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Download Moissis, A.A., and M. Zahn. Boundary Value Problems in Electrofluidized and Magnetically Stabilized Beds, Chemical Engineering Communications 67, 181-204, 1988
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This article was downloaded by: [Massachusetts Institute of Technology, MIT Libraries] On: 18 February 2011 Access details: Access Details: [subscription number 922844579] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 3741 Mortimer Street, London W1T 3JH, UK Chemical Engineering Communications Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713454788 BOUNDARY VALUE PROBLEMS IN ELECTROFLUIDIZED AND MAGNETICALLY STABILIZED BEDS A.A. Moissisa; Markus Zahna a High Voltage Research Laboratory Laboratory for Electromagnetic and Electronic Systems Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts To cite this Article Moissis, A.A. and Zahn, Markus(1988) 'BOUNDARY VALUE PROBLEMS IN ELECTROFLUIDIZED AND MAGNETICALLY STABILIZED BEDS', Chemical Engineering Communications, 67: 1, 181 — 204 To link to this Article: DOI: 10.1080/00986448808940383 URL: http://dx.doi.org/10.1080/00986448808940383 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Chem. Eng. Comm. 1988, Vol. 67, pp. 181-204 Reprints available directly from the publisher. Photocopying permitted by license only. © 1988 Gordon and Breach Science Publishers S.A. Printed in the United States of America Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 BOUNDARY VALUE PROBLEMS IN ELECTROFLUIDIZED AND MAGNETICALLY STABILIZED BEDS A.A. MOISSIS and MARKUS ZAHN High Voltage Research Laboratory Laboratory for Electromagnetic and Electronic Systems Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology Cambridge, Massachusetts 02139 (Received March 14, 1987; in final form August 11, 1987) The stress balance boundary condition at the free surface of a one dimensional electrofluidized or magnetically stabilized bed is reviewed. Conditions for zero interfacial force balance at the top of the bed are investigated. For the case of a uniformly fluidized bed, it is shown that a nonzero particle collision pressure is necessary for a force free bed surface. For the case where the collision pressure is neglected, it is shown that the electrofluidized or magnetically stabilized bed must be nonuniform at equilibrium. Stability criteria for a semi-infinite bed show that a critical field strength is necessary for stabilization. Below this critical condition the bed is convectively unstable. The temporal and spatial responses of a stabilized bed to small signal variations in superficial gas velocity, electrode position, and applied potential are calculated. Specific attention is directed to sinusoidal time variations in superficial gas velocity, where at low frequencies a standing wave character results for the fluidized bed variables. KEYWORDS Electrofluidization Magnetically stabilized beds Fluidization Stabilization Boundary value problems. INTRODUCTION Hydrodynamic analysis has demonstrated that the state of uniform fluidization is unstable to small signal perturbations [1]. For a gas-solid fluidized bed these instabilities result in the formation of gas bubbles. Recent work has shown by analysis and experiment that these bubbles can be eliminated if a suitably strong magnetic or electric field collinear with the direction of the gas flow is applied to a bed of highly magnetizable or polarizable particles [2-10]. Unlike magnetic systems which only have magnetization forces, electric field systems can also have free charge forces described by Coulomb's law. Such space charge effects cause particle chaining resulting in loss of fluidity. To eliminate space charge effects in the electric case, alternating high voltages must be applied at a frequency much greater than the largest reciprocal dielectric relaxation time of the system. Since the polarization forces on dielectric particles will act in exactly the same way as magnetization forces on magnetizable particles, the analysis is presented here for beds of polarizable particles but may also be applied directly to beds of magnetizable particles if we replace the electric field E by the magnetic field H and the bed permittivity e by the bed magnetic permeability u. This paper 181 A.A. MOISSIS AND M. ZAHN 182 presents the governing equations, general equilibrium and perturbation solutions, and stability conditions together with a formulation of appropriate boundary conditions for the various parameters used in a two phase model of a gas fluidized particle bed stabilized against bubbling by an electric field. Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 GOVERNING EQUATIONS Hydrodynamics By introducing the voidage cp(r, r), a 'smoothed' variable defined as the volume fraction of fluid in a volume element of the fluidized bed, we write conservation of mass for the fluid and particle phases ata (CPPt) + V· (CPPt u ) = 0 a at [(1 - . (fluid phase) cP )Ps] + V . [(1- cP )Psv] = 0 (solid phase) (1) (2) where Pt and Ps are the fluid and particle phase mass densities, while u and v are the interstitial fluid and particle phase velocities respectively. For incompressible phases, where Pt and Ps are constants, (1) and (2) may be added giving v . [cpu + (1 - cP )v] = 0 (3) Since we are concerned with the fluidization of a solid by a gas with negligible mass density and viscosity when compared to the solid particles, we write the conservation of momentum for gas and particles in the approximate form -v,(cpJ}i)-p(cp)[u-v]=O (fluid phase) Ps(1- CP) [ av ] at + (v- V)v _ (4) _ = V· [(1- CP)Ts]- V· [(1- CP)P,.I] + (1- CP)Psg- J}VCP + CPP(CP)[u-v] + Fe (solid phase) (5) where Pt is the fluid phase pressure, P, is the solid phase pressure, P(CP) is the Darcy drag coefficient, which for laminar flow is assumed to depend on voidage cP, Ts is the particle viscous stress tensor, and Fe is the body force density acting on the particle phase due to the electric field. If we assume that each phase behaves like a Newtonian fluid we may describe the