A Simplified Calculation Method of Heat Source Model for Induction

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Article A Simpliﬁed Calculation Method of Heat Source Model for Induction Heating

Hongbao Dong 1 , Yao Zhao 1,2,3,*, Hua Yuan 1,2,3, Xiaocai Hu 4 and Zhen Yang 4

1 School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China 2 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai 200240, China 3 Hubei Key Laboratory of Naval Architecture and Ocean Engineering Hydrodynamics (HUST), Wuhan 430074, China 4 Shanghai Waigaoqiao Shipbuilding Co., Ltd., Shanghai 200137, China * Correspondence: [email protected]

Received: 4 August 2019; Accepted: 9 September 2019; Published: 11 September 2019

Abstract: Line heating is used in forming the complex curve plates of ships, and this process is becoming integrated into automated tools. Induction heating equipment has become commonly used in automatic line heating. When applying automated equipment, it is necessary to calculate the relationship between the heating parameters and the temperature field. Numerical methods are primarily used to accomplish the calculations for induction heating. This computation process requires repeated iterations to obtain a stable heat generation rate. Once the heat generation rate changes significantly, a recalculation takes place. Due to the relative position of the coil and plate changes during heating, the grid needs to be frequently re-divided during computation, which dramatically increases the total computation time. In this paper, through an analysis of the computation process for induction heating, the root node that restricts the computation efficiency in the conventional electromagnetic-thermal computation process was found. A method that uses a Gaussian function to represent the heat flux was proposed to replace the electromagnetic computation. The heat flux is the input for calculating the temperature field, thus avoiding the calculation of the electromagnetic analysis during induction heating. Besides, an equivalence relationship for multi-coil was proposed in this paper. By comparing the results of the experiment and the numerical method, the proposed heat source model’s effectiveness was verified.

Keywords: induction heating; heat source model; shape parameters; Gaussian function

1. Introduction Line heating is a typical process in the plate forming of ships. In the past, the operation was mostly manual and included the use of a flame heater. With advancements in research and the development of automated line heating equipment, flame heating by oxyacetylene combustion has shown its limitations because of the difficulty with temperature control. Induction heating is used extensively for the forming of plates because it is easily controlled and has high efficiency. The principle of electromagnetic heating is to generate an eddy current in a workpiece, which is then heated by the Joule heat generated by the workpiece’s resistance. An induction heater includes a power source and an induction coil. The shape of the induction coil varies based on the object to be heated. The induction heating process includes electromagnetic induction and heat conduction. It is tough to obtain the temperature distribution of the workpiece during induction heating through a theoretical method and experiment-based methods are too costly. Therefore, numerical simulation

Materials 2019, 12, 2938; doi:10.3390/ma12182938 www.mdpi.com/journal/materials Materials 2019, 12, 2938 2 of 15 has become an effective solution for obtaining the temperature fields of induction heating. A typical process forMaterials induction 2019, 12, heatingx FOR PEER simulation REVIEW is presented in Figure1. First, time-harmonic electromagnetic2 of 14 analysis is performed with the designated initial conditions and constraint conditions under a given frequencyprocess and power.for induction The Joule heating heat simulation distribution is ofpresented the induction in Figure coil 1 is. calculatedFirst, time-harmonic when a stable heat generationelectromagnetic rate is obtained.analysis is Theperformed temperature with the field designated of the workpiece initial conditions can be calculatedand constraint using the conditions under a given frequency and power. The Joule heat distribution of the induction coil is Joule heatcalculated distribution when asa stable the input heat generation of thermal rate analysis. is obtained. The The thermal temperature analysis field endsof the whenworkpiece a stable can node temperaturebe calculated is obtained. using the If the Joule set heat calculation distribution time as the has input not of been thermal reached, analysis. the The induction thermal analysis coil is then moved toends the when next a position. stable node A temperature new analysis is obtained. is carried If the out set when calculation establishing time has not the been constraint reached, equations the about theinduction coil-workpiece-air coil is then moved interface. to the next position. A new analysis is carried out when establishing the constraint equations about the coil-workpiece-air interface.

Start

Set Geometric Physical Environment

Set Initial Conditions

Set Constraint Conditions

Electromagnetic Calculation

Electromagnetic No Analysis Last Substep of Analysis?

Yes

Joule Heat Distribution No

Heat Transfer Calculation

Thermal No Analysis Last Substep of Analysis?

Yes

Last Step of Analysis?

