Temperature Gradient Vector Research Articles

Modeling of heat conduction through rate equations

AbstractStarting from a classical thermodynamic approach, we derive rate-type equations to describe the behavior of heat flow in deformable media. Constitutive equations are defined in the material (Lagrangian) description where the standard time derivative satisfies the principle of objectivity. The statement of the Second Law is formulated in the classical form and the thermodynamic restrictions are then developed following a variant of the Coleman-Noll procedure where the entropy production too is given by a non-negative constitutive equation. Both the free energy and the entropy production are assumed to depend on a common set of independent variables involving, in addition to temperature, both temperature gradient and heat-flux vector together with their time derivatives. This approach results in rate-type constitutive function for the heat flux that are intrinsically consistent with the Second Law and easily amenable to analysis. In addition to providing already known models (e.g., Maxwell-Cattaneo-Vernotte’s and Jeffreys-like heat conductors), this scheme allows the formulation of new models of heat transport that are likely to apply also in nanosystems. This is consistent with the fact that higher-order time derivatives of the heat flux are in order when high-rate regimes occur.

Aerothermal performance improvement by cross and X ribbed stripes

The incentive to save energy in heat exchangers has led to enhanced thermal performance in ducted flows, achieved via inserts designed to increase the heat transfer efficiency and efficacy. We introduce a novel approach using cross and X stripe inserts, roughened by 90° and 45° ribs, to enhance the structural integrity and heat transfer rate of the duct. Additionally, the incorporation of edged notches aims to improve aerothermal performance by directing flow traction away from the recirculating cell behind a rib, thereby reducing the drag. Utilizing the infrared thermography method, this research quantifies the full-field Nusselt number distributions for 34 enhanced ducts at Reynolds numbers 5,000–25,000. The outcomes of flow simulations elucidate the underlying mechanisms of aerothermal performance enhancement afforded by these inserts. These enhancements result from the combined effects of increased turbulence intensity, fluid mixing, and rib-induced multicellular vortices, which improve the synergy between fluid velocity and temperature gradient vectors, thereby optimizing heat transfer. Quantitatively, the Nusselt numbers (friction factors) for ducts with cross or X ribbed stripes increase significantly. For cross stripes, the ranges increase by 1.25–3.17 times (3.14–21.75 for friction factors), and for X stripes, 1.29–3.39 times (6.58–43.41 for friction factors), compared with the Dittus–Boelter (Blasius) standards. Notably, the aerothermal performance index for ducts featuring notched cross stripes with 45° ribs and a rib pitch-to-height ratio of 10 reaches 1.47 to 1.11, though this decreases with increasing Reynolds number. Furthermore, the study presents empirical correlations to assess the average Nusselt number and friction factor in a square duct equipped with either cross or X ribbed stripes.

An improved theory of thermoelasticity for ultrafast heating of materials using short and ultrashort laser pulses

This paper introduces an improved thermoelastic theory designed specifically to address the challenges posed by fast and ultrafast heating of materials. The proposed theory incorporates a time delay in the temperature gradient and introduces the rate of temperature gradient as an additional term in the heat equation. This term accounts for the phase lag and the precedence of the temperature gradient and heat flux vector. By leveraging the sign of the temperature gradient rate, the theory automatically determines the precedence of either the temperature gradient or the heat flux vector. To verify the effectiveness of the proposed theory, experiments reported in the literature were simulated using several theories, namely the classical, Lord-Schulman (LS), Green-Lindsay (GL), dual-phase-lag (DPL), and the presented theory. The simulations considered the effects of relaxation times and pulse duration on the results. The findings reveal that for pulse durations shorter than 50 ps, the second-order time derivative of the temperature gradient becomes a crucial term in the energy equation for accurately predicting temperature profiles. Additionally, as the pulse duration decreases, the sensitivity to relaxation times increases. Regarding the surface temperature corresponding to the material ablation threshold, it is observed that for an 80 fs laser pulse, the ablation temperature falls between the boiling and critical temperatures, significantly higher than the boiling temperature associated with the ablation by a 270 ps pulse duration. Lastly, the simulation results exhibit a high level of agreement with the experimental data, particularly in terms of displacements, further validating the accuracy of the presented simulations.

