Modeling of conjugated heat transfer in a thick walled enclosure filled with nanofluid

The objective of this paper is to investigate the conjugated heat transfer in a thick walled cavity filled with copper-water nanofluid. The analysis uses a two-dimensional rectangular enclosure under conjugated convective-conductive heat transfer conditions and considers a range of Rayleigh numbers. The enclosure was subjected to a constant and uniform heat flux at the left thick wall generating a natural convection flow. The thicknesses of the other boundaries are assumed to be zero. The right wall is kept at a low constant temperature while the horizontal walls are assumed to be adiabatic. A moveable divider is located at the bottom wall of the cavity. The governing equations are derived based on the conceptual model in the Cartesian coordinate system. The study has been carried out for the Rayleigh number in the range of 10⁵ ≤ Ra ≤ 10⁸, and for the solid volume fraction at 0 ≤ ? ≤ 0.05. Results are presented in the form of streamlines, isotherms, average Nusselt number and input heat absorption by the nanofluid. The effects of solid volume fraction of nanofluids, the location of the divider and also the value of the ambient convective heat transfer coefficient on the hydrodynamic and thermal characteristics of flow have been analyzed. An increase in the average Nusselt number was found with the solid concentration for the whole range of Rayleigh number. In addition, results show that the position of the divider and the ambient convective heat transfer coefficient have a considerable effect on the heat transfer enhancement.     

Influence of exit opening location on mixed convection in a channel with volumetric heat sources

Mixed convection heat transfer in a channel with volumetric heat source and different opening ratio at the exit has been analyzed numerically. Steady, laminar, two-dimensional model based on the finite-volume method is used for solving the mass, momentum and energy transfer governing equations. The flow is injected to the duct with a uniform velocity distribution. The problem is studied in three aspect ratios (A = 1, 3 and 5) and three opening cases as single opening near the ceiling, single opening near the bottom and double openings. The study is also tested for different Richardson numbers between 0.01 and 10. Streamlines, isotherms, Nusselt numbers and temperature profiles were obtained for indicated cases. It is found that both the Richardson number and the locations of exit openings have strong effects on flow and temperature distribution in the presence of volumetric heat sources. The highest heat transfer was formed when the outlet port is located onto top of vertical wall.     

Enhancement of heat exchange in a wavy channel linked to a porous domain; a possible duct geometry for fuel cells

In this paper heat transfer and flow field analysis in a wavy channel linked to a porous Gas Diffusion Layer (GDL) is numerically studied. The domain is very similar to our earlier computations of proton exchange membrane fuel cells (see Khakbaz-Baboli and Kermani (2008)). The fluid temperature at the channel inlet (T_in) is taken less than that of the walls (T_w). The governing equations are numerically solved in the domain by the control volume approach based on the SIMPLE technique (1972). A wide spectrum of numerical studies is performed over a range of Reynolds number Re_H: 100 ≤ Re_H ≤ 1000, wave number β: 0 ≤ β ≤ 10, the wave amplitude α: 0 ≤ α ≤ 0.3 and Darcy number Da: 0.1 ≤ Da ≤ 0.001. Simulations show that heat transfer in channels can enhance up to 100%, depending on the duct α, β and flow Re_H. Computations show excellent agreement with the literature. The present work can provide helpful guidelines to the manufactures of the compact heat exchangers.     

Conjugate natural convection in a square cavity with finite thickness horizontal walls

The effect of conduction of horizontal walls on natural convection heat transfer in a square cavity is numerically investigated. The vertical walls of the cavity are at different constant temperatures while the outer surfaces of horizontal walls are insulated. A code based on vorticity–stream function is written to solve the governing equations simultaneously over the entire computational domain. The dimensionless wall thickness of cavity is taken as 0.1. The steady state results are obtained for wide ranges of Rayleigh number (10³ < Ra < 10⁶) and thermal conductivity ratio (0 < K < 50). The variation of heat transfer rate through the cavity and horizontal walls with Rayleigh number and conductivity ratio is analyzed. It is found that although the horizontal walls do not directly reduce temperature difference between the vertical walls of cavity, they decrease heat transfer rate across the cavity particularly for high values of Rayleigh number and thermal conductivity ratio. Heatline visualization technique is a useful application for conjugate heat transfer problems as shown in this study.     

