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Journal of Electrostatics 71 (2013) 867e874

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Journal of Electrostatics

journal homepage: www.elsevier .com/locate/elstat

Numerical simulation of the continuous operation of a tribo-aero-electrostatic separator for mixed granular solids

Fatima Rahou, Amar Tilmatine, Mihai Bilici, Lucian Dascalescu*

PPRIME Institute, UMP 3346, CNRS, University of Poitiers, ENSMA, EHD Group IUT, 4 Avenue de Varsovie, 16021 Angoulême, France

a r t i c l e i n f o

Article history:Received 23 September 2012Accepted 17 June 2013Available online 29 June 2013

Index terms:Electrostatic separationGranular materialsNumerical simulationTriboelectricity

* Corresponding author. Tel.: þ33 545673245.E-mail addresses: lucian.dascalescu@univ-poitie

(L. Dascalescu).

0304-3886/$ e see front matter � 2013 Elsevier B.V.http://dx.doi.org/10.1016/j.elstat.2013.06.004

a b s t r a c t

Numerical simulation has proved to be a powerful tool in the research and development of new elec-trostatic processes. In a previous paper, the authors have introduced a simple mathematical model forsimulating the outcome of a novel tribo-aero-electrostatic separation process for binary mixtures ofgranular materials. The mathematical model assumed that the probability of a granule to be separatedcan be expressed as a function of the number of impacts with granules belonging to the other class ofmaterials. The process is characterized by the fact that the charging of the granules is produced in afluidized bed device, in the presence of an electric field. The aim of the present paper is to simulate thecontinuous operation of such a device at various feed rates. The evolution in time of the mass of granulescollected at the electrodes has been computed for various compositions of the granular mixture. Theeffect of the walls was taken into account. The computed results were in good agreement with theexperiments. They demonstrate that open-loop continuous operation of the separator is possible for arange of feed-rates that depends on the composition of the materials to be separated.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Since the start of the industrial scale production of syntheticpolymers in the 1940s, the generation rate of plastic solid wasteshas increased considerably. The recycling of such wastes hasbecome a major environmental issue [1]. None of the currentavailable technologies of separating mixed plastics is entirelysatisfactory. This explains the considerable research effort that ismade for the development of novel dry processes that wouldenable the separation of at least part of these plastics, so that tocontribute toward a more effective use of primary resources [2,3].

Some of the existing electrostatic separation technologies formineral ores [4e6] have already found application in the separationof plastics. These technologies involve, as a first and most delicateoperation, the triboelectric charging of the constituents of thegranular mixture [7e9], followed by their separation in the elec-trostatic field generated by a system of high-voltage electrodes. Thepositively- and negatively-charged fractions are recovered indistinct compartments of a collector [10e12].

The wider use of these technologies is hampered by the non-homogeneity of the charge acquired by the granules: some do not

rs.fr, ldascalescu@gmail.com

All rights reserved.

carry enough charge to be separated by the electric field forces. Arecently-patented tribo-aero-electrostatic separation process formixed granular plastics [13,14] gives an original solution to thisproblem: the tribo-charging is produced in a parallelepiped fluid-ized bed device, in the presence of an electric field (Fig.1). This field,perpendicular to the direction of the fluidization air, is generated bytwo electrodes glued to opposite walls of the tribochargingchamber and energized from two DC high-voltage supplies ofopposite polarities. Thus, the granules in the fluidized bed cannotleave the tribocharging zone unless they are enough charged to beattracted to the electrodes.

Numerical simulation techniques already been employed toachieve the optimization of role-type electrostatic separators forthe recycling of metallic and insulating particles from cable wastes[15,16]. In that case, numerical models have been proposed forestimating the charge acquired by the particles and for calculatingtheir trajectories in an electric field [17e19]. The simulationspointed out the effects of the various process control variables:high-voltage, roll-speed, etc.

Tribo-charging is a much more complex phenomenon [20e23]and its mathematical modeling is still in progress [24e29]. Thecomputation of particle trajectories, by taking into account all themechanical, aerodynamic and electrical forces is a very complicatedand time-consuming task. Both researchers and practitioners needa more user-friendly simulation tool for performing the feasibilitystudies that precede the development of a new application.

Fig. 1. Schematic representation of the tribo-aero-electrostatic separator.

