Constant wall heat flux boundary conditions in porous media under local thermal non-equilibrium conditions

Boundary conditions for constant wall heat flux in the absence of local thermal equilibrium conditions are analyzed in this work. Effects of variable porosity and thermal dispersion are also analyzed. Different forms of constant heat flux boundary conditions found in the literature were investigated in this work. The effects of pertinent parameters such as porosity, Darcy number, Reynolds number, inertia parameter, particle diameter and solid-to-fluid conductivity ratio were analyzed. Quantitative and qualitative interpretations of the results are utilized to investigate the prominent characteristics of the models under consideration. Limiting cases resulting convergence or divergence of the models are also considered. Results are presented in terms of the fluid, solid and total Nusselt numbers.     

Free convection boundary layer flow over vertical and horizontal flat plates embedded in a porous medium under mixed thermal boundary conditions

A similarity analysis of the steady free convection boundary layer over vertical and horizontal surfaces embedded in a fluid-saturated porous medium with mixed thermal boundary conditions is performed in this paper. New variables relating the similarity solutions of the Darcian velocity and temperature profiles associated with different values of the mixed thermal boundary condition parameter have been obtained. Boundary layer velocity and temperature profiles have been determined numerically for some values of the mixed thermal boundary condition parameter ε and the similarity exponent m.     

Mass transfer behavior at horizontal cylinder in cross flow under different flow conditions

The rate of liquid-solid mass transfer at a horizontal cylinder placed in a cylindrical vertical column under different hydronamic conditions including gas sparging, single phase liquid flow, and two phase (gas-liquid) flow was studied experimentally by using the electrochemical technique which involves measuring the limiting current of the cathodic reduction of K₃Fe(CN)₆ using a solution containing 0.01 M K₃Fe(CN)₆ and 0.1 M K₄Fe(CN)₆ and a large excess of NaOH as supporting electrolyte. Variables studied were: liquid and gas superficial velocities, cylinder diameter, and solution physical properties. For gas sparging: the data were correlated for the conditions 0.2 < (Re.Fr) < 8.7, and 1253 < Sc < 2778 by the equation:

j=0.11(Re.Fr)−0.247.     

Forced convection in porous microchannels with viscous dissipation in local thermal non-equilibrium conditions

Fully developed, steady-state forced convection, in parallel-plate microchannels, filled with a porous medium saturated with rarefied gases at high temperatures, in local thermal non-equilibrium (LTNE) condition, is investigated for the first-order slip-flow regime (0 ≤ Kn ≤ 0.1). Both velocity and temperature jumps at the walls are accounted for. An analytic solution is proposed for the Darcy–extended Brinkman flow model with assigned uniform heat flux at the microchannel walls and viscous heat dissipation in the fluid phase. The solution for NTLE includes the shear work done by the slipping effects. A closed-form expression of the Nusselt number is derived. A validation analysis with respect to the case of channels filled with saturated porous medium is accomplished. The results show that the internal dissipation increases as the velocity slip increases. In addition, the heat dissipation strongly affects the fluid temperature profiles. The increases in velocity slip and temperature jump lead to decreases of temperature gradients in the fluid and solid along the sections. The heat transfer at channel walls is enhanced due to an increase in the bulk heat transfer.     

Jet impingement cooling of a horizontal surface in a confined porous medium: Mixed convection regime

In the present article the jet impingement cooling of heated portion of a horizontal surface immersed in a fluid saturated porous media is considered for investigation numerically. The jet direction is considered to be perpendicular from the top to the horizontal heated element; therefore, the external flow and the buoyancy driven flow are in opposite directions. The governing parameters in the present problem are Rayleigh number, Péclet number, jet width and the distance between the jet and the heated portion normalized to the length of the heated element. The results are presented in the mixed convection regime with wide ranges of the governing parameters with the limitation of the Darcy model. It is found for high values of Péclet number that increasing either Rayleigh number or jet width lead to increase the average Nusselt number. Narrowing the distance between the jet and the heated portion could increase the average Nusselt number as well.No steady-state solution can be found in some cases; when the external jet flow and the flow due to buoyancy are in conflict for domination. The results from the unsteady governing equations in these cases show oscillation of the average Nusselt number along the heated element with the time without reaching steady state.     

