Steady-state Temperature Distribution Research Articles

Symmetry Breaking and Modal Localization in a System of Parametrically Excited Microbeam Resonators

In this work, we study the nonlinear dynamics of parametrically excited bending vibrations of two weakly coupled beam microresonators under electrothermal excitation. A steady-state harmonic temperature distribution in the volume of the resonators in the frequency domain was obtained. A system of equations for mechanically coupled beam resonators is derived, considering the deposited particle on one of them. Using asymptotic methods of nonlinear dynamics, equations in slow variables were obtained, which were studied by methods of the theory of bifurcations. It is shown that in a perfectly symmetrical system in a certain frequency range, the effect of symmetry breaking is observed – the emergence of a mode with different amplitudes of oscillations of two beam resonators, which can be the basis for a new principle of high-precision measurements of weak disturbances of various physical natures, in particular – measurements of ultra-low masses of deposited particles.

Steady-State temperature field prediction for FPGA chips with reduced number of temperature sensors

Obtaining the steady-state temperature distribution of FPGA chip is the basis to build the thermal monitoring mechanism for reliable system performance. Existing work employs the full-coverage strategy to measure the steady-state temperature distribution of chip. However, many temperature sensors make it more complex to manage and optimize the FPGA’s logic, which would degrade the performance of FPGA. To overcome the problem, this paper proposes a method with a reduction of 60 % sensors to achieve an accuracy close to the steady-state temperature field of the FPGA obtained by the full-coverage strategy. In this method, the temperature field prediction model of the chip is established with fewer measurement data to generate the chip’s steady-state temperature distribution. Experimental results show that the proposed method performs excellently at ambient temperatures from 25 °C to 150 °C, with the average absolute error of 0.49 °C. The proposed temperature sensor placement strategy ensures the measurement accuracy and the LAB utilization rate of FPGA is only 3.06 %. Additionally, the proposed method has good generalization ability for different number and placement of hotspots.

Analytical model of the steady state temperature distribution in a single and multi-layer brake disc

This article focuses on the modeling and analyzing effort for the thermal analysis and design of disc brake for endurance braking scenario. Test cases of bike disc brake rotor are chosen as its unique double layered architecture has been recently proposed for thermal optimization especially. This work aims at the evaluation of the steady state temperature distribution during braking to be studied considering a constant rotation speed and localized heat sources at the brake pad area. The problem is solved analytically using variable separation and a double Fourier - Hankel transforms.The modelling strategy is developed for a single layered rotor as reference cases for various rotation speed and rotor thicknesses. The effect of the rotation speed is underlined using a representative constant heat exchange coefficient. For high rotation speed (greater than 50 rad s-1 for the studied geometry), an effective surface layer can be observed which limits the influence of the rotor thickness on the temperature level. The model is extended to a multi-layered material rotor architecture. The potential advantages of this composite structure for different layer thicknesses and materials are evaluated and discussed. It appears that the aluminium core provides the better results, yet the relative reduction of the temperature compared to a plain stainless steel disc is of about 1 % only.

Thermodynamic Study on the Vortex Teeth of Electric Scroll Compressors Based on Gradient Tooth Height

Through the analysis of the adhesion between the tooth head and axial clearance leakage attributed to vortex tooth shape deformation, an innovative tooth shape design concept has been introduced entitled “progressive change tooth high vortex tooth”. This unique design includes a gradual change in tooth height and an elevated vortex tooth profile, using temperature-sensitive materials to enhance the resolution of temperature loading on the vortex disc. By refining the process of resolving the temperature loading on the vortex disc, the mean temperature function of the fluid domain along the wall of the vortex tooth is calculated, and a steady-state temperature distribution model for the solid domain of the vortex tooth is formulated. Subsequently, a finite element model for the high eddy current disc is constructed using Abaqus 2021 finite element software, which facilitates the calculation of stress–strain distribution within the high eddy current disc under both gas pressure and temperature field loads. The results show that, especially under conditions of low speed and low exhaust pressure, the temperature load mainly influences the maximum deformation and stress distribution of the vortex tooth. Specifically, under the influence of heat-solid coupling, the maximum deformation of the progressive change tooth high vortex tooth occurs in close proximity to the central compression cavity, reaching up to 13 microns. These results provide a crucial theoretical basis for the structural design and performance optimization of the compressor.

