2507 stainless steel coil tube chemical component ,Equivalent Thermal Network Simulation Study of a Rare Earth Giant Magnetostrictive Transducer

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Critical to the design of the giant magnetostrictive transducer (GMT) is fast and accurate analysis of the temperature distribution. Thermal network modeling has the advantages of low computational cost and high accuracy and can be used for GMT thermal analysis. However, existing thermal models have limitations in describing these complex thermal regimes in GMT: most studies focus on stationary states that cannot capture temperature changes; It is generally assumed that the temperature distribution of giant magnetostrictive (GMM) rods is uniform, but the temperature gradient across the GMM rod is very significant due to poor thermal conductivity, the non-uniform loss distribution of the GMM is rarely introduced into the thermal model. Therefore, by comprehensively considering the above three aspects, this document establishes the GMT Transitional Equivalent Heat Network (TETN) model. First, based on the design and principle of operation of the longitudinal vibratory HMT, a thermal analysis is carried out. On this basis, the heating element model is established for the HMT heat transfer process and the corresponding model parameters are calculated. Finally, the accuracy of the TETN model for transducer temperature spatiotemporal analysis is verified by simulation and experiment.
The giant magnetostrictive material (GMM), namely terfenol-D, has the advantages of large magnetostriction and high energy density. These unique properties can be used to develop giant magnetostrictive transducers (GMTs) that can be used in a wide range of applications such as underwater acoustic transducers, micromotors, linear actuators, etc. 1,2.
Of particular concern is the potential for overheating of subsea GMTs, which, when operated at full power and for long periods of excitation, can generate significant amounts of heat due to their high power density3,4. In addition, due to the large coefficient of thermal expansion of GMT and its high sensitivity to external temperature, its output performance is closely related to temperature5,6,7,8. In technical publications, GMT thermal analysis methods can be divided into two broad categories9: numerical methods and lumped parameter methods. The finite element method (FEM) is one of the most commonly used numerical analysis methods. Xie et al. [10] used the finite element method to simulate the distribution of heat sources of a giant magnetostrictive drive and realized the design of the temperature control and cooling system of the drive. Zhao et al. [11] established a joint finite element simulation of a turbulent flow field and a temperature field, and built a GMM intelligent component temperature control device based on the results of the finite element simulation. However, FEM is very demanding in terms of model setup and calculation time. For this reason, FEM is considered an important support for offline calculations, usually during the converter design phase.
The lumped parameter method, commonly referred to as the heat network model, is widely used in thermodynamic analysis due to its simple mathematical form and high calculation speed12,13,14. This approach plays an important role in eliminating the thermal limitations of engines 15, 16, 17. Mellor18 was the first to use an improved thermal equivalent circuit T to model the engine heat transfer process. Verez et al. 19 created a three-dimensional model of the thermal network of a permanent magnet synchronous machine with axial flow. Boglietti et al.20 proposed four thermal network models of varying complexity to predict short-term thermal transients in stator windings. Finally, Wang et al.21 established a detailed thermal equivalent circuit for each PMSM component and summarized the thermal resistance equation. Under nominal conditions, the error can be controlled within 5%.
In the 1990s, the heat network model began to be applied to high-power low-frequency converters. Dubus et al.22 developed a heat network model to describe stationary heat transfer in a double-sided longitudinal vibrator and class IV bend sensor. Anjanappa et al.23 performed a 2D stationary thermal analysis of a magnetostrictive microdrive using a thermal network model. To study the relationship between thermal strain of Terfenol-D and GMT parameters, Zhu et al. 24 established a steady state equivalent model for thermal resistance and GMT displacement calculation.
GMT temperature estimation is more complex than engine applications. Due to the excellent thermal and magnetic conductivity of the materials used, most engine components considered at the same temperature are usually reduced to a single node13,19. However, due to the poor thermal conductivity of HMMs, the assumption of a uniform temperature distribution is no longer correct. In addition, HMM has a very low magnetic permeability, so the heat generated by magnetic losses is usually non-uniform along the HMM rod. In addition, most of the research is focused on steady-state simulations that do not account for temperature changes during GMT operation.
In order to solve the above three technical problems, this article uses the GMT longitudinal vibration as the object of study and accurately models various parts of the transducer, especially the GMM rod. A model of a complete transitional equivalent heat network (TETN) GMT has been created. A finite element model and experimental platform were built to test the accuracy and performance of the TETN model for transducer temperature spatiotemporal analysis.
The design and geometric dimensions of the longitudinally oscillating HMF are shown in Fig. 1a and b, respectively.
Key components include GMM rods, field coils, permanent magnets (PM), yokes, pads, bushings, and belleville springs. The excitation coil and PMT provide the HMM rod with an alternating magnetic field and a DC bias magnetic field, respectively. The yoke and body, consisting of a cap and sleeve, are made of DT4 soft iron, which has a high magnetic permeability. Forms a closed magnetic circuit with the GIM and PM rod. The output stem and pressure plate are made of non-magnetic 304 stainless steel. With belleville springs, a stable prestress can be applied to the stem. When an alternating current passes through the drive coil, the HMM rod will vibrate accordingly.
