The magnetic field and performance calculations for an electromagnetic pump of a liquid metal

Transcription

1 J. Phys. D: Appl. Phys. 3 ( Printed in the UK PII: S ( The magnetic field and performance calculations for an electromagnetic pump of a liquid metal Suwon Cho and Sang Hee Hong Department of Physics, Kyonggi University, Suwon, Kyonggi-Do , Korea Department of Nuclear Engineering, Seoul National University, Seoul 5-742, Korea Received 27 January 998, in final form 24 June 998 Abstract. For an annular-type electromagnetic pump of a liquid metal, the magnetic field is found in closed form by means of the Fourier transform method for two-dimensional field analysis based on an equivalent current sheet model, so that the terms contributing to the normal thrust and the end effect are analytically identified. Analytical solutions for the magnetic vector potentials are compared with numerical ones obtained from the finite difference equation, and they turn out to be in good agreement regardless of the assumption of infinite permeability used in the analytical approach. As a result of closed-form solutions, the electromagnetic body force exerted on a liquid metal flow and the efficiency of the pump are calculated without going through tedious time-consuming numerical work usually accompanied by troublesome convergence problems. The calculated results for estimating the influence of end effects on the performance of the electromagnetic pump clearly show that the end effects give rise to the obstructive force to the liquid metal flow at inlet and outlet ends of the pump. In addition, the efficiency is found to be lowered due to end effects, especially when the speed of the flow is high.. Introduction Electromagnetic pumps or linear magnetohydrodynamic (MHD machines have been used in metal refinery for transporting molten metals and sodium-cooled nuclear reactors for circulating liquid metal coolant [ 7. They have many advantages over conventional mechanical pumps. For example no bearings, no seals and no mechanical moving parts are required [2, 5. In a linear induction electromagnetic pump, the three-phase stator creates a magnetic field travelling along the pump duct, inducing electric current in the conducting liquid metal. The interaction between the resultant current and the magnetic field produces an electromagnetic body force pumping the liquid metal flow through the duct [2, 5, 8. The theoretical study of electromagnetic pumps has been very limited in view of the involvement of both electromagnetic and hydrodynamic problems [9 5. However, assuming that the flow is laminar at a constant speed, the electromagnetic phenomena can be separated from the hydromagnetic ones and they can be treated by analytical and numerical approaches used for the analysis of linear induction motors [2. Yamamura and co-workers have derived analytical solutions of the field equations, but their analysis is limited to the cartesian system [6. Although there is some previous work [4, 9 2, 7 on the annular-type of electromagnetic pumps, rigorous analytical studies have not yet appeared. In an annulartype pump, the travelling magnetic field is expressed as a linear combination of modified Bessel functions if the axial length of the pump is assumed to be infinitely long compared with an annular duct gap [2. However, due to the finiteness of the actual pump length, there are end effects at both inlet and outlet of the pump, which degrade the pump performance. They should, therefore, be included in estimating the performance. In this work, treating the three-phase current of the primary windings as the equivalent current sheet, the magnetic field is obtained in closed form by applying the residue theorem to evaluate the inverse Fourier transform resulting from solving a Maxwell equation coupled with the generalized Ohm s law for an MHD flow of constant velocity. The finite difference method is also used to get numerical solutions, which are compared with analytical ones to confirm their mutual agreement. In addition, the validity of the assumption of infinite permeability of the magnet core, which has been used to obtain analytical solutions, is examined by these comparisons between analytical and numerical solutions. Finally, the performance of a pump is evaluated by calculating the driving force and the pump efficiency, and the end effect on the performance is discussed /98/ $9.5 c 998 IOP Publishing Ltd

