Discrete Heat Research Articles

The effect of Nusselt number on the bi-viscosity fluid subjected to the discrete heating effect

Discrete heating action has been established as an energy efficient method. Nusselt number plays a significant role in the heat transfer rate. Local Nusselt number is useful in analyzing the heat transfer rate along the sections of the side walls with different or same temperature in a heated enclosed cavity. Whereas the average Nusselt number quantifies the total heat change in the closed cavity. While the path of the trajectory of the heat flow is being analyzed by heat function. In the current work, multiple heaters of length , and are assembled at each side of the isosceles triangle. The numerical simulations of the modeled system along with boundary conditions are carried out by using the standard Galerkin Finite Element Method. The nonlinear algebraic system is solved by Newton method. The behavior of fluid inside the closed triangle with is observed. The results are reported in terms of the graphs of the temperature contours, streamlines, average and local Nusselt numbers. The maximum heat transfer rate is observed in convection dominant case. By increasing the magnitude of a bi-viscosity parameter, secondary circulation is observed for The rate of heat transfer has a direct relation with the bi-viscosity parameter. The maximum values of average and local Nusselt numbers are achieved by using the heat generation parameter against the Grashof number.

Three-dimensional finite discrete element-based contact heat transfer model considering thermal cracking in continuous–discontinuous media

This paper proposes a three-dimensional heat transfer model that considers contact heat transfer and thermal cracking in continuous–discontinuous media. The 3D heat transfer model comprises the heat conduction model inside the continuum and the contact heat transfer model between discrete blocks. The model is combined with the 3D Finite Discrete Element (FDEM) for coupled thermo-mechanical calculation, which includes the following three parts: the temperature distribution of the system is obtained by the 3D heat transfer model considering contact heat transfer; then, the thermal stress caused by the temperature is applied to the system equation to perform the mechanical fracture calculation; finally, the heat exchange coefficient of the newly generated broken joint element is updated for heat transfer calculation at the next time. Some examples are given to verify the 3D contact heat transfer model with analytical solutions. Moreover, the influence of the heat exchange coefficient of the joint element and the contact heat transfer coefficient is discussed. Finally, examples of contact heat conduction in 3D granular assemblage, and thermal cracking caused by contact heat transfer are demonstrated. These examples demonstrated excellent capacity of the model in simulating contact heat transfer and thermal cracking in continuous–discontinuous media.

A Fast Computational Method for the Optimal Thermal Design of Anisotropic Multilayer Structures with Discrete Heat Sources for Electrified Propulsion Systems

This work investigates the effect of the anisotropy on the heat transfer properties of layered composite materials with discrete heat sources, that are used in the power electronics of hybrid-electric propulsion systems. An analytical method, based on closed-form expressions structured on Fourier expansion series, is proposed for the solution of the Laplace’s steady-state anisotropic heat equation. The physical presence of electrical components is replaced by externally applied power sources. The method provides an accurate prediction of the temperature distribution and of the heat transfer across perfect layer-to-layer adhesion or finite conductance interfaces; its use is therefore encouraged for the optimization of composite substrates. Code verification has been employed on test cases for which the analytical solution was available.

Multi-objective optimization of a composite phase change material-based heat sink under non-uniform discrete heating

This paper presents a novel heat sink that integrates paraffin/copper foam composite phase change material. The heat sink base is subject to non-uniform heating with multiple discrete heat sources on a circuit board. Numerical simulations are performed using FLUENT software to investigate its thermal transfer characteristics and performance. Three types of paraffin, including RT35HC, RT44HC, and RT54HC, are compared. An optimization scheme that combines Artificial Neural Network and Non-Dominated Sorting Genetic Algorithm-II is proposed to (a) maximize the operation time and (b) minimize the heat sink mass. Operation time is defined as the time for simulated chips to reach set point temperature, 80 °C. The result shows that local overheating caused by non-uniform heating limits operation time. The optimization method shows great robustness and effectiveness. Three sets of non-dominated solutions are obtained, and optimal configurations are determined with further analysis. A figure of merit is defined as the ratio of charging time to heat sink mass. Optimal configuration's figure of merit increases by 1.2–44.0% comparing to original designs.

