Local Thermal Equilibrium Condition Research Articles

Emergence hour-by-hour of r-process features in the kilonova AT2017gfo

The spectral features in the optical/near-infrared counterparts of neutron star mergers (kilonovae, KNe) evolve dramatically on hourly timescales. To examine the spectral evolution, we compiled a temporal series that was complete at all observed epochs from 0.5 to 9.4 days of the best optical/near-infrared (NIR) spectra of the gravitational-wave detected kilonova AT2017gfo. Using our analysis of this spectral series, we show that the emergence times of spectral features place strong constraints on line identifications and ejecta properties, while their subsequent evolution probes the structure of the ejecta. We find that the most prominent spectral feature, the 1 μm P Cygni line, appears suddenly, with the earliest detection at 1.17 days. We find evidence in this earliest feature for the fastest yet discovered kilonova ejecta component at 0.40–0.45c. Across the observed epochs and wavelengths, the velocities of the line-forming regions span nearly an order of magnitude, down to as low as 0.04–0.07c. The time of emergence closely follows the predictions for Sr II because Sr III combines rapidly under local thermal equilibrium (LTE) conditions. The transition time between the doubly and singly ionised states provides the first direct measurement of the ionisation temperature. This temperature is highly consistent with the temperature of the emitted blackbody radiation field at a level of a few percent. Furthermore, we find the KN to be isotropic in temperature, that is, the polar and equatorial ejecta differ by less than a few hundred Kelvin or ≲5%, in the first few days post-merger based on measurements of the reverberation time-delay effect. This suggests that a model with very simple assumptions, with single-temperature LTE conditions, reproduces the early kilonova properties surprisingly well.

Gas Kinematics and Dynamics of Carina Pillars: A Case Study of G287.76-0.87

We study the kinematics of a pillar, namely G287.76-0.87, using three rotational lines of 12CO(5-4), 12CO(8-7), 12CO(11-10), and a fine structure line of [O i] 63 μm in southern Carina observed by SOFIA/GREAT. This pillar is irradiated by the associated massive star cluster Trumpler 16, which includes η Carina. Our analysis shows that the relative velocity of the pillar with respect to this ionization source is small, ∼1 km s−1, and the gas motion in the tail is more turbulent than in the head. We also performed analytical calculations to estimate the gas column density in local thermal equilibrium (LTE) conditions, which yields N CO as (∼0.2–5) × 1017 cm−2. We further constrain the gas’s physical properties in non-LTE conditions using RADEX. The non-LTE estimations result in nH2≃105cm−3 and N CO ≃ 1016 cm−2. We found that the thermal pressure within the G287.76-0.87 pillar is sufficiently high to make it stable for the surrounding hot gas and radiation feedback if the winds are not active. While they are active, stellar winds from the clustered stars sculpt the surrounding molecular cloud into pillars within the giant bubble around η Carina.

Numerical simulation of Darcy–Forchheimer flow of Casson ternary hybrid nanofluid with melting phenomena and local thermal non-equilibrium effects

The Casson fluid flow over a stretching sheet is studied in this work in the presence of porous media with melting heat transfer and chemical reaction, combining nanoparticles of Ti4Al6V,AA7072 and AA7075 with a base fluid of sodium alginate (NaAlg). The porous medium Darcy Forchheimer effect is included in the momentum equation. Through the effects of melting, the process of heat transfer is described. In order to explore the features of heat transmission in the absenteeism of LTECs (local thermal equilibrium conditions), the current study employs a mathematical model that has been simplified. For both the solid and liquid phases, the LTNE model yields two different fundamental thermal gradients. This proposed model compares the YOM (Yamada-Ota model) and XM (Xue model), two well-known trihybrid nanofluid models, in terms of performance. After applying the proper transformations, modulated non-linear PDEs (partial differential equations) are simplified in ODEs (ordinary differential equations) and the bvp4c method is used to solve them numerically. A graphic discussion of the relevant factors' significance on the pertinent fields has been presented. It is observed that the Casson ternary hybrid nanofluid temperature and velocity distributions predominate at higher melting parameter. The solid phase's heat transport rate rises with an upsurge in the interphase heat transfer parameter.