2 Electromagnetic pump performance 2. Analytic calculation of the magnetic field In a typical annular-type linear induction electromagnetic pump with the primary winding coils inserted in the slotted external cores, the combined action of three-phase currents of primary windings produces a travelling magnetic field in the annular liquid metal. For a theoretical model of an annular-type pump as shown in figure, the three-phase external current I of primary windings of N turns having p pole pairs and pole pitch τ, can be represented by an equivalent current sheet at the outer wall (r = r 2 of the pump [6 J s (z, t = J a e i(ωt kz ˆθ ( k = π/τ and J a = 3 2kNI/pτ. The current sheet generates the magnetic field and the induced current in the annular liquid gap in the same form of the travelling field as J s with angular frequency ω and propagation constant k. All fields are assumed to be axisymmetric ( / θ = in view of the cylindrical arrangement of the pump system. For a moving liquid metal with velocity v in an annular channel shown in figure, the magnetic vector potential A satisfies the following equation derived from Ampère s law combined with the generalized Ohm s law [8 ( A 2 A = µσ v ( A (2 t µ and σ are the permeability and the conductivity of the medium respectively. Assuming the vector potential of the form A = A(r,z,tˆθ =R(rZ(z e iωt ˆθ (3 and taking the Fourier transform with respect to z, we have Ã(r,ζ,t=Ã(r, ζ e iωt = R(r Z(ζ e iωt (4 Z(ζ = Z(z e iζz dz. (5 Then we obtain a reduced field equation having the r-dependence only: d 2 R dr 2 + r ( dr dr r + 2 α2 R = (6 α 2 = ζ 2 + iµσ (vζ + ω. (7 The solutions of equation (6 are found in terms of modified Bessel functions of a complex argument Ã(r, ζ = a(ζi (αr + b(ζk (αr. (8 For an electromagnetic pump of the structure described in figure, the vector potentials in three regions are expressed respectively as à (r, ζ = a (ζ I (α r à 2 (r, ζ = a 2 (ζ I (α 2 r + b 2 (ζ K (α 2 r à 3 (r, ζ = b 3 (ζ K (α 3 r. (9 r r 2 r Region 3: Outer core (µ Region 2: Liquid metal (µ 2 Region : Inner core current sheet J s (µ Figure. A theoretical model of an annular-type linear induction electromagnetic pump with an equivalent current sheet. From boundary conditions for the normal component of B and the tangential component of H at interfaces (r = r,r 2, we obtain α I (α r a = I (α 2 r a 2 + K (α 2 r b 2 ( K (α 3 r 2 b 3 = I (α 2 r 2 a 2 + K (α 2 r 2 b 2 ( I (α r a α 2 [I (α 2 r a 2 K (α 2 r b 2 = (2 µ µ 2 α 3 K (α 3 r 2 b 3 α 2 [I (α 2 r 2 a 2 K (α 2 r 2 b 2 = J s (3 µ µ 2 the Fourier transform of the sheet current density J s is given by L J s (ζ = J a e ikz e iζz dz = ij a ζ +k (e i(ζ +kl (4 for a current sheet of length L. Assuming that the permeability µ of the inner and outer magnet cores is infinite, a 2 and b 2 are found to be a 2 = µ 2J s K (α 2 r α 2 K (α 2 r I (α 2 r 2 I (α 2 r K (α 2 r 2 b 2 = µ 2J s I (α 2 r α 2 K (α 2 r I (α 2 r 2 I (α 2 r K (α 2 r 2. (5 Then the transformed vector potential in the annular gap can be put into the form Ã(r, ζ = iµj a e i(ζ +kl ζ + k v L G(r, ζ α(ζh(ζ G(r, ζ = K (αr I (αr + I (αr K (αr z (6 H(ζ = K (αr I (αr 2 I (αr K (αr 2 (7 the subscript 2 in µ 2 and à 2 is deleted. Then à is inversely transformed to give A(r, z = iµj a 2π e i(ζ +kl G(r, ζ ζ + k αh (ζ eikz dζ (8 which can be evaluated by using the residue theorem. 2755

3 S Cho and S H Hong ζ + ζ= -k Im ζ - ζ Figure 2. Integrating paths and locations of poles for dominant terms of the magnetic vector potential. Re ζ From the asymptotic expansion of Bessel functions [9, we find G(r, ζ /αh (ζ approaches zero as ζ. The integration paths for the evaluation of equation (8 by the residue theorem consist of the real axis and semicircles whose radii are infinitely large, as shown in figure 2. Poles of the integrand are determined from ζ + k =, α(ζ =, and H(ζ =. From α = ζ = i 2 µσ v ± 4 µ2 σ 2 v 2 iµσ ω ζ ± ζ s are labelled in a such way that Im[ζ + > and Im[ζ <. Zeros of H(ζ can be found numerically using Muller s method [2. Numerical results show that values of αr and αr 2 are somewhat larger than so that zeros of H can be estimated from the asymptotic relations of modified Bessel functions, i.e. H(ζ sinh(r 2 r α = (9 r r 2 α which yields α nπ (2 r 2 r n is an integer. Hence zeros of α(ζ and H(ζ are approximately given by ζ n = i 2 µσ v ± 4 µ2 σ 2 v 2 iµσ ω ( nπ. 4 r 2 r (2 In actual calculations, ζ n are found numerically and are sorted in such a way that Im[ζ + n+ > Im[ζ n + > and Im[ζ n+ < Im[ζ n <. Noting that Im[ζ ± < Im[ζ n (n =, 2,... and the z-dependence is given by e iζz, we find residues at ζ ±, which come from α =, are dominant terms. Since K (αr and K (αr are not finite when α =, residues at ζ ± should be evaluated carefully. From the ascending series of K and K G α = α 2 r + O(α (22 H = ln r 2 + O(α 2 (23 r so that G αh = α 2 r ln(r /r 2 + O(α = r ln(r /r 2 (ζ ζ (ζ ζ + + O(α (24 which enables one to calculate residues at ζ ±. Finally, the vector potentials in the liquid metal are expressed, respectively, at inlet (z <, annular channel ( z L, and outlet (z >L regions: A(r, z = µj a A, + µj a A,n z< (25 G(r, k A(r, z = µj a α( kh( k e ikz +µj a A 2, + µj a A 2,n + µj a A + 2,n z L (26 A(r, z = µj a A 3, + µj a A 3,n z>l (27 terms from residues at ζ ± are given by µj a e i(k+ζ L e iζ z A, = r ln(r 2 /r ζ + k ζ ζ + (28 µj a A 2, = r ln(r 2 /r ( e i(k+ζ L e iζ z ζ + k ζ ζ + + e iζ + z ζ + + k ζ + ζ (29 µj a e i(k+ζ + L e iζ + z A 3, = r ln(r 2 /r ζ + + k ζ + ζ (3 and contributions from residues at ζ n (n > are obtained as A,n = G(r, ζ n α(ζ n H (ζ n [ e i(ζ n +kl ζ n + k A 2,n = G(r, ζ n [ e i(ζn +kl α(ζn H (ζn ζn + k A + 2,n = G(r, ζ n + [ α(ζ n + H (ζ n + ζ n + + k A 3,n = G(r, ζ n + [ e i(k+ζ n + L α(ζ n + H (ζ n + + k ζ + n e iζ n z (3 e iζ n z (32 e iζ + n z (33 e iζ + n z. (34 Since the z-dependence is e iζz or e iζ(z L, magnitudes of A j,n are maximum, at z = orz=l. The computations of partial sums of A j,n up to n at z = demonstrate that the magnitude of A j,n (n =,2,... is negligible compared with that of A j,. In practice, A j,n /A j, (n =, 2,... is less than 3. The inverse Fourier transform, equation (8, is evaluated via direct numerical integration, 2756

4 Electromagnetic pump performance ~ A Figure 3. Behaviour of the integrand in the inverse Fourier transform for the evaluation of the magnetic vector potential. The full and broken curves represent the real and imaginary parts respectively. Vector Potential with end effects -.3 without end effects ζ pump length (L Figure 4. Magnetic vector potentials calculated from closed-form solutions for two cases with (bold curves and without (normal curves end effects. The full and broken curves represent the real and imaginary parts respectively. giving identical results with the closed-form solutions (25 (27 to the given accuracy. But the numerical integration needs many repeated evaluations of the functions since the range of the integral is infinite and the integrand is oscillatory as shown in figure 3. For the results presented as numerical illustrations in this work, the following values of parameters are used: ω = 2π rad s, v = ms,r = 3 cm, r 2 = 4 cm, L = 2 cm, τ = cm, J a = 4 Am, µ=4π 7 Ns 2 C 2,σ=.4 7 m. The vector potential A(r, z includes three exponential functions, e ikz, e iζ z and e iζ + z. Considering the time dependence e iωt and noting that k>, Re[ζ + < and Im[ζ + >, and Re[ζ > and Im[ζ <, we find that the exp[i(ωt kz term represents a normal wave moving along the +z direction, and the exp[i(ωt + ζ + z term indicates an inlet-end-effect wave propagating and damping in the +z direction, while the exp[i(ωt + ζ z term describes an outlet-end-effect wave moving and attenuating along the z direction. In figure 4, the vector potentials obtained from closed form solutions (25 (27 are drawn along z for the cases with and without the end effects respectively. From B = A, the radial component of the magnetic flux density in each propagating region is found to be B r (r, z = µj a e i(k+ζ L iζ eiζ z r ln(r 2 /r ζ + k ζ ζ + z< (35 B r [T Figure 5. Distribution of the radial magnetic field B r along the pump axis z for the cases with (bold curves and without (normal curves end effects. The full and broken curves represent the real and imaginary parts respectively. B z [T Figure 6. Distribution of the axial magnetic field B z along the pump axis z for the cases with (bold curves and without (normal curves end effects. The full and broken curves represent the real and imaginary parts respectively. B r (r, z = µj a r ln(r 2 /r ( e i(k+ζ L iζ eiζ z ζ + k ζ ζ + + iζ + e iζ + z ζ + + k ζ + ζ ikg(r, k +µj a α( kh( k e ikz z L (36 B r (r, z = µj a e i(k+ζ + L iζ + eiζ + z r ln(r 2 /r ζ + + k ζ + ζ z>l (37 residues at ζ n (n =, 2,... have been neglected in the calculation of the B r component, since they make negligible contribution to the field. The axial components along z are given as B z (r, z = µj a B,n z< (38 P(r, k B z (r, z = µj a α( kh( k e ikz + µj a +µj a B 2,n B + 2,n z L (39 B z (r, z = µj a B 3,n z>l (4 2757

