Heat Flow Vector Research Articles

Mathematical modeling of insulating coating of thermal conductivity including body’s own radiation and non-local spatial effects

Abstract In this paper thermal conductivity analysis of insulating coating is discussed. The general partial differential thermal conductivity equation is used with consideration to nonlocal boundary effects. We relying on the Eringen’s approach involves heat-flow vector transformation using nonlocal influence function and weight coefficients. Also presented model includes own surface radiation of the coating.

Examining basic theorems and plane waves in the context of thermoelastic diffusion using a multi-phase-lag model with temperature dependence

This study employs a novel mathematical formulation of temperature-dependent thermoelastic diffusion with multi-phase delays (TDMT). The TDMT model, which is assumed, is obtained by converting the fundamental rules of Fourier’s and Fick’s laws into higher order time derivatives of the heat flow vector, the temperature gradient, diffusing mass flux, and the chemical potential gradient. Fundamental theorems such as energy, uniqueness, reciprocity theorems, and variational criterion are established using the fundamental equations for the assumed model TDMT. The reciprocity theorem is applied to a particular scenario by using instantaneous concentrated body forces, heat sources, chemical potential sources, and moving heat sources as examples. It is noted that both the variational criterion and these theorems depend on the field variables’ susceptibility as well as the fluctuations in the higher order temporal derivatives’ parameters. Additionally, the assumed model’s two-dimensional (2D) plane wave propagation is described. Three longitudinal waves are identified: the longitudinal (P) wave, the thermal (T) wave, the chemical potential (CP) wave, and the linked transverse (SV) wave. Wave properties (phase velocity and attenuation coefficient) are calculated numerically and shown visually. A few special examples are also investigated and compared with the established findings. The framework for additional research into basic issues in thermoelastic continua under various physical field variables is provided by this work. Numerous applications in material science, geomechanics, soil dynamics, and the electronic sector can be made of the current findings.

Generalized MGT Heat Transfer Model for an Electro-Thermal Microbeam Lying on a Viscous-Pasternak Foundation with a Laser Excitation Heat Source

In this study, the effects of laser light on the heat transfer of a thin beam heated by an applied current and voltage are investigated. Laser heating pulses are simulated as endogenous heat sources with discrete temporal properties. The heat conduction equation is developed using the energy conservation equation and the modified Moore–Gibson–Thompson (MGT) heat flow vector. Thermal and structural analysis of Euler–Bernoulli microbeams is provided with the support of visco-Pasternak’s base with three parameters. Using the Laplace transform method, an approximation of an analytical solution is found for the field variables being examined. A comparison was made of the impacts of laser pulse length, the three foundation coefficients, and the thermal parameters on the responses to changes in measured thermophysical fields, such as deflection and temperature.

Influence of a cubic wood particle orientation in space on the characteristics and conditions of its ignition

The article presents the results of the experimental and theoretical studies of the ignition processes of dry particles of woody biomass with a wide variation in heating conditions. A detailed analysis of the typical frames of the videogram has shown that the temperature of the oxidizing agent affects the nature of the ignition process of a wood particle. Under conditions of moderate ambient temperatures (Tg ≤ 873K), ignition, as a rule, occurs in the gas region under the particle. At the same time, the ignition of wood particles under conditions of high-temperature heating (at Tg > 1073K) proceeds in the gas phase in a small (very thin) neighborhood around the wood particle along its entire perimeter.A model is formulated that describes the ignition under the considered rather typical conditions and differs from the known description of the processes of convection of gaseous pyrolysis products during the induction period. Based on the results of the experiments, the influence of the type of woody biomass on the characteristics and conditions of ignition of fuel particles has been analyzed. It has been established that dry particles of woody biomass made of pine under relatively low ambient temperatures (at Tg≤873 K) ignite much faster compared to particles of similar size from birch, aspen and cedar. Differences in ignition delay times reach 90%. At high ambient temperatures (Tg≥1273K), the ignition delay times of all four investigated fairly typical types of woody biomass differ insignificantly (the deviation of the ignition delay times does not exceed 5%).The analysis of the influence of the orientation of wood fibers in space relative to the direction of the heat flow vector on the characteristics and conditions of ignition has also been carried out. It is shown that the orientation of the wood particle in space does not significantly affect the characteristics and conditions of ignition. This trend has been registered for all four types of wood studied.

