Numerical analysis of convective heat transport in a porous semi‐trapezoidal enclosure with radiation effect

A theoretical and numerical study of natural convection of two‐dimensional laminar incompressible flow in a semi‐trapezoidal porous enclosure in the presence of thermal radiation is conducted. The semi‐trapezoidal enclosure has an inclined left wall that in addition to the right vertical wall is maintained at a constant temperature, whereas the remaining (horizontal) walls are adiabatic. The Darcy‐Brinkman isotropic model is utilized. The governing partial differential equations are transformed using a vorticity stream function and nondimensional quantities and the resulting governing nonlinear dimensionless equations are solved using the finite difference method with incremental steps. The impacts of the different model parameters (Rayleigh number [Ra], Darcy number [Da], and radiation parameter [Rd]) on the thermofluid characteristics are studied in detail. The computations show that convective heat transfer is enhanced with the greater Darcy parameter (permeability). The flow is accelerated with the increasing buoyancy effect (Rayleigh number) and heat transfer is also increased with a greater radiative flux. The present numerical simulations are more relevant to hybrid porous media solar collectors.     

Influence of Viscous Dissipation on non‐Darcy Mixed Convection Flow of Nanofluid

In this paper, the steady fully developed non‐Darcy mixed convection flow of a nanofluid in a vertical channel filled with a porous medium with different viscous dissipation models is analyzed. The Brinkman‐Forchheimer extended Darcy model is used to describe the fluid flow pattern in the channel. The transport equations for a nanofluid are solved analytically using the seminumerical‐analytical method known as differential transformation method, and numerically with the Runge‐Kutta shooting method. Finally, the influence of pertinent parameters, such as solid volume fraction, different nanoparticles, mixed convection parameter, Brinkman number, Darcy number, and inertial parameter on the velocity and temperature fields are shown graphically. The results show that velocity and temperature are enhanced when the mixed convection parameter, Brinkman number, and Darcy number increases whereas solid volume fraction and inertial parameter decreases the velocity and temperature fields. The obtained results show that the nanofluid enhances the heat transfer process significantly.     

Natural convection heat transfer and entropy generation inside porous quadrantal enclosure with nonisothermal heating at the bottom wall

In this work, we study numerically the natural convection heat transfer and entropy generation characteristics inside a two-dimensional porous quadrantal enclosure heated nonuniformly from the bottom wall. The effect of Darcy number is significant in dictating the Nusselt number only for higher values of Rayleigh number and the variation is more profound for larger values of Darcy number. The variation of entropy generation rate is significant with the Darcy number only for higher values of Rayleigh number. The entropy generation due to heat transfer is the significant contributor of irreversibility at low values of Darcy number, while for larger values of Darcy number and Rayleigh number entropy generation due to fluid friction becomes dominant.     

Entropy generation due to conjugate natural convection in enclosures bounded by vertical solid walls with different thicknesses

Entropy generation due to conjugate natural convection heat transfer and fluid flow has been studied inside an enclosure with bounded by two solid massive walls from vertical sides at different thicknesses. Enclosure is differentially heated from vertical walls and horizontal walls are adiabatic. Governing equations which are written in streamfunction-vorticity form solved by finite difference technique for the governing parameters as Rayleigh number, 10³ ≤ Ra ≤ 10⁶, length ratio of solid walls as ₁ (for left vertical wall) and ₂ (for right vertical wall) and thermal conductivity ratio of solid to fluid (k), 1 ≤ k ≤ 10. Entropy generation contours due to fluid friction and heat transfer irreversibility, isotherms, streamlines, Nusselt numbers and velocity profiles were obtained. It is found that entropy generation increases with increasing of thermal conductivity ratio and thicknesses of the walls. Entropy generation due to heat transfer is more significant than that of fluid flow irreversibility for all values of thickness of the solid vertical walls.     

