Effects of charge distribution around bubbles on charge induction and transfer to a ball probe in gas–solid fluidized beds

Journal of Electrostatics 58 (2003) 91–115

Effects of charge distribution around bubbles on charge induction and transfer to a ball probe in gas–solid fluidized beds Aihua Chen, Hsiaotao Bi*, John R. Grace Department of Chemical and Biological Engineering, The University of British Columbia, 2216 Main Mall, Vancouver, BC, Canada V6T 1Z4 Received 8 July 2002; received in revised form 21 October 2002; accepted 27 October 2002

Abstract The model of Park et al. (J. Electrostat. 55 (2002) 135) on charge induction and transfer between charged particles surrounding a rising bubble and a ball probe is modified to take into account the charge buildup on particles remote from the bubble, and the particle charge density distribution in the vicinity of the bubble. With particle properties, such as dielectric constant and conductivity, estimated from existing correlations, the model simulation results change only slightly when both the background charge density and a distribution of charge density around a spherical bubble are introduced into the original model. However, significant improvement in agreement between the model and experimental results is achieved when the contributions from the bubble wake and drift carrying highly charged particles are included. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Bubble; Fluidization; Ball probe; Electrostatic charges; Model; Particles; Induced charge; Gas– solid system

1. Introduction In industrial gas–solid fluidized beds, electrostatic charges can cause agglomeration, nuisance discharge and even the danger of explosion [2]. Collision-type electrostatic probes [3–9] have been used to monitor and measure electrostatic charges in gas–solids transport lines and fluidized beds. However, interpretation of signals is still difficult. *Corresponding author. Tel.: +1-604-822-4408; fax: +1-604-822-6003. E-mail address: [email protected] (H. Bi). 0304-3886/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 8 6 ( 0 2 ) 0 0 2 0 1 - 2

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Nomenclature An defined in Fig. 1 Andrift defined in Fig. 14 Anwake defined in Fig. 14 b distance between centre of probe and image charge q0 (see Fig. 3), m B constant in Eq. (1). c constant C capacitance, mF Cmax maximum magnitude of charge curve (see Fig. 6), C Cmin minimum magnitude of charge curve (see Fig. 6), C Ctransfer transferred charge (see Fig. 6), C d distance between centre of probe and point charge q (see Fig. 3), m ds particle diameter, m DB bubble diameter, m Dd diameter of cylindrical drift following a bubble, m DP ball probe diameter, m E relative modulus expressed by Eq. (30), Pa EP Young’s modulus of elasticity of probe, Pa Es Young’s modulus of elasticity of particle, Pa g acceleration of gravity, m/s2 hd height of cylindrical drift following a bubble, m hePs effective conductivity expressed by Eq. (28), 1/Om2 Iinduced induced current on probe, A Ip current (see Fig. 5), A Itotal current (see Fig. 5), A Itransferred current transferred to probe, A kc constant in Eq. (24), (m/s)/(kg/m3) K constant in Eq. (24), C s3/5 m7/5/kg 0 K constant in Eq. (33), O m17/5 s3/5 L vertical distance between probe centre and tip of bubble injector, m m relative mass expressed by Eq. (29), kg mP mass of probe, kg ms particle mass, kg q point charge, C q0 image charge, C qm specific charge or charge density, C/kg qm0 specific charge of bed particles in dense phase remote from bubble, C/kg qs charge of particle, C qv charge density per unit volume, C/m3 QB total charge, C Qinduced induced charge on probe, C Qp cumulative charge on ball probe, C Qtotal total charge carried by probe, C

A. Chen et al. / Journal of Electrostatics 58 (2003) 91–115

Qtransferred r r0 r r1 ; r2 r0B rd rs R RB RP s t t1 t2 UB UP Us V Vs Vsr x 1 ; y1 ; z 1 x 2 ; y2 ; z 2

93

charge transferred to probe, C distance between field point and source point, m distance between centre of probe and bubble, m relative radius expressed by Eq. (31), m distance between point P on probe and point charge q; q0 (see Fig. 3), m radial distance from centre of bubble, m radial distance from axis of cylindrical drift, m radius of particle, m resistance, O radius of bubble, m radius of spherical probe, m surface area, m2 time, s time, s time, s bubble rise velocity, m/s electrical potential on surface of probe, V electrical potential on surface of a charged particle, V volume, m3 particle velocity, m/s radial component of particle velocity, m/s position of point P on probe (see Fig. 1), m position of point B on bubble (see Fig. 1), m

Greek letters a; b; f; g; l; y angles (see Fig. 3), degrees b0 angle defined in Eq. (17a) or (17b), degrees j wake angle (see Fig. 9), degrees dB thickness of bubble in two-dimensional column, m dq defined in Fig. 1, m e voidage Z correction factor in Eq. (25) gP shear strain of ball probe gs shear strain of particle sP electrical conductivity of ball probe, 1/O m ss electrical conductivity of particle, 1/O m c surface charge density, C/m2 ctotal total surface charge density, C/m2 P permittivity of medium, F/m P0 permittivity of vacuum or air, 8.854  1012 F/m Pr relative permittivity or dielectric constant, i.e. P=P0 rs particle density, kg/m3 Superscript 0 00 000 , , contributed from part a, b, c, respectively

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Recently, a simple charge induction and transfer model was developed [1] to describe induction and transfer from charged particles surrounding a rising bubble to an electrostatic ball probe. The model was found to give reasonable qualitative agreement with single bubble injection experimental data obtained from a ball probe in a two-dimensional column filled with 321 mm glass beads, fluidized by air. In the model [1], bubbles were assumed to be perfectly circular with the particles immediately surrounding the bubble being charged when a bubble rises, while there is zero charge buildup in other regions of the bed. Experimentally, it has been reported [10,11] that particles in a fluidized bed are all charged due to particle mixing and charge accumulation. The model of Park et al. [1] thus needs to be modified to account for this background electrostatic charge buildup, although the specific charge density of particles near the bubble can be higher than in regions remote from the bubble due to the generation of charges in the vicinity of the bubble by vigorous particle collisions. At the same time, real bubbles are not circular, but are kidney-shaped [19] with a wake occupying approximately 30% of the cylindrical volume. Moreover they are followed by a long drift finger [12–14] carrying highly charged particles, which may need to be treated to improve the model. In this paper, we present the simulation results of the modified Park et al. [1] model by incorporating the charge buildup of the bed particles and the charge distribution around bubbles. Predictions are then compared with the same experimental data [1].

