Local Thermal Non-equilibrium Research Articles

High-temperature two-layer integrated receiver storage for concentrating solar power systems

Abstract An integrated receiver storage for a beam-down concentrating solar power (CSP) plant enables simplifying the system layout and reducing the costs. Here a two-layer integrated receiver storage (TLIRS) system design is proposed consisting of a cavity receiver and a two-layer packed-bed storage. The first layer is a porous ceramic and the second one is rocks. To enable performance optimization of the system, models for cavity radiation, two-phase transient heat transfer and local thermal nonequilibrium were derived and validated. The thermal performance of the TLIRS system was assessed and characterized by the absorbing, charging, discharging and overall efficiencies and by the solar-to-exergy conversion ratio. The results indicate that the porous ceramic layer significantly enhances the absorption of solar irradiance and thus improves the thermal efficiencies. For CSP applications, the TLIRS system potentially achieves high thermal performance in repetitive charging and discharging cycles.

Chaotic Cattaneo-LTNE porous convection

The chaotic convection in a porous medium has been studied with a local thermal nonequilibrium (LTNE) model and hyperbolic-type heat transport equation in the solid arising due to Cattaneo heat flux law. The Brinkman-extended Darcy model is considered to describe the flow in a porous medium. A system of autonomous nonlinear ordinary differential equations is derived by using a truncated double Fourier series expansion such that the linear stability theory results are the same as those for the full two-dimensional problem. The stability of equilibrium points of the system is discussed in detail for the governing parameters involved therein. The transition to chaos in different phase planes and the transient behavior of heat transfer are analyzed by solving the model equations numerically using the Runge-Kutta-Fehlberg technique. The effect of increasing scaled solid thermal relaxation time parameter is to speed up the transition to chaos and to decrease the heat transfer.

Improving the thermophotovoltaic performance of a hydrogen-fueled planar combustor with four-corner inlets via partially inserting porous medium

Micro thermophotovoltaic (TPV) generator is a potentially efficient approach to achieve microscale fuel-to-electricity conversion with high-energy–density, high-efficiency, and small-scale portability, where micro combustor provides the high-temperature continuous combustion process as well as a heat source of thermal radiation. In this study, a hydrogen-fueled planar combustor with four-corner inlets by partially inserting porous medium is proposed and investigated by a three-dimensional numerical model of porous medium combustion with detailed reaction mechanism and local thermal non-equilibrium. The combustor performance with and without partially inserting porous medium under various inlet velocities Vin and equivalence ratios ϕ are compared and analyzed to reveal the inherent strengthening mechanism. Moreover, the effects of various porous parameters on the energy conversion performance are examined: the porosity ɛ, thermal conductivity ks, and porous medium filled volume. The results demonstrate that the inserted porous medium can significantly enhance combustion efficiency, combustor wall temperature and its uniformity, and the largest increase in mean wall temperature reaches 79.7 K at Vin = 9 m/s, resulting in a 35.8 % increase in micro-TPV system efficiency. As ɛ and ks decrease, the combustor energy conversion performance increases while the temperature uniformity decreases. A moderate channel width L = 2.0 mm is good for further improving temperature uniformity and energy conversion efficiency. This study paves the way for the improvement of efficient, practical, and portable microscale generators.

Modeling on cryogenic heat exchangers considering variable properties, axial heat conduction and viscous heating

Cryogenic heat exchangers are key components of low temperature refrigerators and determine the ultimate refrigeration performance. Accurate modeling on the cryogenic heat exchangers in sub-Kelvin regime has important theoretical and applied values, yet is still short of study. This study proposes local thermal nonequilibrium models of the heat exchangers in sub-Kelvin regime, including continuous and discrete ones. The effects of the variable properties, axial heat conduction, and viscous heating were investigated. The finite volume method was employed to solve the models numerically; in addition, the temperature and pressure distributions were obtained and analyzed. The proposed models were validated using a comparison of the obtained simulation results with experimental results in the literature. The analysis of the obtained results revealed that: (1) for continuous heat exchangers, the variable Kapitza thermal conductance leads to a large temperature gradient at the entrance. The outlet temperature was negligibly affected by applying different inlet temperature. (2) The axial heat conduction significantly degrades the performance of discrete heat exchangers. The performance was found to be limited by the axial heat conduction when the axial heat conduction parameter was larger than 0.02. (3) For the sintered heat exchanger, both the dimensionless temperature and pressure drop are obviously affected by the geometric impedance when it is greater than 4 × 1014 m−4. Consequently, the proposed mathematical models are useful for the accurate prediction and optimization of cryogenic heat exchangers.