Numerical study on mixed convection heat transfer of copper-water nanofluid in a porous cavity with an isothermal solid block in local thermal non-equilibrium

This manuscript reports a numerical study on mixed convection heat transfer of Cu-water nanofluid in a porous cavity with an isothermal solid block in the state of local thermal non equilibrium. The set of governing equations are solved by finite volume method combined with SIMPLE algorithm. Through streamlines and isothermal contour plots as well as the average Nusselt number, the impact of various parameters like the nanoparticle concentration, Richardson number, Darcy number, and inter phase heat transfer coefficient is highlighted. The main findings indicate that the Local thermal non-equilibrium state (H = 1, 10) produces the maximum heat transfer rate than the local thermal equilibrium state (H = 1000).

Computed Rotational Collision Rate Coefficients for Recently Detected Anionic Cyanopolyynes

We report new results from quantum calculations of energy-transfer processes taking place in interstellar environments and involving two newly observed molecular species: C5N− and C7N− in collision with He atoms and p–H2 molecules. These species are part of the anionic molecular chains labeled as cyanopolyynes, which have been observed over the years in molecule-rich circumstellar envelopes and in molecular clouds. In the present work, we first carry out new ab initio calculations for the C7N− interaction potential with He atoms and then obtain state-to-state rotationally inelastic cross sections and rate coefficients involving the same transitions, which have been observed experimentally by emission in the interstellar medium (ISM) from both of these linear species. For the C5N−/He system, we extend the calculations already published in Biwas et al. to compare more directly the two molecular anions. We extend further the quantum calculations by also computing in this work collision rate coefficients for the hydrogen molecule interacting with C5N−, using our previously computed interaction potential. Additionally, we obtain the same rate coefficients for the C7N−/H2 system by using a scaling procedure that makes use of the new C7N−/He rate coefficients, as discussed in detail in the present paper. Their significance in affecting internal state populations in ISM environments where the anionic cyanopolyynes have been found is analyzed by using the concept of critical density indicators. Finally, similarities and differences between such species and the comparative efficiency of their collision rate coefficients are discussed. These new calculations suggest that, at least for the case of these longer chains, the rotational populations could reach local thermal equilibrium conditions within their observational environments.

DIGITAL СORE: TEMPERATURE FIELD INFLUENCE ON TWO-PHASE FILTRATION OF FLUIDS IN ROCKS

Link for citation: Katanov Yu.E., Yagafarov A.K., Aristov A.I. Digital сore: temperature field influence on two-phase filtration of fluids in rocks. Bulletin of the Tomsk Polytechnic University. Geo Аssets Engineering, 2023, vol. 334, no. 10, рр. 108-118. In Rus. Relevance. Computational fluid dynamics is a powerful tool for studying geological processes. Fluid flow equations in porous media can be solved directly with the resulting three-dimensional X-ray and computer tomographic images. Numerous studies of the geological features of rocks depend on heat transfer in porous materials. These processes include heat conduction in the rock material, convection between matter and the surrounding gas, electrical conductivity, viscous dissipation of fluids, kinetics of chemical reactions, and interfacial heat transfer at the solid/liquid interface. While energy transfer in porous media under conditions of local thermal equilibrium is well studied on the scale of Darcy's law, it is less often considered in porous models. Heat transfer can be accounted for by creating a «heat network» based on a porous network model representing a set of fluid-conducting channels with an appropriate solution to the unified energy balance equation. It is also necessary to take into account convective transfer of thermal energy by the flow of liquid phases (oil, water), each of which is represented by a separate network. Knowing the distribution of the temperature field in the texture of each lithological type of rock, it is possible to determine the probabilistic displacement of the fluid displacement front in the corresponding fluid-conducting space of the rock. Objective: neural network modeling of the effect of the temperature field on the filtration characteristics of hydrocarbons in the fluid-conducting space of rocks. Objects: polymictic sandstones of the Tyumen Formation. Methods. Digital reconstruction of rock texture was performed using artificial intelligence methods and neural networks; neural network algorithms of probabilistic displacement front of two-phase flow (oil, water) were developed in Python programming language; methodical approach to the study of thermal field propagation in rock texture and its influence on two-phase fluid flow was developed using Fourier law, Navier–Stokes equations and similarity criteria of hydrocarbon systems. Results. We developed algorithms for neural network modeling of the temperature field in the fluid-conducting space of a rock (polymictic sandstone) as well as neural network algorithms to estimate the displacement of two-phase filtration front under the corresponding influence of the temperature distribution in the texture of a digital core. The basic mathematical models of these algorithms are outlined. The source code of the algorithms is written in the Python programming language with additional use of non-commercial libraries. The results of neural network modeling have a high degree of reliability, confirmed by experiments with the data obtained in the laboratory of core studies (University of Tyumen, Tyumen), the laboratory of the V.I. Shpilman Research and Analytical Centre for the Rational Use of the Subsoil (Khanty-Mansiysk), laboratory of digital research in oil and gas in the framework of the technological project «Digital core» (Industrial University of Tyumen, Tyumen).

