Toroidal Duct Research Articles

Effect of radius of toroidal square duct on the transition of electromagnetically driven liquid metal flow

Toroidal magnetohydrodynamic flows are important for fusion technology because of the blanket surrounding the core of the fusion machine. The effect of the radius of toroidal square ducts on flow transition of an electromagnetically driven liquid metal is assessed. The flow governing equations are solved numerically using direct numerical simulation. Critical Reynolds (Re) numbers where transition occurs are obtained for Hartmann (Ha) numbers in the range of 3 < Ha < 500. The results are compared with previous experimental and numerical results of toroidal square ducts of various radii. It is shown that, independent of the toroidal duct mean radius (r), the transition occurs in three regimes: in the sub-magnetohydrodynamic regime for very low Ha numbers, in a plateau regime for 10 < Ha < 20 and in a higher Ha regime. By increasing the radius of the torus, a transition occurs at higher Reynolds numbers as turbulent fluctuations are developing inside the boundary layers of the cylindrical side walls. The results indicate that transition occurs at RerH̃a-3.39 in the sub-magnetohydrodynamic regime and at RerH̃a-1.4 in the magnetohydrodynamic one. The boundary layer thickness at the Hartmann walls were found to scale as Ha-0.8.

Optimization of laminar convective heat transfer of oil-in-water nanoemulsion fluids in a toroidal duct

This study presents numerical simulations, back propagation artificial neural networks and genetic algorithms for optimizing laminar convective heat transfer of oil-in-water nanoemulsion fluids having non-Fourier heat conduction characteristics. Firstly, a numerical study has been conducted on laminar flow and forced convective heat transfer of oil-in-water nanoemulsion fluids in toroidal ducts using Eulerian-Lagrangian two-phase approach. New correlations of drag coefficient, effective thermal conductivity and effective viscosity were adopted to improve the accuracy of simulation. Numerical results show that convective heat transfer can be enhanced by oil nanodroplets with thermal conductivity lower than that of the base fluid. Then regression models and artificial neural network models were developed based on simulation results for predicting convective heat transfer performances of nanoemulsions, considering effects of cross-sectional aspect ratio, Reynolds number, oil nanodroplet diameter and concentration. Artificial neural network models can predict mean Nusselt number and pressure drop better than the regression model. Finally, genetic algorithms was used to optimize convective heat transfer of nanoemulsions considering droplet migration. It can be found that low cross-sectional aspect ratio of width to height is beneficial for thermal performance factor. For single-objective optimization, mean Nusselt number reaches the maximum 32.3 at aspect ratio of 0.9677 and thermal performance factor reaches the maximum 1.305 at aspect ratio of 0.3935 under certain conditions. Pareto optimal set was obtained for two-objective optimization. This study would be useful for the optimal design of convective heat transfer of emulsions in toroidal ducts.

Validation and verification of a robust 3-D MHD code

The present paper aims to establish the MHD module of Anupravaha-1, a general purpose solver for fluid-flow, developed by our research group as a versatile computational tool for simulating MHD flows through complex geometric configurations. The code is capable of handling non-orthogonal, multi-block, structured grids and treats a coupled fluid–wall (conjugate) modeling to include thermal and electro-magnetic effects in the presence of duct walls and electro-thermally coupled manifolds. The solver module is thoroughly validated using various benchmark numerical and analytical results, which include, but not limited to the following variety of problems: (a) 2-D analytical and numerical reference cases of pressure driven flows in straight and curved channels, and natural convection in a rectangular enclosure, for Hartmann numbers Ha ≤ 300. (b) Fully-developed flow through a rectangular duct for Hartmann numbers up to Ha = 15,000, considering the duct walls to be all insulating (the Shercliff's case) and also with only the Hartmann walls partially conducting (Hunt's case II). (c) For the 3-D solver validation, comparative results for (1) the flow through a straight duct in the presence of a non-uniform magnetic field, (2) free convection in a cubical enclosure under a uniform magnetic field, and (3) an electro-magnetically driven flow in a toroidal duct for low Hartmann numbers (Ha ≤ 100), are obtained and discussed. (d) In addition, the implementation of a simple wall treatment method for computing high Hartmann number flows, as typically encountered in liquid metal cooled fusion-blanket applications, is tested using the case of flow through a rectangular duct in the presence of a homogeneous magnetic field for Ha = 10,000–30,000. The performance of the code in the cases considered ranged from good to excellent. Lastly, comparison with the ALEX (Argonne Liquid metal EXperiment), for the case of flow through a rectangular duct in a fringing magnetic field is presented, that revealed the sensitivity of the results to the precise description of the magnetic field.