particle viscous stress tensor as (6) where lis is the effective particle shear viscosity and k, is the effective particle bulk viscosity. The difference between the solid phase and fluid phase pressure is termed the collision pressure in the fluidization literature [11J, PcCCP, E) = P, - J} (7) Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 ELECTROFLUIDIZED BEDS 183 The collision pressure accounts for the stress transmitted to the particle phase due to the interaction between individual particles. In a fluidized bed particles are not in permanent contact with their neighbors, however a particle may still interact with its neighbors by colliding with them; this interaction can contribute to the force on particles in a non-uniform bed in which a differential volume of particles experiences more collisions on one surface than on its opposite surface; hence the collision pressure will depend on the voidage cjJ [11). For the case of an electrofluidized bed the polarized particles may interact due to forces of electrical origin without necessarily colliding; hence we will assume that the collision pressure is not only a function of the local voidage, but also of the local electric fleld E. The effective particle shear viscosity /is' used in (6) will also depend on the applied electric field. Measurements conducted by Rhee [8) using a Couette viscometer with an applied AC radial electric field showed that for Rochelle salt particles the effective shear viscosity /is in SI units (Kg/m . s) may be expressed by u, = 441.4 exp[ -2. 24UsolUm! ]exp[1.22Ed (8) where Uso is the equilibrium superficial gas velocity, Um! is the minimum fluidization velocity and E 1 is the normalized electric field E, = (Ep E 2/ Ps dpg)112 (9) while dp and Ep are the particle diameter and the permittivity respectively. The fluid phase and solid phase momentum balance equations may be combined: psCl- cjJ) [aatv+ (v - V)v]= -V· [(1- cjJ)Pc]I-+ l3(cjJ)[u - v] + V . [(1 - cjJ )T,)- (1 - cjJ )p,gix + Fe (combined) (10) The fluid and solid phase pressures are thus eliminated in favor of the collision pressure P: Electric Fields and Forces Maxwell's equations in the electroquasistatic limit with no volume charge are v X E =O~E= -V<I> (11) V·D=O (12) D = E(cjJ)E (13) where E is the curl-free electric field which can thus be represented as the negative gradient of a scalar potential <1>, D is the displacement field, and E(cjJ) is the effective medium macroscopic bed permittivity which is some voidage weighted average of particle and fluid permittivities. For simplicity, we assume that the particles are electrically linear so that the effective permittivity f( cjJ) does not depend on field strength. A.A. MOISSIS AND M. ZAHN 184 The principle of virtual work gives the force density and stress tensor as [6-10], [12] (14) Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 (15) THE STRESS BOUNDARY CONDITION FOR A ONE DIMENSIONAL ELECTROFLUIDIZED BED In order to simplify our problem we assume a one dimensional configuration, illustrated in Figure 1. In this one dimensional problem all variables are assumed to be x dependent only, with all vectors (E, v, u, g) x directed. If we make the assumption that the macroscopic electrofluidized bed parameters are finite everywhere, the fluid phase momentum balance Eq. (4) gives us a condition for the fluid pressure continuity at the time varying top surface of the bed (x = L(t», Pr(L(t), t) = 0 (16) where we assume that ambient pressure is zero. In a uniform bed the equilibrium fluid pressure is found to vary linearly with position through the bed. The combined momentum balance Eq. (10) gives us a condition of force equilibrium at the top surface of the bed (1 - t/> )Pc 11._ - Ps1/(l - t/» :: 11._ + Tx..,(x = L+) - TxAx = L) = 0 (17) where L_ is right below the top surface of the bed, while L+ is right above. The viscous coefficient' 1/ is defined as 1/= k s + (4/3)Jls Ps ~ - ______ f ..... (18) high voltage Vo coswt _ _ _ /electrode TO) 1_/ f electrically grounded distributor f" super ficiol gas velocity Us FIGURE lOne dimensional e1ectrotluidized bed geometry. ELECTROFLUIDIZED BEDS 185 EQUILIBRIUM FORCE BALANCE Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 Nonzero Particle Collision Pressure (Uniform Fluidization) Most previous fluidized bed analysts have assumed the particle and fluid phase pressures to be equal, so that the collision pressure is zero. In static equilibrium (v = 0) (17) then requires the jump in the Maxwell stress tensor across the top surface of the electrofluidized bed to be zero. If we use the expression for the Maxwell stress tensor given by (15) with unity voidage outside the bed, we find that, with the assumption of a uniform steady state voidage, the effective permittivity of the fluidized bed which gives no jump in the Maxwell stress tensor across the interface is EpEO E(cfJ) = Eo(1 _ cfJ) + EpcfJ (series) (19) This effective permittivity corresponds to a series model for the electrofluidized bed [6] where the particles of permittivity Ep are totally packed without voids in layers perpendicular to the applied electric field. Capacitance measurements have suggested [6] that this model does not describe well the effective permittivity of the bed. In fact these measurements seem to agree with a Lorentz Sphere Unit Cell model for the effective permittivity, E( cfJ) = Eo[3E +2cfJ(Eo-E») p p 3Eo + cfJ(Ep - Eo) (Lorentz sphere) (20) It appears therefore that condition (17) with zero collision pressure results in an inconsistency in the equilibrium stress balance at the top surface of the one-dimensional bed if the series permittivity model of (19) is not appropriate. A nonzero collision pressure appears to resolve this inconsistency by balancing the jump in Maxwell stress tensor in (17) at the top of the bed for any other permittivity model. Zero Particle Collision Pressure (Nonuniform Equilibrium Fluidization) With the particle collision pressure P; assumed negligible, Pc=O (21) the condition of force equilibrium at the top surface of the bed (17) can still be satisfied