Theoretical modeling and experimental tests of an electromagnetic fluid transportation system driven...

IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 9, SEPTEMBER 2006 2133

Theoretical Modeling and Experimental Testsof an Electromagnetic Fluid TransportationSystem Driven by a Linear Induction Motor

Sérgio Poitout and P. J. Costa Branco

DEEC/AC-Energia, Instituto Superior Técnico, Portugal

We report on a theoretical and experimental study of the use of a traveling magnetic field generated by a linear induction motor topropel a conducting fluid in a closed channel. We initially developed an analytical model to predict the distribution of density currents,forces, and fluid speed. To verify the validity of our assumptions for the analytical model, we developed a finite-element model andcompared its results to those ones obtained from the analytical model. We built a prototype of the fluid transportation system, usingmercury as the conducting fluid. Here, we present the prototype’s experimental characteristics with various alternating current valuesand compare them with the analytical model’s results.

Index Terms—Electromagnetic devices, magnetohydrodynamics.

I. INTRODUCTION

THE use of electromagnetic forces in fluids can present sev-eral objectives: transport, movement, containment, or even

mould the fluid. The most significant applications appear in themetallurgical industry, where the fluid is a molten metal. In [1],the authors describe other applications where the electromag-netic forces are used to levitate liquid metal and also to separateinclusions in molten metal during its solidification process [2].

Fluid transportation by electromagnetic forces is based on theLaplace force. It is necessary that the two electromagnetic quan-tities related with this force (current density and the magneticflux density) take high values in certain locations of the conductwhere the fluid travels to generate fluid displacement.

Depending on the electric conductivity value of the fluid andthe application considered, the transportation mechanism canbe based on induced currents [3], [4] or currents imposed to thefluid by electrodes located in the channel walls [5]–[8]. In metal-lurgic applications, the currents are induced using for this effectbobines placed externally to the fluid and electrically isolated.There are many configurations of those systems which dependof the structure where the fluid circulates (open channel, tube,etc.) and the required application [9]–[13].

The research concerning applications on this area have incommon the objective of determining the electromagnetic quan-tities and the fluid speed. It is possible to refer some recent re-search where the molten metal flow is analyzed in a channelmoved by electromagnetic forces [18]–[22]. In the majority ofthese papers the metal flow control is analyzed, establishinga relation between system’s configuration and its parameters(voltage, current, frequency) and fluid flow.

Concerning those systems used to move fluid metal in chan-nels, the most common systems are the magnetohydrodynamicpumps based on cylindrical linear induction machines with the

Digital Object Identifier 10.1109/TMAG.2006.880396

fluid circulating in its interior. The analysis of one of thesepumps is, for example, studied in [14] and [21].

Most research in the literature use finite-element models(FEMs) to calculate electromagnetic quantities and the fluidspeed, adapted to the problem or specific situation being ana-lyzed. There are other approaches based on FEMs but these aremore general concerning their application [15]–[17], [19], [20].Some authors do a bidimensional analysis of the system (whenthe system’s configuration allow) due to the less computationaleffort.

Recently, applications of fluid transportation by electromag-netic forces have been applied in areas such as biotechnology,bioelectronics, and in miniaturized systems where magnetohy-drodynamic pumps are used. Some applications are in microflu-idic systems as described in [5]–[7], [18].

In this paper, we are concerned with the analysis and numer-ical approximation of a conducting fluid confined in a closedchannel and driven by a traveling magnetic wave from a linearinduction motor. A theoretical approach of the fluid transporta-tion system is given in Section II. In Section III, we derive afinite-element model of the system and compare its results withthose ones from the analytical model. Section IV is devoted tothe experimental tests effectuated with the fluid transportationsystem and the comparison between them and those results pre-dicted by the analytical model.

II. FLUID TRANSPORT SYSTEM: MODELING

A. System’s Configuration

The configuration and picture of the fluid transportationsystem are shown in Fig. 1. The system consists in a closedchannel made of a nonconductive rigid tube of rectangular crosssection containing an inviscid incompressible and electricallyconducting fluid.

