Laminar and turbulent free convection in a composite enclosure

Turbulent natural convection in a two-dimensional horizontal composite square cavity, isothermally heated at the left side and cooled from the opposing surface, is numerically analyzed using the finite volume method. The composite square cavity is formed by three distinct regions, namely, clear, porous and solid region. The development of a numerical tool able to treat all these regions as one computational domain is of advantage for engineering design and analysis of passive thermal control systems. Governing equations are written in terms of primitive variables and are recast into a general form. It was found that the fluid begins to permeate the porous medium for values of Ra greater than 10⁶. Nusselt number values show that for the range of Ra analyzed there is no significant variation between the laminar and turbulent model solution. When comparing the effects of Ra, k_s/k_f and Da on Nu, results indicate that the solid phase properties have a greater influence in enhancing the overall heat transferred trough the cavity.     

Turbulent free convection in a porous square cavity using the thermal equilibirum model

This work investigates the influence of porosity and thermal conductivity ratio on the Nusselt number of a cavity filed with a fluid saturated porous substrate. The flow regime considered intra-pore turbulence and a macroscopic k-ε model was applied. Heat transfer across the cavity assumed the hypothesis of thermal equilibrium between the solid and the fluid phases. Transport equations were discretized using the control-volume method and the system of algebraic equations was relaxed via the SIMPLE algorithm. Results showed that when using the one energy equation model under the turbulent regime, simulated with a High Reynolds turbulence model, the cavity Nusselt number is reduced for higher values of the ratio k_s/k_f as well as when the material porosity is increased. In both cases, conduction thorough the solid material becomes of a greater importance when compared with the overall transport that includes both convection and conduction mechanisms across the medium.     

Turbulent natural convection in a composite annulus using a novel numerical scheme and the thermal nonequilibrium hypothesis

Composite cavities formed by a clear space, a layer of porous material, and a solid plate can be engineered for controlling the overall heat transfer across the enclosure. Using different layer dimensions, as well as distinct porous and solid materials, the value of the cavity Nusselt number can be modified with regard to traditional Nu?∝?Raⁿ behavior, which is encountered either in completely empty cavities or in cavities fully fitted with porous materials. Motivated by such novel application, this work presents a study about turbulent natural convection in a composite concentric annulus. The annulus is assumed to be two-dimensional and positioned horizontally, being isothermally heated at the inner cylinder and cooled from the outer surface. Laminar flow is considered in addition to the turbulent regime, which is handled via the standard k–ε model. The wall treatment applied is the High Reynolds approach. The Two-Energy Equation Model (2EEM) is utilized in the porous section. The transport equations are discretized using the control-volume method. The system of algebraic equations is relaxed via the Semi Implicit Pressure-Linked Equations (SIMPLE) algorithm. A new numerical methodology is applied to resolve all three layers in a single computational domain by establishing two temperature sets, defined according to the location inside the composite structure. Nusselt number behavior shows that for Rayleigh number up to 10⁴ there is no significant variation between the laminar and turbulence models, although the differences increase when the flow gets more intense and/or the porous material becomes more permeable. When comparing the effects of Rayleigh number, Darcy number, porosity, and thermal conductivity ratio between the solid and the fluid on Nu, the results indicate that the solid-phase properties have a greater influence in enhancing the overall heat transferred through the cavity.     

Laminar heat transfer in a porous channel simulated with a two-energy equation model

Laminar heat transfer in a porous channel is numerically simulated with a two-energy equation model for conduction and convection. Macroscopic equations for continuity, momentum and energy transport for the fluid and solid phases are presented. The numerical methodology employed is based on the control volume approach with a boundary-fitted non-orthogonal coordinate system. Fully developed forced convection in a porous channel bounded by parallel plates is considered. Solutions for Nusselt numbers along the channel are presented for laminar flows. Results simulate the effects Reynolds number Re, porosity, particle size and solid-to-fluid thermal conductivity ratio on Nusselt sumber, Nu, which is defined for both the solid and fluid phases. High Re, low porosities, low particle diameters and low thermal conductivity ratios promote thermal equilibrium between phases leading to higher values of Nu.     

Natural convection heat transfer from a horizontal cylinder embedded in a porous medium

This paper presents the results of an experimental investigation of heat transfer by natural convection from a horizontal cylinder embedded in porous media consisting of randomly packed glass spheres saturated by either water or silicone oil. It is shown that the overall range of the Rayleigh number, Ra, can be divided into two subregions, called ‘low’ and ‘high’, in each of which the Nusselt number, Nu, behaves differently. It is demonstrated that the low Ra region corresponds to Darey flow and the high to Forchheimer flow. Correlation equations for Nu for the Darcy regime are presented that account for viscous dissipation, and others for the Forchheimer regime that involve the first and second Forchheimer coefficients. The variation of properties with temperature and the wall effect on porosity (and consequently on heat transfer) are considered. The paper includes information concerning the resistance to flow in porous media that was obtained in conjunction with the heat transfer study.     

