Steady-state Free Convection Research Articles

Investigating the thermodynamic optimization of naturally convective flow in a corrugated enclosure: The influence of gap spacing and orientation of split baffles

The natural convection in cavities is frequently used in fluid mechanics and heat transfer engineering, such as heat exchangers, electronics, solar collectors, and growing crystals. However, the physics of natural convection flow and heat transfer in cavities with split baffles is least understood. The fundamental aim of this research is to investigate the impact of heated split baffles positioned at various locations on steady-state free convection within a sinusoidal corrugated star cavity. In this model configuration, the outer wavy enclosure is maintained at a constant temperature of Tc, while the inner split baffles are heated at a constant temperature of Th. The finite element method is employed to discretize and solve the governing equations describing the fluid flow and heat transfer within the enclosure. This numerical approach aimed to analyze the effects of baffle inclination angles, baffle spacing, Rayleigh number on the fluid dynamics and convective thermal transport characteristics. The variation in velocity and temperature profile is illustrated through the streamlines and isotherm contours. Moreover, the numerical result is displayed in term Nuavg of the heat transfer, which are analyzed for inside space of baffles and angles of the baffle (θ=00,450,900). The key finding demonstrates that increasing the Rayleigh numbers and the different positions (up, central, down) of inner vertical split baffles enhances the magnitude of the velocity by 88.1%, 85.9 and 89.6% respectively. Furthermore, for the inner rectangular split baffle angles of 0°,45°, and 90°, and within the Rayleigh number range of 104 to 106, the Nuavg exhibits significant variations, with maximum increases of 71.9% , 46.7% and a subsequent decrease of 45.9%.

Exploring the Limits of Empirical Correlations for the Design of Energy Systems With Complex Fluids: Liquid Sulfur Thermal Energy Storage as a Case Study

Abstract Sustainable energy technologies often use fluids with complex properties. As an example, sulfur is a promising fluid for use in thermal energy storage (TES) systems, with highly nonlinear thermophysical properties. The viscosity of liquid-phase sulfur varies by four orders of magnitude due to polymerization of sulfur rings between 400 K and 500 K, followed by depolymerization of long rigid chains, and a decrease in viscosity, as temperature increases. These properties may compromise the accuracy of long-established empirical correlations in the design of TES systems. This work uses computational fluid dynamics to compute steady-state free convection heat transfer coefficients of sulfur in concentric cylinders at temperatures between 400 K and 600 K. The results show that uneven distributions of high and low-viscosity sulfur in the system cause variations in flow patterns and highly nonlinear heat transfer coefficients as temperature gradients increase. As a result, existing empirical correlations for describing system performance become inaccurate. Comparisons of simulation results to predictions from well-established literature correlations show that deviations may surpass 50%. Nusselt versus Rayleigh number correlations for heat transfer are significantly affected by the loss of self-similarity. The analysis proves that existing correlations are not able to capture the complex properties of sulfur in this temperature range, suggesting that alternative modeling techniques are needed for the design and optimization of sulfur TES systems. These challenges are unlikely to be limited to sulfur as a working fluid or TES but will appear in a range of energy systems.

Variable thermal conductivity and thermophoretic aspects of free convection flow of a micropolar fluid due to a permeable cone: Heat source/sink

Boundary layer analysis is introduced for providing a numerical treatment of steady state free convection flow of non-Newtonian fluid of micropolar type through an isothermal vertical permeable cone under impacts of variable thermal conductivity and thermophoretic force. Both heat generation or absorption and viscous dissipation are considered. The resulting altered boundary layer flow equations are numerically tackled by the help of the implicit finite difference approach. Two states are addressed: one corresponds to constant thermal conductivity, and the other is variable thermal conductivity. The skin friction, couple stress coefficients fluctuations, and Nusselt number for miscellaneous significant sundry factors are presented graphically and argued. From the calculated outcomes, the coefficient of couple stress reduces for larger values of vortex viscosity parameter R, while the reverse trend is noticed through the surface temperature exponent m factor. Augmentation in thermophoretic factor ω leads to the thermophoretic forces that encapsulates the mass transmission. As a certain case, the system flow equations of a classical Newtonian liquid are gained, by dropping the micropolar factors consequences.