Mechanics and thermal analyses of microfluidic nerve-cooler system

Recent advances in microelectronics and biomedical technology has led to capabilities for integrating programmable cooling mechanisms into bioelectronic devices for pain management and other important applications in human health. However, achieving localized and precise targeted cooling remains a challenge due to the influence of tissue mechanics during device placement/operation, viscous effects in the flow of coolants, and critical physical parameters that locally affect tissue temperature and mechanical responses. Recent studies demonstrate that bioelectronic devices can be equipped with microfluidic structures designed to support phase-change cooling mechanisms, with cooling power adjustable by control of fluid flow rates. When operated in closed-loop feedback schemes based on measurements of temperature at the tissue interface, these platforms can achieve precise temperature control of regions of peripheral nerves involved in pain signaling. Establishing a theoretical model to understand the effects of these devices on the underlying soft and deformable tissues, along with microfluidic deformations that can alter the heat transfer cooling process in viscous flow, is critical for optimized design and operation. To address this, a theoretical model is developed to describe the influence of remote strain in the tissue and to quantify the spatial temperature distribution of tissue cooled by fluid phase change in deformable microfluidic channels. A scaling law is derived to quantify the steady-state temperature distribution in the tissue under physiologically relevant mechanical loads, fluid viscosity, flow rates, and thermal conductivities, enabling the prediction of liquid and gas flux based on expected temperature changes and saturated vapor pressure in the deformed tissue and microchannel. These findings reveal that within a reasonable physiological range, the influence of fluid viscosity on temperature changes can be neglected, and the ratio of fluid to tissue thermal conductivity minimally impacts the temperature change at relatively high flow rates. Additionally, the model suggests a limit to tissue cooling for a given fluid, which may not necessarily increase with higher fluid volumes.

RELAP5-3D validation studies based on the High Temperature Test facility

In the spring and summer of 2019, experiments were conducted at the High Temperature Test Facility (HTTF) that form the basis of an upcoming high-temperature gas-cooled reactor (HTGR) thermal hydraulics (T/H) benchmark. HTTF is an integral effects test facility for HTGR T/H modeling validation. This paper presents RELAP5-3D models of two of those experiments: PG-27, a pressurized conduction cooldown (PCC); and PG-29, a depressurized conduction cooldown (DCC). These models used the RELAP5-3D model of HTTF originally developed by Paul Bayless as a starting point. The sensitivity analysis and uncertainty quantification code, RAVEN was used to perform calibration studies for the steady-state portion of PG-27. We developed four PG-27 calibrations based on steady-state conditions. These calibrations all used an effective thermal conductivity equal to 36 % of the measured thermal conductivity, but they differed with respect to the frictional pressure drops and radial conduction models. These models all captured the trends in steady-state temperature distributions and transient temperature behavior well. All four calibrations show room for improvement in predicting the transient temperature rise. The smallest error in temperature rise during the transient was a 21 % underprediction, and the largest was a 48 % underprediction. The errors in transient temperature rise are largely a result of a mismatch in power density between the RELAP5-3D model and the experiment due to the location of active heater rods along the boundary between heat structures in the model. The best of these calibrations was applied to PG-29 to model the DCC. Once again, temperatures during the transient were underpredicted but trends in temperature were captured. The RELAP5-3D model captured trends in the data but could not reproduce measured temperatures exactly. This result is not attributed to deficiencies in the experimental data or to RELAP5–3D itself. Rather, this result likely arises due to the some of the assumptions and decisions made when the RELAP5-3D model was first developed, prior to the execution of HTTF experiments. An agreement in prediction of temperature trends but challenges reproducing HTTF temperatures within measurement uncertainty is consistent with previous analyses of HTTF in the literature. Future RELAP5-3D validation activities centered around HTTF may be able to provide greater insight into the code’s capabilities for HTGR modeling with a more finely nodalized model.