The role of a second diffusing component on the Gill–Rees stability problem

The stability of natural convection in a vertical porous layer using a local thermal nonequilibrium model was first studied by Rees (Transp Porous Med 87:459–464, 2011) following the proof of Gill (J Fluid Mech 35:545–547, 1969), called the Gill–Rees stability problem. The aim of the present study is to investigate the implication of an additional solute concentration field on the Gill–Rees problem. The stability eigenvalue problem is solved numerically and some novel results not observed in the studies of double-diffusive natural convection in vertical porous (local thermal equilibrium case) and non-porous layers are disclosed. The possibility of natural convection parallel flow in the basic state becoming unstable due to the addition of an extra diffusing component is established. In some cases, the neutral stability curves of stationary and travelling-wave modes are connected to form a loop within which the flow is unstable indicating the requirement of two thermal Darcy–Rayleigh numbers to specify the stability/instability criteria. Moreover, the change in the mode of instability is recognized in some parametric space. The results for the extreme cases of the scaled interphase heat transfer coefficient are discussed.

Local Thermal Non-Equilibrium Dominant Darcy-Rayleigh-Benard-Magneto-Marangoni Convection in a Composite Layer

The domination by Local Thermal Non-Equilibrium (LTNE) with regard to the onset of Darcy-Rayleigh-Benard-Marangoni (DRBM) convection swayed by magnetic field in a composite layer set-up is studied pertinent to incompressible fluid. The precinct above the fluid layer is presumed to be free and that below the porous layer is presumed to be rigid. Regular perturbation technique is exercised on the acquired problem to achieve the analytical solution taking adiabatic-adiabatic conditions into account at the boundaries. Effects of parameters such as solid phase thermal expansion ratio, solid phase thermal diffusivity ratio and inter-phase thermal diffusivity ratio that influences LTNE are discussed. The impact of change in measures of variables viz. fluid phase thermal expansion ratio, fluid phase thermal diffusivity ratio, porous parameter, thermal ratio, Chandrasekhar number and Marangoni number with respect to both LTNE and LTE situations are together explored and portrayed graphically. It is to be noted that for smaller values of fluid phase thermal expansion ratio and fluid phase thermal diffusivity ratio, the LTNE effects are prominent and cannot be ignored.

Impact of Magnetic Field on the Onset of Double Diffusive Rayleigh-Benard Convection Governed by Local Thermal Non-Equilibrium in a Double Layered System

Onset of Double Diffusive Rayleigh-Benard-Magneto (DDRBM) convection is studied for a double layered system composed of incompressible fluid confined by adiabatic rigid peripheries under the governance of Local Thermal Non-Equilibrium (LTNE). Result of the acquired problem is obtained analytically through the mode of Regular Perturbation. Consequence of physical factors such as solid phase thermal expansion ratio, solid phase thermal diffusivity ratio and inter-phase thermal diffusivity ratio that favours LTNE are being analysed. The outcome of altering the constraints namely fluid phase thermal expansion ratio, solute Rayleigh number, Chandrasekhar number, thermal ratio, concentration ratio, fluid phase solute diffusivity ratio, solute diffusivity ratio in fluid layer and porous layer subject to LTNE set-up are paralleled with that of LTE set-up with diagrammatic representation.

Repercussions of Local Thermal Non-Equilibrium in Conjunction with Uniform/Non-Uniform Thermal Gradients and Magnetic Flux on the Onset of Darcy-Rayleigh-Benard-Marangoni Convection in a Composite Layer

Repercussions of uniform/non-uniform thermal gradients such as linear, parabolic, inverted parabolic, piecewise linear gradient heated from below (PLHB), piecewise linear gradient cooled from above (PLCA) and step function (SF) thermal gradients in conjunction with magnetic flux on the onset of Darcy-Rayleigh-Benard-Marangoni (DRBMM) convection in a composite layer involving incompressible fluid and holding horizontal adiabatic boundaries, rigid beneath the porous layer and free aloft the fluid layer is performed apropos to Local Thermal Non-Equilibrium (LTNE). Regular Perturbation method is exercised to attain the analytical solution of the acquired problem. For all the six thermal gradients, (i) The strength of variables such as solid phase thermal expansion ratio, solid phase thermal diffusivity ratio and inter-phase thermal diffusivity ratio that boosts LTNE are analysed and (ii) The influence of change in measure of physical quantities viz. fluid phase thermal expansion ratio, fluid phase thermal diffusivity ratio, Chandrasekhar number and Marangoni number are studied.

