Effects of multi-waveguide configurations in a cylindrical resonant cavity on temperature uniformity during tire microwave vulcanization

Abstract

This study focuses on the non-uniform temperature distribution during tire microwave vulcanization and establishes a multiphysics coupled model of microwave heating in a cylindrical resonant cavity to systematically investigate the effects of waveguide arrangement, rotation angle, waveguide number, and spatial configuration on temperature-field distribution and microwave utilization efficiency. The results indicate that, for the inner-waveguide configuration, the primary heating regions are concentrated in the tread and bead areas, while variations in the number of inner waveguides have little influence on the heating region. The combined inner–outer waveguide arrangement significantly improves the temperature uniformity in the tread region, and changes in the outer-waveguide position do not alter the main heating area of the tire. For the single-waveguide structure, the 45° configuration exhibits the highest microwave utilization efficiency, with the minimum S11 value reaching −20.327 dB and the fastest heating rate. In contrast, the 90° configuration achieves the most uniform temperature distribution and the optimal heating uniformity. Compared with dual- and four-waveguide configurations, the combined three-waveguide side–upper arrangement demonstrates superior electric-field uniformity and overall heating performance, in which the rotation angle of the upper waveguide significantly affects the tire temperature distribution, and the optimal temperature uniformity is obtained at 90°. The present study systematically reveals the coupling relationship between multi-waveguide configuration parameters and tire temperature-field evolution, and provides theoretical guidance for the structural optimization and uniform microwave heating design of large and complex rubber products.

Keywords

tyre microwave vulcanisation cylindrical resonant cavity temperature distribution

Highlights

1. Clarifies the optimization effect of metal interference (copper waveguide structures) on tire microwave vulcanization. The triple-waveguide combination (upper waveguide rotated by 90°) significantly enhances tread and adjacent sidewall temperature uniformity, achieving the optimal vulcanization effect.

2. Reveals the influence law of metal interference parameters: single waveguide at 45° optimizes microwave absorption; dual waveguides at 180° uniformize tread-sidewall temperature; inner waveguide number does not change heating areas, while inner-outer combination improves tread uniformity.

Introduction

Vulcanisation is an important step in the rubber industry and is usually heated by an external heat source such as superheated water, superheated steam, nitrogen, etc. by means of heat conduction,¹The low thermal conductivity of rubber material leads to the problems of long heating time, uneven heating and high power consumption. Compared with traditional heating, microwave heating does not require a heat transfer medium, with penetration, rapid heating, but also has a selective heating, controllability, heating uniformity and energy saving and high efficiency and so on,^2–4widely used in food, chemical and material processing industries.^5–9 The use of microwave high-frequency characteristics, to achieve synchronous heating inside and outside, to ensure that all parts of the tyre heated uniformly, but also significantly shorten the curing time, improve production efficiency, enhance the strength of the tyre, abrasion resistance and resistance to aging, reduce the rate of defective.

The vulcanization effect of rubber is closely related to the crosslinking network structure. Researchers worldwide have conducted extensive studies on the crosslinking modification and vulcanization processes of rubber materials. Ismayilova et al.¹⁰ investigated the effect of polymeric plasticizers on the crosslinking behavior and properties of nitrile butadiene rubber (NBR, SKN-40) under thermal vulcanization, radiation vulcanization, and thermoradiation vulcanization, and constructed hybrid crosslink networks of NBR via polymeric plasticizer-assisted thermoradiation vulcanization. Khankishiyeva et al.¹¹ modified hydrogenated nitrile butadiene rubber (HNBR) with a novel organosilicon crosslinking agent ethylphenylsilylurea (EPSU), and the results showed that the incorporation of EPSU into the HNBR system could significantly improve the vulcanization kinetic properties of rubber. Mammadov et al.¹² studied the effects of chlorine-containing compounds (HCPX and ABSTS) and epoxy resin on the crosslinking behavior and key properties of HNBR elastomers under thermal and thermo-radiation vulcanization.

