Loss and temperature rise of the skin effect electric heat tracing system

Abstract

In actual operation, the skin effect electric heat tracing system is powered by the inverter circuit, and the higher harmonic components are inevitably introduced. There is not much research on the influence of harmonic current on the electric heat tracing load loss. In this paper, the harmonic distribution under PWM driving mode is analyzed, and the load loss of the electric heat tracing body under harmonic current is derived. The co-simulation method of PWM inverter circuit and skin electric heat tracing body coupling is proposed. The simulation results of current and loss of co-simulation method and sine wave driving method are compared. The results show that the parameters of PWM driving mode increase to different degrees, which proves the correctness of the joint simulation method. At the same time, the influence of the output characteristics and modulation parameters of the PWM inverter circuit on the load loss of the electric heat tracing body is analyzed, which provides a theoretical basis for the determination of the parameters of the inverter circuit. In addition, combined with multi-physics coupling analysis, this paper obtains more accurate temperature distribution results of the electric heat tracing system without scaling the size of the electric heat tracing ontology. Not only can we intuitively understand the material selection and hot spot configuration of the electric heat tracing components, but also further understand the influence of current harmonics on the load loss in actual operation, so as to better realize the efficient heat tracing of the skin electric heat tracing system.

Keywords

Electric heat tracing system harmonic analysis loss calculation co-simulation multi-physics coupling analysis

1. Introduction

Electric heat tracing technology is considered to be the most efficient heating method for deep-water submarine pipelines [1]. The skin effect electric heat tracing (SECT) for metal pipe heating is favored for providing heat to long distance pipelines [2]. The SECT system differs from most devices in that it reduces eddy current losses, which are heated by the losses generated by the eddy currents. In the actual operation of the SECT system, the applied excitation frequently switches with the power device turning on and off, so that the inverter output waveform contains rich harmonic components. Although the harmonic component accounts for a small proportion relative to the fundamental wave, however, the higher harmonics cause the electric heat tracing body to increase the harmonic impedance under the influence of the skin effect, the proximity effect, and the hysteresis effect, resulting in more loss and heat. Therefore, the influence of current time harmonic component [3,4] on the heating effect of the SECT system cannot be ignored. For the SECT system, few scholars consider the problem of system loss calculation under harmonics. Jiang derived the eddy current density distribution equation in the SECT tube based on the electromagnetic field theory, and obtained its eddy current loss, but ignored the influence of harmonic current on its loss during the analysis [5]. In [6], the author analyzed the temperature field in the super heavy oil pipeline based on FLUENT software, but did not consider the influence of harmonic current when calculating the heat of the SECT heating tube.

In addition, the design of the electric heat tracing ontology and control system are often split when designing a traditional SECT system. From the perspective of the SECT ontology designer, they assume that the working circuit of the SECT system is ideal, they only need to complete the structure design and electrical design of the SECT ontology. However, the overall performance of the SECT system in the working circuit and whether the heating power supply is matched cannot be investigated. Moreover, the personnel engaged in power system design idealizes the SECT system, ignoring the facts of skin effect, hysteresis effect, proximity effect and harmonics existing in the ontology model, resulting in less satisfactory control effect. Therefore, it is not necessary to integrate the SECT body and the control method, which is beneficial to the overall optimization and control of the SECT system. At present, the research on field-circuit coupling simulation of the SECT ontology and the inverter circuit is still in a blank stage. For direct electric heating (DEH) of subsea pipeline heating systems, Heggdal optimized the number of new cable cores based on finite element simulation, and applied the ideal current to obtain its temperature distribution, and it focuses on the DEH ontology design and does not consider the effects of harmonics contained in the supply current on the system during actual operation [7]. In view of the SECT system, Pan started from the heat transfer mechanism of the SECT tube, established a two-dimensional model of the SECT tube, and analyzed the influencing factors of the heat tracing effect by finite element analysis, and it focuses on the analysis of the SECT tube structure, and does not consider the impact of the harmonics contained in the supply current on the system during actual operation [8]. In [9], the author used the Matlab/Simulink module to build the inverter model, directly gives the ideal PWM waveform as the IGBT control signal, and applies the inverter output AC square wave to the resistor-inductor module equivalent to the SECT ontology. Obviously, this method not only needs to calculate the equivalent impedance of the system, but also lacks the coupling between the SECT ontology and the inverter circuit. Based on the above research status, an analysis method based on the co-simulation of Ansoft and Simplorer is proposed to simulate the actual operation of the SECT system.

2. Theory and methods

2.1. Harmonic analysis of PWM inverter circuit

At present, the drive control system of the SECT system widely uses a voltage source type PWM inverter circuit, and uses a carrier to modulate a sinusoidal signal wave to generate a harmonic component related to a carrier. The frequency and amplitude of these harmonic components are one of the important indicators to measure the performance of the PWM inverter circuit, so it is necessary to perform harmonic analysis on the PWM waveform [10]. The inverter circuit is shown in Fig. 1.