Combined free and forced convection in an inclined channel with discrete heat sources

Mixed convection is studied in an inclined rectangular channel with three discrete heat sources placed on the bottom surface. The Reynolds and Grashof numbers and the channel inclination are respectively: 1 ≤ Re ≤ 1000, 10³ ≤ Gr ≤ 10⁵, and 0° ≤ γ ≤ 90°. The governing equations are solved using the finite element method, the Penalty and Petrov–Galerkin techniques. The inclination has a stronger influence on the flow and heat transfer for low Reynolds numbers. In general, cases which show the lowest temperature distributions on the modules are those where the inclination angles are 45° and 90°.     

Effect of inlet and outlet location on the mixed convective cooling inside the ventilated cavity subjected to an external nanofluid

In this paper, mixed convection flow and temperature fields in a vented square cavity subjected to an external copper–water nanofluid are studied numerically. The natural convection effect is attained by heating from the constant flux heat source on the bottom wall and cooling from the injected flow. In order to investigate the effect of inlet and outlet location, four different placement configurations of the inlet and outlet ports are considered. In each of them, both the inlet and outlet ports are alternatively located either on the top or the bottom of the sides and external flow enters in to the cavity through an inlet opening in the left vertical wall and exits from another opening in the opposite wall. The remaining boundaries are considered 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 the Reynolds number in the range of 50 ≤ Re ≤ 1000, with Richardson numbers 0 ≤ Ri ≤ 10 and for solid volume fraction 0 ≤ ? ≤ 0.05. Results are presented in the form of streamlines, isotherms, average Nusselt number. In addition, the effects of solid volume fraction of nanofluids on the hydrodynamic and thermal characteristics have been investigated and discussed. The algorithm and the computer code have been also compared with numerical results in order to verify and validate the model.     

Numerical simulation of steady natural convection heat transfer in a 3-dimensional single-ended tube subjected to a nanofluid

In this paper, 3-dimensional numerical simulation of steady natural convective flow and heat transfer are studied in a single-ended tube with non-uniform heat input. Apart from some other applications, it serves as a simplified model of the single-ended evacuated solar tube of a water-in-glass evacuated tube solar water heater. It is assumed that the sealed end of tube to be adiabatic and also the tube opening to be subjected to copper–water nanofluid. Governing equations are derived based on the conceptual model in the cylindrical coordinate system. The governing equations have been then approximated by means of a fully implicit finite volume control method (FVM), using SIMPLE algorithm on the collocated arrangement. The study has been carried out for solid volume fraction 0 ≤ φ ≤ 0.05 and maximum heat flux 100 ≤ q_m ≤ 700. Considering that the driven flow in the tube is influenced by the dimensions and the inclination angle of the solar tube, the flow patterns and temperature distributions are presented on different cross sectional planes and longitudinal sections, when the tube is positioned at different orientations.     

Optimal location of a pair heat source-sink in an enclosed square cavity with natural convection through PSO algorithm

In this paper, the optimum positions of a pair heat source-sink in an enclosure have been studied. The objective of this investigation is to minimize the maximum temperature (T_max) on the heat source with constant heat flux. For this, a Particle Swarm Optimization algorithm (PSOA) has been used. Continuity, momentum and energy equations with the Boussinesq approximation for a laminar and incompressible flow of a Newtonian fluid have been solved by finite volume method. The present study has been carried out for governing parameters like Rayleigh numbers (Ra) from 10³ to 10⁶, the cavity aspect ratio, A = H/L = 1 and the source and sink sizes D₀ from 0.2 to 0.5. Numerical results revealed that the optimum configurations are a function of Rayleigh numbers and the source and sink sizes.     