Effects of thin fin on natural convection in porous triangular enclosures

A two-dimensional solution of natural convection in solid adiabatic thin fin attached to porous right triangular enclosures has been analyzed numerically. The vertical wall of the enclosure is insulated while the bottom and the inclined walls are isothermal. The temperature of the bottom wall is higher than the temperature of the inclined wall. Governing equations, which are written using Darcy model, are solved via the finite difference technique. The Successive Under Relaxation (SUR) method was used to solve linear algebraic equations. Dimensionless location of the thin fin from 0.2 to 0.6, the aspect ratio of triangular enclosure from 0.25 to 1, Rayleigh number from 100 to 1000 and the dimensionless height of the fin from 0.1 to 0.4 are used as governing parameters that are effective on heat transfer and fluid flow. Results for the mean Nusselt number, velocity profiles, the contour maps of the streamlines and isotherms are presented. It is observed that the thin fin can use as a passive control element for flow field, temperature distribution and heat transfer.     

Effects of the inclination angle on natural convection heat transfer and entropy generation in a square porous enclosure

ABSTRACT

In this paper, we analyze numerically the effects of the inclination angle on natural convection heat transfer and entropy generation characteristics in a two-dimensional square enclosure saturated with a porous medium. There is a significant alteration in Nusselt number with the orientation of the enclosure at higher values of Rayleigh number. It reveals that the variation of entropy generation rate with the inclination angle is significant for higher values of Darcy number. The dominant source of irreversibility is due to heat transfer at low values of Darcy number, whereas entropy generation due to fluid flow dominates over that due to heat transfer for larger values of Darcy number.     

Natural convection in an enclosure with localized heating and salting from below

Two-dimensional, double diffusion, natural convection in a rectangular enclosure filled with binary fluid saturating porous media is investigated numerically. Multiple motions are driven by the external temperature and concentration differences imposed across horizontal walls with the simultaneous presence of discrete heat and contaminant sources. The general Brinkman-extended Darcy model is adopted to formulate the fluid flow in the cavity. The fluid, heat and moisture transport through the isotropic porous layer are analyzed using the streamlines, heatlines and masslines, and the heat and mass transfer potentials are also explained by the variations of overall Nusselt and Sherwood numbers. The numerical simulations presented here span a wide range of the main parameters (thermal Rayleigh numbers, strip pitches and Darcy number) in the domain of destabilizing solutal buoyancy forces. It is shown that the heat and mass transfer potential can be promoted or inhibited, depending strongly on the permeability of porous medium, the strip pitch, the thermal and solutal Rayleigh numbers.     

Buoyant Darcy flow driven by a horizontal temperature gradient: A linear stability analysis

The linear stability of a fluid saturated porous layer bounded by two parallel impermeable plane walls is investigated. The lower wall is subject to a uniform heat flux, while the upper wall is subject to a linearly varying temperature in a horizontal direction. Two parameters govern the onset of convection in the porous layer: the vertical Darcy–Rayleigh number, and the horizontal Darcy–Rayleigh number. The objective of this study is to obtain the onset conditions for the instability of the basic parallel flow in the layer. The governing balance equations are written in a dimensionless form and solved on assuming oblique roll disturbances, arbitrarily oriented in the horizontal plane. Mathematically, this leads to a system of two ordinary differential equations to be solved as an eigenvalue problem. The solution, carried out numerically, provides the neutral stability condition. The numerical solution is performed by employing a procedure based on the sixth-order Runge–Kutta method and on the shooting method for satisfying the boundary conditions at the upper boundary wall.     

