Local Thermal Non-equilibrium Model Research Articles

Thermal Transport Analysis of Injected Flow through Combined Rib and Metal Foam in Converging Channels with Application in Electronics Hotspot Removal

The development of miniaturized and more powerful electronic devices and microprocessors has resulted in high heat generation and temperature values in these components. That affects the performance, reliability and lifespan of the devices. In addition, more local hot spots are generated in the components that require even more attention to prevent stress failure and fatigue. As such, the employment of advanced electronics cooling systems is very essential to keep the temperature below the safe temperature. In this work, innovative cooling systems are developed and analyzed employing jet impingement technology, high conductive metal foam, rib structured target surfaces, confined non-uniform small-scale channel, and conductive heat spreader plate. The base of the foam-filled cooling channel is subject to a uniform high heat flux value resembling the electronics device to be cooled. For numerical modeling of thermal transport through foam filled region, the local thermal non-equilibrium model in porous media is utilized resulting in two energy equations for solid and fluid phases. For better understanding of flow and thermal characteristics of jet impingement through the combination of metal foam and rib structured surfaces in confined channels, several effective parameters are studied such as slot and circular jet impingements, the shape and orientation of the ribs, impinging jet velocity, applied heat flux and the thickness of conductive heat spreader plate for hotspot removal. The results show that the fully foam filled channel provides a more efficient cooling in comparison with partially foam filled channel. Furthermore, the results indicate the advantage of utilization of ribs at the stagnation region of the impinging jet, for local thermal treatment of hotspots. The perpendicular cuboid ribs placed at the stagnation zone of the coolant jet impingement provide 13% increase in maximum local Nusselt number while the increase in the pressure drop and required pumping power are as small as 4.2%. Doubling the velocity would result in 35.3% increase in the maximum local Nusselt number, 180% increase in the pressure drop and 460.7% increase in the required pumping power. An increase in the thickness of the conductive heat spreader plate increases the base local temperature but improves local temperature uniformity.

Numerical Study of MHD Free Convection in a Packed Bed Square Enclosure Using Local Thermal Non-Equilibrium (LTNE) Model

In the present study, natural convection of fluid in a square packed bed enclosure is investigated numerically using a uniform magnetic field. The geometry model is heated from left-hand side vertical wall and cooled from opposite wall with adiabatic condition at both the top and bottom walls. Normally to this enclosure an electric coil was set to generate a uniform magnetic field. The Brinkman–Forchheimer extended Darcy model was used to solve the momentum equations, while the energy equations for fluid and solid phase were solved using the local thermal non-equilibrium (LTNE) model. Computations are performed for a range of the Darcy number from 10-5 to10-1, the porosity from 0.3 to 0.9, and Hartmann number from 0 to 75. The results showed that both the strength of applied magnetic field and the packed bed characteristics have significant effect on the flow field and heat transfer.

Numerical analysis on the improved thermo-chemical behaviour of hierarchical energy materials as a cascaded thermal accumulator

The present study aims to improve the thermo-chemical conversion behaviours, including reactive transport processes and output performances of an open thermochemical energy storage (TCES) unit. The local thermal non-equilibrium (LTNE) model and the effect of non-uniform porosity are adopted and considered to better elucidate the conversion processes. Cascading the reaction sub-units filled with different thermochemical materials (TCMs), i.e., zeolite, salt hydrate-based composite sorbent, and pure salt of SrBr2·6H2O, to form an integrated storage bed ameliorates the output performance. The numerical results indicate that the maximum temperature difference ranging from 3.5 to 4.9 °C between heat transfer fluid and solid reactants exists during desorption, and the realistic non-uniform porosity facilitates the reactant conversion near the wall compared to the uniform porosity assumption. The cascaded scheme promotes the charging and discharging processes compared to the cases filled with sole TCM, the time required for charging this 10.8 kWh storage model is 16 h. Increasing the charging temperature from 100 °C to 145 °C, the charging time reduced to 6.5 h, saving 59.4%. Boosting the inlet velocity of airflow also accelerates the charging rate. The cascaded storage unit significantly stabilises the output temperature during discharging, warming up the airflow from 20 °C to 35 °C for 24 h with a tiny temperature fluctuation. Airflow with higher relative humidity facilitates hydration but shortens the stable period. Overall power and thermal efficiency of the “thermal accumulator” in charging are 598 W and 92.8%, 164 W and 92.4% in discharging, with a total COP of 0.71. The satisfying performances suggest that the cascaded TCES unit may provide a strategy and reference in the design and promotion of the low-grade energy storage system.

