Numerical study of the influence of the powder and pipe properties on electrical charging during pneumatic conveying

Powder Technology 315 (2017) 227–235

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Numerical study of the influence of the powder and pipe properties on electrical charging during pneumatic conveying Holger Grosshans* , Miltiadis V. Papalexandris Institute of Mechanics, Materials and Civil Engineering, Université catholique de Louvain, Louvain-la-Neuve 1348, Belgium

A R T I C L E

I N F O

Article history: Received 22 December 2016 Received in revised form 21 March 2017 Accepted 1 April 2017 Available online 6 April 2017 Keywords: Particle-laden flows Multiphase flows Explosion safety Large Eddy Simulation (LES) Triboelectricity

A B S T R A C T During pneumatic transport, powders often experience the build-up of electrostatic charge due to collisions of the particles with the pipe. Since this can lead to hazardous spark discharges with unwanted implications, such as accidental explosions, there is a strong interest in exploring various options to limit this electrification process. In this paper we present the results of numerical simulations that we performed in order to evaluate the influence of the material properties of the powder and pipe, respectively, on this process. In our study, the turbulent flow of the carrier gaseous phase was treated numerically via Large Eddy Simulations while the particles were tracked individually in Lagrangian framework. Four-way coupling between the powder and the gaseous phase was also assumed. Our numerical simulations predicted that the particle Poisson ratio and Young’s modulus, its electrical resistivity, as well as the permittivity represent promising measures to control the charge of the powder. More specifically, an increase of each of these quantities by 50% may lead to a decrease of the powder charge of up to 40%. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Powders often gain electrostatic charge during pneumatic transport, which can lead to hazardous spark discharges. In the past, such discharges have been the cause of numerous dust explosions. On the other hand, dust explosions are especially dangerous since the resulting heat radiation lasts much longer than in typical gaseous explosions [9]. For this reason, a significant research effort has been devoted to the understanding of the physics underlying the charging process of powders. Moreover, in modern technological applications, such as lithography and visual displays, there is a strong need to find methods to influence the charging process in a desired way. In the literature there are several experimental studies (e.g. [7,35,38]) as well as numerical investigations (e.g. [24,41,45]) that examined the effect of the conveying conditions on the powder electrification. Through that, consensus concerning the high sensitivity of the powder charge on their transport velocity was reached. The rapid development of the field of Computational Fluid Dynamics (CFD) made computer simulations a reliable tool to model this kind of complex flows. Also, in a recent study of ours, the charge exchange of a particle with a pipe wall and the subsequent electrostatic field

* Corresponding author. E-mail address: [email protected] (H. Grosshans).

http://dx.doi.org/10.1016/j.powtec.2017.04.012 0032-5910/© 2017 Elsevier B.V. All rights reserved.

was successfully predicted via numerical simulations [13]. In the same study, we also modeled the charge build-up of polymethyl methacrylate (PMMA) powder during pneumatic conveying. This was achieved by implementing dynamic models describing the charge exchange between particles and the pipe and in-between particles. The turbulent flow inside the pipe was treated numerically via the technique of Large Eddy Simulation (LES) according to which the large turbulent structures are directly resolved whereas the structures smaller than the grid resolutions are suitably modeled. The same methodology was employed in a subsequent study [12] to evaluate the influence of the design parameters of a pneumatic system on the powder charge. Therein, the conveying gas velocity was confirmed to be the most important factor to enhance the electrification process. On the other hand, the powder mass flow rate and the pipe diameter were predicted to have a minor effect. Nonetheless, there is a consensus that, besides these aforementioned design parameters, the mechanical and electrical properties of the particle and pipe materials can also significantly affect the charging process. However, up to now, only a few and sporadic investigations have been devoted to this issue. The influence of the type of material on the contact potential difference due to different work functions was already quantified in an early work by Harper [22]. His investigations involved a chromium sphere in contact with another metal sphere of a different kind. This study was later extended by Davies [4] and Murata and Kittaka [34] to charge

