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Multifidelity Uncertainty Quantification for Focused Ultrasound Breast Cancer Therapies Using Reduced Order Models. 聚焦超声乳腺癌治疗的多保真度不确定量化使用降阶模型。
IF 1.8 0 ENGINEERING, MECHANICAL Pub Date : 2025-12-01 Epub Date: 2025-08-19 DOI: 10.1115/1.4069124
Jakob G Bates, Christopher R Dillon, Matthew R Jones, John T Tencer

Focused ultrasound (FUS) is a thermal therapy used to noninvasively destroy diseased tissues. Computational tools are being explored to plan faster, safer, and more effective focused ultrasound treatments by using simulations to predict their outcomes. For simulations to be used with confidence, the uncertainties in their predicted outcomes must be characterized. This is challenging because the simulations have a large computational cost and performing uncertainty quantification (UQ) typically requires evaluating the simulations many times. Multifidelity uncertainty quantification uses techniques that aim to reduce the computational cost of uncertainty quantification. This is done by combining results from computationally expensive and accurate high-fidelity models with lower-fidelity models that sacrifice some accuracy to reduce computational expense. In this work, a multifidelity uncertainty quantification technique using projection-based reduced order models (ROMs) as the low-fidelity model is used on thermal simulations of two focused ultrasound sonications performed as part of breast cancer treatments. The errors in mean response estimates of multiple quantities of interest (QoIs) using this multifidelity uncertainty quantification technique are compared against those using traditional Monte Carlo uncertainty quantification. The mean, standard deviation, and skewness estimated using the multifidelity and Monte Carlo techniques are fit to Pearson Type III distributions to compare their predictions of quantity of interest distributions. It is found that multifidelity uncertainty quantification predicts the mean response of the quantities of interest with up to 50% lower error while maintaining similar accuracy in distribution predictions when compared to Monte Carlo.

聚焦超声(FUS)是一种用于无创破坏病变组织的热疗法。人们正在探索计算工具,通过模拟来预测结果,来计划更快、更安全、更有效的聚焦超声治疗。为了使模拟可信地使用,必须对其预测结果中的不确定性进行表征。这是具有挑战性的,因为模拟具有很大的计算成本,并且执行不确定性量化(UQ)通常需要多次评估模拟。多保真度不确定量化采用旨在减少不确定量化计算成本的技术。这是通过将计算成本高且精确的高保真度模型的结果与牺牲一些精度以减少计算费用的低保真度模型相结合来实现的。在这项工作中,使用基于投影的降阶模型(ROMs)作为低保真度模型的多保真度不确定性量化技术用于作为乳腺癌治疗一部分的两次聚焦超声的热模拟。将该多保真度不确定量化技术与传统蒙特卡罗不确定量化技术在多个兴趣量(qoi)平均响应估计中的误差进行了比较。使用多保真度和蒙特卡罗技术估计的均值、标准差和偏度适合皮尔逊III型分布,以比较它们对利息分布数量的预测。研究发现,与蒙特卡罗相比,多保真度不确定性量化预测感兴趣数量的平均响应误差可降低50%,同时在分布预测中保持相似的精度。
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引用次数: 0
Performance Evaluation of Boiling Chamber With Microchannel Chip and Taper Microgap. 采用微通道芯片和锥形微间隙的沸腾室性能评价。
IF 1.8 0 ENGINEERING, MECHANICAL Pub Date : 2025-12-01 Epub Date: 2025-08-19 DOI: 10.1115/1.4069269
Nooruldeen E Mustafa, Satish G Kandlikar

The increasing trend of power densities in high-performance computing, driven by artificial intelligence, machine learning, and cloud computing, necessitates advanced thermal management solutions to maintain operational stability and energy efficiency. This study examines the effectiveness of cooling a 1.5 U simulated copper microchannel chip compared to a plain chip. Both chip types were tested with and without configurations for dual taper microgaps to enhance the heat transfer performance of a boiling chamber (BC). Experimental investigation was conducted using 500 μm wide × 400 μm deep microchannels separated by 200 μm fins. Varying inlet gaps (0.5-4 mm) and taper lengths (8.25 mm and 16.5 mm) with a taper angle of 3 deg were employed in dual taper configuration. Their impact on critical heat flux (CHF) and subcooled boiling dynamics was investigated. Microchannels provided considerable performance enhancement over a plain surface with or without the dual taper microgap. The findings demonstrate that smaller inlet gaps (0.5-1 mm) and longer taper lengths (16.5 mm, with central liquid inlet) significantly enhance nucleate boiling. These configurations improve vapor escape and delay CHF through subcooled boiling and submerged condensation. However, a lower CHF was noted due to vapor agglomeration within the microgap. The 80% fill ratio microchannel chip exhibited the highest CHF as subcooled boiling increased liquid replenishment and prevented vapor stagnation. Similarly, lower coolant temperatures (20-30 °C) enhanced boiling performance, where submerged condensation accelerated bubble collapse and improved heat dissipation efficiency in lower surface temperatures.

