Numerical investigation of natural convection in a rectangular enclosure due to partial heating and cooling at vertical walls

A comprehensive numerical investigation on the natural convection in a rectangular enclosure is presented. The flow is induced due to the constant partial heating at lower half of the left vertical wall and partial cooling at upper half of the right vertical wall along with rest walls are adiabatic. In this investigation the Special attention is given to understand the effect of aspect ratio and heat source intensity i.e. Rayleigh number, Ra, on the fluid flow configuration as well as on the local and average heat transfer rates. The range of Rayleigh (Ra) and aspect ratio (A) is taken [10³, 10⁶] and [0.5, 4] respectively. The results are presented in terms of stream function (ψ), temperature (θ) and heat transfer rates (local Nusselt numbers Nu_L, and average Nusselt numbers Nu). The numerical experiments show that increasing of Ra implies the enhancement of thermal buoyancy force, which in turn increases the thermal convection in the cavity. As a result, the local as well as average heat transfer rate is expected to increase. The local transfer rate (Nu_L) is increases in the small region near the left vertical wall of the left wall of the cavity and after that it is decreases in the middle portion of heated region. And, it start to increase near to the middle point of left wall. It is also observed that the local heat transfer is increases as increases the aspect ratio. The average heat transfer rate (Nu) is increases as the aspect ratio A increases from 0.5 to 1 and beyond that it is decreases smoothly. It is also found that the heat transfer rate attains its maximum value at aspect ratio one.     

The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition

The forced convection heat transfer resulting from the flow of a uniform stream over a flat surface on which there is a convective boundary condition is considered. In previous papers [5], [6], [7], [8] it was assumed that the convective heat transfer parameter h_f associated with the hot surface depended on x, where x measures distance along the surface, so that problem could be reduced to similarity form. Here it is assumed that this heat transfer parameter h_f is a constant, with the result that the temperature profiles and overall heat transfer characteristics evolve as the solution develops from the leading edge. The heat transfer near the leading edge (small x), which we find to be dominated by the surface heat flux, the solution at large distances along the surface (large x), which dominated by the surface temperature, are discussed. A numerical solution to the full problem is then obtained for a range of values of the Prandtl number to join these two solution regimes.     

Analysis of melting behavior of PCMs in a cavity subject to a non-uniform magnetic field using a moving grid technique

Melting flow and heat transfer of electrically conductive phase change materials subjecting to a non-uniform magnetic field are addressed in a square enclosure. The top and bottom walls of the cavity are adiabatic, and the sidewalls are isothermal at different temperatures. The temperature of the hot wall is higher than the fusion temperature of PCM (T_f), and the cold wall is at the fusion temperature or lower. At the initial time, the cavity is filled with a solid saturated PCM. In the vicinity to the hot wall, there is an external line-source magnet, inducing a magnetic field. The location of the magnetic source (Y₀) can be changed along the hot wall. The cavity domain is divided into two parts of the liquid domain and the solid domain. The moving grid method is utilized to track the phase change interface at the exact fusion temperature of T_f. The governing equations for continuity, flow and heat transfer associated with the Arbitrary Lagrangian–Eulerian (ALE) moving mesh technique are solved using the finite element method. The results are investigated for the melting behavior of PCM by the study of Hartmann number (0 ≤ Ha ≤ 50) and the location of the magnetic source (0 ≤ Y₀ ≤ 1). Outcomes show that the effect of the magnetic field on the melting behavior of PCM is negligible at the initial stages of the melting (Fo < 1.15). However, after the initial stages of the melting, the effect of the presence of a magnetic field becomes significant. Moreover, the location of the magnetic source induces a feeble effect on the melting front at the initial melting stages, but its effect on the shape of the melting front increases by the increase of the non-dimensional time. The location of the magnetic source also significantly affects the streamlines patterns. Changing the position of the magnetic source from the bottom of the cavity (Y₀ = 0.2) to the almost middle of the cavity (Y₀ = 0.6) would decrease the required non-dimensional time of full melting from Fo = 10.4 to Fo = 9.0.     

