Interfacial heat transfer coefficient for non-equilibrium convective transport in porous media

The literature has documented proposals for macroscopic energy equation modeling for porous media considering the local thermal equilibrium hypothesis and laminar flow. In addition, two-energy equation models have been proposed for conduction and laminar convection in packed beds. With the aim of contributing to new developments, this work treats turbulent heat transport modeling in porous media under the local thermal non-equilibrium assumption. Macroscopic time-average equations for continuity, momentum and energy are presented based on the recently established double decomposition concept (spatial deviations and temporal fluctuations of flow properties). Interfacial heat transfer coefficients are numerically determined for an infinite medium over which the fully developed flow condition prevails. The numerical technique employed for discretizing the governing equations is the control volume method. Preliminary laminar flow results for the macroscopic heat transfer coefficient, between the fluid and solid phase in a periodic cell, are presented.     

Analytical characterization of heat transport through biological media incorporating hyperthermia treatment

Modeling and understanding heat transport and temperature variations within biological tissues and body organs are key issues in medical thermal therapeutic applications, such as hyperthermia cancer treatment. The biological media can be treated as a blood saturated tissue represented by a porous matrix. A comprehensive analytical investigation of bioheat transport through the tissue/organ is carried out including thermal conduction in tissue and vascular system, blood–tissue convective heat exchange, metabolic heat generation and imposed heat flux. Utilizing local thermal non-equilibrium model in porous media theory, exact solutions for blood and tissue phase temperature profiles as well as overall heat exchange correlations are established for the first time, for two primary tissue/organ models representing isolated and uniform temperature conditions, while incorporating the pertinent effective parameters, such as volume fraction of the vascular space, ratio of the blood and the tissue matrix thermal conductivities, interfacial blood–tissue heat exchange, tissue/organ depth, arterial flow rate and temperature, body core temperature, imposed hyperthermia heat flux, metabolic heat generation, and blood physical properties. A simplified solution based on the local thermal equilibrium between the tissue and the blood is also presented.     

A general bioheat model at macroscale

We develop a general bioheat transport model at macroscale for biological tissues with the required closure provided. The model shows that both blood and tissue macroscale temperatures satisfy the dual-phase-lagging (DPL) energy equations. Due to the coupled conduction between the blood and the tissue, thermal waves and possibly resonance may appear in bioheat transport. The blood–tissue interaction yields a very rich effect of the interfacial convective heat transfer, the blood velocity, the perfusion and the metabolic reaction on blood and tissue macroscale temperature fields. Examples include: (i) the spreading of tissue metabolic effect into the blood DPL bioheat equation, (ii) the appearance of the convection term in the tissue DPL bioheat equation due to the blood velocity, and (iii) the appearance of sophisticated heat source terms in energy equations for blood and tissue temperatures.     

Non-Darcian effects on natural convection heat transfer in a wavy porous enclosure

A numerical investigation is carried out to analyze natural convection heat transfer inside a cavity with a sinusoidal vertical wavy wall and filled with a porous medium. The vertical walls are isothermal while the top and bottom horizontal straight walls are kept adiabatic. The transport equations are solved using the finite element formulation based on the Galerkin method of weighted residuals. The validity of the numerical code used is ascertained by comparing our results with previously published results. The importance of non-Darcian effects on convection in a wavy porous cavity is analyzed in this work. Different flow models for porous media such, as Brinkman-extended Darcy, Forchheimer-extended Darcy, and the generalized flow models, are considered. Results are presented in terms of streamlines, isotherms, and local heat transfer. The implications of Rayleigh number, number of wavy surface undulation and amplitude of the wavy surface on the flow structure and heat transfer characteristics are investigated in detail while the Prandtl number is considered equal to unity.     

An Analytic Solution of One-dimensional Steady-state Pennes'''' Bioheat Transfer Equation in Cylindrical Coordinates

Based on the Pennes' bioheat transfer equation, a simplified one-dimensional bioheat transfer model of the cylindrical living tissues in the steady state has been set up for application in limb and whole body heat transfer studies, and by using the Bessel's equation, its corresponding analytic solution has been derived in this paper. With the obtained analytic solution, the effects of the thermal conductivity, the blood perfusion, the metabolic heat generation, and the coefficient of heat transfer on the temperature distribution in living tissues are analyzed. The results show that the derived analytic solution is useful to easily and accurately study the thermal behavior of the biological system, and can be extended to such applications as parameter measurement, temperature field reconstruction and clinical treatment.     