Numerical investigation on thermal–hydraulic characteristics of the helically corrugated tube in intermediate heat exchangers for very-high-temperature gas-cooled reactors

A three-dimensional numerical simulation is conducted to investigate the flow and heat transfer characteristics of the helically corrugated tube (HCT) intermediate heat exchanger (IHX) for very-high-temperature gas-cooled reactors (VHTRs). Helium is the working fluid with Reynolds number (Re) ranging from 10,000 to 20,000 and Prandtl numbers of 0.66. The heat transfer enhancement mechanism is analyzed in detail with the use of the ANSYS Fluent 18.0 computational fluid dynamics program. The results show that the helical corrugation disrupts the boundary layer development, strengthens the secondary flow intensity, and improves the synergy between the velocity vector field and temperature gradient vector field, thus enhancing the heat transfer. Moreover, the effect of geometric parameters characterized by dimensionless corrugation height e/d (0.02, 0.04, and 0.06) and dimensionless corrugation pitch p/d (0.5, 1.0, and 1.5) on the thermal–hydraulic performance are examined. The results indicate that increasing e/d and reducing p/d can enhance heat transfer. The Nusselt number of HCTs is increased by 17.25 %-81.46 % compared to a smooth tube, while the friction factor is also increased by 38.01 %-361.69 %. The comprehensive performance represented by the performance evaluation criterion is improved by 3.26 %–11.84 %. In the present study, e/d = 0.04 and p/d = 0.5 are the optimal HCT structures to obtain better comprehensive performance for the IHX design.

Three-dimensional thermoelectroelastic analysis of structures with distributed piezoelectric sensors and actuators with temperature-dependent material properties

In this paper, a geometrically exact hybrid-mixed four-node laminated solid-shell element with piezoelectric patches with temperature-dependent material properties through the sampling surfaces (SaS) formulation is developed. The SaS formulation is based on the choice of SaS, parallel to the middle surface and located at Chebyshev polynomial nodes within the layers, to introduce the temperatures, displacements and electric potentials of these surfaces as basic shell unknowns. The outer surfaces and interfaces are also included into a set of SaS. Such a choice of unknowns with the use of Lagrange polynomials in the through-thickness approximations of temperatures, temperature gradient, displacements, strains, electric potential and electric field leads to a very compact higher-order thermopiezoelectric shell formulation. To implement efficient analytical integration throughout the solid-shell element, the extended assumed natural strain method is employed for all components of the temperature gradient, strain tensor and electric field vector. The developed hybrid-mixed four-node laminated solid-shell element with piezoelectric sensors and actuators is based on the extended Hu-Washizu variational principle and shows superior performance in the case of coarse meshes. This can be useful for the three-dimensional temperature-dependent response of laminated composite shells with piezoelectric patches, since the SaS formulation allows one to obtain the numerical solutions with a prescribed accuracy, which asymptotically approach the exact solutions of thermopiezoelectricity as a number of SaS tends to infinity.

Prediction of Epitaxial Grain Growth in Single-Track Laser Melting of IN718 Using Integrated Finite Element and Cellular Automaton Approach.

The mechanical properties of selective laser melting (SLM) components are fundamentally dependent on their microstructure. Accordingly, the present study proposes an integrated simulation framework consisting of a three-dimensional (3D) finite element model and a cellular automaton model for predicting the epitaxial grain growth mode in the single-track SLM processing of IN718. The laser beam scattering effect, melt surface evolution, powder volume shrinkage, bulk heterogeneous nucleation, epitaxial growth, and initial microstructure of the substrate are considered. The simulation results show that during single-track SLM processing, coarse epitaxial grains are formed at the melt–substrate interface, while fine grains grow at the melt–powder interface with a density determined by the intensity of the heat input. During the solidification stage, the epitaxial grains and bulk nucleated grains grow toward the top surface of the melt pool along the temperature gradient vectors. The rate of the epitaxial grain growth varies as a function of the orientation and size of the partially melted grains at the melt–substrate boundary, the melt pool size, and the temperature gradient. This is observed that by increasing heat input from 250 J/m to 500 J/m, the average grain size increases by ~20%. In addition, the average grain size reduces by 17% when the initial substrate grain size decreases by 50%. In general, the results show that the microstructure of the processed IN718 alloy can be controlled by adjusting the heat input, preheating conditions, and initial substrate grain size.