Thermal convection in a toroidal duct of a liquid metal blanket. Part I. Effect of poloidal magnetic field

We explore the effect of poloidal magnetic field on the thermal convection flow in a toroidal duct of a generic liquid metal blanket. Non-uniform strong heating (the Grashof number up to 1011) arising from the interaction of high-speed neutrons with the liquid breeder, and strong magnetic field (the Hartmann number up to 104) corresponding to the realistic reactor conditions are considered. The study continues our earlier work [1], where the problem was solved for a purely toroidal magnetic field and the convection was found to result in two-dimensional turbulence and strong mixing within the duct. Here, we find that the poloidal component of the magnetic field suppresses turbulence, reduces the flow's kinetic energy and high-amplitude temperature fluctuations, and, at high values of Hartmann number, leads to a steady-state flow. At the same time, the intense mixing by the surviving convection structures remains able to maintain effective heat transfer between the liquid metal and the walls.

Thermal convection in a toroidal duct of a liquid metal blanket. Part II. Effect of axial mean flow

The work continues the exploration of the effect of thermal convection on flows in toroidal ducts of a liquid metal blanket. This time we consider the effect of the mean flow along the duct and of the associated heat transfer diverting the heat deposited by captured neutrons. Numerical simulations are conducted for a model system with two-dimensional (streamwise-uniform) fully developed flow, purely toroidal magnetic field, and perfectly electrically and thermally insulating walls. Realistically high Grashof (up to 1011) and Reynolds (up to 106) numbers are used. It is found that the flow develops thermal convection in the transverse plane at moderate Grashof numbers. At large Grashof numbers, the flow is dominated by the top-bottom asymmetry of the streamwise velocity and stable stratification of temperature, which are caused by the buoyancy force due to the mean temperature growing along the duct. This leads to suppression of thermal convection, weak mixing, and substantial gradients of wall temperature. Further analysis based on more realistic models is suggested.

Plasma distribution properties of vacuum ribbon-like cathodic arc plasma fliter and Raman studies of diamond-like carbon films perpared by it

As is well known, most filtered cathodic vacuum arc deposition technology adopts filters with various geometries to remove macro particles in the last three decades, but almost all of them have a circular cross-section. Compared with the traditional toroidal duct filters, the rectangular graphite cathodic arc source can have a larger area which can be an arc source of a ribbon-like cathodic arc plasma filter, which has a higher coating efficiency due to its larger area arc source and may be more suitable for a larger scale industrial production. Thus, the research on the plasma distribution properties within the vacuum ribbon-like cathodic arc plasma filter is of great significance. In this paper, a rectangular graphite cathodic arc source is used to produce the ribbon-like cathodic arc plasma. Within the filter, a 90 curved magnetic duct with a rectangular cross-section is used as the arc filter. The ribbon-like cathodic arc plasma is transmitted from cathode to the deposition area along the magnetic line produced by external coils. A Faraday cup ion energy analyzer and a Langmuir probe are used to characterize the distribution properties of the filtered plasma at 15 places on the exit plane. Ion energies and ion density at these positions are obtained. For the special retrograde motion of the cathode spot on the rectangular target surface, the ion energies and ion density data are not stable. In order to obtain representative values, the net results are the average value of 3 measurements. Diamond-like carbon (DLC) films are deposited by the ribbon-like cathodic arc plasma filter at the same exit plane and their structures are characterized by Raman shift. To compare the distinctness of the 15 Raman spectrums, each Raman spectrum of the DLC films is normalized and shown in a figure. Meanwhile, the thicknesses of all the DLC films are measured by step profiler. Results show that the ion energies are of Maxwell distributions at all the 15 places on the exit plane. The ion energies vary from 0 to 60 eV, most being in the range from 20 to 30 eV. The arc voltage is 30 eV, which exactly coincides with the ion energies. While Raman spectra of the DLC films show an obvious correspondence relationship with the ion energies as well as the ion density and the DLC film thickness. The nano-hardness of the DLC films lies in a range of 25-43 GPa. Although the ion energies, ion density, DLC film thickness and nano-hardness are slightly different at different locations, they are not significant. Owing to the relatively evenly distributed properties of the ribbon-like arc plasma this may open great opportunities for a large area filtered arc deposition technique.

