Effect of aspect ratio on entropy generation in a rectangular cavity with differentially heated vertical walls

In the present study, entropy generation in rectangular cavities with the same area but different aspect ratios is numerically investigated. The vertical walls of the cavities are at different constant temperatures while the horizontal walls are adiabatic. Heat transfer between vertical walls occurs by laminar natural convection. Based on the obtained dimensionless velocity and temperature values, the distributions of local entropy generation due to heat transfer and fluid friction, the local Bejan number and local entropy generation number are determined and related maps are plotted. The variation of the total entropy generation and average Bejan number for the whole cavity volume at different aspect ratios for different values of the Rayleigh number and irreversibility distribution ratio are also evaluated. It is found that for a cavity with high value of Rayleigh number (i.e., Ra = 10⁵), the total entropy generation due to fluid friction and total entropy generation number increase with increasing aspect ratio, attain a maximum and then decrease. The present results are compared with reported solutions and excellent agreement is observed. The study is performed for 10² < Ra < 10⁵, 10^− 4 < ? < 10^− 2, and Pr = 0.7.     

Entropy generation of turbulent double-diffusive natural convection in a rectangle cavity

Turbulent double-diffusive natural convection is of fundamental interest and practical importance. In the present work we investigate systematically the effects of thermal Rayleigh number (Ra), ratio of buoyancy forces (N) and aspect ratio (A) on entropy generation of turbulent double-diffusive natural convection in a rectangle cavity. Several conclusions are obtained: (1) The total entropy generation number (S_total) increases with Ra, and the relative total entropy generation rates are nearly insensitive to Ra when Ra ≤ 10⁹; (2) Since N > 1, S_total increases quickly and linearly with N and the relative total entropy generation rate due to diffusive irreversibility becomes the dominant irreversibility; and (3) S_total increases nearly linearly with A. The relative total entropy generation rate due to diffusive and thermal irreversibilities both are monotonic decreasing functions against A while that due to viscous irreversibility is a monotonic increasing function with A. More important, through the present work we observe a new phenomenon named as “spatial self-copy” in such convectional flow. The “spatial self-copy” phenomenon implies that large-scale regular patterns may emerge through small-scale irregular and stochastic distributions. But it is still an open question required further investigation to reveal the physical meanings hidden behind it.     

Entropy generation in natural convection in a symmetrically and uniformly heated vertical channel

In this study numerical predictions of local and global entropy generation rates in natural convection in air in a vertical channel symmetrically heated at uniform heat flux are reported. Results of entropy generation analysis are obtained by solving the entropy generation equation based on the velocity and temperature data. The analyzed regime is two-dimensional, laminar and steady state. The numerical procedure expands an existing computer code on natural convection in vertical channels. Results in terms of fields and profiles of local entropy generation, for various Rayleigh number, Ra, and aspect ratio values, L/b, are given. The distributions of local values show different behaviours for the different Ra values. A correlation between global entropy generation rates, Rayleigh number and aspect ratio is proposed in the ranges 10³ ⩽ Ra ⩽ 10⁶ and 5 ⩽ L/b ⩽ 20.     

Entropy generation in Poiseuille–Benard channel flow

The issue of entropy generation in Poiseuille–Benard channel flow is analyzed by solving numerically the mass, momentum and energy equations with the use of the classic Boussinesq incompressible approximation. The numerical scheme is based on Control Volume Finite Element Method with the SIMPLER algorithm for pressure–velocity coupling. Results are obtained for Rayleigh numbers Ra and irreversibility φ ranging from 10³ to 5×10⁴ and from 10⁻⁴ to 10 respectively. Variations of entropy generation and the Bejan number as a function of Ra and φ are studied. The limit value φ_l for which entropy generation due to heat transfer is equal to entropy due to fluid friction is evaluated. It has been found that φ_l is a decreasing function of the Rayleigh number Ra. φ_l varies from 0.0015 to 0.096 when Ra decrease from 5×10⁴ to 10³. Stream lines and entropy generation maps are plotted at six times over one period at Ra =10⁴ and φ=10⁻³. It has been found that the maximum entropy generation is localized at areas where heat exchanged between the walls and the flow is maximum. No significant entropy production is seen in the main flow.     