Fractional Moore-Gibson-Thompson heat transfer model with two-temperature and non-singular kernels for 3D thermoelastic solid

A new model of thermoelasticity based on fractional calculus in combination with Fourier's law of heat and dual temperature theory is presented, including the Moore-Gibson-Thomson equation. In the suggested model, the heat equation is represented in a fractional integral form using the Atangana–Baleanu fractional operators. The presence of the concept of the delay parameter in the heat flow vector leads to a finite velocity of heat wave propagation. The model is then applied to investigate the thermoelastic interaction in a homogeneous thermoelastic 3D elastic solid with a thermally loaded and traction-free surface. In the field of Laplace and double Fourier transforms, analytical solutions for thermal variables were derived, which were then reversed to the time-space by means of the Honig-Hirdes procedure. The classical theories of thermoelasticity and some generalized theories as special examples can be derived from the presented model of thermoelasticity with a two-temperature and fractional derivative. Numerical values of the studied field variables were calculated and presented graphically. In order to explore the influences of relaxation time, temperature discrepancy, and fractional order parameters, certain comparisons of thermophysical variables were presented in the figures.

Morphological and nanomechanical changes in tungsten in high heat flux conditions

Morphological and nanomechanical alteration of tungsten in extreme environments, like those in edge localized modes in nuclear fusion environments, up to 46.3 GWm−2 heat fluxes were experimentally simulated using electrothermal plasma. Surface and subsurface damage to the tungsten is seen mainly in the form of pore formation, cracks, and resolidified melt instabilities. Mirco voids, rosette-type microfeatures, core-shell structure, particle enrichment, and submicron channels all manifest in the damaged subsurface. The formation of voids in the subsurface was determined to originate from the ductile fracture of hot tungsten by plastic flow but not developed to cracking. The voids were preferentially settled in grain boundaries, interfaces. The directionality of elongated voids and grains is biased to the heat flow vector or plasma pathway, which is the likely consequence of the thermally driven grain growth and sliding in the high-temperature conditions. The presence of a border between the transient layer and heat-affected zone is observed and attributed to plasma shock and thermal spallation of fractural tungsten at high temperature. Plasma peening-like hardening effects in tungsten were observed in the range of 22.7–46.3 GWm−2 but least in the case of the lowest heat flux, 12.5 GWm−2.

Second order causal hydrodynamics in Eckart frame: using gradient expansion scheme

In the present work, we develop a causal theory of relativistic non-ideal fluids up to the second order in the Eckart frame using gradient expansion scheme. Keeping the spirit of Mueller–Israel–Stewart formalism, the general forms of bulk viscosity, shear viscosity tensor and the heat flow vector are presented. Since each of the flux quantities explicitly carry curvature terms, we show that our formalism finds application in astrophysics in particular in the strong gravity regime. We elucidate two such applications namely in viscous thick accretion disks also known as Polish doughnuts and in addressing non-rotating equilibrium configuration, like neutron stars.

Generalized mathematical novel model of thermoelastic diffusion with four phase lags and higher-order time derivative

In this work, taking into account the influences of thermal and diffusion, a new four-phase-lag model comprising the macroscopic and microscopic is constructed. The introduced model is an extension of the papers of Nowacki (Bull Acad Pol Sci Ser Sci Tech 22:55–64, 1974; Bull Acad Pol Sci Ser Sci Tech 22:129–135, 1974; Bull Acad Pol Sci Ser Sci Tech 22:257–266, 1974; Proc Vib Prob 15:105–128, 1974), Sherief et al. (Int J Eng Sci 42:591–608, 2004) and Aouadi (J Therm Stress 30:665–678, 2007). In this model, the Fourier’s and the Fick’s laws have been modified to include higher-order time derivatives of heat flow vector, the gradient of temperature, diffusing mass flux and gradient of chemical potential. Many models in the thermoelastic-diffusion field have been deduced as special cases from the current investigation. Using the resulting formulation, we have studied a thermoelastic-diffusion interaction in a half-space exposed to thermal and chemical shock. Also, the sensitivity of the studied field variables to the variation of the parameters of higher-order time derivative has been investigated.

