Jet impingement cooling of a horizontal surface in an unconfined porous medium: Mixed convection regime

Two-dimensional slot jet impingement cooling of an isothermal horizontal surface immersed in an unconfined porous medium is simulated numerically to gain insight into thermal characteristics under mixed convection conditions with the limitation of the Darcy model. The jet direction is considered to be perpendicular from the top to the horizontal heated element; therefore, the jet flow and the buoyancy driven flow are in opposite directions. The results are presented in the mixed convection regime with wide ranges of the governing parameters: Péclet number (1 ? Pe ? 1000), Rayleigh number (10 ? Ra ? 100), half jet width (0.1 ? D ? 0.5), and the distance between the jet and the heated portion (0.1 ? H ? 1.0). It is found that the average Nusselt number increases with increase in either Rayleigh number or jet width for high values of Péclet number. The average Nusselt number also increases with decrease in the distance between the jet and the heated portion. It is shown that mixed convection mode can cause minimum average Nusselt number at two values of Péclet number and a maximum average Nusselt number occurs in between theses two Péclet numbers at higher Rayleigh number due to counteraction of jet flow against buoyancy driven flow. Hence careful consideration must be given while designing a system of jet impingement cooling through porous medium.     

Jet impingement cooling of a constant heat flux horizontal surface in a confined porous medium: Mixed convection regime

In the present study, numerical investigation of jet impingement cooling of a constant heat flux horizontal surface immersed in a confined porous channel is performed under mixed convection conditions with the limitation of the Darcy model. The results are presented in the mixed convection regime with wide ranges of the governing parameters: Péclet number (1 ? Pe ? 1000), Rayleigh number (10 ? Ra ? 100), half jet width (0.1 ? D ? 1.0), and the distance between the jet and the heated portion (0.1 ? H ? 1.0). It is found that the average Nusselt number increases with increase in either Rayleigh number or jet width for high values of Péclet number. The average Nusselt number also increases with decrease in the distance between the jet and the heated portion. The correlation for Nu_avg in the forced convection regime is suggested. It is shown that mixed convection mode can cause minimum average Nusselt number unfavorably due to counteraction of jet flow against buoyancy driven flow. Hence, careful consideration must be given while designing a system of jet impingement cooling through porous medium.     

Unsteady Mixed Convection of a Confined Jet in a Fluid-Superposed High Porosity Medium

Numerical study of a confined jet impingement cooling of a fluid-superposed porous medium heated from below is conducted to investigate the oscillatory mixed convection. The effects of the Rayleigh number (2 × 10⁵ ≤ Ra ≤ 1 × 10⁶) and the Darcy number (1 × 10^?5 ≤ Da ≤ 5 × 10^?4) on the heat transfer are investigated for different Péclet numbers. It is found that, the average Nusselt number increases with the increase in Darcy number or Rayleigh number. The values of average Nusselt number are found to oscillate with time for some combination of Rayleigh numbers (Ra ≥ 4 × 10⁵) and Péclet numbers (200 ≤ Pe ≤ 1000), at which the oscillatory convection occurs. The oscillation of average Nusselt number is investigated for different porous medium height and porous medium-to-fluid heat capacity ratio.     

Mixed convection from two thermal sources in a vertical porous layer

In this paper the steady mixed convection flow adjacent to a vertical surface embedded in a fluid-saturated porous medium, on which two isolated thermal sources are located is investigated theoretically. The thermal sources are taken as long planar sources of finite height and the resulting two-dimensional flow is numerically studied using the finite volume method. The nature and the basic characteristics of the mixed aiding as well as mixed opposing flows that arise are investigated using the Darcy law model. The governing parameters are the Rayleigh number, Péclet number, separation distance between heated elements, their lengths and heat flux ratio in additional to the external flow direction. These parameters are varied over wide ranges and their effect on the heat transfer characteristics is studied in detail.     

Mixed convection on jet impingement cooling of a constant heat flux horizontal porous layer

In the present study, numerical investigation of jet impingement cooling of a constant heat flux horizontal surface immersed in a confined porous channel is performed under mixed convection conditions, and the Darcian and non-Darcian effects are evaluated. The unsteady stream function-vorticity formulation is used to solve the governing equations. The results are presented in the mixed convection regime with wide ranges of the governing parameters: Reynolds number (1 ≤ Re ≤ 1000), modified Grashof number (10 ≤ Gr¹ ≤ 100), half jet width (0.1 ≤ D ≤ 1.0), Darcy number (1 × 10^?6 ≤ Da ≤ 1 × 10^?2), and the distance between the jet and the heated portion (0.1 ≤ H ≤ 1.0). It is found that the average Nusselt number (Nu_avg) increases with increase in either modified Grashof number or jet width for high values of Reynolds number. The average Nusselt number also increases with decrease in the distance between the jet and the heated portion. The average Nusselt number decreases with the increase in Da for the non-Darcy regime when Re is low whereas Nu_avg increases when Re is high. It is shown that mixed convection mode can cause minimum heat transfer unfavorably due to counteraction of jet flow against buoyancy driven flow. Minimum Nu_avg occurs more obviously at higher values of H. Hence the design of jet impingement cooling through porous medium should be carefully considered in the mixed convection regimes.     

