Relativistic heat conduction

The hyperbolic heat conduction equation (HHCE), which acknowledges the finite speed of heat propagation, is based on microscopic evidence from the kinetic theory and statistical mechanics. However, it was argued that the HHCE could violate the second law of thermodynamics. This paper shows that a HHCE-like equation (RHCE) can be derived directly from the theory of relativity, as a direct consequence of space-time duality, without any consideration of the microstructure of the heat-conducting medium. This approach results in an alternative expression for the heat flux vector that is more compatible with the second law. Therefore, the RHCE brings the classical field theory of heat conduction into agreement with other branches of modern physics.     

Thermomechanical coupling and second sound in superfluid flows

The present paper is concerned with the modelling of the influence of the thermomechanical coupling on the propagation of temperature waves (second sound) in superfluids. For an adequate heat transfer analysis in superfluids, finite thermal wave speed must be considered. Besides hyperbolic heat transfer, turbulent flows are generally observed despite the lost of internal friction. Due to the thermo-mechanical coupling, density waves may induce moving heat sources or sinks while temperature waves propagate at a different and independent speed. In particular, a rotational flow in this kind of fluid can strongly affect heat propagation. A general procedure, developed within the framework of thermodynamics of irreversible processes, is proposed to obtain constitutive relations that verify automatically the second law of thermodynamics and the principle of material objectivity. Such a phenomenological continuum approach allows a rational identification of the terms responsible for the thermomechanical coupling in the heat equation, which is a first step to better understand its influence on the superfluid flow.     

Modelling of heat and mass transfer in a tunnel dryer

This work presents a numerical model for heat and mass transfer of granular products in a fixed-bed tunnel dryer. The drying process is simulated under real operating conditions based on a thin layer model and experimental drying kinetics. A simplified heat and mass transfer numerical model is developed based on the governing equations and the drying rate of a thin layer bed of granular products.The obtained system of non-linear partial differential equations is numerically solved by a finite volume method. The turbulent airflow and granular bed convection coefficient as well as the effective conductivity are estimated using the turbulent airflow over flat-plate correlations. Simulations are compared with experimental data from drying of grapes in a thin layer model.In order to study the effects of the air inlet conditions on the relative moisture content and the drying time and therefore to optimise the tunnel dryer operation, the influences of different parameters essentially the air flow characteristics and the fixed-bed dryer length are examined. The numerical code allows establishing the drying front propagation for several operating conditions.     

Analysis of three-dimensional heat transfer in micro-channel heat sinks

In this study, the three-dimensional fluid flow and heat transfer in a rectangular micro-channel heat sink are analyzed numerically using water as the cooling fluid. The heat sink consists of a 1-cm² silicon wafer. The micro-channels have a width of 57 μm and a depth of 180 μm, and are separated by a 43 μm wall. A numerical code based on the finite difference method and the SIMPLE algorithm is developed to solve the governing equations. The code is carefully validated by comparing the predictions with analytical solutions and available experimental data. For the micro-channel heat sink investigated, it is found that the temperature rise along the flow direction in the solid and fluid regions can be approximated as linear. The highest temperature is encountered at the heated base surface of the heat sink immediately above the channel outlet. The heat flux and Nusselt number have much higher values near the channel inlet and vary around the channel periphery, approaching zero in the corners. Flow Reynolds number affects the length of the flow developing region. For a relatively high Reynolds number of 1400, fully developed flow may not be achieved inside the heat sink. Increasing the thermal conductivity of the solid substrate reduces the temperature at the heated base surface of the heat sink, especially near the channel outlet. Although the classical fin analysis method provides a simplified means to modeling heat transfer in micro-channel heat sinks, some key assumptions introduced in the fin method deviate significantly from the real situation, which may compromise the accuracy of this method.     

A numerical study of fluid flow and heat transfer in eccentric curved annuli

In this article fluid flow and heat transfer in curved eccentric annuli are studied numerically. A second order finite difference method based on the Projection algorithm is implemented to solve the governing equations including the full Navier–Stokes, the continuity, and the energy equations in a toroidal coordinate system. For convenience a bipolar based toroidal coordinate system is employed to discretize the governing equations in the annulus domain using a uniform staggered grid which is required in finite difference methods. Considering hydrodynamically and thermally fully developed conditions, the effects of different physical parameters such as eccentricity, Dean number, curvature, Prandtl number on the flow field and thermal characteristics at different thermal boundary conditions are investigated in detail. It is also shown that in contrast to straight eccentric annuli, heat transfer rates can be augmented in the eccentric curved annuli comparing with the straight concentric annuli at the large dean numbers.     