2. Charge induction and transfer model 2.1. Effect of charge distribution around the bubble In the Park et al. [1] model, it was assumed that charges were localized on particles around the bubble surface. In reality, particles in the fluidized bed are all charged, although possibly with higher charge density around the bubble. To simplify the model simulation, several assumptions are adopted with respect to the distribution of specific particle charge density surrounding a bubble: (a) The distribution of specific particle charge density in C/kg surrounding a bubble (shown in Fig. 1) is assumed to follow an exponential decay function given by 0

2

qm ¼ An qm0 eBðrB =RB 1Þ þ qm0 ;

ð1Þ

where qm0 is the particle charge density far from the bubble (i.e. background charge density of bed materials), approximately equal to the average charge density in the bed; rB 0 is the distance from the centre of the bubble; An and B are constants. (b) Particle holdup inside the bubble is neglected, so that there are assumed to be no charges inside the bubble.

A. Chen et al. / Journal of Electrostatics 58 (2003) 91–115

outside bubble

inside bubble

Charge Density, C/kg

0

95

qm0

A n /2qm0

δq /R

B

A n qm0

0

2

A nqm0e -B(rB′ /RB-1) +q m0

1

2

Distance from center of bubble, rB′ /RB

Fig. 1. Distribution of particle charge density around a bubble in a fluidized bed.

(c) The charge density per unit volume surrounding the bubble is related to the charge density per unit mass by qv ¼ qm ð1  eÞrs

ð2Þ

where rs is the particle density and e is the voidage. (d) The charge distribution in Fig. 2 is estimated by the superposition of three simple contributions: (i) a uniformly charged bed with a specific charge qm0 ; (ii) a charged spherical ball with a charge density equal to qm0 and (iii) charged particles surrounding the bubble with an exponentially decaying charge B½ðr0B =RB Þ1 2 distribution, q000 and zero charge density inside the bubble. m ¼ An qm0 e 2.1.1. Charge induction As in the earlier work [1], consider a bubble approaching a grounded ball probe (UP ¼ 0) from below, as illustrated in Fig. 3. The bubble wake and drift are first ignored. The bubble rises at a constant velocity, UB ; with the centre of the probe lying on the axis of the rising bubble. The y coordinate of the bubble centre C is then L þ UB t; where L is the distance between the tip of the bubble injector and the centre of the ball probe and UB is the bubble rise velocity. Using polar coordinates, we can express the positions of point P on the surface of the electrostatic ball probe and point B on the bubble surface, respectively as ðx1 ; y1 ; z1 Þ ¼ ðr cos y cos f; r sin y; r cos y sinfÞ;

ð3Þ

ðx2 ; y2 ; z2 Þ ¼ ðr0B cos b cos g;  L þ UB t þ r0B cos b sin g; r0B sinbÞ:

ð4Þ

The distance between the centre of the probe and point B on the bubble surface is then qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ d ¼ x22 þ y22 þ z22 ;

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Fig. 2. Representation of specific charge distribution by three additive components. Note that qm ¼ q0m þ q00m þ q000 m:

while the distance between point P on the probe surface and point B on the bubble surface is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r1 ¼ ðx1  x2 Þ2 þ ðy1  y2 Þ2 þ ðz1  z2 Þ2 : ð6Þ Instead of the surface charge density assumed by Park et al. [1], the point volume charge density, qv; (in C/m3) is applied and the charge on the control volume around point B is DQB ¼ qv DV ;

ð7Þ

where 0 DV ¼ r02 B cos b Db Dg DrB :

ð8Þ

Substituting Eq. (8) into Eq. (7), we obtain 0 DQB ¼ qv r02 B cos b Db Dg DrB :

ð9Þ

With the dielectric constant assumed to be macroscopically uniform throughout the bed, the method of images [15] is applied to obtain an expression for the induced charge on the probe, with a single image charge, q0 ; added to make the sphere a surface of zero potential. Note that the method of images is not strictly applicable to this case in view of the micro-scale structure surrounding the bubble, causing the

A. Chen et al. / Journal of Electrostatics 58 (2003) 91–115 y

97

y

RP Electrostatic Ball Probe

b

U=0

P

P(x1,y1,z1)

O r

O

λ

x

φθ

r2 q’ z

z

r1

d RB

B(x2,y2 ,z2) q

Bubble

B β

rB’

O’ γ

O’

α

x

rd z z

Wake Drift

Rd hd rd

d,b = distance between centre of probe and image charge q, q' r = distance between point P on probe and centre of probe, O r1, r2 = distance between point P on probe and image charge q, q' rB′ = distance between point B and centre of bubble, O' RB, RP = radius of bubble and spherical probe Rd, hd = radius and height of cylindrical drift rd = distance between point B and axis of drift α,λ,θ,φ,β,γ = angles

Fig. 3. Schematic of two-dimensional bubble vertically aligned with a grounded ball probe.