Fig. 1.

Inverter circuit diagram.

In Fig. 1, the modulation wave u _r is a sine wave, and the carrier u _c is a triangular wave, which is compared with a triangular wave by a sine wave. When the sine wave is larger than the triangular wave, the switching tubes V ₁ and V ₄ are turned on to generate a positive pulse of the PWM wave, and when the sine wave is smaller than the triangular wave, the switching tubes V ₂ and V ₃ are turned on to generate a negative pulse of the PWM. For convenience of analysis, the triangular wave is represented by two piecewise functions, whose slopes are +2U _c∕π and −2U _c∕π; the initial values are +U _c and −U _c, U _c is the amplitude of the carrier triangular wave u _c, and the carrier triangular wave can be as shown in Eq. (1). $\begin{eqnarray}\displaystyle u_{c}=\left\{\begin{array}{@{}l@{}}-({\omega}_{c}t-2{\pi}k-2{\pi}){\displaystyle \frac{2U_{c}}{{\pi}}}-U_{c},2{\pi}k+{\pi}\leq {\omega}_{c}t\leq 2{\pi}k+2{\pi}\\ ({\omega}_{c}t-2{\pi}k){\displaystyle \frac{2U_{c}}{{\pi}}}-U_{c},2{\pi}k\leq {\omega}_{c}t\leq 2{\pi}k+{\pi}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (1)

The expression of the sinusoidal modulated wave is $\begin{eqnarray}\displaystyle u_{r}=U_{r}\cdot \sin ({\omega}_{r}t-{\phi}) & & \displaystyle\end{eqnarray}$ (2) where the modulation ratio M = U _r∕U _c ≤ 1, ω_c is the triangular wave u _c angular frequency (rad/s), and ω_r is the sine wave u _r angular frequency (rad/s), then the carrier ratio N = ω_c∕ω_r ≥ 1, k = 0, 1, 2, 3…, and N is any positive integer. Assume that the DC power supply of the inverter is U _d, J _n is the n-time Bessel function, m represents the harmonic order of the carrier, and n represents the harmonic order of the modulated wave, then the double Fourier series of u ₀ is expressed as $\begin{eqnarray}\displaystyle u_{0} & = & \displaystyle MU_{d}\sin ({\omega}_{r}t-{\phi})+\frac{4U_{d}}{{\pi}}\mathop{\sum }_{m=1,2,3,\ldots }^{\infty }\frac{J_{0}\left(\frac{mM{\pi}}{2}\right)}{m}\sin \frac{m{\pi}}{2}\cdot \cos (mN{\omega}_{r}t)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\frac{4U_{d}}{{\pi}}\mathop{\sum }_{m=1,2,3,\ldots }^{\infty }\mathop{\sum }_{n=\pm 1,\pm 2,\ldots }^{\pm \infty }\frac{J_{n}\left(\frac{mM{\pi}}{2}\right)}{m}\sin \left(\frac{m+n}{2}{\pi}\right)\cdot \cos \left[(mN+n){\omega}_{r}t-n{\phi}-n\frac{{\pi}}{2}\right].\end{eqnarray}$ (3)

From Eq. (3), we can see the harmonic angular frequency of u ₀. $\begin{eqnarray}\displaystyle m{\omega}_{c}\pm n{\omega}_{r}. & & \displaystyle\end{eqnarray}$ (4) The ratio of the harmonic angular frequency to the modulated wave angular frequency is the number of harmonic distributions. $\begin{eqnarray}\displaystyle (m{\omega}_{c}\pm n{\omega}_{r})/{\omega}_{r}=mN\pm n. & & \displaystyle\end{eqnarray}$ (5)

It can be seen that the PWM wave does not contain low-order harmonics, and only includes the angular frequency ω_c and its nearby harmonics, and harmonics such as 2ω_c, 3ω_c, and the like. Among the above harmonics, the angular frequency with the highest amplitude and the greatest influence is the harmonic component of ω_c. When the carrier ratio N is an odd number, it contains only odd harmonics, and the highest harmonic amplitude is the Nth harmonic.

2.2. Load loss at harmonic current

The SECT system mainly has two sources of heat: one is the resistance heat of the current in the conductor; the other is the iron loss heat in the conductor. That is to say, the load loss of the inverter power source includes the resistance loss of the SECT tube and the heating cable, and the eddy current loss and hysteresis loss in the SECT tube. When the iron loss heat is much smaller than the resistance loss under the power frequency condition, the iron loss heat is neglected. However, when considering the current time harmonic, the iron loss cannot be ignored due to the high harmonic component, and the resistance loss is also affected. In order to analyze the temperature rise of crude oil in the oil pipeline, it is necessary to calculate the load loss under harmonic current.