在人工智能、机器学习和云计算的推动下,高性能计算中功率密度的增长趋势需要先进的热管理解决方案来保持运行稳定性和能源效率。本研究考察了与普通芯片相比,1.5 U模拟铜微通道芯片冷却的有效性。对两种类型的芯片进行了有和没有双锥度微间隙配置的测试,以提高沸腾室(BC)的传热性能。实验研究采用500 μm宽× 400 μm深的微通道,由200 μm的鳍片分隔。采用不同的进口间隙(0.5- 4mm)和锥度长度(8.25 mm和16.5 mm),锥度角为3度,采用双锥度结构。研究了它们对临界热流密度(CHF)和过冷沸腾动力学的影响。微通道在有或没有双锥度微隙的平坦表面上提供了相当大的性能增强。研究结果表明,较小的进口间隙(0.5-1 mm)和较长的锥长(16.5 mm,中心液体进口)显著促进了核沸腾。这些结构通过过冷沸腾和浸没冷凝改善蒸汽逸出和延迟CHF。然而,由于微隙内的蒸气聚集,CHF较低。填充率为80%的微通道芯片,由于过冷沸腾增加了液体补充,防止了蒸汽停滞,CHF最高。同样,较低的冷却剂温度(20-30°C)提高了沸腾性能,其中浸没冷凝加速了气泡的破裂,并提高了较低表面温度下的散热效率。
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引用次数: 0
Nonclassical Heat Transfer and Recent Progress. 非经典传热及其最新进展。
0 ENGINEERING, MECHANICAL Pub Date : 2025-03-01 Epub Date: 2024-12-16 DOI: 10.1115/1.4066973
Chuanjin Su, Huan Wu, Lingyun Dai, Zhihan Zhang, Suixuan Li, Yongjie Hu

Heat transfer in solids has traditionally been described by Fourier's law, which assumes local equilibrium and a diffusive transport regime. However, advancements in nanotechnology and the development of novel materials have revealed non-classical heat transfer phenomena that extend beyond this traditional framework. These phenomena, which can be broadly categorized into those governed by kinetic theory and those extending beyond it, include ballistic transport, phonon hydrodynamics, coherent phonon transport, Anderson localization, and glass-like heat transfer. Recent theoretical and experimental studies have focused on characterizing these non-classical behaviors using methods such as the Boltzmann transport equation, molecular dynamics, and advanced spectroscopy techniques. In particular, the dual nature of phonons, exhibiting both particle-like and wave-like characteristics, is fundamental to understanding these phenomena. This review summarizes state-of-the-art findings in the field, highlighting the importance of integrating both particle and wave models to fully capture the complexities of heat transfer in modern materials. The emergence of new research areas, such as chiral and topological phonons, further underscores the potential for advancing phonon engineering. These developments open up exciting opportunities for designing materials with tailored thermal properties and new device mechanisms, potentially leading to applications in thermal management, energy technologies, and quantum science.

