Axial Conduction Research Articles

Axial and radial thermal conductivity measurement of 18,650 Lithium-ion battery

Thermal conductivities of lithium-ion batteries are critical for the thermal management of battery packs. In this work, a novel method and experimental apparatuses are developed to measure the axial and radial thermal conductivities of the 18,650 LiNiCoAlO2 (NCA) lithium-ion battery. For the axial conductivity measurement, the one-dimensional steady conduction principle considering uniform heat loss is used, while the combined method of experiment and computational fluid dynamics (CFD) simulation is used for the radial conductivity measurement. By measuring the conductivity of the stainless steel cylinder, the errors are estimated to be less than 1.45 % and 1.37 % for the axial and radial measurement methods respectively. The axial and radial conductivities of the battery are measured at different open circuit voltage (OCV) and temperatures. Under the experimental conditions, the axial conductivity of the battery ranges from 10.26 to 14.20 W/m/K, and the axial conductivity generally increases with OCV and temperature. The radial conductivity ranges from 2.14 to 2.87 W/m/K if the battery core and the steel shell are considered as a whole; and the radial conductivity of the core ranges from 1.84 to 2.55 W/m/K if the core and the shell are considered separately. In the normal OCV range, there is no obvious relationship among the radial conductivity, temperature and OCV, whether the core and the shell are treated separately or not. This work may contribute towards improved thermal simulation of Li-ion cells.

Magnetic field optimization of hall thruster with large height-radius ratio for high specific impulse operation

Hall thrusters with large height-radius ratio not only have incalculable application values in reducing the volume and weight of thrusters, but also have the potential advantages of higher discharge performance and longer service life. However, the lower propellant density in the main ionization zone and the higher electron temperature in the channel aggravate the loss of propellant and current under high voltage, and significantly reduce the discharge efficiency under high specific impulse mode. To improve the discharge performance of Hall thrusters with large height-radius ratio under high voltage, an optimization scheme of internally loaded magnetic field was proposed in this work. The simulation results show that under the internally loaded magnetic field, both the ionization zone and the acceleration zone move toward the inside of the channel. Although the ion loss on the walls increases, the higher propellant density at the channel upstream greatly promotes the increase of ionization rate and significantly improves the propellant utilization efficiency. The second zone crossed by magnetic field lines in the channel can be established by the internally loaded magnetic field, which enhances the magnetic field intensity on the inner and outer walls, and reduces the electron temperature near the channel outlet significantly. So that the axial conduction of electrons is effectively restrained and the current utilization efficiency is greatly improved. With the introduction of internally loaded magnetic field, the total efficiency of HEP-1350PM can be increased by 7.2% at 400 V. Moreover, the performance optimization effect brought by the internally loaded magnetic field will be gradually amplified with the increase of discharge voltage, which makes the Hall thruster with large height-radius ratio expected to achieve high-efficiency discharge under higher specific impulse.

Thermal Transport in Poly(p-phenylene): Anomalous Dimensionality Dependence and Role of π-π Stacking.

For heat conduction along polymer chains, a decrease in the axial thermal conductivity often occurs when the polymer structure changes from one-dimensional (1D) to three-dimensional (3D). For example, a single extended aliphatic chain (e.g., polyethylene or poly(dimethylsiloxane)) usually has a higher axial thermal conductivity than its double-chain or crystal counterparts because coupling between chains induces strong interchain anharmonic scatterings. Intuitively, for chains with an aromatic backbone, the even stronger π-π stacking, once formed between chains, should enhance thermal transport across chains and suppress the thermal conductivity along the chains. However, we show that this trend is the opposite in poly(p-phenylene) (PPP), a typical chain with an aromatic backbone. Using molecular dynamics simulations, we found that the axial thermal conductivity of PPP chains shows an anomalous dimensionality dependence where the thermal conductivity of double-chain and 3D crystal structures is higher than that of a 1D single chain. We analyzed the probability distribution of dihedral angles and found that π-π stacking between phenyl rings restricts the free rotation of phenyl rings and forms a long-range order along the chain, thus enhancing thermal transport along the chain direction. Though possessing a stronger bonding strength and stabilizing the multiple-chain structure, π-π stacking does not lead to a higher interchain thermal conductance between phenyl rings compared with that between aliphatic chains. Our simulation results on the effects of π-π stacking provide insights to engineer thermal transport in polymers at the molecular level.