if we require the jump in Maxwell stress to be zero. This then requires the voidage to be unity at the top of the bed, cfJ(L(t), t) =1 (22) and we allow the bed to be nonuniform. For steady state conditions a/at = 0 and v = 0, so the governing equations (1), (4), (5) for one dimensional flow with PI = 0 reduce to d(cfJu)/dx = 0 or u = UsN (23) dl}/dx = -f3(cfJ)u f3(cfJ)u - (1- cfJ)Psg +F., =0 (24) (25) A.A. MOlSSIS AND M. ZAHN 186 where Us is the superficial gas velocity. Equations (24) and (25), combined with the expression for the force density of electric origin, leads to a differential equation for cf> [6]. f3( cf> >Us Psg - cf>(I- cf» Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 dcf> dx 1 2eo£o dcf>2 (26) 2[1] 2 2d e( cf» As a model for the interfacial drag we use the Carman-Kozeny drag law, f3( cf> ) = 1501lt (I - cf> )2 (cf>sdp)2 cf>2 (27) where cf>s denotes the particle sphericity and Ilt the effective fluid viscosity. Boundary condition (22) combined with the differential Eq. (26) allows numerical integration backwards from x = L, where cf>(x = L) = 1. For the modest electric fields used in our experiments (y=eo£UpsgL«I) Zahn and Rhee predicted that the voidage is essentially uniform over most of the bed volume, and then rises to unity over a thin transition region near the top (Figure 2). As the electric field strength increases, the width of this transition region was calculated to increase. The values of y = 1l0M~f p.gl: corresponding to the magnetically stabilized analysis [5] were found to be significantly higher (y =0.82) than the values calculated for our electrofluidized bed experiment (y = 0.05) because the magnetic force for easily achievable laboratory magnetic fields (B 1000 Gauss) greatly exceeds the electric force for easily achievable laboratory electric fields (breakdown limited in air to values less than 30 kV fcm). Consequently, the thickness of the transition layer at the top of the magnetic bed is significantly = 1.0,---------------, y 0.8 - = .0 E~o ~ =0.0!5 1!50'1U' {o.r e- o =t¢. dp)2 p. Q = 0.;3 - - - Xo. 6 _ -e- Independent I ~~.::~l~ o4 _ Lorenlz.b 'Oh'" mod., / ' . 0.2- I I I I O:---::-'::------:"-,-------,:!.:----,L-----' o 0.2 04 0.6 0.6 1.0 xl l FIGURE 2 Nonuniform voidage stratification necessary to satisfy zero surface force boundary condition of unity voidage at the top of the bed with collinear electric field and flow. Parameter Q' is a measure of the drag force while y is a measure of the relative electric force, both compared to particle weight. In electrofluidized bed experiments, Q' typically varied from 0.12 at minimum fluidization 10 0.3; y varied from zero to 0.05 16]. Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 ELECfROFLUIDIZED BEDS 187 larger than in the electric case This bed nonuniformity explains anomalous yield stress measurements in a magnetizable fluidized bed which approaches zero at the top bed surface, as the bed approaches unity voidage [5]. Due to this nonuniformity in voidage, the equilibrium fluid pressure distribution will no longer be linear. The pressure will vary linearly with position in the bulk region where the voidage is essentially uniform, however in the transition region near the top of the bed the pressure change with position will be more abrupt. Consequently, the pressure drop across the bed will be less than the weight per unit area of the bed because the particles at the top of the bed are partially supported by the electric (or magnetic) force. Experimental pressure profiles of magnetically stabilized beds did exhibit this theoretical trend [5]. GENERAL SOLUTIONS FOR PERTURBATIONS For the electrofluidized bed, because the parameter y is so small, the bed is essentially uniform over its entire length, except for the thin transition region at the top. Alternatively, we may allow the collision pressure to account for stress equilibrium at the top so that the equilibrium voidage distribution is strictly uniform everywhere. For either reason our continuing analysis assumes an equilibrium state of uniform fluidization where voidage cj>o is constant. One Dimensional Problem For the governing equations presented above, we thus assume the simplest equilibrium solution of uniform fluidization with constant interstitial gas velocity uoix, void age cj>u. with stationary particles (v = 0). With voidage cj> constant, c( cj» and E are constant so that the electric force density Fe in (14) is zero. Then (24) and (25) reduce to dP. (28) -(3(cj»Usfcj> = -(1- cj>o)Psg c:= which tells us that the static fluid pressure in the electrofluidized bed will support the weight of the particles. For the one-dimensional problem of Figure 1, the electrically grounded distributor base corresponds to x = 0, L(t) is the bed height, while the high voltage electrode at sinusoidal rms voltage Vo is placed above the bed at x = T(t), where T> L. For such a one dimensional geometry we find that at equilibrium the fluid pressure and electrostatic potential distributions are given by p. (x) = cj>o)Psg(Lo-x) (O<x < L o) fO 0 (L o< x < To) (29) [(1- Co VoX cuLo + c( cj>o)[ To - L o] <l>o(x) = [ (O<x<L o) coLoVo + c( cj> )[x - LoJVo ( ) Lo<x<To coLo + c( cj>o)[To - L u] (30) 188 A.A. MOISSIS AND M. ZAHN In order to determine the response of the one-dimensional electrofluidized bed system to small signal excitations we introduce perturbed (primed) variables Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 I/>(x, t) = 1/>0 + I/> '(x, t) vex, t) = v'(x, t)ix E(x, t) = Eoix + e'(x, t)ix <I>(x, t) = <l>o(x) + <I>'(x, t) u(x, t) = Uoix + u'(x, t)ix If(X, t) = PjO(x) + P;'(x, t) D(t) = Doix + d'(t)ix L(t) = L o + L'(t) (31) The perturbation in the electric displacement d' will not be a function of x due to the fact that the electrofluidized bed is assumed to be charge free by (12). The perturbed collision pressure of (7) may thus be expanded to linear terms about its equilibrium value (32) Furthermore, from the constitutive law (13) we may relate the perturbations e' and d' d'(t) = e(l/>o)e'(x, t) + Eo: ; 10 I/> '(x, t) (33) If we then