dielectric constant to vary across the bubble. However, this method provides a useful first approximation. The potential at arbitrary point P due to q and q0 is derived using UP ðr; y; fÞ ¼

q q0 þ ; 4pPr1 4pPr2

ð10Þ

where r21 ¼ r2 þ d 2  2rd cos l;

r22 ¼ r2 þ b2  2rb cos l:

ð11Þ

Therefore, on the surface of the grounded ball probe, r ¼ RP and UP ðRP ; y; fÞ ¼ 0 for all y and f: It can be shown [1] that this requires q0 =q ¼ r2 =r1 ¼ RP =d

and

b ¼ R2P =d:

ð12Þ

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The surface charge density is given by qU at r ¼ Rp ; c ¼ P0 qr     1 q qr1 q0 qr2 q 1 qr1 c qr2 þ þ c¼ ¼ ; 4pPr r21 qr r22 qr 4p r21 qr r22 qr

ð13Þ

where c ¼ Rp =d; P0 is the permittivity of vacuum. Pr ¼ P=P0 is the relative permittivity of the fluidized particles and P is the permittivity of medium. For a packed bed of glass beads, Pr was measured to be 3.0 [16] using the capacitance method. Eqs. (6) and (11) are used to calculate r1 ; r2 ; qr1 =qr and qr2 =qr: With the aid of Eq. (7) and letting q be DQB ; Eq. (13) becomes   1 1 qr1 c qr2 þ c¼ qv r02 DV : ð14Þ 4pPr B r21 qr r22 qr Eq. (14) represents the surface charge density induced by the charge on the control volume inside and around the bubble. It is necessary to integrate this expression over the entire field. The simulation was carried out for the three additive components shown in Fig. 2 based on the model equations presented above. The total induced charge is the sum of three terms: Part a: uniformly charged bed. The induced charge should be constant because of the uniform distribution of volume charge density throughout the whole bed. Therefore, Q0induced ¼ constant:

ð15Þ

The constant is taken as zero in the following calculation of overall charge induction because only dynamic changes of induced charge contribute to the current flow. Part b: charged spherical ball. In the second term, there is no charge outside the hypothetical particle-filled bubble. The induced charge is only generated from the charge inside the bubble. Integrating Eq. (14) over the volume of the ball and substituting qv ¼ qm0 ð1  eÞrs we obtain   Z RB Z 2p Z b0 1 1 qr1 c qr2 00 02 þ  qm0 ð1  eÞrs rB cos b 2 ctotal ¼ dbdgdr0B ; 4pPr r1 qr r22 qr 0 b0 0 ð16Þ where sin b0 ¼ 0:5dB =RB

for 2-dimensional bed ;

b0 ¼ p=2 for 3-dimentional bed :

ð17aÞ ð17bÞ

The total volume charge density is then integrated over the surface of the electrostatic ball probe to obtain the total induced charge at any time: Q00induced ¼

a probe surface

ctotal ds:

ð18Þ

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After substituting Eq. (17a) and (17b) and ds ¼ R2P cos y dy df into Eq. (18) the total induced charge can be written as Z Z Z Z Z qm0 ð1  eÞrs R2P RB 2p p=2 2p b0 0 2 Q00induced ¼ rB cos y cos b 4pPr 0 b0 0 p=2 0   1 qr1 c qr2 þ 2 ð19Þ  2 db dg dy df dr0B : r1 qr r2 qr Part c: charge particles surrounding the bubble. In part c, there is no charge inside the bubble. The induced charge is only generated from charged particles around the bubble. Integration of Eq. (14) over the volume outside the bubble and over the surface of the electrostatic ball probe together with the charge distribution function 0 2 qv ¼ An qv0 eBðrB =RB 1Þ and qv0 ¼ qm0 ð1  eÞrs yields Q000 induced ¼

An qm0 ð1  eÞrp R2P 4pPr Z N Z 2p Z p=2 Z 2p Z b0 0 2 0  eBðrB =RB 1Þ rB2 cos y cos b R b0 0 p=2 0 B  1 qr1 c qr2 0 þ 2  2 db dg dy df drB2 : r1 qr r2 qr

ð20Þ

The total induced charge is Qinduced ¼ Q0induced þ Q00induced þ Q000 induced :

ð21Þ

The induced current can then be calculated from Iinduced ¼ 

dQinduced : dt

ð22Þ

2.1.2. Charge transfer In addition to charge induction, direct charge transfer takes place when charged particles collide with the probe. Therefore, the total charge on the probe is given by Qtotal ¼ Qinduced þ Qtransferred :

ð23Þ

Zhu and Soo [3] estimated the electric current through a ball probe due to the collision between the probe and particles in a pneumatic transport line as Itransferred ¼ Kð1  eÞrs Vs8=5 ekc ðrs ð1eÞ=Vs Þ ;

ð24Þ

where Vs is the particle velocity. K is a dimensional constant related to the ball probe characteristics, particle surface characteristics, specific charge of particles and particle properties. Park et al. [1] determined K by fitting the model to their experimental data. Zhu and Soo [3] and Fan and Zhu [14] showed that K can be estimated by  4=5 4:76Z m K¼ ðDP þ ds Þ2 ðUs  UP ÞhePs r3=5 ; ð25Þ ms E

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where Z is a correction factor. Us and UP are electric potential or work function for the particle surface and probe surface, respectively, with qs qm rs pds3 qm rs ds2 Us ¼ ¼ ¼ ; ð26Þ 2pPds 12pP=P0 P0 ds 12Pr P0 UP ¼ 0