Although microwave heating technology is widely used in many fields and has achieved certain development, but for a series of problems arising from the microwave heating process, such as hot spots, heating uniformity and other issues to be further resolved.^13–15 For the microwave heating process heated objects appear hot spot phenomenon which leads to the problem of heating uniformity, the existing research mainly from two aspects, on the one hand, by changing the position of the waveguide, increase the number of waveguide, change the frequency of the waveguide, set up different shapes of mode stirrers in the multimode resonant cavity and other aspects of the research to improve the uniformity of microwave heating^16–18;On the other hand, the temperature uniformity of the object to be heated is improved by changing the position, shape, size, electromagnetic characteristics of the object to be heated or by changing the internal structure of the resonance cavity, for example, by installing a rotating tray inside the resonance cavity to change the position of the object, by mixing wave-absorbing materials with the object to be heated, or by installing differently shaped protruding structures in the inner wall of the resonance cavity.^19,20 Jiang et al. proposed a new method of microwave heating by combining rotation and boundary motion, which can effectively improve the heating efficiency and heating uniformity of the object.²¹ Zhu et al. proposed a new microwave heating method with a rotating radiation structure, proving that the rotating radiation structure has obvious superiority.²² Keangin studied the effect of waveguide position on electric field and temperature distribution in natural rubber gloves.²³ Ye et al. proposed a heating method with slow rotation of heated samples in multiple mould cavities and fast rotation of the mould stirrer to improve heating uniformity and heating efficiency.²⁴ Fu et al. change the input power from 500 W to 900 W, the temperature range of HDPE increases from 161.5°C to 220.2°C, and the average temperature rise rate increases from 2.38°C/min to 3.36°C/min.²⁵ Ke et al. varying the input power from 300 W to 500 W increases the average temperature rise rate of SiC from 0.89°C/s to 7.08°C/s. 2²⁶。Yang et al. proposed a novel microwave continuous-flow heating system in which a 45°inclined waveguide can significantly reduce reflection losses in both input directions by enhancing the microwave resonance effect, thus further improving energy efficiency.²⁷ Cui et al. based on a basic model of a loaded electromagnetic lossy dielectric resonant cavity, analysed the influence of the thickness and type of lossy dielectric on its electromagnetic properties by calculating the first resonant mode, TM01 mode.²⁸ Liao et al. proposed a novel microwave cavity structure based on phase-shift heating. By establishing an electronic mathematical model to predict the temperature distribution and adopting a new calculation method to solve the moving boundary problem in the model simulation, they aim to improve the temperature uniformity of microwave-treated materials.²⁹ Li et al. proposed a method to introduce periodically arranged metallic resonant structures and dielectric shims on the surface of a material to change the microwave absorption capacity of the material and thus improve the heating uniformity.³⁰ Betime et al. studied the influence of absorbing materials on the uniformity of microwave heating of composite materials and found that applying absorbing materials to composite materials with non-uniform geometry shapes can change the temperature distribution, and it is expected to achieve a uniform temperature distribution by better adjusting the absorption effect.³¹ Wang et al. proposed placing an array of eccentrically rotatable copper columns around a cylindrical microwave reactor cavity to form a controllable time-varying electromagnetic boundary to improve microwave heating performance, achieving different heating performance through different boundary adjustment strategies, and improving heating uniformity through a temperature coefficient of variation optimisation algorithm.³²

Most of the existing research focuses on regular heated body, the research on large size and complex shape tyres is still shallow, and the research on the uniformity of tyre curing is even less. Therefore, it is of great research value and practical significance to explore the optimisation of process parameters, improve the uniformity of microwave field, enhance the degree of curing uniformity and shorten the curing time. The main innovation of this work is that a waveguide layout with a 45° rotation angle on the side of the cylindrical resonant cavity is employed for tire microwave vulcanization. On this basis, the influences of waveguide quantity, arrangement position, and internal–external combination mode on the uniformity of the temperature field are systematically investigated, and the optimal layout for highly uniform microwave vulcanization of tires is finally obtained.

Simulation conditions and size calculation

Calculation of resonant cavity size

In the microwave heating system, the resonant cavity as one of the core components, its size design directly affects the distribution of microwave field and heating effect. The resonant frequency of the cylindrical resonant cavity under TE modes is given by:

f_{r} = C \sqrt{{(\frac{μ_{m n}}{2 π R})}^{2} + {(\frac{p}{2 l})}^{2}}

(1)

When the power satisfies Formula (2), the electromagnetic field distribution mode of the cylindrical resonant cavity can be generated within the cavity.

f_{0} - Δ f \leq C \sqrt{{(\frac{μ_{m n}}{2 π R})}^{2} + {(\frac{P}{2 l})}^{2}} \leq f_{0} + Δ f

(2)

Among them:f_r is the resonant frequency of the resonant cavity,f₀ is the operating frequency, $Δ f$ is the frequency width, R is the radius of the cylindrical resonant cavity, l is the height of the cylindrical resonant cavity, m, n, and p are the mode indices, where m ≥ 0, n ≥ 1, and p ≥ 0. Each combination constitutes a working mode, and the number of combinations is the number of modes. $μ_{m n}$ denotes the nth root of the derivative of the m-order Bessel function in TE modes. All the above parameters are in international units.

The tire model selected in this paper is the 7.50R20 inner support outer tire. The radius of the cylindrical resonant cavity is set at 652 mm and the height is set at 470 mm. It is calculated through Formula (2) that the number of resonant modes in the resonant cavity is 9, the degenerality ratio is 0, the electromagnetic field distribution in the resonant cavity is relatively uniform and the loss is low.

Simulation conditions and model simplification

The resonant cavity environment is set to the air domain, and both the resonant cavity and the waveguide are copper media. The heating time is set to 1800 s, the feed frequency is 915 MHz, the mode is TE10, and the microwave input power is 10 kW, which is evenly distributed to each waveguide. The initial temperature is set to 20°C. The tyre rotates at a constant speed of 0.1r/s.This rotation rate is low enough to avoid centrifugal deformation and to keep the Doppler frequency shift far below the 915 MHz operating frequency, while still improving temperature uniformity by redistributing the electromagnetic field.