Table 1Main characteristics of the granular materials employed in this study.

Granule PA PC

Color Blue OrangeForm Quasi cylindrical Quasi cylindricalTypical size [mm] Ø 2.7 � 3.4 Ø 2.9 � 3.5Average mass [mg] 21 24

F. Rahou et al. / Journal of Electrostatics 71 (2013) 867e874868

In a previous work, a simple mathematical model for simulatingthe outcome of the tribo-aero-electrostatic separation process forbinary mixtures of granular plastics was developed [30]. However,the analysis was simplified by assuming that the process workswith samples of constant mass. Therefore, the aim of the presentwork is to develop a mathematical model for simulating acontinuously-operating industrial process, where the granularmixture to be separated is introduced in the fluidized bed at aconstant rate.

2. Experimental procedure

The tribo-electrostatic separation experiments were performedon blue virgin polyamide (PA) and orange polycarbonate (PC)granules, used in the plastics industry (Fig. 2 and Table 1). Thesamples were prepared as binary mixtures with different compo-sitions. The experiments were carried out at fixed air velocityv ¼ 6 m/s under relatively stable ambient conditions: temperatureT ¼ 17e22 �C, relative humidity RH ¼ 44e60%.

Fig. 2. Aspect and size of the polyamide (PA) and polycarbonate (PC) particlesemployed in this study.

The experimental device (Fig.1) consisted in a rectangular prismchamber (115 mm� 85 mm� 400mm), with two opposite verticalwalls made of polycarbonate (PC), the other two consisting inaluminum plates connected to two adjustable DC high-voltagesupplies of positive and negative polarity (model ES60P-20W andES60N-20W, Gamma HV Research Inc., Ormond Beach, FL). Thefluidization air is introduced through a perforated plate at thebottom of the chamber. The granules deposited on this plate aredispersed by the ascending air in the tribocharging chamber, wheremultiple granuleegranule and some granuleewall collisions takeplace [14]. The charged granules are attracted to the electrodes ofopposite polarity and fall into the two collecting hoppers. Theinstantaneous mass of the collected products is measured everyDt¼ 1 s with electronic balances (resolution: 0.01 g), connected to adata acquisition system [14].

3. Mathematical model

The study is focused on the separation of a binary mixture ofgranular materials, denoted A and B. The two classes of granuleshave similar size and mass density, but different tribochargingcharacteristics.

The mathematical model is based on the assumption that theprobability of a granule to be separated can be expressed as afunction of the number of impacts with granules belonging to theother class of materials and with the walls of the fluidized bed. Thenumber of such collisions depends on the concentration of eachclass of materials in the granular mixture.

The effect of the granules-to-wall collisions is similar to thepresence of a third type of granules in the fluidized bed. Thus, inorder to take into account both type of collisions, the total massM(t) of the materials processed at an instant t is expressed as :

M(t) ¼ MA(t) þ MB(t) þ MW (1)

where MA(t) and MB(t) are the masses of the two classes of gran-ules; MW is the fictitious mass of the walls, which is constant intime. Under these circumstances, the respective concentrations ofthe materials A, B and W are:

cA(t) ¼ MA(t)/M(t), cB(t) ¼ MB(t)/M(t), cW ¼ MW/M(t) (2)

In a fluidized bed of known geometry and air velocity, eachgranule experiences N(t) collisions per unit time. For the initialcomposition of thematerials in the fluidized bed, at t¼ 0, N(0)¼N1.The charge exchange in each granule-to-wall collision is differentthan in the case of a contact between two granules, as it depends onthe relative position of thewall materialWand of the twomaterialsA and B in the triboelectric series. If a unit of adimensional chargewere exchanged in a collision between two granules A and B, theadimensional charge exchanged between a granule A (or B) and thewall would be lA (respectively lB). There are several situations thatcan be encountered in practice:

(i) W is between A and B in the triboelectric series, then0 � lA � 1, 0 � lB � 1 (the granule-to-wall collisions have lesseffect than those between two granules);

F. Rahou et al. / Journal of Electrostatics 71 (2013) 867e874 869

(ii) B is between A andW, then lA > 1 (an A granule charges betterin a collisionwith thewallW thanwith a B granule), and lB< 0(the charge B exchanges with the wall has an opposite signthan that exchanged with A);

