Entropy generation due to natural convection in discretely heated porous square cavities

Optimization of industrial processes for higher energy efficiency may be effectively carried out based on the thermodynamic approach of entropy generation minimization (EGM). This approach provides the key insights on how the available energy (exergy) is being destroyed during the process and the ways to minimize its destruction. In this study, EGM approach is implemented for the analysis of optimal thermal mixing and temperature uniformity due to natural convection in square cavities filled with porous medium for the material processing applications. Effect of the permeability of the porous medium and the role of distributed heating in enhancing the thermal mixing, temperature uniformity and minimization of entropy generation is analyzed. It is found that at lower Darcy number (Da), the thermal mixing is low and the heat transfer irreversibility dominates the total entropy generation. In contrast, thermal mixing is improved due to enhanced convection at higher Da. Friction irreversibility is found to dominate the total entropy generation for higher Prandtl number (Pr) fluids at higher Da, whereas the heat transfer irreversibility dominates the total entropy generation for lower Pr fluids. Based on EGM analysis, it is established that larger thermal mixing at high Darcy number may not be always recommended as the total entropy production is quite large at high Darcy number. Overall, it is found that the distributed heating methodology with multiple heat sources may be an efficient strategy for the optimal thermal processing of materials.     

Role of Entropy Generation On Thermal Management During Natural Convection in a Tilted Square Cavity with Isothermal and Non-Isothermal Hot Walls

Entropy generation during natural convection within tilted square cavity inclined with different angles (? = 30°and 75°) for various thermal boundary conditions (case 1: isothermal heating and case 2: non-isothermal heating) has been studied. Simulations are carried out over a range of parameters: Rayleigh number (10³ ≤ Ra ≤ 10⁵) and Prandtl numbers (Pr = 0.025 and 998.24). The numerical results are presented in terms of isotherms (θ), streamlines (ψ), entropy generation due to heat transfer (S _θ ) and fluid friction (S _ψ ). Heating strategy is energy efficient for case 2 (non-isothermal heating) due to its less total entropy generation with reasonable heat transfer rate, irrespective of Pr.     

Effects of Thermal Boundary Conditions on Entropy Generation during Natural Convection

A comprehensive numerical study on entropy generation during natural convection is studied in a square cavity subjected to a wide variety of thermal boundary conditions. Entropy generation terms involving thermal and velocity gradients are evaluated accurately based on the elemental basis set via the Galerkin finite element method. The thermal and fluid irreversibilities during the conduction and convection dominant regimes are analyzed in detail for various fluids (Pr = 0.026,988.24) within Ra = 10³–10⁵. Further, the effect of Ra on the total entropy generation and average Bejan number is discussed. It is observed that thermal boundary conditions significantly affect the thermal mixing, temperature uniformity, and the entropy generation in the cavity. A case where the bottom wall is hot isothermal with linearly cooled side walls and adiabatic top wall is found to result in high thermal mixing and a higher degree of temperature uniformity with minimum total entropy generation.     

Entropy Generation Minimization Versus Thermal Mixing Due to Natural Convection in Differentially and Discretely Heated Square Cavities

Uniform temperature distribution is a key parameter in many thermal processing applications. A considerable amount of additional energy is used to enhance the fluid mixing in order to maintain the temperature uniformity, but that affects the overall efficiency of the process. In this article, an alternate approach is proposed for maintaining uniform temperature via various distributed/discrete heating strategies while maintaining the minimal entropy generation. The system of laminar natural convection in differentially and discretely heated square cavities filled with various materials (molten metals, air, aqueous solutions, oils) is considered, and finite element simulations are performed for a range of Rayleigh numbers (Ra = 10³–10⁵). Entropy generation is evaluated using finite-element basis sets for the first time in this work, and the derivatives at particular nodes are estimated based on the functions within adjacent elements. Analysis of entropy generation in each case is carried out and a detailed investigation of entropy production due to local heat transfer and fluid friction irreversibilities is presented. It is found that a high thermal mixing may not be the optimal situation for achieving uniform temperature distribution based on entropy production. A greater degree of temperature uniformity with moderate thermal mixing may correspond to minimum entropy generation with distributed heating. Further, based on entropy generation minimization approach, it has been thermodynamically established that the distributed heating methodology with multiple heat sources may be the energy efficient strategy for attaining adequate uniform temperature distribution with minimum entropy generation.     