Effect of curvature and magnetic field on MHD flow of a liquid metal in a curved annular duct

Purpose – The steady laminar quasi-3D fully developed magnetohydrodynamic (MHD) flow of a liquid metal in a curved annular channel is studied in order to determine the effect of the magnetic field on the velocity distribution. The paper aims to discuss this issue. Design/methodology/approach – Due to the fluid motion under the effect of the applied transverse external magnetic field, an additional magnetic field and an electric current density are induced. A hybrid formulation is used for the induced electric current density, implementing for its axial component the Ohm’s law and for its transverse components the Ampere’s law. The suggested formulation (denominated h-formulation) is combined with the extended Continuity Vorticity Pressure numerical variational method for MHD flows. Findings – Results are obtained for different values of curvature ratios and Hartmann numbers. It is proved that as the strength of the magnetic field increases, two side regions of velocity jets are formulated in the right and in the left half sides of the inner cylinder in a direction parallel to the external magnetic field. The magnitude of the axial velocity at each region is determined by the balance of the centrifugal and the electromagnetic forces. The results help to better understand the MHD flow in toroidal ducts. Originality/value – The results aim to help to a better understanding of the MHD flow in curved ducts.

Two-dimensional turbulent convection in a toroidal duct of a liquid metal blanket of a fusion reactor

Convection in a horizontal duct aligned with a uniform magnetic field is analysed computationally. The motivation of the study is the concept of a liquid metal blanket for a tokamak fusion reactor, in which ducts are oriented toroidally, i.e. parallel to the main component of the magnetic field. Computations of two-dimensional (streamwise-uniform) flows appearing at very strong magnetic fields and of three-dimensional flows in long domains are conducted. Non-uniform volumetric internal heating is applied, while the walls are maintained at a constant temperature. Two-dimensional or nearly two-dimensional turbulent convection is found at high Grashof and Hartmann numbers typical for fusion reactor conditions. The turbulence results in stronger mixing and more uniform distribution of wall heat flux, indicating promising potential of this concept of the blanket.

Transition of an electromagnetically driven liquid metal flow from laminar to turbulent in a toroidal square duct

The electromagnetically driven liquid metal flow in a toroidal duct of a square cross-section is studied numerically searching for the critical Reynolds (Re) and Hartmann (Ha) numbers where transition occurs. Results are reported for Ha up to 500, building a transition map of critical Reynolds and Hartmann numbers and showing two distinct regimes of transition signified by the destabilization of the secondary flow: The first, for Ha < 18, where the Lorentz force is balanced by fluid momentum diffusion and the second, for Ha ⩾ 18, where it is balanced by fluid inertia. In both cases, the destabilization of the secondary flow occurs primarily near the concave sidewall. For Ha ⩾ 18 the transition follows approximately the relationship Re = 16.803 Ha1.43 for values of Re and Ha in the range 80 < Re/Ha < 224.

Instabilities and turbulence in magnetohydrodynamic flow in a toroidal duct prior to transition in Hartmann layers

AbstractFlow of an electrically conducting fluid in a toroidal duct of square cross-section is analysed. The flow is driven by the azimuthal Lorentz force resulting from the interaction between the radial electric currents created by the difference of electric potential maintained between the cylinder walls and the strong magnetic field imposed in the axial direction. The flow geometry and the value of the Hartmann number correspond to the experiment of Moresco &amp; Alboussière (J. Fluid Mech., vol. 504, 2004, pp. 167–181). The purpose of the analysis is to reveal the flow features at Reynolds numbers below the threshold of transition to turbulence in Hartmann layers. We find that the flow experiences a complex evolution. The laminar base flow experiences the first instability at the Reynolds number significantly smaller than that of the threshold. The instability is axisymmetric and oscillatory. Turbulence appears at a slightly higher Reynolds number. Right up to the Hartmann layer instability, the turbulence remains localized in a layer near the outer cylinder wall. It is demonstrated that the turbulence may affect the transition in the Hartmann layers via unsteady forcing of the outer flow.

Numerical simulation of magnetohydrodynamic flow in a toroidal duct of square cross-section

We present numerical simulation results of the quasi-static magnetohydrodynamic (MHD) flow in a toroidal duct of square cross-section with insulating Hartmann walls and conducting side walls. Both laminar and turbulent flows are considered. In the case of steady flows, we present a comprehensive analysis of the secondary flow. It consists of two counter-rotating vortex cells, with additional side wall vortices emerging at sufficiently high Hartmann number. Our results agree well with existing asymptotic analysis. In the turbulent regime, we make a comparison between hydrodynamic and MHD flows. We find that the curvature induces an asymmetry between the inner and outer side of the duct, with higher turbulence intensities occurring at the outer side wall. The magnetic field is seen to stabilize the flow so that only the outer side layer remains unstable. These features are illustrated both by a study of statistically averaged quantities and by a visualization of (instantaneous) coherent vortices.