Two-temperature thermoelastic model without energy dissipation including higher order time-derivatives and two phase-lags

In the current analysis, we have constructed a new general thermoelastic heat conduction model with two-temperature including higher order time-derivatives and two phase-lags. The thermoelastic models established by Chen and Gurtin Chen, Gurtin 1968. Z. Angew. Math. Phys. 19:614–627, Green and Naghdi of type III Green and Naghdi 1993. J. Elast. 31:189–208 and dual-phase-lag model proposed by Tzou 1995 . ASME J. Heat. Transf. 117:8–16 are extended. The proposed model has been restructured by the theory of fading memory, where the heat flow vector depends on the history of the gradient of thermal displacement. The problem of the infinite medium with a cylindrical cavity exposed to a decaying exponentially laser pulse is treated based on the resulting formulation. To explain the validity and accuracy of the derived model, the impacts of various physical parameters on the field variables are discussed in detailed and presented in tabular and graphical forms.

Nonlinear oscillatory fully-developed rarefied gas flow in plane geometry

The nonlinear oscillatory fully developed rarefied gas flow between parallel plates due to an external harmonic force is investigated by the Direct Simulation Monte Carlo (DSMC) method in terms of the parameters characterizing the flow, namely, the gas rarefaction and oscillation parameters, and the force amplitude. The results are in dimensionless form and include the axial velocity, temperature, pressure, wall shear stress, and heat flow vector, as well as the flow rate, cycle-averaged wall shear stress, space-averaged axial heat flow, and pumping power. Even with large force amplitudes, all macroscopic distributions have a sinusoidal pattern with their fundamental frequency being the same with the driving frequency of the external force without the appearance of other harmonics, except of the axial heat flow where the nonlinearities are responsible for generating oscillatory motion containing several harmonics. Nonlinear effects are becoming more significant in highly rarefied flows and low oscillation frequencies. The temperature profile, including the bimodal shape encountered in steady-state flows in the continuum limit, strongly depends on the gas rarefaction and oscillation parameters. The DSMC results have been compared with the corresponding linear oscillatory results, available in the literature, to find out that at small and moderate external forces, the agreement between DSMC and linear flow rates is very good and always remain less than 10%, while at large external forces, the deviation in the flow rate amplitude reaches about 25%. The cycle-averaged oscillatory pumping power is not zero and smaller than the corresponding linear one, following the trend of the flow rates.

Thermal vorticity and thermal chirality in relativistic thermally conducting fluid

The present work focuses on the study of dissipation caused by heat flow in a Carter’s theory of a heat conducting fluid which is expected to be applicable to a particular astrophysical context like early phases of evolution of stellar objects that are far from thermal equilibrium. The dynamical response of entropy fluid in the interaction of heat flow with the motion of matter part of fluid is investigated on the basis of a pair of Maxwell’s like equations that govern the evolution of a heat conducting fluid pioneered by Carter. It is found that the variation of thermal vorticity flux along thermal vorticity tube is coupled to the twist of fluid’s vortex tube and to the spacelike twist of heat flow lines. The proper time rate of change of the thermal vorticity flux in a stream tube associated with the motion of the entropy fluid is related to the combined effect of fluid’s vorticity and spaclike twist of heat flow lines together with the existence of string like structure formed by the entropy flow lines and -lines associated with a particular evolution equation. A covariant solution of a pair of Maxwell’s like equations governing the motion of a heat conducting fluid is obtained under the assumption that the background spacetime representing the gravitational field of a heat conducting fluid is non-circular axisymmetric and stationary. In this case it is shown that the existence of meridional circulation of matter part of fluid is essential for the survival of toroidal components of heat flow vector. A conserved physical quantity called thermal chirality associated with the non-conserved thermal helicity is shown to exist under the assumption that the entropy fluid is in uniform rotation. The relation between the effective thermal angular momentum per entropon of the entropy fluid and twist of heat flow lines is obtained. It is shown that the heat flow causes the twist of the entropy fluid’s vortex lines if the entropy fluid flows are axisymmetric stationary and without meridional circulations but with uniform rotation.

