Model Optimization of Ice Melting of Bridge Pylon Crossbeams with Built-In Carbon Fiber Electric Heating

Abstract

This paper aims to improve the deicing performance and energy utilization of bridge pylon crossbeams with built-in carbon fiber electric heating (BPB–CFEH). Therefore, a three-dimensional thermal transfer model of BPB–CFEH with one arrangement is established. Two ice-melting regions and two ice-melting stages were set up according to the characteristics of the icing of the crossbeam. The effects of wind speed and ambient temperature on the paving power required to reach the complete melting of the icicles within 8 h were analyzed. The effects of the laying spacing and rated voltage of the carbon fiber heating cable on the melting ice sheet and the thermal exchange of the two regions of the icicle after heating for 8 h were compared. Additionally, its effect on energy utilization of the process from the ice sheet melting stage to the ice column melting stage was analyzed. Ice-melting experiments verified the applicability and reasonableness of the simulated ice-melting calculation formula. The results show that under ambient temperature of −10 °C and wind speed of 4.5–13.5 m/s, the proposed paving power is 817.5–2248.12 W/m². Increasing the rated voltage and shortening the spacing increases the thermal exchange capacity of the two melting regions. The shortening of the spacing improves the energy utilization rate of the melting stage of the ice sheet to the melting stage of the icicle processes. The difference between the melting time obtained from the formula proposed by numerical simulation and the melting time obtained from indoor tests is about 10 min. This study provides a design basis for the electrothermal ice melting of bridge pylon crossbeams.

1. Introduction

Bridges play a vital role in modern cities and have far-reaching implications for the efficiency and sustainability of urban transport. When ambient temperatures are low, fog, rain, and snow can cause bridges and bridge cables to freeze. As the temperature warms, they will fall onto the roads, seriously threatening pedestrians and road safety [1,2,3,4]. And the ice accretion to the variation of bridge properties (mainly mass) can lead to false alarms if a monitoring system is installed [5]. At the same time, bridge pylon crossbeams have a larger surface region compared to bridge cables, and the risk of freezing is more significant [6].

The main objects of current research on bridge ice melting are bridge deck structures and bridge cables. Their ice-melting methods are mainly divided into four categories: laying carbon fiber heating cables inside the bridge deck [7,8]; laying thermal exchange pipes inside the bridge deck (using thermal pumps to transfer thermal from the thermal source to the roadway through circulating fluids) [9,10]; removing the ice buildup by human resources, but this method is inefficient and dangerous to work at high altitudes [11,12,13]; and applying a hydrophobic coating on the surface of the bridge structure to prevent icing, but with a concise service life [14,15,16,17]. Among them, burying carbon fiber heating cables and thermal exchange pipes is the method of melting ice which uses thermal energy [18]. However, due to the high position of the bridge pylon crossbeams, the total trip of laying the thermal exchange pipe is too long and leads to too much thermal loss, and the effect of melting ice is poor. Ice melting by carbon fiber electric heating is not affected by the environment and deployment location and is highly applicable. And there are already engineering applications that use carbon fiber heaters to melt ice and snow. At present, scholars have applied carbon fiber heating wires to bridge cables [19], road pavements [20], tunnels [21], and airport pavements [22]. The research results show that the carbon fiber heating wire can achieve the effect of melting snow and melting ice in practical engineering. At present, there are no examples of carbon fiber release wires applied to bridge pylon crossbeams, only numerical simulations of this aspect [6]. Therefore, this paper chooses to use carbon fiber electric heating as the suspension bridge pylon crossbeam structure of the ice-melting program.