Yes

Temperature Results

Figure 1. Electromagnetic-thermal computation process. Figure 1. Electromagnetic-thermal computation process. In the computation process mentioned above, it is necessary to calculate the electromagnetic In thefield computation and the temperature process field. mentioned The Joule heat above, obtained it is necessaryfrom the electromagnetic to calculate calculation the electromagnetic is the field andinput the temperatureof the heat transfer field. calculation. The Joule It is heat necessary obtained to frequently from the re- electromagneticdivide the grid based calculation on the coil is the input of theposition heat when transfer performing calculation. moving It induction is necessary heating to frequentlyforming simulations, re-divide which the gridincrease based the ontime the coil position whenof computations performing and movingdecrease computation induction heating efficiency. forming simulations, which increase the time of computationsSome and studies decrease have computation investigated methods efficiency. to improve computation efficiency. Monzel [1] used the neural network method to carry out preliminary research on the features of the inductive current Some studies have investigated methods to improve computation efficiency. Monzel [1] used the neural network method to carry out preliminary research on the features of the inductive current distribution during transverse flux induction heating. He gave measures for reducing cable loss by changing each coil’s current to change the heat source distribution. The typical form of heat source Materials 2019, 12, 2938 3 of 15 density was given in the study, but the corresponding simplified numerical method was not mentioned. Bay [2] postulated that the temperature calculation relative to the electromagnetic calculation was a quasi-static process and proposed mathematical and numerical models for the coupled axisymmetric induction heating process. The model simplified the full-coupling process into a weak coupling process, used the eddy current obtained in the electromagnetic calculation as the heat generation rate in the temperature calculation, and presented a preliminary calculation method for the heat generation rate, but the expression for the heat source model was not given. Luo [3–6] carried out finite element calculations on the temperature field and the deformation field during high-frequency induction heating. He compared the relevant results with the experimental results, which avoided the electromagnetic coupling calculation. He gave the corresponding heat source model, but the calculation method for the heat source model and the corresponding relationship of various parameters in the model with the actual coil were not discussed. Liu [7] presented a numerical method for the temperature field of induction heating and studied the law of temperature distribution. When handling the electromagnetic-thermal coupling problem, a heat source model based on an empirical formula was used, but the calculation process for the model was not given. Hu [8] proposed the use of an equivalent heat source to replace the electromagnetic coupling calculation in induction heating and carried out experimental verification of the proposed heat source model, but did not give an explicit expression for the heat source model. Zhang [9] analyzed the similarities and differences of induction heating and flame heating in the line heating of ship plates and thought that induction heating was feasible in heat forming. When handling the induction heating process, he provided a heat source model for induction heating and used it as the initial input for the thermal distribution calculation. Compared with the experimental results, the errors of the calculation results satisfied practical engineering requirements. However, the model was simplified and the meaning of various model parameters and the corresponding numerical methods were not explained. Bae [10] proposed a two-dimensional circular heat input model to simulate the induction heating process and obtained satisfactory results, but the expression of the heat source model was not taken into account. Jeong [11] simplified coupled induction heating into the electromagnetic calculations and thermal calculations. He used statistical methods to describe the correlation between deformation and the input parameters, but the expression of the heat source model was not described in the study. Bai [12] used stepwise analysis to carry out coupling analysis within a typical time to obtain the state of induction heat distribution. He used the distribution as the moving heat source for a subsequent calculation, but the expression for the heat source was not given. Kubota [13] used a three-dimensional coupling and heat source model to perform calculations on the induction heating process and found that the results were consistent with measured results. The heat source model was obtained based on the obtained temperature field. Yang [14] proposed a heat source model based on the characteristics of the induction heating eddy current and used the finite element method (FEM) and relevant tests to verify the effectiveness of the model. The proposed heat source model was based on empirical methods and only targeted a particular coil form without giving the expression form of the corresponding heat source for other coil forms. Through theoretical analysis, Li [15] studied the analytical solution for obtaining the heat generation rate in the semi-infinite space during induction heating and discussed the effects of relevant parameters on the heat generation rate. To compare the difference in residual stress in a plate after single heating and double heating, Aung [16] proposed numerical methods based on a surface heat source and a body heat source. The effectiveness of the heat source models was verified through the experiment. Aung provided the expression form for the heat source model but did not discuss its calculation. Riccio [17] used different numerical models to study the bonding of carbon fiber reinforced polymer components with induction heating. ABAQUS was used to carry out electromagnetic analysis to obtain the energy loss caused by the Joule effect. He provided the finite element calculation model but not the expression for the Joule heat of the coil. In other studies related to the induction heating process and induction coil design [18–20], the full-coupling numerical method was also used for the induction heating calculations. Although the results were satisfactory, the time Materials 2019, 12, 2938 4 of 15 cost was high and the applicability was low for coils of different shapes and sizes or with different processing conditions. Zhang [21] replaced the electromagnetic-thermal coupling calculations with a heat source model corresponding to a discrete form and proposed a simplified calculation method for the high-frequency induction heat source. The computation efficiency for the calculation of a certain specific induction heating parameter was greatly improved, but any change in the induction heating parameters necessitated repeating the entire calculation. The induction heating calculation, especially the direct calculation of the moving induction heating process, results in a long computation time due to the need to continuously change the grid. However, the use of the heat flux to replace the coil can achieve high precision and reduce the computation time. Currently, most studies focus on empirical methods, and there are comparatively few studies that calculate the heat source and investigate the effect of the coil on the model. Based on these, the analytical method was employed in this study to calculate the heat source model for the induction coil. The effects of coil shapes on the heat source were examined. In addition, related experiments were conducted to verify the obtained heat source model for single-coil and multi-coil.