Rotational motion of a stationary axisymmetric relativistic heat conducting fluid configuration

The present work describes some dynamical features associated with the dissipation caused by the heat flow in the context of rotational motion of a heat conducting fluid based on Carter’s two-fluid model. A covariant solutions to a pair of Maxwell’s like equations governing the evolution of a heat conducting fluid are used to determine the heat flow vector and the matter part of fluid’s 4-velocity under the assumption that the space-time configuration representing the gravitational field of a heat conducting fluid is non-circular stationary and axisymmetric. These results are used to derive the rotational velocity in terms of gradients of the effective energy and effective angular momentum per particle. It is demonstrated that the meridional circulations of the matter part of fluid induce the rotational velocity in addition to the usual rotational velocity found in a circular space-time. It is shown that the differential rotation along the thermal –fluid vorticity causes the twist of matter part of fluid’s vortex lines in addition to the other effects .It is found that the gravitational isorotation is balanced by counter rotating thermal isorotation in the case of differentially rotating thermal-fluid surfaces in a frame of reference in which thermal –fluid helicity is conserved. The dynamic interaction between the matter part of fluid and the entropy fluid leads to the mutual exchange between the energy and angular momentum per baryon of the matter part of fluid and the flux of energy and angular momentum per entropon per unit of local temperature coupled to heat flow vector created by the entropy fluid.

Ballistic and Collisional Flow Contributions to Anti-Fourier Heat Transfer in Rarefied Cavity Flow

This paper investigates anti-Fourier heat transfer phenomenon in a rarefied gas confined within a lid-driven cavity using a novel flow decomposition technique in the direct simulation Monte Carlo (DSMC) method proposed by Stefanov and co-workers. An isothermal cavity with different degrees of flow rarefaction from near continuum to mid transition regimes was considered to investigate cold-to-hot heat transfer from ballistic/collision flow decomposition viewpoint. A new cold-to-hot heat transfer indicator in the form of a scalar product of normalized heat flow vector and normalized temperature gradient vector has been introduced for the overall, ballistic and collision parts of these vectors. Using the new indicator, contributions of ballistic and collision flow parts to temperature and heat flux components was investigated with a specific emphasis on the cold-to-hot heat transfer phenomenon. We demonstrated that both ballistic and collision flow parts contribute to the occurrence of cold-to-hot heat transfer. However, it was found out that considered separately both ballistic and collision parts of heat transfer, when related to corresponding ballistic and collision temperature fields, they are ever hot-to-cold for all degrees of flow rarefaction. Thus, cold-to-hot heat transfer is a result of a subtle interplay between ballistic and collision parts in the slip and transition Knudsen regimes.

Overview of thermal conductivity models of anisotropic thermal insulation materials

Currently, the most of existing materials and substances under elaboration are anisotropic. It makes certain difficulties in the study of heat transfer process. Thermal conductivity of the materials can be characterized by tensor of the second order. Also, the parallelism between the temperature gradient vector and the density of heat flow vector is violated in anisotropic thermal insulation materials (TIM).One of the most famous TIM is a family of integrated thermal insulation refractory material («ITIRM»). The main component ensuring its properties is the «inflated» vermiculite. Natural mineral vermiculite is ground into powder state, fired by gas burner for dehydration, and its precipitate is then compressed. The key feature of thus treated batch of vermiculite is a package structure. The properties of the material lead to a slow heating of manufactured products due to low absorption and high radiation reflection. The maximum of reflection function is referred to infrared spectral region.A review of current models of heat propagation in anisotropic thermal insulation materials is carried out, as well as analysis of their thermal and optical properties. A theoretical model, which allows to determine the heat conductivity «ITIRM», can be useful in the study of thermal characteristics such as specific heat capacity, temperature conductivity, and others. Materials as «ITIRM» can be used in the metallurgy industry, thermal energy and nuclear power-engineering.

An accelerated discrete velocity method for flows of rarefied ternary gas mixtures in long rectangular channels

An accelerated discrete velocity method is presented for rarefied three-component gas mixtures flowing through long rectangular ducts. The scheme is developed on the basis of the McCormack linearized kinetic model. Diffusion equations are derived for the velocity and the heat flow vector. The kinetic and diffusion equations are solved in a coupled iteration. Simulations are carried out for He–Ar–Xe mixture to test the computational performance of the scheme for pressure driven flow. It is shown that the accelerated method requires a fewer number of iterations and smaller computational time in a large part of the gas rarefaction from the transition region to the hydrodynamic limit than the non-accelerated one. These quantities are studied as a function of the convergence parameter also. The flow rates as a function of the rarefaction parameter and representative velocity profiles are shown. A good agreement is found between the total flow rate and the corresponding slip flow value at large rarefaction parameters. The present approach can be useful for fast computation of flows of ternary gas mixtures in long rectangular channels.