Galerkin finite element analysis of Darcy–Brinkman–Forchheimer natural convective flow in conical annular enclosure with discrete heat sources

This numerical study is intended for the analysis of thermal convection induced by buoyancy forces generated within a conical annular porous gap. The annulus was vertically positioned, it contains a discrete heat source and is filled with a Single-Walled Carbon Nanotubes-Water (SWCNT-H2O) nanoliquid exposed to a Lorentz force. To describe the porous medium in question, we have adopted the Darcy–Forchheimer model. Galerkin Finite Element Method (GFEM) has been used in this study to predict both thermal and hydrodynamic fields in the physical model. An extensive range of parameters are explored, i.e., the Rayleigh number (103 to 106), Hartman number Ha (1 to 100), and the volume fraction of nanoparticles (0 ≤ϕ≤0.08). For the purpose of exterminating the effects of heat source location on thermal and hydrodynamic fields, three locations (bottom, middle and upper) have been considered. Findings include current lines, isotherms, and Nusselt number evolution according to the previously stated variables. Predictably, our findings prove that heat transfer rate exhibits a decreasing function of Ha and an increasing function of Da and ϕ. Also, it was revealed that the convective regime is preponderant when the heat source was located in the bottom wall.

MHD Double Diffusive Convection of Al2O3-Water Nanofluid in A Porous Medium Filled an Annular Space Inside Two Vertical Concentric Cylinders with Discrete Heat Flux

The magnetoconvection phenomenon of a double diffusive free convection in an annular porous space inside two concentric cylinders saturated by a (Al2O3, water) nanofluid has been investigated in the current study. The transport equations for vorticity, energy and concentration as well as stream function are solved using the finite difference method. At lower temperatures and concentrations, the outer cylinder is sustained, the discrete heat flux with the unheated adiabatic portions as well as the higher concentrations are imposed in the inside walls of the central cylinder. The base walls are insulated and impermeable. In the vertical direction, external magnetic field (MF) with uniform intensity is applied. The results of the obtained numerical simulation are presented to exhibit the consequences of various numbers such: Rayleigh RaTh, Hartmann Ha, Buoyancy forces ratio N and the solid nanoparticles (NPs) volume fraction  on the pattern of the nanofluid flow and the transferred mass and thermal energy in the active wall. It is noted that the transferred mass and thermal flux in the active wall augments as the RaTh and N augments. The thermal energy transferred augments with the growth of , while the transferred mass in the active wall decreases. The greater magnetism declines the rate of the mass and thermal energy transport in the active wall.

Nonlocal thermal diffusion in one-dimensional periodic lattice

Heat conduction in micro-structured rods can be simulated with unidimensional periodic lattices. In this paper, scale effects in conduction problems, and more generally in diffusion problems, are studied from a one-dimensional thermal lattice. The thermal lattice is governed by a mixed differential-difference equation, accounting for some spatial microstructure. Exact solutions of the linear time-dependent spatial difference equation are derived for a thermal lattice under initial uniform temperature field with some temperature perturbations at the boundary. This discrete heat equation is approximated by a continuous nonlocal heat equation built by continualization of the thermal lattice equations. It is shown that such a nonlocal heat equation may be equivalently obtained from a nonlocal Fourier's law. Exact solutions of the thermal lattice problem (discrete heat equation) are compared with the ones of the local and nonlocal heat equation. An error analysis confirms the accurate calibration of the length scale of the nonlocal diffusion law with respect to the lattice spacing. The nonlocal heat equation may efficiently capture scale effect phenomena in periodic thermal lattices.

Numerical Investigation on Two-Phase Flow Heat Transfer Performance and Instability with Discrete Heat Sources in Parallel Channels

With the rapid development of integrated circuit technology, the heat flux of electronic chips has been sharply improved. Therefore, heat dissipation becomes the key technology for the safety and reliability of the electronic equipment. In addition, the electronic chips are distributed discretely and used periodically in most applications. Based these problems, the characteristics of the heat transfer performance of flow boiling in parallel channels with discrete heat source distribution are investigated by a VOF model. Meanwhile, the two-phase flow instability in parallel channels with discrete heat source distribution is analyzed based on a one-dimensional homogeneous model. The results indicate that the two-phase flow pattern in discrete heat source distribution is more complicated than that in continuous heat source distribution. It is necessary to optimize the relative position of the discrete heat sources, which will affect the heat transfer performance. In addition, compared with the continuous heat source, the flow stability of discrete heat sources is better with higher and lower inlet subcooling. With a constant sum of heating power, the greater the heating power near the outlet, the better the flow stability.

