1 Introduction

When a liquid metal flows in a pipe or in an open channel under the influence of thenonuniform magnetic field created by a permanent magnet,eddy currents are induced insidethe liquid. These eddy currents create a Lorentz force opposite to the direction of the meanflow^{[ 1, 2 ]}. Moreover,the eddy currents induce an additional magnetic field which interacts withthe permanent magnet in such a way as to create a force upon the magnet. By measuring theseforces one can determine the velocity of the liquid metal. This noncontact electromagnetic flowmeasurement technique is called the Lorentz force velocimetry and permits flow measurementin hot and aggressive fluids such as liquid aluminum or molten steel^{[ 1, 2, 3, 4, 5, 6 ]}. When designing theflowmeters based on Lorentz force velocimetry,which is often called the Lorentz force flowmeters(LFF),it is important to know how strongly the measured force depends on the velocity profilein order to design the magnet systems in such a way as to minimize this profile-dependence.Whereas this question is reasonably well understood for laminar axi-symmetric pipe flows^{[ 4 ]},our underst and ing of this problem for open-channel flows with a free surface is not sufficientlyadvanced. The goal of the present work is to bridge this gap.

The specific motivation of the present work is illustrated in Fig. 1. The flow direction is per-pendicular to the paper. Bold arrows indicate the magnetic field lines. The Lorentz force actson both magnets and is in the same direction as the mean flow. The objective of the presentwork is to compute this Lorentz force for different flow rates and heights of the free surface.Figure 1 shows a highly simplified representation of the Lorentz force flowmeter used byKolesnikov et al.^{[ 5 ]} to determine the flow of liquid aluminium in secondary aluminium pro-duction. The flowmeter consists of two permanent magnets held together by a yoke which isconnected to a force measurement system(not shown). The force measurement system recordsthe Lorentz force which depends on the unknown volumetric flow rate,on the electrical con-ductivity of the liquid metal and on the spatial distribution of the magnetic field. Here,weare interested in the question how strongly the measured Lorentz force depends on the detailsof the velocity profile for a given flow rate and for a given liquid metal level. This questionis of considerable practical importance because Lorentz force flowmeters are often calibratedby replacing the flowing liquid metal with a moving solid metal bar that has the same cross-section as the liquid metal. It is obvious that in this procedure,which is referred to as drycalibration^{[ 7 ]},the velocity profile of a moving solid body is different from that in a liquid metal.It would therefore be important to know how much a calibration constant(to be defined below)determined from dry calibration differs from the desired “true” calibration constant for a realflow.

To answer this question comprehensively,it would be necessary to numerically solve thefull three-dimensional set of the Navier-Stokes equations and the Maxwell equations for theturbulent free-surface flow. Such a procedure is quite expensive computationally and does notprovide insight into the general dependence of the Lorentz force on the key parameters of ourproblem^{[ 8 ]}. We therefore define in Section 2 a highly simplified problem that is amenable tocomparatively inexpensive numerical simulation and can be used to explore the Lorentz forcefor a wide range of geometry parameters. In Section 3,we present results of computations forthe particular case when the two magnets are very small and can be represented by magneticdipoles. Section 4 is devoted to the more general case of finite-size magnets. In Section 5,wesummarize our conclusion and discuss topics that would be useful to investigate in future.2 Formulation of problem

We simplify the situation shown in Fig. 1 to the greatest possible extent and consider themodel shown in Fig. 2. A liquid or solid metal with width ℓ and height h moves at mean velocityv₀ in the z-direction. The metal is exposed to two symmetrically placed permanent magnetswhose magnetization is along the x-axis.

The movement of a fluid of electrical conductivity σ in a magnetic field of induction Binduces electrical currents of density j,given as follows:

where φ is the electric potential.

The interaction between eddy currents and the applied magnetic field is described by theLorentz forces of density as follows:

The integrated Lorentz force over the metal volume,which is given by

depends on both the magnetic field and the velocity distribution in the fluid. For this reason,it is very difficult to find an analytical expression which relates the Lorentz force to the meanvelocity of the flow v₀. A simplified expression is given as follows:where the calibration coefficient depends on the magnetic field and velocity distribution Cv(v,B).The dependence F(v₀)in(4)is linear in the case of laminar or turbulent pipe flows,where thevelocity distribution can be described as follows:where g(ε)is a non-dimensional shape function^{[ 4 ]}.

(4)becomes nonlinear in the case of open channel flows,as shown by experiment^{[ 9 ]} and numerical simulation^{[ 10 ]}.