(iii) A is between B and W, then lB > 1 (a B granule charges betterin a collision with the wall W than with an A granule), andlA < 0 (the charge A exchanges with the wall has an oppositesign than that exchanged with B);

Thus, during a time period t, the number of unit adimensionalcharges exchanged by an A granule is:

XAðtÞ ¼Zt

0

½cBðtÞ þ lAcWðtÞ�NðtÞdt (3a)

Similarly, the number of unit adimensional charges exchangedby each B granule can be expressed as:

XBðtÞ ¼Zt

0

½cAðtÞ þ lBcWðtÞ�NðtÞdt (3b)

Let P(XA) ¼ p(xA), with xA ¼ XA/N1, be the probability for an Agranule to be collected at the electrode after exchanging XA(t) unitadimensional charges, under well-defined operating conditions (na-ture and size of the two classes of granules, geometry of the fluidizedbed,fluidizedair pressure, high-voltage applied to the electrodes, etc).The probability P(XA) can be assumed to be given by Gauss’s law

P(XA) ¼ P((XA � XAav)/sxA) (4)

where P is the standard normal distribution function, XAav desig-nates the average value and sxA the standard deviation. Thisexpression can be reformulated as follows:

P(XA) ¼ P((xA � xAav)/sxA) ¼ p(xA) (5)

where: xA ¼ XA/N1; xAav ¼ XAav/N1; sxA ¼ sxA/N1. Similar formulascan be written for a B granule, and the masses of materials sepa-rated up to any instant t are:

MAsðtÞ ¼Zt

0

PðXAðtÞÞMAðtÞdt ¼Zt

0

pðxAðtÞÞMAðtÞdt (6a)

MBsðtÞ ¼Zt

0

PðXBðtÞÞMBðtÞdt ¼Zt

0

pðxBðtÞÞMBðtÞdt (6b)

4. Simulation algorithm

An iterative algorithm can be employed for obtaining the esti-mation XAe(i, j) of the number of unit adimensional chargesexchanged at step j by the granules introduced in the fluidized bedat step i, the estimation MAse(j) of the separated mass MAs(t), att ¼ jDt, where i and j are positive integer and Dt is sufficiently smallfor the mass MA(t), the concentration cA(t) and the collision fre-quency N(t) to be considered constant:

MAe(k) ¼ MA((k � 1)$Dt), (7a)

cAe(k) ¼ cA((k � 1))$Dt), (7b)

Ne(k) ¼ N((k � 1)$Dt) (7c)

in any interval (k � 1)Dt � t < kDt, k ¼ 1, 2,., j.Step 1: The estimated number of unit adimensional charges

exchanged by each A granule can be computed from (3a) as follows:

XAe(1,1) ¼ [cB(0) þ lAcW]N(0) (8)

with xAe(1,1) ¼ XAe(1,1)/N(0), the estimated mass MAse(1) of the Agranules separated at Dt is obtained from (6a):

MAse(1) ¼ p(xAx(1,1)) MAe(1) ¼ p(xAe(1,1)) mAe(1,1) (9)

Similarly:

XBe(1,1) ¼ [cA(0) þ lBcW]N(0) (10)

where xBe(1,1) ¼ XBe(1,1)/N(0), and the estimated mass MBse(1) of Bgranules separated at Dt is calculated as follows:

MBse(1) ¼ p(xBe(1,1)) MBe(1) ¼ p(xBe(1,1)) mBe(1,1) (11)

where mAe(1,1) ¼MA(0) andmBe(1,1) ¼MB(0) are the initial massesat t ¼ 0. Remaining masses after the first step are calculated fromthe initial mass:

mAe(1,2) ¼ mAe(1,1) � p(xAe(1,1))mAe(1,1) (12a)

mBe(1,2) ¼ mBe(1,1) � p(xBe(1,1))mBe(1,1) (12b)

In the above formulas, mAe(1,2) and mBe(1,2) are the particlesthat have experienced collisions in the first step and are going toundergo further collisions during the second iteration. Therefore,their probability of separation is greater compared to the newparticles that will be introduced into the fluidized bed at the nextstep.