Outcomes of Uniform as well as Non-Uniform Temperature Profiles on the Onset of Double Diffusive Magneto-Darcy-Rayleigh-Benard Convection in a Two Layer set up in the presence of Local Thermal Non-Equilibrium

Outcomes of uniform as well as non-uniform temperature profiles that is linear, parabolic, inverted parabolic, Piecewise Linear Profile Heated from Below (PLHB), Piecewise Linear Profile Cooled from Above (PLCA) and Step Function (SF) temperature profile on the onset of Double Diffusive Magneto-Darcy-Rayleigh-Benard (DDMDRB) Convection in a two layer set up in the presence of Local Thermal Non-Equilibrium with incompressible fluid horizontally surrounded by adiabatic rigid boundaries. Analytical solution to the attained problem is accomplished by means of regular perturbation technique. The consequences of altering the parameters namely fluid phase thermal expansion ratio, solid phase thermal expansion ratio, solid phase thermal diffusivity ratio, inter-phase diffusivity ratio, solute Rayleigh number, Chandrasekhar number, thermal ratio and porous parameter have been elucidated graphically.

Influence of local thermal non-equilibrium on enhanced geothermal system

Effective development of hot dry rock depends on complex fracture network formed by the enhanced geothermal system. Different scales of mass and heat transfer environment are formed between tight matrix and different fractures, which leads to the complexity of mass and heat transfer process of porous media. In order to analyze the influence of local thermal non-equilibrium on the mass and heat transfer process of complex fracture system, a micro pore scale mass and heat transfer model considering local non-thermal equilibrium is established. The microscopic mechanism of the effect of local non-thermal equilibrium assumption on mass and heat transfer process in matrix-fracture system is revealed. An embedded discrete fracture model considering local thermal non-equilibrium assumption and the solution method are also established. The necessity of considering local thermal non-equilibrium is analyzed by studying the sensitivity of formation physical parameters. The results show that ignoring the local thermal non-equilibrium assumption will underestimate the rock temperature near the fracture zone and overestimate the fluid temperature in the heat transfer front zone. The effect of local thermal non-equilibrium on the enhanced geothermal system is mainly reflected in the early stage of injection. The higher the injection intensity, the lower the matrix fracture permeability, the greater the matrix fracture porosity difference, the greater the rock thermal diffusion coefficient, the smaller the convective heat transfer coefficient, the more it is necessary to consider the local thermal non-equilibrium assumption. The research results can provide reference for the collection and collation of field data.

A macroscopic and mesoscopic model of Newtonian and non-Newtonian nanofluids with a two-energy equation method

We present an updated comprehensive macroscopic model of nanofluids, considering a revisited local thermal non-equilibrium (LTNE) condition to study the temperature difference between carrier fluid and nanoparticles. A new relation for thermal conductivity of solid and liquid phases in the LTNE condition is introduced which considers the possible particle aggregation. This model is thermodynamically consistent and covers the non-Newtonian models of nanofluids, including power-law and viscoplastic ones. A mesoscopic scheme based on the lattice Boltzmann method (LBM) which satisfies the presented macroscopic equations is introduced and derived. This investigation is a further development of our recent studies[G. H. R. Kefayati and A. Bassom, “A lattice Boltzmann method for single and two phase models of nanofluids: Newtonian and non-Newtonian nanofluids,” Phys. Fluids 33, 102008 (2021); G. H. R. Kefayati, “A two- and three-dimensional mesoscopic method for an updated non-homogeneous model of Newtonian and non-Newtonian nanofluids,” Phys. Fluids 34, 032003 (2022).] for simulating and analyzing nanofluids by a two-phase model. To assess the present numerical method, it is studied for a benchmark problem of natural convection in a cavity. The dimensional and non-dimensional macroscopic equations for the mentioned benchmark are defined and the implemented non-dimensional relations of LBM are shown. The present approach is verified with the obtained results of the mixture approach and a previous two-phase model, which demonstrated the accuracy of the presented method. The results including the temperature distributions of the solid and fluid phases, the nanoparticles distributions, and fluid flow behavior as well as the yielded/unyielded sections for the viscoplastic nanofluids are shown and discussed for the defined non-dimensional parameters. It was also demonstrated that the previous proposed thermal conductivity model of nanofluids in the LTNE approach generates a significantly different value compared to experimental results, and the current suggested model produces reliable results to the experimental ones.