The motion is produced by the stator of a linear induction ma-chine which is inserted at the bottom of the closed channel asshown in Fig. 1(a). A piece of ferromagnetic material (iron) is

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2134 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 9, SEPTEMBER 2006

Fig. 1. (a) Configuration of the fluid transportation system—front view and (b) lateral view. (c) Schematic drawing of the fluid transportation system and its maindimensions. (d) Picture with a general view of the system. (Color version available online at http://ieeexplore.ieee.org.)

placed above the fluid channel to increase the vertical compo-nent of the magnetic flux density in the space occupied bythe conducting fluid above the stator of the induction machine.The channel is made parallel to the stator and its length is longerthan the stator one. The channel is filled with fluid to a certainheight, to avoid touching the iron plate, as shown in Fig. 1(b).A top view schematic diagram with the main dimensions of thefluid transportation system built in our laboratory is presentedin Fig. 1(c). A picture with a general view of the experimentalsystem, fluid channel plus the stator of the induction machine,is presented in Fig. 1(d).

The induction machine imposes a sliding linear current den-sity defined by

(1)

where is the angular frequency and is the machine waveconstant. The sliding current density subjects the conductingfluid to a time variable flux density in the form of a trav-eling wave, which gives origin to a current density distributionin the fluid. Internal force densities appear then in the fluid byLaplace forces. These forces modify the fluid speed, changing

POITOUT AND COSTA BRANCO: MODELING AND TESTS OF AN ELECTROMAGNETIC FLUID TRANSPORTATION SYSTEM 2135

TABLE IPARAMETERS OF THE FLUID TRANSPORTATION SYSTEM

TABLE IIFLUID PARAMETERS (MERCURY)

the currents in the fluid due to the appearance of electromotiveforces, thus completing the electromagnetic coupling with theconducting fluid.

The system parameters are those ones of the prototype built inour laboratory. Table I lists the parameters relative to the linearinduction machine, fluid channel, and the iron plate. The fluidparameters are listed in Table II.

The model can be divided in two components: the electro-magnetic and the fluid mechanics component. To begin themodel design, practical and reasonable assumptions on eachcomponent of the fluid transportation system have to be made.In the following, both components and respective modelingassumptions are explained.

B. Simplifying Assumptions: Electromagnetic Component

A set of simplifications were assumed to the electromagneticcomponent of the model.

1) The component of flux density is null.Since the stator windings of the induction machine are

oriented, their current density and also the current densityinduced in the fluid were also assumed axes oriented.Neglecting any border effects in the region of the fluidchannel, the and components of the density current inthe fluid were considered null

(2)

Using (2) in , the magnetic field andflux density in the fluid will have only the andcomponents

(3)

2) The air-gap between stator and fluid is neglected.Although there is a small spacing between stator and

fluid, it was assumed that the fluid is contiguous to thestator due to the thickness of the channel base, smallerthan the fluid height.

3) Stator negligible border effects.

Because the fluid channel section in analysis is muchsmaller in length than the stator of the linear inductionmachine, possible border effects due to changes in thestator magnetic field have been considered negligible.

4) The stator is modeled by a ferromagnetic material blockplus a linear current density sheet.

Because the stator is made from a laminated ferromag-netic material block where there is a three-phase set ofcopper windings embedded, one can assume to the modelthe use of a unique ferromagnetic material block, alsoconsidering that substituting the copper and air regionsby ferromagnetic material will not significantly affect themodel. In the upper face of the ferromagnetic block it isconsidered the presence of a linear density current sheetwith a value equivalent to that one circulating in the statorwindings.

C. Simplifying Assumptions: The Fluid Mechanics Component

Some simplifications were assumed to the fluid mechanicscomponent.

1) Laminar flow.Considering that the system transports a high mass den-

sity conducting fluid with significant viscosity, it was as-sumed a fully developed laminar flow condition.