Numerical prediction of buoyancy driven micropolar fluid flow within uniformly heated eccentric annulus

In this paper, the problem of buoyancy driven micropolar fluid flow within an annulus formed between two circular concentric/eccentric tubes has been numerically investigated using Fourier spectral method. The annulus inner wall is uniformly heated and maintained at constant heat flux while the outer wall is cooled and kept at constant temperature. The full governing equations of linear momentum, angular momentum and energy have been solved to give the details of flow and thermal fields. The heat convection process in the annulus is mainly controlled by modified Rayleigh number Ra, Prandtl number Pr, radius ratio Rr, eccentricity, e and material parameters of Micropolar fluid. The material parameters are dimensionless spin gradient viscosity λ, dimensionless micro-inertia density B and dimensionless vortex viscosity D. The study considered a range of modified Ra up to 10⁵ and is carried out at three values of Pr, namely Pr = 0.1, 1.0 and 7.0, and at three values of parameter D, namely, D = 2, 4, 8 while the eccentricity is varied between −0.65 and +0.65. The radius ratio is fixed at 2.6 while the material parameters B and λ are assigned the value of 1. The effect of the controlling parameters on flow and thermal fields has been investigated with emphasis on the effect of these parameters on local and mean inner wall temperatures. The study has shown that for certain controlling parameters the steady mean temperature of inner wall of the annulus is maximum at a certain eccentricity. The study has also shown that as the parameter D increases the steady mean inner wall temperature increases. Moreover, the study has shown that as the Pr increases the mean inner wall temperature decreases.     

MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical plate is studied numerically, taking into account the effects of Ohmic heating and viscous dissipation. A uniform magnetic field is applied perpendicular to the plate. The resulting governing equations are transformed into the non-similar boundary layer equations and solved using the Keller box method. Both the aiding-buoyancy mode and the opposing-buoyancy mode of the mixed convection are examined. The velocity and temperature profiles as well as the local skin friction and local heat transfer parameters are determined for different values of the governing parameters, mainly the magnetic parameter, the Richardson number, the Eckert number and the suction/injection parameter, f_w. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase owing to suction of fluid, increasing the Richardson number, Ri (i.e. the mixed convection parameter) or decreasing the Eckert number. This trend reverses for blowing of fluid and decreasing the Richardson number or decreasing the Eckert number. It is disclosed that the value of Ri determines the effect of the magnetic parameter on the momentum and heat transfer.     

The influence of heat transfer on peristaltic transport of a Jeffrey fluid in a vertical porous stratum

The peristaltic flow of a Jeffrey fluid in a vertical porous stratum with heat transfer is studied under long wavelength and low Reynolds number assumptions. The nonlinear governing equations are solved using perturbation technique. The expressions for velocity, temperature and the pressure rise per one wave length are determined. The effects of different parameters on the velocity, the temperature and the pumping characteristics are discussed. It is observed that the effects of the Jeffrey number λ₁, the Grashof number Gr, the perturbation parameter N = EcPr, and the peristaltic wall deformation parameter ϕ are the strongest on the trapping bolus phenomenon. The results obtained for the flow and heat transfer characteristics reveal many interesting behaviors that warrant further study on the non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear-thinning reduces the wall shear stress.     

Oberbeck convection through vertical porous stratum

Natural convection through a vertical porous stratum is investigated both analytically and numerically. Analytical solutions are obtained using a perturbation method valid for small values of buoyancy parameterN and the numerical solutions are obtained using Runge-Kutta-Gill method. It is shown that analytical solutions are valid forN < 1 and several features of the effect of large values ofN are reported. The combined effects of increase in the values of temperature difference between the plates and the permeability parameter on velocity, temperature, mass flow rate and the rate of heat transfer are reported. It is shown that higher temperature difference is required to achieve the mass flow rate in a porous medium equivalent to that of viscous flow.     