∙ Resistance loss

In the harmonic current, the resistance loss of the SECT tube and the heating cable is proportional to the square of the effective value of the current [11]. The SECT system can be regarded as an iron core coil circuit of AC excitation [12]; the SECT tube is used as the iron core, the heating cable and the SECT tube are used as an excitation coil. In calculating the resistance loss of the transformer winding under harmonic current, the IEEE Std C57.110:2008 standard directly uses the winding DC resistance as the actual resistance value of the winding under harmonic current. Zhou proposed that the harmonic resistance loss calculation model of IEEE Std C57.110:2008 ignores the influence of the winding skin effect on the loss calculation, so the winding resistance value under the fundamental current is taken as the actual resistance value [13]. Then the resistance loss P _Z under the harmonic can be obtained. $\begin{eqnarray}\displaystyle P_{Z}=I_{rms}^{2}R=\mathop{\sum }_{i=1}^{i_{m}}I_{i}^{2}R. & & \displaystyle\end{eqnarray}$ (6) Among them, I _rms is the load current of SECT ontology under the harmonic current, i is the harmonic current number, i _m is the highest number of harmonic currents, and I _i is i-th harmonic current. R is the resistance of the SECT tube and the heating cable on the fundamental current.

∙ Eddy current loss

Eddy current loss is when a ferromagnetic conductor is connected to an alternating current, an alternating magnetic field is induced in the conductor to form an eddy current and generate thermal energy in the ferromagnetic conductor. The current distribution in the SECT tube is derived from the basic equation of the electromagnetic field. For a cylindrical conductor with an inner radius and an outer radius of r ₂ and r ₃, respectively, the electric field strength and current density are only axial components, the magnetic field strength is only a circumferential component, and has symmetry, and the field amount varies only with the radius r. Therefore, in the cylindrical coordinate system, the Maxwell equations shown below can be obtained [14]. $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\text{d}\boldsymbol{H}_{Q}/\text{d}r+\boldsymbol{H}_{Q}/r={\sigma}\boldsymbol{E}_{\boldsymbol{Z}}\\ \text{d}\boldsymbol{E}_{\boldsymbol{Z}}/\text{d}r=\text{j}{\omega}{\mu}\boldsymbol{H}_{\boldsymbol{Q}}.\end{array} & & \displaystyle\end{eqnarray}$ (7)

Available from Eq. (7) $\begin{eqnarray}\displaystyle (\text{d}^{2}\boldsymbol{E}_{\boldsymbol{Z}}/\text{d}r^{2})+(1/r)(\text{d}\boldsymbol{E}_{\boldsymbol{Z}}/\text{d}r)-\text{j}{\omega}{\mu}{\sigma}\boldsymbol{E}_{\boldsymbol{Z}}=0 & & \displaystyle\end{eqnarray}$ (8) where H _Q is the circumferential component of the magnetic field strength, E _Z is the electric field strength in the axial direction, σ is the electrical conductivity, and μ is the magnetic permeability. Let $k=\sqrt{\text{j}{\omega}{\sigma}{\mu}}$ be able to rewrite Eq. (8) into Eq. (9). $\begin{eqnarray}\displaystyle (\text{d}^{2}\boldsymbol{E}_{\boldsymbol{Z}}/\text{d}(kr)^{2})+(1/kr)(\text{d}\boldsymbol{E}_{\boldsymbol{Z}}/\text{d}(kr))-\boldsymbol{E}_{\boldsymbol{Z}}=0. & & \displaystyle\end{eqnarray}$ (9)

Equation (9) is a deformed Bessel function whose general solution is $\begin{eqnarray}\displaystyle \boldsymbol{E}_{\boldsymbol{Z}}=AI_{0}(kr)+BK_{0}(kr) & & \displaystyle\end{eqnarray}$ (10) where I ₀(kr) and K ₀(kr) are the first-order and second-class modified zero-order Bessel functions, respectively, and the A and B constants can be determined by the boundary conditions [15]. The magnetic field strength can be obtained from the characteristics of Eq. (7) and the Bessel function. $\begin{eqnarray}\displaystyle \boldsymbol{H}_{\boldsymbol{Q}}=(1/\text{j}{\omega}{\mu})\cdot (\text{d}\boldsymbol{E}_{\boldsymbol{Z}}/\text{d}r)=({\sigma}A/k)I_{1}(kr)-({\sigma}B/k)K_{1}(kr). & & \displaystyle\end{eqnarray}$ (11)