Instability of magnetohydrodynamic flow in an annular channel at high Hartmann number

Instability of a flow of an electrically conducting fluid in an annular channel is analyzed. Strong constant magnetic field is imposed in the axial direction. Similarly to toroidal duct experiments, the flow is driven by the azimuthal Lorentz force resulting from the interaction between the magnetic field and the radial electric currents created by a difference of electric potential imposed between the cylinders. The instability of the base flow, while clearly of centrifugal nature, is significantly different from the Dean instability detected earlier in hydrodynamic systems and similar MHD systems at low and moderate magnetic fields. Growing perturbations are oscillating and axisymmetric and consist of counter-rotating toroidal vortices arranged side by side in the radial direction and having meridional cross-sections in the form of elongated ellipses oriented slightly obliquely to the axial direction. Simulations of the secondary flow show an interesting feature of periodic transitions between two symmetric solutions.

Ion current produced by a vacuum arc carbon plasma source

A vacuum arc carbon plasma source is described, in which an arc was ignited between a cathode and an anode having In aperture, by bringing the two electrodes into contact, and parting them while current was flowing. The inter-electrode gap length was varied. A focusing magnetic field was applied in the inter-electrode gap, and a toroidal magnetic field was applied to guide plasma through a toroidal duct. The influence of the arc current (I/sub arc/), gap length (L), and focusing magnetic field (B/sub f/) on the cathode spot motion, plasma behavior, and output ion current was studied. It was shown that with I/sub arc/=150-250 A and relatively weak B/sub f/, the spots moved on the cathode periphery, causing noisy arcing and fluctuation of the output ion current, while at a lower I/sub arc/ (50-100 A) and strong B/sub f/ (/spl ges/28 mT), there were fewer spots on the cathode surface, they tended to move toward the cathode center, the arc noise decreased and the average output ion current increased. With increasing L from 2 to 18 mm, the total ion current at first rapidly increased, and then saturated at L/spl ges/10 mm. Best performance of the carbon plasma source was obtained with an arc current chosen so that with a given field configuration, the cathode spots locate in the center of the cathode surface, and with strong magnetic fields in both the electrode and toroidal filter regions.

Graphite macroparticle filtering efficiency of three different magnetic filter designs used in the filtered cathodic vacuum arc deposition of tetrahedral amorphous carbon films

The graphite macroparticle filtering efficiency of three different magnetic filter designs (straight duct, toroidal duct, and 90° free-standing solenoid) used in the filtered cathodic vacuum arc deposition of tetrahedral amorphous carbon (ta-C) films was investigated. The particle contents in the coatings were measured using scanning electron microscopy (SEM) technique. In addition, during the ta-C deposition, long time exposure photographs were taken to see the graphite macroparticle trajectories and thus to qualitatively evaluate the filtering performance of different magnetic filters. The particle density measurements from the SEM images show that the filtering efficiency of the 90° free-standing solenoid design is 13 times better than the filtering efficiency of the straight duct design. However, the toroidal duct design had only 3 times higher filtering efficiency compared to the straight duct design.

On the lowest radial frequencies of a rigid-walled narrow toroidal duct

For a rigid-walled toroidal duct of rectangular cross section, narrow in the radial direction, the lowest radial characteristic frequency approximation appears to form a sequence proportional to the angular wave number integer [C. C. Lawrenson, L. D. Lafleur, and F. D. Shields, “The solution for the propagation of sound in a toroidal waveguide with driven walls (the acoustitron),” J. Acoust. Soc. Am. 103, 1253–1260 (1998)]. By use of properties of Bessel functions, a more accurate higher-order formula for these frequencies is presented which, moreover, shows that there is, in fact, some deviation from the strict harmonicity.

Inertial effects on electrically conducting fluids in the presence of transverse magnetic fields: An example problem

When an electric current flowing within a conducting fluid crosses magnetic field lines, a body force is created, which may significantly affect the motion of the fluid. This paper treats the flow of a conducting fluid in a toroidal duct. A transverse magnetic field is present, and a portion of one surface of the toroidal duct serves as an electrode. Numerical solutions are presented which illustrate the effect of the electromagnetic body forces and inertial effects on the motion of the fluid. Under the conditions specified, in the absence of a magnetic field, the fluid would remain stagnant, as there is no mechanism to drive the flow.