Considering the errors between the numerical simulation process and the actual experiment, and in order to simplify the microwave vulcanization model of tires and ensure the convergence of the calculation results, reasonable assumptions need to be made for the model:

The outer walls of the resonant cavity and the waveguide are both made of metal materials, and the thickness of the metal walls is much smaller than that of the cavity. Due to the skin effect, the electromagnetic field hardly penetrates the metal walls. Therefore, the walls of the resonant cavity and the waveguide can be set as impedance boundary conditions.

The rubber material of the tire is uniform and isotropic. The initial temperatures of the tires and the air are uniform. The mass transfer process is negligible; Ignore the convective heat transfer between the air and the tires; Ignore the dielectric loss of air; Ignore the heat conduction in the air. Although the aforementioned assumptions help to improve computational efficiency and ensure convergence, they may still have certain effects on the simulation results. Neglecting convective heat transfer between the air and the tire may lead to reduced heat dissipation on the tire surface, resulting in local temperatures slightly higher than those in actual conditions. Moreover, during actual microwave heating, the dielectric properties of rubber materials vary with temperature, whereas constant dielectric properties were adopted in this study, which may affect the accuracy of the electromagnetic field distribution and power deposition. However, the present study primarily focuses on the relative variations in tire temperature distribution and heating uniformity under different waveguide configurations. Since identical material parameters and boundary conditions were applied to all cases, the above simplifications do not compromise the validity of the comparative analysis between different configurations.

Microwave heating control equation

The propagation of electromagnetic waves in tires is calculated using Maxwell’s equations. The general form of Maxwell’s equation is presented by simplification in the electric field through which the tire penetrates. The simplified control equation is as follows:

\nabla \times μ_{r}^{- 1} (\nabla \times E) - k_{0}^{2} (ε_{r} - \frac{j σ}{ω ε_{0}}) E = 0

(3)

Among them: $E$ is the electric field intensity, $ω$ is the angular frequency; $ε_{0}$ is the vacuum dielectric constant (8.85 × 10⁻¹² F/m); $μ_{r}$ is the relative permeability; $ε_{r}$ is the relative dielectric constant; $k_{0}$ is the wave number in free space; $σ$ is electrical conductivity; $k_{0}$ is given by the following formula:

k_{0} = ω \sqrt{ε_{0} μ_{0}} = \frac{ω}{c_{0}}

(4)

Among them: $c_{0}$ is the speed of light in a vacuum.

The complex dielectric constant is a function of the dielectric constant and the dielectric loss factor, which is given by the following expression:

ε^{*} = ε^{″} - j ε^{'} = ε_{0} ({ε_{r}}^{'} - j {ε_{r}}^{″})

(5)

Among them: $ε^{*}$ is the complex dielectric constant (F/m); $ε^{'}$ is the dielectric constant; $ε^{″}$ is the dielectric loss factor; ${ε_{r}}^{'}$ is the relative dielectric constant; ${ε_{r}}^{″}$ is the relative loss factor.

The heat transfer equation couples the microwave field with a Fourier energy balance equation, given by the following expression:

ρ C_{P} \frac{\partial T}{\partial t} + ρ C_{p} u \cdot \nabla T = \nabla \cdot (k \nabla T) + Q

(6)

Q = 2 π f \cdot ε_{0} ε_{r} \cdot \tan δ {| E |}^{2}

(7)

Among them: $ρ$ is the density (kg/m³); $C_{p}$ is the heat capacity at normal pressure (J ⋅kg ⁻¹ ⋅K⁻¹);T is temperature(K);k is the thermal conductivity coefficient (W⋅m⁻¹⋅K⁻¹);E is the electric field intensity vector (V/m); $ε_{r}$ is the dielectric constant;f is the frequency; $δ$ is the dielectric loss Angle;Q is a volumetric heat source.

The assessment of the microwave heating uniformity of tyres in this paper relies on the temperature field profile of the tyre cut at an average tyre body temperature of 120°C and the coefficient of variation (COV) of the temperature as the main reference. The temperature coefficient of variation COV is given by the following equation (8)^33,34:

C O V = \frac{\sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(T_{i} - \bar{T})}^{2}}}{\bar{T}}

(8)

Among them: $T_{i}$ is the point temperature of the temperature field of the material dividing the regional grid, $\bar{T}$ indicating the average temperature of the grid points, and n is the total number of grid points. The smaller the value of COV, the better the heating uniformity of the heating system.

Physical model and mesh-independent validation

The schematic diagram of the microwave curing model of tyre is shown in Figure 1, the waveguide is placed in the centre of the side of the cylindrical resonance cavity, the diameter of the cylindrical resonance cavity is 1304 mm, and the height of the resonance cavity is 470 mm. The outer tyre of 7.50R20 support is simplified into 5 layers, which are the tread, the shoulder, the sidewalls, the airtightness, and the belt bundle, and the tyre is placed in the centre of the cylindrical resonance cavity. The length of the waveguide is set to 100 mm, and according to the operating frequency of 915 MHz, the waveguide model BJ8 is selected. The frequency range of the BJ8 rectangular waveguide model is 0.64-0.98 GHz, and the width is 292.1 mm, and the height is 146.05 mm.³⁵

Figure 1.