Analysis of entropy generation for distributed heating in processing of materials by thermal convection

Analysis of entropy generation has been carried out for square cavities with distributed heated sources filled with various materials involving wide range of Pr(=0.015, 0.7, 10, 1000) during the conduction and convection regime within Ra(=10³ ? 10⁵). Entropy generation terms involving thermal and velocity gradients are evaluated accurately based on elemental basis set via Galerkin finite element method. Local entropy maps are analyzed in detail for various cases and the dominance of thermal and frictional irreversibilities is studied via average Bejan number. The heat transfer irreversibility is found to dominate during conduction regime while the fluid friction irreversibility dominates the entropy generation in the convection regime, except for the low Pr fluid based on the heating configuration of the cavity. Further, the variation of total entropy generation has been observed to be similar for different heating configurations for higher Pr fluids (=10, 1000) whereas, the configuration of cavity has been found to have little effect on total entropy generation for fluids with Pr = 0.7 during both conduction and convection regimes. Thermal mixing and degree of temperature uniformity due to distributed heating in various cases are also reported and optimum cases for processing of various fluids are presented based on minimum entropy generation.     

Heat generation effects in natural convection inside a porous annulus

The interplay between internal heat generation and externally driven natural convection inside a porous medium annulus is studied in detail using numerical methods. The axisymmetric domain is bounded with adiabatic top and bottom walls and differentially heated side walls sustaining steady natural convection of a fluid with Prandtl number, Pr = 5, through a porous matrix of volumetric porosity, ? = 0.4. The generalized momentum equation with Brinkman–Darcy–Forchheimer terms and the local thermal non-equilibrium based two-energy equation model are solved to determine the flow and the temperature distribution. Beyond a critical heat generation value defined using an internal Rayleigh number, Ra_I,cr^?, the convection transits from unicellular to bicellular mode, as the annulus T_max becomes higher than the fixed hot-wall temperature. The Ra_I,cr^? increases proportionately when the permeability based external Rayleigh number Ra_E^? and the solid–fluid thermal conductivity ratio γ are independently increased. A correlation is proposed to predict the overall annulus Nu as a function of Ra_E^?, Ra_I^?, Da and γ. It predicts the results within ± 20% accuracy.     

Investigation of natural circulation in cavities with uniform heat generation for different Prandtl number fluids

Natural convection in enclosures with uniform heat generation and isothermal side walls is studied here. For the rectangular enclosure, two-dimensional conservation equations are solved using SIMPLE algorithm. Parametric studies are conducted to examine the effects of orientation of the cavity, fluid properties (Pr number), and aspect ratio for Rayleigh numbers up to 10⁶. For a horizontal square cavity, the flow becomes periodically oscillating at Ra = 5 × 10⁴ and chaotic at Ra = 8 × 10⁵. With a slight increase in the inclination angle, the oscillations die and for inclination angles greater than 15⁰, the flow attain a steady state over a range of Ra. It is found that for tall cavities (aspect ratio > 1), the steady-state solution is obtained for all values of Ra considered here. However, for wide cavities (aspect ratio < 1), an oscillatory flow regime is observed. The maximum temperature within the cavity is calculated for the range of Ra, aspect ratio and Pr number. Correlations for the maximum cavity temperature is presented here. The values of critical Rayleigh number at which the convection sets in the rectangular cavity are also studied and two distinct criteria are determined to evaluate the critical Rayleigh number. Further, a three-dimensional simulation is performed for a cubic cavity. It is found that the steady state solutions are obtained for all Rayleigh number, except at Ra = 10⁶. This is in contrast to the predictions for a two-dimensional square cavity, which has an oscillatory zone from Ra = 5 × 10⁴ onwards.     

Non-Darcy effects in buoyancy driven flows in an enclosure filled with vertically layered porous media

Natural convection in an enclosure filled with two layers of porous media is investigated numerically. Constant heat flux is imposed on the left vertical wall and the right wall is assumed to be at a low temperature. The focus of the work is on the validity of the Darcy model when various combinations of fluid Rayleigh number, Darcy number and permeability ratios are considered for fixed values of the modified Rayleigh numbers. It is found that the boundary effects (Brinkman term) have significant importance at higher modified Rayleigh numbers (Rayleigh number based on permeability, Ra_m). Calculations are performed for a modified Rayleigh number up to 10⁵. The results showed that, for the investigated range of parameters, the flow structure and heat transfer could be different than what Darcy model predicts. Two circulations are predicted for Ra_f=1×10⁸, for two different cases, Da=1×10⁻³, K_r=1000 and Da=1×10⁻⁴, K_r=100 (K_r=K₁/K₂). For K_r>1, increasing permeability ratio decreases flow penetration from layer 1 to layer 2 while reverse is true for K_r<1. For low Ra_m (Ra_m?10³) and K_r=1000, the heat transfer is conductive in the right layer, while this is true for the left layer for K_r=0.001. It is possible to obtain no-slip velocity boundary conditions both at the walls and at the interface between the porous layers even for very low permeability.     