ð27Þ

and hePs ¼

sP ss ; ds sP þ DP ss

ð28Þ

where subscripts s and p denote the particle and probe, respectively. d and D are diameters and s is electrical conductivity. Also m; E and r are the relative mass, contact modulus and relative radius, such that 1 1 1 ¼ þ ; ð29Þ m ms mP 1 1  g2s 1  g2P ¼ þ ; E Es EP

ð30Þ

1 1 1 ¼ þ ; r rs RP

ð31Þ

where m is mass, r and R radius, g shear strain, and E is Young’s Modulus. Substituting Eqs. (26) and (27) into Eq. (25) gives  4=5 2 0:3967Z m 2 qm rs ds K¼ ðdP þ ds Þ hePs r 3=5 : ð32Þ ms E Pr P0 Let K0 ¼

 4=5 0:3967Z m ðdP þ ds Þ2 rs ds2 r 3=5 =P0 ; ms E

which is a constant in a system of given particles and ball probe. Then K can be expressed as K ¼ K 0 qm hePs =Pr : ð33Þ The constant kc in Eq. (24) is related to the local particle velocity and local solids concentration [17] by   Vs m=s kc ¼ : ð34Þ rs ð1  eÞ kg=m3 Substitution of Eqs. (33) and (34) into Eq. (24) allows the transfer electrical current to be calculated as 8=5

Itransferred ¼ 0:3679K 0 qm hePs =Pr rp VP ð1  eÞ:

ð35Þ

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Davidson and Harrison [18] give the radial component of particle velocity relative to a circular bubble in a two-dimensional fluidized bed:  ð36Þ Vsr ¼ UB 1  R2B =r02 cos y: If the ball probe lies on the axis of the rising bubble, the absolute particle velocity along the path of the collision is then Vs ¼ UB R2B =r02 ;

ð37Þ

where the distance between the centre of the bubble and the probe is r0 ¼ L þ UB t: The bubble rise velocity for isolated bubbles in a two-dimensional fluidized bed [18] is pffiffiffiffiffiffiffiffiffi ð38Þ UB ¼ 0:511 gDB : Substituting Eq. (36) into Eq. (34) and letting Itransferred ¼ dQtransferred =dt; we obtain  8=5 dQtransferred UB R2B 0 Itransferred ¼  ¼ 0:3679K qm hePs =Pr rs ð1  eÞ ; dt ðL þ UB tÞ2 ð39Þ Qtransferred ¼ 

Z 0

t



UB R2B 0:3679K qm hePs =Pr rs ð1  eÞ ðL þ UB tÞ2 0

8=5 dt:

ð40Þ

Charges are assumed to be transferred by collision not only from the thin layer of particles at the surface of the bubble, but also from particles in the dense phase surrounding the rising bubble. As in the calculation of the induced charge, the transfer charge is obtained as the sum of the transfer charges from the three terms shown in Fig. 2. Part a: uniformly charged bed. The entire region has the same charge density, qv0 ; but the particle velocity varies as given by Eq. (37). From Eqs. (39) and (40), before the bubble passes through the probe, i.e. for 2L þ UB to  RB ;  8=5 dQ0transfer UB R2B 0 0 ¼ 0:3679K qm0 hePs =Pr rs ð1  eÞ ; ð41aÞ Itransferred ¼  dt ðL þ UB tÞ2 Q0transferred ¼ 0:1672K 0 qm0 hePsb =Pr rs ð1  eÞ  ðUB R2B Þ1:6 =UB ½L2:2  ðL  UB tÞ2:2 ;

ð42aÞ

while the bubble encloses the probe, i.e. for RB p2L þ UB tpRB ; the particle velocity in the bubble is Vs ¼ UB dQ0transferred 8=5 0 ¼ 0:3679K 0 qm0 hePs =Pr rs UB ð1  eÞ; Itransferred ¼ ð41bÞ dt   L  RB Q0transferred ¼ 0:3679K 0 qm0 hePs =Pr rs UB1:6 ð1  eÞ t UB þ 0:1672K 0 qm0 hePs =Pr rs ð1  eÞðUB R2B Þ1:6 =UB ½L2:2  R2:2 B : ð42bÞ

102

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After the bubble has passed the probe, i.e. for 2L þ UB t > RB ;  8=5 dQ0transfer UB R2B 0 ¼ 0:3679K 0 qm0 hePs =Pr rs ð1  eÞ ¼ ; ð41cÞ Itransferred dt ðL þ UB tÞ2 Q0transferred ¼ 0:1672K 0 qm hePs =Pr rs ð1  eÞðUB R2B Þ1:6 =UB ½ðUB t  LÞ2:2  RB2:2

 0:7358K 0 qm0 hePs =Pr rs UB0:6 ð1  eÞRB þ 0:1672K 0 qm0 hePs =Pr rs ð1  eÞðUB R2B Þ1:6 =UB ½L2:2  R2:2 B : ð42cÞ Part b: charged spherical ball. In the second term, there is no charge around the bubble, while the particle velocity inside the bubble equals the bubble velocity, UB : Therefore, before the bubble passes through the probe, i.e. for 2L þ UB to  RB ; 00 Itransferred ¼ 0; ð43aÞ Q00transferred ¼ 0: While the bubble is enclosing the probe, i.e. for RB p2L þ UB tpRB ; dQ00transfer 8=5 00 Itransferred ¼ 0:3679K 0 qm0 hePs =Pr rs UB ð1  eÞ; ¼ dt   L  RB t : Q00transferred ¼ 0:3679K 0 qm0 hePs =Pr rs UB1:6 ð1  eÞ UB After the bubble has passed the probe, i.e. for 2L þ UB t > RB 00 Itransferred ¼ 0; Q00transferred ¼ 0:7358K 0 qm0 hePs =Pr rs UB0:6 ð1  eÞRB :