On the other hand, it is essential to consider the ice-melting efficiency of the approach, namely the energy usage of the system, to minimize energy wastage during the ice-melting process [23]. Kai Liu et al. [7,24] studied the energy utilization of bridge deck structures and developed an automatic snow-melting system (AS-EHP) that utilizes cable heating and snow melting. Their findings indicate that a shallower burial depth of the cables leads to improved energy utilization in bridge deck structures. AS-EHP’s design and operation strategy is proposed by studying the energy of three factors: pipe burial spacing, burial depth, and wind speed. Huining Xu et al. [25,26] developed a model that combines thermodynamics and mass transfer to analyze hydraulically heated pavements. They also proposed an optimal operation strategy for these pavements. Additionally, they created a numerical model that combines thermodynamics and economics to evaluate a hydraulic snow-melting pavement system. This model was compared with data from the Beijing Daxing International Airport, providing recommendations for the system’s operation strategy. Yong Lai et al. [27] conducted field experiments and numerical simulations to investigate the temperature and energy distribution at various pavement depths. Their research aimed to give empirical references for enhancing the energy efficiency of snow and ice melting on airport pavements.

However, all of the above studies have been conducted on bridge decks, pavements, and airport runways. Referring to the current research results [28,29], when the bridge pylon crossbeam encounters freezing rain, the side of the crossbeam will be covered with water droplets which follow to the bottom; during this process, the water droplets will not directly fall onto the ground but attach to the bottom boundary of the side of the crossbeam, and over time, the bottom of the crossbeam will generate icicles, which provide an extra surface for the water droplets to attach to, and increase the rate of the droplets’ icing. The place where the icicles are generated belongs to the boundary region of the structure, and the heating efficiency in the boundary region of the structure is slower than that in the center region [30]. Bridge pylon crossbeams need to consider the ice-melting performance at the bottom in the process of ice melting compared to the above research objects, and in order to reduce energy loss and improve energy utilization, it is necessary to conduct further research on the energy utilization of bridge pylon crossbeams for ice melting.

This study takes the model of a bridge pylon crossbeam with built-in carbon fiber electric heating (BPB–CFEH) in the pilot base of snow melting and ice melting in Hubei University of Technology. As an example, this study establishes a three-dimensional simulation model of the bridge pylon crossbeam, and puts forward the division method of the melting region and melting stage of BPB–CFEH. We also investigate the effects of environmental factors and heating schemes on the ice-melting performance of bridge pylon crossbeams and the energy utilization of bridge pylon crossbeams when the icicles are completely melted under different heating schemes. A multifactor optimization method is used to propose a calculation method for the pavement power and lateral thermal transfer required for ice melting in BPB–CFEH. Additionally, combined with the local winter weather conditions, the BPB–CFEH ice-melting optimization strategy is proposed to provide a design basis for the bridge pylon beam electrothermal ice melting.

2. BPB–CFEH Structure and Operational Stages

2.1. Structure

There is a risk of ice building up and falling off the bridge pylon crossbeams during the winter. In this regard, we set up a laboratory test model according to the scale reduction of the real bridge pylon crossbeam structure. The physical significance of nomenclature in this paper is shown in

Table 1. Abbreviations used in this paper.