Flow and heat transfer irreversibility in partial filled metal foams

This paper presents the analysis of entropy generation, exergy and heat transfer through metal foams partially and completely filled in a vertical channel. The assembly of heater located in between two aluminum plates is located at the core of the channel and for the purpose of heat transfer augmentation aluminum metal foam of pore density 10 with porosity 0.95 are located on either side of the plates. Four different filling rates of 25%, 50%, 75% and 100%, and three metal foam thicknesses of 10 mm, 20 mm and 30 mm are examined in this research for a fluid velocity ranging from 0.1 to 3 m/s. Darcy Extended Forchheimer (DEF) model and local thermal non-equilibrium (LTNE) model is used for forecasting the flow features and heat transfer through the metal foam. The numerical methodology adopted in this research is confirmed by comparing the results with the literature and found fairly good agreement between them. The flow physiognomies in terms of friction factor, heat transfer performance in terms of Nusselt number and performance factor, exergy transfer in terms of mean exergy based Nusselt number, and entropy generation i.e., total irreversibility in terms of Bejan number and entropy generation number are presented and discussed. Results showed that the effect of partially filled metal foams are found to be better in terms of entropy generation, and exergy transfer, and the effect of higher metal foam thickness is found to be better in terms of exergy and heat transfer, but in terms of total entropy generation resulting in terms of both heat transfer and pressure drop smaller metal foam thickness exhibits better performance.

Analysis of functionally graded metal foams for the accomplishment of heat transfer enhancement under partially filled condition in a heat exchanger

The use of partially filled high porosity graded aluminum and copper foams is explored to satisfy both heat transfer and pressure drop in a heat exchanger. Both positive and negative orientations are accomplished for the enhancement of heat transfer with reduction in the pressure drop. The present research includes three different configurations M1, M2 and M3 (porous layer inner diameter = 0.06 m, 0.04 m, and 0.02 m, respectively, while outer diameter = 0.10 m) partially filled with positive (i.e., increasing, 20/45 PPI) negative (i.e., decreasing, 45/20 PPI) and compound (i.e., 45 Cu/20 Al PPI) graded porous layer thickness. Each configuration involves three different graded porous layer to present the optimum graded porous layer thickness. The thermo-hydrodynamic characteristics are apprehended by using Darcy Extended Forchheimer (DEF) flow and local thermal non-equilibrium (LTNE) models for the partially filled graded porous structure and k-ω turbulence model is accomplished in open passage flow of the conduit. The decreasing graded foam located inside the models M1 and M2 performed 1.68%–12.85% and 13.42%–23.32% higher heat transfer rate compared to without graded metal foam of models M2 and M3, respectively accompanied with 55.43%–84.02% and 35.69%–50.31% lesser pumping power.

Local thermal non-equilibrium bioheat transfer model for interstitial hyperthermia treatment of tumour cell: A numerical approach

Cancer treatment is achieved by destroying the damaged tissue with precise heating, which may be internally or externally on a human body. Thus, tracking the temperature at the targeted site during thermal therapy is essential to avoid unnecessary damage to the neighbouring tissues. Therefore, to avoid difficulties in the experimental in-vivo analysis of the human body, more and more priority has been given to computational modelling. Dual-phase lag bioheat transfer modelling is one that pioneers the biological heat transfer problem to a new horizon where the non-Fourier approach makes the model near realistic. The present paper has developed a numerical model based on the Local Thermal Non-Equilibrium Bioheat Transfer model, as the phase lag values directly depend on the biological tissues' thermophysical properties. Besides the effect of vasodilation and vasoconstriction, metabolic heat generation, as well as muscle shivering, are also considered in the present numerical model. A modified spatial Gaussian heat distribution function has been adapted to model the external heat source and destroy the targeted tissue inside the skin layers. A numerical code is developed using MATLAB in a finite difference approach, which can evaluate the temperature data in an anisotropic medium like human skin. A detailed 2D analysis has been done in different therapeutic conditions, various levels of doses, and different body positions during interstitial hyperthermia treatment. Analysis of biological tissue using the LTNE DPL bioheat transfer equation has not been reported for thermal therapy. Outcomes of the present study give an overview of the range of thermal dose, environmental effect on the treatment of cancer cells, and, most notably, the comparison with Fourier and Local Thermal Equilibrium Non-Fourier models.