Thermal analysis of sinusoidal doubly salient Electro-Magnetic Machine with distributed magnetomotive forces considering complex heat distribution and material thermal characteristics

Sinusoidal doubly salient electro-magnetic machine with distributed magnetomotive forces has DC excitation and AC armature windings in the same stator slot, which leads to the complicated distribution of heat sources and thermal resistances. One of the keys to design this motor is to estimate the temperature rise accurately. In this paper, the thermal performance of the motor materials at different temperatures was tested. According to the complex structure characteristics, the layered and sub-regional equivalent model of the stator slot is proposed. The model not only uses the material thermal data obtained from experiments, but also considers the complex thermal resistances and AC/DC losses distribution in the slots, as well as the influence of winding and insulation processes on the temperature rise of the motor, which is conducive to obtaining the highest point of the motor temperature and ensuring the calculation accuracy. The 3-D global steady-state temperature distribution of sinusoidal doubly salient electro-magnetic machine with distributed magnetomotive forces is studied, and the temperature rise of each part under different current densities is also analyzed. Finally, the temperature experiment of the prototype is carried out. The experiments show that the accurate loss calculation and thermal parameters make the maximum error between the simulation model and the actual measurement is 3.8 °C. The results not only guide the selection of reasonable armature and excitation winding current density of sinusoidal doubly salient electro-magnetic machine with distributed magnetomotive forces, but also provide data and experience support for temperature estimation of motors with complex winding distribution.

Cryogenic vacuum chamber testing of a conductively-cooled, high temperature superconducting rotor for a 1.4 MW electric machine for aeronautics applications

This paper presents the second ever demonstration of a superconducting rotor cooled only via conductive connection to a cryocooler, i.e., without pumped or solid cryogens. The full-scale rotor was operated statically in its intended environment – about 1e-3 torr vacuum with only conductive cooling through mechanical connections to a cryocooler contained on the rotor. Stable operation of the rotor in a representative magnetic environment was demonstrated up to its rated direct current (57.2 A) and the designed temperature limit for the superconductor (62 K). Additional electrical measurements increased confidence that the superconducting coils functioned as intended. A collection of steady state temperature distributions was measured at operating and non-operating conditions. In all but one case, the observed temperature gradient between the cold tip and superconducting coils satisfies the design but with little margin. Opportunities to reduce the gradient are identified.

Thermal and Structural Analysis on Simulated Model of a Beam Dump

This paper deals with the design, thermal and structural analysis for stopping the proton beam. A conical beam dump is being designed to handle the thermal and structural stresses resulting due to high heat load and hydraulic pressure of coolant, while handling 600 KW proton beam power in CW operation at 20 MeV energy. Pressure drop and heat transfer calculations are done considering spiral channel cooling system. The heat flux has been calculated by using Bi-Gaussian distribution density function of particles in beam. Nickel is selected as the beam dump material because of its favorable properties. In order to understand the thermal performance of beam dump a finite element model has been developed to predict the steady state temperature and stress distributions within the beam stop material. Key Word: Beam dump design, FEM, Thermal and Structural stress.

Mass Sensing by Symmetry Breaking and Mode Localization in a System of Parametrically Excited Microbeam Resonators

In this work, we study the nonlinear dynamics of a mode-localized mass detector. A system of equations is obtained for two weakly coupled beam resonators with an alternating electric current flowing through them and taking into account the point mass on one of the resonators. The one-dimensional problem of thermal conductivity is solved, and a steady-state harmonic temperature distribution in the volume of the resonators is obtained. Using the method of multiple scales, a system of equations in slow variables is obtained, on the basis of which instability zones of parametric resonance, amplitude-frequency characteristics, as well as zones of attraction of various branches, are found. It is shown that in a completely symmetrical system (without a deposited particle), the effect of branching of the antiphase branch of the frequency response is observed, which leads to the existence of an oscillation regime with different amplitudes in a certain frequency range. In the presence of a deposited particle, this effect is enhanced, and the branching point and the ratio of the amplitudes of oscillations of the resonators depend on the mass of the deposited particle.