Transient behaviour of heat transfer with complete evaporation process in Porous Channel with localised heating using non-Darcian flow and LTNE model

This study numerically presents the transient behaviour of complete evaporation process inside porous channel. Based on the modified enthalpy formulation under non-Darcy flow and Local Thermal Non-Equilibrium (LTNE) conditions, solutions have been obtained utilizing the finite volume method for various relevant parameters. The results have been validated with the experiments and they show good agreement between them. The results demonstrate that LTNE model should be employed while modelling the transient complete evaporation process in porous media. The non-Darcy effects have considerable influences on the locations of the initiation and termination of the phase change processes as compared to Darcy flow model. Therefore, the behaviours of the two-phase and the superheated vapour zones, when the steady-state solutions is reached, are substantially changed for non-Darcy flow as compared to Darcy model due this assumption provides an additional mechanism for heat transfer that represented by the heat exchange between the solid and fluid phases. It has been observed that the two-phase zone is considerably extended in the axial direction and this zone has not been occupied a large size towards transverse direction due to the axial velocity is noticeably higher than the transverse velocity, caused by reducing the diffusive energy through the two-phase region. The results indicate that non-Darcian effects leads to considerably extend the size of superheated region towards the outlet of the channel, which is not the case for Darcy model. The non-Darican effects become more pronounced for high Reynolds number and heat flux. The influences of varying in Reynolds number, heat flux, porosity, and solid thermal conductivity are substantial near the heated surface, whereas Darcy number has a minor impact on the solution of complete evaporation process. It can be emphasised from the present predictions that the non-Darcy flow along with LTNE condition is indispensable model for the complete evaporation process, especially under high mass flowrate as well heat flux.

Performance evaluation of partially filled high porosity metal foam configurations in a pipe

• Numerical simulation of partially filled metal foams for heat transfer augmentation is studied. • Darcy Forchheimer and Local Thermal Non-Equilibrium models for numerical modeling. • Six different strategies for maximum heat transfer and minimum pressure drop. • 10 to 45 pores per inch and 0.88 and 0.95 porosity are considered. • 30 pores per inch with porosity of 0.92 provides the best performance factor. In this contemporary research, a parametric analysis of partially filled high porosity metallic foams in a horizontal conduit is performed to augment heat transfer with reasonable pressure drop. The investigation includes six different models filled partly with aluminium foam by varying internal diameter of foams from the wall side and external diameter of foam from the core of the tube. The pore density of the foam ranges from 10 to 45 PPI and their porosity varies from 0.90 to 0.95. Flow dynamics are captured using Darcy Extended Forchheimer model for the porous filled region and two-equation turbulence k-ω model employed in clear region of the fluid. The local thermal non-equilibrium assumption is incorporated in porous filled region of the conduit to compute the heat transport characteristics. The results showed that the thermal performance factor of 10 PPI aluminium foam performs close to the 10 PPI expensive copper foam. The performance factor is found to be higher for 30 PPI aluminium foam amongst the PPI’s of the foam considered. However, the performance factor is found to be 2.93, 2.22 and 1.73 for 30PPI, 45 PPI and 20PPI with their porosities of 0.92, 0.90 and 0.90, respectively for the model 1, model 2 and model 3 at lower Reynolds number of 4500 and then it decreases progressively with increasing flow rates of the fluid. The results of average wall temperature, average Nusselt number and Colburn j factor are also evaluated to obtain best possible performance.

