Fourier's Law Research Articles

Numerical simulation of hybrid Casson nanofluid flow by the influence of magnetic dipole and gyrotactic microorganism

The heat and mass transfer investigation of the boundary layer flow of chemically reactive Casson nanoparticles (Ag and MgO) is done by the existence of magnetic dipole and gyrotactic microorganism through a stretching cylinder. The generalized Fourier and Fick law and the thermal radiation effect are taken into account to analyze energy transportation. The stratification conditions are executed on the boundary of the surface of the stretching cylinder. The main idea behind this investigation is to empower the heat transport phenomenon and thermal conductivity of fluid by the inclusion of nanoparticles into the base fluid. The flow equations (PDEs) are changed into nonlinear coupled ODEs by the usage of suitable transformations. The solution of transferred coupled equations is established by the succor of the Bvp4c MATLAB approach. The emerging parameter results are obtained graphically along the velocity, concentration, temperature, and microorganism density profile. The microorganism transport rate and the skin friction coefficient rate are analyzed with the help of tabulated data. It is worth interesting to see that due to the enlargement of (curvature parameter), the velocity of the fluid is boosted, although the decreasing trend is seen in the velocity of fluid by the stronger estimation of slip and magnetic parameters.

MHD flow, radiation heat and mass transfer of fractional Burgers' fluid in porous medium with chemical reaction

Non-Newtonian fluids such as asphalt are widely used in engineering field, but their application will also cause environmental pollution. This paper investigates the MHD flow of this kind of non-Newtonian fluid in porous media by using fractional Burgers' model. The effects of first-order chemical reaction, radiation effects and periodic oscillating boundary condition on fluid flow, heat and mass transfer are considered. The governing equations including a multi-term time fractional derivative are obtained by using the modified Darcy's law, fractional Fourier's law and fractional Fick's law. A convergent and stable L-algorithm, is established for governing equations. The influences of model parameters on the velocity, temperature and concentration distributions are analyzed. Numerical simulation results indicate that fractional derivative α and Darcy number Da have significant effect on velocity distribution. The momentum boundary layer becomes thinner remarkably with fractional derivative α. While the influence of Darcy number Da on the velocity performs conversely.

Challenges in incorporating phase change materials into thermal control units for lithium-ion battery cooling

Keeping cool the lithium-ion battery improves its performance. In this study, focusing on battery cooling, a thermal control unit (TCU) containing metal fins was integrated into the battery. To boost TCU effectiveness, phase change material (PCM) was injected into it. By numerical solution, the temperature distribution in three areas of battery, fins and PCM was determined. Using Fourier's law, TCU heat dissipation to the environment was evaluated at different convection coefficients (5, 10, and 20 Wm2.K). It was observed that the presence of PCM is not always advantageous. The presence of PCM can only be useful if the PCM undergoes phase changes. This happened at 5 Wm2.K, and at best conditions, the battery temperature dropped by 8°C due to the PCM presence. After the melting process, the PCM presence significantly augmented the battery temperature. At 20 Wm2.K battery temperature was increased by 4.8 °C when PCM was loaded into TCU. Therefore, adding PCM to TCU with convection coefficient is not recommended. Increasing the number of fins declined the battery temperature. If the number of fins increases from three to five, the battery cooled by 1 to 1.7°C at convection coefficient range of 5–20 Wm2.K

Enhancement of thermal rectification effect based on zigzag interfaces in bi-segment thermal rectifier using bulk materials

The thermal rectification effect refers to a phenomenon by which heat can be asymmetrically transferred through a system. For bulk materials, the thermal rectification effect can be achieved in a bi-segment thermal rectifier using materials which have thermal conductivities with different temperature dependency. However, the thermal rectification ratio is usually limited by using existing bulk materials. In the present paper, we propose to introduce zigzag-structure interfaces into the bi-segment thermal rectifier. The finite element method is employed to conduct the numerical calculations and the heat flux through the system is calculated based on the Fourier's law. It is found that the zigzag interface can improve the thermal rectification ratio by more than 36% compared a bi-segment thermal rectifier with a flat interface. The enhancement in the thermal rectification ratio is explained based on the theory of the transformation thermotics. The present paper provides a novel approach and analytical tool to design thermal rectifier with meta-materials and enhance the thermal rectification effect in bulk materials.