Step j (j � 2): Let m(j,j) be the mass m added at each step j:

me(j,j) ¼ m; mAe(j,j) ¼ mA; mBe(j,j) ¼ mB (13)

where mAe(j,j) and mBe(j,j) are the masses of the new granulesintroduced into the fluidized bed. The total mass of A granules inthe fluidized bed at step j can be expressed as:

MAeðjÞ ¼Xj

i¼1

mAeði; jÞ (14)

A similar formula can be written for MBe(j). The total estimatedmass at step j is:

Me(j) ¼ MAe(j) þ MBe(j) þ MW (15)

The masses mAe(i,j) and mBe(i,j) that entered at step i and stillpresent in the fluidized bed at the beginning of step j are calculatedseparately:

mAe(i,j) ¼ mAe(i,j � 1) � mAe(i,j � 1)*p(xAe(i,j � 1) (16a)

mBe(i,j) ¼ mBe(i,j � 1) � mBe(i,j � 1)*p(xBe(i,j � 1) (16b)

The concentrations cAe(j) and cAe(j) of the A and B granules canbe determined from (2), using the estimatesMAe(j) andMBe(j) of themasses of the two materials in the fluidized bed:

cAe(j) ¼ MAe(j)/M(j) (17a)

cBe(j) ¼ MBe(j)/M(j) (17b)

0 20 40 60 80 1000

2

4

6

8

10

12

Time [s]

Mas

s [g

]

Delta MAse

Delta MBse

(a)

0 20 40 60 80 1000

50

100

150

200

250

Mas

s [g

]

Time [s]

MAse

MBse

(b)

0 20 40 60 80 10050

100

150

200

Time [s]

Mas

s [g

]

M

(c)

Fig. 3. Separated mass DMAse and DMBse (a) total masses MAse and MBse (b), and themass in the fluidized bed M ¼ M(j) e MW (c), estimated for the case of a 50% A þ 50% Bgranular mixture, lA ¼ 0.8; lB ¼ 0.2, the initial mass and the feed rate being respec-tively M1 ¼ 180 g and m (j,j) ¼ 3 g/s.

F. Rahou et al. / Journal of Electrostatics 71 (2013) 867e874870

Let:

xAe(i,j) ¼ XAe(i,j)/N(0), xBe(i,j) ¼ XBe(i,j)/N(0) (18)

With these notations, the normalized unit adimensional chargesexchanged by the A and B granules of the initial masses mAe(1,1)and mBe(1,1) in the fluidized bed up to the instant t ¼ jDt can beexpressed as follows:

xAe(1,j) ¼ {[cBe(1)þlAcWe(1)] þ [cBe(2) þ lAcWe(2)]Me(2)/Me(1). þ [cBe(j) þ lAcWe(j)]Me(j)/Me(1)}Dt (19a)

xBe(1,j) ¼ {[cAe(1)þlBcWe(1)] þ [cAe(2)þlAcWe(2)]Me(2)/Me(1). þ [cAe(j) þ lBcWe(j)]Me(j)/Me(1)}Dt (19b)

The new particles introduced at each time step Dt have thepossibility to get charged starting with step j:

xAe(j,j) ¼ {[cBe(j) þ lAcWe(j)]Me(j)/Me(1)}Dt (20a)

xBe(j,j) ¼ {[cAe(j) þ lBcWe(j)]Me(j)/Me(1)}Dt (20b)

More generally:

xAe(i,j) ¼ {[cBe(i) þ lAcWe(i)]Me(i)/Me(1) þ [cBe(i þ 1) þ lAcWe(i þ 1)]Me(i þ 1)/Me(1) þ .[cBe(j) þ lAcW(j)]Me(j)/Me(1)}Dt (21a)

xBe(i,j) ¼ {[cAe(i) þ lBcWe(i)]Me(i)/Me(1) þ [cAe(i þ 1) þ lBcWe(i þ 1)]Me(i þ 1)/Me(1) þ .[cAe(j) þ lBcW(j)]Me(j)/Me(1)}Dt (21b)

Consequently, the estimated masses of A and B granules sepa-rated at a step j is obtained from (6a):

DMAse(j) ¼ p(xAe(1,j))mAe(1,j) þ p(xAe(2,j))mAe(2,j) þ . þ p(xAe(i,j))mA(i,j) (22a)

DMBse(j) ¼ p(xBe(1,j))mBe(1,j) þ p(xBe(2,j))mBe(2,j) þ . þ p(xBe(i,j))mB(i,j) (22b)

5. Numerical simulation results

The calculations were carried out with a program written inMATLAB 7.0. The iteration step was Dt ¼ 1 s, the initial mass wastaken M1 ¼ 180 g (with MW ¼ 20 g) and the mass m(j,j) intro-duced at each step was assumed to be 3 g/s, 6 g/s or 9 g/s. Thevalues of xAav, sxA, xBav, sxB were chosen following the procedureand based on the experimental data presented in a previouspaper [30].