At the same time, the magnetic field strength is obtained according to the full current law. $\begin{eqnarray} \displaystyle \begin{array}{@{} l@{}}\boldsymbol{H}_{\boldsymbol{Q}}|_{r=r_{2}}=\boldsymbol{I}/2{\pi}r_{2}=\displaystyle\mathop{\sum }_{i=1}^{i_{m}}\boldsymbol{I}_{\boldsymbol{i}}/2{\pi}r_{2}\\ \boldsymbol{H}_{\boldsymbol{Q}}|_{r=r_{3}}=0 \end{array} & & \displaystyle \end{eqnarray}$ (12) where I _i is i-th harmonic current flowing through the section of the SECT tube. According to the boundary condition of Eq. ((12)), the values of the constants A and B in Eq. ((11)) can be determined, so that the values of E _Z and J can be obtained. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{E}_{\boldsymbol{Z}}=\frac{k\boldsymbol{I}}{2{\pi}{\sigma}r_{2}}\cdot \frac{I_{0}(kr)K_{1}(kr_{3})+K_{0}(kr)I_{1}(kr_{3})}{I_{1}(kr_{2})K_{1}(kr_{3})-K_{1}(kr_{2})I_{1}(kr_{3})}\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{J}={\sigma}\boldsymbol{E}_{\boldsymbol{Z}}=\frac{k\boldsymbol{I}}{2{\pi}r_{2}}\cdot \frac{I_{0}(kr)K_{1}(kr_{3})+K_{0}(kr)I_{1}(kr_{3})}{I_{1}(kr_{2})K_{1}(kr_{3})-K_{1}(kr_{2})I_{1}(kr_{3})}.\end{eqnarray}$ (14)

Under harmonic current, the eddy current loss of the SECT tube needs to consider the skin effect and proximity effect between the SECT tube and the heating cable. The eddy current loss P _E of the SECT tube can be obtained by the above electromagnetic field theory analysis. $\begin{eqnarray}\displaystyle P_{E}=\mathop{\sum }_{i=1}^{i_{m}}(J_{i}^{2}/{\sigma}). & & \displaystyle\end{eqnarray}$ (15) In Eq. (15), J _i is the amplitude of i-th harmonic current density. It can be seen from the expression that the harmonic current has a significant influence on the eddy current loss of the SECT tube.

∙ Hysteresis loss

The hysteresis loss is the friction generated by the magnetic domain in the process of rotating inside the conductor [16], and finally released in the form of thermal energy. The hysteresis loss P _H of the SECT tube is proportional to the loss coefficient 𝛼 of the maximum magnetic flux density B _m. The maximum magnetic flux density B _m is proportional to the applied voltage U. The power supply voltage of the SECT drive control system is known, and the amplitude, frequency, phase and other parameters of the voltages on the inverter side can be obtained by Fourier analysis, and the hysteresis loss of the SECT tube can be obtained. $\begin{eqnarray}\displaystyle P_{H}=\left(\mathop{\sum }_{i=1}^{i_{m}}(U_{i}/iU_{1})\cos {\varphi}_{i}\right)^{{\alpha}}. & & \displaystyle\end{eqnarray}$ (16) Among them, U _i is the i-th harmonic voltage, U ₁ is the fundamental voltage, and φ_i is the i-th harmonic voltage initial phase angle. It can be seen from the expression that the harmonic voltage has a great influence on the hysteresis loss of the SECT tube.

3. Simulation and analysis

In order to obtain the simulated current waveform of the PWM inverter circuit, the PWM inverter circuit is built in Simplorer, and the SECT ontology model is established in Ansoft. The two simultaneously run and interact data in real time to simulate the current waveform of the actual power supply of the inverter, which is closer to the actual situation, and it is more convenient to investigate the influence of the current time harmonic on the SECT ontology loss.

3.1. Establishment of co-simulation model

The field-circuit coupling simulation of the SECT system introduces the electromagnetic model of the SECT ontology into the Simplorer by setting the interface function with the Simplorer in Ansoft. Then, the inverter model, PWM signal generation module, SECT tube resistance leakage inductance module, voltage and current measurement module are established in Simplorer. Finally, the two softwares are simultaneously simulated in the Simplorer environment, and Fig. 2 is the co-simulation model.

Fig. 2.

Co-simulation model. The simulation process of system field coupling is generally as follows. First, check the “Enable transient-transient link with Simplorer” option in Maxwell to start the interface between Maxwell and Simplorer. Then, the SECT ontology electromagnetic model is introduced into Simplorer, and the inverter model, PWM signal generation module, SECT tube resistance leakage inductance module, voltage and current measurement module are established in Simplorer. Finally, the settings for solving the two softwares can achieve synchronous simulation.

In the simulation, the three-phase uncontrollable rectification link is neglected, and the DC voltage source is used instead. The input DC voltage is 500 V, and the single-phase AC power converted into a certain frequency through the inverter circuit supplies power to the SECT tube and the heating cable. The electric current of equal size and opposite direction is formed in the SECT tube and the heating cable conductor, and the crude oil in the pipeline is heated under the influence of the skin effect, the proximity effect and the hysteresis effect [17]. From the perspective of simulation time and computational memory, the two-dimensional electromagnetic field finite element model is usually used to establish the SECT ontology, as shown in Fig. 3. The dimensions of the various parts of the SECT ontology are set according to the size of the field application. The conventional SECT tube is ∅27 × 3 20#steel pipe [18,19]. The heating cable adopts a copper core wire with a cross section of 25 mm², and the oil pipeline is Q235#steel. The reference pipe size in a certain pipeline project is taken as ∅ 219 mm × 7 mm, length is taken as 1000 m [20], and the two-dimensional physical model of the pipeline is established in Maxwell.