2. Simpliﬁcation of Heat Source

2.1. Finite Element Calculation In the induction heating process, the input is alternating current. The alternating current generates an alternating magnetic field in the coil which generates an induced electric field in the workpiece, which in turn generates heat in the workpiece. The electromagnetic field in the induction heating process is given by Maxwell’s equations:

∂D→ H→ = →J + (1) ∇ × ∂t

∂→B →E = (2) ∇ × − ∂t

→B = 0 (3) ∇· D→ = ρ (4) ∇· → where H→ is the magnetic field strength, →J is the conduction current density, ∂D is the displacement − ∂t current density, t represents time, →E is the electric field, →B is the magnetic field, D→ is the electric flux density, and ρ is the volume charge density. The corresponding auxiliary equations are:

D→ = ε→E (5)

→B = µH→ (6)

→J = σ→E (7) where ε represents the dielectric constant, µ is the magnetic permeability, and σ is the electrical conductivity. The eddy current density distribution in the workpiece can be obtained by solving the aforementioned equations and the distribution of the heat generation rate can be obtained using Joule’s law: 2 →J

q = (8) σ Materials 2019, 12, 2938 5 of 15 where q represents the heat generation rate. The heat generation rate is used as the input in the thermal conduction equation for calculating the thermal distribution, and the temperature field of the plate is obtained: ! ! ! ∂T ∂ ∂T ∂ ∂T ∂ ∂T Materials 2019, 12, x FOR PEER REVIEWρc = λ + λ + λ + q 5 of(9) 14 ∂t ∂x ∂x ∂y ∂y ∂z ∂z where 휌 is the density, 푐 is the specific heat, 푇 represents the temperature, and 휆 represents the where ρ is the density, c is the specific heat, T represents the temperature, and λ represents the thermal conductivity. thermal conductivity. The above procedure is the detailed process for the calculation of the induction heating. It is The above procedure is the detailed process for the calculation of the induction heating. It is necessary to continuously update the corresponding parameters with temperature changes during necessary to continuously update the corresponding parameters with temperature changes during the calculation as the workpiece’s electromagnetic and thermal property parameters are related to the calculation as the workpiece’s electromagnetic and thermal property parameters are related to the the temperature, which leads to an extremely time-consuming computation process. When the coil temperature, which leads to an extremely time-consuming computation process. When the coil and the and the workpiece have relative movement, the computation process, as mentioned earlier, must be workpiece have relative movement, the computation process, as mentioned earlier, must be repeated repeated to obtain the temperature field during movement. to obtain the temperature field during movement.

2.2.2.2. SimplificationSimplification ofof HeatHeat SourceSource ModelModel The key to temperature calculation is in obtaining the distribution of the heat generation rate 푞. The key to temperature calculation is in obtaining the distribution of the heat generation rate q. Due to the skin effect, the heat generated by the induction is concentrated on the surface of the Due to the skin effect, the heat generated by the induction is concentrated on the surface of the workpiece so the surface heat flux can be used as the input during induction heating. The heat flux workpiece so the surface heat flux can be used as the input during induction heating. The heat model used in this paper is shown in Figure 2. The cumbersome electromagnetic calculation is flux model used in this paper is shown in Figure2. The cumbersome electromagnetic calculation is unnecessary once the heat flux model is used to represent the coil. unnecessary once the heat flux model is used to represent the coil.

(a)

(b)

FigureFigure 2.2. DiagramsDiagrams ofof computationcomputation modelmodel andand heatheat sourcesourcedistribution: distribution: ((aa)) ComputationComputation modelmodel andand ((bb))Gaussian Gaussian distribution distribution of of the the heat heat source. source.

TheTheheat heat flux flux along along the the surface surface of the of workpiecethe workpiece in the in radial the directionradial direction of a single of coila single was assumedcoil was to follow a Gaussian function: assumed to follow a Gaussian function: r R 2 ( − 0 ) ( ) = − r푟H−푅 q r qme −( 0)2 (10) 푟 (10) 푞(푟) = 푞푚푒 퐻 where q(r) represents the heat flux at a distance r from the center of the heat source, r is the coordinate where 푞(푟) represents the heat flux at a distance 푟 from the center of the heat source, 푟 is the along the radial direction of the coil, qm is the maximum heat flux, R0 represents the radius of the heat coordinate along the radial direction of the coil, 푞푚 is the maximum heat flux, 푅0 represents the source, and rH represents the effective radius of the heat flux. 푟 radiusThe of total the heat power source on the, and workpiece 퐻 represents is equal the to effective the effective radius power of theQ heatof the flux. induction coil, that is: The total power on the workpiece is equal to the effective power 푄 of the induction coil, that is: Z 2π Z + 2휋 +∞∞ Q =푄 = ∫0 R푅00d푑휃θ ∫−∞ 푞q((푟r))푑푟dr = 2휋π√√휋π푞q푚m푅R00푟r퐻H ((11)11) 0 which leads to: −∞ 푄 푞푚 = (12) 2휋√휋푅0푟퐻 Thus:

푟−푅 푄 −( 0)2 푞(푟) = 푒 푟퐻 (13) 2휋√휋푅0푟퐻

Materials 2019, 12, 2938 6 of 15 which leads to: Q qm = (12) 2π √πR0rH Thus: r R 2 Q ( − 0 ) Materials 2019, 12, x FOR PEER REVIEW q(r) = e− rH 6 of 14(13) 2π √πR0rH EquationEquation (13) (13) is is the the heatheat flux flux corresponding corresponding toto the the coil. coil. The The temperature temperature field fieldof a workpiece of a workpiece can be directly calculated after the heat flux is substituted into Equation (9). Since the Joule heat can be directly calculated after the heat flux is substituted into Equation (9). Since the Joule heat distribution is obtained directly from Equation (13) to represent the induction coil, the distribution is obtained directly from Equation (13) to represent the induction coil, the electromagnetic electromagnetic calculation in Figure 1 is unnecessary, and the heat transfer is directly calculated. calculation in Figure1 is unnecessary, and the heat transfer is directly calculated. Moreover, grid Moreover, grid redivision during coil movement is unnecessary, resulting in fast computation. redivision during coil movement is unnecessary, resulting in fast computation. 3. Verifications of Heat Source Model 3. Verifications of Heat Source Model 3.1. Experimental Verification for Single Coil 3.1. Experimental Verification for Single Coil The heat flux model of a coil during induction heating has been previously stated, and a static The heat flux model of a coil during induction heating has been previously stated, and a static induction heating experiment was conducted on a three-axis motion platform to verify the induction heating experiment was conducted on a three-axis motion platform to verify the effectiveness effectiveness of the model. The platform had degrees of freedom in the three directions of X-Y-Z with of the model. The platform had degrees of freedom in the three directions of X-Y-Z with a positional a positional precision of 1 mm and a velocity precision of 1 mm/s. The schematic diagram of the precisioninduction of 1heating mm and experiment a velocity is precision shown in ofFigure 1 mm 3./s. The schematic diagram of the induction heating experiment is shown in Figure3.

(a)

(b)

FigureFigure 3. 3.Schematic Schematic diagram diagram ofof thethe inductioninduction heating experiment: (a ()a )Experimental Experimental platform platform for for inductioninduction heating heating and and (b ()b Positions) Positions of of thethe temperaturetemperature measurement points. points.

The experimental plate was low carbon steel and the material properties were similar to those in the Reference [22], as listed in Tables 1 and 2. The size of the plate was 1000 mm × 800 mm × 16 mm.

Materials 2019, 12, 2938 7 of 15

The experimental plate was low carbon steel and the material properties were similar to those in the Reference [22], as listed in Tables1 and2. The size of the plate was 1000 mm 800 mm 16 mm. × × Table 1. Thermal parameters of low carbon steel.

Heat Conductivity MaterialsTemperature 2019, 12, (x FORC) PEER DensityREVIEW (kg/m3) Speciﬁc Heat (J/(kg C)) 7 of 14 ◦ ·◦ Coeﬃcient (W/(m C)) ·◦ 0Table 7842 1. Thermal parameters 450.36 of low carbon steel. 66.97 50 - 464.6 65.21 Temperature200 (°C) Density (kg/m3) 7822 Specific Heat (J/(kg·°C)) 498.1 Heat Conductivity Coefficient 57.38 (W/(m·°C)) 2500 7842 -450.36 502.2666.97 54.91 30050 - -464.6 514.8265.21 53 400200 7822 7802 498.1 537.4257.38 47.92 250 - 502.26 54.91 450 - 623.64 45.83 300 - 514.82 53 500 - 707.35 43.53 400 7802 537.42 47.92 600 7782 812 39.3 450 - 623.64 45.83 650 - 904.07 36.37 500 - 707.35 43.53 700 - 967.69 34.74 600 7782 812 39.3 800650 - 7761904.07 1026.3236.37 31.02 700 - 967.69 34.74 800 Table7761 2. Mechanical parameters1026.32 of low carbon steel. 31.02

Temperature Young‘s Modulus Heat Expansion Yield Strength Table 2. MechanicalPoisson parameters Ratio of low carbon steel. (◦C) (GPa) Coeﬃcient (1/◦C) (MPa) Temperature Yield Strength Young‘s Modulus (GPa) Poisson Ratio Heat Expansion Coefficient5 (1/°C ) (0°C ) 206 0.267 1.20 10− 235 (MPa) × 5 500 206 196 0.267 0.29 1.25 1.2010 ×− 10−5 - 235 × 5 20050 196 0.3220.29 1.40 1.2510 ×− 10−5 163 - × 5 250200 196 186 0.322 0.296 1.43 1.4010 ×− 10−5 - 163 × 5 300250 186 0.296 0.262 1.47 1.4310 ×− 10−5 - - × 5 400300 186 166 0.262 0.24 1.54 1.4710 ×− 10−5 130 - × 5 450400 166 157 0.2290.24 1.57 1.5410 ×− 10−5 - 130 × 5 −5 500450 157 0.229 0.223 1.59 1.5710 ×− 10 - - × 5 −5 600500 157 135 0.223 0.223 1.64 1.5910 ×− 10 119 - × 5 −5 650600 135 117 0.223 0.223 1.66 1.6410 ×− 10 - 119 650 117 0.223 ×1.66 × 510−5 - 700 112 0.223 1.67 10− - 700 112 0.223 ×1.67 × 510−5 - 800 113 0.223 1.69 10− 109 800 113 0.223 ×1.69 × 10−5 109

A ring-typeA ring-type coil coil was was used used in the in the experiment, experiment, as shown as shown in Figure in Figure3. The 3. The induction induction coil wascoil was copper copper withwith an inner an inner diameter diameter of 90 of mm 90 mm and anand outer an outer diameter diameter of 110 of mm. 110 mm. The cross-sectionThe cross-section of the of coil the was coil a was rectangle with a size of 10 mm 8 mm× and a wall thickness of 1.5 mm. The schematic diagram of the a rectangle with a size of 10× mm 8 mm and a wall thickness of 1.5 mm. The schematic diagram of singlethe coil single is illustrated coil is illustrated in Figure in4 Figure. 4.

FigureFigure 4. Schematic 4. Schematic diagram diagram of the of singlethe single coil. coil.