In order to simulate the effects of the nature of thewallWon theresults of the separation, two situations were considered:

(i) lA ¼ 0.8; lB ¼ 0.2, which means that W is between A and B inthe triboelectric series, with the granule-to-wall collisionshaving less effect than those between two granules;

(ii) lA¼ 1.2; lB¼�0.2, whichmeans that B is between A andW, sothat an A granule charges better in a collision with the wall Wthanwith a B granule, and the charge B exchangeswith thewallW has an opposite sign than that exchangedwith an A granule.

In a first series of simulations, the masses DMAse(j) and DMBse(j)separated at each step j were estimated with (22a) and (22b) andrepresented in Figs. 3a and 4a, for the two cases, considering thatthe granular mixture is composed of 50% Ad þ 50% B (i.e.,cA(0) ¼ cB(0)) and the feed-rate m(j,j) ¼ 3 g/s. The slight differencebetween the separated masses of A and B in the first case (Fig. 3b) is

due to the different effect of the granule-to-wall collisions (strongerfor A than for B, as lA ¼ 0.8 > lB ¼ 0.2).

After a transient regime that lasts less than 60 s, the processattains a stable operation: the estimated values of the massesseparated in any unit of time are equal to the feed rate, and the

0 20 40 60 80 1000

2

4

6

8

10

12

Time[s]

Mas

s [g

]

Delta MAse

Delta MBse

(a)

0 20 40 60 80 1000

50

100

150

200

250

Mas

s [g

]

Time [s]

MAse

MBse

(b)

0 20 40 60 80 10060

80

100

120

140

160

180

200

Time [s]

Mas

s [g

]

M

(c)

Fig. 4. Separated mass DMAse and DMBse (a) total masses MAse and MBse (b), and themass in the fluidized bed M ¼ M(j) e MW (c), estimated for the case of a 50% A þ 50% Bgranular mixture, lA ¼ 1.2; lB ¼ �0.2, the initial mass and the feed rate beingrespectively M1 ¼ 180 g and m (j,j) ¼ 3 g/s.

0 20 40 60 80 1000

2

4

6

8

10

12

14

Time[s]M

ass

[g]

Delta MAse

Delta MBse

0 20 40 60 80 1000

50

100

150

200

250

300

350

Mas

s [g

]

Time [s]

MAse

MBse

0 20 40 60 80 10080

100

120

140

160

180

200

220

Time [s]

Mas

s [g

]

M

(a)

(b)

(c)

Fig. 5. Separated mass DMAse and DMBse (a) total masses MAse and MBse (b), and themass in the fluidized bed M ¼ M(j) e MW (c), estimated for the case of a 50% A þ 50% Bgranular mixture, lA ¼ 0.8; lB ¼ 0.2, the initial mass and the feed rate being respec-tively M1 ¼ 180 g and m (j,j) ¼ 6 g/s.

F. Rahou et al. / Journal of Electrostatics 71 (2013) 867e874 871

mass of the material left in the fluidized bed is constant and equalwith 80 g (Fig. 3c), which is the minimum quantity which ensuresthe operation of the experimental device employed for the vali-dation of the numerical model.

The transient regime is slightly longer for the second case(Fig. 4b). During the first 15 s, the quantity of A separated at each

step were much larger than those of B, as the granule-to-wall col-lisions favor the charging of A (lA¼ 1.2), and have an opposite effecton B (lB ¼ �0.2). The stable operation is not yet attained after 100 s(Fig. 4c), but in more than 3 min.