A Review of the Complex Flow and Heat Transfer Characteristics in Microchannels.

Continuously improving heat transfer efficiency is one of the important goals in the field of energy. Compact heat exchangers characterized by microscale flow and heat transfer have successfully provided solutions for this purpose. However, as the characteristic scale of the channels decreases, the flow and heat transfer characteristics may differ from those at the conventional scale. When considering the influence of scale effects and changes in special fluid properties, the flow and heat transfer process becomes more complex. The conclusions of the relevant studies have not been unified, and there are even disagreements on some aspects. Therefore, further research is needed to obtain a sufficient understanding of flow structure and heat transfer mechanisms in microchannels. This article systematically reviews the research about microscale flow and heat transfer, focusing on the flow and heat transfer mechanisms in microchannels, which is elaborated in the following two perspectives: one is the microscale single-phase flow and heat transfer that only considers the influence of scale effects, the other is the special heat transfer phenomena brought about by the coupling of microscale flow with special fluids (fluid with phase change (pseudophase change)). The microscale flow and heat transfer mechanisms under the influence of multiple factors, including scale effects (such as rarefaction, surface roughness, axial heat conduction, and compressibility) and special fluids, are investigated, which can meet the specific needs for the design of various microscale heat exchangers.

Water flow within and towards plant roots—a new concurrent solution

Abstract Various analytical models that calculate the water flow either around or inside plant roots are available, but a combined analytical solution has not yet been derived. The classical solution of Landsberg and Fowkes for water flow within a root relates the second derivative of xylem water potential to the radial water influx term. This term can be linked to well-known steady state or steady rate-based solutions for computing soil water fluxes around roots. While neglecting lateral fluxes between local depletion zones around roots, we use this link to construct a system of continuous equations that combine root internal and external water flow that can be solved numerically for two boundary conditions (specified root collar water potential and zero distal influx) and one constraint (mean bulk matric flux potential). Furthermore, an iterative matrix solution for the stepwise analytical solution of homogeneous root segments is developed. Besides accounting for soil water flow iteratively, the intrinsic effect of variable axial conductance is accounted simultaneously. The reference and the iterative matrix solution are compared for different types of corn roots, soil textures and soil dryness states, which showed good correspondence. This also revealed the importance of accounting for variable axial conductance in more detail. The proposed reference solution can be used for the evaluation of different morphological and hydraulic designs of single or multiple parallel-connected roots operating in targeted soil environments. Some details of the iterative matrix solution may be adopted in analytical–numerical solutions of water flow in complex root systems.

End‐winding wire‐insulation heat conduction model for precise thermal predictions of permanent magnet synchronous machines

AbstractAccurate thermal calculations of end‐winding temperature rises are the key issue in designing and developing cooling systems of permanent magnet synchronous machines (PMSMs). Due to the complex connections of the end‐windings, it is challenging to precisely predict the heat dissipation capacities through a traditional wire‐insulation equivalent model. Numerical simulations and lumped parameter thermal networks (LPTN) are used by the authors to develop thermal models that illustrate the end‐winding properties of a 2.1‐kW PMSM. As multi‐strand end wires are bent and bound together by ribbons, they have thermal interrelations which can hardly be considered in LPTN models. An equivalent coefficient enhancing the axial heat conduction capacities of end‐windings is proposed for LPTN calculations to model the aforementioned thermal interrelations with circumferential symmetric structures. To correctly model the mutual thermal coupling between various strands of end‐windings, the thermal conductivity of the equivalent insulation sheets between the multi‐strand end‐windings is determined for exact numerical calculations. The external insulation sheet thickness is calculated based on the equivalent thermal conductivity. After developing heat conduction models for LPTN and numerical simulations, LPTN and FEA are used to determine the PMSM's temperature rise characteristics. To verify the approaches, the computation findings are contrasted with experimental ones.