The frequency of induction heating during the experiment was 12 kHz and the input power was 7.5 kW. The distance between the coil and the surface of the plate was 5 mm, which remained unchanged during heating. The heating time was 120 s. The induction heating efficiency was assumed to be 80%. An infrared temperature sensor (with a measurement range of 250 °C –1450 °C) was used to measure the temperature history of the plate. The distances between the sensors and the center of the coil were 0 mm, 30 mm, and 70 mm. A schematic diagram of the temperature measurement points is shown in Figure 3b.

Materials 2019, 12, 2938 8 of 15

The frequency of induction heating during the experiment was 12 kHz and the input power was 7.5 kW. The distance between the coil and the surface of the plate was 5 mm, which remained unchanged during heating. The heating time was 120 s. The induction heating eﬃciency was assumed to be 80%. An infrared temperature sensor (with a measurement range of 250 ◦C–1450 ◦C) was used to measure the temperature history of the plate. The distances between the sensors and the center of the coil were 0 mm, 30 mm, and 70 mm. A schematic diagram of the temperature measurement points is Materialsshown 2019 in Figure, 12, x FOR3b. PEER REVIEW 8 of 14

3.2.3.2. FEM FEM Calculation Calculation and and Comparison Comparison FollowingFollowing experimental experimental conditions, conditions, the the heat heat flux ﬂux function function of of the the coil coil during during heating heating can can be be calculatedcalculated based based on on the the input input parameters parameters from from Equation Equation (13): (13):

푟−0.05 2 5 −(r 0.05 )2 푞(푟) = 5.39 × 105 × 푒 ( −0.02 ) (14) q(r) = 5.39 10 e− 0.02 (14) × × The heat flux obtained was used as the input of the heat transfer calculation to obtain the temperatureThe heat field flux of obtainedthe plate. wasABAQUS used was as the used input to carry of the out heat the heat transfer transfer calculation calculation. toobtain The plate the sizetemperature was 1000 field mm of × the 800 plate. mm ABAQUS× 16 mm. was The used element to carry type out of the the heat plate transfer is DC3D8 calculation. for heat The transfer plate size was 1000 mm 800 mm 16 mm. The element type of the plate is DC3D8 for heat transfer simulation. The finite× element ×model under different grid sizes was established to investigate the effectsimulation. of the element The finite size. element The correlation model under between different the maximum grid sizes wastemperature established and to calculation investigate time the andeffect the of number the element of grids size. inThe the correlationheat affected between zone (HAZ) the maximum was compared. temperature The width and of calculation the HAZ timewas 400and mm. the numberThe grids of of grids the inlength the heat direction affected and zone the (HAZ)width direction was compared. were the The same width sizes, of theand HAZ the thicknesswas 400 mm. direction The grids was ofunified the length intodirection four grids. and All the widthcalculations direction were were performed the same on sizes, the and same the computer:thickness directionWindows was 10, unified64 bits, intofour four cores grids. with All 4.2 calculationsGHz and 8 GB were of performedRAM. The onrelationship the same computer: between theWindows maximum 10, 64 temperature bits, four cores of the with plate 4.2 GHz and andcalculation 8 GB of RAM.time and The the relationship number of between elements the in maximum HAZ is giventemperature in Figure of the5. plate and calculation time and the number of elements in HAZ is given in Figure5.

FigureFigure 5. 5. RelationshipRelationship between between maximum maximum temperature temperature and and computational computational time time and and number number of of elementselements in in the the heat heat affected aﬀected zone zone ( (HAZ).HAZ).

AsAs seen seen from from Figure Figure 5 thatthat the maximum temperature gradually decreases and reaches a stable valuevalue as as the the number number of of grids grids increases increases to to 80 80 (which (which means means the the element element size size is is 5 5 mm). mm). However, However, the the computingcomputing time time increases increases sharply sharply when when the the number number of of grids grids is is over over 100 100 (which (which means means the the element element sizesize is is 4 4 mm). mm). We We can can conclude conclude that that the the usage usage of of an an element element of of size size equal equal to to 5 5 mm mm is is appropriate appropriate in in reasonablereasonable computational computational time time without without losing losing appreciable appreciable accuracy. accuracy. A A variable variable grid grid was was used used for for elementelement division. division. The The grids grids in in the the HAZ HAZ were were 5 5mm mm × 44 mm mm × 4 mm and the elements far far from from the the × × heatingheating zone werewere 5050 mm mm ×20 20 mm mm8 × mm. 8 mm. The totalThe total number number of elements of elements was 68,600 was and68,600 the and number the × × numberof grids of was grids 86,399. was The86,399. ﬁnite The element finite element result is result given is in given Figure in6 .Figure 6.

Figure 6. The finite element result of the heat transfer calculation.

Materials 2019, 12, x FOR PEER REVIEW 8 of 14

3.2. FEM Calculation and Comparison Following experimental conditions, the heat flux function of the coil during heating can be calculated based on the input parameters from Equation (13):

푟−0.05 −( )2 푞(푟) = 5.39 × 105 × 푒 0.02 (14) The heat flux obtained was used as the input of the heat transfer calculation to obtain the temperature field of the plate. ABAQUS was used to carry out the heat transfer calculation. The plate size was 1000 mm × 800 mm × 16 mm. The element type of the plate is DC3D8 for heat transfer simulation. The finite element model under different grid sizes was established to investigate the effect of the element size. The correlation between the maximum temperature and calculation time and the number of grids in the heat affected zone (HAZ) was compared. The width of the HAZ was 400 mm. The grids of the length direction and the width direction were the same sizes, and the thickness direction was unified into four grids. All calculations were performed on the same computer: Windows 10, 64 bits, four cores with 4.2 GHz and 8 GB of RAM. The relationship between the maximum temperature of the plate and calculation time and the number of elements in HAZ is given in Figure 5.