Bounds on natural convective heat transfer in a porous layer with fixed heat flux

Using the background field variational method, bounds on natural convective heat transfer in a porous layer heated from below with fixed heat flux are derived from the primitive equations. The enhancement of heat transfer beyond the minimal conduction value (the Nusselt number Nu) is bounded in terms of the non-dimensional forcing scale set by the ‘effective’ Rayleigh number (Rˆ) according to Nu ≤ 0.354Rˆ^1/2 and in terms of the conventional Rayleigh number (Ra) defined by the temperature drop across the layer according to Nu ≤ 0.125Ra. It is presented that fixing the heat flux at the boundaries does not change the linear dependence between Nusselt number and Rayleigh number at high Rayleigh number region.     

Visualization of Heat Transport during Natural Convection Within Porous Triangular Cavities via Heatline Approach

In this article, natural convection in a porous triangular cavity has been analyzed. Bejan's heatlines concept has been used for visualization of heat transfer. Penalty finite-element method with biquadratic elements is used to solve the nondimensional governing equations for the triangular cavity involving hot inclined walls and cold top wall. The numerical solutions are studied in terms of isotherms, streamlines, heatlines, and local and average Nusselt numbers for a wide range of parameters Da (10^?5–10^?3), Pr (0.015–1000), and Ra (Ra = 10³–5 × 10⁵). For low Darcy number (Da = 10^?5), the heat transfer occurs due to conduction as the heatlines are smooth and orthogonal to the isotherms. As the Rayleigh number increases, conduction dominant mode changes into convection dominant mode for Da = 10^?3, and the critical Rayleigh number corresponding to the on-set of convection is obtained. Distribution of heatlines illustrate that most of the heat transport for a low Darcy number (Da = 10^?5) occurs from the top region of hot inclined walls to the cold top wall, whereas heat transfer is more from the bottom region of hot inclined walls to the cold top wall for a high Darcy number (Da = 10^?3). Interesting features of streamlines and heatlines are discussed for lower and higher Prandtl numbers. Heat transfer analysis is obtained in terms of local and average Nusselt numbers (Nu _l , Nu _t ) and the local and average Nusselt numbers are found to be correlated with heatline patterns within the cavity.     

Numerical study on coupled fluid flow and heat transfer process in parabolic trough solar collector tube

A unified two-dimensional numerical model was developed for the coupled heat transfer process in parabolic solar collector tube, which includes nature convection, forced convection, heat conduction and fluid-solid conjugate problem. The effects of Rayleigh number (Ra), tube diameter ratio and thermal conductivity of the tube wall on the heat transfer and fluid flow performance were numerically analyzed. The distributions of flow field, temperature field, local Nu and local temperature gradient were examined. The results show that when Ra is larger than 10⁵, the effects of nature convection must be taken into account. With the increase of tube diameter ratio, the Nusselt number in inner tube (Nu₁) increases and the Nusselt number in annuli space (Nu₂) decreases. With the increase of tube wall thermal conductivity, Nu₁ decreases and Nu₂ increases. When thermal conductivity is larger than 200 W/(m K), it would have little effects on Nu and average temperatures. Due to the effect of the nature convection, along the circumferential direction (from top to down), the temperature in the cross-section decreases and the temperature gradient on inner tube surface increases at first. Then, the temperature and temperature gradients would present a converse variation at θ near π. The local Nu on inner tube outer surface increases along circumferential direction until it reaches a maximum value then it decreases again.     

Unsteady analysis of natural convection in a carbon nanotube-water filled cavity with an inclined heater

ABSTRACT

A finite element solution has been performed in this work to solve unsteady governing equations of natural convection in a carbon nanotube–water-filled cavity with inclined heater. The temperature of ceiling and left vertical walls is lower than that of the heater while the other walls are adiabatic. The main governing parameters are nanofluid volume fraction and Rayleigh number (Ra). It is found that the heat transfer rate shows different trends based on Rayleigh number and it increases with increase in nanoparticle volume fraction. It has been estimated that average Nusselt number (Nu) is dependent onRa through power regression models with strong positive linear correlation between ln (Nu) and ln (Ra). In particular, for the maximum time, when the solid volume fraction is varied from 0 to 0.1 the dependence between average Nu and linear Ra, on a logarithmic scale, is very high.     