Fig. 3.

The 2D model of SECT ontology.

3.2. Co-simulation analysis

The modulation ratio, carrier ratio, and operating frequency of the SECT system are given in the simulation. The influence of the harmonics contained in the output current of the inverter side on the operating state and loss of the SECT system during PWM power supply is analyzed to reflect the working performance of the SECT system under actual conditions.

The simulation parameters are as follows. In [21], the author pointed out that increasing the power frequency from 50/60 Hz to 100/200 Hz not only reduces the cost, but also improves the heating effect. Therefore, when the operating frequency of the SECT system is set to 200 Hz during simulation, the frequency of the sinusoidal modulation wave in the co-simulation is also set to 200 Hz. Let the carrier ratio N = 7, the triangular wave frequency is 1.4 kHz; the sine wave amplitude is 1, and the triangular wave amplitude is 1.414, that is, the modulation ratio M = 0.7. Considering the simulation time and the memory are occupied by the calculation results, the two-dimensional finite element model is used to simulate the field coupling. Zhao pointed out that when using the two-dimensional finite element model for co-simulation [22], in order to meet the analysis accuracy, it is necessary to consider the leakage inductance at the end of the conductor. Therefore, the end leakage inductance L1 = 4.53 mH of the SECT tube and the system resistance R1 = 3.2Ω are calculated.

In addition, in order to ensure the simulation accuracy, there are at least 10 sampled data points in one cycle, so the simulation step size is 20 us and the simulation time is 0.02 s. After co-simulation, the voltage and current data on the output side of the inverter can be obtained, and the voltage and current data in the simulation result are imported into MATLAB for Fourier analysis, and the proportion of each harmonic can be obtained. Table 1 shows the proportion of each harmonic of the co-simulation current.

Table 1
The proportion of each harmonic of the co-simulation current

Current harmonic order Harmonic frequency (Hz) Harmonic current amplitude (A) Ratio of fundamental current (%)

3 600 0.3 0.87

5 1000 2.59 7.55

7 1400 9.97 29.07

9 1800 1.37 3.98

11 2200 2.13 2.37

13 2600 1.94 6.20

Current harmonic order	Harmonic frequency (Hz)	Harmonic current amplitude (A)	Ratio of fundamental current (%)
3	600	0.3	0.87
5	1000	2.59	7.55
7	1400	9.97	29.07
9	1800	1.37	3.98
11	2200	2.13	2.37
13	2600	1.94	6.20

It can be seen from Table 1 that when the modulation ratio M = 0.7 and the carrier ratio N = 7, the current only contain odd harmonics, and the harmonic with the highest amplitude is 7 times, which is consistent with the theoretical analysis of PWM harmonics. As the number of harmonics increases, the Bessel function decreases rapidly, the harmonic amplitude decreases gradually, and the harmonic amplitude decreases significantly after the 7th harmonic.

In order to systematically analyze the influence of harmonic current on the loss of various components of the SECT ontology, the PWM driving method and the load sine wave driving method are used for comparison. It is known that the amplitude of the current fundamental wave of the co-simulation is 34.3 A, and the amplitude of the sine wave current is set to be consistent with the co-simulation. Figure 4 is the current waveform under the PWM driving and the sine wave driving. In the PWM drive and load sine wave drive mode, the core loss value of the SECT tube changes with time, as shown in Fig. 5.

Fig. 4.

The Current waveform. (a) Current waveform in PWM drive mode. (b) Current waveform in sine wave drive mode.

Fig. 5.

Comparison of the curve of iron loss with time of the SECT tube.

As can be seen from Fig. 5, when the carrier ratio N = 7, that is, the switching frequency is 1.4 kHz, the maximum iron loss in the PWM driving mode is about 971 W, and the maximum iron loss in the sine wave driving mode is about 495 W. In addition, using Eq. (6) and Table 1, the resistance loss of the SECT system can be calculated to increase by 4.8% in the PWM drive mode compared to the sine wave drive mode. It can be seen that the load loss of the SECT ontology under PWM power supply is greatly increased.

In the PWM power supply mode, the increase in the load loss of the SECT ontology is mainly caused by the increase of eddy current loss and hysteresis loss in the SECT tube. Since the PWM inverter output waveform affects the load loss of the SECT ontology, the inverter output waveform depends on the inverter parameters such as modulation ratio and carrier ratio. Based on the above reasons, it is important to study the influence of inverter parameters on the load loss of the SECT ontology to improve the heating efficiency of the system.

3.3. Influence of inverter parameters on iron loss

In order to explore the influence of different PWM parameters on system loss, the co-simulation was used to analyze the iron loss of SECT tube under different modulation ratios and carrier ratios. According to the calculation formula of the load loss under harmonic current and Ansoft Maxwell’s field calculator, the average value of the iron loss of the SECT tube in the last cycle can be obtained [23].