固体中的传热传统上是用傅立叶定律来描述的,它假设局部平衡和扩散传输状态。然而,纳米技术的进步和新材料的发展已经揭示了超越这一传统框架的非经典传热现象。这些现象可以大致分为动力学理论和动力学理论之外的现象,包括弹道输运、声子流体动力学、相干声子输运、安德森局域化和玻璃样传热。最近的理论和实验研究集中在利用玻尔兹曼输运方程、分子动力学和先进的光谱技术等方法来表征这些非经典行为。特别是,声子的双重性质,既表现出粒子状特征,又表现出波状特征,是理解这些现象的基础。这篇综述总结了该领域的最新研究成果,强调了整合粒子和波模型以充分捕捉现代材料中传热复杂性的重要性。新的研究领域的出现,如手性和拓扑声子,进一步强调了推进声子工程的潜力。这些发展为设计具有定制热性能和新器件机制的材料开辟了令人兴奋的机会,可能导致热管理,能源技术和量子科学的应用。
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引用次数: 0
Atmospheric Bubbling Fluidized Bed Risers: Effect of Cone Angle on Fluid Dynamics and Heat Transfer 大气气泡流化床上升器:锥角对流体动力学和传热的影响
N/A ENGINEERING, MECHANICAL Pub Date : 2024-08-08 DOI: 10.1115/1.4066182
H. J. Das, P. Mahanta
In this paper, a comparative study of fluid dynamics and thermal characteristics of sand particles has been carried out numerically and experimentally in bubbling fluidized bed risers for five-cone angles of the riser wall having 0°, 5°, 10°, 15° and 20°. An Eulerian model with a k-e turbulence model is used to explore the numerical analysis, and the findings are compared to those of the experiments. For the study, the inlet air velocity is fixed at 1.5 m/s with sand particles filled up to 30 cm to maintain bubbling conditions in the risers. The results indicate that when the cone angle increases while maintaining the amount of bed materials constant, there is a corresponding reduction in pressure drop. The expansion of particles along the riser is observed to decrease with an increase in cone angle. The radial solid volume fraction profile transforms to a U shape from the W-type profile as the cone angle increases. Correspondingly, the solid velocity is found to have an inverted U-type and W-shaped profile for the risers. The granular temperature is also found to increase with a decrease in the solid percentage at any location. The average bed temperature, interphase, and bed-to-wall heat transfer coefficient at a location of 10 cm axial height also increase with the cone angle. As a result, the conical riser, when designed with a greater cone angle, exhibits more efficiency in terms of heat transfer characteristics.
本文对冒泡流化床立管中砂粒的流体动力学和热特性进行了数值和实验对比研究,立管壁的五个锥角分别为 0°、5°、10°、15° 和 20°。数值分析采用了带有 k-e 湍流模型的欧拉模型,并将分析结果与实验结果进行了比较。在研究中,进气速度固定为 1.5 米/秒,沙粒填充高度为 30 厘米,以保持立管中的气泡条件。结果表明,在保持床层材料数量不变的情况下,当锥角增大时,压降也会相应减小。据观察,随着锥角的增大,颗粒沿立管的膨胀也随之减小。随着锥角的增大,径向固体体积分数剖面从 W 型剖面转变为 U 型。相应地,立管的固体速度也呈倒 U 型和 W 型。在任何位置,颗粒温度也会随着固体百分比的降低而升高。在轴向高度为 10 厘米的位置,床层平均温度、相间和床层到壁面的传热系数也随着锥角的增大而增大。因此,锥形立管在设计时若采用较大的锥角,则在传热特性方面会表现出更高的效率。
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引用次数: 0
Analytic Modelling of 2-D Transient Heat Conduction with Heat Source Under Mixed Boundary Constraints by Symplectic Superposition 用交映叠加法解析混合边界约束下带热源的二维瞬态热传导模型
N/A ENGINEERING, MECHANICAL Pub Date : 2024-07-22 DOI: 10.1115/1.4066031
Dian Xu, Jinbao Li, Zixuan Wang, Sijun Xiong, Qianqiang He, Rui Li
Many studies have been conducted on 2-D transient heat conduction, but analytic modelling is still uncommon for the cases with complex boundary constraints due to the mathematical challenge. With an unusual symplectic superposition method, this paper reports new analytic solutions to 2-D isotropic transient heat conduction problems with heat source over a rectangular region under mixed boundary constraints at an edge. With the Laplace transform, the Hamiltonian governing equation is derived. The applicable mathematical treatments, e.g., the variable separation and the symplectic eigenvector expansion in the symplectic space, are implemented for the fundamental solutions whose superposition yields the ultimate solutions. Benchmark results obtained by the present method are tabulated, with verification by the finite element solutions. Instead of the conventional Euclidean space, the present symplectic-space solution framework has the superiority on rigorous derivations without pre-determining solution forms, which may be extended to more issues with the complexity caused by mixed boundary constraints.