The role of local thermal non-equilibrium on the instability analysis in the presence of variable gravity field with throughflow

The influence of variable gravity on the stability analysis has been investigated in a horizontal porous region with throughflow under a local thermal non-equilibrium (LTNE) situation. Flow inside the porous region is considered to be followed by the Darcy model. Also, the linear varying gravity field with height and nonlinear (parabolic, cubic, and exponential) changing gravity field depth have been considered to address the investigation. The obtained eigenvalue problem for the perturbed state is solved via bvp4c technique in MATLAB. Finally, we investigated the convective boundaries of the system due to the governing parameters, such as gravity variation parameter (λ), porosity-modified conductivity ratio (γ), interphase heat transfer parameter (H), and throughflow parameter (Q). The outcomes assure that the interphase heat transfer parameter has a significant impact on the breakdown of the convection inside of its intermediate range, while the local thermal equilibrium (LTE) state occurred when H converges both the limiting cases, i.e. H→0 and H→∞. It has also been observed that the gravity variation parameter and throughflow parameter have a stabilizing effect, whereas the porosity-modified conductivity ratio destabilizes the flow. Further, it is distinguished that the instability boundaries are unchanged by H when γ≥10. In addition, the flow field is more delayed for exponential changing gravity field, whereas more advanced for cubic varying gravity field.

Compositional analysis of copper and iron-based alloys using LIBS coupled with chemometric method.

The present manuscript deals with the utility of the calibration-free LIBS and calibration curve methods for the compositional study of different alloys using laser-induced breakdown spectroscopy (LIBS). In the process of alloying in the smelting industry, metal concentration in different alloys affects the physical and chemical properties of the final products. Therefore, LIBS can be used as an efficient quantitative analysis tool for online monitoring of the quality of the products. This is because LIBS can be performed online, in situ, without any pre-processing, and need no sample preparation for the compositional analysis of any type of materials present in any phase (solid, liquid, gas or even molten alloys in the industries). In the present study, four alloys (three copper and one iron-based alloy) consisting of Cu, Al, Zn, Ni, Fe, Cr and Mn as major and Sn and Si as minor elements were selected for the study using calibration-free laser-induced breakdown spectroscopy (CF-LIBS) and calibration curve method i.e. partial least square regression (PLSR). For the CF-LIBS method, the temporal delay has been optimized in order to satisfy the optically thin and local thermal equilibrium (LTE) condition of the plasma. For the PLSR method, different regions of the strongest emission lines of constituents have been selected for quantitative analysis. The study of time-resolved LIBS spectra and the variation of plasma parameters with respect to the delay time is also discussed. The utility of the combined technique of CF-LIBS with the PLSR method for rapid monitoring and quality assessment of desired material/products without any sample pretreatment, thus reducing the cost of the analysis, is presented in this paper.

Heat transfer coefficients for bubbly molten salt nuclear reactors

Modeling the heat transfer process in a molten salt nuclear reactor is challenging due to the non-uniform distribution of fuel material and cooling fluid. Developing theoretical tools to calculate effective parameters and temperature distribution is essential for ensuring the safety of reactor design. In this study, we utilized the theory of volume averaging to upscale the local model for heat transport in a fluid fuel reactor, known as the Molten Salt Fast Reactor (MSFR). This upscaled model, which is valid for the bulk of the reactor core, combines the heat transport physics that occurs at the microscale through mathematically derived effective coefficients. These coefficients can be obtained by solving related boundary-value problems. The standalone core of the MSFR with inert gas bubbles was studied and the local thermal equilibrium (non-equilibrium) condition between the fluid and bubbles was found to be achievable for small (large) bubbles volume fractions. We present the numerically calculated values of the effective coefficients in terms of the volume fraction of bubbles under operating conditions and provide correlations to facilitate their use in computational codes.