基于AlN膜结构VCSEL热特性的研究

In this paper,the internal thermal field and heat flow vector distributions of high power vertical-cavity surface-emitting semiconductor lasers(VCSELs),which is based on AlN film and SiO2 film passivation layers,were analyzed using ANSYS finite-element software.The simulation results proved that the AlN film passivation layer has better features than the SiO2 film passivation layers,and can make the device work in a more stable status,which also improves the device characteristics.Through the simulation,it was found that the Rthjc of VCSEL in AlN film was 3.12 K/W and the Rthjc of VCSEL in SiO2 film was 4.77 K/W.Comparison with the experimental values that the AlN film of 3.59 K/W and the SiO2 film of 4.82 K/W shows that simulation results are in good agreement with the experimental results.The proposed research works prove that the AlN film passivation layer has better thermal features than SiO2 film passivation layer.

Counter-rotating Kerr manifolds separated by a fluid shell

We describe a spheroidal fluid shell between two Kerr vacuum regions which have opposite rotation parameters. The shell has a stiff equation of state and a heat flow vector related to the rotational Killing current. The shell description is useful in exploring the significance of counter-rotation in Kerr metric matches.

16-Moment Approximation for Ion Velocity Distribution and its Application in Calculations of Incoherent Scatter Spectra

A sixteen moment approximation based on a bi-Maxwellian that contains the stress tensor and the heat flow vector is applied to describe the ion velocity distribution which influences the incoherent scatter spectra. A discussion is made about the effects on the incoherent scatter spectra caused by different values of the normalized perpendicular drift velocity D, aspect angle ϕ between the magnetic field and the line-of-sight direction, and the ratio α of the ion-neutral collision to ion cyclotron frequency. Numerical results show that the shifting and asymmetry of incoherent scatter spectra appear parallel to E × B and E as the normalized perpendicular drift velocity D increases due to the ion drift velocity, the stress tensor and the heat flow vector respectively. However, the spectrum is always typically double-humped Maxwellian parallel to B. The ion velocity distribution is more distorted from the Maxwellian as the aspect angle ϕ increases from 0° to 90°, and consequently the incoherent scatter spectra is no longer typically double-humped Maxwellian. As α increases, the ion velocity distribution becomes Maxwellian and the incoherent scatter spectra become typically double-humped Maxwellian even with a large value of the normalized perpendicular drift velocity D. It is reasonable to use the sixteen-moment approximation to describe the non-Maxwellian plasma characterized by the large temperature anisotropy.

Application of 20‐Moment Approximation for Ion Velocity Distribution Function in Calculations of Incoherent Scatter Spectra

AbstractAccording to Grad's theory, the ion velocity distribution function is expanded to a 20‐moment approximation based on the Maxwellian distribution. The effects of the stress tensor and the heat flow vector on the ion velocity distributions are discussed. When the electric field is weak, it is reasonable to neglect the heat flow vector and the stress tensor parallel to E × B and E, respectively. The ion temperature anisotropy and asymmetry in the distribution function are caused by the stress tensor and the heat flow terms. Then using Sheffield's theory and the 20‐moment approximation, normalized spectrum is calculated. Finally the incoherent scatter spectra calculated by the 13‐moment approximation and the 20‐moment approximation are compared. When the electric field is weak, the 13‐moment is nearly consistent with the 20‐moment approximation. However it is unreasonable to neglect the heat flow tensor, and the temperature anisotropic is distinct in the case of a large electric field. It this condition using the 20‐moment approximation is able to describe more accurately the non‐Maxwellian plasma characterized by the temperature anisotropy.

EXACT NONSPHERICAL RADIATING COLLAPSE

We study the junction conditions for a nonspherical collapsing radiating star consisting of a shearing fluid undergoing radial heat flow with outgoing radiation. Radiation of the system is described by the plane symmetric version of the Vaidya solution. Junction conditions which match the collapse solutions to an exterior Vaidya metric show that, at the boundary, the pressure is proportional to the magnitude of the heat flow vector. Physical quantities, analogous to spherical symmetry related to the local conservation of momentum and surface redshift, are also obtained. Finally, exact gravitational collapse solutions, for both shear and shear-free cases, have been obtained by integrating a field equation.