differentiate (10) with respect to x and we use (1), (2), (6) and (11)-(15) we obtain a single linear constant coefficient differential equation in perturbation voidage 1/>' 821/>' 181/>' Vw 81/>' V~ 821/>' a31/>' 8t2 +~at+~ 8x -N 2 ax 2 -7Jax2at=O (34) where (35) (36) (37) with 2 A(1/>0) = W- 2( d [ e(1/>0) 1 ] 1/>0)e 1/>0) dl/>2 (38) U,« is the superficial gas velocity at equilibrium. From (34)-(38) we see that N 2 is a measure of the ratio of drag to electrical forces. It should be pointed out that assuming zero collision pressure in this analysis will not alter the form of the differential equation describing the perturbations in voidage; only N will change. General Solution for Perturbation Voidage The governing Eq. (34) may be written in nondimensional form a2;P a;p a;p 1 a2;P _ a3 ;P a? + at + ai - N 2 ai 2 - 7J ai 2 at = 0 (39) ELECTROFLUIDIZED BEDS 189 where we define ij = 7j/V~t", ro = g/(c/Jouo), L = (ro/Vw)L', Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 4>=c/J' c/Jp , t = rot, i = (ro/Vw)x t; = (ro/Vw)L o u=~u' ii=(I-c/Jo) v ' c/JpVw c/Jp Vw ' - = c/Jo(1 - c/Jo)ro, n. (3 V Z p, 'f'p 0 w d= P [E 0 dE do I 'f' 0 n. 'Yp (40) ]-1 d' ci> = E(c/JO) [ Eo:; 10 c/Jp(Vw/ro)f <1>' where c/Jp is an arbitrary normalizing constant for the voidage. Equation (39) has a general solution 4J(i, t) = [4>1 sin kzi + cPz cos kzi]ekIXeS; + c + BeN'x (41) _ NZ k - .,..-:---.:..."..,;;--:: (42) where 1 - and _ kz = and 8, 2[1 + NZijs] I - N4 V4(1 + NZijs)Z Nt(st + s) 1 + NZijs (43) t are constants independent of time and space. Stability of Semi-infinite Bed Previous work solved for the stability of an electrofluidized bed with and without particle viscosity by solving for the natural frequency s. For finite length beds it was necessary to specify boundary conditions on the voidage at the top and bottom of the bed. Without an applied electric field, the natural frequency s was found to have a positive real part so that any disturbance grew exponentially with time and the system was said to be unstable. It is thought that such instability results in bubbles. As the electric field was increased, a critical value was reached where the real part of s was zero; for larger electric field values the real part of s became negative, so that any disturbance decayed with time. The bed was then said to be stabilized. The analysis which will follow in the next sections generalizes earlier work by treating the case of a bounded bed with boundary conditions 011 the other fluidized bed variables u, v, Pf and <1>. Previous stability analysis on a semi-infinitely long bed has not determined whether an unstable electrofluidized bed is convectively or absolutely unstable. In an unstable medium a disturbance (say a pulse) may evolve in time and space in two different ways: The disturbance may grow and propagate away from its source, so that at a fixed point in space the disturbance will decay with time; the instability is then a convective one. Alternatively, the disturbance may grow encompassing more and more space, so that at a fixed point in space the disturbance grows with time: the instability is then absolute. Work has shown that 190 A.A. MOISSIS AND M. ZAHN Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 an unbounded (viscous or inviscid) bed is unstable [2,3,6,8], a bounded inviscid bed is stable, while a bounded viscous bed is unstable [5,7]. These are general characteristics of convective instability stabilized by boundaries or destabilized by losses. In what follows we will show that the instability of an unbounded bed is convective using the stability approach developed by A. Bers which is generally applied to plasmas [13]. We assume that the voidage iP has a space and time dependence of the general form (which is consistent with the form used by Bers in [13]) (44) where k and ware both complex. Using this assumption in (39) we obtain the dispersion relation for the system: D(k, w) = w2+ j(1 + ijP)w - Uk + k 2/N2) = 0 (45) In order for the system to be stable, all complex frequencies wei(,) corresponding to real wavenumbers k = kr must lie in the bottom half complex w plane (Wi < 0) so that any disturbance decays with time. Solving (45) for wei(,) we obtain solutions of the form w(k ) = -}(1 + ijk;) 2 r 2)] [u 'J11 _ 4Uk(1 ++ijk;)2 k:/N r We define Po= (46) 1 + ijk; (47) 2 so that w(kr) may be expressed as wei(,) = -jpo ± Y-P5 + k;/N2 + jk r (48) We may obtain the imaginary part of W, Wi, using 1 Wi =2][51 - w·] (49) where w· denotes the complex conjugate of the complex frequency using (48) in (49) we obtain Wi = -Po ±~ [ - [-P5 + k;/N 2] + Y[ -P5 + k;/N 2]2 + k;T w. Hence, 12 (50) For wei(,) in the bottom complex w plane we require Wi(i(,) <0, or equivalently, Po> ± ~ [ - [-P5 + k;/N2] + Y[ -P5 + k;/N2]2 + k; r (51) Now Po > 0 and the term in the square root is also real and positive. If we square both sides of the inequality and reduce we obtain the simple condition, N 2< (1 + ijk;)2 (52) This is the condition necessary for the stability of a semi-infinite bed. For the inviscid bed (ij = 0) this condition becomes: N 2 < 1 (inviscid bed) (53) ELECfROFLUIDIZED BEDS 191 which agrees with earlier inviscid analysis [8]. For the case of a viscous bed (nonzero ij) the right hand side of the inequality (52) reaches its minimum for kr = 0; thus the condition for stability of the viscous bed for all wavenumbers will also be N2 < 1 (viscous bed) (54) Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 2 Therefore for N > 1 there is a range of real wavenumbers k, for which the corresponding wi(kr ) > 0 so that any disturbance grows with time and the system is unstable. We now investigate whether the electrofluidized bed instability is convective or absolute. We begin once again with the dispersion relation (45). If we now solve the dispersion relation for k( w) we find: ]2 _ _ j ~[ j 4w( w + j) 2k(w) = jijw -1/N 2± jfJw -1/N 2 -jijw -1/N2 The above expression gives us two separate curves for Wi~ +00 we obtain _ ~-. 2k(Wi~ +00) = ±2j --:!