Magnetohydrodynamic mixed convection in a non-Newtonian third-grade fluid flowing through vertical parallel plates: A semianalytical study of flow and heat transfer

Buoyancy assisted and buoyancy opposed mixed convection of a third-grade fluid, which flows through vertically oriented parallel plates, subjected to uniform and constant wall heat fluxes, under the effect of an externally applied magnetic field, are investigated. The coupled, nonlinear conservation equations of momentum and energy are solved employing the collocation method (CM) and velocity and temperature distributions are solved semianalytically. The results produced by the CM and the results of exact solution are compared for the buoyancy assisted and buoyancy opposed flow of a Newtonian fluid through the vertically oriented parallel plates arrangement without the effect of the externally applied magnetic field. An excellent agreement is exhibited by demonstrating the efficacy of the CM. The effects of the third-grade fluid parameter, Hartmann number, and mixed convection parameter on the dimensionless velocity, temperature, and Nusselt number are studied. The results imply that in the case of buoyancy assisted flow, an increment in the non-Newtonian third-grade fluid parameter causes a decrease in the fluid velocity near the plate walls, which finally causes an increase in the velocity in the central core of the plates. In buoyancy opposed flow, the effect of the same parameter is to oppose the flow reversal near the walls and with higher values of this parameter, it can totally prevent the flow reversal near the walls. The results of the present study can be useful in the fields of flow and heat transfer of various grades of polymers, paints, and food processing.     

Free convection flow through a porous medium bounded by two horizontal walls

Steady free convection flow through a porous medium bounded by two horizontal walls with a uniform axial temperature variation along the walls is considered. We give an analytical solution of the velocity field and discuss the effects of the permeability parameter K and Grashof number G on the velocity.     

Prediction of temperature distribution and Nusselt number in rectangular microchannels at wall slip condition for all versions of constant wall temperature

Slip flow in rectangular microchannels heated at constant and uniform wall temperature (H1 boundary condition) is studied. The study is extended to the eight possible thermal versions that are formed of different combinations of heated and adiabatic walls. Integral transform method is applied to derive the velocity and temperature distributions and thus, the average Nusselt number for all the eight thermal versions. It is found that, for microchannels with perfect accommodation for velocity and temperature, the rarefaction has a decreasing effect on heat transfer for all the eight thermal versions. The results of the paper for the special case of non-slip flow agree exactly with the results found for macrochannels in literature.     

Heat transfer in rectangular microchannels

Convection heat transfer in a rectangular microchannel is investigated. The flow is assumed to be fully developed both thermally and hydrodynamically. The H2-type boundary condition, constant axial and peripheral heat flux, is applied at the walls of the channel. Since the velocity profile for a rectangular channel is not known under the slip flow conditions, the momentum equation is first solved for velocity. The resulting velocity profile is then substituted into the energy equation. The integral transform technique is applied twice, once for velocity and once for temperature. The results show a similar behavior to previous studies on circular microtubes. The values of the Nusselt number are given for varying aspect ratios.     

Influence of surface radiation on the transition to unsteadiness for a natural convection flow in a differentially heated cavity

Abstract

The influence of surface radiation on the transition to the unsteady state in natural convection is studied numerically. The configuration of the differentially heated square cavity with adiabatic horizontal walls is chosen to generate an internal natural convection flow. It is known that radiative transfers reduce the temperature difference between the adiabatic walls, which consequently reduces the thermal stratification of the central zone and increases the velocity flow. Many studies have focused on the stationary regime, but few of them have investigated the transition to unsteady flow. For this purpose, the effect of the wall emissivity on the critical Rayleigh number and the associated critical frequency was studied for a given cavity length. The cavity length and mean temperature of isothermal walls are set for the whole study. The results show that all these values are between the values obtained without radiation and those obtained for perfectly conducting horizontal walls. The critical Rayleigh number decreases with emissivity while the associated frequency increases. Moreover, the symmetry of fluctuating properties of the flow is changed when the radiation is taken into account.     

Experimental and numerical study of heat and moisture transfers by natural convection in a cavity filled with solid obstacles

Heat, moisture transfers and airflow by natural convection in a rectangular cavity containing on line cylinders were studied. The work zone was arranged in such a way that 2D transfer and flow were established. At steady state, temperature, velocity and humidity fields on the symmetry plane were measured in un-humidified and humidified cavity. These results were then used to compare with CFD simulation. The thermal stratification and circular air flow in the cavity was observed. Humidification at the bottom face of cavity contributes to increase air velocity. The influence of radiation near the cold and warm walls is significant.     