Jet impingement cooling of a horizontal surface in an unconfined porous medium: Mixed convection regime

Two-dimensional slot jet impingement cooling of an isothermal horizontal surface immersed in an unconfined porous medium is simulated numerically to gain insight into thermal characteristics under mixed convection conditions with the limitation of the Darcy model. The jet direction is considered to be perpendicular from the top to the horizontal heated element; therefore, the jet flow and the buoyancy driven flow are in opposite directions. The results are presented in the mixed convection regime with wide ranges of the governing parameters: Péclet number (1 ? Pe ? 1000), Rayleigh number (10 ? Ra ? 100), half jet width (0.1 ? D ? 0.5), and the distance between the jet and the heated portion (0.1 ? H ? 1.0). It is found that the average Nusselt number increases with increase in either Rayleigh number or jet width for high values of Péclet number. The average Nusselt number also increases with decrease in the distance between the jet and the heated portion. It is shown that mixed convection mode can cause minimum average Nusselt number at two values of Péclet number and a maximum average Nusselt number occurs in between theses two Péclet numbers at higher Rayleigh number due to counteraction of jet flow against buoyancy driven flow. Hence careful consideration must be given while designing a system of jet impingement cooling through porous medium.     

The effect of local thermal non-equilibrium on forced convection boundary layer flow from a heated surface in porous media

A steady two-dimensional forced convective thermal boundary layer flow in a porous medium is studied. It is assumed that the solid matrix and fluid phase which comprise the porous medium are subject to local thermal non-equilibrium conditions, and therefore two heat transport equations are adopted, one for each phase. When the basic flow velocity is sufficiently high, the thermal fields may be described accurately using the boundary layer approximation, and the resulting parabolic system is analysed both analytically and numerically. Local thermal non-equilibrium effects are found to be at their strongest near the leading edge, but these decrease with distance from the leading edge and local thermal equilibrium is attained at large distances.     

Mixed convection on jet impingement cooling of a constant heat flux horizontal porous layer

In the present study, numerical investigation of jet impingement cooling of a constant heat flux horizontal surface immersed in a confined porous channel is performed under mixed convection conditions, and the Darcian and non-Darcian effects are evaluated. The unsteady stream function-vorticity formulation is used to solve the governing equations. The results are presented in the mixed convection regime with wide ranges of the governing parameters: Reynolds number (1 ≤ Re ≤ 1000), modified Grashof number (10 ≤ Gr¹ ≤ 100), half jet width (0.1 ≤ D ≤ 1.0), Darcy number (1 × 10^?6 ≤ Da ≤ 1 × 10^?2), and the distance between the jet and the heated portion (0.1 ≤ H ≤ 1.0). It is found that the average Nusselt number (Nu_avg) increases with increase in either modified Grashof number or jet width for high values of Reynolds number. The average Nusselt number also increases with decrease in the distance between the jet and the heated portion. The average Nusselt number decreases with the increase in Da for the non-Darcy regime when Re is low whereas Nu_avg increases when Re is high. It is shown that mixed convection mode can cause minimum heat transfer unfavorably due to counteraction of jet flow against buoyancy driven flow. Minimum Nu_avg occurs more obviously at higher values of H. Hence the design of jet impingement cooling through porous medium should be carefully considered in the mixed convection regimes.     

Fluid-solids flow with thermal and hydrodynamic non-equilibrium

A model is developed to describe the behaviour of particles in air streams. The equations of particle flow and heat transfer are given in dimensionless form. Two practical applications for suspension flows are solved: (a) the flow past a temperature step and (b) the injection of hot particles in a pipe carrying a gas. In both cases instantaneous velocity and temperature differences for the gas and solids are calculated and the effect of several dimensionless groups (such as Reynolds numbers, loading and dimension ratios) on these two quantities is determined. It was found that non-equilibrium effects are accentuated when bigger particles are in the mixture.     

