. (c) Phase transition curve along the X-axis intercept. (d) Phase transition curve along the Y-axis intercept.

From Figure 8a,b, it can be seen that the temperature difference does not exceed 0.2 K in the mushy zone as well as in the air gap within the truncation distance, whereas in Figure 8b, the temperature difference can be up to more than 3 K within the OPGW truncation distance. As can be seen from Figure 8c,d, the distribution of phase transitions within the mushy zone is different for three sets of values of the half-width of the phase transition interval, dT, with different starting points and widths of the mushy zone.

The above analysis illustrates that different values of the half-width of phase transition interval, dT, have an impact on the phase transition simulation, mainly in terms of the location of the phase transition interface and the temperature rise of the heat source, OPGW, in the model, due to the fact that the temperature interval of the phase transition, as defined by dT, determines the magnitude of the heat required to be absorbed by the melting process of the phase-transitioned material. From Figure 8c,d, it can be seen that when dT is larger, the phase transition region, i.e., the mushy zone, is also wider and the overall temperature in the computational domain is higher, and when dT is too small, the abrupt change in material properties due to phase transition will be distorted, as shown in Figure 8b with dT = 0.25 K.

4.2.2. Simulation Analysis of the Dynamic Process of Ice Melting for Different Mushy Zone Constants

Taking A_m = 10⁵ as an example, the range of cut-off distance of ice layer, mushy zone, air gap, and OPGW is indicated, as shown in Figure 9a,b. The mushy zone positions of the three groups of mushy zone constants are shown in Figure 9c,d.

Figure 9. Temperature profiles and phase transition profiles during ice melting with different mushy zone constants. (a) Temperature profile along the X-axis intercept. (b) Temperature profile along the Y-axis intercept. (c) Phase transition curve along the X-axis intercept. (d) Phase transition curve along the Y-axis intercept.

Figure 9. Temperature profiles and phase transition profiles during ice melting with different mushy zone constants. (a) Temperature profile along the X-axis intercept. (b) Temperature profile along the Y-axis intercept. (c) Phase transition curve along the X-axis intercept. (d) Phase transition curve along the Y-axis intercept.

As can be seen from Figure 9a,b, compared to Figure 8a,b, the temperature difference increases to a maximum of about 1.5 K in the mushy zone as well as in the air gap in the range of the cut-off distances, whereas the temperature difference decreases to within 1 K in the range of the OPGW cut-off distances. As can be seen from Figure 9c,d, the onset of the mushy zone overlaps for the three sets of values of the mushy zone constant A_m compared to Figure 8c,d, even though the distribution of phase transitions within the mushy zone is also different.

The above analysis illustrates that the effect of different mushy zone constants A_m on the phase transition simulation is mainly reflected in the temperature distribution in the mushy zone, air gap region. This is because A_m is characterizing the state of convective motion of the fluid and affects the convective heat transfer process of the fluid in the mushy zone, air gap region. At A_m = 10⁴, 10⁵, the OPGW temperature difference is within 0.25 K; when A_m is too large, the existence of damping term makes it difficult for convective heat transfer to occur in the air gap, which is closer to the simulation process of ice melting without considering the convection in the gap, such as the Y-axis truncated phase transition curve, shown by A_m = 10⁶ in Figure 9d, which shows the step trend. Although the effect of the mushy zone constant A_m on the position of the phase transition interface is small, it can be seen from Figure 6 that, without considering the gap convection, i.e., when the mushy zone constant is not introduced, there is a 3 mm difference in the size of the air gap for the phase transition interface at the same height below the OPGW, so this parameter should not be neglected in the numerical simulation of the phase transition.

5. Experiment Verification of OPGW DC Ice Melting

The artificial icing of OPGW was carried out in the laboratory, and the DC ice melting method was used to carry out the ice melting experiment. The temperature variation law of OPGW during the ice melting process was measured, and the melting process of ice melting was observed and compared with the simulation results of the dynamic process of ice melting.

The OPGW sample used in the experiment is 1000 mm long and the model is the same as that in the simulation model. The PT-100 patch leaded platinum resistor is used as a temperature sensor, which is pre-buried on the outer surface of the OPGW and close to the ground side, as shown in Figure 10.

An acrylic tube with a length of 700 mm and an inner diameter of 43.2 mm was used as the OPGW ice-covering device, and the OPGW sample with a pre-buried temperature sensor was placed in the center of the acrylic tube, which was then filled with water and placed in the cooler. After the outside of the OPGW in the acrylic tube was completely frozen, it was removed from the cooler, the acrylic tube was removed, and DC ice melting experiments were conducted on the ice-covered OPGW. Figure 11 shows the prepared ice-covered OPGW sample.

The experimental wiring is shown on Figure 12. In order to ensure that the temperature of the ice melting experiment is consistent with the ambient temperature of the simulation model, the ice melting experiment is carried out in a temperature-settable constant-temperature freezer, and the temperature of the freezer is set to 268.15 K, i.e., −5 °C, during the ice melting process. The effect of ambient wind speed on ice melting was simulated by an adjustable axial fan, and the wind speed of the fan was set to 5 m/s. The experimental site layout is shown in Figure 13.

The adjustable DC power supply is used as the ice melting power supply, and the ice melting current is set at 200 A. In order to observe the dynamic process of ice melting in OPGW in real time, a high-definition camera is used to film the whole process of ice melting and to observe the changes of the ice layer. The experimentally observed process of OPGW ice melt dynamics is shown in Figure 14. The platinum resistance temperature sensor was placed at the bottom of the outer surface of the OPGW; the measured temperature data were first processed by the digital display instrument, and then through the 485 communication busy line, and through the host computer real-time monitoring and display every 1 min to record the surface temperature value of the OPGW. The measured temperature rise curve of the OPGW DC ice melting is shown as the red solid line in Figure 15.