ð44aÞ

ð43bÞ ð44bÞ

ð43cÞ ð44cÞ

Part c: charged particles surrounding the bubble. For the third term there is no charge inside the bubble and the charge density on the probe is qv ¼ 2 2 0 0 An qv0 eBðjr j=RB 1Þ or qm ¼ An qm0 eBðjr j=rB 1Þ ; with r0 ¼ L þ UB t: Therefore before the bubble reaches the probe, i.e. for 2L þ UB to  RB ; dQ000 000 transferred Itransferred ¼ dt  8=5 2 UB R2B : ¼ 0:3679K 0 An qm0 eB½ðLUB tÞ=RB 1 hePs =Pr rs ð1  eÞ ðL þ UB tÞ2 ð45aÞ 0 Q000 transferred ¼  0:3679K An qm0 hePs =Pr rs ð1  eÞ  8=5 Z t 2 UB R2B  eB½ðLUB tÞ=RB 1

dt: ðL þ UB tÞ2 0

ð46aÞ

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While the probe is enclosed by the bubble, i.e. for rB p2L þ UB tprB ; since there are no particles inside the bubble 000 ¼0 Itransferred

ð45bÞ

0 Q000 transferred ¼  0:3679K An qm0 hePs =Pr rs ð1  eÞ  8=5 Z t1 UB R2B B½ðLUB tÞ=RB 1 2  e dt; ðL þ UB tÞ2 0

ð46bÞ

where t1 ¼ ðL  RB Þ=UB : After the bubble has passed the probe, i.e. for 2L þ UB t > RB ; 000 Itransferred ¼

dQ000 transferred dt 2

¼ 0:3679K 0 An qm0 eB½ðUB tLÞ=rB 1 hePs =Pr rs ð1  eÞ



UB R2B ðL þ UB tÞ2

8=5 ; ð45cÞ

Q000 transferred ¼  0:3679K’An qm0 hePs =Pr rs ð1  eÞ  8=5 Z t 2 UB R2B  eB½ðUB tLÞ=RB 1

dt ðL þ UB tÞ2 t2  0:3679K’An qm0 hePs =Pr rs ð1  eÞ  8=5 Z t1 2 UB R2B  eB½ðLUB tÞ=RB 1

dt; ðL þ UB tÞ2 0

ð46cÞ

where t2 ¼ ðL þ RB Þ=UB ; The total transfer current and charge are then given by 0 00 000 Itransferred ¼ Itransferred þ Itransferred þ Itransferred ;

ð47Þ

Qtransferred ¼ Q0transferred þ Q00transferred þ Q000 transferred :

ð48Þ

2.1.3. Simulation results The total induced and transferred charges on a ball probe caused by passage of a single bubble were simulated based on the above model equations via numerical integration using a computer program written in FORTRAN. A two-dimensional fully circular bubble with a negative specific charge 1.6E–6 C/kg was used in the first simulation. The distance between the tip of the bubble injector and the probe was 0.205 m, as in the experimental set-up by Park et al. [1], with the simulation covering the time interval where the centre of the bubble moves from –0.205 to 0.205 m, with the centre of the probe as the origin. The bubble rise velocity was calculated from Eq. (38) to be 0.358 m/s for a bubble of diameter 50 mm and thickness 22 mm, while the diameter of the ball probe was 3.2 mm. The particles were assumed to be 321 mm

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104

(a)

Induced charge, 10-10 C

0

probe inside bubble -2

δq /RB = 0.2 An = 0.3 DB = 50 mm DP = 3.2 mm ds = 321µm qm0 = -7.12E-5 C/kg

-4

K′=- 1.95E+2 Ωm17/5 s3/5

-6

term a term b term c total

(b)

Charge transfer, 10-10 C

3

0

-3

-6

-9 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Time, s

Fig. 4. Simulated charge transfer and induction versus time as a bubble with charge density distribution around bubble passes the probe (a) induced charge; (b) charge transferred.

glass beads fluidized by air. The bubble was assumed to rise at steady velocity, with constant diameter and a fully circular cylindrical shape throughout the entire interval of interest. Fig. 4a shows the predicted induced charges versus time. It can be seen that the induced charge generated from part b with a negative specific charge has a negative value, while the induced charge generated from part c with dq =RB ¼ 0:2 and An ¼ 0:3 has a small positive value. The lowest peak occurs when the bubble centre passes the probe centre. Fig. 4b shows the predicted transferred charge. This is similar to the result of Park et al. [1]. It can be seen that the total transferred charge mainly originates from the charge generated from part a. The results in Figs. 4a and b demonstrate that a higher charge density around the bubble causes only a small incremental contribution to the total charge induced on, and transferred to, the probe.

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Fig. 5. Schematic of electrostatic ball probe and electric circuit [1].

2.1.4. Comparison with experimental data Experiments were carried out [1] in a two-dimensional Plexiglas column, 0.307 m wide, 0.022 m thick and 1.24 m high. A collision probe of diameter 3.2 mm was employed to explore charge transfer and polarities around bubbles, as shown in Fig. 5. Park et al. [1] estimated the total current from V Itotal ¼ ; ð49Þ 1000R where V is the voltage output across the resistance, R; which is 1 MO, and the total amplification is 1000. However, the current flow through the ball probe, Ip ; should be different from Itotal flowing through the resistor because of the existence of a 10 mF capacitor as shown in Fig. 5. Based on the electric circuit shown in Fig. 5, the total current flowing through the probe, Ip ; should be estimated by Ip ¼ Itotal þ

C dV : 1000 dt

ð50Þ

By definition, the cumulative charge on the ball probe, QP ; can be obtained from Z QP ¼  Ip dt: ð51Þ Traces of voltage, current and charge appear in Fig. 6. The label ‘‘nose’’ indicates alignment of the nose of the bubble with the centre of the probe, whereas ‘‘bubblewake interface’’ represents the moment when the bubble-wake interface contacts the probe. It is seen that the minimum charge, Cmin ; from Park et al. [1] based on Eq. (49) occurs after the bubble passes the probe, whereas the peak from Eq. (50) with the modulation from the capacitor considered occurs when the centre of the bubble passes through the probe. Compared with the charge based on Eq. (49), the minimum charge, Cmin ; is higher based on Eq. (50), but the charge transfer, Ctransfer ; remains the same. It is also found in Fig. 6b that a maximum charge, Cmax ; occurs after the bubble passes through the probe based on the corrected current from Eq. (50).