Numerical investigation of unsteady natural convection heat transfer and entropy generation from a pair of cylinders in a porous enclosure

Abstract

The present study analyses numerically the unsteady heat transfer and entropy generation characteristics in a two-dimensional porous enclosure embedded with two heated circular cylinders at different positions at the vertical mid-plane. The heat transfer is primarily due to conduction for lower values of Darcy number (10^?4), while heat transfer by convection becomes significant for higher values of Darcy number (10^?3, 10^?2). Contrasting features are observed in the variation of time-average Nusselt number with interspacing distance. The major contributor of irreversibility is the entropy generation due to heat transfer for lower values of Darcy number, while for larger values of Darcy number, it varies with Rayleigh number.     

Natural convection slip flow in a vertical microchannel heated at uniform heat flux

Numerical solutions for steady state developing natural convection flow in air, in vertical parallel-plate microchannels are accomplished. An asymmetric heating is considered and the walls are assumed to be at uniform heat flux. A first-order model is used for slip and jump boundary conditions and an analytical solution for the fully developed flow is also given. Results are performed for air, for the heat flux ratio in the 0.0–1.0 range, for Rayleigh, Ra, and Knudsen, Kn, numbers from 10^?1 to 8 × 10³ and from 0.0 to 0.10, respectively. The maximum mass flow rate is always obtained for the highest considered Kn value, whereas the average Nusselt number, Nu, increases for lower Ra (<10) and decreases for Ra > 100. Wall temperature profiles have the lowest values for highest considered Kn value at lower Ra, whereas for the developing flow, they present opposite trends. For developing flow, velocity profiles for asymmetric and symmetric heating are completely different. In developing flow velocity profiles along the wall present the highest increases for asymmetric heating and the highest considered Kn value.     

CFD-based analysis of entropy generation due to mean and fluctuating fields during turbulent natural convection in a square enclosure

The current study aims to numerically investigate the entropy generation during the natural convection flow of air in a square cavity. The governing equations for the conservation of mass, momentum, energy, and turbulence are solved using a control volume-based technique employing the commercial code Fluent. Runs have been performed for both laminar and turbulent flow regimes by varying the Rayleigh number (Ra) from 10³ to 10¹⁰. On the other hand, various viscous distribution coefficients (ϕ = 10⁻⁴, 10⁻³, and 10⁻²) and constant Prandtl number (Pr = 0.71) were considered. Given the conflicting perspectives in the literature regarding the entropy generation under turbulent regimes, more research is needed to better understand the impact that the fluctuating flow has on entropy production. The four terms of entropy generation inherent to turbulent natural convection (entropy generation due to dissipation in the mean and the fluctuating velocity fields in addition to the heat flux due to the mean and the fluctuating temperature) are computed in the present work and compared to calculations based on only mean values of temperature and velocity gradients. It was found that taking into account the fluctuating terms of temperatures and velocities augment the total entropy generation by 10.10%, 14.43%, and, 17.70%, up to 32.60%, respectively, for Ra = 5 × 10⁸, Ra = 10⁹, Ra = 1.58 × 10⁹, and Ra = 10¹⁰. The gain shows the tendency to increase with the Rayleigh number. Thus, the fluctuating terms cannot be neglected particularly for high Rayleigh numbers. Furthermore, unlike entropy production due to the mean flow field, numerical outcomes reveal that the generated irreversibilities due to fluctuating flow are located around the upper hot and the lower cold corners of the heated walls. In addition, a numerical relationship between the first and the second laws of thermodynamics has been derived. A promising result that emerged from this study has shown that the Nusselt number and therefore the first law of thermodynamics is sufficient to estimate the heat part of entropy generation without the necessity of using the second law.     

Simulation of natural convection and entropy generation of MHD non-Newtonian nanofluid in a cavity using Buongiorno's mathematical model