Jet impingement interaction with cross flow in horizontal porous layer under thermal non-equilibrium conditions

In the present article the jet impingement cooling of heated portion of a horizontal surface immersed in a thermally non-equilibrium porous layer is considered for investigation numerically with the presence of a cross flow. The mathematical model is derived for steady, two-dimensional laminar flow based on Darcy model and two-energy equation for fluid and solid phases. A parametric study is carried out by varying the following parameters: cross flow to jet flow velocity ratio parameter (0 ⩽ M ⩽ 1); porosity scaled thermal conductivity ratio parameter (0.1 ⩽ K_r ⩽ 1000); heat transfer coefficient parameter (0.1 ⩽ H ⩽ 1000); Péclet number (1 ⩽ Pe ⩽ 1000) and Rayleigh number (10 ⩽ Ra ⩽ 100). The total average Nusselt number is defined based on the overall thermal conductivity, which is assumed to be the arithmetic mean of the porosity scaled thermal conductivity of the fluid and solid phases. The total average Nusselt number as well as the average Nusselt number for both fluid and solid phases is presented for different governing parameters. It is found that the presence of a weak cross flow in a jet impinging jet may degrade the heat transfer. The results show that the average Nusselt number calculated from the thermal equilibrium model are the maximum possible values and these values can be reproduced by large values of H × K_r.     

Numerical study of mixed convection on jet impingement cooling in an open cavity filled with porous medium

The present study numerically investigates the opposing mixed convection arises from jet impingement cooling of a heated bottom surface of an open cavity in a horizontal channel filled with porous medium. The FeCrAlY foam is considered in the present study with a porosity of 0.867. The heat transfer characteristics are investigated with governing parameters in the range of Rayleigh number (50 ≤ Ra ≤ 150), Péclet number (1 ≤ Pe ≤ 1000) and dimensionless cavity depth (0 ≤ H ≤ 0.4). The results show that, the average Nusselt numbers decreases with the increase in dimensionless cavity depth. The opposing mixed convection is demonstrated to cause deterioration in average Nusselt number for fluid at certain Peclet number. The average Nusselt number for fluid is found to increase with the increase in Rayleigh number but the effect of Rayleigh number become insignificant at high Peclet number (Pe > 500).     

Mixed convection in a square vented enclosure filled with a porous medium

Steady mixed convection flow in a vented enclosure with an isothermal vertical wall and filled with a fluid-saturated porous medium is investigated numerically. The forced flow conditions are imposed by providing an inlet at the bottom surface, and a vent at the top, facing the inlet. The nature and the basic characteristics of the mixed aiding as well as mixed opposing flows that arise are investigated using the Darcy law model. The governing parameters are the Rayleigh number, Péclet number, and the width of the inlet as a fraction of the height of the square enclosure. These parameters are varied over wide ranges and their effect on the heat transfer characteristics is studied in detail.     

Thermally dependent viscosity and non-Newtonian flow in a double-pipe helical heat exchanger

A double-pipe helical heat exchanger was numerically studied to determine the effects of thermally dependent viscosity and non-Newtonian flows on heat transfer and pressure drop for laminar flow. Thermally dependent viscosities were found to have very little effect on the Nusselt number correlations for Newtonian fluids; however significant effects on the pressure drop in the heat exchanger were predicted. Changing the flow rate in the annulus can significantly affect the pressure drop in the inner tube, since the average viscosity of the fluid in the inner tube would change due to the change in the average temperature.The effects of non-Newtonian power law fluids on the heat transfer and the pressure drop were determined for laminar flow in the inner tube and in the annulus. The Nusselt number was correlated with the Péclet number for heat transfer in the inner tube. For the annulus, the Nusselt number was found to correlate best with the Péclet number and the curvature ratio. Pressure drop data were compared by using ratios of the pressure drop of the non-Newtonian fluid to a Newtonian fluid at identical mass flow rates and consistency indices.     

MHD Mixed Convection with Joule Heating Effect in a Lid-Driven Cavity with a Heated Semi-Circular Source Using the Finite Element Technique

A computational fluid dynamics simulation of heat transfer characteristics on the conjugate effect of Joule heating and magnetic field acting normal to the lid-driven cavity with a heated semi-circular source on one wall under constant temperature is investigated. The left wall of the cavity moves in an upward (case I) or downward (case II) direction, and buoyancy forces are also effective. Horizontal walls are adiabatic. The governing mass, momentum, and energy equations along with boundary conditions are expressed in a normalized primitive variables formulation. The finite element method is used in the solution of the normalized governing equations. The study is performed for pertinent parameters such as the Rayleigh number, Hartmann number, and Joule heating parameter. It is found that the average Nusselt number can be decreased with the increasing of the Rayleigh number in the presence of Joule effect. The magnetic field can be a good control parameter for heat transfer and fluid flow.     

