Transient Magnetic Translating Motion Finite Element Model of the Annular Linear Induction Pump Abstract The paper constitutes a study of the startup, steady state operation and dynamic behavior of a double sided Annular Linear Induction Pump (ALIP), based on finite element transient magnetic translation motion coupling models. Using a model of the pump load proportional with the square of the flowrate, the Electromagnetic Pressure Flowrate characteristic was determined and considered further in expressing what a stable pump operating state is. Data regarding the highest admissible jump load are given. The dependence of the pumping efficiency on the flowrate is presented and provides with a view regarding the maximum capabilities of such devices of converting electrical energy into motion. I. INTRODUCTION In the context of transportation liquid sodium as a coolant agent in the heat exchange loops of Sodium Fast Reactors (SFR), many papers are devoted to study the global characteristics [-], and instability issues [-] of Annular Linear Induction Pumps (ALIPs). The exploitation opportunity is a question of balancing their advantages with pumping efficiency [-7]. The reliability of such pumps in comparison with mechanical ones comes from the absence of any shaft or other moving part. Thus the full sealing of the liquid inside the pump is possible. Also these pumps are free of any mechanic vibration and do not require any bearing and lubrication. ALIPs can operate very well with liquid metals at high temperatures and presents the advantage of current generation in the pumping channel contactless, namely by the electromagnetic induction phenomena. The three phased winding which ensure a traveling field generates an induced azimuthal electric current, which interacting with the radial magnetic field gives birth to an axial oriented electromagnetic force acting on the sodium body to pump it further. This work analyses the results of coupled field circuit - translating motion finite element models of a double sided ALIP pump for liquid sodium in the assumption of bloc pumping, models able to study the pump startup, the steady pump operating regime and the dynamic stability at sudden jump loads. a) b) II. FINITE ELEMENT D MODEL OF THE ANNULAR DOUBLE SIDED LINEAR INDUCTION PUMP FOR SODIUM BLOC PUMPING A. Geometry and Mesh Figure a) presents the D axisymetric computation domain that contains the annular sodium region of the pumping channel, the metallic walls around this region, c) Fig.. Geometry, zooms and mesh of the finite element computation domain
the magnetic yokes and the coils of the pump three phase six poles winding. The geometry of the model is characterized by the pole pitch length mm and the inductor airgap. mm. B. Physical Properties To completely describe the model, the needed physical properties are the resistivity and magnetic permeability of the regions sodium and channel walls and the magnetic property of the nonconductive magnetic yokes. The instantaneous values of the two current sources in the circuit model of the double sided ALIP pump, Fig., are: i A (t) = I sin t, i B (t) = I sin ( t - /) () where I is the RMS value of the current related to each inductor slot. Taking into account the value of the slot pitch, the rms value of the current sheet equivalent to the one side inductor winding is. A/mm. The supply frequency f = / = Hz was considered, so that the speed of the traveling electromagnetic field wave, called synchronous speed of the pump, is v s = f =. m/s. armature of the device. The state variable A(r, t) of the electromagnetic field satisfies the equations: curl[(/ )curla] + ( A/ t) = J, diva = () The D axisymmetric model of the annular linear induction pump in the cylindrical coordinates system [r,, z] in this paper takes into account the properties J [, J (r,z,t), ] and A[, A(r,z,t), ] and the bloc motion of sodium region along the pump axis Oz, Fig.. In this context, the second equation () is implicitly satisfied and the vector potential A(r,z,t) satisfies the equation: / z[(/μ) A/ z ] + / r[(/μ) A/ r] - σ( A/ t) = - J () The finite element solution of the unknown vector potential A (k) (r,z) at the current time step t k considers the boundary condition of tangent magnetic field on all contour of the computation domain, Fig. a). With respect to the time t variable, the step by step scheme is applied and the initial condition A () (r,z) = A(r,z,) = is considered. Regarding the motion of the sodium region along the Oz axis, Fig., the velocity v(t) of this region satisfies the equation: m(dv/dt) = F elmg F load () Fig.. Circuit model of the pump C. Electromagnetic Field Circuit Motion Coupling The time and space dependent electromagnetic field characterizing the pump operation, whose source is the current density J (r, t) in the regions of inductor slots, can be expressed in term of the magnetic vector potential A(r, t), so that the magnetic flux density is B = curla. The correlation between the known geometry and the currents () and the current density J, respectively the coupling electromagnetic field - circuit, is ensured by the pump circuit model in Fig.. The time dependence of the electromagnetic field is numerically treated through the step by step in time domain method. Thus, the term representing the induced current density in conductive regions with the