Two-dimensional Heat Research Articles

Preliminary analysis of maximum hypothetical accident for a solid core space nuclear reactor power system

In order to study the safety characteristics of the solid core space nuclear reactor power (SNRP) system under the maximum hypothetical accident, a two-dimensional entire core transient heat transfer analysis model was established, and the key parameters response characteristics of the ultra-small lithium-cooled SNRP under two maximum hypothetical accidents, namely, loss of heat sink (LOHS) and loss of coolant accident (LOCA), were calculated and analyzed. In the maximum hypothetical accidents, the coolant cooling capacity is lost, thus the core decay power is discharged into space only through the radiation of the residual heat removal system and the side surface of the reactor vessel. During the accidents, the heat transfer of the core deteriorated, and the fuel temperature may rise to the melting point, resulting in radioactive leakage. The results show that: (1) In the LOHS accident, the maximum fuel temperature reaches 2150 K at 550 s, and the pressure of the primary system volume accumulator continues to increase to the set system pressure safety limit of 2 MPa, resulting the primary loop overpressure failure. And the fuel matrix temperature is close to the set cladding limit temperature of 2200 K; (2) In the LOCA, the deterioration of heat transfer in the core makes the temperature increase rapidly, reaching a maximum of 3016 K at about 630 s, which is very close to the melting temperature of UN fuel 3123 K. As the decay power decreases, the maximum core temperature decreases to less than 1600 K after 24 h of the accident. The auxiliary cooling system of the solid core SNRP system under the maximum hypothetical accident is optimized, and the design parameters of the auxiliary cooling system meeting the safety requirements are obtained.

An Efficient Method for Solving Two-Dimensional Partial Differential Equations with the Deep Operator Network

Partial differential equations (PDEs) usually apply for modeling complex physical phenomena in the real world, and the corresponding solution is the key to interpreting these problems. Generally, traditional solving methods suffer from inefficiency and time consumption. At the same time, the current rise in machine learning algorithms, represented by the Deep Operator Network (DeepONet), could compensate for these shortcomings and effectively predict the solutions of PDEs by learning the operators from the data. The current deep learning-based methods focus on solving one-dimensional PDEs, but the research on higher-dimensional problems is still in development. Therefore, this paper proposes an efficient scheme to predict the solution of two-dimensional PDEs with improved DeepONet. In order to construct the data needed for training, the functions are sampled from a classical function space and produce the corresponding two-dimensional data. The difference method is used to obtain the numerical solutions of the PDEs and form a point-value data set. For training the network, the matrix representing two-dimensional functions is processed to form vectors and adapt the DeepONet model perfectly. In addition, we theoretically prove that the discrete point division of the data ensures that the model loss is guaranteed to be in a small range. This method is verified for predicting the two-dimensional Poisson equation and heat conduction equation solutions through experiments. Compared with other methods, the proposed scheme is simple and effective.

Experimental and numerical study on temperature control performance of phase change material heat sink

With the increasing power consumption of electronic products, the rising temperature leads to the deterioration of working environment and the reduction of service life. In order to control the temperature rise of electronic products, in this paper, the heat sinks with phase change material (PCM) and honeycomb metal were designed. And their performance was compared with that of the original heat sink (without PCM and honeycomb metal) then studied under various working conditions. Paraffin (n-eicosane) and prepared low melting point alloy were filled into the heat sink with honeycomb metal. The temperature control performance of heat sink was tested experimentally under different heating power. A two-dimensional heat transfer mathematical model of honeycomb metal PCM heat sink was established and solved numerically. The numerical results were compared with experimental data to verify the mathematical model. The results demonstrate that PCM heat sinks effectively controls the temperature of electronic devices and the low melting point alloy makes heat sink better performance than paraffin. Honeycomb metal further decreases the temperature of the paraffin heat sink by up to 3.6 °C, and extends the effective working time of the alloy heat sink by 14 %. The performance of the low melting point honeycomb metal heat sink is superior to that of the paraffin honeycomb metal heat sink at high heating power, and is close to the latter at low heating power. The effective working time of PCM honeycomb metal heat sink is significantly extended when the PCM melting point is close to the limit temperature, especially at high heating power and for the PCM with low thermal conductivity. This study provides valuable reference for temperature control design of electronic products.

