Simulation Analysis and Optimization Design of Electromagnetic Fields in Coaxial Processing Cavities Based on the Theory of Electrostatic Fields

The coaxial processing cavity is an important component of high-frequency electromagnetic water treatment devices. Due to its easy installation, wide applicability, and excellent scale removal effect, it has been widely used in various factories. To study the influence of excitation source waveform, processing cavity length, and anode-cathode radius ratio on the intensity and distribution of the electromagnetic field in the coaxial processing cavity under time-varying fields, this paper derives the numerical calculation formula for electromagnetic energy in the coaxial processing cavity based on the theory of electrostatic fields. The finite element method is used to visualize the intensity and distribution of the electromagnetic field in the coaxial processing cavity. By optimizing the structure of the coaxial processing cavity, the optimal range of the anode-cathode radius ratio is determined. The research results show that, under the conditions of electrostatic field theory, square waves have the maximum electromagnetic energy, the length of the processing cavity does not affect the distribution of the electromagnetic field, and the optimal anode-cathode radius ratio range is 0.375-0.5.


Introduction
Industrial circulating cooling water plays a crucial role as one of the key methods for heat and energy exchange in large-scale mechanical equipment [1][2][3] .It finds widespread application in areas such as steel smelting, metal manufacturing, and thermal power generation, where it assumes an irreplaceable role.However, as the circulating water undergoes continuous concentration, scales start to deposit on the inner walls of pipes, leading to the growth of bacteria.This phenomenon reduces heat transfer efficiency, corrodes the pipes, and adversely affects the normal operation of equipment [4][5][6] .
High-frequency electromagnetic water treatment is an emerging descaling technology.This equipment consists of a processing cavity and a high-frequency signal generator.It offers advantages such as pollutionfree operation, ease of use, and remarkable descaling effects.Therefore, it has found extensive applications in the water treatment systems of various industrial plants.The processing cavity, as a crucial component of electromagnetic water treatment devices, plays a vital role in descaling and scale inhibition.Through a review of relevant literature, it has been found that factors such as the size of the processing cavity, waveform of the excitation source, and anode-cathode radius ratio have an impact on the electromagnetic field inside the coaxial processing cavity, thereby influencing the descaling effects.The temporal variation and complex structure of the electromagnetic field within the coaxial processing cavity introduce numerous influencing factors, leading to the current instability in the descaling performance of the processing cavities [7][8][9] .
This study is based on the theory of electrostatic fields to derive the electromagnetic field calculation formula for the coaxial processing cavity.By utilizing the COMSOL Multiphysics software, a finite element model is established to analyze the influence of the excitation source waveform, the length of the pipes, and the anode-cathode radius ratio on the size and distribution of the electromagnetic field in the coaxial processing cavity.This provides a theoretical basis for the design of the processing cavity in high-frequency electromagnetic water treatment equipment.

Basis of a quasi-static method for coaxial processing cavity
The differential form of the Maxwell equations is ) The phenomenon of electromagnetic waves is intricate and complex.In a processing chamber where a time-varying excitation source is applied, electric and magnetic fields appear alternately, interdependent, and constrained.Maxwell's equations provide a numerical description of electromagnetic waves, where changing electric fields generate magnetic fields, and changing magnetic fields generate electric fields.However, solving such time-varying fields accurately is highly challenging.Based on the works of Xiong Lan, Hermann A. Haus, and others [8,10,11] , the electromagnetic field inside a coaxial processing chamber can be considered a quasistatic field under certain conditions, enabling the obtainment of accurate analytical data.The conditions for a quasistatic field are as follows: within the time frame of our interest, the electromagnetic waves in the studied system are stable and rapidly varying.Thus, the studied problem can be regarded as an electro-quasistatic field, allowing for precise analysis.