On fig. 2 shows the process of heat exchange inside the GMT. GMM rods and field coils are the two main sources of heat for GMTs. The serpentine transfers its heat to the body by air convection inside and to the lid by conduction. The HMM rod will create magnetic losses under the action of an alternating magnetic field, and heat will be transferred to the shell due to convection through the internal air, and to the permanent magnet and yoke due to conduction. The heat transferred to the case is then dissipated to the outside by convection and radiation. When the heat generated is equal to the heat transferred, the temperature of each part of the GMT reaches a steady state.
The process of heat transfer in a longitudinally oscillating GMO: a – heat flow diagram, b – main heat transfer paths.
In addition to the heat generated by the exciter coil and HMM rod, all components of a closed magnetic circuit experience magnetic losses. Thus, the permanent magnet, yoke, cap and sleeve are laminated together to reduce the magnetic loss of the GMT.
The main steps in building a TETN model for GMT thermal analysis are as follows: first group components with the same temperatures together and represent each component as a separate node in the network, then associate these nodes with the appropriate heat transfer expression. heat conduction and convection between nodes. In this case, the heat source and the heat output corresponding to each component are connected in parallel between the node and the common zero voltage of the earth to build an equivalent model of the heat network. The next step is to calculate the parameters of the thermal network for each component of the model, including thermal resistance, heat capacity and power losses. Finally, the TETN model is implemented in SPICE for simulation. And you can get the temperature distribution of each component of GMT and its change in the time domain.
For the convenience of modeling and calculation, it is necessary to simplify the thermal model and ignore the boundary conditions that have little effect on the results18,26. The TETN model proposed in this article is based on the following assumptions:
In GMT with randomly wound windings, it is impossible or necessary to simulate the position of each individual conductor. Various modeling strategies have been developed in the past to model heat transfer and temperature distribution within windings: (1) compound thermal conductivity, (2) direct equations based on conductor geometry, (3) T-equivalent thermal circuit29.
Composite thermal conductivity and direct equations can be considered more accurate solutions than the equivalent circuit T, but they depend on several factors, such as material, conductor geometry and the volume of residual air in the winding, which are difficult to determine29. On the contrary, the T-equivalent thermal scheme, although an approximate model, is more convenient30. It can be applied to the excitation coil with longitudinal vibrations of the GMT.
The general hollow cylindrical assembly used to represent the exciter coil and its T-equivalent thermal diagram, obtained from the solution of the heat equation, are shown in fig. 3. It is assumed that the heat flux in the excitation coil is independent in the radial and axial directions. The circumferential heat flux is neglected. In each equivalent circuit T, two terminals represent the corresponding surface temperature of the element, and the third terminal T6 represents the average temperature of the element. The loss of the P6 component is entered as a point source at the average temperature node calculated in the “Field coil heat loss calculation”. In the case of non-stationary simulation, the heat capacity C6 is given by the equation. (1) is also added to the Average temperature node.
Where cec, ρec and Vec represent the specific heat, density and volume of the excitation coil, respectively.
In table. 1 shows the thermal resistance of the T-equivalent thermal circuit of the excitation coil with length lec, thermal conductivity λec, outer radius rec1 and inner radius rec2.
Exciter coils and their T-equivalent thermal circuits: (a) usually hollow cylindrical elements, (b) separate axial and radial T-equivalent thermal circuits.
The equivalent circuit T has also shown to be accurate for other cylindrical heat sources13. Being the main heat source of the GMO, the HMM rod has an uneven temperature distribution due to its low thermal conductivity, especially along the axis of the rod. On the contrary, radial inhomogeneity can be neglected, since the radial heat flux of the HMM rod is much less than the radial heat flux31.
To accurately represent the level of axial discretization of the rod and obtain the highest temperature, the GMM rod is represented by n nodes uniformly spaced in the axial direction, and the number of nodes n modeled by the GMM rod must be odd. The number of equivalent axial thermal contours is n T figure 4.
To determine the number of nodes n used to model the GMM bar, the FEM results are shown in fig. 5 as a reference. As shown in fig. 4, the number of nodes n is regulated in the thermal scheme of the HMM rod. Each node can be modeled as a T-equivalent circuit. Comparing the results of the FEM, from Fig. 5 shows that one or three nodes cannot accurately reflect the temperature distribution of the HIM rod (about 50 mm long) in the GMO. When n is increased to 5, the simulation results improve significantly and approach FEM. Increasing n further also gives better results at the cost of longer computation time. Therefore, in this article, 5 nodes are selected for modeling the GMM bar.
Based on the comparative analysis carried out, the exact thermal scheme of the HMM rod is shown in Fig. 6. T1 ~ T5 is the average temperature of five sections (section 1 ~ 5) of the stick. P1-P5 respectively represent the total thermal power of the various areas of the rod, which will be discussed in detail in the next chapter. C1~C5 are the heat capacity of different regions, which can be calculated by the following formula
where crod, ρrod and Vrod denote the specific heat capacity, density and volume of the HMM rod.
Using the same method as for the exciter coil, the heat transfer resistance of the HMM rod in Fig. 6 can be calculated as
where lrod, rrod and λrod represent the length, radius and thermal conductivity of the GMM rod, respectively.