Heat transfer model for moving packed-bed particle-to-sCO2 heat exchangers integrated with metal foams

Particle-to-supercritical carbon dioxide (sCO2) heat exchangers (HXs) play a vital role in coupling heat transfer fluid (HTF) from high-temperature thermal receivers to power cycle working fluids (WF). Heat transfer enhancement is essential for adopting particle-based moving packed-bed heat exchangers (MPBHXs) in next-generation thermal energy storage (TES) systems, as MPBHXs usually exhibit low particle bed-to-wall heat transfer coefficients. High-porosity metal foams have shown their effectiveness in heat transfer enhancement. This work presents a continuum heat transfer model to demonstrate the heat transfer enhancement in MPBHXs when the particle bed channel is filled with high porosity metal foams compared to open channel MPBHXs. The presence of metal foams increases the effective thermal conductivity in the particle channel and enhances the interstitial heat transfer coefficient between the moving particle bed and the stationary metal foams. The present model considers coupled two-dimensional (2D) heat transfer in the particle channel with metal foams and 1D heat transfer in the sCO2 channel and the dividing wall. The temperature profiles for the particle bed, metal foam, dividing wall, and sCO2 stream, as well as the local heat flux profiles between those, are studied in detail for various foam porosities, from which the particle bed-to-wall heat transfer coefficient, overall heat transfer coefficient, and total heat transfer rate for the MPBHX are determined. The effects of major MPBHX design and operating parameters on the particle bed-to-wall heat transfer coefficient and overall heat exchange capacity are thoroughly examined; thus, the MPBHX performance improvement with metal foams is quantified for each case. The present heat transfer model can provide valuable insights into the metal-foam MPBHX design and optimization, scale-up, and operating parameters selection.

Nonlinear two-dimensional transient thermoelastic analysis of functionally graded composite plates subjected to localised cooling loads

In this study, for the first time, localized thermal cooling is considered for the transient thermoelastic response of an axisymmetric plate, addressing the underexplored area of localized low-temperature thermoelasticity. Functionally graded composite plates are analyzed which consist of a mixture of stainless steel (SUS304) and low-carbon steel (AISI1020). A two-dimensional transient heat conduction equation is employed to capture localized cooling effects, featuring various adaptive time-dependent thermal boundary conditions which mirror both loading and unloading scenarios, effectively simulating real-world phenomena. Novel parameters are introduced to facilitate a comprehensive understanding of various aspects of thermal load handling and their impact on thermoelastic responses. The heat conduction equations are solved using the Generalized Differential Quadrature Method (GDQM) and the Crank-Nicolson scheme. The nonlinear governing equations, incorporating geometrical nonlinearity through the von Kármán assumption within the First Shear Deformation Theory (FSDT) framework, are solved using the GDQM and the Picard's technique. The model is validated using Finite Element Method (FEM) through Abaqus. A comprehensive analysis is provided that considers influence of the ratio of thermally affected and unaffected plate surface, thermal load magnitude, rapidity of thermal loading and unloading, duration of the cooling load, geometrical nonlinearity, and temperature dependence of material. The study shows that the maximum von Mises stress within the structure remains consistent regardless of the duration of the cooling load, as long as the rate at which the cooling load is applied stays the same. Additionally, the findings reveal that even minor localized thermal loads can produce substantial stress, especially at the intersection of thermally affected and non-affected zones on the exposed surface in some situations.

Nonlinear thermal analysis of two-dimensional materials with memory

A nonlinear hyperbolic heat transport equation has been proposed based on the Cattaneo model without mechanical effects. We analyze the two-dimensional Maxwell-Cattaneo-Vernotte heat equation in a medium subjected to homogeneous and non-homogeneous boundary conditions and with thermal conductivity and relaxation time linearly dependent on temperature. Since these nonlinearities are essential from an experimental point of view, it is necessary to establish an effective and reliable way to solve the system of partial differential equations and study the behavior of temperature evolution. A numerical scheme of finite differences for the solution of the two-dimensional non-Fourier heat transfer equation is introduced and studied. We also investigate the attributes of the numerical method from the aspects of stability, dissipation and dispersive errors.