Impacts of Amplitude and Local Thermal Non-Equilibrium Design on Natural Convection within NanoflUid Superposed Wavy Porous Layers.

A numerical study is presented for the thermo-free convection inside a cavity with vertical corrugated walls consisting of a solid part of fixed thickness, a part of porous media filled with a nanofluid, and a third part filled with a nanofluid. Alumina nanoparticle water-based nanofluid is used as a working fluid. The cavity’s wavy vertical surfaces are subjected to various temperature values, hot to the left and cold to the right. In order to generate a free-convective flow, the horizontal walls are kept adiabatic. For the porous medium, the Local Thermal Non-Equilibrium (LTNE) model is used. The method of solving the problem’s governing equations is the Galerkin weighted residual finite elements method. The results report the impact of the active parameters on the thermo-free convective flow and heat transfer features. The obtained results show that the high Darcy number and the porous media’s low modified thermal conductivity ratio have important roles for the local thermal non-equilibrium effects. The heat transfer rates through the nanofluid and solid phases are found to be better for high values of the undulation amplitude, the Darcy number, and the volume fraction of the nanofluid, while a limit in the increase of heat transfer rate through the solid phase with the modified thermal ratio is found, particularly for high values of porosity. Furthermore, as the porosity rises, the nanofluid and solid phases’ heat transfer rates decline for low Darcy numbers and increase for high Darcy numbers.

Cooling Design for PEM Fuel-Cell Stacks Employing Air and Metal Foam: Simulation and Experiment

A new study investigating the cooling efficacy of air flow inside open-cell metal foam embedded in aluminum models of fuel-cell stacks is described. A model based on a commercial stack was simulated and tested experimentally. This stack has three proton exchange membrane (PEM) fuel cells, each having an active area of 100 cm2, with a total output power of 500 W. The state-of-the-art cooling of this stack employs water in serpentine flow channels. The new design of the current investigation replaces these channels with metal foam and replaces the actual fuel cells with aluminum plates. The constant heat flux on these plates is equivalent to the maximum heat dissipation of the stack. Forced air is employed as the coolant. The aluminum foam used had an open-pore size of 0.65 mm and an after-compression porosity of 60%. Local temperatures in the stack and pumping power were calculated for various air-flow velocities in the range of 0.2–1.5 m/s by numerical simulation and were determined by experiments. This range of air speed corresponds to the Reynolds number based on the hydraulic diameter in the range of 87.6–700.4. Internal and external cells of the stack were investigated. In the simulations, and the thermal energy equations were solved invoking the local thermal non-equilibrium model—a more realistic treatment for airflow in a metal foam. Good agreement between the simulation and experiment was obtained for the local temperatures. As for the pumping power predicted by simulation and obtained experimentally, there was an average difference of about 18.3%. This difference has been attributed to the poor correlation used by the CFD package (ANSYS) for pressure drop in a metal foam. This study points to the viability of employing metal foam for cooling of fuel-cell systems.

Finite element treatment for nanofluids flow within enclosures filled with hydrodynamically and thermally anisotropic porous media: Impacts of sinusoidal heating

AbstractThe local thermal non‐equilibrium model (LTNEM) in case of a sinusoidal heating is applied to investigate the unsteady nanofluid flow and thermal fields inside an inclined enclosure filled with a thermally and hydrodynamically anisotropic porous medium. The Galerkin finite element method (GFEM) together with the Characteristic‐based Split (CBS) Scheme is used to solve the dimensionless governing equations. Impacts of the controlling parameters are studied in two cases, namely, non‐Darcy Regime in which values of the Darcy number and the Rayleigh number are, respectively, and the Darcy regime where . Parametric studies are performed for wide ranges of the permeability ratio , the Nield number , the inclination angle , the nanoparticles volume fraction , the inclination of the principal axes , the amplitude ratio and the phase deviation . The main outcomes revealed that the sinusoidal heating results in multicellular flow components inside the geometry. Also, the increase in the Nield number causes that the system converts from the LTNE state to local thermal equilibrium (LTE) state. For the non‐Darcy regime, the convective process and the rate of the heat transfer are augmented as the inclination of the principal axes is increased.