Role of various moving walls on entropy generation during mixed convection within entrapped porous triangular cavities

The aim of the present investigation is to analyze the effect of the motion of horizontal walls on the entropy generation and heat transfer rates in an entrapped triangular porous cavity during mixed convection. Two different thermal boundary conditions are considered as follows: (i) hot inclined walls and cold horizontal walls and (ii) cold inclined walls and hot horizontal walls. Overall, Re?=?100 may be recommended at Pr_m?=?0.026, 7.2, Gr?=?10⁵, and Da_m?=?10^?4 to 10^?2 within the upper and lower cavities for cases 1 and 2.     

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.     

Heatline based thermal management for natural convection within right-angled porous triangular enclosures with various thermal conditions of walls

Analysis of natural convection in porous triangles have many important energy related applications in geophysical and solar energy fields. A numerical study on heat distribution and thermal mixing during steady laminar natural convective flow inside a right-angled triangular enclosure filled with porous media subjected to various wall boundary conditions is investigated in this study using Bejan’s heatlines approach. Influence of various thermal boundary conditions and inclination angles (φ) on evaluation of complex heat flow patterns are studied as a function of Darcy numbers (Da) for various regimes of Prandtl (Pr) and Rayleigh (Ra) numbers. Studies illustrate that maximum heat transfer occurs at the top vertex for lower top angle (φ=^°15) at higher Da(Da=10⁻³). As φ increases to ^°45, the maximum heat flux at the top vertex decreases and thermal mixing increases irrespective of Da and Pr. The enhanced convection at higher Da significantly affects the heat flow distribution, which is clearly depicted by high local Nusselt numbers at Da=10⁻³. It is also found that isothermal heating of walls enhances the heat distribution and thermal mixing. Overall, it is shown that heatlines provide suitable guideline on thermal management in porous right-angled triangular enclosures with various heating strategies.     

Analysis of entropy generation during natural convection in tilted triangular enclosures with various base angles

ABSTRACT

This paper presents a study of entropy generation during natural convection in a triangular enclosure with various configurations (cases 1 and 2 symmetric about Y-axis, and case 3 symmetric about X-axis) for the linearly heated inclined walls. The detailed analysis and comparison for the various base angles (φ = 45° and 60°) of the triangular enclosures have been carried out for Pr = 0.015 ? 1,000 and Ra = 10³ ? 10⁵. The results show that, case 3 configuration with the tilt angle φ = 60° may be the optimal shape based on the minimum total entropy generation (S_total) with the high heat transfer rate at Ra = 10⁵, irrespective of Pr.     

Effect of Undulations on the Natural Convection Heat Transfer and Entropy Generation Inside a Porous Right-Angled Triangular Enclosure

In the present study, natural convection in a two-dimensional porous right-angled triangular enclosure with one wavy wall is studied numerically. Three cases with one, two, and three undulations on the left wall are studied in this analysis. The stream function-vorticity equations are solved using finite-difference technique and a structured nonorthogonal body-fitted mesh is used for computations. The effect of Rayleigh number (Ra = 10³–10⁶), Darcy number (Da = 10^?4–10^?2) and undulations on the heat transfer, fluid flow, and entropy generation is investigated. It is found that average Nusselt number increases with Darcy number and number of undulations present on the left wall at fixed Darcy number.     

Analysis of entropy generation during natural convection in porous enclosures with curved surfaces

The natural convection is analyzed via the entropy generation approach in the differentially heated, porous enclosures with curved (concave or convex) vertical walls. The numerical simulations have been carried out for various fluids (Prandtl number: Pr_m?=?0.015, 0.7, and 7.2) at various permeabilities (Darcy numbers: 10^?5?≤?Da_m?≤?10^?2) for a high value of Rayleigh number (Ra_m?=?10⁶). The finite element method is employed to solve the governing equations and that is further used to calculate the entropy generation and average Nusselt number. The detailed spatial distributions of S_θ and S_ψ are analyzed for all the wall curvatures. Overall, the case with the highly concave surfaces (case 3) is the optimal case at low Da_m, whereas the cases with the less convex surfaces (cases 1 and 2) are the most efficient cases at high Da_m.     

Natural convection in a square cavity filled with a porous medium: Effects of various thermal boundary conditions

Natural convection flows in a square cavity filled with a porous matrix has been studied numerically using penalty finite element method for uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls. Darcy–Forchheimer model is used to simulate the momentum transfer in the porous medium. The numerical procedure is adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁶, Darcy number Da, 10⁻⁵ ⩽ Da ⩽ 10⁻³, and Prandtl number Pr, 0.71 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous thermal boundary conditions. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. It has been found that the heat transfer is primarily due to conduction for Da ⩽ 10⁻⁵ irrespective of Ra and Pr. The conductive heat transfer regime as a function of Ra has also been reported for Da ⩾ 10⁻⁴. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes the power law correlations between average Nusselt number and Rayleigh numbers are presented.     

