Effect of induced magnetic field on natural convection in vertical concentric annuli

In the present paper,we have considered the steady fully developed laminar natural convective flow in open ended vertical concentric annuli in the presence of a radial magnetic field.The induced magnetic field produced by the motion of an electrically conducting fluid is taken into account.The transport equations concerned with the considered model are first recast in the non-dimensional form and then unified analytical solutions for the velocity,induced magnetic field and temperature field are obtained for the cases of isothermal and constant heat flux on the inner cylinder of concentric annuli.The effects of the various physical parameters appearing into the model are demonstrated through graphs and tables.It is found that the magnitude of maximum value of the fluid velocity as well as induced magnetic field is greater in the case of isothermal condition compared with the constant heat flux case when the gap between the cylinders is less or equal to 1.70 times the radius of inner cylinder,while reverse trend occurs when the gap between the cylinders is greater than 1.71 times the radius of inner cylinder.These fields are almost the same when the gap between the cylinders is equal to 1.71 times the radius of inner cylinder for both the cases.It is also found that as the Hartmann number increases,there is a flattening tendency for both the velocity and the induced magnetic field.The influence of the induced magnetic field is to increase the velocity profiles.     

Natural convection in vertical concentric annuli under a radial magnetic field

Exact solutions for fully developed natural convection in open-ended vertical concentric annuli under a radial magnetic field are presented. Expressions for velocity field, temperature field, mass flow rate and skin-friction are given, under more general thermal boundary conditions. It is observed that both velocity as well as temperature of the fluid is more in case of isothermal condition compared with constant heat flux case when gap between cylinders is less or equal to radius of inner cylinder while reverse phenomena occur when the gap between cylinders is greater than radius of inner cylinder.     

Prandtl number effects on natural convection heat transfer in concentric and eccentric horizontal cylindrical annuli

The influence of various Prandtl numbers on the laminar convection flow between concentric and vertically eccentric cylinders is studied numerically. To overcome the difficulties associated with the complex physical domains a numerical transformation method is used to map this region on a rectangle. Although two independent computer programs which are based on different formulations of the governing equations were used, nearly identical results were obtained. Local heat transfer results are presented for a wide range of Rayleigh numbers for the first time. Local heat transfer rates are found to depend on the Prandtl number in addition to the Rayleigh number dependence.     

Transient natural convection between horizontal concentric cylinders: Natural convection between concentric cylinders

The initial phase of transient natural convection between two horizontal concentric cylinders is investigated. Short-time solutions to the momentum and energy equations are obtained using the method of matched asymptotic expansions. The solutions for the velocity and temperature are expressed in terms of three expansions reflecting the existence of three distinct regions in the initial flow field, and four different patterns of motion may be distinguished, corresponding to four different types of thermal boundary conditions.     

Transient natural convection through vertical porous stratum

An analytical study is made to examine the flow behaviour of a fully developed transient free-convective flow of an incompressible viscous fluid between two heated vertical walls in a porous system. A Laplace transform technique has been employed to obtain the expression for velocity, temperature and skin-friction. The influence of the various parameters, entering into the problem, on the velocity field and skin-friction is discussed in detail. Received on 11 March 1997     

Laminar natural convection of Cu-water nanofluid in concentric annuli with radial fins attached to the inner cylinder

Laminar natural convection of Cu-water nano-fluid between two horizontal concentric cylinders with radial fins attached to the inner cylinder is studied numerically. The inner and outer cylinders are maintained at constant temperature. The governing equations in the polar two-dimensional space with the respective boundary conditions are solved using the finite volume method. The hybrid-scheme is used to discretize the convection terms. In order to couple the velocity field and the pressure in the momentum equations, the well known semi-implicit method for pressure linked equation reformed algorithm is adopted. Using the developed code, a parametric study is undertaken, and the effects of the Rayleigh number, Number of fins, length of the fins and the volume fraction of nano-particles on the fluid flow and heat transfer inside the annuli are investigated. In this study, two cases with different number of fins are considered. It is observed from the results that the average Nusselt number increases with increasing both the Rayleigh number and the volume fraction of the nano-particles. Moreover, the average Nusselt number decreases by increasing the fins’ length and the number of fins. Heat transfer rate increases by increasing the fins’ length at all Rayleigh numbers, but it increases by increasing the number of fins at high Rayleigh numbers.     