Long‐time dynamics of the 2D nonisothermal hyperbolic Cahn–Hilliard equations

In this paper, we consider the hyperbolic relaxation of the nonisothermal Cahn–Hilliard equation based on either Fourier law or Maxwell–Cattaneo law for heat conduction. In the Maxwell–Cattaneo case, we reformulate the problem by using enthalpy instead of relative temperature. In both cases, we prove the existence of the global attractor for the weak solutions of the problem. Moreover, for both laws, we establish that every full trajectory in the global attractor converges to a single stationary point as t → ∞ and another single stationary point as t → −∞. Consequently, we infer that the global attractor is equal to a union of the unstable manifolds emanating from the stationary points.

Blasius-Rayleigh-Stokes Flow of Hybrid Nanomaterial Liquid Past a Stretching Surface with Generalized Fourier's and Fick's Law.

The effect of Stefan blowing on the Cattaneo–Christov characteristics of the Blasius–Rayleigh–Stokes flow of self-motive Ag-MgO/water hybrid nanofluids, with convective boundary conditions and a microorganism density, are examined in this study. Further, the impact of the transitive magnetic field, ablation/accretion, melting heat, and viscous dissipation effects are also discussed. By performing appropriate transformations, the mathematical models are turned into a couple of self-similarity equations. The bvp4c approach is used to solve the modified similarity equations numerically. The fluid flow, microorganism density, energy, and mass transfer features are investigated for dissimilar values of different variables including magnetic parameter, volume fraction parameter, Stefan blowing parameter, thermal and concentration Biot number, Eckert number, thermal and concentration relaxation parameter, bio-convection Lewis parameter, and Peclet number, to obtain a better understanding of the problem. The liquid velocity is improved for higher values of the volume fraction parameter and magnetic characteristic, due to the retardation effect. Further, a higher value of the Stefan blowing parameter improves the liquid momentum and velocity boundary layer thickness.

Implication of Revised Fourier Law of Heat Conduction in Flow of Non-Newtonian Nanoliquid Over a Stretched Surface

Implication of Revised Fourier Law of Heat Conduction in Flow of Non-Newtonian Nanoliquid Over a Stretched Surface

Study on time-dependent Oldroyd-B fluid flow over a convectively heated surface with Cattaneo-Christov theory

The time-dependent two-dimensional Oldroyd-B fluid flow is investigated in the presence of generalized Fourier's and Fick's laws. The mechanism of heat and mass transport is studied using the Cattaneo-Christov (CC) double diffusion theory, which characterizes the thermal and solutal relaxation factors. In addition, the temperature-dependent thermal conductivity of the fluid is taken into consideration. The modified Fourier's and Fick's approach is used to develop a set of partial differential equations for the flow of an Oldroyd-B fluid as well as thermal and solutal transport. By using the suitable similarity transformations, the governing equations are converted into an ordinary differential equation. To visualize the effects of dimensionless physical constraints on flow and energy transport phenomenon, in the domain [0, ) numerical integration is performed with the help of (Midrich scheme) in Maple software to solve the highly non-linear ordinary differential equations. The effects of various arising parameters can be seen in the graphs of velocities, temperature, and concentration. According to the findings, the thermal and solutal penetration depth declines for the growing values of flow controlling parameter and Lewis number, respectively. The current results are very similar to those found in the literature.

Molecular dynamics study of thermal properties of polyamide nanocomposites reinforced with natural biofillers

Bio-based fillers have been used for reinforcements in thermoplastic polymer, predominantly in the past decade, to improve its mechanical and thermal performance. This study has made use of Molecular Dynamics (MD) simulations using Large-scale Atomic/Molecular Massively Parallel Simulator package to investigate the thermal behavior of polyamide 6 nanocomposites when they are reinforced with natural biofillers such as starch, cellulose, and coconut shell nanoparticles. A comparative study is conducted to examine the degree of improvement or deterioration of thermal properties under the effect of respective nanofillers. Density, glass transition temperature (Tg), heat capacity, thermal diffusivity, and thermal conductivity were inspected with simulations run in a temperature range of 200–600 K. Glass transition temperature has been calculated from three parameters, by the dependence of density, dielectric constant, and electrostatic energy on temperature. Non-equilibrium MD simulations using Fourier's law method were followed to calculate thermal conductivity. This study has concentrated on thermal evaluations of bio-based polyamide nanocomposites helpful in determining the flame retardancy of the materials.