In order to analyze the effect of the modulation ratio on the SECT tube loss, the sine wave frequency and the triangular carrier frequency were kept constant at 200 Hz and 1.4 kHz, respectively. The modulation ratio was changed, and the iron loss value of the SECT tube with different modulation ratios was obtained by simulation, as shown in Fig. 6.

Fig. 6.

The effect of M on iron loss.

As can be seen from Fig. 6, as the modulation ratio increases, the iron loss increases, which is because the modulation ratio increases, so that the amplitude of the output harmonic component of the inverter increases. When the modulation ratio M = 0.7, the iron loss of the SECT tube under PWM driving is 1.4 times that under the sine wave driving. Compared with M = 0.9, the iron loss under PWM driving is 1.2 times that under the sine wave driving. Therefore, when the PWM inverter is used to supply power to the system, a lower modulation ratio should be used to increase the iron consumption of the SECT ontology, so the modulation ratio of the PWM inverter circuit is selected to be M = 0.7.

In order to analyze the effect of the carrier ratio on the SECT ontology, the modulation ratio M = 0.7 is kept constant and the sine wave frequency is unchanged at 200 Hz. Changing the size of the triangular carrier frequency also changes the carrier ratio of the inverter parameters, and the switching frequency is consistent with the frequency of the triangular carrier. At present, the switching frequency of the IGBT can reach more than 20 kHz. When considering the switching loss of the power device, the setting of the switching frequency should not be too high, so the adjustment switching frequency is gradually increased from 1.4 kHz to 4.6 kHz. The carrier ratio is changed, and the iron losses of the SECT tube with different carrier ratios are obtained as shown in Fig. 7.

Fig. 7.

The effect of N on iron loss.

In Fig. 7, as the carrier ratio increases, the trend of the iron loss of the SECT tube decreases during PWM driving is consistent with the conclusion in [24], indicating the correctness of the simulation results. However, the iron loss of the SECT tube is gradually increased under the sine wave drive, which is due to the increase of the carrier ratio, so that the amplitude of the harmonic component of the inverter output increases. When the carrier ratio N = 7, the iron loss under PWM driving is 1.4 times that under the sine wave driving. Compared with N = 23, the iron loss under PWM driving is only 7.7 W higher than that under sine wave driving. Therefore, when the PWM inverter is used to supply power to the system, a lower carrier ratio should be used to increase the iron loss of the SECT ontology, so the carrier ratio of the PWM inverter circuit is selected to be N = 7.

Under the PWM driving, the loss of the SECT ontology increases, and the increased loss inevitably leads to an extra temperature rise, so the temperature of the SECT tube and the heating cable is higher in the harmonic environment than when the sine wave is driven. In order to accurately analyze the temperature rise of the SECT tube and the heating cable under harmonic current, the three-dimensional temperature field model of the SECT system is established based on the transient thermal module, and the temperature rises of the SECT ontology under the two driving conditions are compared.

4. Temperature field simulation

Through the field-circuit coupling simulation analysis, the optimal settings of the inverter circuit parameters have been determined, and the load loss of the SECT system is obtained through co-simulation. Then, using the Ansoft, Simplorer and Transient Thermal modules integrated in the workbench working platform, a multi-physics coupling simulation platform for the SECT system is established. At this time, the finite element model of the SECT ontology, the control mode of the inverter circuit and the temperature field analysis are integrated into a simulation model that is closer to the actual operation of the SECT system. Ansoft provides analysis of magnetic fields, Simplorer provides analysis of circuits, and Transient Thermal performs temperature field calculations to achieve multi-physics coupling analysis of the SECT system. Among them, the Simplorer module and the electromagnetic field module have been established in the field-circuit co-simulation model, and then the temperature field analysis of the SECT system is carried out. The finite element temperature field simulation is mainly divided into five steps: model structure establishment, mesh generation, load application, boundary conditions, and solution results [25].

4.1. Model structure establishment and meshing

The long-distance oil pipeline used in the field is up to several kilometres. In the simulation, the calculation time and computer memory are saved. The length of the pipeline is 1000 mm. In order to be closer to the actual operation, a 3D model of the SECT ontology without scaling is established by using the 3D drawing software Solidworks. The material parameters of the components of the SECT ontology are set as shown in Table 2. In addition, since the shape of each component of the SECT ontology is presented as a cylinder and the shape is relatively regular, the Sweep method can be adopted in the mesh generation. Due to the difference in volume and heat of each component, the mesh size is also different, but the basic criterion for meshing is that important components such as SECT tube and the heating cable need to be finely divided, and other components such as insulation layer are roughly divided. The mesh size set for different parts in this paper is shown in Table 3, and the mesh division is shown in Fig. 8.