A quasi-2D thermodynamic model for performance analysis and optimization of liquid hydrogen storage system with multilayer insulation and vapor-cooled shield

Liquid hydrogen (LH2) holds great potential in both aerospace and civil markets due to its high energy density. However, on account of the low boiling point and latent heat of vaporization of LH2, the high performance insulation storage system is the key to its efficient storage. One of the most efficient insulation methods for a LH2 storage system is considered to be Variable Density Multilayer Insulation (VDMLI) coupled with Vapor-Cooled Shield (VCS). This study establishes a quasi-two-dimensional (quasi-2D) thermodynamic model of liquid hydrogen storage system with VDMLI and VCS. The quasi-2D model incorporates the temperature gradient of VCS and its impact on VDMLI. The model can reveal the two-dimensional temperature and heat distribution in insulation systems, predict heat leakage and evaporation rate, and can be utilized for optimizing the design of the VCS structure. Using a NASA's tank as an example, the analysis conducted with the model indicates that the presence of VCS reduces the average temperature gradient of the inner VDMLI. The upper half of the tank exhibits a significantly smaller temperature gradient compared to the lower half, resulting in lower heat leakage. This effect becomes more pronounced as the VCS moves outward of the tank. Additionally, the VCS absorbs more heat in the inlet section than that in the outlet section, and this disparity becomes increasingly significant as the VCS moves outward of the tank. The model can also be employed to optimize the position of the VCS, resulting in a 69.73 % reduction in daily evaporation rate compared to NASA's original design. Lastly, a graphical method for the quick design of VCS structure is proposed which is based on calculated data for daily evaporation rate and apparent thermal conductivity. This method enables the retrieval of the corresponding apparent thermal conductivity based on the desired evaporation rate, facilitates the rapid determination of the range of VCS position, and allows for the quick prediction of the evaporation rate based on the VCS position. In conclusion, this research presents a new approach for the performance analysis and optimization design of liquid hydrogen storage system with VDMLI and VCS.

Investigation of neutronic and thermal–electric coupling phenomenon in a 100 kWe-level nuclear silent heat pipe-cooled reactor

Heat pipe-cooled reactors are attractive because of their compact structure, high power capacity and reliability, and inherent safety features. These reactors are distinguished by the use of high-temperature heat pipes that directly bridge the reactor core and energy conversion devices, thus creating a tightly knit modular system. However, comprehensive coupled analysis of these reactors remains challenging because of the intricate multiphysics interactions involved. To address this, the neutronic and thermoelectric coupled phenomena of heat pipe-cooled reactors are investigated in this study. Models are introduced for the point reactor kinetics, core heat transfer, channel heat transfer, two-phase two-dimensional heat pipes, thermoelectric coupling, and cold plates. The channel heat transfer is refined in a 100 kWe-level nuclear silent heat pipe-cooled reactor and a model for the three-dimensional heat transfer process in its thermoelectric matrix established, which effectively resolves the calculation distortions of the lumped model. The maximum deviation between the calculation results from the proposed model and the verification data for a full-system coupled computation is less than 10 K. The core temperature difference from the three-dimensional model is five times that of the lumped model with a maximum average fuel temperature difference of 225.7 K. The maximum and average thermoelectric conversion efficiencies are 15.56 % and 14.43 %, respectively. The failures of one, five, and nine heat pipes are analysed. The failure of a single heat pipe has negligible effects with a peak fuel temperature surge of only 121 K. In contrast, a failure involving nine heat pipes leads to a peak fuel temperature spike of 778 K and a 47.1 % increase in the maximum heat transfer power, marking a critical operational threshold. It is therefore imperative to consider the safety margin in the potential failure of multiple central heat pipes during the preliminary design stages.