Quasi-static electric field
The empirical method to determine whether the electromagnetic field inside a coaxial processing chamber is electro-quasistatic or magneto-quasistatic involves reducing the frequency of the excitation source to make the field static.In this limiting case, if the magnetic field disappears, then the field is electro-quasistatic.In the present study, when a constant electromotive force source is used to excite the coaxial processing chamber, the chamber becomes charged, generating an electric field.However, in this limiting case, there is almost no current and the magnetic field is weak.Therefore, the electromagnetic field inside the coaxial processing chamber is considered to be electro-quasistatic.
The electric quasi-static law refers to the approximation made by neglecting electromagnetic induction in Maxwell's equations.
In an electro quasistatic field, the electric field intensity E  described by Equation ( 5) is irrotational, meaning it has zero curl.According to Equation ( 6), when the charge density is known, the divergence and curl of the electric field intensity E  can be determined.By solving these equations simultaneously, Equation (9) and Equation ( 10) can be obtained.Taking the divergence of Equation (1.6) eliminates the magnetic field intensity H   and results in Equation (11).
By combining Equations ( 9) -( 11), the current density J  and electric field intensity E  can be determined.Then, Equations ( 6) and ( 8) are used to determine the magnetic field intensity B  .Next, let's analyze the conditions for using the electro quasistatic field.When the "error field" in the electroquasistatic field is much smaller than the derived values in the quasi-static field, the electro-quasistatic field can be used for calculations.According to the literature by Xiong Lan, Hermann A. Haus, and others, when there is only one typical length scale L, the curl and divergence operators can be approximated by 1/L.If the excitation source is time-varying and has a characteristic timeτ (usually the reciprocal of the angular frequency, in this paper, it is taken as the reciprocal of the frequency), Equation ( 7) can be transformed into Equation (12).
By substituting the electric field intensity for the charge density in Equation ( 5) and utilizing Ampere's law, i.e., Equation ( 6), we can solve for the induced magnetic field intensity H   in the free space region.This leads to Equation (13).
By substituting Equation (12) into Equation (13), we obtain: The "error field er E  can be estimated using Equation ( 14), which is obtained by substituting Equation (14) into Equation ( 5) to calculate the magnitude of the "error field": The ratio between the "error field er E  " and the derived value of the quasi-static field E  can be obtained from Equations ( 15) and (12) as: In the equation: er E  represents the electric field intensity generated by the error field.-0 ε represents the vacuum permittivity.-0 μ represents the vacuum permeability.τ represents the characteristic time, which is the reciprocal of time.

SECMP-2023
c represents the speed of light, with a value of this holds, where the ratio / L c represents the time required for electromagnetic waves to propagate through a system of length L , at a speed of c , within a time frame of interest τ , then any quasi-static approximation method is effective.In this article, the maximum typical length L of the system is 0.30 m, and the maximum / L c is 1×10^ (-9) s at a frequency of 100 kHz.Therefore, the characteristic time τ is 1×10^ (-5) s, satisfying the conditions for the quasi-static approximation method, allowing the analysis and calculation of the coaxial treatment chamber using the quasi-static approach.

Numerical analysis of electric field intensity in the coaxial cavity
By examining the design and installation of the coaxial treatment chamber (Figure 3.1a), it can be observed that the chamber effectively forms a cylindrical capacitor.The inner electrode rod and the outer electrode wall are both cylindrical, and there is insulation material and a dielectric medium with a high dielectric constant, such as water, between the two electrodes.Let's assume that the voltage of the excitation source for the coaxial treatment chamber is U .If we consider a Gaussian surface in the form of a coaxial cylinder with a radius of ξ and a length of ' l , the electric flux density D   on this cylindrical surface is uniformly distributed and oriented radially.The charge per unit length on the inner and outer conductors is λ and λ − , respectively.Applying Gauss's law, we have: ( ) ( ) If we set the reference point for the potential on the outer electrode wall of the treatment chamber, i.e., If the excitation voltage is 0 U , then By substituting Equation (11) into Equations ( 7) and ( 8), we obtain the electric field intensity in the insulating layer and the electric field intensity in the water as follows: ( ) ( ) The expression for the equivalent capacitance of a capacitor is given by: The energy stored in a capacitor is given by the expression: According to the principles of electromagnetic field, the electric field energy of a coaxial processing cavity is given by: In the equation: - v represents the volume of water inside the processing cavity.
e W   represents the electric field energy.
-1 ε represents the electric field parameters of the insulating layer.-2 ε represents the electric field parameters of water.eζ  represents the unit vector.

Simulation analysis of electromagnetic field size and distribution in coaxial processing cavity
In this study, the electromagnetic energy in the coaxial processing chamber was analyzed at the time point corresponding to half of a period, specifically at 5 microseconds (5 μs).The analysis was conducted using the COMSOL Multiphysics simulation software with finite element analysis capabilities.1.By observing the electric field mode distribution graphs (a), (b), and (c), the following conclusions can be drawn: The radial distribution of the electric field mode in the coaxial processing chamber is nonuniform, gradually decreasing from the anode rod to the cathode rod.The change in length does not affect the radial electric field mode in the coaxial processing chamber.In this experiment, the maximum electric field mode under different lengths is consistently 2.2×104 V/m.By observing the electric field mode distribution graphs (d), (e), and (f), the following conclusions can be drawn: The axial distribution of the electric field mode in the coaxial processing chamber is uniformly distributed from left to right in the cross-sectional direction.Changing the length of the processing chamber does not affect the magnitude and distribution of the electric field mode.However, as the length increases, the effective volume of the electromagnetic field in the processing chamber also increases.Therefore, under permissible conditions, increasing the length of the processing chamber appropriately will increase the total energy of the electromagnetic field in the chamber, thus achieving better electromagnetic treatment effects.