Schematic diagram of tire microwave vulcanization model.

The mesh quality has a strong influence on the computational accuracy of the finite element model. A free tetrahedral mesh with physical field control is used to verify the mesh independence by monitoring the average tyre temperature. As shown in Figure 2, the average tyre temperature gradually stabilises as the mesh is refined and remains almost constant after the mesh density level G4. This convergence behaviour proves the grid independence beyond this density and confirms the adequacy of the G4 configuration for achieving reliable simulation accuracy. Information on the number of model grids constructed is given by Table 1.

Figure 2.

Grid independence analysis.

Table 1.

Grid number division.

	G1	G2	G3	G4	G5
Number of grids	31,351	58,171	142,901	378,692	979,115

Results and discussion

Effect of single waveguide lateral arrangement on microwave vulcanisation of tyres

Changes in the angle of rotation of the waveguide can affect the various parameters of the tyre microwave heating process, the waveguide rotation for the waveguide centre as the base point, with 0° as the basis for rotation counterclockwise rotation. Cylindrical resonant cavity side of a single waveguide placed at different angles as shown in Figure 3.

Figure 3.

Schematic model of different rotation angles of the waveguide on a cylindrical resonant cavity.

The return loss parameter S11 reflects the efficiency of the port energy input and can be calculated from equation (9).³⁶

S_{11} = 10 \times \lg \frac{P_{out}}{P_{in}}

(9)

Among them: $P_{in}$ is the port input power; $P_{out}$ is the port reflected power.

S₁₁ reflects the proportional relationship between the power reflected back from the port and the incident power of the microwave signal during transmission. The smaller the value of S₁₁, the smaller the power reflected back, which means the higher transmission efficiency of the microwave system.

The effect of different rotation angles of the single waveguide on the microwave curing time of a tyre is shown in Figure 4. Taking the body-averaged temperature of microwave cured tyres as a criterion, when the body-averaged temperature of 140°C is reached, the microwave curing time of the tyre is the shortest when the waveguide is rotated by 45°, and the rate of temperature rise is larger.

Figure 4.

Heating time for the average temperature of the tire body to reach 140°C under different rotation angles of waveguide.

From Table 2 it can be seen that the waveguide rotation angle of 0°–45° changes in the volume average temperature of the tyre increases, the return loss parameter S₁₁ decreases, the electromagnetic loss density increases, 45°–90°of the changes in the parameters with the increase in the rotation angle and decrease, the waveguide angle of 45°when S₁₁ is the smallest. It can be seen that when the waveguide rotation angle of 45°when the cylindrical resonant cavity in the tyre microwave curing process absorbed the maximum microwave energy, the highest microwave utilization.

Table 2.

Relationship between waveguide rotation angle and body average temperature, S₁₁, and microwave utilization during heating for 1200 s.

Waveguide rotation angle	1200 s volume average temperature/°C	S₁₁/dB	Electromagnetic loss density/W
0°	122.92	−6.7777	7890.4
15°	128.67	−7.9338	8380.7
30°	142.24	−14.44	9627.2
45°	143.41	−20.327	9898.2
60°	132.4	−10.193	9036.8
75°	120.43	−6.8956	7951.4
90°	113.2	−5.8319	7385.2

Effect of dual waveguide lateral arrangement on microwave vulcanisation of tyres

Based on the good microwave utilisation when the waveguide rotation angle of 45°is arranged in the cylindrical resonant cavity, the dual-waveguide microwave vulcanisation model with different rotation angles is constructed with a rotation angle interval of 30°and a rotation interval of 30°–180°as shown in Figure 5.

Figure 5.

Model of dual waveguide resonant cavity side arrangement with different rotation angles.

Numerical simulation of the dual waveguide resonant cavity model with six different rotation angles shown in Figure 5 yields a tyre temperature distribution result with an average tyre body temperature of 120°C as shown in Figure 6.

Figure 6.

Tire temperature distribution at a mean body temperature of 120°C.

Figure 6 shows that the heating area of the tyre changes significantly with the rotation angle of the waveguide. When rotated at 30°, the heating is concentrated in the lower sidewall; at 60°, it extends to the tread and lower sidewall but remains unevenly distributed; at 90°, the tread is mainly heated, showing a significant temperature difference between the upper and lower regions; at 120°, the upper tread exhibits a higher temperature; and at 150°, the tread heating is still non-uniform. At a rotation angle of 180°, the temperatures in the tread and adjacent sidewall areas all exceed 120°C, and the heated area is the largest. Although the COV value at 180° (0.4674) in Table 3 is the highest, indicating poor overall temperature uniformity, this is mainly attributed to temperature differences in non-critical regions such as the bead. In the key regions for tire vulcanization (the tread and sidewalls), the temperature distribution is relatively uniform. Therefore, considering both the heated area and the temperature distribution in the critical regions, the 180° configuration remains the optimal choice. Table 3 also shows that the heating time required for the average tire body temperature to reach 120°C gradually decreases from 30° to 90°, and then increases from 90° to 150° (with the longest time at 150°).