Figure 1 shows the case when the size of the magnets is much smaller than the width and height of the moving metal. We are interested in the z-component of the Lorentz forceF integrated over the volume of the conductor(which is equal to the force acting upon themagnets)as a function of the two geometry parameters,namely the non-dimensional fillingparameter

and the nondimensional distance parameterWe shall present our results both in dimensional and non-dimensional form. It has been shownby Thess et al.^{[ 4 ]} that the Lorentz force can be represented in the following form:where Q is the volumetric flow rate(connected to the mean velocity as Q = v0hℓ),B0 isthe magnetic field at a given reference position(the center of the bottom of the conductor inFig. 2),and ℓ is a characteristic length scale of the problem. The dimensionless quantity C,which is the central focus of the present work,is either called calibration constant or calibrationfunction. If the flow takes place in a fixed cross section,the quantity C is indeed constant and its computation is straightforward^{[ 11 ]}. If the cross section of the flow changes as in our presentproblem,C depends on the geometry parameters of the problem. In our case,we have C(ξ,σ),whose determination is the goal of our computation.

In what follows,we will use three kinds of velocity distributions. We start our computationswith the solid body case when the velocity is uniform over the cross section of the conductor.We then perform simulations for an analytically given profile which represents the laminarsolution of the Stokes problem with a free upper surface and no-slip boundary conditions at theside wall and at the bottom. We finally compare the results with a laminar Stokes flow whereall four boundaries are solid walls at which the no-slip condition has to be satisfied.

The computations are limited to the kinematic case,which neglects the effect of the Lorentzforces acting on the flowing liquid. The influence of the magnetic field on the velocity distribu-tion becomes significant when the magnetic interaction parameter N = σℓB₀²/(ρv₀)has valuesmuch greater than one^{[ 2 ]}. This parameter has low values in our simulations: N = 4 × 10^-10 inthe case of dipoles and N = 2.1 in the case of finite size magnets,respectively. The magneticeffects on velocity profiles have been theoretically investigated by Hunt^{[ 12 ]} for laminar ductflows under the influence of uniform magnetic fields. The effects of non-uniform magnetic fieldson turbulent and laminar flows have been numerically investigated in the previous work^{[ 8 ]}.2.1 Solid conductor

The numerical computations are performed with the finite element software COMSOLMultiphysics^{[ 13 ]}. In the first step,numerical simulations are performed for a simple case ofa solid conductor moving at constant velocity v0. The magnetic field produced by two perma-nent magnets and the Lorentz force acting on the moving conductor are computed by usingthe magnetostatic application mode of the AC/DC COMSOL electromagnetic module. Theseelectromagnetic computations have been validated in other simulation work^{[ 14 ]} by comparingnumerical results with experimental data.

The magnetostatics application mode solves the coupled AmpYre’s law and the continuityequation written in terms of the electric potential φ and the magnetic vector potential A,i.e.,

where M is the magnetization vector,v is the conductor velocity,and μ is the magnetic perme-ability. The Lorentz force acting on the conductor is computed by integrating the force densityover the conductor volume.

The transversal cross section of the simulated configuration is presented in Fig. 2. Themagnetization of dipoles is along the x-direction. Two finite size permanent magnets are alsoused in the simulations.

A surrounding air domain of dimensions few times larger than the characteristic dimensionof the magnetic system have been considered in the simulation. The magnetic potential is set tozero at the air domain boundaries A = 0. The electric insulation boundary condition n · J = 0is imposed at the conductor walls.2.2 Laminar flow

For a laminar flow in a duct having a rectangular cross section of dimensions ℓ = 2a and h = 2b in the xy-plane,the velocity distribution is given by^{[ 15 ]}

where αn = π(2n − 1)/2. The mean velocity of the flow obtained by integrating the velocitydistribution over the tube cross section is given bywhere χ is the negative of the pressure gradient along the flow direction(z-axis),ρ is the density,τ is the dynamic viscosity,g is the gravitational acceleration,and θ is the inclination angle ofthe tube. The velocity distribution for the laminar flow in an open channel of width 2a and height h is given by the same equations(11) and (12),where the half length b is replaced byh. The vertical velocity profile on y-axis is half the complete parabolic profile in a laminar flowthrough a closed rectangular channel^{[ 15 ]}.

The Navier-Stokes equations are solved for an incompressible Newtonian fluid by using theCOMSOL Multiphysics. The simulation domain has the dimensions 2a = 0.01m,2b = 0.005m and the length L = 1m. In order to have a fully developed flow,the channel length must begreater than the entrance length Le,which is given by

where ℓ = 2a is the channel width. The Reynolds number is defined aswhere the hydraulic radius Rh is the ratio between the cross-sectional area and the wettedperimeter.

At the inlet boundary,the input velocity is set to v0,and at the outlet boundary the pressureis p = 0. The no-slip boundary condition(v = 0)is imposed along the channel walls and noshear stress condition is used at the free surface.