A new study of laminar natural convection in two concentric vertical cylinders

5 and cylinder aspect ratios of 0.5 to 10. The effects of the changes in these parameters on the variation of the temperature and flow patterns within the cylinder have been studied. The local and average Nusselt numbers correlations of interest in determining the insulating value of an annular cavity formed by two concentric vertical cylinders have been obtained. Received on 24 July 1998     

Developing mixed convection in vertical eccentric annuli

This paper deals with the problem of combined (forced–free) convection in vertical eccentric annuli with simultaneously developing hydrodynamic and thermal boundary layers. A bipolar model has been developed and a numerical algorithm for solving this model is outlined. Results, not available in the literature, are presented for the developing velocity profiles, axial variation of pressure, full development length, and heat transfer parameters under thermal boundary conditions of having one of the annulus boundaries at a constant temperature while the other boundary is insulated. Both aiding and opposing free convection have been considered and possibilities of flow reversal occurrence have also been checked. After a distance from the channel entrance and provided that the value of Gr/Re is sufficiently large, aiding free convection can develop to overcome the fluid friction and the eccentric annular channel eventually works as a diffuser. The value of Gr/Re for which a vertical eccentric annular channel can work as a diffuser decreases as the eccentricity increases. The axial distance from the entrance at which the channel starts to work as a diffuser decreases as Gr/Re increases.     

Laminar mixed convection in horizontal concentric annuli with non-uniform circumferential heating

 Steady, laminar, mixed convection in the fully developed region of horizontal concentric annuli has been investigated numerically for the case of non-uniform circumferential heating. Two heating conditions were studied, one in which a 180^∘ arc encompassing the top half of inner surface of the inner cylinder is uniformly heated while the bottom half is kept insulated, and the other in which the heated and the insulated surfaces were reversed. The fluid flow and heat transfer characteristics were found to be affected by the heating conditions. For the investigated range of the governing buoyancy parameter, the modified Grashof number (Gr^*), it was found that bottom heating arrangement gives rise to a vigorous secondary flow, with the result that the average Nusselt numbers are much higher than those for pure forced convection. On the other hand, the local Nusselt numbers are nearly circumferentially uniform. In the case of top heating arrangement, a less vigorous secondary flow is induced because of temperature stratification, with average Nusselt numbers that are substantially lower than those for bottom heating and with large circumferential variation of the local Nusselt number. Received on 15 March 2000     

Skewed Poiseuille-Couette flows of sPTT fluids in concentric annuli and channels

Analytical solutions have been derived for the helical flow of PTT fluids in concentric annuli, due to inner cylinder rotation, as well as for Poiseuille flow in a channel skewed by the movement of one plate in the spanwise direction, which constitutes a simpler solution for helical flow in the limit of very thin annuli. Since the constitutive equation is a non-linear differential equation, the axial and tangential/spanwise flows are coupled in a complex way. Expressions are derived for the radial variation of the axial and tangential velocities, as well as for the three shear stresses and the two normal stresses. For engineering purposes expressions are given relating the friction factor and the torque coefficient to the Reynolds number, the Taylor number, a nondimensional number quantifying elastic effects (εDe²) and the radius ratio. For axial dominated flows fRe and C_M are found to depend only on εDe² and the radius ratio, but as the strength of rotation increases both coefficients become dependent on the velocity ratio (ξ) which efficiently compacts the effects of Reynolds and Taylor numbers. Similar expressions are derived for the simpler planar case flow using adequate non-dimensional numbers.     