Simultaneous determination of the space-dependent source and initial value for a two-dimensional heat conduction equation

Simultaneous determination of the space-dependent source and initial value for a two-dimensional heat conduction equation

A hybrid numerical method for non-linear transient heat conduction problems with temperature-dependent thermal conductivity

This paper introduces a hybrid numerical technique aimed at effectively addressing two-dimensional (2D) non-linear transient heat conduction problems, featuring temperature-dependent thermal conductivity. This scheme involves integrating the Krylov deferred correction (KDC) method in the temporal discretization and the generalized finite difference method (GFDM) in the spatial discretization. The core of the KDC method introduces a new unknown variable in the form of the first-order time derivative of temperature, leading to a non-linear equation after temporal discretization. Subsequently, the corresponding non-linear equation is solved in the spatial domain using the GFDM, supported by the Jacobian-free Newton–Krylov (JFNK) solver and fourth-order Taylor series expansion. The accuracy and stability of the hybrid approach are verified through two numerical experiments, demonstrating the strong performance of the present algorithm known as the KDC-GFDM for the simulations of the interested problems in long-time intervals.

Exact solutions for permeable wall laminar jet with velocity slip: Momentum and thermal profiles

This study presents exact analytical solutions for the two-dimensional laminar jet flow and heat transport of Casson fluid under velocity slip condition. The laminar wall jet of classical Glauert type is extended to the Casson fluid model considering the flow to be along a flat permeable surface. The roles of velocity slip, wall temperature, and mass transfer on the fluid flow and rate of thermal exchange are investigated. The outcomes for momentum and thermal fields, skin friction and heat transfer gradient are obtained analytically with numerical solutions of the suction-far stream velocity expression. It is noted that mass transfer and velocity slip have significant effects on the flow of fluid and heat transfer. Thermal profile varies significantly for increasing Prandtl values with small suction.

Optimization design of solid-state hydrogen storage device for fuel cell forklift

Solid-state hydrogen storage device using metal hydride have enormous advantages for fuel cell forklifts. In addition to high volume hydrogen storage density, the solid-state hydrogen storage device can also be used as a counterweight for the forklift. At the same time, the solid-state storage device has low hydrogen storage pressure and safety, and the fuel (H2) refueling is convenient and low-cost. In this paper, a two-dimensional heat and mass transfer numerical model has been developed to predicting the performance of the metal hydride tank filled with Ti0.9Zr0.1Cr0.35Mn1.4V0.2Fe0.05 alloy. The validity of this numerical model was tested by comparison with the experimental data of the metal hydride hydrogen storage tank with a hydrogen capacity of 1 kg, achieving a good agreement between all the data. And a solid-state hydrogen storage device with an effective hydrogen capacity of 1.5 kg is optimally designed for 3.5 T fuel cell forklift. The completion rate of hydrogen refueling in the solid-state hydrogen storage device will reach 97.6 % within 30 min, and continuously discharging over 1.5 kg H2 under flow rates of 150 SL/min and 250 SL/min. The optimized solid-state hydrogen storage device was integrated in a power module for 3.5 T fuel cell forklift which allows uninterrupted operation for at least 6 h 8 min under rated operation.

Direct and inverse problems for a 2D heat equation with a Dirichlet–Neumann–Wentzell boundary condition

In this paper we present the inverse problem of determining a time dependent heat source in a two-dimensional heat equation accompanied with Dirichlet–Neumann–Wentzell boundary conditions. The model is of significant practical importance in applications where the time dependent internal source is to be controlled from total energy measurements in the case when the boundary is defined with partially absorbing (Dirichlet), partially reflecting (Neumann), and partially having the capacity for storing heat (Wentzell) conditions. The considered spatial domain is a rectangle which allows the application of a Fourier series analysis to the solution of the problem whose auxiliary spectral problem is a novel eigenproblem. This eigenvalue problem is defined using the two-dimensional Laplace operator where the spectral parameter resides in one of the accompanying boundary conditions. Under this setting we study the well-posedness of both the direct and inverse problems. In addition, we use the results obtained on the well-posedness to establish the numerical solvability of all the problems considered in the paper. In order to achieve this, we make use of a Chebyshev spectral collocation method for the spatial discretisation and a backward differentiation formula for the temporal discretisation. All the numerical results are shown to be consistent with the theory and in good agreement with the analytical solutions.

