Effective Thermal Conductivity Ratio Research Articles

Forced Convection in a Transpired Passage With a Porous Medium Insert

Abstract Forced convective heat transfer in a passage with a porous medium matrix subjected to uniform blowing has been fully analyzed, using a local thermal nonequilibrium approach, in application of transpiration systems on the basis of porous medium. An exact solution was found for the asymptotic case of infinitely high interstitial Biot number. In order to extend the problem to general local nonthermal equilibrium cases, an analytical solution procedure has been developed, introducing six unknown parameters to describe both fluid and solid temperature profiles, which are then determined by solving six independent differential and integral equations based on the two energy equations for individual phases and their corresponding boundary conditions. The important performance index for the problem, namely, the overall cooling effectiveness is found quite sensitive not only to the blowing ratio, but also to the interstitial Biot number and fluid to solid effective thermal conductivity ratio. It rises with the blowing ratio and effective thermal conductivity ratio, while it drops as increasing the interstitial Biot number toward the local thermal equilibrium. The present analysis reveals that the local nonthermal equilibrium analysis is definitely needed for accurate estimation of the transpiration-cooled wall temperature.

Forced convection in a fully-filled bidisperse porous annular duct subject to asymmetric heat fluxes

• Heat transport in bidisperse porous annular duct is theoretically studied. • The bidispersivity triggers critical effective thermal conductivity ratio. • Nusselt number reaches the peak with a heat flux ratio of 0.175. • Wider range of high thermal performance appears by increasing the heat flux ratio. In the present work, convective heat transport through an annular duct fully filled with a bidisperse porous medium (BDPM) is analytically studied by using the two-velocity two-temperature model. The annulus is heated from inner and outer walls by asymmetric heat fluxes. Analytical solutions are deduced for the velocity and temperature distributions of f- and p-phases in the developed and laminar regime. The results show that the bidispersivity benefits the heat transfer enhancement in a lower range of effective thermal conductivity ratio compared to the monodisperse porous medium (MDPM). However, this advantage will vanish once the conductivity ratio exceeds its critical value. The BDPM annular duct having a peak average Nusselt number is found at the heat flux ratio equal to 0.175 under considered parameters.

ANFIS modelling of effective thermal conductivity of ethylene glycol and water nanofluids for low temperature heat transfer application

Nanofluids have been proven to be promising coolants for heat transfer application due to their superior thermal properties. In the present work, experimental studies were carried out to investigate the thermal conductivity of ethylene glycol (EG) and water mixture (35:65 v/v in ratio) based CuO, Al2O3 and TiO2 nanofluids. The study has considered nanofluid containing nanoparticles in the concentration range of 0.2 to 2 wt%. The effect of nanoparticle concentration on thermal conductivity was investigated in terms of effective thermal conductivity (ratio of thermal conductivity of nanofluid to that of base fluid) at low temperatures (5 to 25 °C). Results indicated significant improvement in thermal conductivity at lower temperature of nanofluids. Enhancement of 6.34%, 4.87% and 3.59% in thermal conductivity was obtained for 2 wt% of concentration at 5 °C for CuO, Al2O3 and TiO2 nanoparticles respectively compared to EG:Water. Correlations for effective thermal conductivity were developed by regression considering two input (temperature, concentration) and three input (temperature, concentration and nanoparticles thermal conductivity) parameters using experimental results. An intelligent model, adaptive neuro-fuzzy inference system (ANFIS) approach was used to model the effective thermal conductivity of nanofluids. ANFIS model performed better than the correlations developed.

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.