Visualization of heat flow due to natural convection within triangular cavities using Bejan’s heatline concept

Natural convection and flow circulation within a cavity has received significant attention in recent times. The wide range of applicability of flow inside a cavity (food processing industries, molten metal industries, etc.) requires thorough understanding for cost efficient processes. This paper is based on comprehensive analysis of heat flow pattern using Bejan’s heatline concept. The key parameters for our study are the Prandtl number, Rayleigh number and Nusselt number. The values of Prandtl number (0.015, 0.026, 0.7 and 1000) have been chosen based on wide range of applicability. The Rayleigh number has been varied from 10² to 10⁵. Interesting results were obtained. For low Rayleigh number, it is found that the heatlines are smooth and perfectly normal to the isotherms indicating the dominance of conduction. But as Ra increases, flow slowly becomes convection dominant. It is also observed that multiple secondary circulations are formed for fluids with low Pr whereas these features are absent in higher Pr fluids. Multiple circulation cells for smaller Pr also correspond multiple cells of heatlines which illustrate less thermal transport from hot wall. On the other hand, the dense heatlines at bottom wall display enhanced heat transport for larger Pr. Further, local heat transfer (Nu_l, Nu_t) are explained based on heatlines. The comprehensive analysis is concluded with the average Nusselt number plots. A correlation for average heat transfer rate and Ra has been developed and the range of Rayleigh number is also found, to depict the conduction dominant heat transfer.     

Analytical investigation of heat transfer enhancement in a channel partially filled with a porous material under local thermal non-equilibrium condition

Forced convection through a channel partially filled with a porous medium is investigated analytically in the present work. Thermally developed condition is considered and the local thermal non-equilibrium model is utilized to obtain the exact solutions of both fluid and solid temperature fields for flow inside the porous material as well as for flow in the clear region. Nusselt number is obtained in terms of the porous insert thickness (S), porosity (?) as well as pertinent parameters such as thermal conductivity ratio (k), Biot number (Bi), and Darcy number (Da). The values of S by which the temperature difference between the two phases approach to zero, for different values of Bi, k, and Da number are obtained. It is found that three mechanisms affect the Nu number i: clear fluid conduction ii: internal heat exchange in the porous medium iii: channeling effect in the clear flow. The value of S, which yields the highest Nu number is found to vary linearly from 0.8 to 0.97 as the value of Da decreases from 10⁻³ to 10⁻⁷. At the expense of reasonable pressure drop the optimum thickness of porous material in order to enhance the heat transfer rate is found S = 0.8.     

Local thermal nonequilibrium conjugate natural convection heat transfer of nanofluids in a cavity partially filled with porous media using Buongiorno’s model

The natural convection heat transfer in a cavity filled with three layers of solid, porous medium, and free fluid is addressed. The porous medium and free fluid layers are filled with a nanofluid. The porous layer is modeled using the local thermal nonequilibrium (LTNE) model, considering the temperature difference between the solid porous matrix and the nanofluid phases. The nanofluid is modeled using the Buongiorno’s model incorporating the thermophoresis and Brownian motion effects. The governing equations are transformed into a set of nondimensional partial differential equations, and then solved using finite element method in a nonuniform grid. The effects of various nondimensional parameters are discussed. The results showed that the Brownian motion and thermophoresis effects result in significant concentration gradients of nanoparticles in the porous and free fluid layers. The increase in Rayleigh (Ra), Darcy (Da), the thermal conductivity ratios for the solid wall and solid porous matrix, i.e., K_r and R_k, enhanced the average Nusselt number. The increase in the convection interaction heat transfer parameter between the solid porous matrix and the nanofluid in the pores (H) increases the average Nusselt number in the solid porous matrix but decreases the average Nusselt number in the nanofluid phase of the porous layer.     

Double-diffusive laminar free convection in a porous cavity simulated with the two-energy equation model

Numerical simulations for laminar double-diffusive free convection in a porous square cavity using the Thermal Non-Equilibrium Model were presented. Vertical surfaces were maintained at constant temperature and concentration whereas horizontal walls were kept insulated. The cavity was filled with a rigid and isotropic porous matrix, which was saturated with an incompressible fluid. Transport equations were discretized by means of the control volume method leading to a coupled algebraic equation set that was solved via the SIMPLE method. Results pointed that both Nu_w and Sh_w are dependent on porosity ϕ and on the thermal conductivity ratio k_s/k_f. Nu_w decreases as ϕ decreases or k_s/k_f increases due to enhancement of conduction transport across the cavity. On the other hand, Sh_w and wall mass flux increases as porosity decreases or k_s/k_f increases. Such dependence of Sh_w arises from the intensification of recirculating motion in the cavity as ϕ is reduced or k_s/k_f is of a higher value, which affects heat exchange between phases and, consequently, wall mass fluxes. Finally, this study shows that both average Nusselt and Sherwood numbers diverge from published correlation when k_s/k_f > 1 for same Da value.     