Finite Difference Method for Solving the Two-Dimensional Laplace Equation in Curvilinear Coordinates

Abstract This paper presents a numerical method for solving the two-dimensional Laplace equation in curvilinear coordinates using a finite difference scheme. Laplace equation is a differential equation used to describe various physical phenomena studied within different branches of engineering. The two-dimensional Laplace equation has analytic solutions only for simple shaped domains and simple boundary conditions (first and second kind). Even so, the results are usually complicated to interpret and use. The present paper proposes a method to solve the equation for domains bounded by complex inner and outer curves. The boundary curves are defined by a set of points from which the equations of curves are approximated by using Fourier series. The domain is generated by blending the two curves using a blending function. The results are a set of continuous parametric equations which are defining the domain in angle and radius variables. Following a method presented in the literature, the Laplace equation is transformed from Cartesian coordinates to curvilinear coordinates, as the Fourier series is easily differentiable. Next, the new Laplace equation is approximated using the finite difference method and a MATLAB code is developed to solve the system of linear equations using successive over-relaxation method (SOR). The presented method is used to simulate steady two-dimensional heat conduction in a polygonal shape domain, with uniform and non-uniform Dirichlet boundary conditions. Results are analyzed in order to determine the model’s precision, speed and convergence criteria.

Experimental and theoretical study on cooling performance of membrane roof with circulated water film

Membrane roofs are widely used in buildings for excellent daylight and visual effects. Nevertheless, excessive solar penetration results in an overheated indoor thermal environment. The experimental and theoretical study on the cooling performance of circulated water film technology on membrane roofs was conducted. The scale experiment results indicated that with circulated water film the membrane surface temperatures decreased rapidly to a level slightly above the water temperature, where the polyvinylidene fluoride (PVDF) and ethylene tetrafluoroethylene (ETFE) membranes were 0.4 and 2.4 °C higher than water temperature. While the indoor air temperature gradually decreased in instances where PVDF roofs were utilized, the application of ETFE roofs did not yield a significant decrease in temperature. Subsequently, we derived an analytical solution for the dynamic two-dimensional heat and mass transfer process of the water film flowing on the membrane with incident solar radiation, which was validated by experiment. The analysis of influential factors indicated that the temperature drop increased with absorbed solar radiation and outdoor dry-bulb temperature, and decreased with the ambient wet-bulb temperature. Conversely, the cooling performance exhibited minimal sensitivity to variations in water film thickness. The indoor operative temperature dropped by 0.5–0.8 °C in the membrane roofed atrium with water film. The water film effectively cools the indoor thermal environment with membrane roofs.

Influence of time-dependent wall temperature fluctuation on conjugate mixed convection inside a chamber with an oscillating cylinder

The current work is an attempt to examine the mixed convective heat flow in a square chamber subjected to time-dependent sinusoidal wall temperature containing a heat-conducting rotationally oscillating circular cylinder at the center. The right wall is kept at a cold temperature and the left wall is subjected to time-varying wall temperature, whereas the remaining walls are thermally insulated. A sinusoidal oscillating velocity is applied on the surface of the oscillating cylinder. This problem has practical relevance for several applications, notably solar collectors, microelectronic cooling systems, chemical reactors, radiators, and heaters. The main goal of this study is to promote heat transfer by investigating optimum system characteristics. The pressure-velocity formulation of the Navier-Stokes and heat energy equations is used to model the fluid domain inside the chamber, and heat transfer inside the cylinder is modeled by the two-dimensional heat conduction equation. Those equations are discretized using the Galerkin finite element method after converting them into non-dimensional forms. The instantaneous and time-averaged Nusselt numbers of the heated wall are examined for three different frequencies of the cylinder oscillation and sinusoidal wall temperature variation. Parametric simulation is carried out for three different cases within a mixed convection regime through a combination of Reynolds (31.62–316.23), Grashof (103–105), and Richardson (0.1–10) numbers. For each different combination, the optimum values of cylinder frequency are determined. Due to the thermal boundary layer's repeated contraction and expansion, the instantaneous Nusselt number demonstrates an oscillating pattern over time. Moreover, the maximum wall temperature frequency is found to maximize the heat transfer process.