Differentially heated cubical cavity using energy pathlines and field synergy

The article deals with the natural convective flow of air in a cubical cavity which is analyzed numerically. Isothermal temperature is maintained on the vertical walls where the temperature of the left wall is more than the right wall and all other walls are assumed to be kept insulated. In this present article, upwind, QUICK, SUPERBEE, and self‐filtered central differencing schemes are compared based on their accuracy and computational time with a numerical example. An attempt has been made to analyze the flow behavior inside the cavity using vortex corelines, streamlines, isotherms energy pathlines, and field synergy by varying the Rayleigh number (Ra) from 10³ to 10⁶. In the vicinity of isothermal vertical walls, the velocity, and temperature boundary layers become thinner as Ra increases. The energy pathlines are in oscillating nature when Ra increases to 10⁵ and above. The field synergy principle implies by improving the synergy between the velocity and temperature, the heat transfer gets enhanced with the less increased flow resistance.     

Magnetohydrodynamic chemically reactive viscoelastic fluid flow through an inclined porous layer

Analysis of a time-independent magnetohydrodynamic viscoelastic fluid flow in a deformable inclined porous layer with first-order chemical reaction has been investigated. Walters' fluid model has been used to study viscoelastic fluid. The walls are suctioned/injected at a constant rate. The expression representing the solution for solid displacement, fluid velocity, temperature, and concentration distribution is obtained. The effect of applicable parameters on solid displacement, fluid velocity, temperature, and concentration are discussed graphically, while skin friction, heat transfer, and mass transfer are revealed in a tabular structure. It is noticed that solid displacement, fluid velocity, and temperature profiles decrease when the viscoelastic parameter increase. Solid displacement enhances and the velocity of the fluid reduces owing to the influence of increasing drag parameter, whereas the reverse effect is seen for the volume fraction parameter. Nusselt number at the walls shows the opposite behavior for the viscoelastic parameter and Eckert number. Sherwood number at the walls shows opposite behavior for Reynolds number, Schmidt number, and radiation parameter. Also, the entropy generation number rises as a result of the influence of viscoelasticity and Eckert number.     

Nonequilibrium Gas Flow and Heat Transfer in a Heated Square Microcavity

The flow of a rarefied gas in a square enclosure with one wall at high temperature and the other three walls at the same low temperature is investigated. The flow, characterized by the reference Knudsen number and ratio of the cold over the hot temperatures, is simulated both deterministically, using the nonlinear Shakhov kinetic model, and stochastically, using the DSMC method. Excellent agreement between the two approaches is obtained. It is found that along the side walls the gas velocity, depending on the flow parameters, may be either from cold to hot or from hot to cold regions. Furthermore, it is confirmed that the average heat flux departing from the hot plate exhibits a nonmonotonic behavior with regard to the temperature ratio, deducing a maximum heat flux at a temperature ratio of about 0.3. The flow and heat transfer characteristics are explained by computing the ballistic and collision parts of the total bulk quantities and by investigating the contribution of each part to the overall solution.     

Subsonic compressible flow in two-sided lid-driven cavity. Part I: Equal walls temperatures

This paper presents a numerical study of the laminar, viscous, subsonic compressible flow in a two-dimensional, two-sided, lid-driven cavity using a multi-domain spectral element method. The flow is driven by steadily moving two opposite walls vertically in opposite directions. All the bounding walls have equal temperatures. The results of the simulations are used to investigate the effects of the cavity aspect ratio, the Reynolds number and the Mach number on the flow. At lower Reynolds numbers, the flow pattern consists of two separate co-rotating vortices contiguous to the moving walls. For higher Reynolds numbers, initially a two-vortex flow is formed, which eventually turns into a single elliptical vortex occupying most of the cavity. For a higher aspect ratio, the flow patterns are dissimilar in that the streamlines become more and more elliptic. For aspect ratios as high as 2.5, at high Reynolds numbers, a three-vortex stage is formed. It is found that the compressibility effects are not very significant for Mach numbers less than 0.4. Dissipation of kinetic energy into internal energy changes the temperature field especially near the boundaries. Boundary layer studies suggest that the velocity and temperature boundary layer thicknesses are lower for higher Reynolds numbers. For engineering purposes, these thicknesses can be approximated by the existing flat-plate solutions.     