Characteristics of natural convection in n-eicosane in a square cavity with discrete heat source

The natural convection of phase change material (PCM) in a two-dimensional square cavity is numerically analyzed. The cavity consists of heat surface with a constant total heated area and adiabatic wall. The Grashof and Prandtl numbers for the PCM (n-eicosane with the melting temperature, Tmelt=36°C) in basic LHS system are 9 × 10^5 and 62.7, respectively, at 350.15K. The mass, momentum and energy balance equations of the system were considered. Three basic heating surface strategies were considered (discrete heat sources): single, side-side and side-bottom heating surface. The results show that the transient Nusselt number, mean kinetic energy at the surface and energy storage rate of the fluid are effectively enhanced by proper arrangement of the discrete heat source location and heating from the bottom half of the left and right sides requires the least time for 300 kJ energy storage. The results indicate that optimally placed discrete heat sources in PCM could be a promising alternative for high-efficient thermal energy storage.

MHD double-diffusive mixed convection and entropy generation of nanofluid in a trapezoidal cavity

Double diffusive, Magnetohydrodynamics (MHD), mixed convection flow of Al2O3-water nanofluid inside a trapezoidal enclosure with discrete heat and mass source at the bottom wall of the cavity has been discussed numerically. Together with these entropy generations due to fluid friction, heat transfer, mass transfer and magnetic field are discussed. The upper wall of the cavity is considered to move with a fixed velocity U0 towards positive X-direction. Different inclination angles and different aspect ratios are considered in the present study. Second and fourth order finite difference approximations are used to solve the governing equations by using biconjugate gradient stabilized method (BiCGStab). It is observed that, entropy generation mostly depends on heat transfer and lower values of Richardson number (Ri=0.01) reduces the entropy generation. The highest quantitative value of |Sθ|max is 21,664 for Ri=100,Re=100andϕ=450 and lowest quantative value of |Sθ|max is 5,824 for Ri=0.01,Re=1000andϕ=600 throughout the present study. Also the highest quantitative values of average Nusselt and Sherwood numbers are 0.2597 and 0.4208 for Ri=10,Re=100andϕ=45∘. The main objective of our present study is to minimize entropy generations due to combined effects of fluid flow, heat transfer, mass transfer and the effect of magnetic field so that the loss of energy can be reduced. This study observed that mixed convection flow, low magnetic field and low aspect ratio cavity of rectangular shape is always preferable to reduce total entropy generation and the middle part of the lower wall of the cavity is highly acceptable for heat and mass source for sprinkle of heat and mass throughout the whole of the cavity.

Discrete heat kernel, UV modified Green’s function, and higher derivative theories

We perform the UV deformation of the Green’s function in free scalar field theory using a discrete heat kernel method. It is found that the simplest UV deformation based on the discretized diffusion equation leads to the well-known Pauli–Villars effective Lagrangian. Furthermore, by extending assumptions on the discretized equation, we find that the general higher derivative theory is derived from the present UV deformation. In some specific cases, we also calculate the vacuum expectation values of the scalar field squared in nontrivial background spaces and examine their dependence on the UV cutoff constant.

A review on metal additive manufacturing: modeling and application of numerical simulation for heat and mass transfer and microstructure evolution

Metal additive manufacturing technology has been widely used in prototyping, parts manufacturing and repairing. Metal additive manufacturing is a multi-scale and multi-physical coupling process with complex physical phenomena of heat and mass transfer and microstructure evolution. It is hard to directly observe the dynamic behavior and microstructure evolution of molten pool during additive manufacturing. Therefore, numerical simulation of additive manufacturing process is significant since it can efficiently and pertinently predict and analyze the physical phenomena in the process of metal additive manufacturing, and provide a reference for technological parameters selection. In this review, the research progress of numerical simulation of metal additive manufacturing is discussed. Various aspects of numerical simulation models are reviewed, including: (1) Introduction of basic control method and physical description of numerical simulation models; (2) Comparison of various heat and mass transfer models based on different physical assumptions (heat conduction model; heat flux coupling model; discrete powder particle heat flux coupling model); (3) Applications of various microstructure evolution models [phase field (PF), cellular automata (CA), and Monte Carlo (MC)]. Finally, the development trend of numerical simulation of metal additive manufacturing, including the thermal-flow-solid coupling model and deep learning for numerical model, is analyzed.