The wave shape affects the electromagnetic field distribution of the processing cavity
The three commonly used waveforms, namely square wave, sine wave, and triangular wave, were selected for analysis.The amplitude was set to 60 V. Based on Equation (27), the electromagnetic energy values at various time steps, with a time step of T/10, were calculated.The numerical values of the electromagnetic energy at different time steps are shown in Table 1.The numerical calculation results indicate that the electromagnetic energy density in the coaxial processing chamber fluctuates with the waveform of the excitation source.Under the same frequency, when using a square wave as the excitation source, the coaxial processing chamber reaches its peak energy faster, while the time to reach peak electromagnetic density is almost the same for sine waves and triangular waves.The square wave maintains the peak electromagnetic density for a longer duration, approximately 3/4 of a period.Theoretically, the square wave is expected to have better descaling effects compared to sine waves and triangular waves.To further confirm the stronger electric field energy of the square wave, Equation ( 27) is used to integrate the electromagnetic density over the volume of the processing chamber.The resulting energy distribution at different time points is shown in Figure 3.It can be observed that the total electromagnetic energy in the processing chamber is highest for the square wave, followed by the sine wave, and lowest for the triangular wave.Based on these calculation results, it can be inferred that in practical applications, when the excitation source is a square wave, the electromagnetic field inside the processing chamber is stronger, leading to better water treatment effects.

Effect of radius ratio of the cathode to the anode on electromagnetic field distribution in the treatment cavity
The different ratios of the anode-cathode radius in coaxial processing chambers directly affect the distribution of the electromagnetic field and the efficiency of water treatment.To better determine the influence of the anode-cathode radius ratio on the electromagnetic field in the processing chamber, this study selects three ratios, namely 0.6, 0. As shown in Figure 4. From the simulation results, it can be observed that as the anode-cathode radius ratio increases, both the minimum and maximum electric field magnitudes inside the processing cavity increase to some extent.When the anode-cathode radius ratio increases from 0.2 to 0.6, the maximum electric field magnitude in the processing cavity increases from 2.32×10^4 to 3.06×10^4, resulting in a 31% improvement in energy efficiency.According to the simulation results, theoretically, a larger anode-cathode radius ratio leads to a higher rate of improvement in energy efficiency.However, in practical applications, as the anode-cathode radius ratio increases, the cross-sectional area of the flow inside the processing cavity decreases.In extreme cases, when the anode-cathode radius ratio reaches 1, the processing cavity becomes a solid pipe with overlapping anode and cathode, where there is no water flow through the processing cavity.Therefore, it can be concluded that a larger radius ratio is not always better.To determine the optimal radius ratio, further refined experimental analysis is needed.

Optimization design of processing cavity
Through the simulation analysis in Chapter 2, the author determined that changes in the length of the processing chamber do not affect the distribution of the electromagnetic field.The electromagnetic field intensity is maximum when the excitation source is a square wave.However, the optimal range of the radius ratio is still unclear.To determine the optimal range of the radius ratio, the author selected and designed finite element models of processing chambers with radius ratios ranging from 0.125 to 0.875 in this chapter.The magnetized volume of water passing through the processing chamber was calculated and the total electromagnetic energy absorbed by the water was determined through numerical calculations.The results are shown in Table 2.The data shows that the ratio of anode-cathode radius in the coaxial processing chamber has a significant impact on the distribution of the electromagnetic field.When the radius ratio is between 0.375 and 0.5, the energy absorbed by the water is maximum.Therefore, in practical design, the anodecathode radius ratio of the processing chamber should be set within this range.

Conclusion 1)
Through theoretical analysis and simulation of the coaxial processing chamber, it has been determined that the electric field plays a dominant role inside the coaxial processing chamber.A numerical calculation formula for the energy of the coaxial processing chamber has been derived.
2) Simulation analyses were conducted on the length of the coaxial processing chamber, the waveform of the excitation source, and the ratio of the anode-cathode radius.The results showed that the length of the processing chamber had little impact on the distribution of the electromagnetic field, and the electromagnetic field energy was strongest when using a square wave as the excitation source.The ratio of the anode-cathode radius had a significant effect on the electromagnetic field.
3) Further investigation focused on the electromagnetic energy absorbed by water in the processing chamber with different radius ratios through simulations and numerical calculations.It was concluded that the water absorbed the most electromagnetic energy when the radius ratio was between 0.375 and 0.5, indicating theoretically better descaling performance.

1 E
  and water E  represent the electric field of the insulating layer or water in the processing cavity.
(a) 100 mm radial pattern of electric field mode (b) 200 mm radial pattern of electric field mode (c) 300 mm radial pattern of electric field mode (d) 100 mm profile of electric field mold (e) 200 mm profile of electric field mold (f) 300 mm profile of electric field mold Figure 2 Electromagnetic field distribution of different length processing cavity As shown in Figure 2. The author established finite element models with lengths of 100 mm, 200 mm, and 300 mm.The electric field mode distributions under different lengths are shown in Figure

Figure 3
Figure 3 Electromagnetic density distribution with time under different waveforms Note: The Blue Line represents a square wave, the green line represents a sine wave, and the purple line represents a triangular wave

Figure 4
Figure 4 Electromagnetic field distribution with a different radius ratio

Table 1
Different waveform electric field energy

Table 2
Water absorbs energy with different radius