Multi-Objective Optimization of a Bi-Metal High-Temperature Recuperator for Application in Concentrating Solar Power

Abstract Supercritical CO2 closed Brayton cycles are a major candidate for future power cycle designs in concentrating solar power applications, with high-temperature recuperators playing an essential role in realizing their high thermal efficiency. Printed circuit heat exchangers (PCHEs) are often chosen for this role due to their thermal-hydraulic and mechanical performance at high temperatures and pressures, all while remaining compact. However, PCHEs can be costly because of the high-performance materials demanded in these applications, and the heat exchanger internal geometry is restricted by their manufacturing process. Additively manufactured heat exchangers can address both of these shortcomings. This work proposes a modular bi-metal high-temperature recuperator with integrated headers to be produced with additive manufacturing. Beginning with existing PCHE channel geometries, a 1D heat exchanger model is developed. Then, multi-objective optimization is used to maximize the heat transfer effectiveness of a lab-scale device while limiting its size. Two distinct channel geometries emerge from the optimization. Optimal designs achieve up to 88% effectiveness with negligible pressure drop. Deterioration of effectiveness due to axial conduction of heat in the heat exchanger walls is found to be a notable problem for lab-scale PCHEs, and the optimal designs obtained here minimize its detrimental effects. A sensitivity analysis reveals that the effectiveness of the recuperator is much less sensitive to variation in mass flowrate in off-design operation when axial conduction is significant, while increasing the length of the device easily increases effectiveness.

Thermal parameters of cylindrical power batteries: Quasi-steady state heat guarding measurement and thermal management strategies

In this work, a new quasi-steady state heat guarding measurement method for the thermophysical parameters of cylindrical batteries is proposed. The effectiveness of the heat guarding method is evaluated by the finite element analysis and the measurement of a standard ANSI 304 stainless steel sample. The heat loss is minimized to 2.4% maximum for cylindrical batteries under test based on numerical simulation. With the optimization of the test configuration, the experimental characterization of batteries was conducted to determine the thermal parameters accurately. The experimental results show that the axial thermal conductivity of 18650 and 21700 batteries are within 11.8–15.4 W·m−1 °C−1 and 12.6–16.7 W·m−1 °C−1, respectively, which are much lower than the material test values by the laser flash method. Axial thermal conductivity increases with temperature and SOC, and the specific heat increases linearly with temperature but varies little with SOC. This study demonstrates a fast, cost-effective and nondestructive way to obtain the axial thermal conductivity and specific heat of cylindrical batteries simultaneously and accurately. The subsequent thermal analysis on the thermal design of cylindrical batteries is also conducted based on the measured thermal parameters, which facilitate appropriate thermal management strategies for different types of cylindrical batteries.

Exploring the effect of axial wall conduction on the heat transfer mechanism of hourglass microchannels: Design guidelines for maintaining a constant wall temperature

To accurately estimate the heat transfer mechanisms of microscale thermal devices, it is essential to quantify the effects of axial wall conduction, which can significantly alter the thermal boundary conditions. This study focuses on exploring the effect of axial wall conduction on the thermal mechanism of hourglass microchannels, a topology that has proven useful in several engineering applications. However, there is a lack of research on the impact of axial wall conduction on the heat transfer mechanism of hourglass microchannels, and their potential for maintaining constant temperatures in biological applications has not been explored. To address this issue, a three-dimensional numerical analysis is conducted to investigate the impact of various design variables on the thermal mechanics of hourglass microchannels. An experiment has also been performed to validate the numerical approach employed in the study. The results show that the axial wall conduction effect is insignificant for modified Maranzana number below 0.01. It is further noted that the Nusselt number attains a maximum at an intermediate thermal conductivity ratio. This ratio ranges from 27 to 246 and is moderately dependent on the convergence-divergence angle, width ratio, and thickness ratio. Interestingly, an hourglass shape can maintain a near-isothermal boundary condition with a temperature difference of 1.4K at a lower width ratio. This work also establishes the design criteria for achieving a nearly isothermal condition within prescribed limits for important parameters, such as convergence-divergence angle (8°-16°), width ratio (1–2), thermal conductivity ratio (>646), thickness ratio (>10), and Reynolds number (>300). The established design criteria will pave the way for developing lab-on-chip devices for biological applications, such as PCR, that require the maintenance of constant wall temperature.