Performance-based comparison of Yamada\u2013Ota and Hamilton\u2013Crosser hybrid nanofluid flow models with magnetic dipole impact past a stretched surface

The nanofluid flows play a vital role in many engineering processes owing to their notable industrial usage and excessive heat transfer abilities. Lately, an advanced form of nanofluids namely “hybrid nanofluids” has swapped the usual nanofluid flows to further augment the heat transfer capabilities. The objective of this envisaged model is to compare the performance of two renowned hybrid nanofluid models namely Hamilton–Crosser and Yamada–Ota. The hybrid nanoliquid (TiO2-SiC/DO) flow model is comprised of Titanium oxide (TiO2) and Silicon carbide (SiC) nanoparticles submerged into Diathermic oil (DO). The subject flow is considered over a stretched surface and is influenced by the magnetic dipole. The uniqueness of the fluid model is augmented by considering the modified Fourier law instead of the traditional Fourier law and slip conditions at the boundary. By applying the suitable similarity transformations, the system of ordinary differential equations obtained from the leading partial differential equations is handled by the MATLAB solver bvp4c package to determine the numerical solution. It is divulged that the Yamada–Ota model performs considerably better than the Hamilton–Crosser flow model as far as heat transfer capabilities are concerned. Further, the velocity reduces on increasing hydrodynamic interaction and slip parameters. It is also noted that both temperature profiles increase for higher hydrodynamic interaction and viscous dissipation parameters. The envisioned model is authenticated when compared with an already published result in a limiting case.

New decay rates for a Cauchy thermoelastic laminated Timoshenko problem with interfacial slip under Fourier or Cattaneo laws

The objective of the present paper is to investigate the decay of solutions for a laminated Timoshenko beam with interfacial slip in the whole space subject to a thermal effect acting only on one component modeled by either Fourier or Cattaneo law. When the thermal effect is acting via the second or third component of the laminated Timoshenko beam (rotation angle displacement or dynamic of the slip), we obtain that both systems, Timoshenko–Fourier and Timoshenko–Cattaneo systems, satisfy the same polynomial stability estimates in the L2‐norm of the solution and its higher order derivatives with respect to the space variable. The decay rate depends on the regularity of the initial data. In addition, the presence and absence of the regularity‐loss type property are determined by some relations between the parameters of systems. However, when the thermal effect is acting via the first component of the system (transversal displacement), a new stability condition is introduced for both Timoshenko–Fourier and Timoshenko–Cattaneo systems. This stability condition is in the form of threshold between polynomial stability and convergence to zero. To prove our results, we use the energy method in Fourier space combined with judicious choices of weight functions to build appropriate Lyapunov functionals.

Analytical evaluation of non-Fourier heat pulse experiments on room temperature

The classical constitutive equation for heat conduction, Fourier's law, plays an important role in the engineering practise and can be applied only for homogeneous materials. However, most of the materials consist of heterogeneity, such as porosity, cracks, or different materials are in contact.We used heat pulse measurement method (“flash method”), a standard method in the engineering practise, to measure the thermal diffusivity of each rock sample. During the heat pulse experiment the pulse reaches the front of the sample and the temperature of the rear is measured with a thermocouple. This is a common method to determine the thermal diffusivity of a material on room temperature. We found that the rear side temperature history can deviate from the one predicted by Fourier's law. Therefore, in the evaluation of the experimental data, we used an analytical solution of a non-Fourier model, called Guyer-Krumhansl equation. We present the measurement methodology and data recorded together with the evaluation procedure.