In this paper, natural convection and entropy generation of non-Newtonian nanofluid, using the Buongiorno's mathematical model in a cavity in the presence of a uniform magnetic field has been analyzed by Finite Difference Lattice Boltzmann method (FDLBM). The cavity is filled with nanofluid which the mixture shows shear-thinning behavior. This study has been performed for the certain pertinent parameters of Rayleigh number (Ra = 10⁴ and 10⁵), Hartmann number (Ha = 0, 15, 30), buoyancy ratio number (N_r = 0.1, 1, and 4), power-law index (n = 0.4–1), Lewis number (Le = 1, 5, and 10), Thermophoresis parameter (N_t = 0.1, 0.5, 1), and Brownian motion parameter (N_b = 0.1, 1, 5). The Prandtl number is fixed at Pr = 1. The Results indicate that the augmentation of Hartmann number causes heat and mass transfer to drop. The increase in Rayleigh number enhances heat and mass transfer for various power-law indexes. The alteration of the power-law index changes heat and mass transfer. In addition, the rise of Hartmann number declines the shear-thinning behavior. The increase in the Lewis number augments mass transfer while it causes heat transfer to drop. The rise of the Thermophoresis and Brownian motion parameters ameliorate mass transfer and declines heat transfer significantly. The augmentation of buoyancy ratio number enhances heat and mass transfer. The augmentation of the power-law index declines various entropy generations in different Rayleigh numbers and Hartmann numbers. The increase in Hartmann number declines total entropy generation in different Rayleigh numbers. In addition, the rise of Rayleigh number and Hartmann number causes Bejan number to drop in various power-law indexes. The enhancement of the Lewis number provokes the total irreversibility to rise. Further, the total entropy generation increases as the buoyancy ratio number augments. It was shown that the increase in the Brownian motion and Thermophoresis parameters enhance the total irreversibility.     

Double-diffusive natural convection in an asymmetric trapezoidal enclosure: unsteady behavior in the laminar and the turbulent-flow regime

In the present work the natural convective heat and mass transfer in an asymmetric, trapezoidal enclosure is studied numerically. Such a configuration is encountered in greenhouse-type solar stills, where natural convection in the enclosed humid air due to vertical temperature and concentration gradients between the saline water and the transparent cover, plays a decisive role. In this double-diffusion problem, the relative magnitude of the thermal and the concentration (or solutal) Rayleigh numbers, expressed by their ratio N is a key parameter. The two-dimensional flow equations, expressed here in a stream function-vorticity (Ψ − Ω) formulation, along with the energy and concentration equations are solved. Due to the large values of the Rayleigh numbers encountered under realistic conditions (10⁷ ? Ra ? 10¹⁰), mostly turbulent flow conditions prevail. A two-equation, low-Reynolds number turbulence model has thus been selected and a curvilinear coordinate system is employed, allowing for better matching of the computational grid to the enclosure geometry. The numerical solutions yield a multi-cellular flow field, with the number of cells depending on the Rayleigh number for a fixed Lewis number and geometry. For a positive value of N (N = 1) the solution is qualitatively similar to the case with only thermal buoyancy present (N = 0). However, for negative values (N = −1), more complex unsteady phenomena arise, having a different nature in the laminar and the turbulent flow regime, which are both investigated. Correlations for the mean convective heat and mass transfer coefficients are obtained for a wide range of Rayleigh numbers, and comparisons are made for the different values of N, showing lower values and different rate of increase with Ra for N = −1.     

Numerical simulation of three-dimensional double diffusive free convection flow and irreversibility studies in a solar distiller

Numerical results of double-diffusive natural convection are presented in a three-dimensional solar distiller. The flow is considered laminar and caused by the interaction of the thermal energy and the chemical species diffusions. Equations of concentration, energy and momentum are formulated using vector potential-vorticity formulations in its three-dimensional form, then solved by the finite volume method. The Rayleigh number is fixed at Ra = 10⁵ and the effects of the buoyancy ratio are studied for opposed temperature and concentration gradients, with a particular interest to the three-dimensional aspects and entropy generation.     

Entropy generation analysis of convective airflow in a solar updraft tower power plant

The main objective of this numerical investigation is to interpret the entropy generation for free convection airflow in a solar tower updraft system. The ground surface is subjected to uniform hot temperature and the collector cover is maintained at lower constant temperature while the chimney wall is adiabatic. Two dimensionless equations of steady laminar free convective airflow are discretized using the finite volume approach. Numerical solutions were accomplished for different values of the Rayleigh number. Results are given in terms of isotherms, velocity magnitude, local entropy generation associated with thermal and fluid friction, local total entropy generation and local Bejan number contours for Rayleigh number ranging between 10³ and 10⁸. The reported results show that thermal and frictional irreversibilities are proportional to the Rayleigh number. Also, it was found that, at lower Rayleigh, total irreversibility is attributable to the thermal irreversibilities and occurs essentially in the collector section. At higher Rayleigh, frictional irreversibilities are increased significantly and become the dominant source of irreversibility in the solar tower, and the chimney section is the main contributor in the total irreversibility in the system.     