DOUBLE-DIFFUSIVE CONVECTION IN A POROUS ENCLOSURE WITH COOPERATING TEMPERATURE AND CONCENTRATION GRADIENTS AND HEAT GENERATION OR ABSORPTION EFFECTS

The problem of unsteady, laminar double-diffusive convective flow of a binary gas mixture in a rectangular enclosure filled with a uniform porous medium is considered. A temperature-dependent heat source or sink is assumed to exist within the enclosure boundaries. Transverse cooperating gradients of heat and mass are applied on the two opposing vertical walls of the enclosure while the other two horizontal walls are adiabatic and impermeable to mass transfer. A numerical solution based on the finite-difference methodology is obtained. Representative results illustrating the effects of the inverse Darcy number, the heat generation or absorption coefficient, and the buoyancy ratio on the contour maps of the streamline, temperature, and concentration as well as the profiles of velocity, temperature, and concentration at the midsection of the enclosure are reported. In addition, results for the average Nusselt and Sherwood numbers are presented in tabulated form and discussed for various parametric conditions.     

Filtered Smith predictor with feedback linearization and constraints handling applied to a solar collector field

Feedback linearization (FL) has recently been used to control the temperature in a solar field using the water flow rate as the manipulated variable. Nevertheless, the system shows undesirable control signal oscillation caused by a delay uncertainty. In this paper, the dynamics resulting from applying the FL transformation to the nonlinear process model is approximated by a first-order model plus dead-time, and a filtered Smith predictor (FSP) is proposed as the control strategy. This controller deals with dead-time errors but also includes control signal saturation caused by flow rate constraints. Simulations demonstrate the robustness of the FSP structure in comparison to a simple PID and a Smith predictor (SP) approach when a faster closed-loop is required in a system with delay uncertainties. Real experimental results included show the improved behavior of the proposed control scheme.     

Energy losses and heat transfer enhancement in transversally corrugated pipes

Wall friction, temperature distribution and heat transfer through pipe walls are investigated in forced convection with Newtonian fluids in pressure gradient driven hydrodynamically and thermally fully developed steady laminar flow in transversally corrugated pipes. Novel analytical solutions derived via the epitrochoid conformal mapping are presented for the velocity and temperature fields. Analytical results are compared with numerical solutions obtained using the finite volume technique. The effect of the corrugation amplitude and the number of waves on the friction factor, the temperature distribution and the Nusselt number is discussed.     

Fully-developed, forced convective flow through an annular packed-sphere bed with wall effects

An analysis is performed for a fully-developed, forced convective flow through a packed-sphere bed between concentric cylinders maintained at different temperatures. The radial variations of the porosity and permeability in the bed near the walls, known as wall effects, are approximated by exponential functions. The Brinkman model with variable permeability is used as the momentum equation. An analytical solution based on the method of matched asymptotic expansions is obtained for the velocity distribution. It is shown that velocity overshoots occur in the variable permeability bed near the inner and outer cylinders. Because of the non-uniform porosity variation near the walls, the stagnant thermal conductivity of the bed also varies in the radial direction accordingly. A mixing length theory, proposed recently by Cheng and Vortmeyer for the transverse thermal dispersion, is employed to obtain the radial temperature distribution and the Nusselt number of the annular bed. Computations of the heat transfer characteristics were carried out based on three velocity models, i.e. Brinkman's model with variable and constant permeabilities as well as the plug flow model. It is found that with the mixing length theory, theoretical predictions of the heat transfer characteristics based on the three velocity models are in good agreement with the existing experimental data. The predicted temperature profiles, based on the Brinkman model with a variable permeability, agree the best with temperature data.     

Role of suction/injection and slip flow on hydromagnetic free convective flow in a vertical coaxial cylinder under the influence of radial magnetic field

This investigation attempts to address heat and mass transfer behavior exhibited by a steady fully developed natural convective flow of a viscous, incompressible, and electrically conducting fluid in a vertical porous annulus in the presence of radially applied magnetic field and velocity slip. The motion of the fluid in the annular gap is triggered by the buoyancy forces due to temperature gradient of the inner and outer cylinders. The governing momentum and energy equations responsible for the flow are transformed into dimensionless forms using the appropriate dimensionless parameters. Accordingly, analytical solutions of the energy and momentum fields are derived with the appropriate boundary conditions. The effects of the controlling parameters involved in the flow on the temperature field, velocity field, and drag on the walls of the cylinders are illustrated graphically and with the aid of tables. Findings affirm that fluid temperature can be decreased/increased by increasing suction/injection on the porous wall. Furthermore, the fluid flow in the annular gap can be enhanced by increasing Grashof number, fluid injection, and velocity slip.     