关于二维瞬态热传导的研究很多,但对于具有复杂边界约束的情况,由于数学上的挑战,解析建模仍不常见。本文采用非同寻常的交映叠加法,报告了在边缘混合边界约束下矩形区域热源的二维各向同性瞬态热传导问题的新解析解。通过拉普拉斯变换,得出了哈密顿支配方程。适用的数学处理方法,如变量分离和交映空间中的交映特征向量展开,都是针对基本解实施的,这些基本解的叠加产生了终极解。表中列出了本方法获得的基准结果,并通过有限元解进行了验证。与传统的欧几里得空间相比,本交映空间求解框架在不预先确定求解形式的情况下进行严格推导方面具有优势,可扩展到更多由混合边界约束引起的复杂问题。
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引用次数: 0
Melting Behavior Effect of MXene Nanoenhanced Phase Change Material on Energy and Exergyanalysis of Double and Triplex Tube Latent Heat Thermal Energy Storage MXene 纳米增强相变材料的熔融行为对双管和三联管潜热储能器能量和放能分析的影响
0 ENGINEERING, MECHANICAL Pub Date : 2024-07-20 DOI: 10.1115/1.4065997
Utkarsh Srivastava, Rashmi Sahoo
The impacts of melting behaviour on the thermal performance of TT-TES and DT-TES systems employing cetyl alcohol and 3% v/v. MXene nanoenhanced PCM are compared and numerically evaluated in this work. For both the DT-TES and TT-TES systems, the following were investigated in connection to melting time: system efficiency, discharged energy, heat transfer rate, exergy destruction, entropy generation number, exergetic efficiency, melting fraction, and melting temperature contours. In addition, the effect of Stefan, Rayleigh, and Nusselt numbers on Fourier numbers are compared for the DT-TES and TT-TES systems with MXene NEPCM. MXene-based nano-enhanced PCM melting in TT-TES displayed 6.53% more Stefan number than cetyl alcohol. Pure melting of MXene-based nano-enhanced PCM in a TT-TES had 4.16% higher storage exergy than cetyl alcohol. The entropy generation number of pure melting of MXene-based nano-enhanced PCM in TT-TES is 7.93% lower than that of cetyl alcohol. Pure cetyl alcohol has 76.99% optimal system efficiency at 5400 seconds melting time and MXene NEPCM 77.04% at 4800 seconds in DT-TES. The charging temperature for pure cetyl alcohol PCM in TT-TES is 0.7% lower than in DT-TES. Furthermore, pure melting of MXene-based nano-enhanced PCM in a TT-TES has 1.95% lower storage energy than cetyl alcohol. For a given volume of MXene-based nano-enhanced cetyl alcohol PCM, melting occurs more rapidly in a TT-TES system.
熔化行为对采用十六醇和 3% v/v. MXene 纳米增强 PCM 的 TT-TES 和 DT-TES 系统热性能的影响。MXene 纳米增强型 PCM 的热性能的影响进行了比较和数值评估。对于 DT-TES 和 TT-TES 系统,研究了与熔化时间相关的以下内容:系统效率、排出的能量、传热速率、放能破坏、熵生成数、放能效率、熔化率和熔化温度等值线。此外,还比较了使用 MXene NEPCM 的 DT-TES 和 TT-TES 系统的斯蒂芬数、瑞利数和努塞尔数对傅里叶数的影响。在 TT-TES 中熔化的 MXene 纳米增强 PCM 的斯特凡数比鲸蜡醇高出 6.53%。在 TT-TES 中,MXene 纳米增强型 PCM 的纯熔化比十六醇高出 4.16% 的储能。在 TT-TES 中纯熔化的 MXene 纳米增强型 PCM 的熵产生数比十六醇低 7.93%。在 DT-TES 中,纯十六醇在 5400 秒熔化时间内的最佳系统效率为 76.99%,而 MXene NEPCM 在 4800 秒熔化时间内的最佳系统效率为 77.04%。纯十六醇 PCM 在 TT-TES 中的加料温度比 DT-TES 低 0.7%。此外,在 TT-TES 中纯熔化的 MXene 纳米增强 PCM 的存储能比鲸蜡醇低 1.95%。对于一定体积的 MXene 纳米增强鲸蜡醇 PCM,在 TT-TES 系统中熔化的速度更快。
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引用次数: 0
Experimental and Numerical Evaluation of the Film Cooling Characteristics of the Multi-cavity Tip with Inclined Film Holes 带倾斜膜孔的多腔尖端薄膜冷却特性的实验和数值评估
0 ENGINEERING, MECHANICAL Pub Date : 2024-05-16 DOI: 10.1115/1.4065515
Zhe Jia, Feng Li, Weixin Zhang, Zhao Liu, Zhenping Feng
Due to the complex flow field and the considerable heat load on the turbine blade tip, film cooling is essential to protect the tip from being overheated. In this paper, an experimental work was conducted to compare the film cooling distributions of four tip structures (cavity numbers are one, two, three, and four) with two film hole configurations (perpendicular and 45 degrees inclined to the cavity floor) under three coolant blowing ratios. By using pressure sensitive paint technique, the distributions of film cooling effectiveness were measured. Moreover, a computation with careful validation was executed to obtain the cooling traces in the tip region and compare the aerodynamic performance of these multi-cavity tips. The results showed that the value and uniformity of film cooling effectiveness were improved by the inclined configuration. The tip film cooling was enhanced when using the multi-cavity tips. The aerodynamic loss of the tested tips was compared as well.