Heat transfer and viscous polymer melting capacity correlation in self-controlled torsion induced extrusion

A novel design concept of torsion screw configuration has been proposed to improve polymer melting ability. The heat transfer and melt flow behavior of a two-phase viscous polymer, induced by torsion elements in a screw compression section, were analyzed with the computational fluid dynamics simulation. It has been found that screws with added torsion elements have larger Nusselt number and liquid mass fraction, and shorter melting zone compared to that of a conventional screw. The heat transfer and melting capacity improve with the increasing number of torsion elements and their preferred arrangement in the screw. A simplified simulation model of torsion and standard elements has been developed to predict the heat transfer and melt flow mechanism, and the simulation results are in consistent with the field synergy theory. The existence of torsional flow in the region of torsion channels increases the perturbation of the fluid, improves interaction between temperature gradient and velocity vectors, resulting in improved mass and heat transfer and rapid melting of polymers. Furthermore, the paper proposed a qualitative melt flow model of polymers in the channel of torsion element.

Analytical study of the Ekman angle for the Benard–Marangoni convective flow of viscous incompressible fluid

The paper considers the convective flow of a viscous incompressible fluid over a rotating surface. It studies the angle between the fluid velocity vector in the upper layer and the temperature gradient vector on the free surface. For the study, an analytical solution to the Oberbeck–Boussinesq equations is constructed, which describes the stratified Ekman flow taking into account two components of the Coriolis force. The temperature gradient and the conditions of Marangoni thermocapillary convection are set at the upper (free) boundary, and the condition of fluid adhesion is set on the lower (solid) boundary. The representation of velocities in the form of linear functions of horizontal coordinates is used. It is shown that, when the flow depth tends to infinity, the angle between the upper layer fluid velocity vector and the temperature gradient vector tends to π/2 .

Effect analysis on performance enhancement of a hydrogen/air non-premixed micro combustor with sudden expansion and contraction structure

Improvement of combustion stability and efficiency is still the main challenge of micro combustors. In this work, a non-premixed combustion model of the hydrogen/air in a dimensionless micro combustor with sudden expansion and contraction structure at the entrance is established. The effects of different micro combustor diameters, inlet velocities and wall materials on non-premixed combustion performance enhancement of the hydrogen/air in the micro combustor is investigated. By comparing the temperature distribution and combustion efficiency of the micro combustor, the combustion characteristics of the hydrogen/air under different influence factors are obtained. The results show that the appropriate diameter of the micro combustor and the inlet velocity are of a great influence on the stable combustion. Furthermore, the field synergy analysis between temperature gradient and velocity vector fields under different inlet velocities of hydrogen/air is carried out, and an enhanced combustion method is proposed in this work. The calculation results show that the inlet velocity decrescence of the hydrogen can significantly improve combustion stability under constant inlet velocity of air condition, which will be useful for stabilizing non-premixed combustion in the micro combustor.

Coupled thermoelectroelastic analysis of thick and thin laminated piezoelectric structures by exact geometry solid‐shell elements based on the sampling surfaces method

AbstractA hybrid‐mixed exact geometry four‐node thermopiezoelectric solid‐shell element through the sampling surfaces (SaS) formulation is proposed. The SaS formulation is based on the choice of an arbitrary number of SaS within layers parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the temperatures, displacements and electric potentials of these surfaces as basic shell unknowns. Due to the variational formulation, the outer surfaces and interfaces are also included into a set of SaS. Such choice of unknowns with the use of Lagrange polynomials in the through‐thickness approximations of temperatures, temperature gradient, displacements, strains, electric potential, and electric field leads to a very compact higher‐order thermopiezoelectric shell formulation. To implement efficient analytical integration throughout the solid‐shell element, the extended assumed natural strain method is employed for all components of the temperature gradient, strain tensor, and electric field vector. The developed hybrid‐mixed four‐node thermopiezoelectric solid‐shell element is based on the Hu–Washizu variational principle and shows a superior performance in the case of coarse meshes. It can be useful for the three‐dimensional thermoelectroelastic analysis of thick and thin doubly curved laminated piezoelectric shells under thermal loading, since the SaS formulation allows one to obtain the numerical solutions with a prescribed accuracy, which asymptotically approach exact solutions of the theory of thermopiezoelectricity as a number of SaS tends to infinity.