A new LTNE analysis process for predicting the steady-state temperature distribution in a fixed bed of randomly packed mono-sized rough spheres based on pore-scale simulations

A new analysis process for the LTNE (local thermal non-equilibrium) model to solve for steady-state, fully-developed convection in mono-sized-sphere packed bed is proposed to predict the temperature distribution inside a fixed bed. For an actual mono-sized-sphere packed bed, the voidage lies between those of the regularly arranged beds of FCC, BCC, and SC. Through an interpolation/extrapolation process based on the pore-scale computation results of the SC, BCC, and FCC packings, the heat transfer parameters of a randomly packed bed can be calculated. To reliably estimate the local heat transfer rate, the regional mass flow rates in a packed bed are accurately predicted first. Then, the regional solid-phase thermal conduction can be independently evaluated without interference from the regional fluid-solid convection heat transfer. Additionally, a finite-thickness wall is constructed into the computational model to significantly improve the accuracy of the predicted radial temperature profiles. With realistic temperature self-adjustment within the wall, there is no need to set unrealistic boundary conditions of either isothermal, isoflux or their combinations at the outer face of the porous zone, as many conventional methods in the literature do. The solution process and necessary parameters are summarized into flowcharts. Finally, the accuracy of the new LTNE analysis process in predicting the temperature distribution throughout the packed bed is validated with the literature experiment. In contrast, the conventional methods fail to provide the correct trend of the temperature variation.

Simultaneous estimation of SAR, thermal diffusivity, and damping using periodic power modulation for MRgFUS quality assurance

Purpose: A crucial aspect of quality assurance in thermal therapy is periodic demonstration of the heating performance of the device. Existing methods estimate the specific absorption rate (SAR) from the temperature rise after a short power pulse, which yields a biased estimate as thermal diffusion broadens the apparent SAR pattern. To obtain an unbiased estimate, we propose a robust frequency-domain method that simultaneously identifies the SAR as well as the thermal dynamics. Methods: We propose a method consisting of periodic modulation of the FUS power while recording the response with MR thermometry (MRT). This approach enables unbiased measurements of spatial Fourier coefficients that encode the thermal response. These coefficients are substituted in a generic thermal model to simultaneously estimate the SAR, diffusivity, and damping. The method was tested using a cylindrical phantom and a 3 T clinical MR-HIFU system. Three scenarios with varying modulation strategies are chosen to challenge the method. The results are compared to the well-known power pulse technique. Results: The thermal diffusivity is estimated at 0.151 mm2s–1 with a standard deviation of 0.01 mm2s–1 between six experiments. The SAR estimates are consistent between all experiments and show an excellent signal-to-noise ratio (SNR) compared to the well established power pulse method. The frequency-domain method proved to be insensitive to B 0-drift and non steady-state initial temperature distributions. Conclusion: The proposed frequency-domain estimation method shows a high SNR and provided reproducible estimates of the SAR and the corresponding thermal diffusivity. The findings suggest that frequency-domain tools can be highly effective at estimating the SAR from (biased) MRT data acquired during periodic power modulation.

The axial temperature distribution of the channel in the linear reactivity model

Employing a linear reactivity burnup-dependent model, we investigate how the new axial flattened power densities (Pontes and Driscoll models) affect the steady-state axial temperature distributions of the sub-channel in pressurized water reactors (PWR). Two approaches were used: analytical, using mean parameters, and numerical, using appropriate empirical correlations. The Pontes distribution showed advantages in all evaluated regions except in the centerline of the fuel rod, outperformed by the Driscoll distribution. The results indicate that the linear reactivity model (LRM) is extremely advantageous in the evaluated reactor, as it reduces axial temperatures on hot spots and has the potential to enhance the reactor’s safety, efficiency, and longevity.