由于涡轮叶尖上存在复杂的流场和巨大的热负荷,薄膜冷却对保护叶尖不被过热至关重要。本文通过实验比较了四种叶尖结构(腔数分别为 1、2、3 和 4)和两种膜孔结构(垂直于腔底和与腔底倾斜 45 度)在三种冷却剂喷射比下的膜冷却分布。通过压敏涂料技术,测量了薄膜冷却效果的分布。此外,还进行了仔细的验证计算,以获得尖端区域的冷却轨迹,并比较这些多腔尖端的空气动力性能。结果表明,倾斜配置提高了薄膜冷却效果的数值和均匀性。使用多腔针尖时,针尖薄膜冷却效果增强。此外,还比较了测试喷嘴的空气动力损耗。
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引用次数: 0
Construction of Short-Time Heat Conduction Solutions in One-Dimensional Finite Rectangular Bodies 构建一维有限矩形体中的短时热传导解决方案
Pub Date : 2024-05-06 DOI: 10.1115/1.4065449
Filippo de Monte, K. Woodbury, Hamidreza Najafi
The concept of both penetration and deviation times for rectangular coordinates along with the principle of superposition for linear problems allow short-time solutions to be constructed for a one-dimensional rectangular finite body from the well-known solutions of a semi-infinite medium. Some adequate physical considerations due to thermal symmetries with respect to the middle plane of a slab to simulate homogeneous boundary conditions of the first and second kinds are also needed. These solutions can be applied at the level of accuracy desired (one part in 10A, with A = 2, 3, …, 15) with respect to the maximum temperature variation (that always occurs at the active surface and at the time of interest) in place of the exact analytical solution to the problem of interest.
根据矩形坐标的穿透时间和偏离时间概念,以及线性问题的叠加原理,可以从已知的半无限介质解构建一维矩形有限体的短时解。此外,还需要对板坯中间平面的热对称性进行一些充分的物理考虑,以模拟第一和第二种均质边界条件。这些解法可根据所需的精度水平(10A 中的一部分,A = 2、3、......、15)应用于最大温度变化(始终发生在活动表面和相关时间),以取代相关问题的精确分析解法。
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引用次数: 0
Experiments On Gasketed Plate Heat Exchangers with Segmented Corrugation Pattern 带分段波纹图案的板式热交换器实验
Pub Date : 2024-05-06 DOI: 10.1115/1.4065453
Matheus Strobel, L. Beckedorff, Giovani Martins, J. Oliveira, K. Paiva
Gasket plate heat exchanger (GPHE) is among the most used heat exchanger types, known for its high effectiveness and compact design. Its remarkable feature is the corrugated plate geometry, typically a Chevron pattern. This work aims to analyze another corrugation pattern, which has segments with different angles to the vertical. The strengths and weaknesses of the segmented plate are still unclear, as the studies on this pattern are scarce. To fill this gap, we experimentally assess the pressure drop and heat transfer in a GPHE composed of 31 segmented plates. The plates have four quadrants, and the combination of low-angle and high-angle plates can form up to six channel types. Pressure and temperature data are acquired in 144 sets of experiments. In the pressure drop results, we observe a considerable discrepancy between the two streams, which leads to a discussion of a relevant phenomenon: the elastic deformation of the plates. If the inner pressure of the streams is not equal, the pressure gradient causes the plates to deform and change the channel geometry. The stream with the higher pressure has its channels expanded, while the lower pressure channels will be strangled. This phenomenon is rarely reported in the literature and strongly affects the pressure drop. Moreover, we present friction factor correlations for six channel types using flow data. Based on the generalized Lévêque analogy in the heat transfer experiments, we argue that the plates' deformation also affects the heat transfer.
