Electromagnetic coupling phenomena in co-axial rectangular channels

Highlights

• The internal channel is less affected by the coupling phenomena.

• The external channel is dominated by the coupling showing high-velocity reverse jets on the shared side walls.

• A correction factor between pressure gradient of coupled and uncoupled channel is defined.

Abstract

In the Water-Cooled Lithium Lead (WCLL) blanket, the eutectic alloy lithium-lead (PbLi) is used as tritium breeder and carrier, neutron multiplier and heat transfer medium. The liquid metal is distributed to and collected from the breeding zone through a compact poloidal manifold composed of two co-axial rectangular channels. The external channel, tasked with distribution, and the internal one, assigned to the collection, are co-flowing and share an electrically conductive wall ($c_w=0.1$). The liquid metal, interacting with the reactor magnetic field, leads to the arising of MagnetoHydroDynamic (MHD) effects that are expected to significantly modify the flow feature and electrically couple the external and internal channels. In this work, the general-purpose CFD code Ansys CFX 18.2 is used to study the coupling phenomena in a wide range of magnetic fields (up to $Ha=2000$) for a prototypical square co-axial channel. Characteristic flow features and their evolution with increasing magnetic field and varying mass flow rate between the channels are discussed and compared with the uncoupled case, which is in turn composed by a rectangular electro-conductive annulus (external) and a square electro-conductive duct (internal). A correlation is found linking the pressure loss in the studied configuration and an equivalent square channel through a corrective factor ε, which exhibits an asymptotic behavior for $Ha>1000$.

Introduction

The Breeding Blanket (BB) is one of the key components of a nuclear fusion reactor, since it has the tasks of producing and extracting the tritium and thermal power generated therein. In the Water-Cooled Lead Lithium (WCLL) concept, PbLi is employed as tritium breeder/carrier and neutron multiplier, whereas pressurized water cools the system. The current design (2018), based on DEMO 2017 specifications and derived from R&D activities conducted in the framework of the EUROfusion Programme, relies on the Single Module Segment approach, with a breeding element repeated along the poloidal direction [1]. The liquid metal is distributed to the elementary cells composing the Breeding Zone (BZ) through a compact poloidal manifold constituted by two long co-axial rectangular channels that fulfill both the distribution and collection task (Fig. 1A).

A liquid metal that flows through a magnetic field, as the one in the fusion reactor, leads to the appearance of MagnetoHydroDynamic (MHD) effects, which significantly influence the flow features. Indeed, the magnetic field induces electric currents in the metal that, in turn, interact with the magnetic field and generate a Lorentz force, which changes the force balance in the fluid. Moreover, the Lorentz force is not uniformly distributed on the channel cross-section but, rather, it is dependent on the overall magnitude and paths of the current inside the fluid, which, in turn, depends on the electric conductivity of the walls. When two or more channels are electrically connected by a common conductive wall, currents generated in one fluid body can be exchanged with adjacent ones and, therefore, influence their flow features. These “leakage currents” make such channels electromagnetically coupled that is fluid motion in one channel is no longer univocally determined by the current field internally generated (Madarame effect).

The first study on MHD coupled flow is the seminal work by Madarame et al. [2] (1985) that demonstrated how pressure loss in a simple square orifice can be greatly magnified by the interaction with nearby channels. The first studies considering this effect on the manifold are those of McCarthy and Abdou [3] (1991) and Molokov [4] (1993), that investigated the global influence of leakage currents in a manifold composed by square channels and found that flow distribution is also affected. Recently, Bluck and Wolfendale [5] have analytically investigated a coupled array of ducts, in both co- and counter-flow configurations, stacked parallel to the applied magnetic field, where these ducts have insulated side walls and conducting Hartmann walls. Mistrangelo and Buhler [6] numerically studied a similar configuration with an arbitrary orientation of the magnetic field and direction of driving pressure gradients. Swain et al. [7] (2018) have been performed numerical simulation and experiments at high magnetic fields for PbLi flow in coupled co- and counter-flow configuration, L- and U-type bend. Studies dealing with a co-axial coupled channel are scarcer compared with duct arrays. One of the few examples of note is Chen et al. [8] that investigates the MHD phenomena in an assembly composed by three co-axial counter-flowing rectangular channels electrically decoupled or coupled from each other through multi-layer flow channel inserts of arbitrary conductivity.

In this paper, the MHD electro-coupled forced convection flow in co-axial and co-flowing annular configuration is studied. Numerical simulations are performed for a wide range of magnetic field intensity ($Ha=100÷2000$) and with a finite electrical conductivity for both the internal and external walls, as foreseen in the WCLL current design, using the general CFD code ANSYS CFX 18.2. A similar model has been used in [9] to study the basic phenomena in an uncoupled annular channel and it has been updated to account for the inner channel effect in this work that can be considered as an extension of [9].

A correct estimate of the MHD pressure drop is critical to design the WCLL PbLi loop and, therefore, an accurate prediction of the losses in the manifold is required since these account for the bulk of the blanket contribution [10]. The pressure gradient calculated by the CFD code for both the external annular channel ($\nabla p_e^n$) and the internal one ($\nabla p_i^n$) $,$ is compared with the analytical value calculated for the fully developed flow neglecting coupling phenomena. For the annular channel, the theoretical value ($\nabla p_e$) is computed from an equivalent square duct, whereas it is directly calculated for the internal one ($\nabla p_i$). Engineering correction factors (${\epsilon }_{e,i}=\nabla p_{e,i}^n/\nabla p_{e,i}$) are defined to express the relationship between the flow in co-axial, coupled, channels and the flow in square, uncoupled, conduits. Then, it is demonstrated that these correction factors are invariant at $Ha>{10}^3$. This allows to extrapolate our results at higher field intensity, thus making possible to calculate the co-axial manifold pressure loss from analytical correlation.