Figure 5. Relationship between maximum temperature and computational time and number of elements in the heat affected zone (HAZ).

As seen from Figure 5 that the maximum temperature gradually decreases and reaches a stable value as the number of grids increases to 80 (which means the element size is 5 mm). However, the computing time increases sharply when the number of grids is over 100 (which means the element size is 4 mm). We can conclude that the usage of an element of size equal to 5 mm is appropriate in reasonable computational time without losing appreciable accuracy. A variable grid was used for element division. The grids in the HAZ were 5 mm × 4 mm × 4 mm and the elements far from the heating zone were 50 mm × 20 mm × 8 mm. The total number of elements was 68,600 and the number ofMaterials grids2019 was, 12, 293886,399. The finite element result is given in Figure 6. 9 of 15

MaterialsMaterials 2019 2019, ,12 12,Figure ,x x FOR FOR PEER FigurePEER 6. The REVIEW REVIEW 6. fTheinite ﬁnite element element result result of of theheat heat transfer transfer calculation. calculation. 99 ofof 1414

ComparisonsComparisons of ofofthe thethe experiment experimentexperiment results resultsresults and andand ﬁnite finitefinite element elementelement calculation calculationcalculation using usingusing the heat thethe source heatheat source modelsource are shown in Figures7–9, the relative locations of C ,C , and C are shown in Figure3. modelmodel areare shownshown inin FiguresFigures 77––9,9, thethe relativerelative locationlocation1 ss ofof2 CC11,, CC22,3, andand CC33 areare shownshown inin FigureFigure 3.3.

Figure 7. Comparison of the temperature histories of C1. Figure 7. Comparison of the temperature histories of C1..

Figure 8. Comparison of the temperature histories of C2. Figure 8. Comparison of the temperature histories of C2..

FigureFigure 9. 9. Comparison Comparison of of the the temperature temperature histories histories of of C C33. .

ItIt cancan bebe seenseen fromfrom FiguresFigures 77––99 thatthat thethe temperaturetemperature historieshistories obtainedobtained usingusing thethe heatheat sourcesource modelmodel forfor calculationcalculation areare inin agreementagreement withwith thethe experimentexperiment resultsresults.. TableTable 33 summarizedsummarized thethe relativerelative errorerror betweenbetween simulatedsimulated resultsresults andand experimentalexperimental results.results. ItIt cancan bebe seenseen thatthat thethe maximummaximum relativerelative errorerror waswas 5.42%5.42% andand thethe averageaverage relativerelative errorerror waswas 3.64%.3.64%. TheThe comparisoncomparison indicatesindicates thatthat thethe temperaturetemperature fieldfield ofof inductioninduction heatingheating cancan bebe accuratelyaccurately simulatedsimulated usingusing aa heatheat sourcesource model.model.

Materials 2019, 12, x FOR PEER REVIEW 9 of 14

Comparisons of the experiment results and finite element calculation using the heat source model are shown in Figures 7–9, the relative locations of C1, C2, and C3 are shown in Figure 3.

Figure 7. Comparison of the temperature histories of C1.

Materials 2019, 12, 2938 10 of 15 Figure 8. Comparison of the temperature histories of C2.

Figure 9. Comparison of the temperature histories of C3..

It cancan bebe seen seen from from Figures Figures7– 97 that–9 that the temperaturethe temperature histories histories obtained obtained using using the heat the source heat modelsource formodel calculation for calculation are in agreementare in agreement with the with experiment the experiment results. results Table. 3Table summarized 3 summarized the relative the relative error betweenerror between simulated simulated results results and experimental and experimental results. results. It can It be can seen be thatseen the that maximum the maximum relative relative error waserror 5.42% was and5.42% the and average the average relative errorrelative was error 3.64%. was The 3.64%. comparison The comparison indicates that indicates the temperature that the

ﬁeldtemperatureMaterials of induction2019, 12field, x FOR heatingof inductionPEER REVIEW can be heatingaccurately can be simulated accurately using simulated a heat sourceusing a model. heat source model.10 of 14

TableTable 3.3. ComparisonComparison betweenbetween simulatedsimulated resultsresults andand experimentalexperimentalresults. results.

SensorSensor Maximum RelativeRelative ErrorError Average Average Relative Relative Error Error Minimum Minimum Relative Relative Error Error

C1 C1 3.55%3.55% 2.43% 2.43% 1.15% 1.15% C2 C2 3.84%3.84% 2.42% 2.42% 1.16% 1.16% C3 C3 5.42%5.42% 3.64% 3.64% 2.46% 2.46%

The computation efficiency using the heat source model was compared with that using the full- The computation eﬃciency using the heat source model was compared with that using the coupling method. COMSOL Multiphysics was used to carry out the coupling calculations. The full-coupling method. COMSOL Multiphysics was used to carry out the coupling calculations. The maximum element size of the model was 1 mm. The total number of elements was 34,274. The maximum element size of the model was 1 mm. The total number of elements was 34,274. The coupling coupling model is shown in Figure 10. model is shown in Figure 10.