Nomenclature	Physical Significance
EC	The total energy consumption (KJ)	EL	The energy lost in the environment (KJ)	T	Temperature (°C)
SEMI1	Stored melted icicle energy (KJ)	SEMI2	Stored melted ice sheet energy (KJ)	t	The heating time (h)
ELR	The proportion ratio of the EL (%)	SEMI1R	The proportion ratio of the SEMI1 (%)	AT	The ambient temperature (°C)
SEMI2R	The proportion ratio of the SEMI2 (%)	$Q_u$	Thermal exchange in region 1 (KJ)	P	Paving power (W/m2)
$M_{isnow(i=\mathrm{1,2})}$	1: weight of the ice sheet (kg) 2: weight of the icicle (kg)	$Q_a$	Thermal exchange in region 2 (KJ)	TI	The thickness of the ice sheet (cm)
$S_{i(i=\mathrm{1,2})}$	1: Area of region 2 (cm2) 2: Area of region 1 (cm2)	$Q_1$	The energy required for the stage transformation of accumulated snow liquefaction (KJ)	IC	Ice coverage (%)
$t_1$	Time for the average temperature of the region 2 to reach 0 °C (h)	$Q_2$	The energy required for the temperature increase in the accumulated snow (KJ)	LI	The average length of the icicle (cm)
$t_2$	Time for the average temperature of the region 1 to reach 0 °C (h)	$Q_{ea}$	Energy required to melt the ice sheet in the experimental (KJ)	${\rho }_{snow}$	Density of ice (kg/m3)
$t_{ea}$	Predicted elapsed time for complete melting of the ice sheet in the experiment (min)	$Q_{ea}$	Energy required to melt the icicles in the experimental (KJ)	$\rho$	Density of concrete (kg/m3)
$t_{eu}$	Predicted elapsed time for complete melting of the icicles in the experiment (min)	$\overline{n}$	Vectors in the direction of temperature conduction	$x,y,z$	The coordinate in the model
$h_c$	Convective thermal transfer coefficient (W m−2 K−1)	N	Number of records	$H\left(t\right)$	Thermal source
$L_1$	Length of heating cable buried in the model (m)	$L_2$	Total length of heating cable (m)	L	Length of crossbeam (m)
$p$	Heating cable power per unit (W/m)	$U$	Rated voltage (V)	$X_{sim,i}$	Simulated value
$R$	Unit cable resistance (Ω/m)	$v$	Wind velocity (m/s)	$X_{mod,i}$	experimental
$\phi$	Energy utilization ratio	$C_{snow}$	Specific thermal capacity of ice (J/kg °C)	$C$	Specific thermal capacity of concrete (J/kg °C)

. The BPB–CFEH has a structural dimension of 2 m × 0.4 m × 0.2 m. Inside the structure is a carbon fiber hot-wire cable arranged in a multi-U-type configuration. This configuration consists of four parallel hot-wire cables and three bending hot-wire cables, as depicted in Figure 1a. The distance between each parallel heating cable is 10 cm, and three K-type thermocouple cables are positioned on each heating cable as temperature measurement locations, as depicted in Figure 1b,c. BPB–CFEH consists of C50 concrete and is positioned on two concrete pedestals with a height of 40 cm, as depicted in Figure 1d.

2.2. Operational Stages and Ice-Melting Regions

In PBB-CFEH, there are two main regions of ice melting: region 1 involves the melting of icicles, and region 2 involves the melting of ice sheets. The heating surface of PBB-CFEH is shown in Figure 2a. Regions 1 and 2 are divided on the heating surface, as shown in Figure 2b. When the average temperature of the heating surface of the test model reaches 0 °C, the ice begins to melt. Therefore, region 2 of the test model is heated to an average temperature of 0 °C in stage 1, and region 1 is heated to an average temperature of 0 °C in stage 2. The times for the first and second stages are $t_1$ and $t_2$, respectively.

3. Thermal Transfer Modelling of BPB–CFEH

3.1. BPB–CFEH Physical Model

In this part, concerning the scaled-down model of the bridge pylon crossbeam [29], a 3D simulation model with the exact dimensions is constructed, as shown in Figure 3a. The model is meshed with tetrahedral cells, and the carbon fiber heating cable is meshed with hexahedral cells, as shown in Figure 3b. The total number of nodes is 1,994,535, and the total number of finite elements is 10,013,528, as shown in Figure 3. The minimum orthogonal mass and maximum skewness are 0.15 and 0.84, respectively, both of which are within the acceptable range. The material parameters involved in the simulation model are shown in

Table 2. Material physical parameters.

Material Parameter	Concrete	Carbon Fiber Heating Cable	Polythene
Thermal conductivity (W/m °C)	2.2	6.8	0.4
Densities (kg/m3)	2500	1200	930
Specific thermal capacity (J/kg °C)	970	1440	2300

.

3.2. Basic Assumptions and Boundary Conditions for the BPB–CFEH Thermal Transfer Model

Many different thermal transfer processes are involved in the heating and ice-melting process of the bridge pylon crossbeams, as shown in Figure 4. In order to obtain better simulation results, the following assumptions are made in this study:

(1) In the thermal transfer process, the model is surrounded by convective boundary conditions, and the solar radiation process is not considered on the surrounding surfaces because the indoor tests are carried out in a laboratory with a constant temperature without solar radiation. The model has an adiabatic boundary in the direction along which the heating cable is arranged. This is because the cross-sectional region of the heating cable is much smaller than that of the beams, and the temperature change is minimal along the direction where the heating cable is located [31].