Steady State Free Convection through some Variable Fin Geometries

Thermal performances on some typical variable geometries were conducted in this work which can be reported into two phases. In the first phase of this work, fins of rectangular, cylindrical, and conical geometries having the same total surface area were studied. It is found that the cylindrical fin is the least efficient and least effective among the three. In the second phase, design modification in the form of grooves is made on cylindrical fins. Cutting grooves on cylindrical fins increase its surface area. The three grooved cylindrical fins are namely- fully grooved cylindrical fins, bottom half grooved cylindrical fins, and top half grooved cylindrical fins. All three grooved fins have the same volume. It was noticed hat, on cutting groove the on fin the quality of convective heat transfer, as well as the effectiveness, improves, but the efficiency degrades. Keywords: Finite Element Method, thermal steady state, fin effectiveness, fin efficiency, free convection

New application of MOL-PACT for simulating buoyancy convection of a copper-water nanofluid in a square enclosure containing an insulated obstacle

<abstract> <p>In this study, we have simulated transient and steady state free convection flow and heat transfer inside a square enclosure filled with a copper-water nanofluid of spherical shape nanoparticles following Tiwari-Das model. The cavity containing an insulated rectangular obstacle of height ranging from 0% to 50% of the cavity side-length. The vertical sides of the enclosure are kept at different temperatures, while the flat sides are assumed to be adiabatic as the obstacle. The combined method of lines/penalty-artificial compressibility technique (MOL-PACT) has been applied to solve the dimensional time dependent mathematical model after converting it into a non-dimensional structure. The combined method of lines/penalty-artificial compressibility technique is recently successfully applied to simulate free convection of MHD fluid in square enclosure with a localized heating. The extension of this promising technique for studying heat transfer of nanofluids is one of the objectives of this paper. Another objective of the study is to inspect the impact of several model parameters such as, the obstacle height, nanoparticles volume-fraction, nanoparticles radius and Rayleigh number on streamlines, temperature distribution and Nusselt number as an expression of heat transfer inside the enclosure. The results have been discussed and shown graphically. Comparisons with former results for related cases in the literature are made and reasonably good agreements are observed.</p> </abstract>

Numerical modeling for steady-state nanofluid free convection involving radiation through a wavy cavity with Lorentz forces

In this research, steady-state magnetized nanofluid flow under radiative wavy cavity is demonstarted. The porosity term is modeled by employing the non-Darcy’s theory of porosity. The working nanofluid is based on water-Al2O3 type nano-powder with the concnetration of 4%. The numerical modeling is estimated via control volume finite element (CVFE) technique. The results against the distinct emerging constraints namely magnetic force, Rayleigh number, porosity factor, shape factors, and shape of nanoparticles values on flow streamlines, average Nusselt number (Nuavg) and isotherms are sketched and reported indetail. The sequence of radition on nanoraticles is discussed by considering distinct nanoparticles forms. The results show that the fluid advection decays with the enhancing trend of the media porosity and the phenomenon of conduction dominates over the process of convection. The nanoparticle shape insignificantly influences the Nuavg for incrementing trend of magnetic and buoyancy forces.