- (55) k (w). In the limit where (56) t) indicating that the two curves for k(w) lie on opposite sides of the real k axis. As the imaginary part of w is reduced these two curves eventually approach each other and touch at a point referred to as a pinch point [13]. The instabilities will be absolute jf we obtain pinch points in the upper half complex w plane. In order to obtain the pinch points of the dispersion relation D(k, w) we set - Dtk, w) = 0 aD\ and - ak = 0 (57) wconst Conditions (57) lead to a fourth order polynomial equation for W, 4fJ2w4 + 4ij[fJj - 2/N2]W 3 + (4/N 2)[I/N2 - 2fJilw2 + [4j/N 4-jfJ +2fJ]w -1/N 2=0 (58) For the inviscid case (fJ = 0) this simplifies to w 2 + j w - N 2/4=0 (59) which has solutions for W, _ j 2 1,~ w = - - ± - v lV - - 1 2 (60) For an unstable inviscid system (N 2 ) 1) the instabilities are thus convective, since no pinch points exist in the upper half w plane. We have thus determined that a semi-infinite electrofluidized bed will be stable 2 for N < 1. The instabilities of an inviscid fluidized bed were found to be convective. To determine whether the instabilities of a viscous bed are convective or absolute, one would have to solve for the four roots (pinch points) of w in 2 (58). If (for N > 1) all pinch points are found in the lower half w plane (Wi < 0), the instability is convective. Any pinch points in the upper half w plane (Wi> 0) would represent an absolute instability. 192 A.A. MOISSIS AND M. ZAHN General Solutions for Other Electrofluidized Bed Parameters Given the solution for the perturbation in voidage ;p of (41) we may then express all other variables, namely the interstitial velocities, pressure and electrostatic potential, in terms of ;Po Thus, from the mass conservation equation for the fluid phase (1), we may express the interstitial gas velocity ii in terms of the voidage ;po ail ax Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 -= alP (1'_. alP ---=-at (61) ax' Furthermore, from the mass conservation equation for the solid phase (2), we may also express the interstitial particle velocity ii in terms of the voidage ;p: aii alP ax = ai (62) In order to find an expression for the fluid phase pressure distribution in terms of the voidage, we look at the normalized momentum balance equation for the fluid phase, which is derived from (4), ~ ax + (1 - I/>o)il - I/>oii + [(1'(1 - 21/>0) - 1]1/>- = 0 If we differentiate w.r.t. of ;p we obtain x and we use (61) and (62) to eliminate il and ;j2p alP alP ~ = ai - [(1'1/>0 + 1] ax (63) ii in favor (64) Finally, in order to express the electric potential in terms of the voidage, we use (33) expressed in the form aet>' - ax = Eo: ; 10 I/>'(x, t) - d'(t) -e'(x, t) e( 1/>0) (33) In normalized terms this takes the form, ali> - ax = I/> - - d (65) The non-dimensional perturbation displacement field d, defined in (40), depends on voidage as well as, on perturbations in applied voltage and in electrode position. Using the general voidage solution of (41) in (61), (62), (64), and (65) we obtain general solutions for the electrofluidized bed parameters. By integrating (61) we obtain an expression for the perturbation in the interstitial fluid velocity ELECTROFLUIDIZED BEDS 193 where a(i) is a time-dependent constant in space which comes about from the integration in i. In a similar manner, from (62) we find an expression for the perturbation in the mean particle velocity _ _ se ii _... _ 1. 2 [(k,4>, + k 24>2)sin k2i + (k,4>z - kz4>,)cos k2i]e k ,X (67) A v(i, t) = vet) +P Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 1+ _ _... _... _ _ 2 Here again v(i) is a constant in space that results from the integration in i. From (64) we find that the perturbation in fluid pressure takes the form, __ [[s[cPI(kf-kD+2k,k2cP2] Pt(x, t)= kf+q -. _.] + (1 + «4>o)(kl4>1 + k 24>2) - sin k 2x + - (1 + «4>0)(k 2cPI - I., cP2) ]cos k 2i] [S[cPzCkT- kD-2k,k 2cP d k- 2 -2 1 + kz ~k:e;~ + xp,(i) + P2(i) + (1 + «4>0)(B/N 2)e N 2X (68) where PI(i), P2(i) are constants of the integrations in space. The constants of integration u(i), v(i), p,(i) and P2(i) are in fact related. If we use (41) and (66)-(68) in the fluid phase momentum balance equation (4) and the combined momentum balance equation (10) we obtain compatibility equations p,(i) + (1 - 4>o)a(i) - 4>ov(i) + [«(1- 24>0) -1]C = 0 dv(t) rz:: (1 - 4>o)u(t) - 4>ov(t) + [«(1 - 4>0) - - I]C (69) (70) Finally from (65) we find that the perturbations in electrostatic potential will take the form, «(>(x, i) = ei'[[(cP,k, + cP zk2)sin k 2x .. _ .. _ _ ek 1X + (4)2 k, - 4>,k 2)cos kzx] -2 -2 k l +kz 2)e N 2X - d(i)i + (jj / N + Cx (71) Boundary/Jump Conditions In order to solve for the perturbations in voidage, fluid and solid phase velocities as well as fluid pressure and electrostatic potential we also need to introduce boundary/jump conditions. Using (3) we may relate the particle and fluid velocities to the superficial gas velocity 4>u + (1- 4»v = Us (72) in terms of (normalized) perturbations this gives as(t) = «C + u(i) + v(i) (73) 194 A.A. MOISSIS AND M. ZAHN We assume that the particles at the bottom of the bed are constrained by the ground electrode to remain stationary Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 v(x = 0, t) = v'(x = 0, t) = ° (74) Continuity of the electric displacement vector D and of the electric potential 4> across the time varying top surface of the bed (by (11)-(12» allows us to relate the perturbation in the electric displacement to the perturbation in bed height L' as well as to the perturbations in the amplitude of the applied voltage V' and the position of the top electrode T' [6] d'(t) = EoR(cf>o)L'(t) - GMo)V'(t) - E OG2(cf>0)T'(t) (75) where for the specific geometry of Figure 1 we obtain 1 - cf>o ] [E(r/lO)[E(cf>O) - Eo] + Eo dE I ] EoL o + E(cf>o)[7o - L o] 1 - cf>o dcf> 0 EoE(cf>o) E(cf» G1(cf>0) = EuL u + e(cf>u)[T _ L ; G2(cf» = ~ G1(cf» o o] R(cf>o) = [ (76) Finally the stress balance requirements at the top surface of the bed were given by (16) and (17). RESPONSES TO SMALL SIGNAL EXCITATIONS Nonzero Collision Pressure We now focus on the problem of a generalized small signal excitation in the