Non-Darcy natural convection of a non-Newtonian fluid in a porous cavity

A numerical study of non-Darcy natural convection in a porous enclosure saturated with a power-law fluid is presented. Hydrodynamic and heat transfer results are reported for the configuration in which the enclosure is heated from a side-wall while the horizontal walls are insulated. The flow in the porous medium is modeled using the modified Brinkman–Forchheimer-extended Darcy model for power-law fluids, which accounts for both inertia and boundary effects. The results indicate that when the power law index is decreased, the circulation within the enclosure increases leading to a higher Nusselt number and these effects are enhanced as the Darcy number is increased. Consequently as the power law index decreases, the onset of the transitions from Darcy regime to Darcy–Forchheimer–Brinkman regime to asymptotic convection (boundary layer) regime shift to higher corresponding values of the Darcy number. An increase in Rayleigh number produces similar effects as a decrease in power law index.     

Laminar conjugate mixed convection in a vertical channel with heat generating components

Conjugate mixed convection arising from protruding heat generating ribs attached to substrates (printed circuit boards) forming channel walls is numerically studied. The substrates with ribs form a series of vertical parallel plate channels. Assuming identical disposition and heat generation of the ribs on each board, a channel with periodic boundary conditions in the transverse direction is considered for analysis. The governing equations are discretised using a control volume approach on a staggered mesh and a pressure correction method is employed for the pressure–velocity coupling. The solid regions are considered as fluid regions with infinite viscosity and the thermal coupling between the solid and fluid regions is taken into account by the harmonic thermal conductivity method. Parametric studies are performed by varying the heat generation based Grashof number in the range 10⁴–10⁷ and the fan velocity based Reynolds number in the range 0–1500, with air as the working medium. Results are obtained for the velocity and temperature distributions, natural convection induced mass flow rate through the channel, the maximum temperatures in the heat sources and the local Nusselt numbers. The natural convection induced mass flow rate in mixed convection is correlated in terms of the Grashof and Reynolds numbers. In pure natural convection the induced mass flow rate varies as 0.44 power of Grashof number. The maximum dimensionless temperature is correlated in terms of pure natural convection and forced convection inlet velocity asymptotes. For the parameter values considered, the heat transferred to the working fluid via substrate heat conduction is found to account for 41–47% of the heat removal from the ribs.     

HYDROMAGNETIC NATURAL CONVECTION FROM AN INCLINED POROUS SQUARE ENCLOSURE WITH HEAT GENERATION

The problem of unsteady, laminar, two-dimensional hydromagnetic natural convection heat transfer in an inclined square enclosure filled with a fluid-saturated porous medium in the presence of a transverse magnetic field and fluid heat generation effects is studied numerically. The walls of the enclosure are maintained at constant temperatures. The flow in the porous region is modeled using the Brinkman-extended Darcy's law to account for the no-slip conditions at the walls. The control volume method is used to solve the governing balance equations for different values of the Darcy number, Hartmann number, and the inclination angle. Favorable comparisons with previously published work are performed. These comparisons confirmed the correctness of the numerical results. The obtained numerical results are presented graphically in terms of streamlines and isotherms as well as velocity and temperature profiles at midsections of the cavity to illustrate interesting features of the solution.     

Control of buoyancy-induced temperature and flow fields with an embedded adiabatic thin plate in porous triangular cavities

Natural convection has been performed in an insulated horizontally thin plate embedded in a triangular enclosure filled with fluid saturated porous medium, numerically. Bottom and inclined wall of triangular enclosure are isothermally heated and cooled, respectively. Vertical wall of enclosure is adiabatic. Steady, two-dimensional, laminar governing equations, which are written with Darcy model, were solved with finite-difference method. Calculations are conducted for different lengths and locations of thin plate, different aspect ratios, and Darcy-modified Rayleigh number. Prandtl number was chosen as 0.71. It is observed that the change on plate location in vertical and horizontal axes makes small effect on heat transfer; however flow field and temperature distribution strongly affected from these parameters.     

