Theory of a magnetohydrodynamic flow regulator for liquid metal

A theory is developed for unsteady flow of liquid metal in a de MHD flow regulator channel for small magnetic Reynolds numbers. It is shown that it is possible to use the quasistationary approximation for calculating the integral flow parameters.MHD methods, along with their applications in direct transformation of thermal energy into electrical energy, in recent years have been ever more widely used in nuclear energy and metallurgy for transporting and for measuring the parameters of liquid metal flows [1, 2]. Complete sealing, operation over a wide temperature range, and simplicity of control and automation are among the unquestionable advantages of all MHD devices.The existing theory of MHD devices is limited primarily to the stationary regimes of operation of electromagnetic pumps and magnetic flow-meters.The most characteristic operating regimes for the MHD regulator are the transient regimes, which are defined primarily by the hydrodynamics of the liquid metal flow.The present article is devoted to the study of the unsteady flow of liquid metal in the channel of a dc MHD flow regulator with independent excitation. The magnetic Reynolds are low (R _m =U ₀ b1).     

K-ε-T model of dense liquid-solid two-phase turbulent flow and its application to the pipe flow

To predict the characteristics of dense liquid-solid two-phase flow, K-ε-T model is established, in which the turbulent flow of fluid phase is described with fluid turbulent kinetic energy K_f and its dissipation rate ε_f, and the particles random motion is described with particle turbulent energy K_p and its dissipation rate ε_p and pseudothermal temperature T_p. The governing equations are also derived. With K-ε-T model, numerical study of dense liquid-solid two-phase turbulent up-flow in a pipe is performed. The calculated results are in good agreement with experimental data of Alajbegovic et al. (1994), and some flow features are captured.     

Hall effects on MHD flow in a rotating system with heat transfer characteristics

Closed-form solutions are derived for the steady magnetohydrodynamic (MHD) viscous flow in a parallel plate channel system with perfectly conducting walls in a rotating frame of reference, in the presence of Hall currents, heat transfer and a transverse uniform magnetic field. A mathematical analysis is described to evaluate the velocity, induced magnetic field and mass flow rate distributions, for a wide range of the governing parameters. Asymptotic behavior of the solution is analyzed for large M ² (Hartmann number squared) and K ² (rotation parameter). The heat transfer aspect is considered also with Joule and viscous heating effects present. Boundary layers arise close to the channel walls for large K ², i.e. strong rotation of the channel. For slowly rotating systems (small K ²), Hall current parameter (m) reduces primary mass flow rate (Q _x /R ρ v). Heat transfer rate at the upper plate (d θ/d η) _η=1 decreases, while at the lower plate (d θ/d η) _η=−1 increases, with increase in either K ² or m. For constant values of the rotation parameter, K ², heat transfer rate at both plates exhibits an oscillatory pattern with an increase in Hall current parameter, m. The response of the primary and secondary velocity components and also the primary and secondary induced magnetic field components to the control parameters is also studied graphically. Applications of the study arise in rotating MHD induction machine energy generators, planetary and solar plasma fluid dynamics systems, magnetic field control of materials processing systems, hybrid magnetic propulsion systems for space travel etc.     

Heat transfer in a flow past a semi-in-finite plate with oscillating temperature

The response of laminar boundary layer flow past a semi-infinite flat plate to harmonic oscillations in the plate temperature in the form of a travelling wave convected in the direction of the free-stream has been studied here. Series solutions in terms of the small amplitude and the small oscillations to the non-linear system have been derived and the resulting nonlinear ordinary equations due to usual similarity transformations are solved numerically. The function affecting the temperature is shown on a graph. Due to greater viscous dissipative heat the function K ₁, increases and it decreases with increasing Prandtl number. Also the time averaged heat flux function K ₁(0) increases with Prandtl number and decreases due to greater viscous dissipative heat.     

Fractographic analysis of fatigue cracking in spheroidal graphite cast irons

This paper is concerned with the influence of R-ratio on the fatigue threshold ΔK_th, in two spheroidal graphite cast irons. The microstructures, viz, ferritic and pearlitic, exhibited significant and consistent R-ratio effects on ΔK_th. Intergranular failure facets were observed and they tended to reach a maximum amount when the reversed plastic zone size approached grain size dimensions. Reasonable correlation of experimental data required the increment of crack closure effects in the analytical model. At R-ratio values approaching zero, both microstructures showed ΔK_th, values that were markedly greater than those recorded for steels even when the data was corrected or normalised to a constant yield stress and grain size for a specific microstructure. The presence of graphite nodules may cause an increase in crack closure which in turn promoted high ΔK_th values.     