Numerical analysis of natural convection for a porous rectangular enclosure with sinusoidally varying temperature profile on the bottom wall

Numerical investigations of steady natural convection flow through a fluid-saturated porous medium in a rectangular enclosure with a sinusoidal varying temperature profile on the bottom wall were conducted. All the walls of the enclosure are insulated except the bottom wall which is partially heated and cooled. The governing equations were written under the assumption of Darcy-law and then solved numerically using finite difference method. The problem is analyzed for different values of the Rayleigh number Ra in the range 10 ≤ Ra ≤ 1000, aspect ratio parameter AR in the range 0.25 ≤ AR ≤1.0 and amplitude λ of the sinusoidal temperature function in the range 0.25 ≤ λ ≤ 1.0. It was found that heat transfer increases with increasing of amplitude λ and decreases with increasing aspect ratio AR. Multiple cells were observed in the cavity for all values of the parameters considered.     

Predictions of buoyancy-induced flow in various across-shape concave enclosures

The present study was conducted to numerically investigate the steady laminar buoyancy-driven and convection heat transfer characteristics within three different across-shape concave enclosures for the Prandtl number of 0.71 and 4, the Grashof number range 10⁴ ≤ Gr ≤ 2 × 10⁵_, and the gap range 0 ≤ H₁/H₂ ≤ 0.25. The steady Navier-Stokes equations, governing the flow under Boussinesq approximation, are solved with the dimensionless stream function-vorticity formulation in terms of curvilinear coordinates using the finite difference method. The results show that the effects of various shapes, the strength of the vortex is relatively bigger in the rectangular-rectangular concave enclosure than in the rectangular-circular concave enclosure at the same Grashof number. Heat transfer from the different across-shape concave enclosures is evaluated, and flow and heat transfer characteristics are discussed.     

Natural convection in an open square cavity with slots

In this paper, we investigate heat transfer by natural convection in an open cavity in which a uniform heat flux is applied to the inside active wall facing the opening with slots. Conservation equations are solved by finite difference–control volume numerical method. The relevant governing parameters are: the Rayleigh numbers from 10³ to 10⁶, the Prandtl number, Pr = 0.7, constant for air, the cavity aspect ratio, A = L/H = 1. Number of slots N is varied from 2 to 8 and the dimensionless opening ratio OR from 0.1 to 0.6. We found that the Nusselt number and the volume flow rate are both increasing functions of the Rayleigh number; they are a decreasing function of the number of slots and increasing function of the opening ratio, though there is an optimum opening ratio at high Rayleigh numbers.     

Effect of nano-particles on forced convection in sinusoidal-wall channel

In this paper heat transfer and flow field in a wavy channel with nano-fluid is numerically studied. The temperature of input fluid (T_c) is taken less than that of the wavy horizontal walls (T_w). The governing equations are numerically solved in the domain by the control volume approach based on the SIMPLE technique. Copper–water nano-fluid is considered for simulation. A wide spectrum of numerical simulations has been done over a range of Reynolds number, Re_H, 5 ≤ Re_H ≤ 1500, nano-fluid volume fraction, ?, 0 ≤ ? ≤ 20% and the wave amplitude, α, 0 ≤ α ≤ 0.3. The effects of these parameters are investigated on the local and average Nusselt numbers and the skin friction coefficient. Simulations show excellent agreement with the literature. From this study, it is concluded that heat transfer in channels can enhance by addition of nano-particles, and usage of wavy horizontal walls. These can enhance the heat transfer by 50%. The present work can provide helpful guidelines to the manufacturers of the compact heat exchangers.     