Table 3.

Parameters of microwave curing results for lateral dual waveguide tires.

Waveguide rotation angle	30°	60°	90°	120°	150°	180°
COV	0.3347	0.2841	0.2884	0.3469	0.3269	0.4674
Heating time/s	1020	990	920	970	1560	1020

As shown in Figure 7, the temperature difference on the tyre surface shows a trend of decreasing and then increasing. As the angle increases, the maximum temperature of the tyre surface decreases and then increases, while the minimum temperature continues to decrease, indicating that the larger the angle of the dual waveguide, the lower the minimum surface temperature.

Figure 7.

Maximum and minimum tire surface temperatures at different rotation angles.

Effect of three-waveguide lateral arrangement on microwave vulcanisation of tyres

Based on the dual-waveguide 180° sidelobe structure (optimal), this study further constructs a triple-waveguide sidelobe model (Figure 8). On the basis of maintaining the symmetrical arrangement of the dual waveguide, the third waveguide is added by rotating in the range of 0°–150° in 30° increments, and its effect on the microwave vulcanisation process of tyres is systematically investigated.

Figure 8.

Three-waveguide side arrangement model.

By numerical simulation of five sets of lateral three-waveguide models (Figure 8) with different rotation angles, the results of the temperature distribution when the average temperature of the tyre body reaches 120°C were obtained (Figure 9).

Figure 9.

Temperature distribution of tires arranged on the side of three waveguides.

From Figure 9, it can be seen that when the waveguide rotation angle from 30° to 150°, due to the high temperature area heating area is small and the overall distribution is poor, so the overall heating effect is poor.

From Figure 10,it can be seen that with the increase of the side waveguide rotation angle of the tyre microwave heating time first decreases and then increases, the waveguide rotation angle of 90° when the microwave heating time is the shortest. It can be seen that the maximum and minimum temperatures of the tyre surface show opposite trends.

Figure 10.

Relation between the rotation angle of the waveguide and the maximum and minimum temperature of the tire.

From Table 4, it can be seen that the waveguide rotation angle of 60° and 120° of the temperature coefficient of variation is the smallest, but the waveguide rotation of 60° and 120° temperature distribution is not uniform, the rotation of the angle of the COV is small due to the small temperature difference between the surface of the tyre, but due to the heating area of the tyre is less but not conducive to the microwave vulcanisation of the tyre.

Table 4.

Microwave vulcanization result parameters of three-waveguide lateral arrangement tire.

Waveguide rotation angle	30°	60°	90°	120°	150°
COV	0.3660	0.3196	0.3566	0.3177	0.3838
Heating time/s	1100	1050	990	1050	1100

The effect of a combined triple waveguide side and upper side arrangement on the microwave vulcanisation of tyres

Adding a waveguide in the centre part of the upper side of the cylindrical resonant cavity, considering the effect of the rotation angle of the upper side waveguide, the upper side waveguide is rotated counterclockwise around the centre point on a 0° rotation basis, as shown in Figure 11.

Figure 11.

Schematic diagram of the three-waveguide side and upper side arrangement.

The upper lateral triple waveguide model simulations for different rotation angles are shown in Figure 12.

Figure 12.

Tire temperature distribution combined with the side and upper side of the three-waveguide.

The high-temperature zone is concentrated in the tread under all working conditions, and the temperature of the bead is always low. 0°, the heating zone is located in the tread and the lower sidewall, and the high-temperature zone is formed from the upper shoulder to the middle of the sidewall, but the uniformity is not good; the temperature distribution is similar between 15° and 45°, and the high-temperature zone expands to the tread and the middle of the sidewall, and the uniformity of the tread is the best at 30°; the high-temperature zone is contracted to the upper half of the tread from 60° to 75°; the heating zone covers the whole tread and extends to the upper sidewall, and the temperature uniformity is the best at 90°. At 90°, the heating zone covers the whole tread and extends to the upper sidewall, with the best temperature uniformity.

The introduction of the upper side waveguide significantly improves the uniformity of tire heating. Among them, the 30° and 90° configurations perform exceptionally well, not only effectively expanding the heating area to the tread and adjacent sidewalls, but also greatly enhancing the temperature uniformity. Compared with the traditional double-waveguide side-placed structure, this three-waveguide arrangement scheme significantly expands the effective heating area while maintaining the vulcanization temperature, providing a better solution for the microwave vulcanization process of tires.

As shown in Table 5 and Figure 13, when a single waveguide is arranged in the centre of the upper side of the cylindrical resonance cavity, the heating time of the tyre rises sharply with the change of the waveguide rotation angle and then decreases gently: the heating time increases sharply from 0° to 15°, and the influence of the rotation angle of the waveguide on the upper side on the heating time decreases after exceeding 15°.

Table 5.

Tire vulcanization result parameters under different rotation angles on the upper side waveguide of the three-waveguide resonator.