Numerical study on buoyant convection and thermal radiation in a cavity with various thermal sources and Cattaneo-Christov heat flux

The rate of change of thermal and flow transfer inside an enclosed box under the effect of thermal radiation and convective flow driven by buoyancy force with discrete heating and cooling is analyzed here. The “Cattaneo-Christov heat flux’ (CCHF) pattern is implemented in thermal energy model. The study is performed by varying the partial thermal sources namely the heater and cooler along the sidewalls of the region in three different positions: bottom, center and top. The inactive parts of the box, namely vertical sidewalls and the horizontal flat walls are maintained adiabatic. Numerical computations to examine the distinctiveness of liquid flow & heat transport are calculated by method of finite volume. The solutions are displayed as graphs. Thermal radiation effects inside the cavity illustrate a drift from linear to nonlinear increase in rate of change of heat. This is observed when the Grashof value is altered from 103 to 106. The results obtained show that middle-middle location of the partial heater/cooler in the cavity produce enhanced heat transfer rate and top-bottom position showed low rate of heat transfer.

ИСПОЛЬЗОВАНИЕ ОБМЕННОЙ ЭНЕРГИИ В ФОРМИРОВАНИИ ТЕПЛОВОГО СОСТОЯНИЯ ОРГАНИЗМА КРУПНОГО РОГАТОГО СКОТА

Scientific advances in biological sciences make it possible to significantly increase the energy efficiency of productive livestock. For life, as the highest form of existence of matter, thermal energy is of particular importance. It does not only connect the actions and interactions of all types of matter, it creates order from the chaotic movements of discrete heat sources, determining the measure of irreversible energy dissipation (entropy) and the change gradient of metabolic processes, “outflow and inflow of energy”, the state of saturation and deficiency of nutrients in the body. Metabolic energy is the energy of nutrients entering the tissues and cells of the body from the digestive tract. In the process of intracellular metabolism, substances are converted into new compounds, energy is released and accumulated. Approximately half of the energy is used in the electrochemical reactions of the synthesis of substances inherent in this organism. Heredity, age, environment, condition of animals influence their quantity and quality. The second half of the energy generated in the basic metabolism is “dissipated” and released into the internal and external environment. This part of the energy, in the thermoregulation process, provides isothermal state of the body of animals. Thermal homeostasis, the range of fluctuations in body temperature within the physiological norm is a significant part of the metabolic energy consumption. The article presents results of studying such consumptions when adapting to feeding factors and changes of weather conditions of cattle of different age and productivity.

Natural convection from discrete reactions on the bottom wall of an enclosure

Natural convection resulting from discrete reactive heat sources on the bottom wall of an enclosure is investigated. The rest of the bottom wall apart from the heat sources, the top wall, and the vertical walls are kept at the surrounding temperature. The remarkable findings, which have not been reported in any study, are that the flow field and temperature distribution are quite distinct depending on the odd and even number of reactive heat sources on the bottom wall. For odd numbers of heat sources, the vortices have a quite sharp corner near the center of the middle source and the base of the thermal plume is in the middle heat source. Contrary to this, for even numbers of reactive heat sources, the vortices have a blunt corner near the middle two heat sources and it seems that the thermal plume evolves from the coalescence of the middle two heat sources. Whatever the number of heat sources, for increasing the Rayleigh number, the maximum value of the stream function increases and the maximum temperature decreases. However, both of them are increased for higher values of the Frank–Kamenetskii number. It is also observed that the heat loss to the environment through the walls of the enclosure is stronger with the increase in the Rayleigh number and Frank–Kamenetskii number.

Modeling and Analysis of Temperature Compensation for Multi-temperature Zone Sintering Furnace Temperature Sensing

The surface temperature of workpieces in a multi-temperature zone sintering furnace is an important parameter to characterize the performances of a multi-temperature zone system. Due to the practical structural properties of the sintering furnace, however, the conventional way of temperature measurement cannot detect the exact surface temperature of the workpieces directly, making it difficult to control the multi-temperature zone system performances precisely. To address such an issue, this paper proposes, for the first time to the best of our knowledge, a temperature compensation algorithm based soft-measurement technique to compensate for the temperature of the measuring points. Specifically, we analyze the heat-transfer mechanism within the electric heating surface and the workpiece surface and establish a mathematical model for it first, and then calculate the radiative heat transfer coefficient between the diffuse gray surface within sintering furnace using the discrete radiation heat transfer method. Finally, the heat transfer mechanism based soft measurement technique is proposed and applied to compensate for the temperature. Both simulations and experiments with a practical sintering furnace are conducted to verify the correctness and effectiveness of the proposed temperature compensation algorithm in different cases. Results show that the proposed algorithm could help maintain the temperature difference within a range limit of ±5°, which is much better as compared with the conventional temperature measurement methods.