An experiment carried out in conditions similar to thoseconsidered for the simulation do not totally confirmed these pre-dictions. As a matter of fact, the transitory regime is longer (roughly

F. Rahou et al. / Journal of Electrostatics 71 (2013) 867e874872

5 min), because after less than 30 s the mass remained in the flu-idized bed is less than 80 g, which is not enough for the operation ofthe fluidized bed, and the quantities of separated materials willthen fluctuate several times before stabilizing at values equal to thefeed-rate. From this, it can be concluded that the operation of theseparator at low feed-rate should be avoided.

0 20 40 60 80 1000

5

10

15

Time[s]

Mas

s [g

]

Delta MAse

Delta MBse

0 20 40 60 80 1000

50

100

150

200

250

300

350

Mas

s [g

]

Time [s]

MAse

MBse

0 20 40 60 80 100100

120

140

160

180

200

220

Time [s]

Mas

s [g

]

M

(a)

(b)

(c)

Fig. 6. Separated mass DMAse and DMBse (a) total masses MAse and MBse (b), and themass in the fluidized bed M ¼ M(j) e MW (c), estimated for the case of a 50% A þ 50% Bgranular mixture, lA ¼ 1.2; lB ¼ �0.2, the initial mass and the feed rate beingrespectively M1 ¼ 180 g and m (j,j) ¼ 6 g/s.

Therefore, in a second series of simulations the feed-rate wasassumed to be m(j,j) ¼ 6 g/s. The results displayed in Figs. 5 and 6are in many respects similar to those obtained at lower feed-rate.It seems that the separator attains faster the stabilized operation(50 s, in the case lA ¼ 0.8> lB¼ 0.2), i.e. the estimated values of themasses of granules A and B that are collected in any unit of time are

0 20 40 60 80 1000

5

10

15

Time[s]M

ass

[g]

Delta MAse

Delta MBse

0 20 40 60 80 1000

50

100

150

200

250

300

350

400

450

500

Mas

s [g

]

Time [s]

MAse

MBse

0 20 40 60 80 100120

140

160

180

200

220

240

Time [s]

Mas

s [g

]

M

(a)

(b)

(c)

Fig. 7. Separated mass DMAse and DMBse (a) total masses MAse and MBse (b), and themass in the fluidized bed M ¼ M(j) e MW (c), estimated for the case of a 50% A þ 50% Bgranular mixture, lA ¼ 0.8; lB ¼ 0.2, the initial mass and the feed rate being respec-tively M1 ¼ 180 g and m (j,j) ¼ 9 g/s.

0 20 40 60 80 1000

2

4

6

8

10

12

Time[s]

Mas

s [g

]

Delta MAse

Delta MBse

0 20 40 60 80 1000

50

100

150

200

250

300

350

400

450

500

Mas

s [g

]

Time [s]

MAse

MBse

(a)

(b)

Fig. 8. Separated mass DMAse and DMBse (a) and total masses MAse and MBse (b),estimated for the case of a 30% A þ 70% B granular mixture, lA ¼ 0.6; lB ¼ 0.4, theinitial mass and the feed rate being respectively M1 ¼ 180 g and m (j,j) ¼ 9 g/s.

0 20 40 60 80 1000

2

4

6

8

10

12

Time[s]

Mas

s [g

]

Delta MAse

Delta MBse

0 20 40 60 80 1000

50

100

150

200

250

300

350

400

450M

ass

[g]

Time [s]

MAse

MBse

(a)

(b)

(c)

F. Rahou et al. / Journal of Electrostatics 71 (2013) 867e874 873

equal to the rate at which they are introduced in the separator (6 g/s). The mass in the fluidized bed never diminishes below 90 g, at-tains a maximum of less than 220 g, and stabilizes at values rangingfrom 120 g (Fig. 5c) to somewhat less than 140 g (Fig. 6c),depending on the tribocharging characteristics of the materialswith respect to the wall.

Fig. 9. Experimentally recorded instantaneous values of the masses of PA (solid line)and PC(dotted line) granules collected at the two electrodes. for a granular mixture:30% PA þ 70% PC.

These predictions have been confirmed qualitatively by anexperiment conducted in conditions similar to those considered forthe simulation. After 2 min of operation, the masses of separatedgranules A and Bwere respectively 408 g and 362 g, with a quantityof 126 g still in the fluidized bed (several grams were lost duringoperation due to the poor sealing of the fluidized bed chamber).