Natural Convection Heat Transfer in an Inclined Enclosure Filled with a Water-Cuo Nanofluid

This article presents the results of a numerical study on natural convection heat transfer in an inclined enclosure filled with a water-CuO nanofluid. Two opposite walls of the enclosure are insulated and the other two walls are kept at different temperatures. The transport equations for a Newtonian fluid are solved numerically with a finite volume approach using the SIMPLE algorithm. The influence of pertinent parameters such as Rayleigh number, inclination angle, and solid volume fraction on the heat transfer characteristics of natural convection is studied. The results indicate that adding nanoparticles into pure water improves its heat transfer performance; however, there is an optimum solid volume fraction which maximises the heat transfer rate. The results also show that the inclination angle has a significant impact on the flow and temperature fields and the heat transfer performance at high Rayleigh numbers. In fact, the heat transfer rate is maximised at a specific inclination angle depending on Rayleigh number and solid volume fraction.     

Natural Convection Heat Transfer and Entropy Generation in a Porous Trapezoidal Enclosure Saturated with Power-Law Non-Newtonian Fluids

Abstract

In the present study, natural convection heat transfer and its associated entropy generation in a porous trapezoidal enclosure saturated with a power-law non-Newtonian fluid has been numerically investigated. Horizontal walls of the enclosure are assumed to be adiabatic while the side walls are considered to be kept at a constant temperature. A continuum-based approach is adapted here to model the fluid flow through porous media and the Darcy’s law is modified to account for non-Newtonian rheological behavior of the fluid. The obtained governing equations are discretized using the finite volume method and a detailed parametric study is undertaken to account for the effects of various relevant parameters of the problem on the heat transfer and entropy generation rates. It was shown that the impact of the power-law index on both entropy generation and heat transfer significantly intensifies in a convection-dominated flow regime inside the enclosure, especially for a shear thinning liquid. Moreover, heat transfer rate and entropy generation increase as the sidewall angle is elevated.     

Soret and Dufour effects on free convection along a vertical wavy surface in a fluid saturated Darcy porous medium

In this article, free convection of heat and mass transfer along a vertical wavy surface in a Newtonian fluid saturated Darcy porous medium is studied by considering cross diffusion (namely the Soret and the Dufour effects) in the medium. The vertical wavy wall and the flow governing equations are transformed to a plane geometry case by using a suitable transformation. Then a similarity solution to this problem is presented under the large Darcy–Rayleigh number assumption. The governing partial differential equations are reduced to a set of ordinary differential equations that are integrated using numerical methods to study the nature of the non-dimensional heat and mass transfer coefficients in the medium. The results are presented for a range of the flow governing parameters such as the diffusivity ratio parameter, the buoyancy ratio parameter, the Soret parameter, the Dufour parameter and the amplitude of the wavy surface.     

Combined convection heat transfer in a porous lid-driven enclosure due to heater with finite length

A numerical work is performed to analyze combined convection heat transfer and fluid flow in a partially heated porous lid-driven enclosure. The top wall of enclosure moves from left to right with constant velocity and temperature. Heater with finite length is located on the fixed wall where its center of location changes along the walls. The finite volume-based finite-difference method is applied for numerical experiments. Parameters effective on flow and thermal fields are Richardson number, Darcy number, center of heater and heater length. The results are shown that the best heat transfer is formed when the heater is located on the left vertical wall.     

Numerical Investigation on Natural Convection Inside the Side Ventilated Square Enclosure with Vertical Mid-Partition

A finite volume-based computational study of steady laminar natural convection inside the square enclosure with cold partition wall centrally placed on top and bottom is presented. The fluid considered is air with Prandtl number 0.71. Except the partition walls, all other walls were assumed as hot. The heights of inlet and outlet ports are constantly fixed as 20% of height of the enclosure. The height of the opening in the partition walls were 10%, 20%, and 30% of height of enclosure. The buoyancy-driven heat transfer mechanism inside the domain is influenced by the percentage of opening in the partition, Rayleigh number, and geometrical position of inlet and outlet. The cold partition walls attract the flow that subsequently influences the thermal modifications around the partition. The fluid flow and heat transfer were investigated for 81 cases with different positions of inlet and outlet and varying height of openings for Ra = 10³, 10⁴, and 10⁵. The cross flow between the vertical walls through the opening in the central partition wall was considered in this study. The formation of vortices and their sizes depends on the configuration of inlet and outlet ports. The higher temperature gradient occurs near the inlet and outlet port of vertical walls. Local Nusselt number is maximum just below the inlet for all cases. Due to the dominance of buoyancy forces, heat transfer rate increases when Rayleigh number increases for all the cases. The hydrodynamic block effect by the openings on the partition wall has significant effect on the velocity profile than on the heat transfer.     