Table 2
Material properties at normal temperature

Project Density Specific heat Thermal conductivity Dynamic viscosity coefficient

(kg ⋅ m⁻³) (J ⋅ kg⁻¹ ⋅ K⁻¹) (W ⋅ m⁻¹ ⋅ K⁻¹) (kg ⋅ m⁻¹ ⋅ s⁻¹)

Copper core 8933 385 400 —

20#steel 7850 468.9 51.9 —

Oil 862.4 2140 0.14 0.04

Steel pipeline 7800 500 48 —

Insulating layer 60 700 0.04 —

Project	Density	Specific heat	Thermal conductivity	Dynamic viscosity coefficient
Copper core	8933	385	400	—
20#steel	7850	468.9	51.9	—
Oil	862.4	2140	0.14	0.04
Steel pipeline	7800	500	48	—
Insulating layer	60	700	0.04	—

Table 3

The mesh size of each component

Name	Mesh size (m)	Division method
SECT tube	0.0005	Sweep
The heating cable	0.0006	Sweep
Oil pipeline	0.001	Sweep
Insulating layer	0.002	Sweep
Oil	0.003	Sweep
Other	0.005	Sweep

Fig. 8.

Meshing of the SECT system.

4.2. Load calculation and boundary condition setting

Thermal loads in transient thermal analysis include heat flow rate and internal heat generation. The load loss of the SECT tube under the harmonic current accounts for the main part of the heat generation of the SECT ontology, and the heat generation in the cell is simulated in the form of internal heat generation. Internal heat generation as a body load can only be applied to the volume, i.e. the volumetric heat generation rate. According to the harmonic loss value calculated by Ansoft Maxwell, divided by the volume of the corresponding component, the volumetric heat generation rate of each component in the simulation model can be obtained [26]. $\begin{eqnarray}\displaystyle q=P_{loss}/V. & & \displaystyle\end{eqnarray}$ (17) Among them, P _loss is the loss value of each component of the SECT ontology, W; V is the volume of each component, m³.

Boundary conditions in thermal analysis include constant temperature, thermal convection, thermal radiation, etc., where a constant temperature is applied as a degree of freedom constraint to a boundary where temperature is known. Convective heat transfer occurs through the fluid contact surface and can only be applied to the surface. The heat exchange between the inner surface of the oil pipeline and the fluid is convective. Thermal radiation relies on the surface of the object to emit visible or invisible rays for heat transfer. Compared with convective heat transfer, the radiation heat transfer between the components of the SECT is much smaller, so the heat radiation is ignored during the simulation.

The heat transfer between the inner surface of the oil pipeline and the fluid inside the pipeline is regarded as forced convection, and the convection coefficient is calculated by the experimental correlation [27]. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle h=({\lambda}/d)\cdot N_{u}=({\lambda}/d)\cdot 0.023\cdot R_{e}^{0.8}P_{r}^{1/3}\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle R_{e}=ud/({\mu}/{\rho})\end{eqnarray}$ (19) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{r}=C_{p}{\mu}/{\lambda}\end{eqnarray}$ (20) where h is the convection coefficient of crude oil, W∕(m² ⋅ K), N _u is the Nusselt number of the inner surface of the oil pipeline, 𝜆 is the thermal conductivity of crude oil, w/(m ⋅ K), d is the inner diameter of the oil pipeline, m. R _e is the fluid flow Reynolds number, P _r is the fluid Prandtl number; u is the crude oil flow rate, m/s; μ is the crude oil dynamic viscosity coefficient, kg/(m ⋅ s); C _p is the specific heat of the crude oil, J/(kg ⋅ K). The temperature of the deep seawater is maintained at 4 °C, and the temperature and flow rate of the crude oil in the pipeline are 28 °C and 0.5 m/s, respectively, corresponding to a convection coefficient of 95.47 W∕(m² ⋅ K) between the pipeline and the crude oil.

4.3. Analysis of simulation results

The transient electromagnetic field simulation was carried out based on the Ansoft software embedded in the Ansys Workbench platform. The transient electromagnetic field analysis results were imported into Transient Thermal to analyze the heating capacity of the SECT system. In the PWM driving and sine wave driving mode, the heat generation of each component of the SECT ontology can be obtained according to Eq. (17), as shown in Table 4.

Table 4
Unit heat generation rate of each component under two driving modes

Driving mode Component name Heat generation rate (×10⁴ W∕m³)

Sine wave SECT tube 1.123

The heating cable 1.78

PWM SECT tube 1.76

The heating cable 1.997

Driving mode	Component name	Heat generation rate (×10⁴ W∕m³)
Sine wave	SECT tube	1.123
	The heating cable	1.78
PWM	SECT tube	1.76
	The heating cable	1.997

Based on the heat generation rate calculated above, the mesh is divided according to the Sweep method in the Transient Thermal module, and the convection coefficient between the oil pipeline and the crude oil is given under the condition that the crude oil inlet flow rate is 0.5 m/s, thereby setting the boundary conditions. The ambient temperature is set as 4 °C to simulate a deep water environment. In the PWM and sine wave driving modes, the temperature field distribution of the SECT system can be obtained by multi-physics analysis, as shown in Fig. 9.