Numerical Investigation of a Thermal Ablation Porous Media-Based Model for Tumoral Tissue with Variable Porosity

Thermal ablation is a minimally or noninvasive cancer therapy technique that involves fewer complications, shorter hospital stays, and fewer costs. In this paper, a thermal-ablation bioheat model for cancer treatment is numerically investigated, using a porous media-based model. The main objective is to evaluate the effects of a variable blood volume fraction in the tumoral tissue (i.e., the porosity), in order to develop a more realistic model. A modified local thermal nonequilibrium model (LTNE) is implemented including the water content vaporization in the two phases separately and introducing the variable porosity in the domain, described by a quadratic function changing from the core to the rim of the tumoral sphere. The equations are numerically solved employing the finite-element commercial code COMSOL Multiphysics. Results are compared with the results obtained employing two uniform porosity values (ε = 0.07 and ε = 0.23) in terms of coagulation zones at the end of the heating period, maximum temperatures reached in the domain, and temperature fields and they are presented for different blood vessels. The outcomes highlight how important is to predict coagulation zones achieved in thermal ablation accurately. In this way, indeed, incomplete ablation, tumor recurrence, or healthy tissue necrosis can be avoided, and medical protocols and devices can be improved.

Semi-analytical study of impingement cooling of metal foam heat sinks of CPUs with air and hydrogen jets under LTNE condition

In this paper, a mathematical model for heat transfer in a heat sink covered by an open-cell metal foam under impinging jet flow is presented. Hydrogen and air were considered as the cooling fluids. The Darcy–Brinkman–Forchheimer relationship for the momentum equation was employed, while the two-equation energy model under the local thermal non-equilibrium (LTNE) condition for the energy equation was used. The mathematical model of similarity solution was first validated with analytical and experimental data from literature, and numerical data from Ansys-Fluent®. Then, the differences between numerical predictions under the local thermal equilibrium (LTE) and LTNE on flow and temperature distributions were presented and analyzed. It was shown that when the effective thermal conductivity of solid is very low, the difference of temperature of solid and gas cannot be neglected and LTNE should be considered. In other words, when effective thermal conductivity ratio, β, declines, the deviation between the LTNE and LTE models increases considerably. Finally, the effects of critical dimensionless parameters on the flow and heat transfer characteristics were investigated. The results indicate that cooling performance of the metal foam can be improved by increasing the Reynolds and Darcy numbers while decreasing the aspect ratio of the heat sink. Besides, the properties of metal foam and gas were studied. Hydrogen, as a cooling gas, has better thermal performance than air.

Relationship Between the Nonlocal Effect and Lagging Behavior in Bioheat Transfer

Abstract Lots of generalized heat conduction models have been developed in recent decades, such as local thermal nonequilibrium model, phase lagging model, and nonlocal heat conduction model. But no attempt was made to prove which model is better (or worse) than others, or whether there is a certain relationship between these different models. With this inspiration, we establish the nonlocal bioheat transfer equations with lagging time, and the two and three-temperature bioheat transfer equations with considering all the carrier's heat conduction effect are also constructed. Comparing the two (or three)-temperature equation model with the nonlocal bioheat transfer models with lagging time, one may obtain: the lagging time of temperature gradient τtand the nonlocal characteristic length λq in the space derivative items of heat flux have the same effect on heat transfer; when the heat transport occur among N energy carriers with considering the conduction effects of all carries, the heat transfer processes are dependent upon the high-order effect of τqN-1, τtN-1 and λt(2N-1) in nonlocal dual phase lag bioheat transfer model. This phenomenon is very important for biological and medical systems where numerous carriers may exist on the cellular level.