(2) The carbon fiber heating cable is a cylindrical thermal source encased in a PVC layer. The PVC, carbon fiber heating cable, and concrete form a homogeneous medium, and their thermal properties remain constant regardless of temperature fluctuations. And the carbon fiber hot wire, PVC and concrete are in close contact, without considering the contact thermal resistance.

The thermal transfer of BPB–CFEH is a three-dimensional non-stationary thermal transfer process with coordinates $x-y-z$. Defining the thermal generating cable as the thermal source term $H\left(t\right)$, the thermal transfer process follows Fourier’s law as in Equation (1):

$$\frac{\partial T}{\partial t}=a\left(\frac{{\partial }^{2}T}{\partial X^2}+\frac{{\partial }^{2}T}{\partial Y^2}+\frac{{\partial }^{2}T}{\partial Z^2}\right)+\frac{H\left(t\right)}{Cp}$$

(1)

The boundary conditions around the simulation model (excluding the two faces along the direction of the heating cable arrangement) conform to Equation (2):

$$-\lambda \frac{\partial T}{\partial \overrightarrow{n}}=h\left(T-AT\right)$$

(2)

Adiabatic boundary conditions are set along the two faces in the direction of the heating cable arrangement, in accordance with Equation (3):

$${\left.\lambda \frac{\partial T}{\partial X}\right|}_{X=0}=0,{\left.\lambda \frac{\partial T}{\partial X}\right|}_{X=L}=0$$

(3)

In order to stream the cable to the boundary conditions of current thermal exchange, this study assumes that the ambient temperature remains constant and does not undergo any changes.

The thermal exchange coefficient (h_c, W·m⁻² K⁻¹) is defined as shown in Equation (4):

$$h_c=\left\{\begin{array}{c}6+4v\left(v\le 5\mathrm{m}/\mathrm{s}\right)\\ 7.41v^{0.78}\left(v\ge 5\mathrm{m}/\mathrm{s}\right)\end{array}\right.$$

(4)

3.3. Model Validation

The 10th set of test data from the indoor modeling tests, which had the most significant effect on the internal temperature field, was selected as the control group for the simulation data [29]. The duration of this group of test ice melting is 8 h, the ambient temperature is constant at −15 °C, and the wind speed is 4.5 m/s. The test is carried out at the same time. The initial internal temperature of the BPB–CFEH was measured at −9.99 °C using a Keysight thermometer before testing. A carbon fiber heating wire was used to heat the crossbeam during the experiment. Real-time monitoring of temperature data was conducted at three measurement stations labeled A, B, C, and D. The same test conditions as in group 10 were set up in the simulation. However, the internal initial temperature of the BPB–CFEH in the indoor modeling test was only determined based on the average of 12 temperature measurement points, which does not accurately measure its internal initial temperature. The initial temperature of the internal concrete part set by BPB–CFEH in the simulation is consistent with the ambient temperature of −15 °C, and the initial temperature of the carbon fiber heating cable and polyethylene is −9.99 °C.

The difference between the average value of twelve temperature measurement points on heating cables A, B, C, and D and the finite element simulation results was selected as the error evaluation criterion. According to the Root Mean Square Error (RMSE) Equation (5), the Root Mean Square Error (RMSE) of the average value of the experimental data and simulation data is 0.243, which indicates that the simulation accuracy is high and can be used as a reference:

$$RMSE=\sqrt{\frac{{\sum }_{i=1}^N{(X_{sim,i}-X_{mod,i})}^2}{N}}$$

(5)

Figure 5 shows the simulated values were consistently lower than the experimental values until 4 h. This is due to the fact that the average temperature of the concrete portion of the model modeled in the tests under the initial conditions is higher than the initial temperature set in the simulation. After 4 h, the simulated value gradually exceeded the test value. This is because, during the ice-melting process in the test, the upper part of the beam side where the ice melts into water does not directly enter the ground. Instead, it slowly flows towards the bottom of the model and remains attached to the bottom for a certain period before finally flowing into the ground. Throughout the procedure, the water would persistently assimilate the internal temperature of the model, resulting in the occurrence above.