Conjugate natural convection in air filled tube inserted a square cavity

Numerical analyses of fluid flow and heat transfer due to buoyancy forces in a tube inserted square cavity filled with fluid were carried out by using control volume method in this study. The cavity was heated from the left wall and cooled from the right isothermally and horizontal walls were adiabatic. A circular tube filled with air was inserted into the square cavity. The case that the inside and outside of the tube were filled with the same fluid (air) was examined. Varied solid materials were chosen as the tube wall. Results were obtained for different Rayleigh numbers (Ra = 10⁴, 10⁵ and 10⁶), thermal conductivity ratio of the fluid to the tube wall (k = 0.1, 1 and 10) and different location centers of the tube (c (0.25 ≤ x ≤ 0.75, 0.25 ≤ y ≤ 0.75)). Comparison with benchmark solutions of the natural convection in a cavity was performed and numerical results gave an acceptable agreement. It was found that varied location of the tube center can lead to different flow fields and heat transfer intensities which are also affected by the value of Rayleigh number.     

Numerical study of double diffusive natural convective heat and mass transfer in an inclined rectangular cavity filled with porous medium

Two-dimensional double-diffusive natural convective heat and mass transfer in an inclined rectangular porous medium has been investigated numerically. Two opposing walls of the cavity are maintained at fixed but different temperatures and concentrations; while the other two walls are adiabatic. The generalized model with the Boussinesq approximation is used to solve the governing equations. The flow is driven by a combined buoyancy effect due to both temperature and concentration variations. A finite volume approach has been used to solve the non-dimensional governing equations and the pressure velocity coupling is treated via the SIMPLER algorithm. The results are presented in streamline, isothermal, iso-concentration, Nusselt and Sherwood contours for different values of the non-dimensional governing parameters. A wide range of non-dimensional parameters have been used including, aspect ratio (2 ≤ A ≤ 5), angle of inclination of the cavity (0 ≤ ? ≤ 85), Lewis number (0.1 ≤ Le ≤ 10), and the buoyancy ratio (− 5 ≤ N ≤ 5).     

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.     

Natural convection heat transfer from a heat sink with hollow/perforated circular pin fins

Experiments were performed on natural convection heat transfer from circular pin fin heat sinks subject to the influence of its geometry, heat flux and orientation. The geometric dependence of heat dissipation from heat sinks of widely spaced solid and hollow/perforated circular pin fins with staggered combination, fitted into a heated base of fixed area is discussed. Over the tested range of Rayleigh number, 3.8 × 10⁶ ≤ Ra ≤ 1.65 × 10⁷, it was found that the solid pin fin heat sink performance for upward and sideward orientations shows a competitive nature, depending on Rayleigh number and generally shows higher heat transfer coefficients than those of the perforated/hollow pin fin ones in both arrangement. For all tested hollow/perforated pin fin heat sinks, however, the performance for sideward facing orientation was better than that for upward facing orientation. This argument is supported by observing that the augmentation factor was around 1.05–1.11, depending on the hollow pin diameter ratio, D_i/D_o. Meanwhile, the heat sink of larger hollow pin diameter ratio, D_i/D_o offered higher heat transfer coefficient than that of smaller D_i/D_o for upward orientation, and the situation was reversed for sideward orientation. The heat transfer performance for heat sinks with hollow/perforated pin fins was better than that of solid pins. The temperature difference between the base plate and surrounding air of these heat sinks was less than that of solid pin one and improved with increasing D_i/D_o.     

High-resolution simulations of magnetohydrodynamic free convection in an enclosure with a transverse magnetic field using a velocity–vorticity formulation

The differential quadrature method (DQ) is employed to simulate the effect of a transverse magnetic field on buoyancy-driven magnetohydrodynamic (MHD) flow in an enclosure. The DQ numerical procedure is adopted for solving the velocity–vorticity form of Navier–Stokes equations in two dimensions. These equations together with respective (appropriate) boundary conditions are solved numerically using a DQ method by a coupled algorithm for the velocity–vorticity–temperature coupled together with a bi-conjugate gradient iterative solver technique. The velocity–vorticity formation is properly utilized to obtain results in the range of Grashof numbers (10⁴ ≤ Gr ≤ 10⁵), Hartmann numbers (0 ≤ Ha ≤ 100), and Prandtl numbers (0.01 ≤ Pr ≤ 10), as well as aspect ratios A = L/H varying from 1 to 3 in a differentially heated cavity with a transverse magnetic field. The algorithm is then employed to compute the average Nusselt numbers and flow parameters for all the cases. With support from the present simulations of the heat transfer characteristics for a constant value of Gr within the cavity, the heat transfer rate is at its maximum for higher Pr and in the absence of MHD effects (Ha = 0), while it is lower with increase in external magnetic field strength in the lower region of the Prandtl number.     