About the applicability of the theory of porous media for the modelling of non‐isothermal material injection into porous structures

AbstractIn this contribution we investigate the relevance of the theory of porous media for the non‐isothermal modelling of material injection into porous structures. In particular, we provide a model describing the injection of cement during percutaneous vertebroplasty, which is derived by consistently following the theory of porous media. We demonstrate numerically that this model elicits unphysical behaviour under local thermal non‐equilibrium conditions. No distinct unphysical behaviour is observed under local thermal equilibrium conditions. We conclude that heuristic modifications of the model equations are necessary and suspect the unphysical behaviour to be caused by contradictory modelling assumptions.

Melting performance enhancement of febro-hydrodynamic carreau non-Newtonian PCM in porous media: A geometrical evaluation

In this paper melting of a carreau non-Newtonian PCM in the space between two concentric horizontal tubes, which is partially filled with porous material with different shapes but same area, is investigated numerically. A magnetic source is located in the center of the geometry for melting process of PCM to occur in the presence of ferro-hydrodynamic effects. Porous material is made of Cu that covers the cross-section of the inner tube. In addition, the space between inner and outer tubes is saturated with paraffin-wax PCM. Flow of melted paraffin-wax is considered as a Carreau non-Newtonian, laminar and incompressible flow with viscous dissipation that is evaluated in a specific time interval. Boussinesq approximation is valid for the PCM. Also local thermal equilibrium condition is assumed between the porous and the PCM. Galerkin finite element method has been utilized to solve the problem. Results showed that melting rate is higher for the third model in comparison other models. Also effects of the magnetic number depends on the shape of the porous medium. Therefore, that increase in the magnetic number, increasingly enhances the progress of the melting front in the second case. Moreover, effects of Carreau index, Stefan number, and porosity on the melting process are studied.

HEAT AND MASS TRANSPORT IN A NANOFLUID LAYER USING A THERMAL NONEQUILIBRIUM MODEL CONFINED WITHIN A HELE-SHAW CELL UNDER THE EFFECT OF GRAVITY MODULATION

The local thermal nonequilibrium (LTNE) condition is the temperature difference between the base fluid and the nanoparticle. In the present research article, linear analysis was done to know about the onset of convection in the system, and a weakly nonlinear stability analysis was done to know about heat and mass transport in the system for both the unsteady and steady case. Here we have taken temperature to be constant and nanoparticle flux to be zero at the upper and lower boundaries of the system. The normal mode technique is used for linear analysis, and the truncated Fourier series method is used for nonlinear analysis; plot streamlines, isotherms, and isohaline are used to visualize the conduction, convection, and steady state. We found that the behavior of Hele-Shaw cell is the same in the case of LTNE and local thermal equilibrium (LTE). The effect of Hele-Shaw number, interphase heat transfer coefficient, modified thermal capacity ratio, thermal diffusivity ratio, amplitude, and frequency of modulation on the onset of convection, heat, and mass transfer are depicted graphically. We found that the effect of LTNE can be seen only for the intermediate values of the interphase heat transfer coefficient, and this region is called the LTNE region. We also discuss the result of thermal Nusselt number, streamlines, and isothermals of fluid and particle phase for steady case and plot the graphs with respect to the Hele-Shaw cell Rayleigh number. Rate of heat transfer for the particle phase is higher than the fluid phase for both the unsteady and steady state. In this research paper we find the result for both LTE and LTNE conditions with unsteady and steady cases, while in the previous study we analyzed only for the LTE condition.

DOUBLE-DIFFUSIVE CONVECTION-REACTION FLOW IN A SQUARE ENCLOSURE FILLED WITH POROUS MEDIUM

In this work, we analyze the double-diffusive convection-reaction of a binary mixture in a square enclosure filled with a porous media, with constant temperatures and concentrations at the two vertical walls and adiabatic and impermeable horizontal walls. The boundaries of the enclosure are under chemical equilibrium. The porous medium is assumed to be in local thermal equilibrium (LTE) state and the solubility of the dissolved salts depends on temperature. The two-dimensional steady-state flow rate is determined by a non-Darcy (Darcy-Brinkman) model, and the entire governing equations are solved using the usual SIMPLE-R finite-volume methodology. It has been found that as the reaction rate increases in the system the salt precipitation also increases, causing a decrease in overall mass transfer rate. Also, the chemical reaction tends to decrease the overall heat transfer-rate.