Analytical solution of the parabolic and hyperbolic heat transfer equations with constant and transient heat flux conditions on skin tissue

In this article, the parabolic (Pennes bioheat equation) and hyperbolic (thermal wave) bioheat transfer models for constant, periodic and pulse train heat flux boundary conditions are solved analytically by applying the Laplace transform method for skin as a semi-infinite and finite domain. The bioheat transfer analysis with transient heat flux on skin tissue has only been studied by Pennes equation for a semi-infinite domain. For modeling heat transfer in short duration of an initial transient, or when the propagation speed of the thermal wave is finite, there are major differences between the results of parabolic and hyperbolic heat transfer equations. The non-Fourier bioheat transfer equation describes the thermal behavior in the biological tissues better than Fourier equation. The outcome of transient heat flux condition shows that by penetrating into the depths beneath the skin subjected to heat, the amplitude of temperature response decreases significantly. The blood perfusion rate can be predicted using the phase shift between the surface temperature and transient surface heat flux. The thermal damage of the skin is studied by applying both the parabolic and hyperbolic bioheat transfer equations.     

Conjugate heat transfer analysis of a heat generating vertical plate

This paper mainly deals with conjugate heat transfer problem pertinent to rectangular fuel element of a nuclear reactor dissipating heat into an upward moving stream of liquid sodium. Introducing boundary layer approximations, the equations governing the flow and thermal fields in the fluid domain are solved simultaneously along with two-dimensional energy equation in the solid domain by satisfying the continuity of temperature and heat flux at the solid–fluid interface. The boundary layer equations are discretized using fully implicit finite difference scheme so as to adopt marching technique solution procedure, while second-order central difference scheme is employed to discretize the energy equation in the solid domain and the resulting system of finite difference equations are solved using Line-by-Line Gauss–Seidel iterative solution procedure. Numerical results are presented for a wide range of parameters such as aspect ratio, A_r, conduction–convection parameter, N_cc, heat generation parameter, Q, and flow Reynolds number, Re. It is concluded that there exist an upper or a lower limiting value of these parameters above or below which the temperature in the fuel element crosses its allowable limit. It is also found that an increase in Re results in considerable increase in overall heat dissipation rate from the fuel element.     

Maximum heat transfer density rate enhancement from cylinders rotating in natural convection

In this paper we investigate the thermal behaviour of an assembly of consecutive cylinders in a counter-rotating configuration cooled by natural convection with the objective of maximizing the heat transfer density rate (heat transfer rate per unit volume). A numerical model is used to solve the governing equations that describe the temperature and flow fields. The spacing between the consecutive cylinders is optimised for each flow regime (Rayleigh number) and cylinder rotation speed. It was found that the optimized spacing decreases as the Rayleigh number increases and the heat transfer density rate increases, for the optimized structure, as the cylinder rotation speed is increased. Results further shows that there is an increase in the heat transfer density rate of the rotating cylinders over stationary cylinders.     

Convective heat transfer from a rotating cylinder with inline oscillation

In this study convective heat transfer from a rotating cylinder with inline oscillation is studied using a finite element method based on the Characteristic Based Split method (CBS) to solve governing equations consisting of continuity, full Navier–Stokes, and energy equations. Employing the Arbitrary Lagrangian-Eulerian (ALE) formulation, the dynamic unstructured triangular grid used here is accompanied by lineal and torsional spring analogy to consider large boundary movements. Simulations are conducted to study convective heat transfer past a rotating cylinder with inline oscillation at Reynolds numbers of 100, 200 and 300. Different rotational speeds of the cylinder in the range of 0–2.5 are considered at various oscillating amplitudes and frequencies with three different Prandtl numbers of 0.7, 6 and 20. Effects of oscillation and rotation of cylinder on the temperature and flow field, vortex lock-on, mean Nusselt number, and the pattern of vortex shedding are investigated in detail at constant temperature boundary condition on the cylinder surface. It is found that similar to the fixed cylinder, beyond a critical rotating speed, the vortex shedding is strongly suppressed. Furthermore, as the rotational speed of the cylinder increases, both the Nusselt number and the drag coefficient decrease rapidly. In the vortex lock-on region, the Nusselt number increases rapidly.     