Cooling of high-performance electronic equipment using graphite foam heat sinks

In the present research, highly-conductive graphite foams have been employed to effectively dissipate the heat generated in electronic components. The heat sinks suggested have been configured from staggered foamed-baffles arranged either in parallel or perpendicular to the air paths through the slots in between to reduce the pressure drop resulted while improving the heat dissipation. The performance of the currently proposed heat sinks has been examined numerically based on the volume averaging concept of porous media, with employing the local thermal non-equilibrium model to account for the interstitial heat exchange between the foam solid matrix and the fluid particles flowing across. The Simcenter STAR-CCM+ CFD commercial code has been utilised to implement the iterative solution based on the SIMPLE algorithm. A wide range of design parameters have been tested including the heat sink configuration along with geometrical characteristics of the graphite foam used. The impact of operating conditions, including the inlet airflow strength and the heat flux applied, has been inspected as well. The currently proposed heat sinks have been found efficient to meet the extremely thermal demands of high-performance electronic equipment and sweep away the heat generated there with a reasonable cost of pressure drop, where hot spots can be eliminated entirely with proper manipulation of design conditions.

Pennes\u2019 bioheat equation vs. porous media approach in computer modeling of radiofrequency tumor ablation

The objective of this study was to compare three different heat transfer models for radiofrequency ablation of in vivo liver tissue using a cooled electrode and three different voltage levels. The comparison was between the simplest but less realistic Pennes’ equation and two porous media-based models, i.e. the Local Thermal Non-Equilibrium (LTNE) equations and Local Thermal Equilibrium (LTE) equation, both modified to take into account two-phase water vaporization (tissue and blood). Different blood volume fractions in liver were considered and the blood velocity was modeled to simulate a vascular network. Governing equations with the appropriate boundary conditions were solved with Comsol Multiphysics finite-element code. The results in terms of coagulation transverse diameters and temperature distributions at the end of the application showed significant differences, especially between Pennes and the modified LTNE and LTE models. The new modified porous media-based models covered the ranges found in the few in vivo experimental studies in the literature and they were closer to the published results with similar in vivo protocol. The outcomes highlight the importance of considering the three models in the future in order to improve thermal ablation protocols and devices and adapt the model to different organs and patient profiles.

Linear Stability on the Local Thermal Nonequilibrium Model of Mixed Convection Boundary Layer Flow over a Moving Wedge in a Porous Medium: Viscous Dissipation and Radiation Effects

Abstract This paper studies the local thermal nonequilibrium (LTNE) model for two-dimensional mixed convection boundary-layer flow over a wedge, which is embedded in a porous medium in the presence of radiation and viscous dissipation. It is considered that the temperature of the fluid and solid phases is not identical; hence, we require two energy equations: one for each phase. The motion of the mainstream and wedge is approximated by the power of distance from the leading boundary layer. The flow and heat transfer in the LTNE phase is governed by the coupled partial differential equations, which are then reduced to nonlinear ordinary differential equations via suitable similarity transformations. Numerical simulations show that when the interphase rate of heat transfer is large, the system attains the local thermal equilibrium (LTE) state and so is for porosity scaled conductivity. When LTNE is strong, the fluid phase reacts faster to the mainstream temperature than the corresponding solid phase. The state of LTE rather depends on radiation and viscous dissipation of the model. Further, numerical solutions successfully predicted the upper and lower branch solutions when the velocity ratio is varied. To assess which of these solutions is practically realizable, an asymptotic analysis on unsteady perturbations for a large time leading to linear stability needs to be performed. This shows that the upper branch solutions are always stable and practically realizable. The physical dynamics behind these results are discussed in detail.

An improved analytical model for heat flow in cancerous tumours to avoid thermal injuries during hyperthermia.