A. Chen et al. / Journal of Electrostatics 58 (2003) 91–115

106 6

-10 -20

-40 0.5 bubble

(b)

0

0.0

Ctransfer = charge transfer

-2 -3 bubble

injection

0.0

Equation (50) from [1]

bubble-wake interface

nose

-5

0.5

-0.5 -1.0

Cmin Cmin

-4 period

1.0

1.5

Ctransfer

0.0

-0.2

9

-8

Cmax

-1

0.2

10-8 C

-30

0.4

Charge,

0

Charge, 10 C

Voltage, V

0 2

-2 1

Charge, 10 C

-8

10

Current, 10 A

20

(a) 4

-0.4 2.0

2.5

3.0

3.5

Time, s

-1.5 2.0

Time, s

Fig. 6. Experimental results with and without correction for capacitor modulation for single bubble injection in a two-dimensional bed. (a) Voltage output and current; (b) charge induced and transferred.

1

0

experiment model with An = 0.3, δq /R B =0.2 q m0=-7.12E-5 C/kg,

-3

-4

0.0

bubble-wake interface

nose 0.5

1.0

K′=-1.95E+2Ω m17/5 s 3/5

1.5

0.2

-8

-2

-5

0.4

DB=50 mm DP=3.2 mm ds= 321µm

Charge, 10 C

Charge, 10-8 C

-1

charge transfer 0.0

-0.2

-0.4 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5

Time, s

2.0

Time, s Fig. 7. Comparison of simulated total charge with experimental data [1] for a single bubble injection in a two-dimensional bed.

Fig. 7 compares the corrected experimental results with the current model simulation, with parameters qm0 and K 0 obtained by solving two equations simultaneously, matching experimental values of Cmin and Ctransfer to the model. It is seen that the model simulation shows good agreement with the experimental data,

A. Chen et al. / Journal of Electrostatics 58 (2003) 91–115

(a)

15

-5

-qm0 , 10 C/kg

20

107

DB = 50 mm DP = 3.2 mm ds = 321 µm

10

δq /R

B

5

2

-K′, 10 Ω m17/5s

3/5

2.0

0 0.1 0.2 0.3 0.4 0.5

(b)

1.5 1.0 0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

An

Fig. 8. (a) Specific charge of fluidized bed particles, qm0 ; (b) charge transfer coefficient, K 0 ; fitted from experimental data as a function of charge density distribution of particles around bubble, as represented by An and dq :

especially in the region before the bubble passes the ball probe. The peak minimum charge in Fig. 7 occurs while the centre of bubble pass through the probe. However, the model fails to predict the region around the maximum peak after the bubble passes the probe. Figs 8a and b show the effect of charge density distribution around the bubble (represented by the two parameters, An and dq ) on the fitted parameters qm0 and K 0 : It is seen that as the specific charge, An ; increases, the highly charged particle layer around the bubble, dq ; increases, qm0 increases, while K 0 decreases when the simulation results are matched with the experimental values of Cmin and Ctransfer : The variation in the distribution of charge density around the bubble, however, results in almost the same profile of the total charge, which still cannot predict a maximum peak and the region around the maximum peak, suggesting that the contribution from bubble wake and drift may also need to be considered in the model. 2.2. Effect of bubble wake Real bubbles in gas–solids fluidized beds are not circular, but are kidney-shaped with a solids-containing wake [19]. To simplify the simulation, it is assumed here that the particle charge density around a bubble is uniform, while particles inside the wake carry the same or opposite charges to the particles around the bubble, as shown in Fig. 9. A bubble wake angle, j; of 1181 is chosen based on Rowe and Partridge [19]. The velocity of particles in the wake, Vs ; is assumed to be equal to the bubble rise velocity, UB : Similarly, the charge distribution around the bubble is estimated by superposition of three simple contributions: (a) a uniformly charged bed with a specific charge qm0 ; (b) charged sphere with charge density equal to qm0 ;

A. Chen et al. / Journal of Electrostatics 58 (2003) 91–115

108

Fig. 9. Representation of specific charge distribution including the wake by three additive components. Note that qm ¼ q0m þ q00m þ q000 m:

and (c) charged cap shape with a charge density equal to Anwake qm0 : The simulation was carried out for each component as in the previous section. 2.2.1. Induced charge Part a: uniformly charged bed. Eq. (15) can be employed. Part b: charged spherical ball. Eq. (19) is used again. Part c: charged wake. Outside wake Q000 induced ¼ 0;

ð52aÞ

inside wake

Q000 induced ¼

Anwake qm0 ð1  eÞrs R2P 4pPr Z RB Z 2p Z p=2 Z 2p Z  p=2 0 db dg dy df dr0B : 0



0

b0 b0

r0B 2 cos y cos b



1 qr1 c qr2 þ r21 qr r22 qr

 ð52bÞ

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109

2.2.2. Charge transferred Part a: uniformly charged bed. Eqs. (42a)–(42c) are used. Part b: charged spherical ball. Eqs. (44a)–(44c) are employed. Part c: charged wake. Before the wake touches the probe Q000 transferred ¼ 0:

ð53aÞ

While the wake encloses the probe 0 1:6 Q000 transferred ¼ 0:3679K Anwake qm0 hePs =Pr rs UB ð1  eÞ   L þ RB cosð180  jÞ t :  UB

ð53bÞ

After the wake has passed the probe 0 1:6 Q000 transferred ¼ 0:3679K Anwake qm0 hePs =Pr rs UB ð1  eÞ   RB ½cosð180  jÞ  1

:  UB

ð53cÞ

Charge transfered,10-10C Induced charge, C 10-8 C

2.2.3. Simulation results and comparison with experimental data Fig. 10 shows the predicted induced charge and charge transfer versus time for Anwake ¼ 0:5: Fig. 11 compares the experimental results with the model simulation, with qm0 and K 0 fitted with experimental values of Cmin and Ctransfer : The variation is seen to be insignificant when the bubble is negatively charged (i.e. Anwake ¼ 0:5), neutral (i.e. Anwake ¼ 0) or positively charged (Anwake ¼ 0:5). The peak minimum charge occurs slightly later for Anwake ¼ 0:5; i.e. the charge density in the wake has an opposite sign of charge in the bed, while the peak occurs a little bit earlier for Anwake ¼ 0:5; i.e. the charge density in the wake 0 -1

(a) DB = 50 mm Dp = 3.2 mm ds = 321 µm qm0 = -5.95E-5 C/kg,

-2 -3

K′ = -3.66E+2 Ωm17/5s3/5 A nwake=-0.5

-4 -5 5

term a term b term c total

(b)

0 -5 -10 -15 0.0

0.5

1.0

1.5

2.0

Time, s

Fig. 10. Simulated (a) induced charge; (b) charge transferred versus time as a single bubble with the wake passes the probe.

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110

1

Total charge, 10-8 C

0

DB = 50 mm DP = 3.2 mm ds = 321 µm

-1

-2 model -8 2 Anwake -qm0 , 10 C/kg -k ′, 10 Ωm17/5 s3/5 -0.5 5.95 3.66 0.0 6.54 2.34 0.5 7.06 1.67 experiment

-3

-4

-5 0.0

0.5

1.0

1.5

2.0

Time, s Fig. 11. Simulated total charge versus time as a single bubble with wake passes the probe.

has the same sign as the charge in the bed. Overall, the inclusion of the contribution from the bubble wake is still insufficient to predict the region around the maximum. 2.3. Effect of bubble drift As shown above, the model with a higher charge density distribution around the bubble or a wake behind the bubble cannot predict the peak region of the measured charge vs. time curve which occurs after a bubble passes the probe. The contribution from bubble-induced solids drift drawn upwards behind the bubble is considered next. Fig. 12 shows the charge distribution around the bubble and in the drift. It is assumed that the charge density around the bubble remains uniform while the drift is approximated as being of cylindrical shape with different charge density. Particles in the drift volume are assumed to have a velocity of Vs ¼ 0:38UB [20]. Again as shown in Fig. 12, the charge distribution is estimated by superposition of three simple contributions: (a) a uniformly charged bed with a specific charge qm0 ; (b) a charged circular cylindrical shape with a charge density equal to qm0 ; and (c) a charged cylindrical column with charge density equal to ðAndrift  1Þqm0 : The simulation was carried out for these three additive components based on the equations presented above, except that Eq. (4) is modified to account for the drift geometry shown in Fig. 3. 2.3.1. Induced charge Part a: uniformly charged bed. Eq. (15) is used. Part b: charged spherical ball. Eq. (19) is used.

A. Chen et al. / Journal of Electrostatics 58 (2003) 91–115

111

Fig. 12. Representation specific charge distribution including drift due to three additive components. Note that qm ¼ q0m þ q00m þ q000 m:

Part c: charged drift. For point B in the drift, according to Fig. 3, Eq. (4) is adjusted to ðx2 ; y2 ; z2 Þ ¼ ðrd cos a; y2 ; rd sin aÞ

ð54Þ

and Eq. (14) is integrated over the volume of the drift. In the drift: qv ¼ ðAndrift  1Þqm0 ð1  eÞrs

ð55Þ

DV ¼ rd da drd dy2 :

ð56Þ

and

Therefore, c000 total ¼

Z

Rd

Z

yo þhd

Z

2p

Andrift  1 qm0 ð1  eÞrs rd 4pPr   1 qr1 c qr2 þ 2  2 da dy2 drd ; r1 qr r2 qr 0

y0

0

where y0 ¼ L þ UB t  hd  ðR2B  r2d Þ0:5 :

ð57Þ

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112

The total volume charge density, Eq. (57), is then integrated over the surface of the electrostatic ball probe to give Q000 induced

Z Z Z Z Z ðAndrift  1Þqm0 ð1  eÞrs R2P Rd 2p p=2 y0 þhd 2p ¼ rd cos y 4pPr 0 0 p=2 y0 0   1 qr1 c qr2  2 þ da dy2 dy df drd : r1 qr r22 qr