Crosswise stream of hydrogen-oxide (H2O) through a porous media containing copper nanoparticles

Transport theories in porous media are quite operative to analyse heat transferral phenomenon in biological tissues, reducing bio convective flow instabilities by means of porous media and many more. Inspired by these remarkable features, the present study is conducted to analyse heat transfer phenomenon for obliquely striking nanofluid through a porous media. Copper (Cu) nanoparticles are suspended in traditional Hydrogen Oxide based fluid. Scaling group of transformations is conveniently employed to reduce governing transport equations and is tackled numerically afterwards. Influence of nanoparticles volume fraction, stretching ratio and porosity parameter on physical measures of concern such as normal and tangential skin friction and corresponding heat flux at wall is portrayed. Streamline patterns are traced out to discover the influence of porosity factor on actual flow behavior. It was observed that solid volume fraction of copper nanoparticles enhanced the skin friction coefficients and heat flux. Increasing the porosity parameter leads to greater heat flux and tangential skin friction co-efficient.     

An analytic solution of one-dimensional steady-state Pennes’ bioheat transfer equation in cylindrical coordinates

Based on the Pennes’ bioheat transfer equation, a simplified one-dimensional bioheat transfer model of the cylindrical living tissues in the steady state has been set up for application in limb and whole body heat transfer studies, and by using the Bessel’s equation, its corresponding analytic solution has been derived in this paper. With the obtained analytic solution, the effects of the thermal conductivity, the blood perfusion, the metabolic heat generation, and the coefficient of heat transfer on the temperature distribution in living tissues are analyzed. The results show that the derived analytic solution is useful to easily and accurately study the thermal behavior of the biological system, and can be extended to such applications as parameter measurement, temperature field reconstruction and clinical treatment.     

A general bioheat transfer model based on the theory of porous media

A volume averaging theory (VAT) established in the field of fluid-saturated porous media has been successfully exploited to derive a general set of bioheat transfer equations for blood flows and its surrounding biological tissue. A closed set of macroscopic governing equations for both velocity and temperature fields in intra- and extravascular phases has been established, for the first time, using the theory of anisotropic porous media. Firstly, two individual macroscopic energy equations are derived for the blood flow and its surrounding tissue under the thermal non-equilibrium condition. The blood perfusion term is identified and modeled in consideration of the transvascular flow in the extravascular region, while the dispersion and interfacial heat transfer terms are modeled according to conventional porous media treatments. It is shown that the resulting two-energy equation model reduces to Pennes model, Wulff model and their modifications, under appropriate conditions. Subsequently, the two-energy equation model has been extended to the three-energy equation version, in order to account for the countercurrent heat transfer between closely spaced arteries and veins in the circulatory system and its effect on the peripheral heat transfer. This general form of three-energy equation model naturally reduces to the energy equations for the tissue, proposed by Chato, Keller and Seiler. Controversial issues on blood perfusion, dispersion and interfacial heat transfer coefficient are discussed in a rigorous mathematical manner.     

Review on thermal transport in high porosity cellular metal foams with open cells

Thermal transport in metal foams has received growing attention in both academic research and industrial applications. In this paper the recent research progress of thermal transport in metal foams has been reviewed. This paper aims to provide the comprehensive state-of-the-art knowledge and research results of thermal transport in open celled cellular metal foams, which covers the effective thermal conductivity, forced convection, natural convection, thermal radiation, pool boiling and flow boiling heat transfer, solid/liquid phase change heat transfer and catalytic reactor. The forced convection and thermal conductivity have been extensively investigated, while less research were performed on two-phase (boiling and solid/liquid phase change heat transfer) and thermal radiation in metal foams. Also most research still treats the metal foam as one type of effective continuous porous media, very few researchers investigated the detailed thermal behaviours at the pore level either by numerical or experimental approaches.     

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.     

On natural convection in enclosures filled with fluid-saturated porous media including viscous dissipation

Care needs to be taken when considering the viscous dissipation in the energy conservation formulation of the natural convection problem in fluid-saturated porous media. The unique energy formulation compatible with the First Law of Thermodynamics informs us that if the viscous dissipation term is taken into account, also the work of pressure forces term needs to be taken into account. In integral terms, the work of pressure forces must equal the energy dissipated by viscous effects, and the net energy generation in the overall domain must be zero. If only the (positive) viscous dissipation term is considered in the energy conservation equation, the domain behaves as a heat multiplier, with an heat output greater than the heat input. Only the energy formulation consistent with the First Law of Thermodynamics leads to the correct flow and temperature fields, as well as of the heat transfer parameters characterizing the involved porous device. Attention is given to the natural convection problem in a square enclosure filled with a fluid-saturated porous medium, using the Darcy Law to describe the fluid flow, but the main ideas and conclusions apply equally for any general natural or mixed convection heat transfer problem. It is also analyzed the validity of the Oberbeck–Boussinesq approximation when applied to natural convection problems in fluid-saturated porous media.     