The present study highlights an analytical hybrid scheme consisted of a shift of variables and finite integral transform for analysing a local thermal non-equilibrium (LTNE) bioheat model. This model can have utilised to be a betterment of prediction of the temperature field in the localised hyperthermia therapy (LHT) for the treatment of cancer patients. As the hyperthermia treatment is only the application in living tissues, an appropriate initial condition for the therapeutic thermal response is proposed instead of a constant temperature taken in the previous studies based on the 1-D heat flow. The present analysis suggests the therapeutic exposure time of 7776.8s (2.16 h) with constant heat flux and the exposure time of 10969.9s (3.06 h) with a sinusoidal heat flux within the usual temperature range of the hyperthermia (in a combination of thermal ablation and medium temperature hyperthermia) to be more effective in the treatment protocol. The presented results show that fatal injuries (tissue trauma, thermal burn, etc.) of internal organs might be possible to avoid by the current therapeutic condition. Therefore, this study may nullify the adverse effect of the existing model with the constant heating and consequently, the repercussion of the several therapeutic variables is to estimate with the development of a thermal profile for the suitability of a therapeutic condition. On the other hand, the present study well matches with the published analysis in case of both the theoretical and experimental (live tissues of the pig due to unavailability of real-time data on the human body) studies and it found the maximum deviation of the thermal response as 2.26% and 2.66%, respectively.

Three-dimensional numerical investigation on melting performance of phase change material composited with copper foam in local thermal non-equilibrium containing an internal heater

This work aims to numerically study the 3D melting heat transfer of latent heat thermal energy storage (LHTES) systems with phase change material (PCM) embedded in copper foam inside an internal heated cubic cavity. The enthalpy-porosity method to model the phase change process is employed and the Darcy-Forchheimer law and local thermal non-equilibrium (LTNE) model are assumed for the metal foam. The numerical code of proposed model shows good agreement for copper foam composited PCM melting with carefully designed experiments. Special attentions are given to detect the effects of Rayleigh number (Ra), porosity (ε), Darcy number (Da) and heater size on evolvement of solid-liquid interface, temperature stratification and full melting. The results show that increasing Ra enhances convective heat transfer in the upper part of the porous cavity; though the completely melting time remains almost unchanged. For a fixed porosity, decrease of Da reduces the melting rate in the medium melting stage, and the temperature distribution of low Da is dominated by the heat conduction in porous media at the later stage, but Da shows a negligible effect on the complete melting time. For a fixed Da, increase of porosity decelerates the complete melting time. Apart from that, the heater size also plays an important role in the melting performance of composited PCM.

Thermal management of power electronics with liquid cooled metal foam heat sink

In this paper, a combined experimental and numerical study of liquid cooled aluminum foam (AF) heat sink applicable for high power electronics cooling was conducted. AF heat sinks with porosity (0.88) and pore densities (10 PPI and 20 PPI) are fabricated and the empty channel is employed for comparison. Experimental measurements were conducted subjected to non-Darcy flow regime. Numerical analysis was based on the Brinkman-Forchheimer extended Darcy momentum model, together with the energy equations of local thermal non-equilibrium (LTNE) and local thermal equilibrium (LTE) models. The permeability and inertia coefficient of the AF samples were correlated based on the measured pressure drop. New correlations of interfacial heat transfer coefficient between water and AF solid matrix were also determined. Numerically predicted local surface temperature and temperature field distributions were presented and validated with the experimental measurements. The results reveal that the permeability decreases and the inertial coefficient increases with increasing pore density. The thermal performance of AF heat sinks is appreciably superior to the empty channel, under a given flow rate, the Num of the AF heat sink can be 1.40–1.76 times of the empty channel. Heat transfer enhancement by 20 PPI foam is inappreciable, approximately 5%, compared to that of 10 PPI. The LTNE model demonstrates more accurate for predicting the spatial temperature distributions than the LTE model. The results also indicate that the non-equilibrium heat transfer phenomenon in water-cooled AF heat sink is not pronounced, and the non-equilibrium effect is declined with increasing velocity.