Analysis of plane elasticity problems using the dual mesh control domain method

The dual mesh control domain method (DMCDM) proposed by Reddy combines the advantages of the finite element method (interpolation of variables) and the finite volume method (satisfaction of the global form of the governing equations). Currently, the method has been used to solve two-dimensional heat transfer, fluid and solid mechanics problems on structured meshes of linear right triangular and bilinear rectangular elements. However, such meshes are not useful from the standpoint of most practical problems which are posed on irregular domains. The present study extends the DMCDM for the analysis of plane elasticity problems involving non-rectangular domains meshed with arbitrarily shaped elements. The approach makes use of the isoparametric formulation, in which the primal mesh used for the approximation of the geometry is also used for approximating the dependent unknowns. Several boundary value problems of elasticity are solved using the DMCDM, and the results are compared with those obtained using the finite element method (FEM). The results show that the DMCDM exhibits excellent convergence and accuracy while requiring less computational cost when compared with the FEM.

Response spatiotemporal correlation and transient temperature field direct reconstruction for heat conduction system

In this paper, according to a spatiotemporal correlation characteristic of the temperature response of a transient heat conduction system, a response spatiotemporal correlation (RSTC) model of the temperatures between measurement space and estimation space of the system is proposed. Furthermore, a RSTC based direct reconstruction (RSTC-DR) method is developed for a transient temperature field reconstruction problem of a heat conduction system with unknown heat sources. In the offline stage, firstly, mapping eigenvectors of a heat conduction system are determined by the transfer equations without specific heat source information. Then, according to the correlation degree between the mapping eigenvectors, a finite number of representative spatial points (RPs) are extracted. Meanwhile, a fuzzy membership relationship between the mapping characteristics of the RPs and that of the temperature field is determined. Namely, effective mapping characteristic coverage of the RPs to the whole temperature field is realized. Moreover, a RSTC model between the measurement spatial points (MPs) and the RPs is obtained by the step response model of the temperature field. In the online stage, based on the RSTC model, the temperature responses of the RPs are estimated by employing the time series vector of the temperature measurements of the MPs. Next, according to the fuzzy membership relationship, the temperature field can be reconstructed in real time by synthesizing the temperature estimations of the RPs. The RSTC-DR method is employed to achieve the non-destructive reconstruction of the transient temperature field of a two-dimensional heat conduction system. The temperature field is reconstructed by the numerical experiment to verify the effectiveness of the proposed method and to discuss its adaptability. Furthermore, the practical effectiveness of the method is demonstrated by applying the existing experimental data.

Study on Sublimation Drying of Carrot and Simulation by Using Cellular Automata

Vacuum freeze-dried products exhibit properties characteristic of porous media, rendering them superior in both drying and rehydration capabilities. However, the process of sublimation drying is constrained by its substantial time and energy costs. To comprehensively grasp its technological process and identify the optimal process parameters, the cellular automata method was employed for sublimation process simulation. Carrot slices, measuring 10 mm in thickness and 40 mm in radius, were selected for both simulation and experimentation. The sublimation process was characterized using a two-dimensional heat and mass transfer equation, inclusive of a dusty gas model. Additionally, a cellular automaton model was applied to simulate the mass transfer process, temperature, and moisture content changes in the sublimation drying stage. Then, the accuracy of the model was verified through experimentation. There was a remarkable alignment between simulation and experimental outcomes, with determination coefficients R2 of 99.4% for moisture content and 97.6% for temperature variations.

A numerical investigation of 2D transient heat conduction problems in anisotropic FGMs with time-dependent conductivity

A class of two-dimensional (2D) transient heat conduction problems for anisotropic Functionally Graded Materials (FGMs) of space–time dependent conductivity and coefficient of variable separable forms is considered. The problems are reduced to a boundary-only integral equation which can be solved numerically using a standard Boundary Element Method (BEM). Some examples are considered to show the validity of the analysis and to examine the accuracy of the numerical method.