垫片板式热交换器(GPHE)是最常用的热交换器类型之一,以其高效和紧凑的设计而著称。其显著特点是波纹板的几何形状,通常为雪佛龙图案。本研究旨在分析另一种波纹模式,即与垂直方向不同角度的分段。由于对这种模式的研究很少,分段板的优缺点尚不清楚。为了填补这一空白,我们通过实验评估了由 31 块分段板组成的 GPHE 的压降和传热情况。板有四个象限,低角度板和高角度板的组合可形成多达六种通道类型。在 144 组实验中获得了压力和温度数据。在压降结果中,我们观察到两股气流之间存在相当大的差异,由此引出了一个相关现象的讨论:板的弹性变形。如果两股水流的内部压力不相等,压力梯度会导致板变形并改变通道的几何形状。压力较高的水流会使其通道扩张,而压力较低的通道则会被勒住。这种现象在文献中很少报道,但对压降影响很大。此外,我们还利用流量数据给出了六种水道类型的摩擦因数相关性。根据传热实验中的广义 Lévêque 类比,我们认为板的变形也会影响传热。
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引用次数: 0
Genetic Algorithm as the Solution of Non-Linear Inverse Heat Conduction Problems: a Novel Sequential Approach 遗传算法作为非线性反热传导问题的解决方案:一种新颖的序列方法
Pub Date : 2024-05-06 DOI: 10.1115/1.4065452
Dominic Allard, Hamidreza Najafi
Direct measurement of surface heat flux could be extremely challenging, or impossible, in numerous applications. In such cases, the use of temperature measurement data from sub-surface locations can facilitate the determination of surface heat flux and temperature through the solution of the inverse heat conduction problem (IHCP). Different techniques have been developed for solving IHCPs. Inspired by the filter coefficient approach, a novel method is presented in this paper for solving one-dimensional IHCPs in a domain with temperature-dependent material properties. A test case is developed in COMSOL Multiphysics where the front side of a slab is subject to known transient heat flux and the temperature distributions within the domain are calculated. The IHCP solution in the form of filter coefficients is defined and a genetic algorithm is used for the calculation of filter matrix. The number of significant filter coefficients required to evaluate surface heat flux at each time step is determined through trial and error and the optimal number is selected for evaluating the solution. The structure of the filter matrix is assessed and discussed. The resulting filter coefficients are used to evaluate the surface heat flux for several cases and the performance of the proposed approach is assessed in detail. The results showed that the presented approach is robust and can result in finding optimal filter coefficients to accurately estimate various types of surface heat flux profiles using temperature data from a limited number of time steps and in a near real-time fashion.
在许多应用中,直接测量表面热通量可能极具挑战性,甚至是不可能的。在这种情况下,利用来自地下位置的温度测量数据,可以通过解决反热传导问题(IHCP)来确定表面热通量和温度。目前已开发出不同的 IHCP 求解技术。受过滤系数方法的启发,本文提出了一种新方法,用于求解具有温度相关材料特性的域中的一维 IHCP。在 COMSOL Multiphysics 中开发了一个测试案例,板的正面受到已知瞬态热通量的影响,并计算域内的温度分布。定义了滤波系数形式的 IHCP 解决方案,并使用遗传算法计算滤波矩阵。通过试验和误差来确定评估每个时间步长的表面热通量所需的重要滤波系数数量,并选择最佳数量来评估解决方案。对滤波矩阵的结构进行了评估和讨论。得出的滤波系数用于评估几种情况下的表面热通量,并详细评估了所提方法的性能。结果表明,所提出的方法是稳健的,能够找到最佳滤波系数,从而利用有限时间步数的温度数据,以接近实时的方式准确估算各种类型的表面热通量剖面。
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引用次数: 0
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ASME journal of heat and mass transfer
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