superficial gas velocity, or in the applied voltage, or in the position of the top electrode (with particular emphasis on perturbations in superficial gas velocity). The excitations in superficial gas velocity, voltage and top electrode position are given in the general form: usCi) = UsDC + Us exp(si) v(i) = vDC + if exp(si) (77) t(i) = t DC + t exp(si) where s is the driving frequency (s = jw for sinusoidal excitation) while UsDC' VDC , t DC correspond to the DC components of the excitations. From the analysis presented thus far the following equations may be derived: The boundary condition of zero mean particle velocity at the bottom of the bed (74) applied to (67) gives: v(i) = (78) The boundary conditions for the electromagnetic fields give us the condition (75) which takes the nondimensional form (79) ELECfROFLUIDIZED BEDS 195 where (80) Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 (81) From the boundary condition of zero fluid pressure at the top of the bed (16) we obtain using (68), cA 3 pl + C.Ap2 + Lopli) + P2(i) + Coll = CsL(i) (82) where (83) (84) (85) - - - JAJ(L - - , t) - - ii -afbl J2d(t) o at with = i. (86) 0 s, = _1_[(1 _ cjJo) aPe I + aTx x I _aTu I ] e( cjJo) de J2 = Eo dcjJ aE 0 I0 (Jd p, V 2w); aE Lo+ aE (87) Lo- (88) Bed particle mass conservation implies that the integral of (1 - cjJ) over the entire (time varying) length of the bed is a constant. This leads to the condition [6], L(i) = J.e'ifb! + Jse,ifb2 + J6 (; + 8 1 B (89) (90) (91) A.A. MOISSIS AND M. ZAHN 196 Finally, the combined mass conservation equation of (3) gave us (73), while were expressed by compatibility relations between f>tCi), u(i), v(i), and (69)-(70). We may express the steady state values of these variables in response to the excitations of (77) in terms of a DC and a time varying term at the frequency of excitation s t Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 Ii (i) = Uoc + Ct exp(Si); L(i) = L oc + L exp(si); PI(i) = PIDC + PI exp(si); + fl exp(si) d(i) = doc + exp(si) pz(i) =P2DC + pz exp(si) v(i) = voc a (92) In particular we will focus on the case of an excitation in the superficial gas velocity with zero change in the applied voltage (V' = 0) or the top electrode position (T' = 0). Using (92) in (69)-(70), (73), (78), (82), (86), and (89) we obtain a system of eight equations for the eight DC terms, voc, UOC' PIDC, PZOC' t, E, doc, and L oc and a system of eight equations for the eight coefficients of the time varying steady state terms, Ct, fl, PI' Pz, cP" cPz, and L. In order to solve these systems we need to know the dependence of the collision pressure P; on the electric field E and on the voidage </>. Using (75) we conclude that at equilibrium a, P (A. c 'f'O> E) = _ E~ [e( </>0)[e(</>0) - eo] 0 2eo 1 - </>0 + de I ] eo d</> 0 (93) Given the partial derivatives of the collision pressure Pc with respect to the void age and the electric field, which enter in (32) and hence in (37), (17), (88) and also (86), we would thus be able to solve for the first order perturbations in Pc and therefore to determine the response of the system. Since we do not know the exact dependence of the collision pressure Pc ( </>, E) to the voidage and the electric field, we cannot proceed until the dependence of Pc on </> and E is established. It should be pointed out that in order to find Pc (</>, E) in general it is incorrect to simply replace </>0 by </> and Eo by E in (93). Just because functions are equal in equilibrium, does not imply that they will be equal for perturbations about this equilibrium. The solutions to the system of eight equations for the DC terms suggest an additional constraint on the collision pressure. Solving the system of equations consisting of the DC parts of (69)-(70), (73), (78)-(79), (82), (86), (89) we find c = (1 - </>o)usoc R 1lz16 - 13 _ B = 1 exp[NZLo]- R l lzBI (1 - </>o)usoc UDC = [1 - a(1 - </>O)]UsDC L oc = 16 (1 - </>o)usoc + BI B dDC = R,1 6 (1 ~ </>o)usoc + R,B,E VDC =0 (94) 3 P IDC = a</>o(1 - </>o)usoc P2DC = [GsBI - G6 ]E (95) We therefore find that the DC components of the responses to an excitation in superficial gas velocity are proportional to the excitation UsDC. As should be expected, the mean particle velocity of the response is found to have a DC ELECfROFLUIDIZED BEDS 197 component of zero. These solutions would correspond to the steady state part of the response of a stable electrofluidized bed system to a step in superficial gas velocity of amplitude UsDC' For such a step excitation in superficial gas velocity we would find from (41) in the limit t~oo Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 cP(i, i~ (0) = iJe N2x+ C (96) which implies that the new steady state void age distribution could be nonuniform. Experimental observations, as well as previous fluidized bed analysis suggest that the steady state voidage inside an electrofluidized bed is uniform. This implies that iJ = 0 (97) For the case of a conventional fluidized bed where the electric field is zero, the need for making any such assumption did not arise. For zero electric field, the coupling constant N 2 goes to infinity and consequently the steady state solution of the differential equation for the perturbed voidage in (41) gives us uniform voidage without any assumptions necessary. Setting B equal to zero is also consistent with our original assumption that at equilibrium the voidage is uniform. From equation (94) we see that in order for iJ to be zero, R )J2J6 - J3 = 0 (98) which is a constraining relation for the derivatives of the collision pressure, since these enter in the expressions for J2 and J3 • If we now use the expressions for J2 and J3 given by (88) with N 2 given by (36)-(38) we obtain ( QI <Po, Ell) 8~1 8E 0 + (1- <Po) 8~1 8<p 0 = Qi<Po, Eo) (99) where QI(<PO, Eo) and Q2(t/JO, Eo) are generally nonzero and depend on the equilibrium voidage and electric field (100) (101) We therefore find that the assumption of a uniform steady state electrofluidized bed void age imposes constraints on the dependence of the collision pressure on the void age and the electric