Design of Bioinspired Highly Aligned Bamboo-Mimetic Metamaterials With Structural and Functional Anisotropy

Metamaterials have received widespread attention for their unique properties and broad application prospects. With inspiration from bamboo, metamaterials can be rationally designed to achieve anisotropic functionality. Using a combination of 3D printing and subsequent hot pressing of dual-curing composites, we describe a simple method for creating well-aligned SiC whisker (SiCw) composites that mimic bamboo and have achievable anisotropic thermal conductivity (TC), dielectric properties (DP), and flashover performances (FP). The anisotropic structure is proved by scanning electron microscope and X-ray diffraction observations. At the low loading of 8 vol% SiCw, the axial thermal conductivity of the structured composites can reach 0.655 W m-1 K-1, whereas the radial thermal conductivity is only 0.281 W m-1 K-1, with a high anisotropic TC ratio of 2.33. Strong anisotropy in the dielectric constant was also observed with the permittivity of 134 along the axial direction versus that of 65 along the radial direction at 1 kHz, with a high anisotropic DP ratio of 2.06. Notably, metamaterials exhibit significant anisotropy in both their surface conductivity and flashover voltage, with a high conductivity ratio of ~10,000. The flashover voltage in the diametrical direction is 6 times that in the axial direction and 60% higher than that of the neat resin. This study shows that this approach is anticipated to offer a new path for the creation of high-performance functionally anisotropic materials to meet the diversified requirements of modern electronics.

Transferring heat accumulated by asphalt pavement from inside to outside through carbon fibers

On the Qinghai-Tibet Plateau, intense solar irradiation will cause serious heat accumulation in asphalt pavement. Downward heat transfer will disturb the permafrost under subgrade, thus leading to a series of subgrade and pavement diseases. This makes construction of asphalt pavement in the permafrost regions of the Qinghai-Tibet Plateau a world-class challenge. This paper proposes for the first time a transversal heat transfer channel in asphalt pavement, which can quickly transfer heat accumulated in pavement to road shoulders. Finite element software ABAQUS was used to simulate the rutting plate with high axial thermal conductivity carbon fibers inserted transversely. Under the constant heat load of 500 W/m2, the heat flux near the carbon fibers showed the characteristic of convergence. The maximum transverse heat flux in the carbon fibers reached 4.97×107W/m2, which was 14 times that in the asphalt mixture. Meanwhile, the maximum vertical heat flux in the carbon fibers reduced to 2.62×106W/m2, which was 12 % less than the asphalt mixture. The effective thermal conductivity of composite specimens with different carbon fiber parameters was studied through control experiments. When the axial thermal conductivity, volume fraction and number of carbon fiber bundles increased to 100 W/m·°C, 10.72 % and 4 bundles respectively, the transverse ETC of composite specimens increased to 4.05 W/m·°C, 7.09 W/m·°C and 16.18 W/m·°C respectively. They were 3.12 times, 5.45 times and 12.45 times of the control group. The radial thermal conductivity of carbon fibers has a significant effect on the vertical ETC. When it decreased to 0.2 W/m·°C, the vertical ETC decreased by 6.15 % compared with the control group. The temperature variation trend of specimens were measured by a laboratory irradiation experiment. The maximum temperature difference between the control group and the experimental group at 0 mm was 5.81 °C, and the average temperature difference within 2 h was 5.14 °C. The maximum temperature difference at 50 mm was 3.84 °C and the average temperature difference within 2 h was 3.24 °C. The results showed that the transverse heat transfer channel can reduce downward heat transfer. The application of the channels will enhance transverse heat release and relieve heat accumulation of asphalt pavement. This can create a low temperature state for pavement structure and has a positive effect on maintaining the stability of the permafrost regions on the Qinghai-Tibet Plateau.

Investigating Thermally Developing Gas Slip Flow Inside a Micro-annulus Including Viscous Dissipation and Axial Conduction Effects Using the Lattice Boltzmann Method

Investigating Thermally Developing Gas Slip Flow Inside a Micro-annulus Including Viscous Dissipation and Axial Conduction Effects Using the Lattice Boltzmann Method

Highly reliable measurement of temperature fluctuation in near-wall turbulence by a sophisticated cold-wire technique for considering the frequency response and spatial resolution