Steady slow rotation of a sphere in a visco-elastic liquid

Summary In this paper, we obtain the flow due to slow steady rotation of a sphere in a visco-elastic liquid characterized by the constitutive relation given by Rivlin. The non-Newtonian effects are strongly dependent on a non-dimensional parameter K independent of the angular velocity of the sphere. If 1<K3, we notice four vortices symmetrically placed around the sphere. When K lies outside this range, the direction of the flow pattern is the same as that in the Newtonian case but displaced towards the sphere as K decreases. Also the expression for the couple on the sphere has been obtained which depends on K.     

Direct simulation of MHD flows in dual-coolant liquid metal fusion blanket using a consistent and conservative scheme

Direct simulation of 3-D MHD (magnetohydrodynamics) flows in liquid metal fusion blanket with flow channel insert (FCI) has been conducted. Two kinds of pressure equilibrium slot (PES) in FCI, which are used to balance the pressure difference between the inside and outside of FCI, are considered with a slot in Hartmann wall or a slot in side wall, respectively. The velocity and pressure distribution of FCI made of SiC/SiC_f are numerically studied to illustrate the 3-D MHD flow effects, which clearly show that the flows in fusion blanket with FCI are typical three-dimensional issues and the assumption of 2-D fully developed flows is not the real physical problem of the MHD flows in dual-coolant liquid metal fusion blanket. The optimum opening location of PES has been analyzed based on the 3-D pressure and velocity distributions.     

Fatigue crack extension behaviour in porous plasma spray 80%Ni---20%Cr material: The influence of R-ratio

This paper attempts to describe the fatigue crack growth response of a plasma spray 80%---20%Cr material, utilised in the corrosion protection of engineering components, whose microstructure consisted of (i) an austenitic matrix, (ii) a secondary dispersion of a chromite non-metallic inclusion phase and (iii) regions of closed and connected porosity.It was demonstrated that little or no effect of R-ratio was observed on the threshold stress intensity range ΔK_th, which was attributed to both the materials fine to intermediate grain size and probable plain stress testing conditions which significantly decrease crack closure effects. At intermediate fatigue crack growth rates high ratio results were an order of magnitude faster than the low R-ratio data. This was the result of the high R-ratio case seeking out more regions of porosity which then increased the local ΔK on the remaining ligaments leading to accelerated crack growth rates.Porosity was shown to significantly decrease the value of ΔK_th and the extent of porosity observed on fatigue fracture surfaces increased with ΔK level and was well in excess of that of 5% recorded by metallography. Hence the growing fatigue crack preferentially sought out regions of porosity as they represented locations of low fracture energy.     

Higher-order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution

Steady flow of an incompressible fluid takes place in a porous formation of spatially variable hydraulic conductivityK. The latter is regarded as a lognormal stationary random space function and Y=ln(K/K _G ), whereK _G is the geometric mean ofK, is characterized by its variance ² and correlation scale I. Exact results are known for the effective conductivityK _eff in one- and two-dimensional flows. In contrast, only a first-order term in a perturbation expansion in ² has been derived exactly for the three-dimensional flow. A conjecture has been made in the past onK _eff for any ², but it was not yet proved exactly. This study derived the exact nonlinear correction, i.e. the termO(⁴) ofK _eff, which is found to be the one resulting from the conjecture, strengthening the confidence in it. It is also shown that the self-consistent approximation leads to the exact results for one-dimensional and two-dimensional flows, but underestimates the nonlinear correction ofK _eff for in the three-dimensional case.     

Subsonic and transonic flow field in two-dimensional de laval nozzles

This paper presents solutions of subsonic and transonic flow fields in two-dimensional De Laval nozzles with preassinged contraction ration ₁, expansion ration ₂, and throat wall radiusR _*. The effects of the contraction and the expansion angle on nozzle flow, the transformation of flow pattern of a De Laval nozzle in the throat region, and the conditions of occurrence and the governing parameters of the supersonic bubbles are discussed.     