Analytical and Numerical Study on Thermally Developing Forced Convective Flow in a Channel Filled with a Highly Porous Medium Under Local Thermal Non-Equilibrium

An analytical and numerical study was made on thermally developing forced convective flow in a channel filled with a fluid-saturated porous medium, subject to constant heat flux, under local thermal non-equilibrium. The Brinkman-extended Darcy model was employed to investigate the combined effects of the Darcy number, Biot number, effective thermal conductivity ratio and Graetz number on the longitudinal thermal development in a channel filled with a highly porous medium. 3D convective regime maps were constructed for wide ranges of the parameters, using a 3D coordinate system based on the Biot, Darcy and Graetz number coordinates. The 3D convective regime maps proposed in this study enable one to identify the three distinctive thermal regions possible along the longitudinal coordinate in a channel, namely the entrance developing region, transition region and fully developed region, depending on the relative thicknesses of the fluid and solid thermal boundary layers. It has been found that the Darcy number strongly influences the heat transfer rate in the developing region, yielding a higher Nusselt number for a lower Darcy number, whereas both Biot number and the effective thermal conductivity ratio markedly influence the level of the fully developed Nusselt number. The heat transfer performance evaluation carried out under equal pumping power for the aluminum foam embedded channel saturated with air reveals that a substantial heat transfer argumentation is possible by filling the channel with an aluminum foam.

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.

Pulsating flow in a channel filled with a porous medium under local thermal non-equilibrium condition: an exact solution

The present work investigates analytically the problem of forced convection heat transfer of a pulsating flow, in a channel filled with a porous medium under local thermal non-equilibrium condition. Internal heat generation is considered in the porous medium, and the channel walls are subjected to constant heat flux boundary condition. Exact solutions are obtained for velocity, Nusselt number and temperature distributions of the fluid and solid phases in the porous medium. The influence of pertinent parameters, including Biot number, Darcy number, fluid-to-solid effective thermal conductivity ratio and Prandtl number are discussed. The applied pressure gradient is considered in a sinusoidal waveform. The effect of dimensionless frequency and coefficient of the pressure amplitude on the system’s velocity and temperature fields are discussed. The general shape of the unsteady velocity for different times is found to be very similar to the steady data. Results show that the amplitudes of the unsteady temperatures for the fluid and solid phases decrease with the increase in Biot number or thermal conductivity ratio. For large Biot numbers, dimensionless temperatures of the solid and fluid phases are similar and are close to their steady counterparts. Results for the Nusselt number indicate that increasing Biot number or thermal conductivity ratio decreases the amplitude of Nusselt number. Increase in the internal heat generation in the solid phase does not have a significant influence on the ratio of amplitude-to-mean value of the Nusselt number, while internal heat generation in the fluid phase enhances this ratio.

Forced convective heat and mass transfer in a bidisperse porous parallel-plate channel with a first order reaction on the wall

The fully developed forced convective heat and mass transfer in a parallel-plate channel occupied by a bidisperse porous medium is investigated under the Darcy flow model and the constant heat flux boundary condition. The first order catalytic reaction is assumed to be taken place on the walls. Analytical solutions are sought for the temperature and concentration of two phases using the direct decoupling approach. The effects of Darcy velocity ratio, effective thermal conductivity ratio, inter-phase mass transfer coefficient, Biot, Damkohler and Soret numbers are investigated on the temperature and concentration distributions.

Heat transfer characteristics of thermally developing forced convection in a porous circular tube with asymmetric entrance temperature under LTNE condition

This paper presents a theoretical study on the developing forced convection heat transfer performance in a porous circular tube with asymmetric entrance temperature under LTNE condition. The Nusselt number is derived analytically, and its dependence on the pertinent parameters is investigated in details. It is found that the distribution characteristics of the Nusselt number with the circumstantial coordinate is determined predominantly by the asymmetric form of the entrance temperature function. Besides, the Nusselt number is strongly dependent on the values of the Péclet number, the Biot number, and the effective thermal conductivity ratio. The results obtained in this work are of benefit to further understand the thermal behavior of forced convection in a porous circular tube with asymmetric inlet temperature under LTNE condition. Meanwhile, the presented analytical solution can serve as a benchmark for calibrating numerical solutions of the studied problem.