0 20 40 60 80 10080

100

120

140

160

180

200

Time [s]

Mas

s [g

]

M

Fig. 10. Separated mass DMAse and DMBse (a) total masses MAse and MBse (b), and themass in the fluidized bed M ¼ M(j) � MW (c), estimated for the case of a 50% A þ 50% Bgranular mixture, lA ¼ 0.8; lB ¼ 0.2, the initial mass M1 ¼ 180 g, the feed-rate being m(j,j) ¼ 3 g/s for the first 15 s, then m (j,j) ¼ 9 g/s.

F. Rahou et al. / Journal of Electrostatics 71 (2013) 867e874874

In the third series of simulations the feed-rate was assumed tobe m(j,j) ¼ 9 g/s. The results displayed in Fig. 7 point out that at ahigher feed-rate the influence of the wall is less important and thatthe stabilized regime is likely to be attained faster. However, themass in the fluidized bed is expected to surpass the threshold of220 g that was experimentally determined as the upper limit forthe optimal operation of the separator. Having a larger mass ofgranules in the separation chamber may disturb the formation ofthe fluidized bed and induce instabilities that would require thestop of the machine. An experiment conducted at this high feed-rate had to be stopped after less than 30 s as the air-flow was notenough to keep the granules in fluidized state.

The fourth set of simulations were done for the case of a 30%Aþ 70% B granular mixture, lA ¼ 0.6; lB¼ 0.4, the feed rate beingm(j,j) ¼ 6 g/s (Fig. 8). In the case of the 30% A þ 70% B granularmixtures, the stable operation is attained after a longer transitoryregime: about 3 min (Fig. 4). At first, the A granules, which are inminority, are separated at a much faster rate than the B granules, asthey have more opportunities to exchange charges by collisionswith bodies of different nature. However, in less than 20 s theoutput rate of A granules slows down to a value that slightly os-cillates around the value of their feed rate (i.e., 6 [g/s] � (30/100) ¼ 1.8 [g/s]).

The output rate of B granules slowly increases during the first30 s to attain a maximum close to 5 g/s, then e after a couple ofoscillations of smaller amplitude e stabilizes at a value imposed bythe feed rate (i.e., 6 [g/s] � (70/100) ¼ 4.2 [g/s]). The balance be-tween the input and output rates guarantees that the mass ofmaterial in the fluidized bed is maintained quasi-constant. This is avery important practical conclusion: no feed-back is necessary tokeep the separation process under control.

These predictions are confirmed by the experimental resultsgiven in Fig. 9. The curves recorded by the virtual instrument aresimilar to those obtained by numerical simulation. The minor dis-crepancies that can be detected between the experimental andsimulated curves are due to the fact that part of the parametersemployed for the numerical computations were obtained from aslightly different experiment described elsewhere [30].

In a final set of simulations, the feed-rate was assumed to bem (j,j) ¼ 3 g/s for the first 15 s, then increased to m (j,j) ¼ 9 g/s.Using this control strategy, the stable operation of the separatorwas attained in about 1 min (Fig. 10a and b), and the mass of thegranules in the fluidized bed was maintained between the limitssuggested as acceptable by the experiments, i.e. between 100 gand 200 g. Using a linear variation of the feed-rate from lower tohigher values during the first 10e20 s of operation might lead toeven better results. Various other control strategies could betested using the simulation program before their physical imple-mentation.

6. Conclusions

(1) The continuous operation of a tribo-aero-electrostatic sepa-rator can be accurately predicted by numerical simulation,based on a simple mathematical model

(2) The operation of the separator depends on the composition ofthe mixture. The granules that are in minority get chargedfaster and are easily separated from the mixture, while themajority granules have to spend a longer time in the fluidizedbed prior to being collected at the electrodes.

(3) However, no matter what is the composition of the granularmixture, no feed-back is necessary to maintain the processunder control, as the stabilized operation is easily attained inopen-loop operation: after a short transitory regime, theoutput rate becomes equal to the input rate.

Acknowledgment

The authors gratefully acknowledge fruitful discussions withDr.-Ing. Smaïl Bachir on the numerical model. One of the authors(F.R.) benefited of a research scholarship from behalf of the AlgerianGovernment for the completion of a PhD thesis at the University ofPoitiers, France.

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