Modeling and simulation of the temperature field of selective laser sintering

Selective Laser Sintering (SLS) is a kind of powder-bed additive manufacturing technology, which can directly shape parts with complex geometry. The heat transfer process has an important influence on the forming quality of the SLS process. Computational methods are useful to study the heat transfer in the process since the temporal and spatial scale is extremely tiny. This paper presented a model, coupling the discrete element method and heat transfer equations, to calculate the powder-applying and heat-transfer phenomenon. Firstly, the discrete element method and the basic framework of the heat pipe model were introduced, and then the process of manufacturing a mini propeller blade by SLS was simulated.

An approach based on the porous media model for numerical simulation of 3D finned-tubes heat exchanger

Developing efficient and compact heat exchangers is of important significance to improve the energy utilization efficiency of entire society. 3D finned-tubes, as a kind of passive enhanced heat exchange technology, are widely used for improving the thermal-hydraulic performance of the heat exchanger. In the numerical study of 3D finned-tube heat exchanger, on account of the limit by computing capacity, it is very hard to carry out the full-size numerical simulation, so the method based on porous media model becomes a common approach for numerical simulation of the heat exchanger. In the present study with the discrete 3D finned-tube heat exchanger as the study object, a kind of approach based on the porous media model under the cylindrical coordinate system is proposed. In this approach, only the fin region on the surface of base tube is simplified to the annular porous media zone while the region beyond the fin tips is still regarded as the pure gas flow region. The resistance coefficients of annular porous media, which are used to represent the retardation effects of the fin on the fluid motion, are given under the cylindrical coordinate system. Based on serialized numerical simulations of the fin configuration model seen by the air when it flows in the circumferential, axial and radial direction, the viscous resistance coefficients and inertial resistance coefficients of the annular porous media zone in each direction are obtained through fitting. Through comparison of simulated values between the porous media model and 3D finned-tube practical physical model, it is found that the pressure drop calculated by the method based on porous media model proposed in this study agree well with that based on the real physical model. The pressure drop predicted by the porous media model also agree well with the experimental results. In the approach based on porous media model, the grid number and calculation cost are smaller than those in practical physical model. Especially in case of tube bundles, number of grids applied in the simulation based on porous media model is far less than the grid quantity in the practical physical model simulation, but the pressure drop obtained by the porous media model is almost equal to the result obtained from the practical physical model. Besides, in the present method, only the fin zone is deemed as the porous media, so the flow field in the region beyond the fin tips will still be solved as same as that in the full-size simulation. Thus more flow details within the heat exchanger can be captured. The approach based on porous media model under the cylindrical coordinate system proposed can be used in the simulation study of the internal flow field of 3D finned-tube heat exchangers in actual scale. On this basis, this approach can contribute to optimization design of the 3D finned-tube heat exchanger.

Numerical Investigation of Rayleigh–Benard Natural Convection and Entropy Generation in a Cubic Cavity With Discrete Heat Sources

Abstract Three-dimensional (3D) natural convection with isothermal discrete heat sources in a cubical cavity has been carefully studied using the 3D vector potential–vorticity formulation. Based on the finite volume method, the governing equations are solved with a homemade computational code (written in Fortran). Assuming that all cavity vertical walls are adiabatic, the upper wall of the cavity is kept at a cold temperature. However, in the bottom face, heat sources are placed under different configurations. The size of the discrete sources, their positions, and their numbers are varied for different Rayleigh numbers. The Prandtl number is fixed at 0.71. Three-dimensional distribution of the temperature iso-surfaces, the heat transfer rate, and entropy generation is evaluated. It is found that heat transfer and entropy generation are strongly affected by the arrangement of the discrete heated sources. In conclusion, the heat transfer rate is maximized, and the entropy generation is minimized for the inline arrangement of more than two heaters compared to the diagonal one.

Large time behaviour for the heat equation on [formula omitted], moments and decay rates

The paper is devoted to understand the large time behaviour and decay of the solution of the discrete heat equation in the one dimensional mesh Z on ℓp spaces, and its analogies with the continuous-space case. We do a deep study of the moments of the discrete gaussian kernel (which is given in terms of Bessel functions), in particular the mass conservation principle; that is reflected on the large time behaviour of solutions. We prove asymptotic pointwise and ℓp decay results for the fundamental solution. We use that estimates to get rates on the ℓp decay and large time behaviour of solutions. For the ℓ2 case, we get optimal decay by use of Fourier techniques.