Waveguide rotation angle	0°	15°	30°	45°	60°	75°	90°
COV	0.3382	0.4259	0.4219	0.4222	0.4319	0.4329	0.3608
Heating time/s	710	1330	1320	1310	1300	1300	1290

Figure 13.

Relationship between the upper side waveguide rotation angle and the maximum and minimum temperatures.

Tyre surface temperature extremes show regular changes: the highest temperature reaches a maximum at 15°, the minimum at 90°; the lowest temperature at 75°maximum, 90°down to the minimum. After the rotation angle exceeds 30°, the surface temperature increases and then decreases, which reveals the consistency of the influence of waveguide angle on the temperature distribution mechanism in this interval.

Temperature uniformity analysis shows that the COV value is the smallest at 0°, reaching 0.3382. At 90°, the COV value is 0.3608. Although the COV value is higher than that at 0°, it can be seen from Figure 12 that the uniformity of temperature distribution of the tyre at 90° of the rotation angle of the upper flanks is much better than that at 0°. The overall heating performance is optimal.

Three-waveguide microwave curing model study found that the pure side arrangement of the three-waveguide model for microwave heating of tyres, although in the side waveguide rotation of 30° and 150° when the heating area of the tyre increased, but the high temperature region is concentrated in the sidewall, tread temperature is lower, the overall heating effect is poor. In contrast, the three-waveguide using the upper side of the centre and the side of the dual-waveguide combination of the arrangement, significantly expanding the high-temperature area, and the temperature distribution is more uniform, basically to meet the requirements of the tyre curing temperature. In particular, when the upper side waveguide is rotated at an angle of 90°, the tyre temperature distribution is optimal and the overall temperature difference is minimal, which is more conducive to the implementation of the curing process. Although the heating time to reach an average temperature of 120°C in the tyre body under these conditions is 1290 s, an increase of 190 s compared to the purely lateral arrangement, the advantage of its excellent temperature uniformity is outstanding.

Effect of combined four-waveguide upper and lower lateral and lateral arrangement on microwave vulcanisation of tyres

Based on the three-waveguide model with the best microwave vulcanisation effect, a waveguide is added to the lower side of the cylindrical resonance cavity The lower side waveguide has the same angle as the upper side waveguide at the initial time, and the rotation angle is based on 0°, as shown in Figure 14.

Figure 14.

Diagram of the four-waveguide model with different rotation angles of the lower side waveguide.

By numerically simulating seven sets of four-waveguide models (Figure 14) with upper and lower sides arranged in combination with the sides at different rotation angles, the simulation results for an average tyre body temperature of 120°C are shown in Fig.

As can be seen from Figure 15, the temperature distribution of the tyre changes significantly as the rotation angle of the waveguide increases. When the lower side waveguide is rotated from 0° to 30°, the temperature distribution of the tread becomes more and more uniform, but the low-temperature area of the upper and lower sidewalls increases significantly, and the heating area decreases gradually. When the lower side waveguide changes from 45° to 90°, the temperature distribution of the tyre is roughly similar, and the heating area is concentrated in the middle part of the tread and lower sidewall, the temperature of the upper part of the tread rises, the colour deepens, and the uniformity of the temperature distribution deteriorates,and the location of the low-temperature area in the bead and nearby sidewalls of the tyre remains essentially unchanged (Table 6).

Figure 15.

Tire temperature distribution under different rotation angles of the lower side waveguide.

Table 6.

Results of microwave vulcanization of four waveguides under different rotation angles of lower side waveguides.

Waveguide rotation angle	0°	15°	30°	45°	60°	75°	90°
COV	0.2387	0.3229	0.3753	0.3749	0.3784	0.3459	0.3243
Heating time/s	1370	1370	1380	1400	1430	1450	1530

Figure 16 shows that the maximum temperature of the tyre surface increases with the rotation angle of the waveguide and then decreases, reaching a peak of 201°C at 60°; the minimum temperature decreases and then increases. The heating time increases monotonically with the angle, and the increase is intensified after 30°; the COV value increases first and then decreases, 0°–30° rises significantly, and 30°–60° tends to be stable. 30° is the key turning point, the extreme value of the temperature and the heating time are lower when the angle is smaller than this angle, and there are significant changes in the parameters after exceeding this angle. 0°configuration shows the best comprehensive performance, which has the largest heating area, the shortest time, the smallest COV value and the best temperature uniformity, providing a basis for the optimization of the four-waveguide model. The 0° configuration shows the best overall performance with the largest heating area, shortest time, smallest COV and best temperature uniformity, which provides a basis for the optimisation of the four waveguide model.

Figure 16.

Relationship between the rotation angle of the waveguide and the maximum and minimum tire temperature.

In the four-waveguide model, the arrangement of waveguides in the centre of the upper and lower sides of the cylindrical resonance chamber can increase the minimum temperature of the tyre surface, but compared with the optimal three-waveguide model, this arrangement reduces the heating area, decreases the temperature uniformity of the tread, and prolongs the microwave heating time.