Ion current distribution within a toroidal duct of a filtered vacuum arc deposition system

Vacuum arcs were established on a 90-mm-diameter Ti cathode in a deposition apparatus consisting of a spacer, 122 mm-diameter annular anode, quarter-torus magnetic macroparticle filter, and a deposition chamber. A toroidal magnetic field generally parallel to the torus walls of up to 20 mT was applied. The ion current in various cross-sections of the toroidal duct was measured using: 1) a disc probe of 130-mm diameter, oriented normal to the torus axis used to measure the transmitted ion current, and 2) a hollow cylindrical probe of 135-mm diameter and 25-mm height, whose axis coincided with the torus axis, used to measure ion current losses to the duct wall. The distribution of ion current loss was studied using an 8-segment hollow cylindrical multiprobe, where the individual probes were equally distributed on the circumference of a 130-mm-diameter circle. It was shown that: 1) the ratio of ion currents collected on the cylindrical and disc probes at first decreases with increasing the toroidal field, and then becomes approximately constant; 2) the presence of the large-diameter disc probe does not influence the value of the ion current on the cylindrical probe; and 3) the maximum ion current density near the torus walls is located in the +g direction and displaces in the -(B/spl times/g) direction with increasing the toroidal field, where g and B are the vectors of the centrifugal acceleration and the magnetic field, respectively.

The numerical calculation of plasma beam propagation in a toroidal duct with magnetized electrons and unmagnetized ions

Electron-magnetized vacuum arc plasma transport in a magnetic toroidal duct is calculated numerically taking in account electron - ion collisions, electron and ion temperatures, and the high conductivity of the duct wall. The longitudinal magnetic field in the duct, the fully ionized plasma density and the electric potential distribution at the torus entrance are given, while the plasma density, electrical field and current, and macroscopic plasma velocity across the magnetic field inside the duct are calculated. Toroidal coordinates are used to describe plasma beam propagation. A Runge - Kutta routine is used for the calculations along the torus while a finite difference method is used across the torus cross section. It is found that plasma loss due to particle flux to the duct wall depends on the electron and ion temperatures and the plasma density distribution at the torus entrance cross section. With an electron temperature of , 30 000 K and 50 000 K, an ion temperature and a Gaussian distribution of plasma density at the torus entrance with a maximum value , we found that the duct efficiency was less than 10% for longitudinal magnetic field strengths of 10 mT and 20 mT. In the case where only the electrons are magnetized, filter efficiency depends only weakly on the magnetic field strength, on , and on .

First- and higher-order effects of curvature and torsion on the flow in a helical rectangular duct

A series expansion method is employed to determine the first-order terms in curvature ε and torsion η of fully developed laminar flow in helical square ducts and in helical rectangular ducts of aspect ratio two. The first-order solutions are compared to solutions of the full governing equations. For toroidal square ducts with zero pitch, the first-order solution is fairly accurate for Dean numbers,De=Reε1/2, up to about 20, and for straight twisted square ducts the first-order solution is accurate for Germano numbers,Gn= ηRe, up to at least 50 whereReis the Reynolds number. Important conclusions are that the flow in a helical duct with a finite pitch or torsion to the first order (i.e. with higher-order terms in ε and η neglected) is obtained as a superposition of the flow in a toroidal duct with zero pitch and a straight twisted duct; that the secondary flow in helical non-circular ducts for sufficiently smallReis dominated by torsion effects; and that for increasingRe, the secondary flow eventually is dominated by effects due to curvature. Torsion has a stronger impact on the flow for aspect ratios greater than one. A characteristic combined higher-order effect of curvature and torsion is an enlargement of the lower vortex of the secondary flow at the expense of the upper vortex, and also a shift of the maximum axial flow towards the upper wall. For higher Reynolds numbers, bifurcation phenomena appear. The extent of a few solution branches for helical ducts with finite pitch or torsion is determined. For ducts with small torsion it is found that the extent of the stable solution branches is affected little by torsion. Physical velocity components are employed to describe the flow. The contravariant components are found useful when describing the convective transport in the duct.

Vacuum arc plasma jet propagation in a toroidal duct

A two fluid magneto-hydrodynamic theory of vacuum arc plasma jet propagation in a magnetized toroidal duct is developed. The physical mechanisms of jet transverse displacement and plasma losses are analyzed and the centrifugal force on the ions is shown to play the principle role in these processes. Optimal conditions for jet propagation occur when the centrifugal force is balanced by the electrical force on the ions. An analytical solution of the nonlinear problem of plasma beam transport through a toroidal duct is obtained for the two cases where ions are magnetized or not magnetized. The ion mass current decreases with the azimuthal distance along the torus as (1+φ/φ0)−1 where φ0 is a characteristic angular distance, for the case when ions are magnetized, and exponentially when the ions are not magnetized. Numerical calculations show that the decrease of plasma density leads to a longitudinal electric field and current. This current, together with the current due to the centrifugal drift, form a current loop which is closed through the plasma and structures outside the torus. Moreover if there are optimal conditions for jet propagation at the torus entrance, they are approximately conserved along the length of the torus.