In turbulence measurement of the thermal field in a non-isothermal turbulent boundary layer, it is necessary to use a small-scale temperature sensor that can provide high spatial resolution. Thus, a fine-wire temperature sensor commonly called “cold-wire” is indispensable for capturing small-scale temperature fluctuations near the wall. However, the length-to-diameter ratio of the cold-wire need generally exceed 1000 to minimize the deterioration in dynamic response due mainly to the axial heat conduction of the thin-wire sensor. To meet such a conflicting requirement, the wire diameter of the cold-wire must be extremely thin. In the present study, we have tested experimentally the performance of several cold-wires of different geometrical features, and examined the effects of the temporal and spatial resolutions of the cold-wire on the temperature fluctuation measurements—root-mean-square (rms) values and instantaneous signal traces—in a canonical turbulent thermal boundary layer. As a result, it was found that the previously-reported response compensation technique for the cold-wire outputs enables us to realize sufficiently accurate temperature fluctuation measurement in the immediate vicinity of the wall while avoiding the deterioration in the spatial resolution of the cold-wire. Furthermore, to improve the spatial resolution of the cold-wire probe, we propose a novel temperature sensor consisting of a fine tungsten wire bent at its center to make an apex angle less than about 30 degrees and have named this sensor a “V-shaped” cold-wire. We have verified its measurement performance in reference to the results obtained by a standard cold-wire probe. As a result, it is shown that the V-shaped cold-wire naturally provides high spatial resolution especially for temperature fluctuation measurement near the wall and can take full advantage of the response compensation technique for the normal cold-wire probe.

Effect of thermal dispersion and turbulent conduction on the heat transfer behavior of coaxial pulse tube cryocooler

In this article, a numerical analysis is conducted for a miniature coaxial pulse tube cryocooler including the effect of an axial conduction enhanced term. This term is a representative of supplementary energy loss, which happens because of turbulent conduction, mixing, and thermal dispersion. The complex physical domain of cryocooler is modeled intelligently in one-dimensional model by splitting the total computational domain into two subsections. The individual subsections of the cryocooler have been discretized into a number of small control volumes. Both subsections are merged together to make a complete computational domain for the cryocooler. The conservative form of the nonlinear, coupled partial differential equations demonstrating its heat and fluid flow aspects have been discretized into algebraic equations by utilizing the finite volume technique. Final discretized algebraic equations have been solved iteratively to predict its heat and fluid flow behaviors. Suitable relaxation factors have been incorporated to accelerate the convergence of the simulation. Moreover, the variation of mean cyclic energy, enthalpy, and exergy flow have been visualized along its physical domain with and without fluid axial conduction enhancement loss processes. Influences of different length of inertance tube, outer to inner radius ratio of annular passage, and length to radius ratio of pulse tube on no-load temperature is studied for with and without fluid axial conduction heat loss.

Thermal-oxidation coupled analysis method for unidirectional fiber-reinforced C/SiC composites in air oxidizing environments below 1000 °C

The thermal analysis method for unidirectional fiber-reinforced C/SiC composites (C/SiC-UFRC) in air oxidizing environment below 1000 °C is developed, including a mathematical theoretical model and a finite element model. The mathematical model (MM) is established by combining the thermal resistance network method and the oxidation kinetics theory. The influence of oxidation characteristics on C/SiC-UFRC axial thermal conductivity is firstly introduced into MM, which provides a reference for the theoretical calculation method of C/SiC-UFRC axial thermal conductivity under oxidation conditions. The finite element model (FEM) is constructed on the representative volume element (RVE) of C/SiC-UFRC and the numerical heat transfer method. The FEM numerically simulated the microstructure in the oxidation process and firstly be applied to study the effects of oxidation on the micro temperature field and heat transfer of C/SiC-UFRC. Through the mutual confirmation of MM and FEM, the prediction of thermal conductivity of CMC in air oxidation environment is verified. The effects of oxidation time, oxidation temperature and random distribution of initial cracks are investigated to reveal their influences on the heat transfer behavior of C/SiC-UFRC.

Rheological impact on thermofluidic transport characteristics of generalized Maxwell fluids through a soft nanopore