The numerically computed velocity profiles on the vertical y-axis compared with the analyt-ical predictions given by(11)for closed and open channel flow,are shown in Fig. 3. The velocityprofiles are computed in the absence of the magnetic field. The small differences between nu-merical and analytical results are due to the graphical plot of(11). This equation contains aninfinite sum which cannot be plotted with our graphics software. The agreement is improvedwhen more terms are added in our function plot. Therefore,both the numerical computations and (11)can be used to predict the velocity distribution in laminar closed and open channelflow. The flow simulations for the unidirectional profile have been carried out for the purposeof the verification of the code.

3 Numerical results for magnetic point dipoles

Two small dipoles of dimensions 0.000 4m×0.000 2m×0.000 2m are used in our simulations.In this way,the magnetic field inhomogeneity due to the shape and magnets dimensions areminimal. The magnetization of dipoles in the x-direction is M_x = 10⁶ A/m. The conductorhas the width ℓ = 0.01m and the length L = 0.1m. The distance between the conductor and dipoles(w) and the conductor height(h)are varied over the ranges: w = 0.001m-0.01m(η = 0.1-1) and h = 0.001m-0.02m(ξ = 0.1-1). The Lorentz forces are computed at constantvelocity v₀ = 1m/s,then the dimensionless calibration coefficient is computed by using(8),where the electrical conductivity of the solid body is σ = 10⁶ Q^-1 · m^-1,the volumetric rateis Q = v₀ℓh,and B0 is the x component of the magnetic induction in a point located at thebottom surface of the conductor(see Fig. 2).

The results obtained by varying the conductor height h(dimensionless quantity ξ) and thedistance w of the magnetic dipoles from the conductor(dimensionless quantity η)are plottedin Fig. 4. The calibration coefficient first increases to a maximum value,then decreases whenthe conductor height is augmented beyond a critical value. The existence of a maximum Cversus ξ indicates that for this particular filling level a Lorentz force flow meter has its highestsensitivity. This observation implies that the distance between the magnets of the flow metersketched in Fig. 1 should be chosen in such a way that the most probable filling level correspondsto the filling level at which the sensitivity is maximum.

The existence of a maximum of C(ξ)is a consequence of the fact that C = 0 both for ξ = 0 and for ξ → ∞. The first fact,i.e.,C(0)= 0,follows from the observation that the dominantsource of the eddy currents is the electromotive force proportional to the x-component of themagnetic field and the z-component of the velocity. As ξ → 0,these currents become confined and tend to zero as h2. The decay of C for ξ → ∞ is due to the fact that for increasing fluidheight an increasing part of the conductor is located in a region where the magnetic field isvery weak. Since the calibration constant is a weighted average of contributions from all h,thecontributions from the distant parts of the conductor lead to the asymptotic decay of C. Thevalue of ξ corresponding to a maximum of C increases with increasing distance between themagnets and the conductor(see Fig. 4).

Figure 4 permits a systematic evaluation of the influence of the velocity profile on thecalibration coefficient. Figure 4(a)shows that the calibration coefficient of a moving solidbody(blue line)is roughly twice as high as those of the laminar flow of a fluid. Figures 4(b)-4(d)indicate that this difference becomes smaller when the distance between the magnet and conductor increases. This is due to the fact that for small η the magnets “feel” the velocitygradient near the wall stronger than for high η. These features indicate that a dry calibrationprocedure does not in general provide calibration coefficients that are close to those in a laminarflow. However,it could also be noted that the ratio between the solid and liquid curves in Figure 4 are roughly independent of ξ. Hence,it could be possible to recalculate calibration coefficientsfor liquid metals from dry calibration experiments. Another aspect in favor of dry calibrationis that most flows in metallurgy are turbulent and their velocity profiles are in between thoseof a solid body and a laminar flow. Hence,the difference between dry calibration and liquidcalibration for turbulent flows is smaller than the curves in Fig. 4.4 Numerical computations for finite size magnets

The influence of the magnet dimensions on the calibration function is analyzed by increasingthe magnets size. The simulations are performed for the configuration shown in Fig. 5(a),withtwo cubic magnets of 0.01m length size,distanced by d = 0.03m,and having the magnetizationMx = 2 · 10⁶ A/m. The channel has the dimensions 0.01m × 0.01m × 0.7m in the x-,y-,and z-directions. The centers of the magnetic system,the channel(in the xy-plane),and thecoordinate system coincide. The magnetic field distribution in the channel is non-uniform inthe x-,y-,and z-directions(see Fig. 5(b)).