Analysis of Heat Transfer in a Triangular Enclosure Filled with a Porous Medium Saturated with Magnetized Nanofluid Charged by an Exothermic Chemical Reaction

This paper intends to numerically study the steady-state free convection heat transfer in the presence of an exothermal chemical reaction governed by Arrhenius kinetics within a right-angled enclosure of triangular shape filled by porous media saturated with magnetized nanofluid. An approximation named as Darcy–Boussinesq approximation along with a nanofluid model mathematically propounded by Buongiorno has been implemented to model physical phenomenon representing fluid flow, heat transfer, and nanoparticle concentration. The mathematical equations in a dimensionless form describing the stream function for circulation of the fluid, the energy equation for heat, and nanoparticle volume fraction for concentration are solved using the finite difference method. The validity of the numerical procedure is established by comparing present results with the formerly available works in both statistical and graphical approaches. Streamlines, isotherms, and isoconcentrations are plotted and discussed for the various parametric regimes. The graphical description depicts that the average Nusselt and Sherwood numbers are the decreasing function of the Rayleigh number. The study revealed the accountable influence of model parameters such as thermophoresis and Brownian diffusion on the local Sherwood number, whereas a minimum impact on the local Nusselt number is observed.

Free convection of power-law fluids in enclosures with partially heating from bottom and symmetrical cooling from sides

Steady-state laminar free convection of power-law fluids in a square enclosure with partial heating from below and symmetrical cooling from sides has been investigated based on numerical simulations. The partially heating is conducted by a centrally heated heat source on the bottom wall for different values of normalised heat source length l/L (ranging from 0.2 to 0.8); power-law index n (from 0.6 to 1.8); nominal Rayleigh number (Ra) (from 103 to 106) and nominal Prandtl number Pr (from 10 to 103). An increase in the power-law index n leads to weakening of thermal advection owing to the augmentation of viscous resistance. This is reflected as an increment in the mean Nusselt number Nu‾ with decreasing value of n. Additionally, Nu‾ increases with an increase in Ra and l/L for shear thinning (i.e. n<1), Newtonian (i.e. n=1) and shear thickening (i.e. n>1) fluids. The observed influences of Ra,Pr,nandl/L on Nu‾ have been explained by physical and scaling arguments and parameterized by a correlation for the mean Nusselt number.

Numerical Study of Viscoelastic Micropolar Heat Transfer from a Vertical Cone for Thermal Polymer Coating

Abstract A mathematical model is developed to study laminar, nonlinear, non-isothermal, steady-state free convection boundary layer flow and heat transfer of a micropolar viscoelastic fluid from a vertical isothermal cone. The Eringen model and Jeffery’s viscoelastic model are combined to simulate the non-Newtonian characteristics of polymers, which constitutes a novelty of the present work. The transformed conservation equations for linear momentum, angular momentum and energy are solved numerically under physically viable boundary conditions using a finite difference scheme (Keller Box method). The effects of Deborah number (De), Eringen vortex viscosity parameter (R), ratio of relaxation to retardation times (λ), micro-inertia density parameter (B), Prandtl number (Pr) and dimensionless stream wise coordinate (ξ) on velocity, surface temperature and angular velocity in the boundary layer regime are evaluated. The computations show that with greater ratio of retardation to relaxation times, the linear and angular velocity are enhanced whereas temperature (and also thermal boundary layer thickness) is reduced. Greater values of the Eringen parameter decelerate both the linear velocity and micro-rotation values and enhance temperatures. Increasing Deborah number decelerates the linear flow and Nusselt number whereas it increases temperatures and boosts micro-rotation magnitudes. The study is relevant to non-Newtonian polymeric thermal coating processes.

Mathematical Modeling of Natural Convection in a Third-Grade Viscoelastic Micropolar Fluid from an Isothermal Inverted Cone

A mathematical model is developed to investigate the nonlinear, isothermal steady-state free convection boundary layer flow in an incompressible third-grade viscoelastic micropolar fluid from a vertical isothermal cone. The non-Newtonian fluid exhibits both viscoelastic and micro-structural characteristics and is representative of certain polymers. The transformed conservation equations for mass, linear momentum, angular momentum and heat are solved numerically subject to realistic boundary conditions using the second-order accurate implicit finite-difference Keller-box method. The numerical code is validated with previous studies and also with a Nakamura tridiagonal method. Detailed interpretation of the computations is included. The present simulations are of interest in chemical engineering systems and solvent and low density polymer materials processing.