The effect of a combined four-waveguide side and upper side arrangement on the microwave vulcanisation of tyres

A waveguide is added to the side of the cylindrical resonant cavity to construct a four-waveguide resonant cavity model, which is rotated clockwise with a 15°rotation angle based on the reference axis at the centre of the resonant cavity. Considering the placement angle of the waveguide on the side of the resonant cavity, the new waveguide is added at a base rotation angle of 30°, and the rotation angle is incremented by 15° until it reaches 150°, and the model schematic is shown in Figure 17.

Figure 17.

Schematic diagram of a four-waveguide model in which the side waveguide rotates around the central axis of the model.

By numerically simulating the four-waveguide model (Figure 17) with nine sets of different rotational angles of the lateral and upper lateral combined arrangement, the simulation results are shown in Figure 18.

Figure 18.

Tire temperature distribution at different angles when the side waveguide rotates around the central axis of the model.

Figure 18 shows that the change of the waveguide rotation angle makes the tyre surface temperature change greatly, from the simulation results, the waveguide rotation angle is less than 60°and more than 120° when the heating area of the tyre is larger, in which the side waveguide rotation of 135° tyre is not only a large area of heating but also a more uniform temperature distribution. It can be seen that the waveguide rotation angle can be changed by changing the electromagnetic field distribution target heating specific areas of the tyre, 90° in the heating uniformity and coverage of the best performance, for the optimization of the microwave curing process to provide an important basis (Figure 19).

Figure 19.

Relationship between side waveguide rotation angle and maximum and minimum tire temperature.

The data in Table 7 show that, in the four-waveguide (side three-waveguide and upper side) combined arrangement, the change of the rotation angle of the side waveguide does not have a clear trend of influencing the maximum and minimum temperatures of the tyre surface as well as the heating time, but the microwave heating time and the temperature difference of the tyre are at a lower level at some specific rotation angles such as 45°and 135°.

Table 7.

Results of microwave vulcanization of four waveguide tires under different rotation angles of side waveguide.

Waveguide rotation angle	30°	45°	60°	75°	90°	105°	120°	135°	150°
COV	0.1727	0.1975	0.2103	0.3663	0.3235	0.3732	0.3357	0.2107	0.4559
Heating time/s	1270	1150	1230	1310	1160	1360	1260	1250	1320

In terms of temperature uniformity, the COV was smaller at 30°, 45°, 60° and 135°; at 30°, the heating area was the largest, the low temperature area was the smallest and the COV value was the smallest, but the temperature distribution uniformity was not good; at 135°, the temperature distribution of the tyre tread and the upper sidewall was more uniform.

Analysis of results on the number of optimal waveguides

From Figure 20, it can be seen that the heating area of the optimal three-waveguide is large, the temperature distribution effect is the best, and the temperature distribution is uniform at the tread and in the area of the upper and lower sidewalls near the tread.

Figure 20.

Optimal temperature distribution of different number of waveguides.

As can be seen in Figure 21, when the double waveguide is arranged symmetrically on the side, the resonance cavity has a large area of obvious high electric field area on the upper and lower and left and right sides, and there is a uniform long low electric field area in the centre; in the optimal three waveguide electric field distribution, the electric field intensity increases in the centre, and there is no obvious high electric field area on the right and left sides, and the colour is uniformly distributed; in the optimal four-waveguide electric field distribution, the upper and lower sides have an obvious irregular high electric field area in the centre of the left side, and the right side has a poor electric field uniformity. The uniformity of the electric field on the right side is poor. In summary, the uniformity of the electric field distribution of the three-waveguide is better than that of the two-waveguide and four-waveguide.

Figure 21.

Optimal electric field distribution of different number of waveguides.

It can be known from Figure 22 that the increase in the number of waveguides causes the maximum temperature on the tire surface to decrease and the minimum temperature to increase. The maximum temperature of the three-waveguide model is moderate, neither too high to cause overheating problems nor too high to ensure sufficient vulcanization effect. As can be seen from Figure 23, although the heating time of the three-waveguide model is slightly longer, it performs better in terms of temperature distribution and the maximum temperature of the tire.

Figure 22.

Maximum and minimum tire temperatures for different number of waveguides.

Figure 23.

Microwave heating time of different number of waveguides.

Effect of medial waveguide on microwave vulcanisation of tyres

The microwave vulcanization model of the inner waveguide is shown in Figure 24.

Figure 24.

Schematic diagram of microwave vulcanization model of inner waveguide.

Thanks to the special structure of the tire, this section proposes a form of arranging the waveguide on the inner side of the resonant cavity (as shown in Figure 24) to discuss the influence of the waveguide located on the inner side of the resonant cavity on the microwave vulcanization of the tire.

The temperature distribution of the tyre in Figure 25 shows that when the waveguide resides on the inner side, the temperature distribution of the tyre is similar, and the heating area is mainly concentrated in the tread and bead portion, and there are obvious low-temperature areas on the upper and lower sidewalls.

Figure 25.

Temperature distribution of microwave vulcanization of inner waveguide tire.