A numerical study on heat transfer and MHD flow of a Newtonian through a porous channel—Local thermal nonequilibrium model

Abstract Vast numbers of studies concentrate on the thermal equilibrium state whereas in many real‐world applications the model exists in the nonequilibrium state. Also, local thermal non‐equilibrium precisely represents the thermohydroflow characteristics. Therefore, the current study examines the heat transfer and fluid flow characteristics of the magnetohydrodynamic flow of a Newtonian fluid through a local thermal non‐equilibrium (LTNE) porous channel in the presence of the induced magnetic field. The mathematical model of the prescribed flow encloses the coupled nonlinear equations which are difficult to approach analytically. Hence, they are solved numerically using the shooting method with the Newton–Raphson method. The implications of various physical parameters of the problem on fluid flow, induced magnetic field, current density, temperature profiles, and heat transfer are elucidated with the aid of plots and tables. From the examination, it is clear that the porous medium significantly influences the characteristics of the fluid flow. That is, the least value of the Darcy number is related to a higher momentum field. Another interesting phenomenon is that the induced magnetic field remarkably enhances when the Darcy number is high, whereas the process is contrary to the current density. The effect of LTNE on the flow characteristics and heat transfer ceases for higher values of inter‐phase heat transfer coefficient and the ratio of thermal conductivities, which gives rise to the local thermal equilibrium (LTE) situation. Furthermore, the amount of heat transport is maximum in the LTE case compared to that of the LTNE case.

Fully coupled thermo-hydro-mechanical modeling and simulation of a fluid-saturated porous medium under local thermal non-equilibrium condition

• Fully coupled THM model of porous media under LTNE condition is presented. • T s is always higher than T f due to the LTNE effect. • Dependences of multi-physical fields on the pertinent parameters are investigated. This paper presents a fully coupled thermo-hydro-mechanical (THM) model of a fluid-saturated and compressible porous medium under local thermal non-equilibrium (LTNE) condition and carries out the corresponding numerical simulation. First, the coupling among the temperature field, the pore fluid pressure field, and the solid stress/displacement field is fulfilled by means of strong explicit coupling of governing equations in this study. Then, the presented model is applied to the THM coupling problem in an infinite porous medium with a cylindrical borehole subjected to uniform fluid pressure and temperature at the borehole boundary. Subsequently, the partial differential equations (PDEs) module in COMSOL Multiphysics software is employed to numerically solve the studied problem. Consequently, the finite element solutions of the pore fluid pressure field, the strain/stress fields and the two temperature fields of pore fluid and skeleton solid are obtained, and the spatial-temporal distribution characteristics of the above multi-physical fields are investigated. Finally, the corresponding parametric studies are carried out. The numerical results show that the stress decreases with the increase of the bulk volumetric thermal expansion coefficient of porous media. And the pore pressure decreases when the pore fluid mobility increases. Meanwhile, it is found in this work that the solid temperature is always higher than that of pore fluid due to the LTNE effect. Besides, with increasing the specific solid–fluid interfacial heat transfer coefficient, the solid temperature decreases, while the temperature of pore fluid increases. Accordingly, the temperature difference between the solid and the pore fluid diminishes. In addition, the thermal conductivity of solid grains has a greater impact on the solid temperature but has a limited impact on the temperature of pore fluid, and vice versa. The findings in this work are of benefit to obtain in-depth understanding of THM fully coupling mechanism and LTNE effect in a fluid-saturated porous medium.