2) Analysis centered on the average plan of the fluid channel.If the fluid has a high mass density, its volume consid-

ered is high, and also the channel width is large enough,one can assume that the effects of the friction forces at thelateral channel walls are negligible to the characterizationof fluid flow.

3) Fluid displacement only due to the average value of theelectromagnetic force.

Since the conducting fluid has an high mass density andtaking into account that the electromagnetic force oscil-lates with 50 Hz frequency, it was assumed that instantchanges of force would not significantly affect the fluiddisplacement but only its average value would take effect.

4) The and components of fluid speed are null.Due to a high mass density value of the conducting

fluid, it was considered that the gravity force and the reac-tion forces from the channel base will be enough to equi-librate the electromagnetic forces that have a positive av-erage value in the direction such that . The com-ponent of fluid speed is assumed zero due to the drivingcondition and channel geometry without obstacles of thefluid transportation system. The resulting direction will bethat of the force density, neglecting any pressure gradientbeing applied in the direction.


D. Bidimensional Analysis

From previous assumptions, the component of the flux den-sity at the fluid is zero, , and the current density in thefluid has only the component, . Using (4), theseconditions are applied in (5) to obtain the density force distribu-tion in the fluid (6). This result indicates that the componentof fluid speed can be considered null, , because

is zero

(4)

(5)

(6)

Since the simplifications assumed allow neglecting the ef-fects in the fluid speed distribution of friction forces at the lat-eral walls, and also it allows neglecting the lateral effects in thelinear induction machine, we have established that the electro-magnetic variables and the fluid speed were not functions of the

direction. These considerations mean that a bidimensional ap-proach for the system’s modeling could be used.

1) Electromagnetic Component: Now that the physical na-ture and bidimensional characterization of the fluid transporta-tion system have been described, the magnetic vector poten-tial can be employed to obtain the electromagnetic equationsneeded for a complete description of the system.

Distribution of the magnetic vector potential. The distri-bution of the magnetic vector potential is given by

(7)

Taking into account the set of simplifying assumptions as-sumed before, the components of vector (7) will be only re-sumed to the component giving

(8)

In this equation, fluid speed is function of - and -coor-dinates, making it very difficult to solve an analytical solutionof (8). A numerical solution solving simultaneously the electro-magnetic equation (8) and the fluid mechanical one (31) wouldgive an exact solution. To surpass this difficulty and obtain anapproximated solution to the magnetic potential vector, hasbeen assumed for this calculus a global average fluid speed, thusbeing a constant value which allowed us the following solutionto .

The solution of (8) can assume the form given by (9), whichis similar to the current density imposed by the linear inductionmachine

(9)

Substituting (9) in (8) results in the differential equation

(10)

The general solution of this equation is given by

(11)

where is obtained from the characteristic equation and definedas

(12)

and and are constants to be obtained from the boundaryconditions. The equation giving the vector potential distributionbecomes defined as

(13)

where two boundary conditions are required.First boundary condition: This condition is obtained at

which separates the fluid from the stator block, as shown inFig. 2(a). At this position, there is a discontinuity condition re-lated with the tangent component of the magnetic field andcaused by the stator imposed sheet of linear current density .To obtain , we use the contour shown in Fig. 2(a) en-closing . Integrating Ampere’s law over yields

(14)

The contour can be chosen small enough that the magnetic fielddoes not vary appreciably over its length , and the contouronly links the density current associated to the linear inductionmachine. The integrals associated with the lateral segments canbe set null since their widths can be made very small comparedwith the superior and inferior segments. The tangent componentof associated with the inferior segment is zero consideringthat the magnetic permeability of the stator material is madeinfinity. Based on these assumptions, (14) results in

(15)

The magnetic flux density component is related with thecomponent of by

(16)


Fig. 2. Application of Ampere’s law for obtaining the (a) first boundarycondition at y = 0, (b) second boundary condition including the densitycurrents in the fluid from y = 0 to y = h to the system with a channelcompletely fulfilled, and (c) the second boundary condition for a channelpartially fulfilled. (Color version available online at http://ieeexplore.ieee.org.)