Modeling of turbulent heat transfer and thermal dispersion for flows in flat plate heat exchangers

In this paper, heat transfer and dispersion for both laminar and turbulent regimes in heat exchangers and nuclear cores are considered. Such hydraulic systems might be seen as spatially periodic porous media. The existence of a turbulent flow within a porous medium structure suggests the use of a spatial average operator, combined to a statistical average operator. Previous works [M.H.J. Pedras, M.J.S. De Lemos, Macroscopic turbulence modeling for incompressible flow through undeformable porous media, Int. J. Heat Mass Transfer 44 (2001) 1081–1093; F. Kuwahara, A. Nakayama, H. Koyama, A numerical study of thermal dispersion in porous medium, J. Heat Transfer 118 (1996) 756–761] have applied a double average procedure to the thermal balance equation, which led to a macroscopic turbulent transport and a subsequent macro-scale equation featuring dynamic dispersion. Considering the heat flux at the solid surfaces as a boundary condition for the fluid energy balance, the model proposed in this paper allows one to take into account this dispersion as the sum of two contributions. The first one is the classical dispersion due to velocity heterogeneities [G. Taylor, Dispersion of solute matter in solvent flowing slowly through a tube, Proc. Roy. Soc. Lond. A 219 (1953) 186–203] and the second one is due to wall heat transfer. Applying Whitaker up-scaling method [S. Whitaker, Theory and applications of transport in porous media: the method of volume averaging, Kluwer Academic Publishers, 1999], a “closure problem” is then derived for a representative elementary volume, using the so-called Boussinesq approximation to account for small scale turbulence. The model is used to compute macro-scale heat transfer properties for turbulent flows inside a flat plate heat exchanger. It is shown that, for such flows, both dispersive fluxes strongly predominate over the macroscopic turbulent heat flux.     

A bioheat transfer model: Forced convection in a channel occupied by a porous medium with counterflow

An illustrative model for bioheat transfer is developed. An analytical solution is obtained for forced convection in a parallel plate channel occupied by a layered saturated porous medium with counterflow, the dominant feature that distinguishes bioheat transfer from other forms of heat transfer. The case of asymmetrical constant heat-flux boundary conditions is considered and the Brinkman model is employed for the porous medium. It is found that the Nusselt number Nu is zero when the mean velocity is zero, and negative values can be attained.     

Intelligent networks for crosswise stream nanofluidic model with Cu–H2O over porous stretching medium

The porous media transport theories are thoroughly operative to analyse transferral phenomenon in reducing the bio-convective flow instabilities and biological tissues. The present study is designed to investigate the heat transfer phenomena in nanofluidic system involving Cu ? H₂O over the stretched porous media with the strength of stochastic solver via Levenberg-Marquardt backpropagation networks. The mathematical model of physical phenomena is described in PDEs that are reduced to system of ODEs through scaling group transformations. The datasets are determined through explicit Runge-Kutta numerical method and used as a target parameter for the development of continuous neural networks mapping. The training, testing and validation processes are utilized in learning of neural network models based on backpropagation of Levenberg-Marquardt technique to determines the solution of different scenarios constructed on the various values of porosity parameter along with six different cases based on the stretching ratio values. Validation and verification of neural network model to find the solution of nanfluidic problem is endorsed on the assessment of achieved accuracy through mean squared error, error histograms and regression studies.     

The Radiation Element Method Coupled with the Bioheat Transfer Equation Applied to the Analysis of the Photothermal Effect of Tissues

The purpose of this study is to develop the radiation element method by ray emission method (REM²) code appropriate for coupling with the bioheat transfer equation, and to clarify the photothermal effect of various parameters inside biological tissues. First, the REM², which involves the air-tissue interface effect, is validated with the existing literature. In order to clarify the effects of optical and thermophysical properties of biological tissues, a one-dimensional tissue model of CW light transport and bioheat transfer is employed. The present study provides nondimensional results, which are obtained by varying refractive index, extinction coefficient, scattering albedo, blood perfusion parameter, and conduction-radiation parameter, show valuable guidance for understanding the coupled light and bioheat transport in tissues.     