Finite element based heatline approach to study mixed convection in a porous square cavity with various wall thermal boundary conditions

A penalty finite element method based simulation is performed to analyze the influence of various walls thermal boundary conditions on mixed convection lid driven flows in a square cavity filled with porous medium. The relevant parameters in the present study are Darcy number (Da = 10^?5 ? 10^?3), Grashof number (Gr = 10³ ? 10⁵), Prandtl number (Pr = 0.7–7.2), and Reynolds number (Re = 1–10²). Heatline approach of visualizing heat flow is implemented to gain a complete understanding of complex heat flow patterns. Patterns of heatlines and streamlines are qualitatively similar near the core for convection dominant flow for Da = 10^?3. Symmetric distribution in heatlines, similar to streamlines is observed irrespective of Da at higher Gr in natural convection dominant regime corresponding to smaller values of Re. A single circulation cell in heatlines, similar to streamlines is observed at Da = 10^?3 for forced convection dominance and heatlines are found to emanate from a large portion on the bottom wall illustrating enhanced heat flow for Re = 100. Multiple circulation cells in heatlines are observed at higher Da and Gr for Pr = 0.7 and 7.2. The heat transfer rates along the walls are illustrated by the local Nusselt number distribution based on gradients of heatfunctions. Wavy distribution in heat transfer rates is observed with Da ? 10^?4 for non-uniformly heated walls primarily in natural convection dominant regime. In general, exponential variation of average Nusselt numbers with Grashof number is found except the cases where the side walls are linearly heated. Overall, heatlines are found to be a powerful tool to analyze heat transport within the cavity and also a suitable guideline on explaining the Nusselt number variations.     

A Peclet number based analysis of mixed convection for lid-driven porous square cavities with various heating of bottom wall

Mixed convection flows in a lid-driven square cavity filled with porous medium are studied numerically using penalty finite element analysis for uniform and non-uniform heating of bottom wall. The relevant parameters in the present study are Darcy number (Da = 10^− 5− 10^− 5), Grashof number (Gr = 10³− 10⁵), Prandtl number (Pr = 0.026−10) and Reynolds number (Re = 1−10²). The influence of convection is analyzed with Peclet number (Pe = Re.Pr). It is observed that the temperature profiles are symmetric for low values of Pe or Pr even in the presence of asymmetric flow fields irrespective of Da. The flow distribution affects significantly temperature distributions at high Pe irrespective of Da. Effect of Peclet numbers have been further investigated for both natural convection and forced convection dominant regimes at high Da. Strong coupling between flow fields and temperature are observed at high Pe. It is interesting to observe that large isothermal mixing zone at Pr = 10 reduces the overall flow strength compared to Pr = 0.026 case. Local Nusselt numbers show almost uniform and low values for low Peclet numbers and localized enhanced heat transfer rates are observed for high Peclet numbers at Da = 10^− 3.     

Entropy Generation Due to Natural Convection in a Partially Open Cavity with a Thin Heat Source Subjected to a Nanofluid

This article presents a numerical study of natural convection cooling of a heat source mounted inside the cavity, with special attention being paid to entropy generation. The right vertical wall is partially open and is subjected to copper–water nanofluid at a constant low temperature and pressure, while the other boundaries are assumed to be adiabatic. The governing equations have been solved using the finite volume approach, using SIMPLE algorithm on the collocated arrangement. The study has been carried out for a Rayleigh number in the range 10³ < Ra < 10⁶, and for solid volume fraction 0 <? <0.05. In order to investigate the effect of the heat source and open boundary location, six different configurations are considered. The effects of Rayleigh numbers, heat source and open boundary locations on the streamlines, isotherms, local entropy generation, Nusselt number, and total entropy generation are investigated. The results indicate that when open boundary is located up, the fluid flow augments and hence the heat transfer and Nusselt number increase and total entropy generation decreases.     