The calibration coefficients are computed in the case of a solid conductor and the laminarflow in closed and open channels. The liquid height is varied over the range h = 0.002m−0.02m(see Fig. 6). The input velocity in the computations is the mean velocity of the flow v0,which inlaminar open channel flow is related to the liquid height h by(12). The computations performedby accounting for the laminar flow effects show significant differences in the resulting Lorentzforces and calibration coefficients in comparison with the solid body case. The maximum errorabout the estimation of the calibration function by using the simplified solid body computations,is δC ≈ 38%. However,these errors are much smaller as compared with computations performedfor magnetic dipoles(see Fig. 4).

The Lorentz forces and the calibration coefficients carried out from computations includingvelocity effects are smaller than the same quantities carried out from solid body computations.This can be explained by different patterns of induced eddy currents,which depend on boththe velocity and magnetic field distribution in the liquid metal. These effects are analyzed byplotting the velocity profiles,eddy currents and Lorentz forces computed for a solid conduc-tor(see Fig. 7),and a duct laminar flow(see Fig. 8). The simulations are performed for theconfiguration shown in Fig. 5(a). The magnets are located at z = 0 along the flow direction(z-axis). The mean velocity of the flow is v0 = 0.02m/s,which corresponds to a Reynoldsnumber Re = 500(ρ = 2 370 kg/m³,τ = 0.001N · s/m²).

In the case of a solid conductor,the velocity is constant in the x-,y-,and z-axes(seeFig. 7(a)). The induced eddy currents are plotted in Fig. 7(b). The color maps show eddy cur-rents distribution in the transversal xy-plane and the longitudinal yz-plane. The magnets areshown in the xy-section. The component of eddy currents parallel to the y-axis contributes tothe measureable Fz Lorentz force for a LFF device. The negative jy currents induce negative(damping)F_z forces,while positive j_y currents induce positive(accelerating)F_z forces. Eddycurrents are closed in the longitudinal yz-plane,due to the magnetic field non-homogeneity.The component j_y is negative in the region with high magnetic field and positive in the regionwhere the magnetic field vanishes. The x,and y-axes plots show only negative values of j_y,which means that there is no closed path of currents in the transversal xy-cross section. TheLorentz force density plotted in Fig. 7(c)shows a predominant negative F_z component.

The velocity profiles,eddy currents,and Lorentz forces carried out from laminar flow com-putations are plotted in Fig. 8. The velocity profiles have a parabolic shape in the x- and y-axes(see Fig. 8(a)). The plot along the flow direction(z-axis)shows an increasing velocityover the entrance length(Le ≈ 0.3m). Velocity boundary layers appearing near the walls createa supplementary difference of the electrical potential in the transversal plane. Therefore,eddycurrents can be closed in the transversal xy-cross section,as shown in Fig. 8(b). The x plotshows negative currents in the core region of the channel,and positive currents in the wallboundaries of about 0.001 5m width. The z plot shows also the closure of eddy currents in thelongitudinal yz-plane,as for a solid conductor. Eddy currents which turn in the transversalcross section create positive Fz forces,as shown in Fig. 8(c). Finally,the integrated(negative)F_z force is significantly reduced for a laminar flow(F = −1.54 × 10⁻⁵ N)as compared withthe solid body case(F_S = −1.612×10⁻⁵N). The relative error about solid body computations ofthe Lorentz forces given by δF =(FS −F)/FS is δF = 4.5% for the simulated case. Therefore,we conclude that the velocity effects would be accounted for in the LFF numerical calibrationprocedure.5 Conclusions

The influence of the laminar velocity profile on the calibration function of a Lorentz forceflowmeter is numerically investigated by using COMSOL Multiphysics. Computations per-formed for laminar flows in closed and open rectangular channels are compared with a simplecase of a conductor moving at the mean velocity of the flow.

The laminar flow computations are first validated by comparing the numerical velocity pro-files with theoretical predictions. Then,velocity effects are accounted for in the numericalcomputations of the Lorentz forces. It is found that the laminar velocity profiles has a signifi-cant effect on the resulting Lorentz forces. These effects are explained by different patterns and densities of the induced eddy currents in laminar flows and solid body computations. Numericalcomputations of the calibration function show significant errors if the simplified solid body com-putations are used in the calibration procedure. These errors increase up to δC = 75% in thecase of dipoles and δC = 38% in the case of finite size magnets. Finally,one can conclude thatthe calibration function of a Lorentz force flowmeter is significantly influenced by the velocitydistribution in laminar closed and open channel flow. Therefore,laminar flow computationsshould be included in the LFF calibration procedure.

Acknowledgements The authors are grateful to the German Research Foundation(DeutscheForschungsgemeinschaft)for supporting the work in frame of the Research Training Group(Graduiertenkolleg)“Lorentz force velocimetry and Lorentz force eddy current testing” at IlmenauUniversity of Technology. The authors also acknowledge helpful discussion with Prof. S. MOLOKOVof Coventry University.