Well-posedness and exponential decay for a laminated beam with distributed delay term

"In this paper, we study the well-posedness and the asymptotic behavior of a one-dimensional laminated beam system with a distributed delay term in the first equation, where the heat conduction is given by Fourier's law efective in the rotation angle displacements. We first give the well-posedness of the system by using the semigroup method. Then, we show that the system is exponentially stable under the assumption of equal wave speeds."

COMPUTATIONAL AND EXPERIMENTAL STUDY OF THE THERMAL PROCESS IN AN INDIVIDUAL CYLINDRICAL PARTICLE

he paper proposes a one-dimensional mathematical model of thermal conductivity in a cylinder under conditions of convective heat exchange with an external gas medium based on the difference formulation of Fourier's law of thermal conductivity. Parametric identification was per-formed for the proposed model based on the use of known data and empirical regularities for the material constants of the process. The mentioned data made it possible to adapt a mathematical model to describe the process of heat treatment of granular fuel particles. The verification of the operability of the physical and mathematical model proposed in this way was performed by com-paring the calculated forecasts obtained with the data of a full-scale experiment. For a full-scale experiment fuel particles were prepared in such a way that thermocouple junctions were placed attwo points inside each of them. The presence of thermocouple junctions directly inside the particles made it possible to fix local temperature values of the material during its heat treatment. Particles with thermocouples were placed in the apparatus withthe active hydrodynamic regime of a heating gas medium. Thus, during the computational and experimental study, kinetic characteristics of heating thermally massive cylindrical bodies were obtained under three different hydrodynamic regimes. The calculatedforecasts and experimental data are in good agreement for engineering calculations, which indicates sufficient predictive effectiveness of the proposed physical and math-ematical model and makes it possible to consider it as a reliable basis for constructing computer methods for calculating heat transfer processes. The proposed mathematical model can also serve as an element for assembling more complex discrete models of heat and mass transfer processes in a single particle and/or models of the functioning of technological equipment for heat treat-ment of bulk media.

Mathematical aspects of non-Fourier heat equations

Due to technological advancement, as materials with complex structures (e.g., metamaterials and foams) appear in practice there is a need to develop advanced thermal models. These are called non-Fourier equations, and all have particular mathematical properties differing from the conventional attributes of Fourier's law. The present paper discusses the thermodynamic origin of non-Fourier equations and their consequences. The second law of thermodynamics influences the relations among the material parameters, and therefore, it restricts how the temperature-dependent properties can be included in the model. Furthermore, we present the properties of initial and boundary conditions, since these are crucial in solving any practical problems and are different from the usual interpretation used for the Fourier equation.

Dynamic modeling of the heat transfer process in rotary kilns with indirect oil heating: Parametric analysis of gypsum calcination case

This work proposes a mathematical modeling and numerical simulation of a gyp-sum rotary kiln with indirect oil heating in a 3-D transient regime. The mathematical model was based on Fourier's law as a constitutive relationship and the principle of energy conservation, applied to a control volume in cylindrical co-ordinates. Furthermore, a bed homogenization model was used to represent the most realistic condition of the physical phenomenon since some rotary kilns have internal fins that aim at homogenizing the gypsum temperature during calcination. This work intends to fill the gap found in heat transfer processes on rotary kilns in transient regime considering 3-D positions, to have an accurate projection of the temperature profile of the kiln and also, given by the numerical model, the possibility of a tool that can be used to the optimization of the control system of rotary kilns considering the actual demand of the material in production, leading to the best energy performance of the equipment's activation source, as well as reaching the temperatures and processing time of the product. The numerical simulation results revealed reasonable agreement with the experimentally deter-mined calcination process in rotary kilns. Furthermore, a parametric analysis of the influence of the mixture on the temperature fields and the calcination time was carried out to verify the energetic balance of the rotary kiln.