Transient natural convection in a cylindrical enclosure at high Rayleigh numbers

The transient state of natural convection in a vertical cylindrical enclosure is studied numerically for water at high Rayleigh numbers, extending into values characteristic of the turbulent flow regime. Several two-equation turbulence models are used for this purpose. Heating is provided along the cylindrical surface at a constant heat flux, with the horizontal bounding surfaces being adiabatic and the development of stratification is studied. Such a configuration is very relevant to thermal storage tanks or solar thermal system vessels and the study aims at providing insight into the behavior of the system at the boundary between laminar and turbulent flow so that the appropriate numerical treatment may be adopted in future studies. The main aspect ratio considered is L/D=1 and the Rayleigh number (based on the length L) varies in the range 10¹⁰?Ra?10¹³ for laminar flow and 5×10¹³?Ra?10¹⁵ for turbulent flow, values for which previous data in the literature are all but non-existent. The attainment of a quasi-steady state is achieved after the fluid undergoes an oscillating pattern where secondary flows alternately appear and vanish. These patterns affect the development of stratification in the vessel. Low-Reynolds k-? models predict eventually a relaminarization at large times, but models employing the high-Re form of the k-? model obtain sustained or very slowly decaying turbulence instead. Comparisons are made with experimental results where applicable.     

Thermal lattice-BGK model based on large-eddy simulation of turbulent natural convection due to internal heat generation

To simulate turbulent convection at high Rayleigh number (Ra), we propose a new thermal lattice-BGK (LBGK) model based on large eddy simulation (LES). Two-dimensional numerical simulations of natural convection with internal heat generation in a square cavity were performed at Ra from 10⁶ to 10¹³ with Prandtl numbers (Pr) at 0.25 and 0.60. Simulation results indicate that our model is fit to simulate high Ra flow for its better numerical stability. At Ra = 10¹³, a global turbulent has occurred. With a further increase in Ra, the flow will arrive in a fully turbulence regime. The Nusselt–Rayleigh relationship is also discussed.     

Natural convection effects on heat and mass transfer in a curvilinear triangular cavity

Finite element method is used in this study to analyze the effects of buoyancy ratio and Lewis number on heat and mass transfer in a triangular cavity with zig-zag shaped bottom wall. Buoyancy ratio is defined as the ratio of Grashof number of solutal and thermal. Inclined walls of the cavity have lower temperature and concentration according to zig-zag shaped bottom wall. Enclosed space consists mostly of an absorber plate and two inclined glass covers that form a cavity. Both high temperature and high concentrations are applied to bottom corrugated wall. Computations were done for different values of buoyancy ratio (?10 ? Br ? 10), Lewis number (0.1 ? Le ? 20) and thermal Rayleigh number (10⁴ ? Ra_T ? 10⁶). Streamlines, isotherms, iso-concentration, average Nusselt and Sherwood numbers are obtained. It is found that average Nusselt and Sherwood numbers increase by 89.18% and 101.91% respectively as Br increases from ?10 to 20 at Ra_T = 10⁶. Also, average Nusselt decreases by 16.22% and Sherwood numbers increases by 144.84% as Le increases from 0.1 to 20 at this Rayleigh number.     

The effect of surface thermal radiation on entropy generation in an open cavity with natural convection

In this work, a numerical study on entropy generation in a square open cavity with natural convection and surface thermal radiation is presented. The overall continuity, momentum, and energy equations for the gas phase in the open cavity were solved numerically by means of the finite-volume method. Temperature-dependent fluid properties were considered. During the calculations, the values of the Rayleigh number (Ra) were set in the range of 10⁴–10⁶. The temperature difference between the hot wall and the bulk fluid (ΔT) was varied between 50 and 400 K, and was represented by a dimensionless temperature difference (φ) for the purpose of generalization of the present study. The results of this investigation indicate that surface thermal radiation increases the overall entropy generation rate between 33.52% and 560.87%, and thus cannot be neglected in the analysis of this type of system.     

Experimental study of natural convection heat transfer of air in a cube below atmospheric pressure

Experimental study was carried out on the temperature profile of natural convection of air in a 50 mm cube with the temperature difference of 30 K between two opposing vertical walls. The objective is to investigate the essential aspect of heat transfer in the wide range of Rayleigh number (Ra) by lowering the pressure. The pressure was varied from 5.40 kPa (40.5 mmHg) to 99.99 kPa (750.0 mmHg). These correspond to Ra=1.04×10³ to 3.56 × 10⁵. The results show that the temperature distribution and Nusselt number approach those of conduction state as the pressure decreases.