COMPUTATIONAL MODELING FOR THE PERFORMANCE PREDICTION OF QUAD-SECTOR ROTARY AIR PREHEATER AND COMPARATIVE STUDY WITH TRI-SECTOR AND BI-SECTOR ROTARY AIR PREHEATERS

Rotary air preheaters are mainly used to improve the economy of a steam power plant by preheating the air with the help of hot flue gases from the boiler. Modern steam power plants are preferring tri-sector and quad-sector rotary air preheaters over bi-sector due to their better performance and lower leakage. In the present work, thermal behavior of a quad-sector rotary air preheater is the focus using computational fluid dynamics. The porous media approach is used for numerical simulation assuming local thermal equilibrium condition between fluid and solid. The performance of the air preheater is accessed by effectiveness and pressure drop. Porosity is estimated for four different cell shapes, which are used in the rotary preheater core for numerical investigation. The study reveals that the core consisting of a triangular cell has the highest effectiveness. The effect of porosity on performance is also investigated for various cell shapes. The effect of important operating parameters, such as rotor speed, aspect ratio, and mass flow rate of fluids on the effectiveness, as well as pressure drop is investigated for a quad-sector rotary air preheater. The study ends with a comparative performance analysis between bi-, tri-, and quad-sector rotary air preheaters. The performance of quad-sector air preheater is found to be better than other two preheaters in terms of effectiveness, whereas its pressure drop is maximum.

Analysis of heat transfer using Local thermal non‐equilibrium conditions for a non‐Newtonian fluid flow containing Ti6Al4V and AA7075 nanoparticles in a porous media

AbstractThe accurate forecasting of thermal processes is one of the primary concerns of those working in the area of heat transfer as well as those in the energy research community. The present investigation makes use of a simplified mathematical model as an illustration in order to investigate the properties of heat transport under the lack of local thermal equilibrium conditions. The LTNE model results in two distinct primary thermal gradients, one for the fluid phase and another for the solid phase. The non‐Newtonian fluid flow containing Ti6Al4V and AA7075 nanoparticles with a base fluid sodium alginate over a stretching sheet in a porous media with magnetic effect is studied here. The shooting approach is used to transform the resultant equations of boundary value problems into initial value problems, which are subsequently solved using Runge–Kutta–Fehlberg 45 process. The impact of pertinent parameters on the involved fields have been discussed graphically. Results reveal that the increase in porosity and magnetic parameters value decreases velocity and increases thermal performance. An upsurge in inter‐phase heat transfer parameter increases the heat transport rate of solid phase but, decreases the liquid phase heat transport rate.

Validation of ablation model for polyethylene using pulsed x-ray and proton exposures

The surface erosion of polyethylene is interrogated using pulsed x rays at the Z Machine (Sandia National Laboratories) and with proton beams at the Gamble II generator (Naval Research Laboratory) to validate a coupled model for volumetric thermal ablation, photoionization, finite-rate decomposition, and molecular recombination of radicals. The intense radiation pulses (up to ∼1014W/m2 over tens of nanoseconds) are used to generate one-dimensional vapor flows with low ionization fractions and a simplified geometry compared to typical laser ablation, allowing for evaluation of the model under local thermal equilibrium conditions. Areal momentum carried by the ensuing uniaxial hydrodynamic shock is used to indicate the extent of ablation. The threshold fluence for ablation is found to be in close correspondence with the bulk melt transition, and reasonable agreement with the model is obtained for peak temperatures in polyethylene up to 5500 K and heating rates up to 1011K/s where thermal decomposition reactions are also active.

Laser induced breakdown spectroscopy of aluminum incorporated with metallic nanoparticles

Laser-Induced Breakdown Spectroscopy is a promising spectroscopic technique with a vast spectrum of applications in fields concerned with identification and detection of elements. But it faces some limitations due to self-absorption, noise due to matrix effect and line broadening resulting in low emission signal. This research proposes LIBS signal enhancement by incorporation of metal nanoparticles (Cu, Mg, Au) on Al surface and compares their effect. The successful optical emissions enhancement is achieved as the emission intensities of Al- and Na- lines of three coated samples are compared with those of uncoated Al. The Electron Temperature has been evaluated by Boltzmann plot and an increase in Electron Temperature has been observed with the incorporation of nanoparticles to the aluminum surface as compared to the untreated aluminum, due to more plasma emissions. The Electron Number Density of the aluminum plasma did not have much effect with the incorporation of Nanoparticles. The Local Thermal Equilibrium condition has been satisfied and checked by Mc Whirter’s Criterion. The incorporation of metal nanoparticles can be declared as an effective method not only for LIBS signal enhancement but also better detection of trace elements which were not observed without the use of Nanoparticles.