Method for solving linearized thermal problems taking into account the phenomenon of heat flow relaxation

A method is proposed for computational modeling of the features of motion in a material of a thermal wave within the framework of a linearized problem. Such problems arise when applying heat transfer models with allowance for heat flux relaxation. It is assumed that the proportional relationship between the temperature gradient and the heat flux (Fourier's law) cannot appear instantly when the temperature field changes in the medium. It takes time to change the heat flow. This approach refers to the description of the time variations of the heat flux in the Cattaneo and Vernotte models. The method for solving thermal problems within this scope is reflected in a large number of publications, but they deal mainly with the redistribution of heat in the material. However, there are tasks the solution of which requires understanding of how exactly the change in heat fluxes in the medium occurs. This is especially important when the data are used to find solutions to the problems of determining stress (depending not only on temperature and strain, but also on the value of the heat flux) fields in media. The paper shows how to construct a system of equations convenient for calculations the temperature changes and heat flux fields. It has been established that in a linear problem the initial equations can be transformed into a system of two hyperbolic equations with developed solution algorithms. As an example, computational modeling of the emergence and movement of a thermal wave during the ion-plasma treatment of a material surface is considered. The conditions required for solving the boundary value problem are formulated. The important results of the calculations were the pattern of formation, detachment from the surface of the sample after the completion of the impulse, and the beginning of the movement of a thermal wave into the depth of the material during ion-plasma treatment. The dependence of solutions on the characteristic relaxation time of the heat flux is studied.

Thermalization of a rarefied gas with total energy conservation: Existence, hypocoercivity, macroscopic limit

<p style='text-indent:20px;'>The thermalization of a gas towards a Maxwellian velocity distribution with the background temperature is described by a kinetic relaxation model. The sum of the kinetic energy of the gas and the thermal energy of the background are conserved, and the heat flow in the background is governed by the Fourier law.</p><p style='text-indent:20px;'>For the coupled nonlinear system of the kinetic and the heat equation, existence of solutions is proved on the one-dimensional torus. Spectral stability of the equilibrium is shown on the torus in arbitrary dimensions by hypocoercivity methods. The macroscopic limit towards a nonlinear cross-diffusion problem is carried out formally.</p>

Power generalization of the linear constitutive equations of heat and mass transfer and the variants of writing the equations of momentum transfer, heat and diffusion arising from them

Currently, when solving problems of heat and mass transfer, linear constitutive equations are used - in hydrodynamics, the viscous stress tensor is proportional to the strain rate tensor (Newton's rheological ratio), in heat transfer, the heat flux density is linearly related to the temperature gradient (Fourier's heat conduction law), in mass transfer, the diffusion flux density proportional to the concentration gradient (Fick's law). When writing these linear governing equations, proportionality coefficients are used, which are called the viscosity coefficient, thermal conductivity coefficient and diffusion coefficient, respectively. Such constitutive equations are widely used to describe the processes of heat and mass transfer in a laminar flow regime. For turbulent flows, these equations are unsuitable, it is necessary to introduce into consideration the empirical turbulent coefficients of viscosity μt, thermal conductivity λt and diffusion Dt. However, to describe turbulent flows, it is possible to go in another way - to modify the linear constitutive relations by giving them a nonlinear power-law form. Two-parameter power-law generalizations of Newton's, Fourier's and Fick's formulas for shear stress, heat flux density and diffusion, which, depending on the value of the exponents, can be used to describe the processes of heat and mass transfer both in laminar and turbulent fluid flow. Also, this generalization can be used to describe the behavior of power-law fluids and flows of polymer solutions exhibiting the Toms effect.

Unsteady flow of Casson nanofluid through generalized Fourier's and Fick's law for heat and mass transfer

The purpose of this paper to explain the role and importance of fractional derivatives for mass and heat transfer in Casson nanofluids including clay nanoparticles. These particles can be found in water, kerosene, and engine oil. The physical flow phenomena are illustrated using PDE and thermophysical nanoparticle properties, and this paper addresses the Casson fractional fluid along with chemical reaction and heat generation. The heat and mass fluxes are generalized using the constant proportional Caputo fractional derivative. The present flow model are solved semi-analytically using the Laplace transform. We generated several graphs to understand how various flow factors affect velocity. The acquired results reveal that fractional parameters have dual behavior in velocity profiles. Velocity and temperature are also compared to previous studies. Compared to the other fractional derivatives results, field variables and proposed hybrid fractional derivatives showed a more decaying trend. Furthermore, significant results of clay nanoparticles with various base fluids have been obtained.