MHD-mixed convection from a vertical slender cylinder

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical slender cylinder is studied numerically. A uniform magnetic field is applied perpendicular to the cylinder. The resulting governing equations are transformed into the non-similar boundary layer equations and solved using the Keller box method. The velocity and temperature profiles as well as the local skin friction and the local heat transfer parameters are determined for different values of the governing parameters, mainly the transverse curvature parameter, the magnetic parameter, the electric field parameter and the Richardson number. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase, increasing the Richardson number, Ri (i.e. the mixed convection parameter), electric field parameter E₁ and magnetic parameter Mn.     

The effect of conjugate heat transfer on MHD mixed convection about a vertical slender hollow cylinder

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical slender hollow cylinder is studied numerically, under the effect of wall conduction. A uniform magnetic field is applied perpendicular to the cylinder. The non-similar solutions using the Keller box method are obtained. The wall conduction parameter, the magnetic parameter and the Richardson number are the main parameters. For various values of these parameters the local skin friction and local heat transfer parameters are determined. The validity of the methodology is checked by comparing the results with those available in the open literature and a fairly good agreement is observed. Finally, it is determined that the local skin friction and the local heat transfer coefficients increase with an increase the magnetic parameter Mn and buoyancy parameter Ri and decrease with conjugate heat transfer parameter p.     

Mixed convection of heat generating nanofluid in a lid-driven cavity with uniform and non-uniform heating of bottom wall

Steady state two-dimensional mixed convection in a lid-driven square cavity filled with Cu  –water nanofluid is investigated numerically in the presence of internal heat generation. In the present investigation, bottom wall is uniformly and non-uniformly heated while two vertical walls are fixed and they are thermally insulated. The top wall is moving from left to right at a constant speed. The governing equations are normalized and solved numerically with boundary conditions by finite volume approach using third order accurate upwinding scheme (deferred QUICK). Effects of the pertinent physical parameters are investigated in terms of the flow and temperature fields, as well as Nusselt number distributions. The presented results show that the solid volume fraction plays a significant role on the flow and thermal fields and the Nusselt number distributions for different flow configurations. It is found that Richardson number strongly affect the fluid flow and heat transfer in the cavity. For Ri<1$\mathit{Ri}<1$, the forced convection becomes dominant in the entire cavity, the natural convection relatively weak.     

Group method analysis of combined heat and mass transfer by MHD non-Darcy non-Newtonian natural convection adjacent to horizontal cylinder in a saturated porous medium

The group theoretic method is applied for solving problem of combined magneto-hydrodynamic heat and mass transfer of non-Darcy natural convection about an impermeable horizontal cylinder in a non-Newtonian power law fluid embedded in porous medium under coupled thermal and mass diffusion, inertia resistance, magnetic field, thermal radiation effects. The application of one-parameter groups reduces the number of independent variables by one and consequently, the system of governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The ordinary differential equations are solved numerically for the velocity using shooting method. The effects of magnetic parameter M, Ergun number Er, power law (viscosity) index n, buoyancy ratio N, radiation parameter R_d, Prandtl number Pr and Lewis number Le on the velocity, temperature fields within the boundary layer, heat and mass transfer are presented graphically and discussed.     

Transient nonlinear optically-thick radiative–convective double-diffusive boundary layers in a Darcian porous medium adjacent to an impulsively started surface: Network simulation solutions