For the longitudinal vibration GMT studied in this article, the remaining components and internal air can be modeled with a single node configuration.
These areas can be considered as consisting of one or more cylinders. A purely conductive heat exchange connection in a cylindrical part is defined by the Fourier heat conduction law as
Where λnhs is the thermal conductivity of the material, lnhs is the axial length, rnhs1 and rnhs2 are the outer and inner radii of the heat transfer element, respectively.
Equation (5) is used to calculate the radial thermal resistance for these areas, represented by RR4-RR12 in Figure 7. At the same time, Equation (6) is used to calculate the axial thermal resistance, represented from RA15 to RA33 in Figure 7.
The heat capacity of a single node thermal circuit for the above area (including C7–C15 in Fig. 7) can be determined as
where ρnhs, cnhs, and Vnhs are the length, specific heat, and volume, respectively.
The convective heat transfer between the air inside the GMT and the surface of the case and the environment is modeled with a single thermal conduction resistor as follows:
where A is the contact surface and h is the heat transfer coefficient. Table 232 lists some typical h used in thermal systems. According to Table. 2 heat transfer coefficients of thermal resistances RH8–RH10 and RH14–RH18, representing the convection between the HMF and the environment in fig. 7 are taken as a constant value of 25 W/(m2 K). The remaining heat transfer coefficients are set equal to 10 W/(m2 K).
According to the internal heat transfer process shown in Figure 2, the complete model of the TETN converter is shown in Figure 7.
As shown in fig. 7, the GMT longitudinal vibration is divided into 16 knots, which are represented by red dots. The temperature nodes depicted in the model correspond to the average temperatures of the respective components. Ambient temperature T0, GMM rod temperature T1~T5, exciter coil temperature T6, permanent magnet temperature T7 and T8, yoke temperature T9~T10, case temperature T11~T12 and T14, indoor air temperature T13 and output rod temperature T15. In addition, each node is connected to the thermal potential of the ground through C1 ~ C15, which represent the thermal capacity of each area, respectively. P1~P6 is the total heat output of GMM rod and exciter coil respectively. In addition, 54 thermal resistances are used to represent the conductive and convective resistance to heat transfer between adjacent nodes, which were calculated in the previous sections. Table 3 shows the various thermal characteristics of the converter materials.
Accurate estimation of loss volumes and their distribution is critical to performing reliable thermal simulations. The heat loss generated by the GMT can be divided into the magnetic loss of the GMM rod, the Joule loss of the exciter coil, the mechanical loss, and the additional loss. The additional losses and mechanical losses taken into account are relatively small and can be neglected.
The ac excitation coil resistance includes: the dc resistance Rdc and the skin resistance Rs.
where f and N are the frequency and number of turns of the excitation current. lCu and rCu are the inside and outside radii of the coil, the length of the coil, and the radius of the copper magnetic wire as defined by its AWG (American Wire Gauge) number. ρCu is the resistivity of its core. µCu is the magnetic permeability of its core.
The actual magnetic field inside the field coil (solenoid) is not uniform along the length of the rod. This difference is especially noticeable due to the lower magnetic permeability of the HMM and PM rods. But it is longitudinally symmetrical. The distribution of the magnetic field directly determines the distribution of magnetic losses of the HMM rod. Therefore, to reflect the real distribution of losses, a three-section rod, shown in Figure 8, is taken for measurement.
The magnetic loss can be obtained by measuring the dynamic hysteresis loop. Based on the experimental platform shown in Figure 11, three dynamic hysteresis loops were measured. Under the condition that the temperature of the GMM rod is stable below 50°C, the programmable AC power supply (Chroma 61512) drives the field coil in a certain range, as shown in Figure 8, the frequency of the magnetic field generated by the test current and the resulting magnetic flux density are calculated by integrating voltage induced in the induction coil connected to the GIM rod. The raw data was downloaded from the memory logger (MR8875-30 per day) and processed in MATLAB software to obtain the measured dynamic hysteresis loops shown in Fig. 9.
Measured dynamic hysteresis loops: (a) section 1/5: Bm = 0.044735 T, (b) section 1/5: fm = 1000 Hz, (c) section 2/4: Bm = 0.05955 T, (d ) section 2/4: fm = 1000 Hz, (e) section 3: Bm = 0.07228 T, (f) section 3: fm = 1000 Hz.
According to literature 37, the total magnetic loss Pv per unit volume of HMM rods can be calculated using the following formula:
where ABH is the measurement area on the BH curve at the magnetic field frequency fm equal to the excitation current frequency f.
Based on the Bertotti loss separation method38, the magnetic loss per unit mass Pm of a GMM rod can be expressed as the sum of the hysteresis loss Ph, the eddy current loss Pe and the anomalous loss Pa (13):
From an engineering perspective38, anomalous losses and eddy current losses can be combined into one term called total eddy current loss. Thus, the formula for calculating losses can be simplified as follows:
in the equation. (13)~(14) where Bm is the amplitude of the magnetic density of the exciting magnetic field. kh and kc are the hysteresis loss factor and the total eddy current loss factor.