ð58Þ

2.3.2. Charge transferred Part a: uniformly charged bed. Before the bubble reaches the probe, Eq. (42a) is employed.While the bubble encloses the probe, Eq. (42b) is used. After the bubble has passed the probe, but drift encloses the probe, the particle velocity in the drift is Vs ¼ 0:38UB : Therefore,   L þ RB t Q0transferred ¼ 0:3679K 0 qm0 hePs =Pr rs ð0:38UB Þ1:6 ð1  eÞ UB  0:7358K 0 qm0 hePs =Pr rs UB0:6 ð1  eÞRB þ 0:1672K 0 qm0 hePs =Pr rs ð1  eÞðUB R2B Þ1:6 =UB ½L2:2  R2:2 B : ð59aÞ After the drift cylinder has passed the probe Q0transferred ¼ 0:1672K 0 qm hePs =Pr rs ð1  eÞðUB R2B Þ1:6 =UB ½ðUB t  LÞ2:2  ðRB þ hd Þ2:2  0:3679K 0 qm0 hePs =Pr rs ð0:38UB Þ1:6 ð1  eÞ



hd UB



 0:7358K 0 qm0 hePs =Pr rs UB0:6 ð1  eÞRB þ 0:1672K 0 qm0 hePs =Pr rs ð1  eÞðUB R2B Þ1:6 =UB ½L2:2  R2:2 B : ð59bÞ Part b: charged spherical ball. Eqs. (44a)–(44c) are used here. Part c: charged drift. Before the drift reaches the probe Q000 transferred ¼ 0:

ð60aÞ

While the drift cylinder is enclosing the probe   L þ RB 1:6 0 ¼ 0:3679K ðA  1Þq h =P r ð0:38U Þ ð1  eÞ  t : Q000 ndrift m0 ePs r s B transferred UB ð60bÞ

A. Chen et al. / Journal of Electrostatics 58 (2003) 91–115

113

After the drift cylinder has passed the probe Q000 transferred

 hd ¼ 0:3679K ðAndrift  1Þqm0 hePs =Pr rs ð0:38UB Þ ð1  eÞ : ð60cÞ UB 0

1:6



Charge transferred, 10-10 C Induced charge,10-8 C

2.3.3. Simulation results and comparison with experimental data Fig. 13a shows simulated induced charge versus time as a single bubble with drift passes the probe. It can be seen that the drift has little influence on the region at the front of the bubble, with both the value and position of minimum induced charge remaining essentially the same. However, the charge in the drift with Andrift ¼ 1:26 contributes significantly to the maximum induced charge after the bubble passes the probe. Fig. 13b presents the three components which contribute to the total charge transfer. Fig. 14 compares the experimental results with the model simulation, with parameters qm0 ; K 0 and Andrift obtained by matching experimental values of Cmin ; Ctransfer and Cmax : It is seen that the prediction is improved significantly in the region around the maximum peak when the contribution from the bubble drift is considered. Unlike previous results without considering the effect of bubble drift as shown in Figs. 7 and 11, the total charge shown in Fig. 14 is predicted to reach a maximum positive value after the bubble passes through the probe, consistent with the experimental results, Cmax : The simulated maximum peak shifts to the right with increasing length of the drift cylinder, while the drift has little influence on the position of the minimum peak. Simulation with hd =RB ¼ 6 gives a good match with

2

(a)

DB = 50 mm DP = 3.2 mm ds = 321µm qm0 = -7.13E-5 C/kg,

0 -2

K′= -1.24E+2 Ω m17/5 s3/5 A ndrift = 1.26 hd = 4RB Dd = 0.5DB

-4

(b) 5

term a term b term c total

0 -5 -10 -15 0.0

0.5

1.0

1.5

2.0

Time, s

Fig. 13. Simulated (a) induced charge; (b) charge transferred versus time as a bubble with drift passes the probe.

A. Chen et al. / Journal of Electrostatics 58 (2003) 91–115

114

1

Charge, 10-8 C

0

-1

-2

DB = 50 mm -3 DP = 3.2 mm

-4

ds = 321µm Dd = 0.5DB

bubble-wake interface

nose -5 0.0

experiment model -5 3/5 hd /RB A ndrift - qm010 C/kg -K′ 10 2 Ω m17/5s 0 6.54 2.34 4 1.26 7.13 1.24 6 0.81 7.00 1.00 8 0.59 6.92 0.83

0.5

1.0

1.5

2.0

Time, s Fig. 14. Simulated total charge versus time as a bubble with drift passes the probe.

respect to the location of the peak. The more gradual change of the experimental curve following the bubble passage suggests that the contribution from the particle motion in the drift and nearby region induced by the bubble motion cannot be simply represented as a cylindrical drift region of limited length and diameter. Discrepancies between the predictions and experimental measurements after the bubble passes the probe may thus be due to the assumption in the model that the drift has a cylindrical shape with a uniform charge density. In practice, the drift region is known to exist as a sharp figure shape [12]. Numerical method of integration to solve the model may also account for the oscillation shown in Fig. 14 for the model.

3. Conclusions The model of Park et al. [1] has been modified by (a) applying a specific charge density to fluidized particles, with a higher charge density layer around the bubble; (b) including the effect of the bubble wake and drift. The effect of the capacitor in the electric circuit used in the experiment of Park et al. [1] is taken into account to estimate the total charges induced and transferred to the probe. The corrected experimental data were in good agreement with the model prediction in the region before the bubble passes the probe. The higher charge density particle layer around the bubble has some effect on the total charge induction and transfer to the ball probe. With a higher charge density around the bubble, both the specific charge of bed particles and charge transfer coefficient, qm0 and K 0 ; fitted from experimental data, are lower. For a thicker highcharge-density particle layer around the bubble, the fitted qm0 is higher while the

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fitted K 0 is lower. Bubble wake and drift has little influence on the region ahead of the wake and the drift. However, the particle charge in the drift region contributes significantly to the region around the maximum. To fully understand the charge transfer behaviour in gas–solids bubbling fluidized beds, the distribution of charges in regions around and behind the bubble needs to be measured.

Acknowledgements Financial assistance from Japan Polychem Inc. and the Natural Sciences and Engineering Research Council of Canada (NSERC) is acknowledged with gratitude.

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