Thermal Conductivity of Grade UPV-1 Pyrolytic Graphite at 1900–2950 K

The need to account for a misalignment between the heat flux density and temperature gradient vectors when studying thermal conductivity of anisotropic materials was analyzed. A method for measuring thermal conductivity of pyrolytic graphite (grade UPV-1) in the direction parallel to the precipitation plane was proposed. The advantage of the proposed method is the possibility to determine thermal conductivity of pyrolytic graphite in the direction parallel to the precipitation plane while accounting for a misalignment between the heat flux density and temperature gradient vectors. The test samples were shaped as hollow cylinders with the pyrolytic graphite precipitation plane located along the radius of the cylinder. The heat flux density was determined based on the radiation heat flux emitted from the outer surface of the sample, and the temperature gradient was calculated along the radius, which made it possible to maintain the alignment between the heat flux density and temperature gradient vectors. A comparative analysis of the thermal conductivity values obtained in this study (parallel to the precipitation plane) and those reported in the reference sources was performed. The studied temperature range was extended into the higher temperature region by 450 K and constitutes 1900–2950 K.

Three-Dimensional Ballistic-Diffusive Heat Transport in Silicon: Transient Response and Thermal Conductivity

Abstract Phonons are the main contributors to thermal energy transfer in thin films. The size dependence of the thermal transport characteristics alters the film properties such as thermal conductivity. Hence, in the present study, three-dimensional, transient phonon transport in dielectric material is studied through the Equation of Phonon Radiative Transport (EPRT) to assess the size dependence of thermal conductivity. The numerical scheme is introduced solving the EPRT in three dimensions and the governing algorithm is described in detail. A parametric study is carried out examining the effect of the Kn \mathrm{Kn} number on the thermal energy transport characteristics in three-dimensional thermally excited film. The formulation and estimation of the effective thermal conductivity tensor is presented and discussed, thereby extending, to some extent, the one-dimensional results obtained earlier. We demonstrate that thermal conductivity changes in all directions, depending on the size effect. In addition, the directions of the temperature gradient and heat flux vectors differ as the Kn \mathrm{Kn} number approaches unity.

The relative orientation between the magnetic field and gradients of surface brightness within thin velocity slices of 12CO and 13CO emission from the Taurus molecular cloud

ABSTRACT We examine the role of the interstellar magnetic field to modulate the orientation of turbulent flows within the Taurus molecular cloud using spatial gradients of thin velocity slices of 12CO and 13CO antenna temperatures. Our analysis accounts for the random errors of the gradients that arise from the thermal noise of the spectra. The orientations of the vectors normal to the antenna temperature gradient vectors are compared to the magnetic field orientations that are calculated from Planck 353 GHz polarization data. These relative orientations are parameterized with the projected Rayleigh statistic and mean resultant vector. For 12CO, strongly parallel and strongly perpendicular relative orientations are found in 28 percent and 39 percent of the cloud area respectively. For the lower opacity 13CO emission, strongly parallel and strongly perpendicular orientations are found in 7 per cent and 43 per cent of the cloud area, respectively. For both isotopologues, strongly parallel or perpendicular alignments are restricted to localized regions with low levels of turbulence. If the relative orientations serve as an observational proxy to the Alfvénic Mach number then our results imply local variations of the Alfvénic Mach number throughout the cloud.

Thermoelastic Response of a Rotating Hollow Cylinder Based on Generalized Model with Higher Order Derivatives and Phase-Lags

Generalized thermoelastic models have been developed with the aim of eliminating the contradiction in the infinite velocity of heat propagation inherent in the classical dynamical coupled thermoelasticity theory. In these generalized models, the basic equations include thermal relaxation times and they are of hyperbolic type. Furthermore, Tzou established the dual-phase-lag heat conduction theory by including two different phase-delays correlating with the heat flow and temperature gradient. Chandrasekharaiah introduced a generalized model improved from the heat conduction model established by Tzou. The present work treats with a novel generalized model of higher order derivatives heat conduction. Using Taylor series expansion, the Fourier law of heat conduction is advanced by introducing different phase lags for the heat flux and the temperature gradient vectors. Based on this new model, the thermoelastic behavior of a rotating hollow cylinder is analyzed analytically. The governing differential equations are solved in a numerical form using the Laplace transform technique. Numerical calculations are displayed tables and graphs to clarify the effects of the higher order and the rotation parameters. Finally, the results obtained are verified with those in previous literature.