Numerical illustrations of 3D tangent hyperbolic liquid flow past a bidirectional moving sheet with convective heat transfer at the boundary

Convective heat transfer plays a central role in the numerous industrial devices because it perturbs the mechanical behavior of a system along with its thermodynamics. Keeping such applications in mind, analysis of heat transportation in three‐dimensional tangent hyperbolic fluid flow is investigated here. Convective heat transportation at the boundaries is considered. Rosseland's approximation has been used for the radiation effects. Closed form analytical solutions for the governing equations are difficult to obtain even after the use of similarity transformations. Therefore, the numerical solutions are presented through the Runge‐Kutta‐Fehlberg forth‐fifth method. Graphical analysis of the numerical results has been carried out. Roles of sundry constraints on flow are studied. It is also noted that the rates of heat transportation and skin‐friction are higher in the presence of convective heat transfer near the boundary.     

SPECIALLY TAILORED TRANS FINITE-ELEMENT FORMULATIONS FOR HYPERBOLIC HEAT CONDUCTION INVOLVING NON-FOURIER EFFECTS

The phenomenon of hyperbolic heat conduction in contrast to the classical (parabolic) form of Fourier heat conduction involves thermal energy transport that propagates only at finite speeds as opposed to an infinite speed of thermal energy transport. To accommodate the finite speed of thermal wave propagation, a more precise form of heat flux law is involved, thereby modifying the heat flux originally postulated in the classical theory of heat conduction. As a consequence, for hyperbolic heat conduction problems, the thermal energy propagates with very sharp discontinuities at the wave front. The primary purpose of the present paper is to provide accurate solutions to a class of one-dimensional hyperbolic heat conduction problems involving non-Fourier effects that can precisely help understand the true response and furthermore can be used effectively for representative benchmark tests and for validating alternate schemes. As a consequence, the present paper purposely describes modeling/analysis formulations via specially tailored hybrid computations for accurately modeling the sharp discontinuities of the propagating thermal wave front. Comparative numerical test models are presented for various hyperbolic heat conduction models involving non-Fourier effects to demonstrate the present formulations.     

The effect of geometrical parameters on heat transfer characteristics of microchannels heat sink with different shapes

The effect of geometrical parameters on water flow and heat transfer characteristics in microchannels is numerically investigated for Reynolds number range of 100–1000. The three-dimensional steady, laminar flow and heat transfer governing equations are solved using finite volume method. The computational domain is taken as the entire heat sink including the inlet/outlet ports, wall plenums, and microchannels. Three different shapes of microchannel heat sinks are investigated in this study which are rectangular, trapezoidal, and triangular. The water flow field and heat transfer phenomena inside each shape of heated microchannels are examined with three different geometrical dimensions. Using the averaged fluid temperature and heat transfer coefficient in each shape of the heat sink to quantify the fluid flow and temperature distributions, it is found that better uniformities in heat transfer coefficient and temperature can be obtained in heat sinks having the smallest hydraulic diameter. It is also inferred that the heat sink having the smallest hydraulic diameter has better performance in terms of pressure drop and friction factor among other heat sinks studied.     

Effects of unequal finite piston speeds on the optimal performances of endoreversible Carnot refrigeration and heat pump cycles

The performances of endoreversible Carnot refrigeration and heat pump cycles with loss of heat resistance and finite piston speeds are analysed and optimized by using the combination of finite time thermodynamics, finite speed thermodynamics and direct method. The unequal finite piston speed model on four branches is adopted. Expressions of cooling load of endoreversible Carnot refrigeration cycle and of heating load of endoreversible Carnot heat pump cycle are derived with a fixed cycle period and unequal finite piston speeds on the four branches. Numerical examples show that there exist optimal expansion ratios, which lead to maximum cooling load and maximum heating load for the fixed coefficient of performance (COP), respectively. The maximum cooling load, maximum heating load, optimal ratios of finite piston speeds and optimal hot- and cold-side working fluid temperatures versus COP characteristics for the endoreversible Carnot refrigeration and heat pump cycles are obtained. Moreover, the effects of design parameters on the performances of the two cycles are discussed.     