5 S Cho and S H Hong and B,n = P(r,ζ n α(ζ n H (ζ n ( e i(ζ n +kl ζ n + k B 2,n = P(r,ζ n ( e i(ζn +kl α(ζn H (ζn ζn + k B + 2,n = P(r,ζ+ n ( α(ζ n + H (ζ n + ζ n + + k B 3,n = P(r,ζ+ n ( e i(k+ζ n + L α(ζ n + H (ζ n + + k ζ + n e iζ n z (4 e iζ n z (42 e iζ + n z (43 e iζ + n z (44 P(r,ζ = α[k (αr I (αr I (αr K (αr. (45 The distributions of B r and B z along z are plotted for the cases with and without end effects in figures 5 and 6, in which considerable end effects on B r appear, but negligible ones on B z near the inlet and outlet ends (z =,.2 in this example. This is due to the fact that end effects of B z come from residues only at ζ n, since the r-dependence of residues at ζ ± is /r and B z, = ( / r(a =. 3. Numerical solutions The straightforward application of the finite difference method [8, 2 gives numerical solutions which are used to verify the closed-form solutions obtained by the analytical method for the present work. As shown in figure 7 for vector potential distributions resulting from two approaches, two methods show almost identical results. The numerical approach could be extended to realistic pumps with complex discrete structures of primary winding currents and outer cores, but it is expected to require tedious time-consuming computations for its iterative procedure and a considerable storage requirement in solving the system of linear equations. In this respect, the present analytical solutions in closed form give a relatively simpler way of calculation and retain the degree of flexibility that cannot be found in the numerical approach. The permeability of the iron core has been assumed to be infinite in obtaining closed-form solutions. The validity of this assumption in the analytical method is examined by comparing the numerical results computed with different values of the permeability. Figure 8 apparently indicates that there is little difference between vector potentials computed for finite and infinite values of µ, but virtually no difference if the relative permeability is taken to be greater than some values of about. This suggests that it is not necessary to consider the nonlinear relation of B and H in obtaining reasonably accurate solutions. 4. Performance calculation The electromagnetic body force per unit volume exerted on the liquid metal in the annular gap is f = J B E = ( t/ A = iωa ˆθ, J = σ(e +v B = Vector Potential Figure 7. Comparisons of magnetic vector potentials obtained from the analytical method (normal curves with those from the numerical method (bold curves. The full and broken curves represent the real and imaginary parts respectively. Vector Potential K= K= 2 K= 3 K= 4 K=infinity Figure 8. Comparisons of magnetic vector potentials along the pump axis obtained from numerical solutions with different values of the relative permeability K for the core. Only the real parts of the potentials are displayed. σ( iωa+vb r ˆθ, so that its time-averaged volume density in the z-direction is given as f = 2 Re[σ(iωA vb rbr. (46 A numerical illustration is given in figure 9 for the driving electromagnetic body force resulted from the closed-form solutions for the cases with and without end effects respectively. If the electric input power per unit volume is given as p = 2 Re[J E = 2 Re[σ( iωa + vb riωa (47 the pump efficiency can be defined as the ratio of the mechanical pumping power of liquid metal flow to the input electric power, i.e. η = L r2 r L fvrdrdz r2 r pr dr dz. (48 When the end effects are neglected, it is easy to show that the pump efficiency is given as the ratio of the flow velocity v to the synchronous velocity ω/k of the fields, i.e. η = kv/ω = s s is the slip. The pump efficiency can be computed using numerical integration when the end effects are included. The driving electromagnetic 2758