Flow patterns of n-hexadecane–CO2 liquid–liquid two-phase flow in vertical pipes under high pressure

To promote a better understanding of liquid–liquid two-phase flow behavior, particularly under high pressure, flow patterns of n-hexadecane–CO₂ liquid–liquid two-phase upward flow in vertical stainless steel pipes were experimentally investigated. Observations were made in two 0.0015 m I.D. pipes of different lengths (0.068 m and 0.5 m) under high pressure varying from 10.3 to 29.6 MPa using a high pressure visualization system. The total flow rate was fixed at 2.0 × 10⁻⁶ m³/min, while the flow rate ratio (φ) varied from 0.05 to 19. Bubbly flow, plug flow, slug flow, annular flow, and near-one-phase flow regions were found in both pipes, while stratified flow was observed only in the 0.068 m pipe. Flow pattern maps were constructed in the flow rate ratio versus pressure graph, which demonstrates significant impacts of flow rate ratio, pipe length, and pressure on flow patterns. These impacts are discussed in detail. To the authors’ best knowledge, this work is the first attempt to observe complex liquid–liquid two-phase flow behavior with flow pattern transitions under high pressure, and contributes to a better understanding of liquid–liquid two-phase flow behavior.     

A note on co-current laminar flows of two immiscible elastico-viscous liquids

Summary An analysis is presented of steady (isothermal) co-current laminar flows of two immiscible elasticoviscous liquids in cylindrical channels to include (i) unidirectional stratified flow with ripple-free, plane liquid interface, and (ii) concentric-layered swirling flow with ripple-free cylindrical liquid interface. The general conditions are derived for such two-phase channel flows to be physically realizable. It is shown that, whereas (under certain circumstances)single-phase laminar flows are physically possible,two-phase flows, on the other hand, of liquids of the same class may not be. But liquids of theRoberts type (Roberts 1953), with a normal stress difference equivalent to an extra simple tension along the streamlines in simple shearing, are capable of steady unidirectional flowin all circumstances (whether in single or two-phase flow), though they are not in a privileged position so far astwo-phase swirling flows are concerned.     

Effects of anisotropy in permeability on the two-phase flow and heat transfer in a porous cavity

This paper reports on the results of a numerical study of convection flow and heat transfer in a rectangular porous cavity filled with a phase change material under steady state conditions. The two vertical walls of the cavity are subject respectively to temperatures below and above the melting point of the PCM while adiabatic conditions are imposed on the horizontal walls. The porous medium is characterized by an anisotropic permeability tensor with the principal axes arbitrarily oriented with respect to the gravity vector. The problem is governed by the aspect ratioA, the Rayleigh numberRa, the anisotropy ratioR and the orientation angle θ of the permeability tensor. Attention is focused on these two latter parameters in order to investigate the effects of the anisotropic permeability on the fluid flow and heat transfer of the liquid/solid phase change process. The method of solution is based on the control volume approach in conjunction with the Landau-transformation to map the irregular flow domain into a rectangular one. The results are obtained for the flow field, temperature distribution, interface position and heat transfer rate forA=2.5,Ra=40, 0≤θ≤π, 0.25≤R≤4. It was found that the equilibrium state of the solid/liquid phase change process may be strongly influenced by the anisotropy ratioR as well as by the orientation angle θ of the permeability tensor. First, for a given set of parametersA,Ra andR, there exists an optimum orientation θ_max for which the flow strength, the liquid volume and the heat transfer rate are maximum. There also exists an orientation θ_min=θ_max+π/2 for which these quantities are minimum. Second, when an anisotropic medium is oriented along the optimum direction θ_max, an increase of the permeability component along that direction will increase the flow and heat transfer rate in a same order while an increase of the other permeability component only has a negligible effect. For the parameter ranges considered in the present study, it was found that the optimum direction is lying between the gravity vector and the dominant flow direction.     