Thermal conductivity and viscosity models of metallic oxides nanofluids

For over ten years, investigators focused on determining and modelling the effective thermal conductivity and viscosity of nanofluids. Lately, many theoretical and experimental investigations on convective heat transfer have been performed on the augmentation of heat transfer by utilizing suspensions of nanometer-sized solid particle materials (metallic or nonmetallic) in base fluids. The main purpose of this work is to determine the thermal conductivity and viscosity of various types of metallic oxides (Al2O3, CuO, SiO2 and ZnO) for nanoparticle concentrations of 1–5 vol% at temperatures of 300–320 K and nanoparticle shapes (blades, platelets, cylindrical, bricks, and spherical). The results illustrate that the effective thermal conductivity and thermal conductivity ratio of metallic oxide nanofluids increase with temperature and nanoparticles volume fraction but decreases nanoparticle size intensifies. Besides that, the results of effective viscosity and viscosity ratio obtained indicate a considerable rise with the increase of nanoparticles concentration. Thus, optimum nanoparticle concentration is essential to be determined in forming nanofluids that can enhance thermal systems performance. Finally, it is found that nanoparticles shape has great impact on the thermophysical properties of nanofluids.

Convective heat transfer in a porous-medium micro-annulus with effects of the boundary slip and the heat-flux asymmetry: An exact solution

Flow and heat transfer in a micro-annulus with the porous material is analytically investigated using local thermal equilibrium (LTE) and local thermal non-equilibrium (LTNE) models with effects of flow/thermal slips and asymmetric heat fluxes. Analytical solutions for velocity and temperatures are obtained. Effects of key factors on convective heat transfer are examined. An increase in the Knudsen number and the inverse of a modified Darcy number can homogenize the velocity. When heat flux (HF) ratio is –R2, the temperatures of solid and fluid show linear profiles. LTNE model must be considered for small duct scale, lower effective thermal conductivity ratio or poor local convective heat transfer between solid and fluid phases. A limit exists for heat transfer enhancement by homogenizing the flow field. Critical HF ratios corresponding to the asymptotic line are verified. The opposite effects of flow slip and thermal slip on heat transfer lead to the maximum heat transfer. This analytical solution predicts the thermal performance of porous medium micro-annulus in wide ranges of radius ratio, Knudsen number, and HF ratio.

Numerical investigation of heat transfer in a micro-porous-channel under variable wall heat flux and variable wall temperature boundary conditions using local thermal non-equilibrium model with internal heat generation

The phenomenon of forced convective heat transfer with internal heat generation in a micro-channel filled with a porous material saturated with a rarefied gas is studied using finite element method by varying wall heat flux and wall temperature ratio under local thermal non-equilibrium (LTNE) conditions. The Darcy–Brinkman model is used to describe the flow inside the porous material. Numerical solutions are obtained for both the fluid and solid temperature distributions by considering the case A for variable wall heat fluxes, case B for variable wall temperature ratio and case C for temperature jump coefficient. The Nusselt number for the fluid at the channel walls is calculated. The effect of significant parameters such as Biot number, wall temperature ratio, fluid and solid internal heat generation, fluid to solid effective thermal conductivity ratio are discussed. Comparative study on Nusselt number is carried out in case A with constant wall heat flux as well as variable wall heat flux. The Nusselt numbers obtained for case B indicate a strong dependency on the variation of internal heat generation for Bi=0.5. The velocity slip coefficient is found to have negligible effect on the Nusselt numbers for all the three cases.

Heatline visualization of natural convection in a thick walled open cavity filled with a nanofluid

Natural convection of an alumina-water nanofluid in a partially open rectangular cavity with a left heat-conducting solid wall of finite thickness and conductivity has been studied numerically. Governing equations formulated in dimensionless stream function and vorticity variables on the basis of a single-phase nanofluid model with corresponding boundary conditions have been solved by finite difference method. Analysis of the influence of Rayleigh number (Ra=103–106), thermal conductivity ratio (1⩽K⩽20), solid wall thickness (0.1⩽δ⩽0.3) and nanoparticles volume fraction (0⩽ϕ⩽0.05) on streamlines, heatlines and isotherms as well as average Nusselt number at solid-fluid interface and fluid flow rate has been carried out. It has been revealed that for the considered models for an effective thermal conductivity ratio and dynamic viscosity an increase in the nanoparticles volume fraction leads to the heat transfer reduction and decrease of the fluid flow rate.