4. Multi-Factor Impact Analysis of Ice-Melting Performance and Energy Utilization Based on BPB–CFEH

In order to assess the influence of various environmental conditions and heating programs on the melting efficiency of BPB–CFEH, the melt performance is quantified by measuring the amount of energy needed to achieve stage 2 within 8 h, as well as the total thermal exchange in regions 1 and 2 during this time.

In order to evaluate the impact of different heating schemes on the energy utilization of the BPB–CFEH, its operational stages and energy distribution were analyzed. According to the thermal field distribution and stage division, the total energy consumption (EC) of the heating cables during stage 1 and stage 2 can be divided into three parts: energy lost to the environment (EL), stored melted icicle energy (SEMI1) and stored melted ice sheet energy (SEMI 2), as shown in Figure 6. The above energy can be expressed by the following Equation (6):

$$EC=EL+SEMI1+SEMI2$$

(6)

EC is determined by the heating power, burial spacing, and $t_1$, $t_2$ as shown in Equations (7) and (8):

$$EC=p\times \left(t_2-t_1\right)\times L_1$$

(7)

$$p=\frac{U^2}{{\mathrm{L}}_2\times R}$$

(8)

The formula L₂ is the total cable length of 14 m. R is unit wire resistance (in this study, 48 K carbon fiber thermal wire is chosen, and the wire resistance is 8 Ω/m).

The energy stored in the BPB–CFEH can be expressed by Equation (9):

$$SEMIi=C{V}_i\rho ∆T$$

(9)

Changing the rated voltage and burial spacing will change the EC. Therefore, the percentage of EL, SEMI2, and SEMI 1 is proposed to analyze the energy utilization, which can be represented by Equation (10):

$$ELR+SEMI1R+SEMI2R=\frac{EL+SEMI1+SEMI2}{EC}$$

(10)

4.1. Analysis of the Effect of Wind Speed and Ambient Temperature on Ice-Melting Performance

In this section, the analysis of the effects of wind speed and ambient temperature on the ice-melting performance of bridge pylon crossbeams is carried out, considering three different wind speeds of 4.5 m/s, 9 m/s, and 13.5 m/s, respectively, and three different ambient temperatures of −5 °C, −10 °C, and −15 °C, respectively, for a total of nine sets of working conditions. The heating time was limited to within 8 h, the same as the model test time, to reduce the negative impact of a heating time that is too long to avoid impacting the structural durability [24].

The paving power required to reach the second stage in 8 h in nine groups of working conditions is shown in Figure 7. When the wind speed is 4.5 m/s, the paving power increases by 272.48 W/m² and 476.9 W/m² when the ambient temperature decreases from −5 °C to −10 °C and −15 °C, respectively, and when the wind speed is 13.5 m/s, the paving power increases by 816 W/m² and 1157.35 W/m² when the ambient temperature decreases from −5 °C to −10 °C and −15 °C, respectively. A decrease in ambient temperature leads to an increase in paving power required to melt ice. The change in laying power required to achieve the same melting effect for the same ambient temperature varies for different wind speeds because the increase in wind speed results in more significant energy loss in convective thermal transfer.