New benchmark solutions for transient natural convection in partially porous annuli

Purpose – The purpose of this paper is to focus on the numerical analysis of transient free convection heat transfer in partially porous cylindrical domains. The authors analyze the dependence of velocity and temperature fields on the geometry, by analyzing transient flow behavior for different values of cavity aspect ratio and radii ratio; both inner and outer radius are assumed variable in order to not change the difference ro-ri. Moreover, several Darcy numbers have been considered. Design/methodology/approach – A dual time-stepping procedure based on the transient artificial compressibility version of the characteristic-based split algorithm has been adopted in order to solve the transient equations of the generalized model for heat and fluid flow through porous media. The present model has been validated against experimental data available in the scientific literature for two different problems, steady-state free convection in a porous annulus and transient natural convection in a porous cylinder, showing an excellent agreement. Findings – For vertically divided half porous cavities, with Rayleigh numbers equal to 3.4×106 for the 4:1 cavity and 3.4×105 for the 8:1 cavity, the numerical results show that transient oscillations tend to disappear in presence of cylindrical geometry, differently from what happens for rectangular one. The magnitude of this phenomenon increases with radii ratio; the porous layer also affects the stability of velocity and temperature fields, as oscillations tend to decrease in presence of a porous matrix with lower value of the Darcy number. Research limitations/implications – A proper analysis of partially porous annular cavities is fundamental for the correct estimation of Nusselt numbers, as the formulas provided for rectangular domains are not able to describe these problems. Practical implications – The proposed model represents a useful tool for the study of transient natural convection problems in porous and partially porous cylindrical and annular cavities, typical of many engineering applications. Moreover, a fully explicit scheme reduces the computational costs and ensures flexibility. Originality/value – This is the first time that a fully explicit finite element scheme is employed for the solution of transient natural convection in partially porous tall annular cavities.

Three-dimensional numerical modeling of free convection in sloping porous enclosures

Three-dimensional (3D) numerical simulations are carried out to study steady state free convection in a sloping porous enclosure heated from below. The model is based on Darcy’s law and the Boussinesq approximation. Two different approaches to solve this problem are compared: primitive variables and vector potential. Although both numerical models lead to equivalent results in terms of the Nusselt number and convective modes, the vector potential model proved to be less mesh-dependent and also a faster algorithm. A parametric study of the problem considering Rayleigh number, slope angle and aspect ratio showed that convective modes with irregular 3D geometries can develop in a wide variety of situations, including horizontal porous enclosure at relatively low Rayleigh numbers. The convective modes that have been described in previous 2D studies (multicellular and single cell) are also present in the 3D case. Nonetheless the results presented here show that the transition between these convective modes follows an irregular 3D geometry characterized by the interaction of transverse and longitudinal coils.

Steady-state free convection in right-angle porous trapezoidal cavity filled by a nanofluid: Buongiorno’s mathematical model

A numerical study of the steady free convection in a right-angle porous trapezoidal cavity filled by a nanofluid using Buongiorno’s mathematical model has been performed. The analysis has been done for different values of the governing parameters Nb, Nt, Nr, Le, A and Rayleigh number Ra in the range 50⩽Ra⩽1000 using a finite difference method. It is found that single cell is formed inside the cavity independent of the governing parameters. An addition of the nanoparticles can lead to both an intensification and attenuation of the convective flow and heat transfer in dependence on Nb, Nt, Nr, Le, A and Ra. The results indicate that there exist significant changes in the flow and temperature fields as compared with those of a differentially heated square porous cavity. These results lead, in particular, to the prediction of a position of minimum heat transfer across the cavity, which is of interest in the thermal insulation of buildings and other areas of technology.