Derivation and analysis of a 1D porous medium flow solver embedded in a two-domain model for 2D and 3D transpiration cooling

• A 1D porous medium flow solver is derived and analyzed. • The 1D solver is embedded in a two-domain model for 2D/3D transpiration cooling. • Differences between results of the new model and a full 2D/3D model are negligible. • Transpiration cooling problems are solved significantly faster with the new model. For the numerical investigation of transpiration cooling problems, often two-domain models are used. Different systems of governing equations are solved in a hot gas and a porous medium domain, and both domains are coupled with each other by imposing boundary conditions on the interface. This work presents a novel approach which splits the porous medium domain into 1D problems for Darcy-Forchheimer flow under local thermal nonequilibrium. The fluid and solid temperature solutions are computed analytically, and only one ordinary differential equation has to be solved to determine fluid density and Darcy velocity. Unique solutions to the equations of the 1D model are verified to exist under physically reasonable assumptions. The solutions of the 1D problems are assembled to one 2D or 3D solution. In a postprocessing, slip and adiabatic side walls are modeled despite being nonexistent in 1D. Numerical 2D and 3D test cases together with experimental data verify the suitability of the new assembled-1D approach for the simulation of transpiration-cooled hot gas flows with orthogonal cooling gas injection. Furthermore, the test cases demonstrate the significant computational speed-up related to a full-2D/3D approach.

Natural convection in a fluid saturating an anisotropic porous medium in LTNE: effect of depth-dependent viscosity

Thermal convection in a fluid saturating an anisotropic porous medium in local thermal nonequilibrium (LTNE) is investigated, with specific attention to the effect of variable viscosity on the onset of convection. Many fluids show a remarkable dependence of viscosity on temperature that cannot be neglected. For this reason, we take into account a fluid whose viscosity decreases exponentially with depth, according to Straughan (Acta Mech. 61:59–72, 1986), Torrance and Turcotte (J. Fluid Mech. 47(1):113–125, 1971). The novelty of this paper is to highlight how variable viscosity coupled with the LTNE assumption affects the onset of convection. A numerical procedure shows the destabilising effect of depth-dependent viscosity. Moreover, it comes out that the LTNE hypothesis makes the influence of viscosity more intense. Linear instability analysis of the conduction solution is carried out by means of the Chebyshev-tau method coupled to the QZ algorithm, which provides the critical Rayleigh number for the onset of convection in a straightforward way. The energy method is employed in order to study the nonlinear stability. The optimal result of coincidence between the linear instability threshold and the global nonlinear stability threshold is obtained. The influence of anisotropic permeability and conductivity, weighted conductivity ratio, and interaction coefficient on the onset of convection is highlighted.

Stability Analysis of Double Diffusive Convection in Local Thermal Non-equilibrium Porous Medium with Internal Heat Source and Reaction Effects

Abstract The internal heat source and reaction effects on the onset of thermosolutal convection in a local thermal non-equilibrium porous medium are examined, where the temperature of the fluid and the solid skeleton may differ. The linear instability and nonlinear stability theories of Darcy–Brinkman type with fixed boundary condition are carried out where the layer is heated and salted from below. The D 2 {D^{2}} Chebyshev tau technique is used to calculate the associated system of equations subject to the boundary conditions for both theories. Three different types of internal heat source function are considered, the first type increases across the layer, while the second decreases, and the third type heats and cools in a nonuniform way. The effect of different parameters on the Rayleigh number is depicted graphically. Moreover, the results detect that utilizing the internal heat source, reaction, and non-equilibrium have pronounced effects in determining the convection stability and instability thresholds.

Impact of Thermal Nonequilibrium on Flow Through a Rotating Disk with Power Law Index in Porous Media Occupied by Ostwald-de-Waele Nanofluid

Abstract The current study is made to analyze the impact of local thermal nonequilibrium (LTNE) on the steady, incompressible, and viscous Ostwald-de-Waele nano-liquid over a rotating disk in a porous medium with the various power law index, due to many remarkable applications, such as aeronautical systems, rotating machineries, air cleaning machineries, electrical power-generating systems, heat exchangers, gas turbines, centrifugal pumps. To describe the modeling of the nano-liquid, Brownian movement and thermophoresis are employed with the passive control boundaries. Three temperature model is adopted to distinguish the temperature among the fluid, particle, and solid. The governing transport equations have been converted to a system of nonlinear coupled ordinary differential equations by employing von Karman transformation. Numerical results of the flow and heat and transfer characteristics of the fluid, particle, and solid are obtained by applying Runge–Kutta–Fehlberg method (RKF) together with the shooting technique. The numerical results in the present work are compared with the published results for the case of thermal equilibrium and found that they are in good agreement. It is observed that the temperature profile significantly varies with the fluid-particle, fluid-solid interphase heat transfer coefficients and the modified thermal capacity ratios.