Figure 10. Coupling calculation model. Figure 10. Coupling calculation model. The comparisons between the coupling model and the experiment are shown in Figures 11–13. The comparisons between the coupling model and the experiment are shown in Figures 11–13. The comparison of relative error between coupling model and the experiment are listed in Table4. The comparison of relative error between coupling model and the experiment are listed in Table 4.

Figure 11. Comparison of the temperature histories of C1 between the coupling model and the experiment.

Figure 12. Comparison of the temperature histories of C2 between the coupling model and the experiment.

MaterialsMaterials 20192019,, 1212,, xx FORFOR PEERPEER REVIEWREVIEW 1010 ofof 1414

TableTable 3.3. ComparisonComparison betweenbetween simulatedsimulated resultsresults andand experimentalexperimental resultsresults..

SensorSensor MaximumMaximum RelativeRelative ErrorError AverageAverage RelativeRelative ErrorError MinimumMinimum RelativeRelative ErrorError CC11 33.55.55%% 22.43%.43% 11.15%.15% CC22 3.84%3.84% 22.42%.42% 11.16%.16% CC33 55.42%.42% 33.64%.64% 22.46%.46%

TheThe computationcomputation efficiencyefficiency usingusing thethe heatheat sourcesource modelmodel waswas comparedcompared withwith thatthat usingusing thethe fullfull-- couplingcoupling method.method. COMSOLCOMSOL MultiphysicsMultiphysics waswas usedused toto carrycarry outout thethe couplingcoupling calculations.calculations. TheThe maximummaximum elementelement sizesize ofof thethe modelmodel waswas 11 mm.mm. TheThe totaltotal numbernumber ofof elementselements waswas 34,274.34,274. TheThe couplingcoupling modelmodel isis shownshown inin FigureFigure 10.10.

FigureFigure 10.10. CouplingCoupling calculationcalculation model.model.

MaterialsTheThe2019 comparisonscomparisons, 12, 2938 betweenbetween thethe couplingcoupling modelmodel andand thethe experimentexperiment areare shownshown inin FiguresFigures 1111– of–13.13. 15 TheThe comparisoncomparison ofof relativerelative errorerror betweenbetween couplingcoupling modelmodel andand thethe experimentexperiment areare listedlisted inin TableTable 4.4.

FigureFigure 11. 11. ComparisonComparisonComparison ofof of thethe the temperaturetemperature temperature historieshistories histories ofof ofCC11 Cbetweenbetween1 between thethe thecouplingcoupling coupling modelmodel model andand andthethe experiment.theexperiment. experiment.

Figure 12. Comparison of the temperature histories of C2 between the coupling model and the Figure 12. ComparisonComparison of of the the temperature temperature histories histories of ofC2 Cbetween2 between the thecoupling coupling model model and andthe Materialsexperiment.theexperiment. 2019 experiment., 12, x FOR PEER REVIEW 11 of 14

Figure 13. Comparison of the temperature histories of C3 between the coupling model and the Figure 13. Comparison of the temperature histories of C3 between the coupling model and experiment.the experiment.

Table 4. Comparison of coupling model and experiment.

SensorSensor Maximum Maximum Relative Relative Error Error Average Average Relative Relative Error Error Minimum Minimum Relative Relative ErrorError C1 8.65% 7.55% 6.38% C1 8.65% 7.55% 6.38% C2 C2 2.71%2.71% 1.68% 1.68% 0.50% 0.50% C3 C3 1.61%1.61% 1.17% 1.17% 0.27% 0.27%

As can be seen from Figures 11–13, the temperature histories obtained by the coupling model are in agreement with the experiment results. The relative error between the simulated results and experimental results, listed in Table 4, showed that the average relative error was 7.55% at most, which means the chosen simulation method was correct for calculating induction heating. Table 5 shows the number of degrees and computing time between the coupling model and the heat source model. The number of degrees of freedom and the computation time during coupling computing was higher than the heat source model. It can be considered that the heat source model has sufficient accuracy and efficiency for induction heating calculation.

Table 5. Comparison of computation efficiency.

Number of Degrees of FreedomComputation Time/Min Full-coupling model 230,917 6.7 Heat source model 86,399 3.7

3.3. Application to Multi-Coil It can be seen from Equation (13) that the Gaussian function that represents the heat source has a relationship with the project area of the coil. To further verify the effectiveness of the proposed heat source model, the previously described coil, shown in Figure 3b, was used to conduct a multi-coil experiment. A static induction heating experiment of multi-coil was conducted on the previously described experimental platform. The schematic diagram of the multi-coil is shown in Figure 14. The heating power was 25 kW and the heating time was 100 s. The bottom of the coil was 5 mm from the surface of the plate. The size of the plate and positions of the temperature measurement points were the same as those in the previously described experiment. The experimental results were compared with the finite element results calculated using the heat source model.

Materials 2019, 12, 2938 12 of 15

As can be seen from Figures 11–13, the temperature histories obtained by the coupling model are in agreement with the experiment results. The relative error between the simulated results and experimental results, listed in Table4, showed that the average relative error was 7.55% at most, which means the chosen simulation method was correct for calculating induction heating. Table5 shows the number of degrees and computing time between the coupling model and the heat source model. The number of degrees of freedom and the computation time during coupling computing was higher than the heat source model. It can be considered that the heat source model has suﬃcient accuracy and eﬃciency for induction heating calculation.