fleld. Zero Collision Pressure Even though we cannot proceed in the nonzero collision pressure analysis any further at this time, it is still interesting to examine the consequenses of a zero collision pressure analysis that neglects the inconsistencies in equilibrium stress balance. Several analysts of fluidized and electrofluidized beds have in fact 198 A.A. MOISSIS AND M. ZAHN neglected this stress balance inconsistency in their analysis. A zero collision pressure analysis, although incomplete, does help us to understand the frequency dependence of the spatial distributions of the various parameters used in a macroscopic model of an electrofluidized bed. It also gives us some insight to the effects of varying the applied voltage, effective viscosity, as well as electric field. We thus assume in this section that Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 (102) with N given by (37). It should be pointed out that if we were to use the series model (19) for the effective permittivity of the electrofluidized bed, we would find no electromechanical coupling, since in that case A (<I» in (38) is zero and therefore N is infinite, which is equivalent to setting the applied electric field to zero. Such a bed would not be stabilized against bubbling. With Pc = 0 (and with a model other than the series model used for the permittivity), instead of (17) we use an arbitrary boundary condition that sets the perturbation in voidage to be zero at the top of the bed, <I>'(L, t) =0 (103) We may thus solve the system (69), (70), (73), (78), (79), (82), (89), (103) for the coefficients of the time varying terms 11, fI, PI, Pz, L, <PI and <Pz. We may then use these in (41), (49), (51), (52), and (55) in order to obtain expressions for the time and space dependencies of <P(i, t), u(i, t), ii(i, t), P(i, t), and <i>(i, t). These distributions are plotted for various frequencies in Figures 3-7 for a applied voltage Vo = 5 kV, equilibrium bed height L o = 2 em, with top electrode position To = 2.4 cm and with the ratio R = Uso/ Urn! = 1.2. The equilibrium interstitial gas velocity is Uo = 0.45 m/s while the equilibrium pressure at the a, f 00.0 I Hz f 00.003 Hz ~ ~ "6- z-: "6- 0'::- o -" x/Lo <p~a. - - = 0.23, u~/Uso La 0 2cm FIGURE 3 Spatial distribution of the perturbation in voidage in response to a 2% sinusoidal excitation in superficial gas velocity about its equilibrium value. ELECTROFLUIDIZED BEDS 199 f'O.OIHz f ' 0.03Hz ~ " _E ::l .' ::l OLf,0.03Hz Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 -' o <flo f ' 0 I Hz IV-o L- ---' OL- -' o o u :-nax u~ / Usa 0.45mfs, lo'2cm FIGURE 4 Spatial distribution of the perturbation in interstitial gas velocity in response to a 2% sinusoidal excitation in superficial gas velocity about its equilibrium value. bottom of the bed is 196 Pa. The effective bed viscosity used was given by the empirical formula (8) with Ep = 100, Ps = 1.79 X 103 Kg/rn", dp = 550 Jim for Rochelle salt. The Lorentz sphere model was used to describe the effective bed permittivity. The frequency response of the amplitude of the expansion in bed length was found to have a low pass characteristic whereby the bed height change becomes very small above a cutoff frequency which depends on applied electric field and effective viscosity [9]. f '0.003 Hz f '0.01 Hz ~ " • E ,> > O~~===J o x/La o ~===:::::=:J o v' ~ '0.00185, lo ' 2cm us/ Usa FIGURE 5 Spatial distribution of the perturbation in particle velocity in response to a 2% sinusoidal excitation in superficial gas velocity about its equilibrium value. 200 A.A. MOISSIS AND M. ZAHN f· 0.03Hz f • 0.01 Hz Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 f • 0.03Hz 0':- -" o p' f ma x Us I u~o 195 Pnscols , La ' 2cm FIGURE 6 Spatial distribution of the perturbation in fluid pressure in response to a 2% sinusoidal excitation in superficial gas velocity about its equilibrium value. The results presented above were obtained with the assumption that the electrofluidized bed is stable so that a sinusoidal steady state can be achieved. This requires that Eo exceed the critical value for stabilization. Determination of this critical value was presented earlier for a bed of semi-infinite extent and remains for near future work for a finite length bed. 6 "".01 Hz K C - E ~ -0& FIGURE 7 Spatial distribution of the perturbation in electric potential in response to a 2% sinusoidal excitation in superficial gas velocity about its equilibrium value. ELECfROFLUIDIZED BEDS 201 Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 CONCLUSIONS The introduction of a collision pressure term in the governing equations for the stabilized electrofluidized bed appears to resolve past inconsistencies in the force balance at the top surface of a uniform fluidized bed. Given the dependence of the collision pressure on the voidage and on the electric field, we would be able to determine the response of a one dimensional fluidized bed stabilized against bubbling by an electric field to a sinusoidal small signal excitation in the superficial gas velocity. This dependence of P; on the voidage and electric field has not been established at this point. A simplified zero collision pressure analysis that neglects the inconsistency in the stress balance suggested that the response of the bed voidage to a sinusoidal small signal excitation in superficial gas velocity has a standing wave character, with the apparent wavelength increasing with the frequency of excitation. For frequencies above a certain cutoff, diffusive effects appeared to dominate the response; consequently, for high frequencies the voidage perturbation decreased with frequency and the standing wave pattern was lost. The frequency response of the amplitude of the expansion in bed length was found to have a low pass characteristic. NOMENCLATURE Variables subscripted with zero (e.g. <Po, Eo, uo, ... as in Eq. (31» indicate equilibrium variables. Variables superscripted with a prime (e.g. <p', v', e', ... as in Eq. (31» indicate perturbation variables. Variables superscripted with a tilde (e.g, 4>, of, t, p, ... as