In this paper, the study focuses on the electric field distribution in the resonant cavity, excluding the electric field interference in the rectangular waveguide. Figure 25 shows that the main heating area of the tyre is concentrated at the tread and bead when the medial waveguide is arranged, and the number of medial waveguides does not affect the temperature distribution of the tyre. From Figure 26, it can be seen that the distribution of the electric field inside the resonant cavity is similar for different numbers of medial waveguides, and the change of the number of medial waveguides hardly affects the distribution of the electric field.

Figure 26.

Electric field distribution of the inner waveguide.

Effect of combined inner and outer waveguide arrangement on microwave vulcanisation of tyres

The microwave vulcanisation model of a tyre with a combined inner and outer waveguide arrangement is shown in Figure 27.

Figure 27.

Microwave vulcanization model of inner and outer waveguides.

In order to discuss the possible influence of the outer waveguide on the inner waveguide tyre microwave vulcanisation process, the electric field distribution in the resonant cavity and the temperature distribution of the tyre are explored by the arrangement of the inner waveguide in combination with the outer waveguide (as shown in Figure 27) when the waveguide is arranged simultaneously in the inner and outer parts of the resonant cavity.

Figure 28 shows that the temperature distribution of the tyre is similar when the waveguide is combined on the inner and outer sides and when the waveguide is arranged only on the inner side, but the temperature distribution of the tread is more uniform. Moreover, the change of the angle between the inner and outer waveguides does not affect the heating area of the tyre, but the heating effect is best when the angle between the inner and outer waveguides is 180°(i.e., the microwave input direction is the same) in terms of the uniformity of the temperature distribution in the heating area of the tyre.

Figure 28.

Temperature distribution of the inner and outer waveguides combined with tires.

As shown in Figure 29, the electric field distribution characteristics reveal that the 0° and 180° angles and the 45° and 135° angles show similar electric field patterns. The 45° and 135° configurations have a more uniform distribution of the high electric field region (maximum field strengths of 18,000 V/m and 14,100 V/m, respectively), while the 0° and 180° configurations (maximum field strengths of 9660 V/m and 7750 V/m) show an obvious field strength attenuation effect, especially at 180°, which is significantly smaller than that of the high electric field region at 0°.

Figure 29.

Screenshot of electric field distribution in the inner and outer waveguide bonded resonators.

Comparative analysis shows that although the combined inner and outer side arrangement can improve the tread heating uniformity, there is still the problem of low-temperature zone on the sidewall. Compared with the optimal three-waveguide model, this configuration has obvious defects: (1) Uneven electric field distribution (high field strength areas concentrated on both sides of the cavity); (2) The sidewalls of the tyres are poorly heated; (3) Limited space for inner waveguide adjustment due to cavity dimensions.

Conclusion

In this study, single, dual, triple and quadruple waveguide cylindrical resonant cavity models are constructed to investigate the effects of waveguide placement angle, number and arrangement position on the microwave vulcanisation of tyres and to analyse the vulcanisation characteristics when the waveguide resides on the inner side. The numerical simulation results show that:

(1) In the single waveguide model, the average temperature of the tire body and the microwave utilization rate show a trend of first increasing and then decreasing with the increase of the waveguide placement Angle, reaching the peak at 45°, indicating that the tire has the optimal absorption of microwave energy at this Angle.

(2) For the dual-waveguide side arrangement, the rotation angle of the waveguides significantly influences the tire heating region. The 180° configuration provides the largest heated area in the tread and adjacent sidewall, with critical-region temperatures all exceeding 120°C. Despite a higher overall COV (mostly caused by non-critical areas like the bead), the 180° configuration is the best when both the heated area and critical-region temperature distribution are considered.

(3) When the three waveguides are arranged on the side, different rotation angles correspond to different heating areas. At 30°, 90°, and 150°, the lower die sidewall, tread, and upper die sidewall are heated respectively. Although the tread is heated intensively at 60°and 120°, the temperature distribution is uneven. When the side and upper side are arranged in combination, the rotation Angle of the waveguide on the upper side significantly affects the temperature distribution. At 90°, the temperature distribution in the heating area is the most uniform and the vulcanization effect is the best.

(4) When the upper and lower sides of the four waveguides are arranged in combination with the sides, the rotation Angle of the lower side waveguide changes the temperature distribution. Between 0° and 30°, the tread temperature tends to be uniform but the low-temperature area on the sidewall increases. Between 45° and 90°, the temperature distribution is similar but the uniformity is poor. When the side and top sides are combined and arranged, the temperature difference of the tires is the smallest at 135°, the heating area is large and evenly distributed.

(5) When the waveguide is located on the inner side of the resonant cavity, the heating area of the tire is fixed on the tread and the bead, and does not change with the number of waveguides, but the uniformity is poor. When the inner and outer waveguides are arranged in combination, the heating area is the tread and part of the sidewall. The temperature uniformity is significantly improved, and the position of the outer waveguide does not affect the heating area.

(6) Of all the structured cylindrical resonance cavities, the three-waveguide resonance cavity with the upper side waveguide rotated by 90° has the best microwave vulcanisation of tyres.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Tao Li

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