In three sets of working conditions with an ambient temperature of −10 °C and wind speeds of 4.5 m/s, 9 m/s, and 13.5 m/s, it took 7.33 h, 7.5 h, and 7.5 h, respectively, to reach stage 2. Its paving power increases by 681.25 W/m² and 749.37 W/m² when the wind speed increases from 4.5 m/s to 9 m/s and 9 m/s to 13.5 m/s, respectively. This is due to increased wind speed, which leads to more significant energy loss in convective thermal transfer. Moreover, the increase in laying power is very close for these three sets of conditions. In the three sets of working conditions with ambient temperature −5 °C and wind speeds of 4.5 m/s, 9 m/s, and 13.5 m/s, it took 2.83 h, 3.66 h and 5.833 h to reach stage 2, and the paving power increased by 272.48 W/m², 476.9 W/m² and 476.9 W/m² when the wind speed was increased from 4.5 m/s to 9 m/s and 9 m/s to 13.5 m/s, respectively. The increase in paving power varies considerably. The paving power is also affected by the time taken to reach stage 2.

4.2. Analysis of the Effect of Burial Spacing and Rated Voltage on Ice-Melting Performance

This section analyses the effects of burial spacing and rated voltage on region 1 ($Q_u$) and region 2 ($Q_a$) thermal transfer of the bridge pylon beam. Three types of burial spacing were considered: 5 cm, 7.5 cm, and 10 cm. Three kinds of rated voltage were used: 220 V, 260 V, and 300 V, for a total of nine groups of working conditions. In order to reduce energy consumption, the simulation was set at an ambient temperature of −5 °C, a wind speed of 4.5 m/s, and a heating time of 8 h. The design parameters for different pipe spacing values are detailed in

Table 3. Design parameters of BPB–CFEH with different pipe spacing values.

d/cm	Number of Heating Cables	Distance of Heating Cables from the Boundary/cm	Total Length of Heating Cable/m
10	4	10	7
7.5	6	6	10.4
5	8	5	13.8

.

As shown in Figure 8 and Figure 9, $Q_u$ and $Q_a$ increase with increasing U and decreasing d during heating for 8 h. When the voltage is 220 V, the distance changes from 10 cm to 7.5 cm and 7.5 cm to 5 cm; $Q_u$ changes by 25 KJ and 47.26 KJ, while $Q_a$ changes by 262.46 KJ and 104.97 KJ, respectively. The amount of change in $Q_u$ and $Q_a$ was 9.6% and 45.02%, respectively. When the distance is 10 cm, and the voltage is changed from 220 V to 260 V and from 260 V to 300 V, $Q_u$ changes by 78.65 KJ and 61.46 KJ and $Q_a$ by 1739.64 KJ and 811.36 KJ, respectively. The amount of change in $Q_u$ was 4.52%, and 3.39% of the change in $Q_a$, respectively. It was found that changing U was effective in increasing $Q_u$ and changing d had less of an effect on $Q_a$. Changing U and d had almost the same effect on $Q_u$.

Quantitative analyses were carried out to analyze the effect of different spacing levels and different rated voltages on the amount of heat emitted from regions 1 and 2. Multiple regression was used to analyze the relationship between $Q_u$, $Q_a$ and, d and U, as shown in Equations (11) and (12):

Q_u = −17.425d + 1.941U − 113.373 R² = 0.978

(11)

Q_a = −135.196d + 48.482U − 5063.629 R² = 0.992

(12)

4.3. Analysis of the Effect of d and U on the Energy Utilization of the Process from Stage 1 to Stage 2

This section analyses the energy distribution of the stage 1 to stage 2 process based on the nine sets of working conditions detailed in Section 3.2. The effect of different spacing and rated voltage on SEMI1R and SEMI2R is analyzed. The energy utilization ratio is also presented for analysis.

As shown in Figure 10, both SEMI1R and SEMI2R increase with increasing U, but the growth of SEMI1R is not significant. With the same burial spacing, changing the rated voltage from 220 V to 260 V slowed the growth rate of SEMI1R compared to changing the rated voltage from 260 V to 300 V. Increasing the voltage below the rated voltage of 260 V increases SEMI1R more efficiently than increasing the spacing, whereas increasing the rated voltage to the maximum results in increasing SEMI1R less efficiently.