which can be used to obtain

(17)

Second boundary condition: This condition is obtained atusing again Ampere’s law as indicated in Fig. 2(b). For this

purpose, we use the contour which is perpendicular to thecoordinate and encloses the imposed linear current densityuntil the upper limit of the fluid channel delimited by the

iron plate. Using Ampere’s law in the contour , assumingan infinite magnetic permeability for the material composingthe stator and the iron plate, and making the vertical segmentslocated below and above the fluid limits much smaller than the

height , we obtain

(18)

For this expression, is obtained fromwhich gives and thus

(19)

The current density follows from whichgives

resulting in

(20)

The second equation for computation of constants andis determined using the results (19) and (20) in (18) yielding

(21)

Thus, resolving the system of (17) and (21) for and gives

(22)

which can be used to obtain the magnetic vector potentialdistribution

(23)

Electromagnetic quantities. As a consequence of our deduc-tion concerning the vector potential distribution (23), we can ob-tain the magnetic field distribution (24) and the current density(25)

(24)

(25)

In these equations, parameter depends of the stator and fluidparameters as verified in (12). Any change in the fluid speed willmodify this parameter.

The force density distribution is obtained from (6) where onlythe real part of the electromagnetic quantities is considered,yielding

(26)


Fig. 3. Analytical model, transient regime: distribution of magnetic flux density obtained from the analytical model. This pattern remains constant during thetransient time interval.

(27)

To determine the average value of the force density , theproduct of and must be analyzed separating the terms con-tributing to the average force density and those ones having anull average value. Doing this, becomes (28), shown at thebottom of the page.

2) Fluid Mechanics Component: The Navier–Stokes equa-tion (29) describes the fluid behavior. In this equation, isthe substantive derivative, is the fluid mass density, is thefluid speed, is the gravity acceleration, is the pressure, and

is the fluid viscosity. Equation (29) takes into account thetotal force density on the fluid, where is the magnetic forcedensity, is the gravity force density, term denotes themechanical force density, and the last term is the forcedensity component related with the shear stresses due to the fluidviscosity

(29)

The fluid is considered incompressible and thus . Usingthis condition in (29) yields

(30)

Since we are considering a laminar flow, the component offluid speed can be set as being zero, . The and compo-nents of gravity are null, . The component of fluidspeed is assumed not to change in direction, ,due to our analysis made in a vertical plane in the middle of theduct, which was considered far enough away from the lateralwalls.

Because the fluid displacement is related to the average valueof the magnetic force, we can consider . Thecomponent of fluid speed is considered null, , because thegravity force is considered equilibrating the total force densityin the direction. Also, since there is not an external pressuregradient in the direction, . Using these assump-tions in the and components of (30), results in

(31)

3) Approximated Solution: Steady-State and AverageRegime: The steady-state solution of (31) was obtained con-sidering the average value of each variable instead of itsinstantaneous value: the average value of the componentof fluid speed, , the average value of the component of

(28)


Fig. 4. Analytical model, transient regime: evolution ofB amplitude as function of y coordinate; (a) flux densityB, (b) component B , and (c) component B .

Fig. 5. Analytical model, transient regime: distribution of current density amplitude as function of y coordinate.

magnetic force density, , and the average value of pressure,, all obtained for each -coordinate. Assumptions for the

steady-state solution is that the derivative of with respect foris not significant when compared with the derivative of

with respect for ; also . Hence, (31) can be rewritten as

(32)

Two operating conditions were considered for the pressuregradient term : null, and a condition intended to be rep-

resentative of the prototype system’s geometry with the pressuregradient being a function of the fluid speed, contact surface, andthe channel curvature.

Null pressure gradient: In this condition, . To ob-tain a solution for (32), it is convenient to give an estimation of

as function of coordinate. As our first approximation, wehave estimated the force density evolution in steady-state andwith null pressure gradient being given by an inverse exponen-tial relation as

(33)


Fig. 6. Analytical model, transient regime: distribution of force density ^F in the fluid.