A boundary-layer model is described for the two-dimensional nonlinear transient thermal convection heat and mass transfer in an optically-thick fluid in a Darcian porous medium adjacent to an impulsively started vertical surface, in the presence of significant thermal radiation and buoyancy forces in an (X¹,Y¹,t¹) coordinate system. An algebraic approximation is employed to simplify the integro-differential equation of radiative transfer for unidirectional flux normal to the plate into the boundary-layer regime, by incorporating this flux term in the energy conservation equation. The conservation equations are non-dimensionalized into an (X,Y,T) coordinate system and solved using the Network Simulation Method (NSM), a robust numerical technique which demonstrates high efficiency and accuracy. The transient variation of non-dimensional streamwise velocity component (u) and temperature (T) and concentration (C) functions is computed for various selected values of Stark number (radiation–conduction interaction parameter) and Darcy number. Transient velocity (u) and steady-state local skin friction (τ_X) are also studied for various thermal Grashof number (Gr), species Grashof number (Gm), Schmidt number (Sc) and Stark number (N) values. These computations for the infinite permeability case (Da → ∞) are compared with previous finite difference solutions [Prasad et al. Int J Therm Sci 2007;46(12):1251–8] and shown to be in excellent agreement. An increase in Darcy number is seen to accelerate the flow and boost velocity. A decrease in Stark number (corresponding to an increase in thermal radiation heat transfer contribution) is shown to increase the velocity values. Temperature function is observed to fall in value with a rise in Da and increase with decrease in N (corresponding to an increase in thermal radiation heat transfer contribution). Applications of the study include rocket combustion chambers, astrophysical flows, spacecraft thermal fluid dynamics in debris-laden environments (cosmic dust), heat transfer in forest fire spread, geochemical contamination and ceramic materials processing.     

Steady magnetohydrodynamic flow in a diverging channel with suction or blowing

An analysis is made of steady two-dimensional divergent flow of an electrically conducting incompressible viscous fluid in a channel formed by two non-parallel walls, the flow being caused by a source of fluid volume at the intersection of the walls. The fluid is permeated by a magnetic field produced by an electric current along the line of intersection of the channel walls. The walls are porous and subjected to either suction (k > 0) or blowing (k < 0) of equal magnitude on both the walls. It is found that when the Reynolds number for the flow is large and the magnetic Reynolds number is very small, boundary layers are formed on the channel walls such that a sufficient condition for the existence of a unique boundary layer solution (without separation) in the case of suction is N > 2, N being the magnetic parameter. When k = 0, boundary layer exists without separation only when N > 2. Further, it is found that the necessary and sufficient condition for the existence of a unique solution for boundary layer flow (without separation) even in the presence of blowing (k < 0) is N > 2. For given value of k, velocity at a point increases with increase in N. It is also shown that when N > 2, blowing makes the boundary layer thinner. A similarity solution for steady temperature distribution in the divergent flow is also presented when the channel walls are held at variable temperature. It is found that for fixed value of wall suction, temperature at a point decreases with increase in N. It is further shown that when N > 2, steady distribution of temperature exists even in the case of blowing at the walls.     

Forced convection over a semi-infinite flat plate

The problem of forced convection heat transfer over a semi-infinite flat plate is treated by the method of series truncation, so as to yield results valid from leading edge to far downstream (0 < -R ₀< ∞). Results are presented for Prandtl number Pr = 0.1, 0.7, and 10. It is found that the effect of leading edge on heat transfer is smaller than on skin friction.     

Mixed convection of a viscous dissipating fluid about a vertical flat plate

In this study, the effect of the viscous dissipation in steady, laminar mixed convection heat transfer from a heated/cooled vertical flat plate is investigated in both aiding and opposing buoyancy situations. The external flow field is assumed to be uniform. The governing systems of partial differential equations are solved numerically using the finite difference method. A parametric study is performed in order to illustrate the interactive influences of the governing parameters, mainly, the Richardson number, Ri (also known as the mixed convection parameter) and the Eckert number, Ec on the velocity and temperature profiles as well as the friction and heat transfer coefficients. Based on the facts the free stream is either in parallel or reverse to the gravity direction and the plate is heated or cooled, different flow situations are identified. The influence of the viscous dissipation on the heat transfer varied according to the situation. For some limiting cases, the obtained results are validated by comparing with those available from the existing literature. An expression correlating Nu in terms of Pr, Ri and Ec is developed.     