Steady flow of a liquid metal in a rectangular MHD channel

The known experimental studies of steady flows of a liquid metal in magnetohydrodynamic (MHD) channels of rectangular section [1–4] were performed only for a few values of the Reynolds number, which does not permit a clear delineation of the fundamental governing laws of the flow in the zone of transition from laminar to turbulent flow. In addition, the study of turbulent MHD flows has been limited to two-dimensional channels.Below we present some results of experimental studies of the effect of a transverse magnetic field on the resistance coefficient for mercury flow in an MHD channel with side ratio 1 to 2.5. The choice of a channel with this side ratio was dictated by the need for studies of the intermediate case between flows in two-dimensional and square channels, which differ significantly from one another because of the different effect of the walls parallel to the magnetic field. In our studies, for each value of the Hartmann number the investigations were made for 30–50 values of the Reynolds number.Notation B₀ flux density of the applied magnetic field - M Hartmann number - R Reynolds number - _tm resistance factor of turbulent MHD flow - _* critical value of the resistance factor - geometric parameter of channel - the component of resistance factor in ordinary hydrodynamics due to pulsations - normed function - electric conductivity of metal - viscosity of metal - R₀ shydraulic radius - N smagnetic field parameter     

倾斜非均匀磁场下导电方管磁流体管流的数值模拟研究

热核聚变反应堆液态金属包层应用中的一个重要问题是液态金属在导电管中流动和强磁场相互作用产生的额外的磁流体动力学压降.这种磁流体动力学压降远远大于普通水力学压降.美国阿贡国家实验室ALEX研究小组,对非均匀磁场下导电管中液态金属磁流体动力学效应进行了实验研究,其实验结果成为液态金属包层数值验证的标准模型之一.液态金属包层在应用中会受到不同方向的磁场作用,本文以ALEX的非均匀磁场下导电方管中液态金属管流实验中的一组参数为基础,保持哈特曼数、雷诺数和壁面电导率不变,采用三维直接数值模拟的方法,研究了外加磁场与侧壁之间的倾角对导电方管内液态金属流动的速度、电流和压降分布的影响.研究结果表明:沿流向相同横截面上的速度、电流以及压力分布均随磁场的倾斜而同向旋转.倾斜磁场均匀段,横截面上的高速区位于平行磁场方向的哈特曼层和平行层交叉位置,压力梯度随磁场倾角的增大先增大后减小.倾斜磁场递减段,在三维磁流体动力学效应作用下,横截面上的高速射流位置向垂直磁场方向偏移.磁场递减段的三维磁流体动力学压降随磁场倾角的增大而增大.随磁场倾斜,截面上的射流峰值逐渐减小,二次流增强,引发层流向湍流的转捩.      

A Numerical Study of Micro-Heterogeneity Effects on Upscaled Properties of Two-Phase Flow in Porous Media

Commonly, capillary pressure–saturation–relative permeability (P ^c–S–K _r) relationships are obtained by means of laboratory experiments carried out on soil samples that are up to 10–12 cm long. In obtaining these relationships, it is implicitly assumed that the soil sample is homogeneous. However, it is well known that even at such scales, some micro-heterogeneities may exist. These heterogeneous regions will have distinct multiphase flow properties and will affect saturation and distribution of wetting and non-wetting phases within the soil sample. This, in turn, may affect the measured two-phase flow relationships. In the present work, numerical simulations have been carried out to investigate how the variations in nature, amount, and distribution of sub-sample scale heterogeneities affect P ^c–S–K _r relationships for dense non-aqueous phase liquid (DNAPL) and water flow. Fourteen combinations of sand types and heterogeneous patterns have been defined. These include binary combinations of coarse sand imbedded in fine sand and vice versa. The domains size is chosen so that it represents typical laboratory samples used in the measurements of P ^c–S–K _r curves. Upscaled drainage and imbibition P ^c–S–K _r relationships for various heterogeneity patterns have been obtained and compared in order to determine the relative significance of the heterogeneity patterns. Our results show that for micro-heterogeneities of the type shown here, the upscaled P ^c–S curve mainly follows the corresponding curve for the background sand. Only irreducible water saturation (in drainage) and residual DNAPL saturation (in imbibition) are affected by the presence and intensity of heterogeneities.     

Forced convective flow and heat transfer of upward cocurrent air–water slug flow in vertical plain and swirl tubes

This experimental study comparatively examined the two-phase flow structures, pressured drops and heat transfer performances for the cocurrent air–water slug flows in the vertical tubes with and without the spiky twisted tape insert. The two-phase flow structures in the plain and swirl tubes were imaged using the computerized high frame-rate videography with the Taylor bubble velocity measured. Superficial liquid Reynolds number (Re_L) and air-to-water mass flow ratio (AW), which were respectively in the ranges of 4000–10000 and 0.003–0.02 were selected as the controlling parameters to specify the flow condition and derive the heat transfer correlations. Tube-wise averaged void fraction and Taylor bubble velocity were well correlated by the modified drift flux models for both plain and swirl tubes at the slug flow condition. A set of selected data obtained from the plain and swirl tubes was comparatively examined to highlight the impacts of the spiky twisted tape on the air–water interfacial structure and the pressure drop and heat transfer performances. Empirical heat transfer correlations that permitted the evaluation of individual and interdependent Re_L and AW impacts on heat transfer in the developed flow regions of the plain and swirl tubes at the slug flow condition were derived.     