In this equation, parameters and depend on the stator cur-rent, system geometry, and the conducting fluid characteristics.Parameter can be obtained from (28) using and pa-rameter must be estimated.

Using (33) in (32) and solving this equation, the fluid speedbecomes given by

(34)

where constant is determined substituting (34) in (32) togive . The other constants are obtainedusing the two boundary conditions

and (35)

resulting in constants andto obtain (36) the fluid speed distribution

(36)

Pressure gradient as a function of fluid speed and channelgeometry: This pressure gradient condition is intended to berepresentative of the laboratorial system’s geometry. The trans-portation system is divided in two parts: that one being modeledand the part of the closed channel where there are no electro-magnetic forces on the fluid.

The external pressure gradient applied is composed by the con-tribution of two terms as shown in (37). The first term is re-


Fig. 7. Analytical model, transient regime: (a) evolution of the x component of the mean force density ^F obtained from the analytical model and (b) thecorresponding fluid speed v̂ .

lated with the friction forces (due to the viscosity) exerted on thefluid. The second term represents the additional force resis-tant to the fluid flow caused by the curvatures of the fluid channel

(37)

Using relation (33) to define the average force value and sub-stituting it in (32) yields

(38)

The solution of (38) is obtained, as before, by the sum of ahomogenous and particular solution, using the boundary con-

ditions to calculate the constants. The fluid speed assumes thefollowing form:

(39)

III. NUMERICAL RESULTS

In this section, the transient and steady-state results obtainedusing the analytical model are analyzed and compared with


Fig. 8. Analytical model, steady-state regime: (a) distribution of flux density B, (b) component B , and (c) component B as a function of y coordinate.

those results obtained from a FEM simulation of the fluidtransportation system using the program Ansys.

The analytical model has used the equation (32). Despitethe FEM simulation also consider average quantities, the mainaspect concerning the finite-element model is that the fluidspeed distribution at the channel output in one iteration istaken as the speed distribution at the channel input in the nextiteration. It has been assumed that after leaving the statorregion, the fluid speed remains constant during its circulationin the channel.

Although the analytical and FEMs are intended to be rep-resentative of the same system, they present some significantdifferences.

• In the analytical model, it is estimated that all fluid particlesare subject to electromagnetic forces since ; in theFEM, not all fluid particles are subject to electromagneticforces since the fluid channel modeled is larger than thestator of the induction machine;

• In the FEM, there are border effects due to the stator endwindings. These effects have been neglected in the analyt-ical model;

The conducting fluid considered on both models has beenmercury (as also used in the laboratorial system), the pressuregradient between the channel input-output was considered to benegligible, and it has been set a stator current value of 10 A.

A. Analytical Model

1) Transient Regime: To obtain the electromagnetic and me-chanical quantities in the fluid, the average force density andfluid speed have been calculated in an iterative way usingan integration step of 1 ms, equal to that one used in the FEMsimulation. To find the solution of (32), the fluid is at rest in

and the boundary conditions used were and

. The results in this section show how quan-tities and are distributed through the fluid, also pre-senting their evolution in time.

The distribution of is shown in Fig. 3. The figure showsthat the pattern of remains constant. However, the am-plitude changes during the transient period. This is shown inFig. 4(a)–(c) where the amplitude and its - and - com-ponents are plotted until 9.3 s, time interval considered for thesystem to be in steady-state regime. The results show effectivelya very small difference in the amplitude values of during thetransient time interval.

Fig. 5 presents the evolution in time of the current density am-plitude until steady-state. The figure shows that is maximumin the initial time stage, with the amplitude decreasing progres-sively as the fluid reaches the steady-state regime. The modelshows that in steady-state there is a significant current densityin the bottom of the fluid. This condition is related with the fluidviscosity and the fact that the fluid speed is zero at . Fig. 5shows however that the current density can increase for highervalues of . This condition, which is visible for s, is relatedwith the evolution of the fluid speed , as shown in Fig. 7(b).Despite the values of decrease with , as shown in Fig. 4,there are small fluid speed values occurring in the upper regionof the fluid, thus increasing the local density currents.