Experimental and numerical analysis for thermal-hydraulic characteristics in circular tubes with twisted tape and wire coil inserts

Thermal-hydraulic characteristics in circular tubes with twisted tape (Y=4.762, 7.143 and 9.524) and wire coil (p=20mm, e=2mm) inserts were studied experimentally and numerically. It was found from experimental analysis that Nusselt number and friction factor in tubes with twisted tape inserts were 38%-57% and 148%-202%, respectively, higher than those of plain tubes .When wire coil was inserted in plain tubes, the Nusselt number and friction factor grew by 75%-94% and 685%-792%, respectively. In particular, Nusselt number and friction factor in tubes with twisted tape and wire coil inserts were 113%-139% and 836%-874%, respectively, greater than those of plain tubes. Furthermore,PECvalue of tubes with the twisted tape and wire coil inserts was larger than all others. Numerical analysis shows that a large tangential velocity can be generated after inserting a twisted band or coil in the tube. The angle between the temperature gradient vector and velocity vector was significantly less than 90°. So collaborative degree between temperature field and velocity field was improved.

A Study on the Mechanism of Convective Heat Transfer Enhancement Based on Heat Convection Velocity Analysis

This paper explores the mechanism of convective heat transfer enhancement in a new perspective. In this paper, a new parameter called heat convection velocity is proposed based on the field synergy principle. It is defined as the velocity projection on the temperature gradient vector and reflects the magnitude of the velocity component that contributes to heat convection. Three typical cases are taken into consideration to investigate the influence factors of Nusselt number theoretically. The results indicate that the Nusselt number can be enhanced by increasing the mean heat convection velocity and the dimensionless mean temperature difference. Through theoretical analysis, three suggestions are found for designing heat transfer enhancement components: (a) the overall synergetic effect should be improved; (b) the fluid with lower temperature gradient should be guided to the region where the temperature gradient is higher; (c) temperature distribution should be an interphase distribution of hot and cold fluid. Besides, the heat convection velocity is used to investigate the mechanism of convective heat transfer in the smooth tube. It is found that the increase of Nusselt number is due to the increase of heat convection velocity. In addition, according to design suggestions, a new insert is invented and inserted in the circular tube. With heat convection velocity analysis, it is found that there is much potential of increasing heat convection velocity for enhancing the convective heat transfer in the circular tube.

The turbulent Kármán vortex

This work focuses on the temperature (passive scalar) and velocity characteristics within a turbulent Kármán vortex using a phase-averaging technique. The vortices are generated by a circular cylinder, and the three components of the fluctuating velocity and vorticity vectors, $u_{i}$ and $\unicode[STIX]{x1D714}_{i}$ ($i=1,2,3$), are simultaneously measured, along with the fluctuating temperature $\unicode[STIX]{x1D703}$ and the temperature gradient vector, at nominally the same spatial point in the plane of mean shear at $x/d=10$, where $x$ is the streamwise distance from the cylinder axis and $d$ is the cylinder diameter. We believe this is the first time the properties of fluctuating velocity, temperature, vorticity and temperature gradient vectors have been explored simultaneously within the Kármán vortex in detail. The Reynolds number based on $d$ and the free-stream velocity is $2.5\times 10^{3}$. The phase-averaged distributions of $\unicode[STIX]{x1D703}$ and $u_{i}$ follow closely the Gaussian distribution for $r/d\leqslant 0.2$ ($r$ is the distance from the vortex centre), but not for $r/d>0.2$. The collapse of the distributions of the mean-square streamwise derivative of the velocity fluctuations within the Kármán vortex implies that the velocity field within the vortex tends to be more locally isotropic than the flow field outside the vortex. A possible physical explanation is that the large and small scales of velocity and temperature fields are statistically independent of each other near the Kármán vortex centre, but interact vigorously outside the vortex, especially in the saddle region, due to the action of coherent strain rate.