Jet Impingement Cooling of a Finite Thickness Solid Layer Conjugated with Porous Medium

The present study numerically investigates the mixed convection-conduction problem of impingement cooling of a finite thickness solid wall conjugated with a porous medium. The heat transfer is investigated over wide ranges of governing parameters: Rayleigh number (50 ≤ Ra ≤ 100), Péclet number (1 ≤ Pe ≤ 10³), solid wall thickness (0.05 ≤ H ≤ 0.25), heat transfer coefficient parameter (0.1 ≤ H _v  ≤ 10), and solid wall thermal conductivity (0.24 ≤ k _w  ≤ 240). The results shows that the total average Nusselt numbers can be increased with the decrease in H and increase in k _w and H _v . Opposing mixed convection is demonstrated to occur and the minimum value of average Nusselt numbers for fluid are found except for a low value of k _w .     

Natural convection heat transfer in horizontal and vertical closed narrow enclosures with heated rectangular finned base plate

Natural convection heat transfer and fluid flow characteristics in horizontal and vertical narrow enclosures with heated rectangular finned base plate have been experimentally investigated at a wide range of Rayleigh number (Ra) for different fin spacings and fin lengths. Quantitative comparisons of finned surface effectiveness (ε) and heat transfer rate between horizontal and vertical enclosures have been reported. In comparison with enclosure of a bare base plate, insertion of heat conducting fins always enhances heat transfer rate. Optimization of fin-array geometry has been addressed. The results gave an optimum fin spacing at which Nusselt number (Nu_H) and finned surface effectiveness (ε) are maximum. It has been found that: (1) increasing fin length increases Nu_H and ε; (2) increasing Ra increasesNu_H for any fin-array geometries and (3) for any fin-array geometry and at Ra > 10,000, increasing Ra decreases ε while for fin-array geometries of large fin spacing and at Ra < 10,000, increasing Ra increases ε. Useful design guidelines have been suggested. Correlations of Nu_H have been developed for horizontal and vertical enclosures. Correlations predictions have been compared with previous data and good agreement was found.     

Use of porous-continuum and continuum models for determining the permeability of porous cavities under turbulent free convection

The aim of this work is to estimate the permeability of porous enclosures for numerical solutions of turbulent natural convection in a square cavity. The motivation is that available permeability correlations were proposed based on force rather than natural convection through permeable media. Although commonly seen as a medium property, permeability is measured with a flow through the permeable structure and, as such, its value may carry a flow type dependency. Here, it is assumed that a fixed amount of a solid conducting material is distributed within the cavity and two mathematical models are used and compared when calculating the cavity Nusselt number. First, a porous-continuum model is considered based on the assumption that the solid and the fluid phases are observed as a single medium, over which volume- and time-averaged transport equations apply. Second, a continuum model is used to solve local momentum and energy equations, in both the solid and void spaces, through a conjugate heat transfer solution. The average Nusselt number at the hot wall obtained from the porous-continuum model for several Darcy numbers are compared with those obtained with the continuum model using up to N?=?1,024 obstacles within the cavity. When comparing the two methodologies, this study shows that the average Nusselt number calculated by each approach differs by as much as 32% when the number of obstacles N is increased to 1,024. Based on that, an adjustment on the used correlation for calculating the porous medium permeability is proposed to match the Nusselt numbers calculated with the two models. Results indicate that the use of the new correlation gives results for Nu that differ less than about 4% for the range 4?N?相似文献   

Forced convection heat transfer in microchannel heat sinks

This paper presents an analysis of forced convection heat transfer in microchannel heat sinks for electronic system cooling. In view of the small dimensions of the microstructures, the microchannel is modeled as a fluid-saturated porous medium. Numerical solutions are obtained based on the Forchheimer–Brinkman-extended Darcy equation for the fluid flow and the two-equation model for heat transfer between the solid and fluid phases. The velocity field in the microchannel is first solved by a finite-difference scheme, and then the energy equations governing the solid and fluid phases are solved simultaneously for the temperature distributions. Also, analytical expressions for the velocity and temperature profiles are presented for a simpler flow model, i.e., the Brinkman-extended Darcy model. This work attempts to perform a systematic study on the effects of major parameters on the flow and heat transfer characteristics of forced convection in the microchannel heat sink. The governing parameters of engineering importance include the channel aspect ratio (α_s), inertial force parameter (Γ), porosity (ε), and the effective thermal conductivity ratio (k_r). The velocity profiles of the fluid in the microchannel, the temperature distributions of the solid and fluid phases, and the overall Nusselt number are illustrated for various values of the problem parameters. It is found that the fluid inertia force alters noticeably the dimensionless velocity distribution and the fluid temperature distribution, while the solid temperature distribution is almost insensitive to the fluid inertia. Moreover, the overall Nusselt number increases with increasing the values of α_s and ε, while it decreases with increasing k_r.     