Magnetohydrodynamic Rotating Flow of a Generalized Burgers’ Fluid in a Porous Medium with Hall Current

This study concentrates on the unsteady magnetohydrodynamics (MHD) rotating flow of an incompressible generalized Burgers’s fluid past a suddenly moved plate through a porous medium. Modified Darcy’s law for generalized Burgers’s fluid in a rotating frame has been used to model the governing flow problem. The closed form solution of the governing flow problem has been obtained by employing Laplace transform technique. The integral appearing in the inverse Laplace transform has been evaluated numerically. The influence of various parameters on the velocity profile has been delineated through several graphs and discussed in detail. It was found that the fluid is decelerated with increasing Hartmann number M and porosity parameter K. However, for large Hall parameter m, the real part of velocity decreases and the imaginary part of velocity increases.     

Visualization study of evaporation of single n-pentane drops in water

A laser shadowgraph system was constructed to enable successive filming of a drop or a bubble rising or falling in an immiscible liquid confined within a vertical column. The assembly was applied to a study of the evaporation of n-pentane drops in a stagnant medium of water. The liquid/vapor two-phase bubble evolving from each pentane drop was observed together with its wake, the morphology and the dynamics of which are our primary concern in considering the mechanism of the medium-to-bubble heat transfer.List of symbols a minor axis of ellipsoidal two-phase bubble - b major axis of ellipsoidal two-phase bubble - D ₀ diameter of saturated-liquid drop set to vaporize - Re Reynolds number based on instantaneous, volume-equivalent spherical diameter and rise velocity of two-phase bubble and kinematic viscosity of the continuous phase - t time lapse after the start of evaporation - T ^* excess of undisturbed continuous-phase temperature above the temperature at which the sum of the saturated vapor pressures of the dispersed- and the continuous-phase fluids is equal to the pressure at the position where evaporation starts - opening angle of wake-covered region on bubble surface - _w zenith angle at flow-separation ring on bubble surface - mass fraction of vapor in two-phase bubble     

Hysteretic heat transfer study of liquid–liquid two-phase flow in a T-junction microchannel

Liquid–liquid two-phase flow in microchannels is capable of boosting the heat removal rate in cooling processes. Formation of different two-phase flow patterns which affect the heat transfer rate is numerically investigated here in a T-junction containing water-oil flow. For this purpose, the finite element method (FEM) is applied to solve the unsteady two-phase Navier–Stokes equations along with the level set (LS) equation in order to capture the interface between phases. It is shown that the two-phase flow pattern in microchannels depends on the flow initial condition which causes hysteresis effect in two-phase flow. In this study, the hysteresis is observed in flow pattern and consequently in the heat transfer rate. The effect of wall contact angle on the hydrodynamics and heat transfer in the microchannel is investigated to gain useful insight into the hysteresis phenomenon. It is observed that the hysteresis is significant in super-hydrophilic microchannels, while it disappears at the contact angle of 75^°. The effect of water to oil flow rate ratio (Q_wat/Q_oil) on the heat transfer is also studied. The flow rate ratio has a negligible effect on the Nusselt number (Nu) in the dripping regime, while the Nu decreases with an increase of Q_wat/Q_oil in the co-flow regime. The thickness of the oil film, velocity, and temperature distribution are studied in the co-flow regime. It is revealed that the normalized slip velocity reduces at higher values of Q_wat/Q_oil, which causes a reduction in the averaged Nu. In dripping regimes, higher flow rate ratios lead to a more frequent generation of droplet/slugs at a smaller size. The passage of the slugs or droplets increases the local Nu. Larger droplets generated at lower flow rate ratios cause a larger increase in the local Nu than smaller droplets. The temperature and velocity field around the droplets are also illustrated to investigate the heat transfer improvement. The generated vortex at the tip of the oil jet causes an increase in the velocity and Nu on the water side.