A Fully-Mixed Finite Element Method for then-Dimensional Boussinesq Problem with Temperature-Dependent Parameters

AbstractIn this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection ofn-dimensional fluids,n∈{2,3}{n\in\{2,3\}}, with temperature-dependent viscosity and thermal conductivity. The mathematical model is given by the coupling of the equations of continuity, momentum (Navier–Stokes) and energy by means of the Boussinesq approximation, as well as mixed thermal boundary conditions and a Dirichlet condition on the velocity. Because of the dependence on the temperature of the fluid properties, several additional variables are defined, thus resulting in an augmented formulation that seeks the rate of strain, pseudostress and vorticity tensors, velocity, temperature gradient and pseudoheat vectors, and temperature of the fluid. Using a fixed-point approach, smallness-of-data assumptions and a slight higher-regularity assumption for the exact solution provide the necessary well-posedness results at both continuous and discrete levels. In addition, and as a result of the augmentation, no discrete inf-sup conditions are needed for the well-posedness of the Galerkin scheme, which provides freedom of choice with respect to the finite element spaces. In particular, we suggest a combination based on Raviart–Thomas, Lagrange and discontinuous elements for which we derive optimal a priori error estimates. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical rates of convergence are reported.

Heat-transfer enhancing mechanisms induced by the coherent structures of wall-bounded turbulence in channel with rib

The detailed coherent structures and closely-related heat-transfer characteristics were investigated along the wall of channel with rib tabulator by direct numerical simulation (DNS). The three-dimensional Navier-Stokes equations were solved using the density-based algorithm and strongly-coupled with the heat-transfer equation. Two typical Reynolds numbers defined by the rib height were set at 500, 1200, and the Mach numbers was 0.2, respectively. The heat-transfer enhancing mechanisms incurred by the wall-bounded turbulence coherent structures were investigated and clarified. The DNS results suggest that the bypass transition occurs in the presence of rib structure which plays a role of strong source of disturbance. A large-scale spanwise vortical structure emerges firstly and stretches into a Λ-shaped vortex, and then evolves into a hairpin vortex. As a result, the quasi-streamwise vortice are induced downstream. The Reynolds stress and turbulent kinetic energy are much larger in the turbulent boundary layer (TBL) downstream than those in the laminar boundary layer upstream. The locations of higher Nusselts number (Nu) is consistent with the regions of higher Reynolds stress and turbulent kinetic energy where the coherent structures are active and dominant, giving rise to the phenomena of ejection and sweep. The ejections, sweeps and vortex rotations are the major forms of turbulence motions that enhance the coordination levels between temperature gradient vectors and velocity vectors, which provides the inherent mechanisms to enhance the convective heat-transfer in the rib flow structure.

Phase field simulation of dendrite evolution during powder-based laser metal deposition

An algorithm for simulation of laser metal deposition (LMD) processes is developed, which implements the concept of multi-scale modeling of crystallization. The macro-level model takes into account the coupled dynamics of the free surface (melt-gas) and the energy and mass transfer of the multiphase mixture, which is applied on scales of the order of a few to tens of millimeters. The micro-level model describes the evolution of dendritic structures using the phase-field method (PFM) at scales of the order of a few to tens of micro-meters. Wherein, the resulting macroscopic temperature distribution in the system defines the boundary and initial conditions of the micro-model, which, in turn, control the growth patterns and evolution of the dendrites. In local sub-region, the evolution of the dendrite in an advantageous position was simulated, i.e. having favorable thermodynamic conditions of its growth. Before solving the micro-scale problem with the PFM, the 3D mesh conversion procedure to the new 2D mesh is performed in the sub-region of interest. To do this, we select the plane in which the local temperature gradient vector lies. The test dynamic profiles of the phase field and temperature in different spatial regions of the formed bead are calculated. Simulations used the values of physical quantities for iron. The results of calculations showed that the developed numerical model qualitatively describes the physics of crystallization and the formation of dendrites in the LMD process. The effect on the growth dynamics of dendritic structures of such PFM parameters as thermal noise and the anisotropy amplitude is analyzed. It is shown that thermal fluctuations strongly affect the branching of dendrites. With an increase in the fluctuation amplitude, the phase boundary becomes unstable with the formation of secondary and tertiary branches.