6 Electromagnetic pump performance Force [N/m Without end effects With end effects Figure 9. The time-averaged driving force exerted on a liquid metal flow along the axial direction z for the cases with (bold curves and without (normal curves end effects. Efficiency..9.8 without end effects With end effects slip Figure. Comparison of pump efficiencies between two cases with and without end effects when the pump operates with different flow velocities expressed in terms of the slip. force density and the pump efficiency are presented in figures 9 and for the cases without and with end effects. When end effects are involved in the pump operation, its efficiency increases as the speed of the liquid metal flow increases until the slip decreases up to a certain value, but reduces remarkably as slip decreases further to zero. 5. Conclusions For laminar channel flow of a liquid metal in an annulartype linear induction electromagnetic pump, analytical solutions of the field equation are obtained and compared with numerical solutions. It has been found that results of both methods are in good agreement and the assumption of the infinite permeability used for the analytical model results in reasonably accurate solutions. Terms contributing to the normal thrust and the end effect are analytically identified with closed-form solutions, and then the electromagnetic body force and the pump efficiency are easily calculated with the help of those analytical solutions. The electromagnetic driving force is found to be in the opposite direction to the flow motion in some regions, which causes the pump to have poor efficiency, especially when the speed of the flow is high. This is due to the end effects arising from the finite size of the pump axial length. When the discrete structures of stator coils and cores and the inhomogeneous flow speed are considered, the numerical method seems to be the only resource for analysing the pump system. With the equivalent current sheet model, however, the analytical method can be extended to profiles of the variable speed if the channel is divided into concentric layers of constant speed [2, although this requires a formidable amount of work when the end effects are included. Nevertheless, analytical results of the simple model in this work can be used in estimating the field distributions and the pump performance as well as checking numerical results of complicated models. Acknowledgments This work has been partly supported by the Korea Ministry of Science and Technology and the Korea Ministry of Commerce and Industry. References [ Oto S 995 Nucl. Tech. 96 [2 Nakazak M, Taguchi J, Katuki K, Sato N, Fujii T, Madarme H and Fujii Y 99 Proc. Int. Conf. on Fast Reactions and Related Fuel Cycles vol 3 (Tokyo: Atomic Energy Society of Japan p 2.6 [3 Kwant W, Fanning A W, Brohaugh T R, Patel MRand Dahl L R 99 Proc. Int. Conf. on Fast Reactions and Related Fuel Cycles vol 3 (Tokyo: Atomic Energy Society of Japan p 9.3 [4 Yang C C and Kraus S 977 Nucl. Eng. Des [5 Namba M, Mukaida H, Taguchi J and Miyagi T 978 Toshiba Rev. 6 7 [6 Andreev A M, Bezgachev E A, Karasev B G, Kirllov I R, Ogorodnikov A P, Preslitskii G V and Chvartatskii R V 988 Magn. Gidrodin [7 Andreev A M, Karasev B G, Kirillov I R, Ogorodnikov A P, Ostapenko V P and Semikov G T 978 Magn. Gidrodin [8 Gotoh T, Yamagata M, Suzuoki A and Kazawa Y 983 J. Atom. Energy Soc. Japan [9 Hughes W F and McNab I R 983 Progress in Astronautics and Aeronautics 85 (New York: AIAA p 287 [ Hasebe S and Kano Y 982 Trans. IEE Japan 2-B 77 [ Kim H R, Im K H and Hong S H 995 Energy Eng. J. (Korean Soc. Energy Eng [2 Boldea I and Nasar S A 985 Linear Motion Electromagnetic Systems (New York: Wiley [3 Polovko Yu A and Tropp E A 986 Magn. Gidrodin [4 Polovko Yu A 989 Magn. Gidrodin [5 Kirillov I R, Ogorodnikov A P, Ostapenko V P and Semikov G T 98 Magn. Gidrodin. 9 5 [6 Yamamura S 979 The Theory of Linear Induction Motors (New York: Wiley [7 Eastham J F and Alwash J 972 Proc. IEE 9 79 [8 Nasar S A and Boldea I 976 Linear Motion Electric Machines (New York: Wiley [9 Abramowitz M and Stegun I A (eds 968 Handbook of Mathematical Functions (New York: Dover [2 Press W H, Flannery B P, Teukolsky S A and Vetterling W T 986 Numerical Recipes (Cambridge: Cambridge University Press [2 Binns K J and Lawrenson P J 973 Analysis and Computation of Electric and Magnetic Field Problems (Oxford: Pergamon 2759