Numerical investigation on latent thermal energy storage in shell and corrugated internal tube with PCM and metal foam

A numerical investigation on Latent Heat Thermal Energy Storage System (LHTESS) based on an aluminum foam totally filled with phase change material (PCM) is accomplished. The PCM used is a pure paraffin wax with melting over a range of temperature and a high latent heat of fusion. The LHTESS geometry under investigation is a vertical shell and tube. The corrugated internal surface of the hollow cylinder is assumed at a constant temperature above the PCM melting temperature. The other external surfaces are assumed adiabatic. The paraffin wax phase change process is modelled with the enthalpy-porosity theory, while the metal foam is considered as a porous media obeying to the Darcy-Forchheimer law. Local thermal non-equilibrium (LTNE) model is assumed to analyze the heat transfer in the metal foam. The governing equations are solved employing the Ansys-Fluent code. The numerical simulations results, reported as a function of time, and concerning the LHTESS charging phase, are compared in terms of melting time, average temperature and energy storage rate. The corrugated internal surface effect is analyzed with respect to the wavelength and wave amplitude of the corrugation.

Numerical Investigation of Thermally Developing Non-Darcy Forced Convection in a Porous Circular Duct with Asymmetric Entrance Temperature Under LTNE Condition

This paper numerically investigates the heat transfer performance of thermally developing non-Darcy forced convection in a fluid-saturated porous medium tube under asymmetric entrance temperature boundary conditions. The Brinkman flow model and the local thermal non-equilibrium (LTNE) model are employed to establish the mathematical model of the studied problem to predict the forced convective heat transfer. Then, the mathematical model is numerically solved using COMSOL Multiphysics. Consequently, the fluid velocity field, the solid temperature field, the fluid temperature field and the Nusselt number are obtained. Moreover, the dependences of the Nusselt number on some key parameters are analyzed in detail. The results show that the distribution characteristics of the Nusselt number are strongly dependent on the form of the entrance temperature function. Meanwhile, it is found that the Nusselt number increases first and then tends to approach an asymptotic value with the increase in the Darcy number and the Biot number. The Nusselt number monotonously increases with increasing the Peclet number. On the contrary, the Nusselt number decreases first and then tends to be an asymptotic value owing to the increase in the thermal conductivity ratio and the viscosity ratio. This study is of benefit to provide in-depth insights into the non-Darcy forced convective heat transfer in porous tubes with asymmetric inlet temperature under LTNE condition.

Effect of thermal boundary conditions on forced convection under LTNE model with no-slip porous-fluid interface condition

This paper analytically investigates a fully developed forced convection in a channel partially filled with two porous layers symmetrically arranged at inner walls. The local thermal non-equilibrium (LTNE) model and Brinkman-extended Darcy model are employed as energy and momentum equations respectively in porous region. Three thermal boundary conditions (models A, B and C) and no-slip condition are adopted at porous-fluid interface. Explicit expressions of solid and fluid temperatures and Nusselt number are derived under each thermal boundary condition and variances between the three thermal boundary conditions in predicting thermal behaviors are discussed. It is shown that with present parameters, the temperature bifurcation phenomenon occurs in most cases under model B, while this phenomenon cannot be observed under models A and C. The LTNE model holds in almost any conditions under model B, but under models A and C the LTNE model holds with a low hollow ratio (S) and Biot number (Bi), a large effective thermal conductivity ratio (K) and Darcy number (Da). The increase of interfacial convective heat transfer coefficient (Hs) makes the Nusselt number (Nu) of model C more close to but not exceed the Nu of model A, while the Nu under model B is lower than which under models A and C. By using different thermal boundary conditions, the stress jump coefficient (β) effect on heat transfer can be ignored, and with a high Da, a low β and S the Nu number predicted by stress jump interface condition has slight difference from that by stress continuity interface condition.