Experimental and Numerical Analysis of Forced Convection in a Horizontal Tube Partially Filled with a Porous Medium under Local Thermal Equilibrium Conditions

The objective of the present work is to analyze experimentally and numerically the laminar forced convection flow in a horizontal pipe partially filled with a porous medium under constant heat flux and to study the influence of the eccentricity of the porous medium on the results. In a numerical analysis, the governing equations are solved in three dimensions. To simplify the grid generation and the satisfaction of the boundary conditions, conformal mapping is applied to convert the cross-section of the tube in the fluid domain (space between two eccentric circles) into a rectangle, and the equations are solved in a computational domain in this domain. The Darcy–Brinkman–Forchheimer model is applied to simulate the hydrodynamic behavior of the flow in the porous region. Thermal equilibrium between solid and fluid is assumed for the energy equation. A FORTRAN program was developed to solve the equations using the finite volume method and the SIMPLE algorithm. Velocity profile, pressure drop and average Nusselt number are studied in a wide range of Darcy numbers, thickness of porous mediums and eccentricities. The results show that the eccentricity of the porous material reduces the heat transfer coefficient and the pressure drop simultaneously; of course, the reduction in the heat transfer coefficient is less noticeable when the thickness of the porous medium is smaller. For example, at RP = 0.5, when the eccentricity of the porous medium increases up to E = 0.4, the average Nusselt number decreases by 66%, and this reduction for a smaller porous thickness decreases to 11%. The maximum pressure drop reduction for Da = 10−5 and E = 0.4 is 25%.

Gaseous slip flow in a porous two-dimensional rectangular microchannel subjected to inclined magnetic field

In this study, two-dimensional, steady, laminar, and compressible gaseous flow of Newtonian fluid in a rectangular porous microchannel affected by an electromagnetic field, and under local thermal equilibrium condition was analytically investigated in the slip flow region (0.001<Kn < 0.1). The porous microchannel is entirely affected by uniform inclined magnetic field with an angle α from the positive x-axis, the range of this angle is 0 ≤ α ≤ 90°. Also, the porous microchannel effected by uniform electric field which points in the positive z-direction. Thermophysical properties of the fluid were assumed to be constant. The normalized governing equations were asymptotically solved and expressions of the flow velocity components, the pressure, and the temperature were provided using first-order velocity slip and temperature jump boundary conditions. The effects of various parameters on the flow were studied, such as Darcy number, the porosity, the thermal conductivity ratio, the electrical conductivity ratio, Hartmann number, Knudsen number, the electric field to magnetic field ratio (0 ≤ K ≤ 1), and the magnetic field inclination angle. It was found that increasing the permeability of the porous medium increases the flow velocity but decreases its temperature. Further, the results reveal that applying a stronger magnetic field in absence of electric field decreases the flow velocity but increases its temperature. On the other hand, the stronger the electric field, the higher the flow velocity and its temperature. Also, as the magnetic field inclination angle increases the flow velocity decreases. The effects of Darcy number and the electrical conductivity ratio on the heat flux were studied as well. A comparison of the analytical solutions with the numerical results were presented as a validation for this study. The comparison showed that the analytical results were in good agreement with the numerical results.

Mixed convection phenomenon in packed beds: A comprehensive review

This review summarises first the studies that have been performed to date for establishing a suitable model for mixed convection phenomenon in packed beds with working fluids include some simplifying assumptions such as local thermal equilibrium ( L T E ) condition, neglect of channelling effect, and neglect of thermal dispersion effect. It was found that most of these studies are based on Darcy’s model for the flow and on an averaged single equation model ( L T E ) for the energy equation with using the Boussinesq approximation to represent the variation in fluid density. Theoretical, numerical, or experimental studies, or studies which combine these approaches, are reviewed and for various 2- or 3-dimensional geometries such as horizontal, vertical or inclined parallel plate channels, ducts, cylinders, annuli, etc. In addition, the literature showed that there has been resealable efforts exerted to report works on the non-Darcian mixed convection, in which the effects of boundary, inertia, porosity variation, and inclusion of thermal dispersion under local thermal equilibrium condition ( L T E ) or under thermal non-equilibrium condition ( L T N E ). • This article reviews firstly the studies done about mixed convection phenomenon in packed beds using Darcy and (LTE) energy models. • Studies that include non-Darcian effects such as boundary, inertia, and thermal dispersion, are then summarised. • Works about utilising (LTNE) energy model are lastly reported.