Developing turbulent flow in concentric annuli: An analytical and experimental study

In this paper, the problem of the developing turbulent flow in concentric annuli is studied from an integral view-point based on a modified model ofReichardt's expression for the eddy diffusivity of momentum. The analytical results are compared with the experimental data based on the measurement of local flow conditions for air flow through four concentric annuli for a Reynolds number range of about 20,000 to 110,000. In the analysis, it was assumed that the flow is turbulent everywhere and in the experimental work, the flow was tripped at the starting position of the boundary layers.     

Transient natural convection between concentric and vertically eccentric spheres with mixed boundary conditions

Transient analysis has been investigated numerically to determine heat transfer by natural convection between concentric and vertically eccentric spheres with constant heat flux on the inner wall and a specified isothermal temperature on the outer wall. The governing equations, in terms of vorticity, stream function and temperature are expressed in a spherical polar coordinate system. The alternating direction implicit method and the successive over-relaxation techniques are applied to solve the finite difference form of governing equations. A physical model is introduced which accounts for the effects of fluid buoyancy as well as eccentricity of the outer sphere. Transient solutions of the entire flow field are obtained for a range of modified Rayleigh number (10³<Ra?<5×10⁵), for a Prandtl number of 0.7 and a radius ratio of 2.0, with the outer sphere near the top and bottom of the inner sphere (ε=±0.625). Results of the parametric study conducted further reveal that the heat and flow fields are primarily dependent on the modified Rayleigh number and the eccentricity of the spherical annulus. The results of average Nusselt numbers are also compared with the results obtained for flow between two isothermal spheres.     

A Cartesian grid technique based on one‐dimensional integrated radial basis function networks for natural convection in concentric annuli

This paper reports a radial basis function (RBF)‐based Cartesian grid technique for the simulation of two‐dimensional buoyancy‐driven flow in concentric annuli. The continuity and momentum equations are represented in the equivalent stream function formulation that reduces the number of equations from three to one, but involves higher‐order derivatives. The present technique uses a Cartesian grid to discretize the problem domain. Along a grid line, one‐dimensional integrated RBF networks (1D‐IRBFNs) are employed to represent the field variables. The capability of 1D‐IRBFNs to handle unstructured points with accuracy is exploited to describe non‐rectangular boundaries in a Cartesian grid, while the method's ability to avoid the reduction of convergence rate caused by differentiation is instrumental in improving the quality of the approximation of higher‐order derivatives. The method is applied to simulate thermally driven flows in annuli between two circular cylinders and between an outer square cylinder and an inner circular cylinder. High Rayleigh number solutions are achieved and they are in good agreement with previously published numerical data. Copyright © 2007 John Wiley & Sons, Ltd.     

Transient natural convection along vertical cylinder with Heat and Mass transfer

A numerical solution for the transient natural convection flow over a vertical cylinder under the combined buoyancy effect of heat and mass transfer is presented. The velocity, temperature and concentration profiles, local and average skin-friction, Nusselt number and Sherwood number are shown graphically. It is observed that time taken to reach steady state increases with Schmidt number and decreases as combined buoyancy ratio parameter N increases. Stability and convergence of the finite difference scheme are established. Received on 8 July 1997     

Transient natural convection flow of a nanofluid over a vertical cylinder

An analysis is performed to study unsteady free convective boundary layer flow of a nanofluid over a vertical cylinder. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The governing equations are formulated and a numerical solution is obtained by using an explicit finite-difference scheme of the Crank-Nicolson type. The solutions at each time step have been found to reach the steady state solution properly. Numerical results for the steady-state velocity, temperature and nanoparticles volume fraction profiles as well as the axial distributions and the time histories of the skin-friction coefficient, Nusselt number and the Sherwood number are presented graphically and discussed.     

Heat transfer to non-Newtonian fluids in laminar flow through concentric annuli with or without suction

Rheologica Acta - In this paper we have investigated the heat transfer aspects of the flow of a power law fluid in an annulus with porous walls. The case of no suction (solid walls annulus) is also...     