Fig. 9.

Temperature field distribution of the SECT system. (a) Overall temperature distribution with PWM driving. (b) Temperature distribution of the SECT tube with PWM driving. (c) Overall temperature distribution with sine wave driving. (d) Temperature distribution of the SECT tube with sine wave driving.

As can be seen from Fig. 9(a) and (c), the crude oil is heated to 28.833 °C in one minute after the PWM drive, and the temperature rise effect is remarkable compared to the sine wave power supply. However, due to the point contact, the heat transfer effect of the SECT tube is still greatly affected. Therefore, it is common to increase the thermal conductivity by adding solder joints or applying heat transfer cement in engineering. Figure 9(b) and (d) compare the temperature distribution of the SECT tube with PWM drive and sine wave drive. The temperature of the SECT tube is higher under PWM driving. This is due to the introduction of a large amount of current time harmonics under PWM driving, which causes the eddy current loss on the SECT tube to increase, resulting in an increase in temperature. It can be seen that the existence of current time harmonics cannot be ignored, and it is necessary to make full use of the N-order components of current harmonics to achieve efficient operation of the SECT system.

In the electric heating oil and gas gathering pipeline operating in the oil field, because there is no corresponding theoretical calculation as the basis, the electric heating pipeline has the situation that the design heating power is too large. In this paper, the field-circuit coupling analysis method of the SECT system is closer to the actual operating state, and the influence of current time harmonics is fully considered to calculate the required heating power. Not only can the initial investment cost be reduced, but also the operation management is more scientific and reasonable.

5. Conclusion

In order to reduce the initial investment of the equipment and facilitate the scientific management of the SECT system, the calculation of the temperature field and heating power of the SECT system in the electric heating design of the oil and gas gathering pipeline is particularly urgent. In this paper, a field-circuit co-simulation method of PWM inverter circuit and SECT ontology coupling is proposed to simulate the actual operation of the SECT system. Driven by the inverter circuit, it will affect the current, voltage, loss, etc. in the SECT. Firstly, the harmonic distribution under the PWM driving mode is analyzed, and the load loss of the SECT ontology under the harmonic current is derived. Then, the parameters of current, voltage, loss and other parameters of the electric heating body are compared by PWM driving method and sinusoidal driving method. The results show that the parameters of PWM driving mode increase to different degrees. At the same time, the influence of inverter parameters on the load loss of the SECT ontology is studied. The iron losses of the SECT tube are analyzed under different modulation ratios and carrier ratios, which provide a theoretical basis for the determination of the parameters of the inverter. Finally, the temperature field simulation model of the SECT system is established by using the three-dimensional temperature field software. The electromagnetic field analysis results obtained by the PWM driving method and the sine wave driving method are used as the heat source, and the thermal analysis module is respectively introduced into the multi-physics coupling simulation. Thereby, the temperature rise distributions of the components of the SECT ontology under the two driving modes are obtained. According to the analysis, the overall temperature rise of the system under the PWM driving mode is remarkable. Therefore, in the actual operation, the existence of current time harmonics cannot be ignored, and the combined simulation results are used as the theoretical basis to reverse the heating power of the electric heating oil pipeline to achieve the purpose of energy saving and high efficiency heating.

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Loss and temperature rise of the skin effect electric heat tracing system

Abstract

Keywords

1. Introduction

2. Theory and methods

2.1. Harmonic analysis of PWM inverter circuit

3.1. Establishment of co-simulation model

Table 1 The proportion of each harmonic of the co-simulation current Current harmonic order Harmonic frequency (Hz) Harmonic current amplitude (A) Ratio of fundamental current (%) 3 600 0.3 0.87 5 1000 2.59 7.55 7 1400 9.97 29.07 9 1800 1.37 3.98 11 2200 2.13 2.37 13 2600 1.94 6.20

4.1. Model structure establishment and meshing

Table 4 Unit heat generation rate of each component under two driving modes Driving mode Component name Heat generation rate (×104 W∕m3) Sine wave SECT tube 1.123 The heating cable 1.78 PWM SECT tube 1.76 The heating cable 1.997

References

Table 1
The proportion of each harmonic of the co-simulation current

Current harmonic order Harmonic frequency (Hz) Harmonic current amplitude (A) Ratio of fundamental current (%)

3 600 0.3 0.87

5 1000 2.59 7.55

7 1400 9.97 29.07

9 1800 1.37 3.98

11 2200 2.13 2.37

13 2600 1.94 6.20

Table 4
Unit heat generation rate of each component under two driving modes

Driving mode Component name Heat generation rate (×10⁴ W∕m³)

Sine wave SECT tube 1.123

The heating cable 1.78

PWM SECT tube 1.76

The heating cable 1.997