From natural to mixed convection in horizontal and differentially heated annular ducts: Linear stability analysis

Linear stability analysis of a fully developed mixed convection flow of air in an annular horizontal duct is numerically investigated for the radius ratio R = 1.2, a Péclet and a Rayleigh number less than 200 and 6000, respectively. An iterative method is developed to enable the convergence of the dimensionless parameters to their marginal values at the transition. New mixed convection flows are highlighted that are highly correlated with those obtained in natural convection problems under the assumption of two dimensionality. The synthesis of our results on the transitions permits us to build the map of stability for the steady and established mixed convection flows and clearly shows the occurrence of multiplicity of solutions for some couples of Rayleigh and Péclet numbers.     

Analytical function theory approach to the heat transfer problem of a cylinder in cross-flow at small Péclet numbers

The theory of analytical functions is used to study the heat transfer from a uniformly heated cylinder with large length to diameter ratio in cross-flow, in the limit of small Péclet numbers. The energy conservation equation is solved in Fourier’s space, and inverted by means of the residue theorem to obtain an analytical expression of the average Nusselt number in closed form. The result agrees with other theoretical solutions of the same problem existing in the literature.     

Mixed convection with viscous dissipation and pressure work in a lid-driven square enclosure

Buoyant laminar flow in a square lid-driven enclosure is analysed. The vertical sides are kept isothermal at different temperatures, while the horizontal sides are insulated. Assisting mixed convection flow due to uniform motion of the top side is considered. The governing balance equations are solved numerically by employing a Galerkin finite element method. The effects of viscous dissipation and pressure work are taken into account. In order to investigate the influence of these effects, the Nusselt number is evaluated with respect to the heat fluxes at both vertical sides, for different values of the Rayleigh number and of the Péclet number based on the lid velocity. Two sample fluids are considered: a gas and a highly viscous liquid. In the framework of the Oberbeck–Boussinesq approximation, a comparison is made between three different energy balance models: (A) enthalpy formulation (pressure work and viscous dissipation are included); (B) internal-energy formulation (viscous dissipation is included); (C) both pressure work and viscous dissipation are neglected. It is shown that, in the absence of a lid motion, the three models yield substantially the same predictions. On the other hand, when the forced flow induced by the lid motion becomes sufficiently large, the three models yield discrepant results, thus implying that pressure work and viscous dissipation are not negligible. Moreover, it is shown that, in this case, model (A) yields unphysical results, while model (B) leads to reasonable predictions.     

Natural convection heat transfer of water in a horizontal circular gap

An experimental study on the natural convection heat transfer on a horizontal downward facing heated surface in a water gap was carried out under atmospheric pressure conditions. A total of 700 experimental data points were correlated using Rayleigh versus Nusselt number in various forms, based on different independent variables. The effects of different characteristic lengths and film temperatures were discussed. The results show that the buoyancy force acts as a resistance force for natural convection heat transfer on a downward facing horizontal heated surface in a confined space. For the estimation of the natural convection heat transfer under the present conditions, empirical correlations in which Nusselt number is expressed as a function of the Rayleigh number, or both Rayleigh and Prandtl numbers, may be used. When it is accurately predicted, the Nusselt number is expressed as a function of the Rayleigh and Prandtl numbers, as well as the gap width-to-heated surface diameter ratio; and uses the temperature difference between the heated surface and the ambient fluid in the definition of Rayleigh number. The characteristic length is the gap size and the film temperature is the average fluid temperature.     

A numerical study of laminar natural convective heat transfer around a horizontal cylinder inside a concentric air-filled triangular enclosure

A numerical investigation of steady-state laminar natural convective heat transfer around a horizontal cylinder to its concentric triangular enclosure was carried out. The enclosure was filled with air and both the inner and outer cylinders were maintained at uniform temperatures. The buoyancy effect was modeled by applying the Boussinesq approximation of density to the momentum equation and the governing equations were iteratively solved using the control volume approach. The effects of the Rayleigh number and the aspect ratio were examined. Flow and thermal fields were exhibited by means of streamlines and isotherms, respectively. Variations of the maximum value of the dimensionless stream function and the local and average Nusselt numbers were also presented. The average Nusselt number was correlated to the Rayleigh number based on curve-fitting for each aspect ratio. At the highest Rayleigh number studied, the effects of different inclination angles of the enclosure and various cross-section geometries of the inner cylinder were investigated. The computed results indicated that at constant aspect ratio, both the inclination angle and cross-section geometry have insignificant effects on the overall heat transfer rates though the flow patterns are significantly modified.     