The force density distribution is plotted in Fig. 6 to four timeinstants: s, s, s and s. The resultsshow that as the fluid approaches the steady-state regime at 7s, significant force density values remain only near the surfacebottom of the fluid channel.

Fig. 7(a) presents the density force evolution until thesystem reaches the steady-state regime at s. The fluidspeed is shown in Fig. 7(b) as the fluid starts moving at1 s and reaches the steady-state regime in 9.3 s. The figureshows how the fluid viscosity and the friction at the bottomof the fluid channel affects the fluid speed profile. At the


Fig. 9. Analytical model, steady-state regime: (a) density current J amplitude and (b) the resultant force density ^F distribution.

bottom of the fluid channel the fluid speed is zero, rapidlychanging its value until reaching the maximum value between1 and 2 mm. After, the fluid speed follows the evolution ofthe density force .

2) Steady-State Regime: The steady-state results of the fluidtransportation system has been obtained using (36), which con-siders a null pressure gradient. Parameters and used in theforce function (33) were obtained from the average force den-sity curve for s in Fig. 7(a). Using the fluid speed valuescalculated from (36), Fig. 8(a) shows the amplitude values ofthe flux density and Fig. 8(b) and (c) displays the com-ponents and as function of coordinate. The amplitudevalues of the density current are shown in Fig. 9(a) and theresultant force density distribution is plotted in Fig. 9(b). This

figure shows that the significant force values are located at thebottom of the fluid channel where the fluid speed is slower thanthe synchronous speed.

B. FEM

The electromagnetic and mechanical variables of the FEMare calculated in the central area of the fluid channel. For eachcoordinate, the results were analyzed to obtain the amplitude ofeach variable, since , and are sinusoidal signals. In thissection, plots are presented for the magnetic flux density, force,and fluid speed distribution for different time instants until thesteady-state regime. For all variables, we compare the analyticalmodel and FEM results.


Fig. 10. FEM, transient regime: distribution of flux density B which pattern remains the same during the transient time interval.

Fig. 11. FEM, transient regime: (a) flux density B, (b) component B , and (c) component B .

1) Transient Regime: This section shows the FEM resultsof the fluid transportation system during transient regime. Thesimulation corresponds to a time interval of 10.8 s, with thefluid stopped at . However, the results will be presentedcomparing the amplitude values only during the time intervalbetween s and s. In the other time instants there areperturbations appearing in the FEM caused by the stator bordereffects which were not considered in the analytical model. Plotsof the flux density amplitude and its components and ,force amplitude and components and , and also thefluid speed distribution for some time instants are presented.For comparison with the analytical model, the values obtainedfrom the FEM are registered through a vertical line near the fluidchannel output at m.

Fig. 10 shows the flux density distribution at s, re-maining with the same pattern during the transient time as alsoshown by the analytical model in Fig. 3. Fig. 11(a) to (c) showsthe amplitude values of the flux density and components deter-mined from the FEM and the analytical model. It can be verifiedthat there is a small but constant error between the two results,mainly in the component of which affects the total flux den-

sity. This error is related with the FEM discretization that affectsthe numerical results.

Fig. 12 shows the force density distribution as the fluid accel-erates until it reaches the steady-state regime. The evolution issimilar to the analytical model in Fig. 6, showing that the forcedensity remains significant only near the bottom surface of thefluid channel.

Fig. 13(a)–(c) shows the amplitude of the force density ob-tained from the FEM and analytical model. Fig. 13(d) showsthe fluid speed evolution obtained from the FEM and analyticalmodel. The almost constant error in both results comes againfrom the discretization level used in the FEM, which affects thenumerical results.

2) Steady-State Regime: For the steady-state regime, the re-sults are compared with the analytical model considering a nullpressure gradient. Fig. 14 shows the components of the flux den-sity obtained from the FEM and analytical model.