Conjugate heat transfer in inclined open shallow cavities

Conjugate heat transfer in inclined open shallow cavities has been numerically studied. A thick wall facing the opening is heated by a constant heat flux, sides perpendicular to the heated wall are insulated and the opening is in contact with a fluid at constant temperature and pressure. Conjugate heat transfer by conduction and natural convection is studied by numerically solving equations of mass, momentum and energy. The governing parameters were: Rayleigh numbers, Ra from 10⁶ to 10¹², conductivity ratio, k_r from 1 to 60, cavity aspect ratio, A=H/L from 1 to 0.125, dimensionless wall thickness, ?/L from 0.05 to 0.20 and the inclination angle, ? from 0° to 45° from the horizontal. Isotherms and streamlines are produced, heat and mass transfer is calculated. It is found that volume flow rate, is an increasing function of Ra, A, ?/L, ?, and a decreasing function of k_r. Heat transfer, Nu is an increasing function of Ra, ?/L, and a decreasing function of k_r. A mixed pattern is found with respect to A and ?. In the former, Nu is an increasing function of the aspect ratio up to a critical Rayleigh number, above which the relationship changes and it becomes a decreasing function of A. In the latter case, Nu is a decreasing function at low Raleigh numbers and an increasing one at high Rayleigh numbers.     

Hydromagnetic effect on mixed convection in a lid-driven cavity with sinusoidal corrugated bottom surface

The present numerical simulation is conducted to analyze the mixed convection flow and heat transfer in a lid-driven cavity with sinusoidal wavy bottom surface in presence of transverse magnetic field. The enclosure is saturated with electrically conducting fluid. The cavity vertical walls are insulated while the wavy bottom surface is maintained at a uniform temperature higher than the top lid. In addition, the transport equations are solved by using the finite element formulation based on the Galerkin method of weighted residuals. The implications of Reynolds number (Re), Hartmann number (Ha) and number of undulations (λ) on the flow structure and heat transfer characteristics are investigated in detail while, Prandtl number (Pr) and Rayleigh number (Ra) are considered fixed. The trend of the local heat transfer is found to follow a wavy pattern. The results of this investigation illustrate that the average Nusselt number (Nu) at the heated surface increases with an increase of the number of waves as well as the Reynolds number, while decreases with increasing Hartmann number.     

Analysis of microchannel heat sinks for electronics cooling

This paper presents an analytical and numerical study on the heat transfer characteristics of forced convection across a microchannel heat sink. Two analytical approaches are used: the porous medium model and the fin approach. In the porous medium approach, the modified Darcy equation for the fluid and the two-equation model for heat transfer between the solid and fluid phases are employed. Firstly, the effects of channel aspect ratio (α_s) and effective thermal conductivity ratio (k?) on the overall Nusselt number of the heat sink are studied in detail. The predictions from the two approaches both show that the overall Nusselt number (Nu) increases as α_s is increased and decreases with increasing k?. However, the results also reveal that there exists significant difference between the two approaches for both the temperature distributions and overall Nusselt numbers, and the discrepancy becomes larger as either α_s or k? is increased. It is suggested that this discrepancy can be attributed to the indispensable assumption of uniform fluid temperature in the direction normal to the coolant flow invoked in the fin approach. The effect of porosity (ε) on the thermal performance of the microchannel is subsequently examined. It is found that whereas the porous medium model predicts the existence of an optimal porosity for the microchannel heat sink, the fin approach predicts that the heat transfer capability of the heat sink increases monotonically with the porosity. The effect of turbulent heat transfer within the microchannel is next studied, and it is found that turbulent heat transfer results in a decreased optimal porosity in comparison with that for the laminar flow. A new concept of microchannel cooling in combination with microheat pipes is proposed, and the enhancement in heat transfer due to the heat pipes is estimated. Finally, two-dimensional numerical calculations are conducted for both constant heat flux and constant wall temperature conditions to check the accuracy of analytical solutions and to examine the effect of different boundary conditions on the overall heat transfer.