Non-gray radiative convective conductive modeling of a double glass window with a cavity filled with a mixture of absorbing gases

Coupled radiation and natural convection heat transfer occurs in vertical enclosures with walls at different temperatures filled with gas media. In glass window thermal insulation applications in hot climates, infrared absorbing gases appear as an alternative to improve their thermal performance. The thermal modeling of glass windows filled with non-gray absorbing gases is somewhat difficult due to the spectral variation of the absorption coefficients of the gases and the phenomena of natural convection. In this work, the cumulative wavenumber (CW) model is used to treat the spectral properties of mixtures of absorbing gases and the radiative transport equation is solved using CW model and the discrete ordinates method. Due to the range of temperature variation, the mixture of gases is considered as homogeneous. The absorption coefficients were obtained from the database HITRAN. First, the natural convection in a cavity with high aspect ratio is modeled using a CFD code and the local and global Nusselt numbers are computed and compared with available empirical correlations. Also, the flow pattern for different Rayleigh numbers is analyzed. Then, the heat transfer in the gas domain is approximated by a radiative conductive model with specified heat flux at boundaries which is equivalent to convective transport at the walls surroundings. The energy equation in its two-dimensional form is solved by the finite volume technique. Three types of gas mixtures, highly absorbing, medium and transparent are investigated, to determinate their effectiveness in reducing heat gain by the gas ambient. Reflective glasses are also considered. The numerical method to solve radiative heat transport equation in gray and non-gray participant media was validated previously. The temperatures distributions in the gas and the glass domain are computed and the thermal performance of the gas mixtures is evaluated and discussed. Also, comparison with pure radiative conductive model is shown.     

Secondary instabilities on double-diffusive convection in an anisotropic porous layer with Soret effect

In an anisotropic porous matrix with a Soret coefficient, the onset of double-diffusive convection is investigated analytically using weakly nonlinear analysis. The momentum equation is expressed using a generalized Darcy model with a time derivative term. The Newell–Whitehead–Segel equation is acquired, thereby examining the Eckhaus and zigzag secondary instabilities. Nusselt and Sherwood numbers are used to examine convection onset by quantifying heat and mass movement. Heat and mass transmission dynamics are graphically depicted as a consequence of several parameters. An increase of the positive value of the Soret parameter enhances heat transport, whereas, an increase of the negative value of the Soret parameter reduces it.     

Convection heat transfer with internal heat generation in porous media: Implementation of thermal lattice Boltzmann method

Evaluation of lattice parameters for convection heat transfer in porous media with internal heat generation from physical and macroscale properties was described. A hierarchical process was defined to implement thermal Lattice Boltzmann Method (LBM) to investigate convection heat transfer with internal heat generation in different geometries; from a simple geometry (flow channel) to complex ones (porous media). In this regard, seven different without any obstacle cases with different geometries were designed and the detailed information about how thermal LBM should be implemented for these cases are addressed. Going from one case to the next, the cases with more complex physics and/or geometries were examined. The results showed that LBM is an appropriate method to predict heat transfer with internal heat generation in porous media.     

Heat transfer and phase change of liquid in an inclined enclosure packed with unsaturated porous media

In this paper, we present a mathematical model to describe the simultaneous heat and mass transfer with liquid phase change in unsaturated porous media. Two-dimensional natural convective flow in an inclined rectangular enclosure with porous material unsaturated with fluid is analyzed numerically. The parameter variations are considered for the tilted angle, the aspect ratio and the Darcy–Rayleigh number. Local and global Nusselt numbers are presented as functions of those parameters. Compared with the saturated porous material, the heat transfer characters in the unsaturated case are discussed for the identical aspect ratio and Darcy–Rayleigh number, The discussion is also made for the field synergy of fluid velocity and heat flow in natural convection.     

Thermal analysis of natural convection and radiation in porous fins

In this study, the effects of radiation and convection heat transfer in porous media are considered. The geometry considered is that of a rectangular profile fin. The porous fin allows the flow to infiltrate through it and solid-fluid interaction takes place. This study is performed using Darcy's model to formulate heat transfer equation. To study the thermal performance, three types of cases are considered viz. long fin, finite length fin with insulated tip and finite length fin with tip exposed. The theory section addresses the derived governing equation. The effects of the porosity parameter S_h, radiation parameter G and temperature ratio C_T on the dimensionless temperature distribution and heat transfer rate are discussed. The results suggest that the radiation transfers more heat than a similar model without radiation.