Free convection in a triangular cavity filled with a porous medium saturated by a nanofluid

Purpose – Steady-state free convection heat transfer in a right-angle triangular porous enclosure filled by a nanofluid using the mathematical nanofluid model proposed by Buongiorno has been numerically analyzed. The paper aims to discuss this issue. Design/methodology/approach – The nanofluid model takes into account the Brownian diffusion and thermophoresis effects. The governing equations formulated in terms of the vorticity-stream function variables were solved by finite difference method. Findings – It has been found that the average Nusselt number is an increasing function of the Rayleigh and Lewis numbers and a decreasing function of Brownian motion, buoyancy-ratio and thermophoresis parameters. At the same time the average Sherwood number is an increasing function of the Rayleigh and Lewis numbers, Brownian motion and thermophoresis parameters and a decreasing function of buoyancy-ratio parameter. Originality/value – The present results are new and original for the heat transfer and fluid flow in a right-angle triangular porous enclosure filled by a nanofluid using the mathematical nanofluid model proposed by Buongiorno. The results would benefit scientists and engineers to become familiar with the flow behaviour of such nanofluids, and the way to predict the properties of this flow for possibility of using nanofluids in advanced nuclear systems, in industrial sectors including transportation, power generation, chemical sectors, ventilation, air-conditioning, etc.

Free convection in a tilted triangle porous cavity filled with Cu-water nanofluid with flush mounted heater on the wall

Purpose – Steady-state free convection heat transfer and fluid flow of Cu-water nanofluid is investigated within a porous tilted right-angle triangular enclosure. The paper aims to discuss these issues. Design/methodology/approach – The flush mounted heater with finite size is placed on one right-angle wall. The temperature of the inclined wall is lower than the heater, and the rest of walls are adiabatic. The governing equations are obtained based on the Darcy's law, and the nanofluid model adopted is that by Tiwari and Das. The transformed dimensionless governing equations were solved numerically by finite difference method, and the solution for algebraic equations was obtained through successive under relaxation method. Findings – Investigations were made as the tilted angle of the cavity varies within under different values of Rayleigh number for a porous medium with and solid volume fraction parameter of Cu-water nanofluid with. It is found that the maximum value of the average Nusselt number is achieved with the highest Rayleigh number when the tilted angle of the cavity is 150°, while the minimum value of the average Nusselt number is obtained with the lowest Rayleigh number when the tilted angle of the cavity locates at 240°. As soon as the flow convection in the cavity is not significant, increasing can improve the value of, but opposite effects appear when flow convection becomes stronger. Originality/value – The present results are new and original for the heat transfer and fluid flow in a porous tilted triangle enclosure filled by Cu-water nanofluid. The results would benefit scientists and engineers to become familiar with the flow behaviour of such nanofluids, and the way to predict the properties of this flow for possibility of using nanofluids in advanced nuclear systems, in industrial sectors including transportation, power generation, chemical sectors, ventilation, air-conditioning, etc.

Numerical and experimental study of steady state free convection generated by constant heat flux in tilted hemispherical cavities

Steady-state natural convection taking place in hemispherical air-filled cavities is presented in this work. The circular base of the hemispherical cavity is the hot active wall subjected to a constant heat flux. The closing dome is maintained isothermal at a lower temperature and acts as a cold wall. This cavity can be inclined at an angle α between 0° (hot wall horizontal) and 90° (hot wall vertical) in steps of 15°. The problem is studied both experimentally and numerically. Measurements of thermal variables are taken on an experimental assembly at steady state in order to characterize the heat exchanges at the hot wall. The Rayleigh number resulting from the experimental parameters varies between 3.44×105 and 2.83×107. The numerical study covers a larger Ra range, between 104 and 5×107. Calculations performed using the finite volume method complement the experimental results by examining also the dynamic aspects. The mathematical model used is validated by measurements. Differences between the two approaches are found to be relatively low, always within the uncertainties of the experimental data. The comparison with previous published works dealing with the horizontal cavity is also satisfactory. The resulting thermal and dynamic fields are presented for all the angles treated. Correlations between average Nusselt and Rayleigh numbers are proposed to quantify the convective exchanges for engineering applications.