Free convection boundary layer flow over a horizontal cylinder of elliptic cross section in porous media saturated by a nanofluid

This work studies the free convection boundary layer flow over a horizontal cylinder of elliptic cross section in porous media saturated by a nanofluid with constant wall temperature and constant wall nanoparticle volume fraction. The effects of Brownian motion and thermophoresis are incorporated into the model for nanofluids. A coordinate transformation is performed, and the obtained nonsimilar governing equations are then solved by the cubic spline collocation method. The effects of the Brownian motion parameter and thermophoresis parameter on the profiles of the temperature, nanoparticle volume fraction and velocity profiles are presented. The local Nusselt number is presented as a function of the thermophoresis parameter, Brownian parameter, Lewis number and the aspect ratio when the major axis of the elliptical cylinder is vertical (slender orientation) and horizontal (blunt orientation). Results show that the local Nusselt number is increased as the thermophoresis parameter or the Brownian parameter is decreased. The local Nusselt number increases as the buoyancy ratio or the Lewis number is decreased. Moreover, the local Nusselt number of the elliptical cylinder with slender orientation is higher than those of the elliptical cylinder with blunt orientation over the lower half cylinder.     

Choking flow modeling with mechanical and thermal non-equilibrium

The mechanistic model, which considers the mechanical and thermal non-equilibrium, is described for two-phase choking flow. The choking mass flux is obtained from the momentum equation with the definition of choking. The key parameter for the mechanical non-equilibrium is a slip ratio. The dependent parameters for the slip ratio are identified. In this research, the slip ratio which is defined in the drift flux model is used to identify the impact parameters on the slip ratio. Because the slip ratio in the drift flux model is related to the distribution parameter and drift velocity, the adequate correlations depending on the flow regime are introduced in this study. For the thermal non-equilibrium, the model is developed with bubble conduction time and Bernoulli choking model. In case of highly subcooled water compared to the inlet pressure, the Bernoulli choking model using the pressure undershoot is used because there is no bubble generation in the test section. When the phase change happens inside the test section, two-phase choking model with relaxation time calculates the choking mass flux. According to the comparison of model prediction with experimental data shows good agreement. The developed model shows good prediction in both low and high pressure ranges.     

Analysis of free convection about a horizontal cylinder in a porous media using a thermal non-equilibrium model

The thermally non-equilibrium model is used to study the free convection from a horizontal cylinder immersed in porous media. The governing equations are transformed to dimensionless form by introducing the boundary layer dimensionless variables. The resultant parabolic system of differential equations is solved by using an implicit finite difference method based on Keller box algorithm. The results of the developed code are validated with different mesh sizes, which can be used as benchmark results for thermally equilibrium condition. Numerical results are obtained for non-equilibrium model to analyze the effect of the governing parameters, which are the heat transfer coefficient between the solid and fluid phases H and the porosity scaled thermal conductivity ratio K_r. The results show that increasing H or K_r leads to increase in the total average Nusselt number. The value of the average Nusselt number for both fluid and solid phases as well as the total average Nusselt number have approached the corresponding values of the thermally equilibrium model at high value of H × K_r.     

Jet impingement cooling of a constant heat flux horizontal surface in a confined porous medium: Mixed convection regime

In the present study, numerical investigation of jet impingement cooling of a constant heat flux horizontal surface immersed in a confined porous channel is performed under mixed convection conditions with the limitation of the Darcy model. The results are presented in the mixed convection regime with wide ranges of the governing parameters: Péclet number (1 ? Pe ? 1000), Rayleigh number (10 ? Ra ? 100), half jet width (0.1 ? D ? 1.0), and the distance between the jet and the heated portion (0.1 ? H ? 1.0). It is found that the average Nusselt number increases with increase in either Rayleigh number or jet width for high values of Péclet number. The average Nusselt number also increases with decrease in the distance between the jet and the heated portion. The correlation for Nu_avg in the forced convection regime is suggested. It is shown that mixed convection mode can cause minimum average Nusselt number unfavorably due to counteraction of jet flow against buoyancy driven flow. Hence, careful consideration must be given while designing a system of jet impingement cooling through porous medium.     

An integrated model for interaction between melt flow and non-equilibrium solidification in thermal spraying

In this paper, a micro/macro-integrated model based on the VOF scheme is presented that accounts for free surface movement, thermal contact resistance, and fluid instability. A sub-model is developed to include the non-equilibrium solidification phenomena at the solid/liquid interface. The melt flow is incorporated into the microscopic model through prescribing a velocity profile that is obtained from the interpolation of melt velocities on the macroscopic grids near the interface. To improve the efficiency of the integration between the melt flow and the microscopic model, a relational database is developed and applied to the integrated micro/macro model. Three velocity profiles, e.g., linear, parabolic, and cubic velocity profiles, are considered and the results are compared with those obtained from the diffusion model.     