Analysis of flow and heat transfer of viscous fluid between contracting rotating disks

The present work investigates the effects of disks contracting, rotation and heat transfer on the viscous fluid between heated contracting rotating disks. By introducing the Von Kármán type similarity transformations through which we reduced the highly nonlinear partial differential equation to a system of ordinary differential equations. This system of differential equations with appropriate boundary conditions is responsible for the flow behavior between large but finite coaxial rotating and heated disks. It is important to note that the lower disk is rotating with angular velocity Ω while the upper one with SΩ, the disks are also contracting and the temperatures of the upper and lower disks are T₁ and T₀, respectively. The agents which driven the flow are the contraction and also the rotation of the disks. On the other hand the velocity components and especially radial component of velocity strongly influence the temperature distribution inside the flow regime. The basic equations which govern the flow are the Navier Stokes equations with well known continuity equation for incompressible flow. The final system of ordinary differential equations is then solved numerically with given boundary conditions. In addition, the effect of physical parameters, the Reynolds number (Re), the wall contraction ratio (γ) and the rotation ratio (S) on the velocity and pressure gradient, as well as, the effect of Prandtl number (Pr) on temperature distribution are also observed.     

Similarity solutions for mixed convection boundary layer flow over a permeable horizontal flat plate

The effects of suction and injection on steady laminar mixed convection boundary layer flow over a permeable horizontal flat plate in a viscous and incompressible fluid is investigated in this paper. The similarity solutions of the governing boundary layer equations are obtained for some values of the suction and injection parameter f₀, the constant exponent n of the wall temperature as well as the mixed convection parameter λ. The resulting system of nonlinear ordinary differential equations is solved numerically for both assisting and opposing flow regimes using a finite-difference scheme known as the Keller-box method. Numerical results for the reduced skin friction coefficient, the reduced local Nusselt number, and the velocity and temperature profiles are obtained for various values of the parameters considered. Dual solutions are found to exist for the opposing flow.     

Analyses on 3-D gas flow and heat transfer in ladle furnace lid

In order to verify the reasonableness of off-gas pressure and wall temperature, a mathematical model for gas flow and heat transfer in ladle furnace (LF) lid is developed based on 3-D Navier–Stokes equations and k–ε two equation turbulent models as well as energy conservation equation. The gas velocity vector distribution of skirt clearance between the top edge of ladle and furnace lid and electrode gaps between three graphite electrodes and furnace lid, the gas flow line distribution, pressure and temperature distribution on the furnace lid wall are simulated. Simulation results show that appropriate off-gas pressures are 200 Pa, 200 Pa and 150 Pa when electric arc emerges from molten steel surface and alloy hole is unsealed, electric arc emerges from molten steel surface and alloy hole is closure, electric arc immerges into molten steel surface and alloy hole is closure, respectively. The maximum temperature presents in the middle of LF lid in all of heating conditions, and the temperature value are 563, 603 and 343 K. Finally, the relations between gas volume and off-gas pressure are analyzed in different width of skirt clearance, and some relevant mathematical expressions are obtained. By comparing both simulation results and practical data, the advice on reducing off-gas pressure is proposed, and the maximum temperatures of furnace lid wall have good agreement with actual data.     

Application of the similarity theory including variable property effects to a complex benchmark problem

An asymptotic method to account for variable property effects, recently described in this journal, is now applied to a complex benchmark geometry. It is a room which is ventilated by forced convection through inlet and outlet slit nozzles at the top and bottom of the side walls. Four heating elements standing on the ground floor add heat with constant heat flux density of varying strength. CFD solutions with the full coverage of all property temperature dependencies of air and SF₆ are compared with asymptotic results (ACFD), applied for these fluids. ACFD results are given as systematic expansions with respect to a heat transfer parameter e{\varepsilon} which serves as perturbation parameter. First and second order asymptotic results of the Nu?elt number at the surface of the heating elements are shown as well as temperature distributions along the adiabatic walls of the room. Special attention is given to the reference Nu?elt numbers of zero order (e = 0){(\varepsilon=0)} which are those for constant properties only for pure forced convection.