Table 5. Comparison of computation eﬃciency.

Number of Degrees of Freedom Computation Time/Min Full-coupling model 230,917 6.7 Heat source model 86,399 3.7

3.3. Application to Multi-Coil It can be seen from Equation (13) that the Gaussian function that represents the heat source has a relationship with the project area of the coil. To further verify the eﬀectiveness of the proposed heat source model, the previously described coil, shown in Figure3b, was used to conduct a multi-coil experiment. A static induction heating experiment of multi-coil was conducted on the previously described experimental platform. The schematic diagram of the multi-coil is shown in Figure 14. The heating power was 25 kW and the heating time was 100 s. The bottom of the coil was 5 mm from the surface of the plate. The size of the plate and positions of the temperature measurement points were the same as thoseMaterials in the 2019 previously, 12, x FORdescribed PEER REVIEW experiment. The experimental results were compared with the ﬁnite12 of 14 element results calculated using the heat source model.

Figure 14. Schematic diagram of sizes of multiple coils. Figure 14. Schematic diagram of sizes of multiple coils. During the experiment, the heat flux was more diffused along the radial direction of the coil and During the experiment, the heat flux was more diffused along the radial direction of the coil and its scope of influence had become larger. Therefore, the action radius of the heat flux was assumed to its scope of influence had become larger. Therefore, the action radius of the heat flux was assumed to be rH = 0.04, the position of the heat flux center was at R0 = 0.085 in the middle coil, and the heat flux be 푟 = 0.04, the position of the heat flux center was at 푅 = 0.085 in the middle coil, and the heat was given퐻 by: 0 flux was given by: 2 5 ( r 0.085 ) q(r) = 5.28 10 e −0.04 (15) − 푟−0.085 × × −( )2 푞(푟) = 5.28 × 105 × 푒 0.04 (15) Comparisons of the simulation results and the experiment results are shown in Figures 15 and 16. The errorComparisons comparison betweenof the simulation simulation results results and and the experimental experiment result resultss are listedshown in in Table Figures6. 15 and 16. The error comparison between simulation results and experimental results are listed in Table 6.

Figure 15. The temperature histories of C3 during the multi-coil experiment and simulation.

Figure 16. The temperature histories of C2 during the multi-coil experiment and simulation.

Table 6. Comparison of simulation and experiment.

Sensor Maximum Relative Error Average Relative Error Minimum Relative Error C2 18.63% 12.72% 4.82% C3 15.42% 7.65% 0.15%

Materials 2019,, 12,, x FOR PEER REVIEW 12 of 14

Figure 14. Schematic diagram of sizes of multiple coils.

During the experiment, the heat flux was more diffused along the radial direction of the coil and its scope of influence had become larger. Therefore, the action radius of the heat flux was assumed to be 푟퐻 = 0..04,, the position of the heat flux center was at 푅0 = 0..085 in the middle coil, and the heat flux was given by:

푟−0..085 2 5 −( )2 푞(푟) = 5..28 × 105 × 푒 0..04 (15)

MaterialsComparisons2019, 12, 2938 of the simulation results and the experiment results are shown in Figures 1513 ofand 15 16. The error comparison between simulation results and experimental results are listed in Table 6.

Figure 15. The temperature histories of C3 duringduring the multi multi-coil-coil experiment and simulation.

Figure 16. The temperature histories of C2 duringduring the multi multi-coil-coil experiment and simulation. Table 6. Comparison of simulation and experiment. Table 6. Comparison of simulation and experiment. Sensor Maximum Relative Error Average Relative Error Minimum Relative Error Sensor Maximum Relative Error Average Relative Error Minimum Relative Error C2 C2 18.63%18.63% 12.72% 12.72% 4.82% 4.82% C3 C3 15.42%15.42% 7.65% 7.65% 0.15% 0.15%

It can be seen from Figures 15 and 16 and Table6 that the results of the heat source model are in agreement with the results obtained in the experiment in terms of the trend with a maximum relative error of 18.6%. It means that the proposed heat source model is also applicable for use with multi-coil induction heating, which expands the application of the heat source model.

4. Conclusions In this paper, the heat flux of the coil during induction heating was studied. The expression of the heat flux was obtained and the accuracy was verified through calculation and experiment. The following conclusions were drawn: 1. The heat flux during induction heating can be simplified into a Gaussian function, and the values of the Gaussian function are related to the size of the induction coil and the heating conditions. 2. The heat source model can be used to calculate the temperature distribution of induction heating. The proposed model can improve computation efficiency while ensuring precision. Fast computation of moving induction heating can be achieved using this model. 3. The proposed heat source can also be used in multi-coil induction heating, as the goodness of the finite element results and the experimental result is within an acceptable range. Materials 2019, 12, 2938 14 of 15

Author Contributions: Y.Z. contributed to the discussions and reviewed the paper; H.D. carried out the experiments, performed the numerical simulations and wrote the paper; H.Y. assisted in the experiment; X.H. guided the experiment; Z.Y. reviewed the paper. Funding: This work was supported by the Ministry of Science and Technology of the People’s Republic of China (Grant No.: 2012DFR80390) and Ministry of Industry and Information Technology of the People’s Republic of China (Grant No.:2016545). Conﬂicts of Interest: The authors declare no conﬂicts of interest.

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