in Eqs. (39)-(40» indicate nondimensional variables. Variables subscripted with DC (e.g. usoc, Vo c , as in Eq. (77» indicate DC components of the perturbation variables. Variables with superscript carets (e.g. Us> V, t, ... as in Eq. (77» indicate complex amplitudes of perturbations. Variables with subscript max (e.g. <P;"ax> u;".., v;".x> p/ m .,,, <1>;" ax in Figs. (3)-(7» indicate the maximum value of the perturbation. t-: ... A( <Po) B) jj, C d' dp D(k, w) D = e(<p)E voidage, permittivity parameter defined in Eq. (38) parameter defined in Eq. (91) integration constants for perturbation voidage solutions introduced in Eq. (41) perturbation displacement field defined in Eq. (31) particle diameter dispersion relation defined in Eq. (45) displacement field defined in Eq. (13) 202 e' E Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 Eo E, = [Ep E 2/ (Ps dpg)]'12 g G t ( CPo), G2( CPo) GJC CPo). G4 ( CPo), Gs( CPo), G6 ( CPo) j A.A. MOISSIS AND M. ZAHN perturbation electric field defined in Eq. (31) electric field defined in Eq. (11) rms electric field due to applied voltage normalized electric field that affects particle viscosity defined in Eq. (9) electric force density on particles defined in Eq. (14) acceleration of gravity parameters defined in Eq. (76) parameters defined in Eqs. (83)-(85) identity tensor, unit dyadic j yC1 i.. J2 • J3 parameters defined in Eqs. (86-88) parameters defined in Eqs. (90-91) nondimensional wavenumber parameters defined in Eqs. (42)-(43) real part of the non-dimensional wavenumber effective particle bulk viscosity introduced in Eq. (6) bed length magnetization of magnetically saturated particle non-dimensional electric field and collision pressure parameter defined in Eq, (37) collision pressure defined in Eq. (7) perturbation fluid pressure parameter defined in Eq. (47) fluid pressure solid pressure parameters defined in Eqs. (99)-(101) J4 , Js • J6 kt , k2 Pc=P.-Pt PI Po PI r, Qt(CPo, Eo). Q2(CPO, Eo) ro = g I( CPouo) R( CPo) R I( CPo), R 2( CPo), R3 ( CPo) s normalizing time parameter defined in Eq. (40) parameter defined in Eq. (76) parameters defined in Eqs. (80)-(81) nondimensional complex frequency for perturbation variables introduced in Eq. (41) time Maxwell stress tensor components defined in Eq. (15) solid's viscous stress tensor defined in Eq. (6) ELECTROFLUIDIZED BEDS height of electrode distributor T(I) above 203 electrically grounded Maxwell stress tensor component of force perpendicular to top surface of bed used in Eq. (17) interstitial fluid velocity Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 minimum fluidization velocity superficial gas velocity particle velocity applied voltage amplitude velocity parameter defined in Eq. (36) spatial coordinate x non-dimensional flow parameter defined in Fig. (2) defined in Eq. (61) Darcy drag coefficient used in Eq. (4) and given for Carmau-Kozcny law in Eq. (27) y { e oE5!(psgL ) lloM~/(Psg L) = nondimensional electric or magnetic force parameter used in Fig. (2) o i. »i. = Kronecker delta {1 1= J permittivity of free space particle permittivity effective medium permittivity given for series model in Eq. (19) and Lorentz sphere model in Eq. (20) e( cf» IJ = k, + (4/3)lls viscous coefficient defined in Eq. (18) V del vector operator Ilo III magnetic permeability of free space effective fluid viscosity in Carman-Kozeny drag law of Eq. (27) Ils effective particle shear viscosity in Eq. (6) .. = cf>ouo/g cf> = Uso/ g time constant defined in Eq. (35) voidage tP,. tPz nondimensional perturbation voidage amplitudes defined in Eq. (41) arbitrary normalizing constant for void age defined in Eq. (40) particle sphericity used in Carrnan-Kozeny law of Eq. (27) PI fluid mass density 204 p, <I> Downloaded By: [Massachusetts Institute of Technology, MIT Libraries] At: 19:34 18 February 2011 w A.A. MOISSIS AND M. ZAHN solid mass density scalar electric potential defined in Eq. (11) angular frequency of applied voltage in Fig. (1) non-dimensional complex frequency for perturbations with real part W, and imaginary part Wi complex conjugate of W REFERENCES I. Anderson, T.B., and Jackson, R., "A Fluid Mechanical Description of Fluidized Beds-Stability of the State of Uniform Fluidization," Ind. Eng. Chern. Fundamentals, 6,527 (1967). 2. Rosensweig, R.E., "Fluidization: Hydrodynamic Stabilization with a Magnetic Field," Science, 204,57-60, Apr. 6 (1979). 3. Rosensweig, R.E., "Magnetic Stabilization of the State of Uniform Fluidization," Ind. Eng. Chern. Fundamentals, 18,260-269, Aug. (1980). 4. Rosensweig, R.E., Ferrohydrodynamics, Chap. 9, Cambridge Univ. Press, Cambridge (1985). 5. Rosensweig, R.E., Zahn, M., Lee, W.K., and Hagan, P.S., "Theory and Experiments in the Mechanics of Magnetically Stabilized Fluidized Solids," Theory of Dispersed Multiphase Flow, R. Meyer Ed., Academic Press (1983). 6. Zahn, M., and Rhee, S.W., "Electric Field Effects on the Equilibrium and Small Signal Stabilization of Electrotluidized Beds," IEEE Trans. Ind. Appl., IA·ZO, 137-147, Jan./Feb. (1984). 7. Zahn, M., and Rhee, S.W., "One-Dimensional Small Signal Waves in Electrotluidized Beds," IEEE Trans. Ind. Appl., IA·20, 1591-1597, Nov./Dec. (1984). 8. Rhee , S.W., "Fluid Dynamics and Stability of Fluidized Particle Systems with an Electric Force." Ph.D. thesis in Chemical Engineering, M,LT., pp. 90-102., (1984). 9. Moissis, A.A., and Zahn, M" "Electrotluidized Bed Responses to Small Signal Excitations," Conf. Rec. IEEE Trans. Ind. Appl. Soc., 86CH2272-3, Oct. (1986), pp. 1397-1403. 10. Moissis, A.A., "Electrotluidized Bed Responses to Small Signal Excitations," S.M. Thesis, MIT Dept. of E.E. & C.S., Jan. 1987. 11. Jackson, R., "The Mechanics of Fluidized Beds: Part I: The Stability of the State of Uniform Fluidization," Trans, lnstn. Chern. Engrs., 41, 14 (1963). 12. Melcher, l.R., Continuum Electromechanics, Chap. 3. M.LT. Press, Cambridge, Mass., 1981. 13. Bers, A., "Space-Time Evolution of Plasma Instabilities-Absolute and Convective," Handbook of Plasma Physics, M.N. Rosenbluth, R.Z. Sagdeev, Ed., North-Holland, 1983.