Combined convection and radiation heat transfer to thermally developing laminar droplet flow in concentric annuli

This paper presents numerical results for combined convection and radiation heat transfer to a laminar mist flow in the thermal entrance region of a concentric annulus with a heated core at constant wall temperature and an insulated outer wall. The saturated droplets in the mist flow are considered as equivalent heat sinks distributed in the superheated vapor stream. Numerical calculations are performed for the variations of droplet size, mean vapor velocity, and the local Nusselt number in the streamwise direction until the single-phase fully-developed condition is reached. The important role of the saturated droplets on combined convection and radiation heat transfer to mist flow is clearly demonstrated.

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Kombinierte Wärmeübertragung durch Konvektion und Strahlung im thermischen Einlauf einer laminaren Tröpfchenströmung in einem konzentrischen Ringspalt

Zusammenfassung Dieser Artikel stellt numerische Ergebnisse für kombinierte Wärmeübertragung durch Konvektion und Strahlung im thermischen Einlauf einer laminaren Tröpfchenströmung in einem konzentrischen Ringraum mit beheiztem Kern bei konstanter Wandtemperatur und isolierter Außenwand dar. Die gesättigten Tröpfchen wirken als verteilte Wärmesenken im überhitzten Dampfstrom. Numerische Berechnungen werden unter Variation des Tröpfchendurchmessers, der durchschnittlichen Dampfgeschwindigkeit und der Nusselt-Zahl durchgeführt, bis eine einphasige vollausgebildete Strömung erreicht ist. Der wichtige Einfluß der gesättigten Tröpfchen auf die kombinierte Wärmeübertragung durch Konvektion und Strahlung wird klar gezeigt.

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Nomenclature A liquid loading parameter, defined in Eq. (3) - A _d heat transfer area of droplets per unit volume of vapor - A _w heat transfer area of heated wall per unit volume of vapor - C wall superheat parameter, defined in Eq. (5) - C _p specific heat of vapor - D dimensionless droplet diameter,d/d ₀ - D _h hydraulic diameter, 2(r ₀–r _i) - d droplet diameter - d ₀ droplet diameter at thermal entrance (x=0) - E dimensionless parameter, defined in Eq. (6) - H dimensionless parameter, defined in Eq. (7) - F _w–d geometric view factor - h _d heat transfer coefficient for evaporating droplets - h _p0 heat transfer coefficient of non-evaporating droplet or solid sphere with diameter ofd ₀ - k thermal conductivity of vapor - n droplet number density (number of droplets per unit volume of vapor) - n ₀ droplet number density at thermal entrance (x=0) - Nu _x local Nusselt number, defined by Eq. (17) - Pr Prandtl number of vapor,C _p/k - Q _r radiative heat transfer to droplets (per unit volume of vapor) - q _w heat flux at the inner wall - R dimensionless radial position,r/r _i - Re Reynolds number of vapor, 2 _v V₀ r _i/ - r radial position - r _i radius of inner tube - r _o radius of outer tube - S heat sink parameter, defined in Eq. (4) - T temperature of vapor - T _m bulk mean temperature of vapor - T _s saturated temperature - T _w inner wall temperature - V mean vapor velocity - V fully-developed vapor velocity, given in Eq. (12) - V ₀ mean vapor velocity atx=0 - x axial position in thermal entrance region - X dimensionless axial position, (x/r _i)/(Re·Pr) - z ₀ flow quality atx=0 Greek symbols ₀ vapor void fraction atx=0 - ratio of radius,r _i/r₀ - _d emissivity of droplets - _w emissivity of inner heated wall - dimensionless vapor temperature, defined in Eq. (9) - _m dimensionless vapor mean temperature, given by Eq. (14) - _wi dimensionless inner wall temperature - _wo dimensionless outer wall temperature - dynamic viscosity of vapor - _l liquid density - _v vapor density - Stefan-Boltzmann constant