Transient two-dimensional model of heat and mass transfer in a PEM fuel cell membrane

In the present work, a numerical study of heat and mass transfer within the membrane of a proton exchange membrane fuel cell is presented. The electrolyte membrane is considered an isotropic porous medium and ideal insulator for electrons and reactants. The adopted model in this study is based on the assumption of single-phase and multi-spices flow, supposed two-dimensional and unsteady. For the water transport, the major considered forces are; the convective force, resulting from the pressure gradient, the osmotic force, due to the concentration gradient and the electric force caused by the proton migration from the anode to the cathode. Based on a one-dimensional model, found in the literature, a transient two-dimensional one was proposed. The set of governing equations, written in velocity–pressure formulation, is solved by the implicit finite difference method. An alternating Direct Implicit scheme was used for the calculation. The numerical resolution gives the time- and space-dependent temperature and water concentration. The main focus lies on the influence of different cases of boundary conditions on water concentration and heat transfer variation with the intention of testing the reliability of the proposed computational fluid dynamic (CFD) code.     

Numerical investigation on the heat transfer and flow in the mini-fin heat sink for CPU

Fluid flow and heat transfer in the mini-rectangular fin heat sink for CPU of PC using de-ionized water as working fluid are numerically investigated. Based on the real PC operating conditions, the three-dimensional governing equations for fluid flow and heat transfer characteristics are solved using finite volume scheme. The standard k–ε turbulent model is employed to describe the flow structure and behavior. The predicted results obtained from the model are verified by the measured data. There is a reasonable agreement between the predicted results and experiments. The results of this study are expected to lead to guidelines that will allow the design of the cooling system with improved cooling performance of the electronic equipments increasing reliable operation of these devices.     

Analysis of Thermoelastic Waves in Functionally Graded Hollow Spheres Based on the Green-Lindsay Theory

In this article, the Green–Lindsay theory of thermoelasticity is employed to study the thermoelastic response of functionally graded hollow spheres. This generalized coupled thermoelasticity theory admits the second sound phenomena and depicts a finite speed for temperature wave propagation. The materials of the hollow sphere are assumed to be graded through its thickness in the radial direction while a symmetric thermal shock load is applied to its boundary. The Galerkin finite element method via the Laplace transformation is used to solve the coupled form of governing equations. A numerical inversion of the Laplace transform is employed to obtain the results in time domain. Using the obtained solution, the temperature, displacement, radial stress, and hoop stress waves propagation are studied. Also the material distribution effects on temperature, displacement and stresses are investigated. Finally, the obtained results for the Green–Lindsay theory are compared with the results of classical thermoelasticity theory.     

Numerical studies on flow and heat transfer in membrane helical-coil heat exchanger and membrane serpentine-tube heat exchanger

The flow and heat transfer characteristics of synthesis gas (syngas) in membrane helical-coil heat exchanger and membrane serpentine-tube heat exchanger under different operating pressures, inlet velocities and pitches are investigated numerically. The three-dimensional governing equations for mass, momentum and heat transfer are solved using a control volume finite difference method. The realizable k-ε model is adopted to simulate the turbulent flow and heat transfer in heat exchangers. There flows syngas in the channels consisting of the membrane helical coils or membrane serpentine tubes, where the operating pressure varies from 0.5 to 3.0 MPa. The numerically obtained heat transfer coefficients for heat exchangers are in good agreement with experimental values. The results show that the syngas tangential flow in the channel consisting of membrane helical coils is significant to the heat transfer enhancement to lead to the higher average heat transfer coefficient of membrane helical-coil heat exchanger compared to membrane serpentine-tube heat exchanger. The syngas tangential velocity in the membrane helical-coil heat exchanger increases along the axial direction, and it is independent of the gas pressure, increasing with the axial velocity and axial pitch rise and decreasing with the radial pitch rise.     

Effect of radiation on natural convection flow on a sphere in presence of heat generation

Effect of radiation on natural convection flow around a sphere in presence of heat generation has been investigated. The governing equations are transformed into dimensionless non-similar equations by using a set of suitable transformations and solved numerically by the finite difference method along with Newton's linearization approximation. We have focused our attention on the evaluation of velocity profiles, temperature profiles, shear stress in terms of local skin friction and the rate of heat transfer in terms of local Nusselt number for different values of heat generation parameter, radiation parameter and the Prandlt number and the numerical results have been shown graphically.     

MHD convective flow adjacent to a vertical surface with prescribed wall heat flux

The steady magnetohydrodynamic (MHD) mixed convection flow adjacent to a vertical surface with prescribed heat flux is investigated. The governing partial differential equations are transformed into a system of ordinary differential equations, which is then solved numerically by a finite-difference method. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Both assisting and opposing flows are considered. It is found that dual solutions exist for the assisting flow, besides that usually reported in the literature for the opposing flow.