Thermally developing flow inside a porous-filled channel in the presence of internal heat generation under local thermal non-equilibrium condition: A perturbation analysis

In the present work, a perturbation analysis is performed to study the thermally developing forced convection heat transfer inside a channel filled with a porous material under local thermal non-equilibrium (LTNE) condition. Internal heat generations within the solid and fluid phases are considered. Channel walls are subjected to a constant heat flux. It is assumed that there is a small temperature difference between the fluid and the solid phases of the porous material. So, performing a perturbation analysis enables us to avoid utilizing models for the constant wall heat flux boundary condition to investigate the hydrothermal behavior of the system. Therefore, analytical solutions are developed for temperature difference between the solid and the fluid phases as well as the local Nusselt number in the porous medium. Effects of pertinent parameters such as dimensionless axial length, Biot number, effective thermal conductivity ratio and dimensionless heat generation parameters on the Nusselt number are discussed. To further clarify the validity of the solution provided, the obtained results are compared with the solutions for two primary approaches (Models A and B) for the constant wall heat flux boundary condition. Results show that both the Nusselt number and the thermal entry length increase with the increase of thermal conductivity ratio. The Nusselt number and the thermal entry length are found to decrease with the increase of the internal heat generation of the solid phase. It is further observed that the Nusselt number and the thermal entry length are less sensitive to the solid internal heat generation at high Biot numbers. Finally, it is found that when the effective thermal conductivity ratio tends to infinity, the thermal entry length tends to zero for high Biot numbers.

Microchannels enhanced by porous materials: Heat transfer enhancement or pressure drop increment?

The analytical heat transfer performance study (i.e. HTP, introduced as the ratio of the heat transfer rate enhancement to the pressure drop increment) for microchannels enhanced by a porous material is performed. The Darcy–Brinkman equation of motion and the two-energy equation model for heat transfer within the porous material (i.e. the local thermal non-equilibrium condition, LTNE) are used to find the Poiseuille (Po) and Nusselt (Nu) numbers in the slip-flow regime (i.e. when the Knudsen number, Kn, <0.1). To cover a wide range of heat exchangers, effects of hydrodynamical characteristics of the porous material (i.e. porosity, ϕ, and Darcy number, Da), the effective thermal conductivity ratio (k), the interphase convective heat exchange between the two phases (Bi), the velocity-slip and temperature-jump coefficients (α and β), the dimensionless thermal resistance of the walls (Rw), and the internal heat generations within the solid and fluid phases of the porous material (ωf and ωs) are considered. HTP maps are presented for a wide variation range of all pertinent parameters to design or to select a micro-heat exchanger according to the heat transfer enhancement with respect to the pressure drop cost. Results reveal that the HTP may effectively be increased by inserting rarefied porous inserts. The present analysis suggests the implementation of porous inserts within flow passages especially when 0.001<Kn<0.1 (the slip-flow regime).

Forced convection gaseous slip flow in a porous circular microtube: An exact solution

Rarefied phenomena can occur when a gas flows through a microchannel. However, most available convective solutions were obtained under the local thermal equilibrium condition. In this study, gaseous slip flow in a circular microtube filled with a porous medium is analytically investigated under the local thermal non-equilibrium condition. The first-order velocity slip and temperature jump conditions at the tube wall are invoked in order to account for the rarefaction effects. Rigorous analytical solutions are obtained for the velocity and temperature distributions as well as the average Nusselt number. Theoretical predictions are then compared to those of existing limiting cases in the literature. Results indicate that the degree of rarefaction, represented by the Knudsen number ranging from 10−3 to 10−1, has a significant effect on the velocity, temperature, pressure drop and heat transport within the microtube for various combinations of pertinent parameters such as the porosity, effective thermal conductivity ratio, Biot number and porous media shape factor.