Fig. 15 shows the components of the force density in steady-state. The values obtained are lower than those ones obtainedduring the transient regime (for example, the force density inthe fluid surface at steady-state are almost zero, while during


Fig. 12. FEM, transient regime: distribution of force density F in the fluid as it accelerates and reaches the steady-state regime.

the transient regime the force values were about 5 10 N/m ).At the bottom there is still a high force density value since thefluid speed remains inferior to the synchronous speed. It can beverified in Fig. 15 that the curves obtained do not correspond toan exponential function, presenting a small error relative to thecurve obtained from the analytical model at m. Thiserror is related with the fact that the computed speed from theFEM to this coordinate value be higher than the synchronousspeed.

Fig. 16 presents the fluid speed curves for the analytical andthe FEMs. They show a small error in the upper and in thebottom regions mainly due to the discretization level used bythe FEM in these regions.

IV. EXPERIMENTAL SYSTEM: TEST RESULTS

The results obtained with the fluid transportation systemmounted in our laboratory are presented in this section, wherethe experimental results are compared with the results esti-mated from the analytical model. Fig. 17 shows a detail of thetransportation system where the stator of an induction machineis located under the channel which contains mercury as con-ducting fluid. This figure shows two straight lines marked inthe channel to allow measurement of fluid speed. The channelhas not been completely filled with fluid because the channelis open and also to avoid the contact between fluid and theiron plate. Because of this constraint, the magnetic potential


Fig. 13. FEM, transient regime: (a) total force density ^F , (b) evolution of ^F , and (c) ^F components of the force density from the FEM and analytical model,and (d) the corresponding fluid speed v̂ .

vector has been recalculated to the condition of a fluid channelpartially fulfilled.

A. Potential Vector: Fluid Channel Partially Fulfilled

The solution of the potential vector (8) considers that therelative magnetic permeability of the mercury is one. The firstboundary condition is considered in , as before, and resultsin (17). To the second boundary condition, Fig. 2(c) shows thecontour that encloses and the surface S to include thedensity current in the fluid with height . Integrating Am-

pere’s law over yields

(40)

For this expression, is given by (19). Using this equationin (40), the left term results in

(41)


Fig. 14. FEM, steady-state regime: (a) flux densityB, (b) component B , and (c) component B . (Color version available online at http://ieeexplore.ieee.org.)

Fig. 15. FEM, steady-state regime: (a) force density F, (b) component F , and (c) component F . (Color version available online at http://ieeexplore.ieee.org.)

Theright term isobtainedusing(20)as

(42)

Substituting (41) and (42) in (40) yields

(43)

Hence, constants and can be obtained from (17) and(43). Using and in (13) gives the magnetic potentialvector when the fluid channel is partially fulfilled [see (44) atthe bottom of the next page].


Fig. 16. FEM and analytical model, steady-state regime: the corresponding fluid speed v̂ .

Fig. 17. Picture detail of the fluid transportation system that shows the statorlocated under the channel filled with mercury. Two straight lines were markedto allow measuring the fluid speed.

Using (44), the magnetic field , current density , and den-sity forces and have been obtained and used in the followingresults.

B. Analytical Model and Experimental Results: Analysis

In all tests, the induction machine was excited with phasecurrents from 2 to 5 A. These limits have been defined sincefor currents lower than 2 A the fluid remains halted, while forcurrents higher than 5 A a turbulent regime appears. The averagespeed in the fluid surface and in steady-state has been measuredusing a movie which was analyzed by a video edition program.The results obtained are listed in Table III together with the fluidheight where the speed has been measured.

The change of fluid height is related with the total force inthe fluid, which causes fluid accumulation in the curve regionof the channel (higher the force, more fluid is accumulated in

TABLE IIIAVERAGE FLUID SPEED VALUES FROM THE EXPERIMENTAL SYSTEM

Fig. 18. Picture showing the fluid accumulation case in the curve region of thechannel.

the curve and smaller is the fluid height on the stator region).Fig. 18 shows the curve region with fluid accumulation when in

(44)


Fig. 19. Fluid speed v̂ profile for different currents I : results from the analytical model using the pressure gradient with one component proportional to thefluid speed and other component constant; experimental results presenting the average speed value at the fluid surface for different currents (note that the fluidheight changes).