Joule heating in magnetohydrodynamic flows in channels with thin conducting walls

Joule heating in liquid metal magnetohydrodynamic flows is investigated with reference to self-cooled liquid metal blankets for tokamaks. Pressure-driven flow of an electrically conducting fluid confined between two parallel, infinite walls with a transverse magnetic field is studied. The walls are electrically conducting, which implies strong currents flowing within the thin conducting walls. The problem is solved both analytically and numerically.It is shown that the Joule heat cannot be neglected in certain range of parameters relevant to fusion blanket applications. The magnitude of the Joule heat released inside the channel and the walls depends on the thermal conductivity of the outside surface of the channel walls. For thermally conducting outside surface of the walls the Joule heat can become significant for high values of the Hartmann number and moderate average velocity. The effect is even more pronounced for thermally insulating outside surface of the walls. For example, for lead–lithium flow with stainless steel walls the temperature increase along the flow exceeds 200 °C over the length of the blanket, which is almost three times higher than that for thermally conducting outside surface of the walls.The main reason for such a strong rise in temperature is the heat released inside the walls. The heat produced in the fluid region is quickly convected towards the exit from the channel. The heat released inside the walls can only leave the domain by diffusion into the fluid region and thus is accumulated along the channel length.     

Combined convection heat transfer in a porous lid-driven enclosure due to heater with finite length

A numerical work is performed to analyze combined convection heat transfer and fluid flow in a partially heated porous lid-driven enclosure. The top wall of enclosure moves from left to right with constant velocity and temperature. Heater with finite length is located on the fixed wall where its center of location changes along the walls. The finite volume-based finite-difference method is applied for numerical experiments. Parameters effective on flow and thermal fields are Richardson number, Darcy number, center of heater and heater length. The results are shown that the best heat transfer is formed when the heater is located on the left vertical wall.     

Entropy Analysis of a Coupled‐Convection Flow through a Vertical Channel Partially Filled by a Porous Medium with Injection/Suction and Slip Boundary Conditions

Flow fields, thermal fields, and entropy generation have been investigated for fully developed mixed convection flow between two vertical porous plates. The vertical channel is partially filled by a porous medium, and channel walls are subjected to a constant injection velocity at the left wall and constant suction velocity at the right wall. The viscous dissipation effects and velocity slip for the longitudinal component of the velocity at the channel walls are also taken into account. The momentum and energy equations for the mixed convection problem in the vertical channel are solved by means of the perturbation series method, by taking perturbation parameter proportional to the Brinkman number. For the present problem, numerical solution is also obtained and compared with the analytical solution. The effects of various pertinent parameters on the velocity distribution, temperature distribution, entropy generation rate, and Bejan number are investigated and discussed graphically.     

Influence of differentially heated horizontal walls on the streaming shape and velocity in a standing wave resonator

Influence of differentially heated horizontal walls on shape and amplitude of acoustic streaming velocity field inside a gas-filled rectangular enclosure subject to acoustic standing wave are experimentally investigated. The synchronized particle image velocimetry (PIV) technique has been used to measure the streaming velocity fields. The results show that the temperature difference between the top and the bottom walls deforms the symmetric streaming vortices to the asymmetric ones. As the temperature difference increases, the amplitude of streaming velocity increases.     

Hydromagnetic free convection of a radiating fluid

Natural convection of an electrically conducting and radiating fluid in the presence of an external magnetic field is investigated numerically. The two opposing side walls are differentially heated with a temperature difference specified, while the top and bottom walls are insulated. The coupled momentum and energy equations associating with the electromagnetic retarding force as well as the buoyancy force terms are solved by an iterative procedure using the SIMPLER algorithm based on control volume approach. Steady-state conditions are assumed. The finite-volume method is utilised to solve the radiation transport adopting the same computational grid as used in solving the flow field, with which the radiating fluids in an enclosure are assumed to be radiatively opaque, transparent and participating, respectively. After validating the numerical procedures, the changes in the buoyant flow patterns and temperature distribution affected by combined radiation and a magnetic field are focused mainly. Comparative results for the velocity profiles and the heat transfer rates are presented too. Based on the results of this study, it was found that the radiation played a significant role in developing the hydromagnetic free convective flow in a differentially heated enclosure.