Model-based battery thermal parameter optimization using symbolic regression

Lithium-ion battery packs are commonly used in electromobility and distributed energy storage. However, keeping both cell temperature and inter-cell temperature differences within operating ranges is important to avoid a negative impact on performance and lifespan. For this reason, modeling the thermal behavior of battery packs in different operating conditions is crucial for both design and practical use.In this work, we optimize the parameters of a battery thermal model using a symbolic regression method based on evolutionary algorithms. We obtain power-type expressions for the drag coefficient, friction factor, and heat transfer correlation (Nusselt number), as a function of the Reynolds number and the cell spacing factor, plus the Prandtl number in the case of the Nusselt number. The model is adjusted with CFD simulations of battery modules containing up to 102 cells to obtain the steady-state temperature distribution. Compared to CFD simulations, the proposed model gets a mean absolute percentage error of 2.39 % in estimating the temperature in a 53-cell battery pack. The adjusted parametric thermal model is validated with experimental data for cooling down a battery pack and a single cell, obtaining a mean absolute percentage error of 1.26 % and 4.68 %, respectively, in estimating the dynamic temperature balance.The adjusted parametric thermal model and the mathematical expressions obtained for the drag coefficient, friction factor, and Nusselt number, can be useful for designing battery thermal management systems in electromobility and distributed energy storage.

Transient non-Fourier behavior of large surface bodies

The variety and complexity of heterogeneous materials in the engineering practice are continuously increasing, open-cell metal foams filled with phase change materials are typical examples. These are also having an impact on the recent developments in the energy industry. Earlier room temperature heat pulse experiments on macroscale foam samples showed non-Fourier over-diffusive behavior on a particular time scale. Since there is a need to investigate such complex structures on larger spatial scales and extend the one-dimensional analysis on two-, and three-dimensional settings, here we develop a two-dimensional analytical solution for the Guyer-Krumhansl and Jeffreys-type heat equations in cylindrical coordinates to investigate the transient thermal behavior of large bodies. We provide the steady-state and transient temperature and heat flux distributions for a space-dependent heat source. The solutions presented here will be helpful for the thermal characterization of complex materials and for the validation of numerical methods.

Speed limits to information erasure considering synchronization between heat transport and work cost

With the rapid development of the information technology (IT), heat transport and work cost in the information processing system gain significant scientific interest, but their synchronization problem has not been investigated hitherto. By the virtue of the finite-time Landauer principle, we here study the scenarios wherein one-dimensional Fourier heat transport synchronizes with the work cost of non-quasistatic information erasure. It is demonstrated that in such scenarios, the steady-state temperature distribution and harmonic temperature wave will respectively impose certain speed limits to information erasure, which give upper bounds on the amount of bits erased per unit time. The underlying physics of these speed limits are respectively the local well-definedness of the absolute temperature and the unsteadiness of the harmonic temperature wave. In engineering, the speed limit imposed by the steady-state temperature distribution can be understood as a performance limitation concomitant with the temperature stabilization, which quantitatively reveals the cost of the temperature stabilization.

Mechanical-thermal coupling model of solidnitrogen cryostat for electrodynamic suspension system

High-temperature superconducting magnets wound by REBCO (rare-earth-barium-copper-oxide) coated conductors are promising candidates for electrodynamic suspension system due to their high current-carrying capability, excellent mechanical tolerance and low cooling cost. Solid nitrogen as coolant to protect HTS magnets can significantly promote their dynamic performance and reduce the use of auxiliary equipment, which reduce the system dimension and weight. In this study, we propose a mechanical-thermal coupling model of solid nitrogen cryostat for electrodynamic suspension system. This model is capable of calculating the temperature distribution during cooling process and the transient behavior for the rewarming process of the solid nitrogen cryostat. The cooling capacity map for the two-stage RDK-415D cryocooler was extended for accuracy. The steady state temperature distribution of cooling process and the thermal evolution of three different rewarming processes, including cryocooler reserved, 5 W losses applied and cryocooler detached, were calculated and verified experimentally, followed by the investigation of the structural stresses and displacement of the solid nitrogen cryostat due to thermal displacement among different components. The developed model can serve as a reference and guide for the design and optimization of solid nitrogen cryostat for electrodynamic suspension system.