electric conductivity, ( A/ t), is finite differences approximated, [A (k) - A (k-) ] / t, where k- and k define two successive moments t k-, t k and t = t k - t k-. In case of problems taking into account the motion of a region of the computation domain, the position of such a region inside the domain is changing from the time t k- to t k. This is the case for example of transient magnetic problems with motion, respectively of coupled electromagnetic field - motion models, for the study of linear induction motors operation, or of bloc pumping in linear induction pumps. These problems consider the translation motion between the inductor and the secondary where m is the mass of the sodium region in motion, F elmg is the axial component of the Laplace force generated by the electromagnetic field in the sodium region and F load is the load force of the pump. A model of the pump load proportional with the square of sodium flowrate was considered, respectively F load = k l v. The initial condition related the velocity unknown is v() =. The value v (k) of the sodium region at the time step t k permits the computation of the displacement v (k) t of this mobile region along the Oz axis, respectively of the new position of the sodium region in the computation domain for the next time step t k+ = t k + t. The main interest of the electromagnetic field motion coupling represented by the system of equations () and () is the evaluation of steady state solution, characterised by constant amplitude of the local and global quantities with periodical variation in time. Consequently, the integration with respect the time variable is stopped when this condition is achieved. III. NUMERICAL APPLICATIONS. RESULTS ANALYSIS Figures - present results related the transient of the pump startup when the load dependence on the speed v of the sodium is expressed by the formula F load = v. (kn)......... Fig.. Time variation of electromagnetic force acting on the sodium region
One can observe that when the steady state pump operation is reached, the mean value of electromagnetic force time variation, Fig.,.7 kn, is equal with the mean value of the load force, Fig.. The mean value of the speed characterizing the sodium bloc pumping, Fig., is. m/s. Load (kn),,,..... Fig.. Time variation of sodium region speed,,,,,, From the time variations of the Joule power in the sodium region, Fig. 7 and in the channel walls it result the mean values of the these two global quantities, P sodium = 9.9 kw and P walls =. kw. Active Power (kw),,,,, Fig.. Time variation of the active power in the channel walls A. Local Quantities of the Electromagnetic Field Figures 9- present the significant electromagnetic field quantities for the time step. s, when the instantaneous value of sodium region speed is. m/s. The magnetic field lines that indicate the six poles of the pump are presented in Fig. 9. Fig.. Time variation of pump load, v Fig. presents the time variation of the two components and of the module of the Laplace force. (kn).... Radial Component Axial component Module -...... Fig.. Time variation of the components and module of the Laplace force Active Power (kw)..... Fig. 7. Time variation of the active power induced in the sodium region c) outlet area zoom b) whole pump regions b) inlet area zoom zoom Fig. 9. Magnetic field lines The maps of the magnetic flux density in different regions of the computation domain are presented in Fig..
The interaction between the important value of the current density at the pump outlet, near the magnetic yoke end, Fig, and the no negligible magnetic field in this area, Fig. b) has as result an important breaking effect, Fig., of the electromagnetic field in this area of the channel pump. Fig. shows maps of the induced current density in different sections of the channel metallic walls. The important field entrainment is revealed at the inlet region, with low induced currents. This length is about one sixth of the total pump length. a) inlet area zoom c) outlet area zoom a) in the magnetic cores b) in the airgap and slots Fig.. Maps of the magnetic flux density Fig. presents the maps of the induced current density in the sodium region. Fig.. Maps of the current density in the channel walls The axial component of electromagnetic force volume density along an axial path in the middle of the sodium region is presented in Fig.. It shows the nonuniformity along the channel of the local pumping effect, the effect of the slot-tooth alternation and the important braking effect mainly at the pump outlet. Volume Density (kn/m) - - - -,,, Axial Coordinate (m) Fig.. Axial component of the electromagnetic force volume density along a path in the middle of the pumping channel a) whole sodium region b) outlet area zoom Fig.. Maps of the induced current density in the sodium B. Electromagnetic Pressure Flowrate Characteristic and the Pumping Efficiency Characteristic The dependence Electromagnetic Pressure vs. Pumping Flowrate of the pump in Fig. is split in an ascendant and a descendant branch by the point of maximum pressure,. bar -.9 m /s. The shielding effect of the metallic walls explains why the sodium region speed.9 m/s or the pump flowrate.7 m /s corresponding to null electromagnetic force / electromagnetic pressure are lower than the synchronous values. m/s, respectively.9 m /s (in magenta in Fig. ).