The time-dependent electroosmotic flow (EOF) and heat transfer characteristic of a generalized Maxwell fluid through the polyelectrolyte layer (PEL) grafted nanopore are investigated while considering different permittivity between the PEL and electrolyte solution. The ion partitioning effects arise due to the different permittivity among these regions. Taking the ion partitioning effects, the analytic solution for the induced potential is established within and outside the PEL from the modified Poisson–Boltzmann equation assuming the Debye–Hückel approximation for a low surface charge. The Cauchy momentum equation with a suitable constitutive equation for fractional Maxwell fluids is derived, and the corresponding analytic solution is presented to provide the axial fluid flow distribution in the full domain. The energy fluxes that have major contributions to the energy equation mainly depend on axial conduction, convection due to electrolyte transport, and Joule heating effects for the external electric field. The analytical solutions of the energy equation for hydro-dynamically fully developed flow with constant thermophysical properties are presented to provide the temperature distribution considering constant heat flux at the nanopore wall. The influence of several important factors for characterizing heat transfer behavior is investigated in the present study. The maximum fluid velocity occurs when the permittivity between the PEL and electrolyte region is the same. The increasing values of fluid velocity imply higher convective heat transfer and make the Nusselt number higher. This study makes a conscious effort toward highlighting the modality controlling the heat transfer characteristics for the ion partitioning effects.

Improving the solidification performance of deionized water using magnetically oriented CNT by Fe3O4 nanoparticles as magnetic agents

In this work, directionally distributed Multi-walled Carbon Nanotube (MWCNT) were applied for supercooling suppression and improving the thermal conductivity of deionized water during the solidification. Specifically, Fe3O4 nanoparticles coated MWCNT are magnetized which would be directionally arranged along with the direction of extra magnetic field. A comparative study of solidification characteristics of the water based MWCNT nanofluids was carried out. The results indicated that 0.1% magnetic MWCNT nanofluid shows 31% enhancement in the thermal conductivity under the magnetic field intensity of 150 mT for fully display their axial thermal conductivity, and it exhibits ignorable reduction in the latent heat compared to the pure water. As a result, the magnetic MWCNT nanofluid has the optimal solidification characteristics under the magnetic field with 68% reduction in supercooling and 36% advance in the total solidification time. Moreover, the dynamic characteristics of solid/liquid interface were discussed, and then the growing velocity of ice crystal is obtained.

Method to Measure Radial Thermal Conductivity for Cylindrical Samples.

Anisotropy is a prevailing property in most substances in the real world. The thermal conductivity characteristic of anisotropy must be determined for utilizing geothermal resources and assessing battery performances. Most core samples were primarily obtained by drilling and intended to be cylindrical in shape, with the cores resembling quantities of familiar batteries. Although Fourier's law could be used to measure the axial thermal conductivity of square or cylindrical samples, there is still a need to develop a new method to measure the radial thermal conductivity of cylindrical samples and evaluate their anisotropy. Thus, we established a testing method for cylindrical samples using the theory of complex variable functions following the heat conduction equation and implemented a numerical simulation to determine the difference between this method and typical ones via a finite element model for various samples. Results show that the method could perfectly gauge the radial thermal conductivity of cylindrical samples with more powerful availability.

Thermal entry problem for Vόcadlo fluid model bounded within passive tube and channel with axial conduction and viscous dissipation: A Graetz-Nusselt problem

Thermal entry problem for Vόcadlo fluid model bounded within passive tube and channel with axial conduction and viscous dissipation: A Graetz-Nusselt problem

Investigation and Optimization of Fast Cold Start of 18650 Lithium-Ion Cell by Heating Film-Based Heating Method

In this paper, based on the multi-scale multi-domain (MSMD) battery modeling approach, the NTGK model was used to model the 18650 cylindrical lithium-ion single battery on the electrochemical sub-scale. The model was successful, as it was able to fit the experimental voltage and temperature of the battery at different temperatures. Lithium-ion battery discharge capacity and energy output can be improved during cold starting by preheating and insulation, as demonstrated by a comparison of the impacts of heat transfer coefficient and preheating duration at −20 °C ambient temperature. For the traditional heating method, the heating model of heating film (HF) and liquid-cooled plate (LCP) is constructed in this paper, and the heating performance of both is compared by Fluent. Analysis of the energy balance of Li-ion battery at low temperatures has been presented, showing that Li-ion battery requires a suitable start-up temperature to maximize energy output. Taking care of the problem of excessive temperature difference inside the battery due to excessive heating power, we investigated the effects of axial thermal conductivity, heating power, and heating area on the heating uniformity of the battery in this paper. Finally, a multi-stage stepped power (MSP) heating method was proposed to improve the temperature control accuracy of HF. A level orthogonal test L16(43) without interaction was designed to determine the degree of influence of each parameter on the temperature control performance and the optimal level combination, revealing that the optimized maximum temperature and temperature control rate were reduced by 4.09% and 40.53%, respectively, when compared to direct heating.