Natural convection heat transfer in partially open inclined square cavities

A numerical study has been carried out on inclined partially open square cavities, which are formed by adiabatic walls and a partial opening. The surface of the wall inside the cavity facing the partial opening is isothermal. Steady-state heat transfer by laminar natural convection in a two dimensional partially open cavity is studied by numerically solving equations of mass, momentum and energy. Streamlines and isotherms are produced, heat and mass transfer is calculated. A parametric study is carried out using following parameters: Rayleigh number from 10³ to 10⁶, dimensionless aperture size from 0.25 to 0.75, aperture position at high, center and low, and inclination of the opening from 0° (facing upward) to 120° (facing 30° downward). It is found that the volume flow rate and Nusselt number are an increasing function of Rayleigh number, aperture size and generally aperture position. Other parameters being constant, Nusselt number is a non-linear function of the inclination angle. Depending on the application, heat transfer can be maximized or minimized by selecting appropriate parameters, namely aperture size, aperture position and inclination angle at a given operation Rayleigh number.     

The effects of different entrance sections lengths and heating on free and forced convective heat transfer inside a horizontal circular tube

An experimental study was done for hydrodynamically fully developed and thermally developing laminar air flows in a horizontal circular tube has a 30 mm inside diameter and 900 mm heated length (L/D = 30) under a constant wall heat flux boundary condition, with different aluminum entrance section pipes (calming sections) having the same inside diameter as test section pipe but with variable lengths of 600 mm (L/D = 20), 1200 mm (L/D = 40), 1800 mm (L/D = 60), and 2400 mm (L/D = 80). The Reynolds number ranged from 400 to 1600 and the heat flux is varied from 60 W m^− 2 to 400 W m^− 2. This paper examines the effects of the entrance sections lengths and heating on the free and forced convection heat transfer process. The surface temperature data were measured and heat transfer rates at different heat flux levels as well as different Reynolds numbers were calculated and correlated in the form of relevant parameters. The buoyancy force has a significant effect on the heat transfer and the combined convection factor was approximately varied form 0.13 ≤ Gr/Re² ≤ 7.125. It was found that the surface temperature increases as the entrance section length increases. It was inferred that the heat transfer decreases as the entrance section length increases due to the flow resistance and the mass flow rate. The proposed correlation was compared with available literature and with laminar forced convection and showed satisfactory agreement.     

Numerical study of mixed convective cooling in a square cavity ventilated and partially heated from the below utilizing nanofluid

A numerical investigation of mixed convection flows through a copper–water nanofluid in a square cavity with inlet and outlet ports has been executed. The natural convection effect is attained by heating from the constant flux heat source which is symmetrical located at the bottom wall and cooling from the injected flow. 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 the Reynolds number in the range 50 ≤ Re ≤ 1000, with Richardson numbers 0 ≤ Ri ≤ 10 and for solid volume fraction 0 ≤ ? ≤ 0.05. The thermal conductivity and effective viscosity of nanofluid have been calculated by Patel and Brinkman models, respectively. Results are presented in the form of streamlines, isotherms, average Nusselt number and average bulk temperature. In addition, the effects of solid volume fraction of nanofluids on the hydrodynamic and thermal characteristics have been investigated and discussed. The results indicate that increase in solid concentration leads to increase in the average Nusselt number at the heat source surface and decrease in the average bulk temperature.