Unsteady/Steady Free Convective Couette Flow of Reactive Viscous Fluid in a Vertical Channel Formed by Two Vertical Porous Plates

This paper presents unsteady as well as steady-state free convection Couette flow of reactive viscous fluid in a vertical channel formed by two infinite vertical parallel porous plates. The motion of the fluid is induced due to free convection caused by the reactive nature of viscous fluid as well as the impulsive motion of one of the porous plates. The Boussinesq assumption is applied, and the nonlinear governing equations of motion and energy are developed. The time-dependent problem is solved using implicit finite difference method, and steady-state problem is solved by applying regular perturbation technique. During the course of computation, an excellent agreement was found between the well-known steady-state solutions and transient solutions at large value of time.

Free convection in a triangle cavity filled with a porous medium saturated with nanofluids with flush mounted heater on the wall

Steady-state free convection heat transfer behavior of nanofluids is investigated numerically inside a right-angle triangular enclosure filled with a porous medium. The flush mounted heater with finite size is placed on the left vertical wall. The temperature of the inclined wall is lower than the heater, and the rest of walls are adiabatic. The governing equations are obtained based on the Darcy’s law and the nanofluid model proposed by Tiwari and Das [1]. The transformed dimensionless governing equations were solved by finite difference method and solution for algebraic equations was obtained through Successive Under Relaxation method. Investigations with three types of nanofluids were made for different values of Rayleigh number Ra of a porous medium with the range of 10 ≤ Ra ≤ 1000, size of heater Ht as 0.1 ≤ Ht ≤ 0.9, position of heater Y p when 0.25 ≤ Y p ≤ 0.75, enclosure aspect ratio AR as 0.5 ≤ AR ≤ 1.5 and solid volume fraction parameter ϕ of nanofluids with the range of 0.0 ≤ ϕ ≤ 0.2. It is found that the maximum value of average Nusselt number is obtained by decreasing the enclosure aspect ratio and lowering the heater position with the highest value of Rayleigh number and the largest size of heater. It is further observed that the heat transfer in the cavity is improved with the increasing of solid volume fraction parameter of nanofluids at low Rayleigh number, but opposite effects appear when the Rayleigh number is high.

Empirical correlations at high Ra for steady-state free convection in 2D air-filled parallelogrammic enclosures with isothermal discrete heat sources

New Nu–Ra correlations are proposed to evaluate the steady-state natural convection heat transfer taking place in air-filled enclosures of parallelogrammic section. The thermal conditions and dimensions of the cavities treated in the present work lead to high values of the Rayleigh number, reaching 3 × 10 9. The cavities considered are formed by two vertical active walls connected by a closing adiabatic channel. The cold wall is maintained isotherm at temperature T c while the hot one consists of 5 bands, alternately isotherm at temperature T h and adiabatic. The upper and lower walls of the channel are inclined at an angle α with respect to the horizontal. This angle is either positive (hot wall below the level of the cold one) or negative, giving rise respectively to a conducting or insulating cavity in the convective sense of the term. A detailed steady-state 2D numerical study is carried out with the finite volume method and is accompanied with surface temperature experimental measurements. A good agreement between the experimental and numerical results is found in terms of the global heat exchanges and hence of the convective transfer. The dependence of the mean Nusselt number is presented as a function of the inclination angle α and of the Rayleigh number Ra and confirm the convective diode effect taking place. Correlations of the type Nu ¯ ̲ ̲ α = k ( α ) Ra n , suitable for −60° ⩽ α ⩽ +60°, are proposed for the range 5 × 10 3 ⩽ Ra ⩽ 3 × 10 9 corresponding to diverse engineering applications.

Scales of free convection around a vertical cylinder

The natural scales of the laminar steady-state free convection flow regime surrounding an isothermal vertical cylinder are established. It is shown that nondimensionalizing the momentum and energy equations in terms of the Rayleigh or Boussinesq numbers allows the use of the Prandtl number as a criterion to establish whether the motive buoyancy force is mainly balanced by inertia or by friction. Instead, in the conventional formulation based on the Grashof number, this distinction is not possible.