Entropy generation of viscous dissipative flow in thermal non-equilibrium porous media with thermal asymmetries

The effect of thermal asymmetrical boundaries on entropy generation of viscous dissipative flow of forced convection in thermal non-equilibrium porous media is analytically studied. The two-dimensional temperature, Nusselt number and entropy generation contours are analysed comprehensively to provide insights into the underlying physical significance of the effect on entropy generation. By incorporating the effects of viscous dissipation and thermal non-equilibrium, the first-law and second-law characteristics of porous-medium flow are investigated via various pertinent parameters, i.e. heat flux ratio, effective thermal conductivity ratio, Darcy number, Biot number and averaged fluid velocity. For the case of symmetrical wall heat flux, an optimum condition with a high Nusselt number and a low entropy generation is identified at a Darcy number of 10−4, providing an ideal operating condition from the second-law aspect. This type of heat and fluid transport in porous media covers a wide range of engineering applications, involving porous insulation, packed-bed catalytic process in nuclear reactors, filtration transpiration cooling, and modelling of transport phenomena of microchannel heat sinks.

Analysis of gaseous slip flow in a porous micro-annulus under local thermal non-equilibrium condition – An exact solution

The phenomenon of rarefaction in a micro-annulus filled with a porous medium is analyzed in the slip-flow regime. A local thermal non-equilibrium (LTNE) model is utilized to represent the energy transport within the porous medium. Exact solutions are derived for both the fluid and solid temperature distributions within the annulus. Two distinct cases of the thermal boundary conditions are considered, namely a constant heat flux at the outer wall and adiabatic inner wall (Case I) and vice versa (Case II). By eliminating the temperature difference between the fluid and solid phases, the local thermal equilibrium (LTE) model is theoretically proved to be a special case of the LTNE counterpart. Analytical predictions indicate that although the rarefaction leads to a reduction in the heat transfer, the effects of other thermophysical parameters such as the Biot number, the effective thermal conductivity ratio, the porous media shape factor and the annulus aspect ratio also play an important role. The results suggest that the configuration of Case II is superior to that of Case I from the heat transfer point of view.

Constant wall heat flux boundary condition in micro-channels filled with a porous medium with internal heat generation under local thermal non-equilibrium condition

Forced convection heat transfer in a micro-channel filled with a porous material saturated with rarefied gas with internal heat generation is studied analytically in this work. The study is performed by analysing the boundary conditions for constant wall heat flux under local thermal non-equilibrium (LTNE) conditions. Invoking the velocity slip and temperature jump, the thermal behaviour of the porous-fluid system is studied by considering thermally and hydrodynamically fully-developed conditions. The flow inside the porous material is modelled by the Darcy–Brinkman equation. Exact solutions are obtained for both the fluid and solid temperature distributions for two primary approaches models A and B using constant wall heat flux boundary conditions. The temperature distributions and Nusselt numbers for models A and B are compared, and the limiting cases resulting in the convergence or divergence of the two models are also discussed. The effects of pertinent parameters such as fluid to solid effective thermal conductivity ratio, Biot number, Darcy number, velocity slip and temperature jump coefficients, and fluid and solid internal heat generations are also discussed. The results indicate that the Nusselt number decreases with the increase of thermal conductivity ratio for both models. This contrasts results from previous studies which for model A reported that the Nusselt number increases with the increase of thermal conductivity ratio. The Biot number and thermal conductivity ratio are found to have substantial effects on the role of temperature jump coefficient in controlling the Nusselt number for models A and B. The Nusselt numbers calculated using model A change drastically with the variation of solid internal heat generation. In contrast, the Nusselt numbers obtained for model B show a weak dependency on the variation of internal heat generation. The velocity slip coefficient has no noticeable effect on the Nusselt numbers for both models. The difference between the Nusselt numbers calculated using the two models decreases with an increase of the temperature jump coefficient.

Analysis of thermally developing flow in porous media under local thermal non-equilibrium conditions

In the present study, analytical solutions for thermally developing flows in porous media under local thermal non-equilibrium (LTNE) condition are derived for different fundamental models. The analytical solutions are validated by comparison with the numerical results. Based on the analytical solutions, the local Nusselt number is obtained, and the dimensionless thermal entry length ξc is predicted. The results are presented in terms of the pertinent physical parameters such as the Biot number and the effective thermal conductivity ratio of the fluid to solid phases.