As shown in Figure 14, during the ice melting process, due to the effect of gravity, the ice layer gradually moves downward after forming an air gap, gradually forming an elliptical air gap, which makes the distance between the outer surface of the OPGW and each specific point on the inner surface of the ice layer inconsistent. The thermal resistance of the air gap in different heat transfer directions is also inconsistent, thus making the heat absorbed by the ice layer in different heat transfer directions, different, which leads to different melting rates of ice layers at different locations. Meanwhile, considering the gap convection heat transfer, the airflow velocity at the angle between the two sides of the OPGW and the inner surface of the ice layer is the highest, and the convective heat transfer coefficient is also large. The faster the heat transfer rate, the faster the ice layer melting rate, and the melting rate of the inner surface of the ice layer farthest from the OPGW surface is the slowest. Therefore, the shape of the phase transition interface obtained from simulation and experiment is elliptical.

This phenomenon is consistent with the shape of the phase change cross-section demonstrated by the simulation results in Figure 6b,d, and also with the results of the ice melting test carried out by Jiang Xingliang of Chongqing University at the Xuefengshan natural ice-covered test station in the literature [29], which shows that it is necessary to consider the interstitial convection during the numerical simulation of ice melting, and verifies the accuracy of the OPGW ice melting model and the computational results in this paper.

The temperature rise curve of OPGW measured in the DC ice melting experiment is consistent with the calculated temperature rise curve, which shows the trend of temperature increase and then decrease. This is due to the fact that with the downward movement of the ice layer, the thickness of the OPGW to the top upper surface of the ice layer is gradually reduced, resulting in the reduction of the ice layer’s thermal resistance, and at the same time, the air gap is also gradually increased, the loss of heat to the external environment is increased, and the temperature is gradually lowered.

The measured OPGW DC melting ice temperature rise curves are in good agreement with simulation results, and the relative standard deviation is 5.45%, with a maximum difference of about 3.5 K in temperature and about 10 min in ice melting time, but the overall trend is consistent, which further verifies the accuracy of the OPGW ice melting model established in this paper and the calculation results.

The error between the two is because the ground wire span often reaches several hundred meters in practical engineering. Therefore, the influence of OPGW axial heat transfer is neglected in the simulation model, and the OPGW geometric model is simplified accordingly. In this experiment, the axial length of OPGW is short, and the geometric shape is woven by strands. At the same time, there are uncontrollable differences in wind speed and wind direction on the ice surface in the experiment, which leads to certain errors.

6. Conclusions

Aiming at the dynamic physical process of OPGW DC ice melting, considering the gap convection heat transfer, the corresponding numerical analysis model is established, and the dynamic process of ice phase transition and OPGW temperature rise during ice melting is simulated. The following conclusions are drawn:

(1)

Whether or not to consider gap convective heat transfer during ice melting mainly affects the temperature distribution in the air gap region and has no significant effect on the temperature change in the OPGW. After considering the gap convection heat transfer, the temperature in the air gap area is lower, and at the same time, near the bottom of the OPGW, the maximum difference in the width of the air gap at the same phase change interface position of the ice layer is about 3 mm, and the phase change cross-section of the ice layer is elliptical. This is in line with the results of the actual melting test, and it shows that it is necessary to consider the gap convection heat transfer in the process of DC ice melting.

(2)

In the numerical simulation of the phase transition process, the values of the half width of the phase transition interval dT and the constant A_m of the mushy zone have a direct influence on its correctness, the former mainly reflects the position of the phase transition interface and the temperature rise of the heat source OPGW in the model, while the latter reflects the temperature distribution in the mushy zone and the air gap region, which has a smaller influence on the determination of the position of the phase transition interface and the temperature rise of the OPGW, but it can be seen that this parameter is still not neglected in the numerical simulation of phase transition. From Conclusion (1), it can be seen that this parameter is still not negligible in the numerical simulation of phase change.

(3)

When the half-width of phase transition interval dT is too small, the sudden change in material properties caused by phase transition will be distorted or even the singularity will not converge; when the mushy zone constant A_m is too large, the convective heat transfer in the mushy zone and the air gap is difficult to occur due to the existence of the damping term. At this time, it will approach the ice melting simulation process without considering the gap convection. Combined with the calculation and analysis results, the half-width of phase transition interval dT and the mushy zone constant A_m can be selected as 0.5 K and 10⁴, respectively.

(4)

The shape of the elliptical phase transition cross-section formed during the OPGW DC ice melting experiment is consistent with that of the ice melting simulation model, and is also consistent with the results of the ice melting experiment carried out by the literature [26] at the Xuefengshan natural ice-covered experimental station. Due to the assumption and simplification of the model, as well as the uncontrollable factors, such as wind speed and wind direction in the experiment, the temperature rise curves of OPGW DC ice melting measured in the experiment have some errors with the simulation results, and the relative standard deviation is 5.45%, with a maximum difference of about 3.5 K in temperature and 10 min in ice melting time. But, the overall trend is the identical, and the temperature rise curves are all shown as increasing first and then decreasing, which verifies the accuracy of the OPGW DC ice melting model and the calculation results.

The refined numerical analysis model of OPGW DC ice melting constructed in this paper can provide technical support for evaluating or predicting the real-time ice melting status of overhead ground lines, and then provide reference for the reasonable selection of ice melting current and ice melting time, which is of important engineering practical value.