Two-Dimensional FEM Approach of Metabolic Effect on Thermoregulation in Human Dermal Parts During Walking and Marathon

The physiological mechanisms conduction, convection, and radiation exchange the heat energy in bi-directional routes between the body and the temperature field. Metabolism and evaporation are the uni-directional routes for the exchange of heat energy. In the metabolic process, the body creates internal heat energy, whereas the body loses excess heat energy through the evaporation process and maintains the body temperature. This study has shown steady and unsteady state temperature distribution in three skin layers: epidermis, dermis, and subcutaneous tissue, during walking and marathon. The results have analyzed that each skin layer temperature is higher during a marathon compared with walking due to more metabolic effects. The computation has been carried out for the two-dimensional Pennes’ bio-heat equation using a finite element approach. The generated results have been exhibited graphically.

An efficient thermal model of chiplet heterogeneous integration system for steady-state temperature prediction

As the silicon-based CMOS technology faces significant challenges in terms of physical principles and process limitations by reducing the device dimensions to improve integration, the chiplet heterogeneous integration (CHI) system has become an important solution to continue Moore's law. The wafer-scale CHI systems have also emerged nowadays to achieve more powerful performance for artificial intelligence (AI) applications. Due to the high power and small space, the thermal problems are big challenges. Those problems can lead to severe damages if they are not well solved. In this paper, an efficient thermal model is proposed to predict the steady-state temperature distribution of CHI 2.5-D systems. Specifically, since an oversimplified heat sink can result in huge inaccuracy, the heat sink is modeled according to its actual shape. In addition, the RDL, TSVs and bumps are modeled in an equivalent manner to make this model more accurate. Those efforts enable steady-state thermal simulation of the CHI system. In order to extend the thermal model to wafer-scale CHI systems, an efficient method is developed to transform the circular components into regular shape. Finally, this thermal model is evaluated to be highly accurate with errors <1.8 % and quite fast. Based on this verification, several cases are analyzed to illustrate the applicability of our methodology in describing the thermal dissipation behaviors. The present thermal model is an instructive simulation tool which can be used to perform large scale thermal simulation of heterogeneous integration systems. The proposed methodology of the present simulation work is expected to help the designers to detect the potential temperature-dependent hotspots and improve the reliability and robustness of CHI systems in the early-stage of the design flow.

Steady state thermal analysis of a porous fin with radially outwards fluid flow

Fins are used ubiquitously in engineering devices and systems for enhancement of heat transfer and energy storage. While traditional fins are made of non-porous materials, porous fins with natural convection porous flow orthogonal to the fin direction have also been studied. In contrast to these, there is a lack of work on porous fins in which the fluid flow may be along the direction of the fin. In such a fin, porosity may increase advective heat removal due to increased flow rate but may also impede conductive heat removal due to reduction in effective thermal conductivity. Due to these competing trade-offs, there is a need for comprehensive analysis of thermal performance of such a porous fin. This work derives a solution for the steady-state temperature distribution in a porous fin with advection along the fin direction. It is shown that temperature distribution in such a porous fin is governed by a convection-diffusion-reaction equation. A solution for the temperature distribution is derived in the form of modified Bessel functions of non-zero order. Two distinct fin performance parameters are defined and derived in order to characterize porous fin performance. It is found that thermal properties of the fin as well as ambient convective conditions strongly impact the relationship between fin porosity and fin performance. While in some cases, it is found that an optimum porosity exists that maximizes heat removal, in other cases, the use of a porous fin is found to be not desirable at all. The analysis presented here helps fully understand these trade-offs, and provides useful guidelines for porous fin design for maximum heat removal.