Double diffusive convection within a horizontal porous layer salted from the bottom and heated horizontally

Double-duffusive natural convection within a horizontal porous layer is studied both analytically and numerically. The enclosure is heated and cooled along vertical walls by uniform heat fluxes and a solutal gradient is imposed vertically. In the formulation of the problem use is made of the Darcy model and the density variation is taken into account by the Boussinesq approximation. The governing parameters of the problem are the thermal Rayleigh number, R_T, Lewis number, Le, and buoyancy ratio N. The existence of multiple steady state solutions, for a given set of the governing parameters, is demonstrated. The analytical solution, based on the parallel flow approximation, is found to be in good agreement with a numerical solution of the full governing equations.     

Numerical Investigation of Unsteady Natural Convection in an Inclined Square Enclosure With Heat-Generating Porous Medium

The unsteady laminar natural convection in an inclined square enclosure with heat-generating porous medium whose heat varies by a cosine function is investigated by a thermal equilibrium model and the Brinkman–Darcy–Forchheimer model numerically, with the four cooled walls of closure as isothermal. The numerical code based on the finite-volume method has been validated by reference data before it was adopted. Influence of dimensionless frequency and inclination angle on heat transfer characteristics in a square enclosure, such as flow distribution, isotherm, averaged Nusselt number on each wall, and time-averaged Nusselt number, are discussed, with specified value for Rayleigh number = 10⁸, Darcy number = 10^?4, Prandtl number = 7, porosity = 0.4, and specific heat ratio = 1. It is found that when the internal heat source varies by cosine, the Nusselt numbers of the four walls oscillate with the same frequency as the internal heat source; however, phase difference occurs. Moreover, frequency has little impact on time-averaged Nusselt number of the four walls, which is different from the phenomenon discovered in natural convection with suitable periodic varying wall temperature boundary condition. Moreover, inclination angle plays an important role in the heat transfer characteristics of the walls studied.     

Soret effect on the boundary layer flow regime in a vertical porous enclosure subject to horizontal heat and mass fluxes

The combined thermo- and double-diffusive convection in a vertical tall porous cavity subject to horizontal heat and mass fluxes was investigated analytically and numerically using the Darcy model with the Boussinesq approximation. The investigation focused on the effect of Soret diffusion on the boundary layer flow regime. The governing parameters were the thermal Rayleigh number, R_T, the Lewis number, Le, the buoyancy ratio, N, the Soret parameter, M, which characterized the Soret effect, and the aspect ratio of the enclosure, A_r. The results demonstrated the existence of a boundary layer flow solution for which the Soret parameter had a strong effect on the heat and mass transfer characteristics. For M ≠ 1 and M ≠ −1/Le, the profiles of the vertical velocity component, v, temperature, T, and solute concentration, S, exhibited boundary layer behaviors at high Rayleigh numbers. Furthermore, as R_T increased, the temperature and solute concentration became vertically and linearly stratified in the core region of the enclosure. The thermo-diffusion effect on the boundary layer thickness, δ, was discussed for a wide range of the governing parameters. It was demonstrated analytically that the thickness of the boundary layer could either increase or decrease when the Soret parameter was varied, depending on the sign of the buoyancy ratio. The effect of R_T on the fluid flow properties and heat and mass transfer characteristics was also investigated.