Electromagnetic Pressure (bar)............. Flowrate (m/s) Fig.. Electromagnetic Pressure Flowrate characteristic The pump efficiency defined as the ratio between the electromechanical power and the sum of electromechanical power and Joule power in coils, sodium region and channel walls, is presented in Fig. in function of the pump flowrate. The maximum efficiency of the pump,. %, corresponds to the flowrate value.9 m /s, respectively to the sodium region mean speed value. m/s. Efficiency (%),,,,,,,, Flow rate (m/s) Fig.. Dependence on flowrate of the pumping efficiency C. Dynamic Answer of the Pump for Sudden Load Increase This section studies the dynamic answer for a sudden increase of the load, when a) the operation point of the pump belongs to the descendant branch of the characteristic Electromagnetic Pressure - Flowrate, Fig., and b) the operation point belongs to the ascendant branch. ) Dynamic answer for the operating point on the descendant branch: Figure shows how starting from zero speed and load the pump reaches a steady operating regime with the mean value in time of the load,.7 kn, respectively the mean speed. m/s. At the time. s the pump load suddenly increases at the value kn. After a new transient regime a new steady state pump operation, characterized by the mean value kn of the electromagnetic force, Fig. 7, and the mean value of the sodium region speed. m/s, Fig., is established. If instead kn the sudden load is kn, higher than the maximum of the electromagnetic force,. kn, Fig., the pump is not able to pass from the previous steady regime characterized by the mean value in time of the load,.7 kn, into another steady regime. The mean value, toward which the generated force tends, Fig. 9, is lower then the one required by the new load. Fig. shows a descending tendency of the mean speed variation; after a certain time the speed will be null, respectively the pump will stop. Load (kn) (kn) (kn)........... Fig.. Load jump at the moment. s...... Fig. 7. Time variation of electromagnetic force...... Tim e (s) 7 Fig.. Time variation of sodium region speed...... Fig. 9. Time variation of electromagnetic force...... Fig.. Time variation of sodium region speed
) Dynamic answer for the operating point on the ascendant branch: Figure shows how for the load model F load = v the pump reaches a steady operating regime characterized by the mean value in time.7 kn of the electromagnetic force and of the load, respectively. m/s of the speed. At the moment. s the pump load suddenly increases at kn, lower than the maximum of the electromagnetic force. kn, Fig.. The time evolution of the electromagnetic force, Fig., shows that the pump is not able to compensate the load increase, so that the mean sodium speed, Fig., will rapidly decrease to zero. Load (kn) (kn)........7.. Fig.. Load jump at the moment. s..7............. Fig.. Time variation of electromagnetic force 7........ Fig.. Time variation of sodium region speed For any operating point belonging to the ascendant branch of the Electromagnetic Pressure vs. Flowrate characteristic, Fig., and no matter how small the load jump is, the pump is not able to ensure a new steady operating regime. For this reason, any operating point on this branch represents an unsteady operating point. If the pump it is in an operating point belonging to the descendant branch, for any jump of load lower than the maximum available electromagnetic force, the pump will ensure a new steady regime. For this reason, any point on the descendant branch represents a steady operating point. IV. CONCLUSIONS A transient magnetic model of electromagnetic field - circuit - translating motion coupling type was developed for the study of a double sided Annular Linear Induction Pump for liquid sodium. There were analyzed finite element results corresponding to the steady state pump operation and significant instantaneous quantities of the electromagnetic field. The dependences of Electromagnetic Pressure and Pumping Efficiency on the Pump Flowrate were determined. The braking ends effect and the shielding effect of the metallic walls were highlighted. An analysis regarding the pump dynamic answer for sudden load jump was done. Based on these results it was determined which points of the Electromagnetic Pressure - Flowrate characteristic constitute stable operating points and which do not. For the stable operating points it was determined the maximum admissible jump of the pump load so that after the load jump the pump reaches a new stable operating regime. REFERENCES [] H. Ota, K. Katsuki, M. Funato, J. Taguchi, A.W. Fanning, Y. Doi, N. Nibe, M. Ueta and J. Inagaki, Development of m/min Large Capacity Sodium-Immersed Self-Cooled Electromagnetic Pump, Nuclear Science and Technology (), Vol., n. [] I.R. Kirillov, D.M. Obukhov, Two dimensional model for analysis of cylindrical linear induction pump characteristics: model description and numerical analysis, Energy Convertion and Management () 7-97. [] C.A. Borghi, A. Cristofolini and M. Fabbri, Study of the Design Model of a Liquid Metal Induction Pump, IEEE Transactions on Magnetics, vol., no., 99. [] H. Araseki, I.R. Kirillov, G.V. Preslitsky, A.P. Ogorodnikov, Magnetohydrodynamic instability in annular linear induction pump. Part I. Experiment and numerical analysis, Nuclear Engineering and Design 7 () 9-. [] H. Araseki, I.R. Kirillov, G.V. Preslitsky, A.P. Ogorodnikov, Magnetohydrodynamic instability in annular linear induction pump. Part II.Suppression of instability by phase shift, Nuclear Engineering and Design () 9-97. [] Suwon Cho, Sang Hee Hong, The magnetic field and performance calculations for an electromagnetic pump of a liquid metal, 99 J. Phys. D: Appl. Phys. 7-79. [7] J. Werner, H. Adkins, Electromagnetic pump fabrication and predicted performance, Proceedings of Nuclear and Emerging Technologies for Space, 9.