A boundary layer analysis of heat transfer by free convection from permeable horizontal cylinders of elliptic cross-section in porous media using a thermal non-equilibrium model

This work uses a thermal non-equilibrium model to study the free convection boundary layer flow driven by temperature gradients near a permeable horizontal cylinder of elliptic cross-section with constant wall temperature in a fluid-saturated porous medium. A coordinate transformation is used to obtain the nonsimilar boundary layer equations. The transformed boundary layer equations are then solved by the cubic spline collocation method. Results for the local Nusselt numbers are presented as functions of the porosity scaled thermal conductivity ratio, the heat transfer coefficient between solid and fluid phases, the transpiration parameter, and the aspect ratio when the major axis of the elliptical cylinder is vertical (slender orientation) and horizontal (blunt orientation). An increase in the porosity scaled thermal conductivity ratio or the heat transfer coefficient between the solid and fluid phases increases the heat transfer rates. Moreover, the use of suction (positive transpiration parameter) tends to increase the heat transfer rates between the porous medium and the surface.     

Effect of rotation on the onset of thermal convection in a sparsely packed porous layer using a thermal non-equilibrium model

Linear and nonlinear stability of a rotating fluid-saturated sparsely packed porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium. The extended Darcy–Brinkman model that includes the time derivative and Coriolis terms is employed as a momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for both stationary and oscillatory convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of rotation and thermal diffusion that causes the convection to set in through oscillatory mode rather than stationary. The rotation inhibits the onset of convection in both stationary and oscillatory mode. The Darcy number stabilizes the system towards the oscillatory mode, while it has dual effect on stationary convection. Besides, the effect of porosity modified conductivity ratio, Darcy–Prandtl number and the ratio of diffusivities on the stability of the system is investigated. The nonlinear theory is based on the truncated representation of Fourier series method. The effect of thermal non-equilibrium on heat transfer is brought out. The transient behavior of the Nusselt number is investigated by using the Runge–Kutta method. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study.     

Extended Oberbeck-Boussinesq approximation study of convective instabilities in a porous layer with horizontal flow and bottom heating

A study of the onset of convective instabilities in a porous layer with a horizontal basic flow is performed by including the effects of viscous dissipation and pressure work in the energy balance. Firstly, the so-called extended Oberbeck-Boussinesq approximation, i.e. the model based on the enthalpy formulation of the energy balance, is adopted. Then, the results for the marginal stability condition are compared with those obtained by the so-called Chandrasekhar approximation, i.e. the model based on the internal-energy formulation of the energy balance. It is shown that a marked discrepancy occurs between the two approaches, that becomes specially evident for high values of the Gebhart number. According to the extended Oberbeck-Boussinesq approximation, the effects of the viscous dissipation and of the pressure work result in a stabilization of the basic flow. On the contrary, the Chandrasekhar approximation predicts a destabilization of the basic flow induced by the viscous dissipation. The destabilization can be so intense that the onset of convective rolls may occur even in the absence of a boundary temperature difference, i.e. with a vanishing Darcy-Rayleigh number.     

A local thermal non-equilibrium model for two-phase flows with phase-change in porous media

The assumption of local thermal equilibrium for describing macroscopic heat transfer in a porous medium subjected to a liquid-vapor flow with phase change has been often investigated. Under certain circumstances, this assumption appears to be too restrictive and fails to be valid. In this paper, the method of volume averaging is used to derive a three-temperature macroscopic model considering local thermal non-equilibrium between the three phases. A closed form of the evaporation rate at the macroscopic level is obtained depending on the macroscopic temperatures and the effective properties. Six pore-scale closure problems are proposed, which allow to determine all the effective transport coefficients for representative unit cells. These closure problems are solved for simple unit cells and analytical results are presented in these cases. For these simplified unit cells, a comparison between averaged temperatures obtained from direct pore-scale simulations and averaged temperatures obtained from the three-equation model has been carried out for purely diffusive phase-change processes. A good agreement is obtained between the theory and the pore-scale calculations. This confirms the validity and the practical interest of the proposed approach.