Numerical Study of Laminar Natural Convection in a Side-Heated Trapezoidal Cavity at Various Inclined Heated Sidewalls

Steady natural convection of air flow in a two-dimensional side-heated trapezoidal room was investigated numerically using a non-orthogonal, collocated finite-volume grid system. The considered geometry has an inclined left heated sidewall, a vertical right cooled sidewall, and two insulated horizontal upper and lower walls. Computations are performed for seven values of the heated sloping wall angle, three different values of aspect ratio, and five Rayleigh number values. Results are displayed in terms of streamlines, isotherms, and both local and average Nusselt number values. The principal result of this work is the great dependence of the flow fields and the heat transfer on the inclination angle, the aspect ratio, and the Rayleigh number. A correlation between the average Nusselt number, Rayleigh number, heated sloping wall angle, and aspect ratio is proposed.     

Isotherms in a horizontal differentially heated cavity at intermediate Rayleigh numbers

Rayleigh-Benard convection in a horizontal, differentially heated, high aspect ratio fluid layer is considered. Experiments with three Rayleigh numbers (13900, 34800 and 51800) corresponding to different flow regimes have been performed. The temperature field in the fluid has been recorded in the form of interferometric projections, the fringe patterns representing isotherms. For steady and quasi-steady flow, the three dimensional temperature field has been obtained using an iterative algebraic reconstruction technique. Results show the formation of longitudinal rolls in the cavity at a Rayleigh number of 13900. The flow is unsteady at a Rayleigh number of 34800, but a strong indication of the formation of cubic rolls is seen. At a Rayleigh number of 51800, the flow field is completely unsteady with no discernible pattern. The local ray-averaged Nusselt numbers at each of the heated and cooled walls have been determined at the two lower Rayleigh numbers. The systematic variation of Nusselt number over each of the solid surfaces confirms the flow models proposed at these Rayleigh numbers. The average Nusselt number qualitatively matches the published correlations.     

Numerical investigation of laminar natural convective heat transfer from a horizontal triangular cylinder to its concentric cylindrical enclosure

A numerical simulation was conducted to investigate the steady laminar natural convective heat transfer for air within the horizontal annulus between a heated triangular cylinder and its circular cylindrical enclosure. The Boussinesq approximation was applied to model the buoyancy-driven effect and the governing equations were solved using the finite volume method. Four different Rayleigh numbers and four different radius ratios were considered, and four different inclination angles for the inner triangular cylinder were investigated as well. The computed flow and temperature fields were demonstrated in the form of streamlines and isotherms. Variations of the maximum stream function and the local and average Nusselt numbers were displayed as functions of the above-mentioned parameters. Correlations of the average Nusselt number were proposed based on curve fitting. At constant radius ratio, inclination angles of the inner triangular cylinder are found to have negligible effects on the average Nusselt number.     

A Parametric Study of Prandtl Number Effects on Laminar Natural Convection Heat Transfer From a Horizontal Circular Cylinder to Its Coaxial Triangular Enclosure

A parametric study of Prandtl number effects on laminar natural convection heat transfer in a horizontal equilateral triangular cylinder with a coaxial circular cylinder is conducted. The Prandtl number is varied over a wide range from 10^?2 to 10⁵, which corresponds to a variety of working fluids. The governing equations with the Boussinesq approximation for buoyancy are iteratively solved using the finite volume approach. It is shown that the flow patterns and temperature distributions are unique for low-Prandtl-number fluids (Pr ≤ 0.1), and are nearly independent of Prandtl number when Pr ≥ 0.7. In addition, the inclination angle of the triangular enclosure is found to noticeably affect the variations of the local Nusselt number, and to have insignificant influence on the average Nusselt numbers for low Rayleigh numbers when Pr ≥ 0.7.     

Magnetohydrodynamic mixed convection in a horizontal channel with an open cavity

The development of magnetic field effect on mixed convective flow in a horizontal channel with a bottom heated open enclosure has been numerically studied. The enclosure considered has rectangular horizontal lower surface and vertical side surfaces. The lower surface is at a uniform temperature T_h while other sides of the cavity along with the channel walls are adiabatic. The governing two-dimensional flow equations have been solved by using Galarkin weighted residual finite element technique. The investigations are conducted for different values of Rayleigh number (Ra), Reynolds number (Re) and Hartmann number (Ha). Various characteristics such as streamlines, isotherms and heat transfer rate in terms of the average Nusselt number (Nu), the Drag force (D) and average bulk temperature (θ_av) are presented. The results indicate that the mentioned parameters strongly affect the flow phenomenon and temperature field inside the cavity whereas in the channel these effects are less significant.