Changing the rated voltage greatly affects SEMI2R when the burial spacing is 5 cm. This is because when the burial spacing is 5 cm, there are more carbon fiber heating cables in region 2 compared to when the burial spacing is 7.5 cm and 10 cm. This leads to more significant thermal accumulation, as indicated in Table 3. On the other hand, the SEMI1R value is lowest when the burial spacing is 5 cm. This is because a shorter spacing results in higher overall energy consumption, with most of it dispersed into the environment.

In this study, EL is treated as the energy of melting ice, and SEMI1 directly affects the ice column in region 1 and is considered the effective energy of the melt. SEMI2 has relatively no effect on ice columns in region 1. Then, for a better understanding of the impact of quantified voltage and deposit interval on the energy utilization of the stage 1 to stage 2 processes, the energy usage ratio $\phi$ is presented; see the following formula:

$$\phi =\frac{ELR+SEMI1R}{SEMI2R}$$

(13)

Based on the value of $\phi$, it is known that the burial spacing of 5 cm is the optimum for this study, as shown in Figure 11. Changing the voltage has little effect on $\phi$ when the burial spacing is 10 cm compared to 5 cm. The smaller the spacing, the larger the $\phi$. The larger the rated voltage, the smaller the $\phi$. This is because most of the energy generated by decreasing the spacing is used to enhance the EL thermal transfer rate and decrease the SEMI2R, while most of the energy generated by increasing the rated voltage is used to increase the SEMI2R, as shown in Figure 10.

5. Ice-Melting Strategy of BPB–CFEH

This section aims to make Equations (11) and (12) applicable in practical engineering scenarios and to provide a suitable ice-melting solution for bridge pylon crossbeams in cold regions. Based on the data recorded from the icing test of the crossbeam and the designed energy calculation formula, the energy required for melting the ice layer and the energy required for melting the icicle were calculated and compared with the thermal exchange calculated by Equations (11) and (12). One set of indoor icing and ice-melting tests was designed to record the thickness of the ice sheet (TI), the ice coverage (CR), and the average length of the icicle (LI) on the side of the beams, and the designed equation for the energy required for ice melting was divided into two parts: the energy required for the liquefaction of the ice layer by stage change $Q_1$, and the energy required for the warming of the ice layer by $Q_2$, which can be calculated using Equations (15) and (16) [7]:

$$M_{isnow}=TI\times S_i\times CR\times {\rho }_{snow}$$

(14)

$$Q_1=M_{isnow}\times H_{if}$$

(15)

$$Q_2=M_{isnow}\times \left[C_{snow}\times \left(0-AT\right)\right]$$

(16)

The formula ${\rho }_{snow}$ is the density of ice, 920 kg/m³, $H_{if}$ is the latent heat of ice liquefaction, 333,697 J/kg, $C_{snow}$ is the thermal capacity for ice, 2100 J/kg.

It is not easy to clearly distinguish between the energy expended in melting the ice on the sides of the beams during the ice-melting test. However, it is possible to distinguish between completely melting the ice on one side of a beam and completely melting an icicle. In order to verify the accuracy of Equations (11) and (12), the time consumed for the complete melting of region 1 and region 2 was extrapolated by the percentage of complete ice-melting energy calculated in the test to the energy calculated in Equations (11) and (12), which was calculated as follows:

$$\frac{Q_{ea}}{Q_a}=\frac{t_{ea}}{t}$$

(17)

$$\frac{Q_{eu}}{Q_u}=\frac{t_{eu}}{t}$$

(18)

A complete dataset of ice-melting process data was recorded, and $Q_{eu}$ was 35.65 KJ, and $Q_{ea}$ was 620.65 KJ as calculated by Equations (14)–(16) (see

Table 4. Measured data from beam icing tests and calculated ice cover weights.