Fig. 20. Fluid speed v̂ for different currents I and same y coordinate: results with the analytical model using the pressure gradient with one componentproportional to the fluid speed and other component constant; experimental results.

steady-state. Table III shows that the fluid speed values are sig-nificantly inferior to the synchronous speed, indicating that theanalytical model with a null pressure gradient had no correspon-dence with the system’s configuration mounted in the labora-tory. From the experimental results we can estimate a significantpressure gradient, opposed to the force , acting on the fluid.Therefore, the analytical model uses now the pressure gradientfunction (37) which has been previously established looking fora better correspondence with the operating conditions in the ex-perimental system.

The results obtained from the analytical model depend of thevalues attributed to constants and that characterize the

pressure gradient, as well as parameters and . The first ap-proach has searched for a numerical solution of the analyticalmodel until a steady-state regime has been reached. The advan-tage of this approach is not use parameters and to definethe electromagnetic force function. Fig. 19 shows the fluid speedcurves obtained using the analytical model for different valuesof current, also showing the experimental values. The analyticalcurves were plotted using the parameters

and also the values listed in Table IV functionof current . The experimental results, although showing a sim-ilar evolution for the low speed range when compared with theanalytical values, the results for high current values diverged.


Fig. 21. Fluid speed v̂ profile for different currents I : results from the analytical model using the pressure gradient with one component proportional to thefluid speed and other component constant; experimental results presenting the average speed value at the fluid surface for different currents (note that the fluidheight changes). The results were obtained with the analytical model using parameters K and q determined from the previous simulations in Fig. 19.

TABLE IVK VALUES USED BY THE ANALYTICAL MODEL

The same conclusions can be verified in Fig. 20 where the fluidspeed estimated from the analytical model and correspondingexperimental values are plotted. Note that the analytical modelhas used the experimental fluid height ( coordinate) and equalcurrent values.

The results obtained with the analytical model using (39) andparameters and determined from the previous simulationare presented in Fig. 21. The experimental and analytical modeldifferences are related with the assumptions assumed previouslyfor the analytical model, and also caused by the operating condi-tions in the experimental prototype. The more significant were:the fluid channel is not completely fulfilled, the air-gap betweenstator coils and the channel, and also the curves in the fluidchannel geometry that being closer to the region where is lo-cated the stator gives origin to fluid accumulation which signif-icantly affects the fluid speed profile.

V. CONCLUSION

Concerning the results comparing the analytical model andthe finite-element model of the fluid transport system, we haveconcluded the following.

a) A general correspondence between the results has beenverified. However, in transient regime, there were signifi-cant model differences. Those became less significant asthe fluid has approached its steady-state regime. The dif-ferences were caused by the presence due to the presenceof border effects in the finite-element model, which werenot considered in the analytical model, hence causingchanges in the electromagnetic force at the channelinput/output.

b) The results obtained with the finite-element model insteady-state were equivalent to those ones estimated withthe analytical model. However, a finer grid has been usednear the bottom of the fluid channel to allow more preciseresults.

Concerning the results comparing the experimental tests andthe speed values estimated from the analytical model, we haveconcluded the following.

a) The fluid speed found from the experimental tests hadvalues significantly lesser than those ones estimated usingthe analytical model when this considers a null pressuregradient condition. This difference led us to conclude thata significant pressure gradient condition exists opposed tothe average force density .

b) The use of the analytical model of a pressure gradientcomposed by two components—one component constantand other one proportional to the average speed—allowedus to obtain fluid speed values near to those ones measuredin the prototype system if adequate values for the con-stants are used. The analytical model allows to partiallyestimate the system’s behavior with an approximated con-figuration to that used in the laboratory, but it does notshow a complete correspondence in the results.


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Manuscript received March 27, 2006; revised June 10, 2006. Correspondingauthor: P. J. Costa Branco (e-mail: [email protected]).

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