Analysis of distributed thermal management policy for energy-efficient processing of materials by natural convection

Natural convection is the governing phenomena in many material processing applications. The conventional method of uniform heating at the bottom wall of an enclosure may result in inadequate thermal mixing and poor temperature distribution leading energy wastage. In this work, an alternative, energy-efficient method of distributed heating of the cavity is studied and compared with the isothermal bottom wall heating case in enhancing the thermal mixing and improving the temperature distribution in the cavity. Steady laminar natural convection of various fluids of industrial importance (Pr = 0.015, 07, 10, 1000) in the range of Ra = 10³–10⁵ is studied in a differentially heated cavity and in two cases of discretely heated square cavities. Detailed analysis is carried out by visualizing the heat flow by heatlines. The thermal mixing and temperature uniformity in each case are quantified in terms of cup-mixing temperature and root-mean square deviation (RMSD), respectively. It is found that thermal management policy of distributed heating significantly influences the thermal mixing and temperature uniformity in the enclosures. In a case with multiple discrete heat sources, a remarkable uniformity in temperature across the cavity is achieved with moderate thermal mixing.     

Lattice Boltzmann Simulation of Natural Convection in an Inclined Square Cavity with Spatial Temperature Variation

In this paper, natural convection heat transfer in an inclined square cavity filled with pure air (Pr = 0.71) was numerically analyzed with the lattice Boltzmann method. The heat source element is symmetrically embedded over the center of the bottom wall, and its temperature varies sinusoidally along the length. The top and the rest part of the bottom wall are adiabatic while the sidewalls are fixed at a low temperature. The influences of heat source length, inclination angle, and Rayleigh number (Ra) on flow and heat transfer were investigated. The Nusselt number (Nu) distributions on the heat source surface, the streamlines, and the isotherms were presented. The results show that the inclination angle and heat source length have a significant impact on the flow and temperature fields and the heat transfer performance at high Rayleigh numbers. In addition, the average Nu firstly increases with γ and reaches a local maximum at around γ = 45°, then decreases with increasing γ and reaches minimum at γ = 180° in the cavity with ? = 0.4. Similar behaviors are observed for ? = 0.2 at Ra = 10⁴. Moreover, nonuniform heating produces a significant different type of average Nu and two local minimum average Nu values are observed at around γ = 45° and γ = 180° for Ra = 10⁵ in the cavity with ? = 0.2.     

Numerical Optimization for Nanofluid Flow in Microchannels Using Entropy Generation Minimization

This study presents the numerical simulation of the three-dimensional incompressible steady and laminar fluid flow of a trapezoidal microchannel heat sink using nanofluids as a cooling fluid. Navier–Stokes equations with a conjugate energy equation are discretized by the finite-volume method. Numerical computations are performed for inlet velocity (W _in = 4 m/s, 6 m/s, and 10 m/s), hydraulic diameter D _h  = 106.66 μm, and heat flux (q″ = 200 kW/m². Numerical optimization is demonstrated as a trapezoidal microchannel heat sink design which uses the combination of a full factorial design and the genetic algorithm method. Three optimal design variables represent the ratio of upper width and lower width of the microchannel (1.2 ≤ α ≤ 3.6), the ratio of the height of the microchannel to the difference between the upper and lower width of the microchannel (0.5 ≤ β ≤ 1.866), and the volume fraction (0 ≤ φ ≤ 4%). The dimensionless entropy generation rate of a trapezoidal microchannel is minimized for fixed heat flux and inlet velocity. Numerical results for the system dimensionless entropy generation rate show that the system dimensionless friction entropy generation rate increases with Reynolds number; on the contrary, the higher the Reynolds number, the lower the system dimensionless thermal entropy generation rate. The results below show that the two-phase model gives higher enhancement than the single-phase model assuming a steadily developing laminar flow.     

On the Validity of the Isothermal Model for Natural Convection Flow around Blocks Submitted to Volumetric Heat Generation in a Horizontal Channel

This work presents numerical results of natural convection in a horizontal channel provided with heating blocks periodically mounted on its lower adiabatic surface. The upper surface of the channel is maintained cold at a constant temperature. The parameters of the study are the ratio of solid blocks to fluid thermal conductivities (0.1 ≤ k* = k _s /k _a ≤ 200), the Rayleigh number (10⁴ ≤ Ra ≤ 10⁷), and the relative blocks height (1/8 ≤ B ≤ 1/2). Two models are considered in this study depending on whether the blocks are submitted to uniform heat generation (model 1), or maintained isothermal (model 2) at the average temperature calculated using model 1. The effect of the thermal conductivities ratio and the other controlling parameters on the validity of the isothermal model is examined. It is found that when multiple steady solutions are possible, some of the solutions obtained with the isothermal model may not reproduce the results of the model with blocks submitted to volumetric heat generation, even at very large conductivities ratio.