Group	v/m·s	AT/°C	TI/cm	IC/%	LI/cm	${\mathit{Q}}_{\mathit{e}\mathit{a}}$/KJ	${\mathit{Q}}_{\mathit{e}\mathit{u}}$/KJ
a	0	−5	0.35	70	2.3	620.65	35.65

). The thermal released in 8 h was calculated according to Equations (11) and (12) and the design parameters of the indoor experimental model (d = 10 cm, U = 220 V), and the thermal energy released in 8 h was $Q_u$ = 139.14 KJ, and $Q_a$ = 4250.48 KJ. The energy required for the melting of the icicle and the melting of the side ice in group A accounted for 25.62% and 14.6% of $Q_u$ and $Q_a$, respectively. The analysis of the percentage of energy calculated by Equations (17) and (18) shows that the predicted time for the complete melting of the icicle is 122.97 min, and the predicted time for the complete melting of the side ice layer is 70.08 min. In the indoor test, the time for the complete melting of the icicle is 132 min, and the time for the complete melting of the side ice layer on the crossbeam is 82 min, as shown in Figure 12. The difference between the simulated and tested time for the complete melting of the icicle is 9.03 min. The difference between the simulated and tested time for the complete melting of the lateral ice layer is 11.92 min. The agreement between the tested and simulated values is good, and the error is within the acceptable range, which verifies the accuracy of Equations (11) and (12).

The average temperature in January 2023 in Wuhan was noted to be −1 °C, and the minimum temperature was −5 °C. Concerning the bridge pylon beam scaling model of this study, it is assumed that the beams are covered with 100 percent ice, the ice thickness is 1 cm, and the average length of the icicles is 20 cm. Based on Equations (14)–(16), the values obtained are $Q_{ea}$ = 2341.25 KJ and $Q_{eu}$ = 126.54 KJ. These values are then used in Equations (17) and (18). If it is essential to melt within 2 h completely, $Q_{ea}$ requires $Q_a$ = 9365 KJ. According to Equation (12), d = 5 cm and U = 311.55 V are obtained to satisfy this ice-melting condition. If $Q_{eu}$ is to be completely melted in 2 h, $Q_{eu}$ = 506.16 KJ is required, and according to Equation (11), d = 5 cm and U = 364.06 V are obtained to satisfy the melting condition.

6. Conclusions

This paper uses the model of the bridge pylon crossbeams with built-in carbon fiber electric heating (BPB–CFEH) as an example to establish a thermal transmission simulation model of the same size. We examined the necessary paving power for 8 h under various environmental circumstances to achieve complete melting of the icicle. The impact of various heating methods on the energy release of the melted ice sheet and the melted icicle region, and energy efficiency during the melting ice sheet to melting icicle were investigated. Ultimately, the calculation formula is introduced, and the BPB–CFEH ice-melting optimization strategy is suggested in conjunction with the melting ice test. The main conclusions are as follows:

(1) As the ambient temperature decreases, the wind speed increases, resulting in an increased demand for heating power in the process of heating the bridge pylon. The time it takes to reach the complete melting stage of the icicle will also affect the size of the laying power; the shorter the time it takes, the greater the required laying power.

(2) By increasing the quota voltage and reducing the storage intervals, the thermal exchange of the ice-melting region across the bridge pylon can be enhanced, given an environmental temperature of −5 °C, wind speed of 4.5 m/s, and heating period of 8 h. Changes in voltage and spacing have almost the same effect on the thermal exchange in melting icicle regions. Modifying the voltage, as opposed to altering the intervals, has a more significant impact on thermal exchanges in the regions of the melted ice layer.

(3) The utilization of energy by φ characterizing BPB–CFEH from the ice sheet’s melting stage to the ice sheet’s melting stage is presented. The test results showed that the smaller the laying interval, the higher the energy utilization rate. The larger the rated voltage, the lower the energy usage rate.

(4) The thermal exchange calculation formula for melted ice layers and melting icicles was studied, and the formula’s accuracy was verified through indoor melt-ice tests. Finally, the method of using the formula is presented. It provides the design basis for the thermal melting of ice on the bridge pylon.

The study is limited in its applicability because it is limited to a specific geographic location and set of environmental parameters. The results cannot be generalized to other regions with distinct meteorological characteristics, such as extreme negative temperatures. Finding the ideal configuration can be accomplished by extrapolating the application of the finite element model to different configurations. However, this might be viewed as a possibility for a future research direction, much like the preceding argument.