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exchanges during contact between a metal and an insulator. Later developments [1,25,27] took into account the dependency of the charge exchange between two insulators on the materials involved. Based on this information, Matsusaka et al. [31] proposed to maintain the charge of powder during pneumatic transport at a low level by combining pipes of different materials. To test their idea, they connected alternately one-meter long pieces of steal and brass pipes. With this configuration, the particles gained positive charge during collisions with the steal pipe but negative charge with the brass pipe. As a result, the specific charge of the powder remained within certain limits. Instead, in a subsequent study, Matsusaka et al. [30] considered a pipe made of combinations of two different materials. By doing so, the charge on the particles reached its limiting value without the fluctuations that necessarily appear when using pipes of different materials connected in series. The possibility of reducing the powder charge via the addition of anti-static powders, such as Larostat-519, was explored by Wang et al. [44]. Subsequently, Zhu et al. [50] reported that the charging of a non-conductive pipe is significantly reduced by adding a small amount (0.5% by weight) of this agent. They put forward that Larostat-519 forms a thin layer on the particle surface and on the pipe walls, resulting in a reduction of the contact potential. Interestingly, in an earlier study without this agent Zhu et al. [51] had observed a particular flow pattern which they referred to as annular capsule flow. However, in their subsequent study in which they added the agent, Zhu et al. [50] reported that this pattern disappears when the afore-mentioned agent is added. This suggests that powder charging has an influence of the flow patterns that are developed during pneumatic conveying. All of the above studies were focused on modifying the electrical properties of the system consisting of the two-phase mixture and the pipe walls. More recently, Sow et al. [40] measured the charge transfer between spheres of different materials and latex rubber sheets, which were subjected to mechanical strain. They observed that strain can reverse the direction of the charge transfer which they attributed to an alternation of the physico-chemical properties of the material surface. Furthermore, they reported a decrease of the transferred amount of charge when strain is applied. They attributed this behavior to the decrease in the material elasticity under strain. This produces a reduction to the contact area between a spherical particle and the sheet which, in turn, causes the decrease in charge transfer. In the literature there only a few studies available that deal with the influence of the electrical and mechanical properties of the powder and the pipe on the charge build-up process. While an exhaustive study of this subject is not available yet, the research presented herein aims to filling this gap by means of numerical investigations. More specifically, the mechanical properties under study are the Poisson ratio and Young’s modulus of both the pipe and the particle material. Further, the electric properties involve the resistivity of the particle and the permittivity of the system. The paper is organized as follows. The mathematical model and its validation are elaborated in Sections 2 and 3. Afterward (Section 4), the investigated properties are detailed. In the end, the results are discussed (Section 5) and final conclusions are given (Section 6).

2. Mathematical model According to our approach, the description of the flow of the carrier gas is based on the Navier-Stokes equations with constant diffusivities, whereas the motion of each particle is tracked individually according to the Discrete Element Method (DEM). The governing equations for the gaseous phase are solved in Eulerian framework whereas those of the powder are solved in a Lagrangian framework. The suitability of this approach to treat pneumatic conveying

processes has been demonstrated and discussed in detail by Zhu et al. [49] and Zhou et al. [48]. Further, as regards the motion of powder, we assume four-way coupling [6]; in other words, we take into account momentum exchange between the gas and the particles via aerodynamic drag and also momentum exchange due to particle collisions. The complete mathematical model used in the study is given in detail by Grosshans and Papalexandris [13] and is outlined below. Special attention of our description is paid on the presence of the material properties in the equations that describe the charge exchange process. By spatially filtering the Navier-Stokes equations, we obtain the equations that are used in LES. According to this technique, the largescale turbulent structures of the flow field, which have a leading effect on the particle dispersion, are directly resolved on the grid. The momentum fluxes caused by the action of the small, unresolved turbulent structures are modeled as additional stresses according to the Smagorinsky model. The Smagorinsky constant is calculated by the dynamic approach of Germano et al. [8] using the least-square technique and averaging in the streamwise direction as proposed by Lilly [26]. In order to reduce the requirements on the grid in the near-wall region the wall model by Grötzbach [20] is introduced. The governing equations for the gaseous phase are discretized via finite differences. As regards spatial discretization, the convective terms are approximated up to fifth-order and the diffusive and pressure terms up to fourth-order of accuracy. On the other hand, time integration of the governing equations is performed via an implicit second-order backward scheme. The reader is referred to Gullbrand et al. [21] for further details concerning the numerical implementation. Each individual particle is assumed to be isolated, rigid and spherical. The acceleration of each particle is given in the Lagrangian framework by dup = f ad + f el + f g + f col , dt

(1)

where fad , fel , fg and fcol denote the acceleration due to the aerodynamic, electric field, gravitational and collisional forces, respectively. The collision forces include both, inter-particle and particle-wall collisions. The acceleration of a particle due to the electric field forces, fel , is calculated by f el =

QE mp

(2)

where Q and mp are the charge and the mass of the particle, respectively. The electric field strength, E, is the gradient of the electric potential, 0, E = −∇0 .

(3)

This, in turn, is described by the Poisson equation, ∇20 = −

qel . e0

(4)

which is also solved numerically via finite differences. In the above equation e0 is the permittivity of the gaseous phase and its value is set at e0 = 8.854 • 10 −12 F m −1 . The electric charge density, denoted by qel , is calculated by taking into account the individual positions and charges of the present particles. Charge exchange, as well as momentum exchange, takes place when a particle collides either with the wall of the pipe or with another particle. With regard to momentum exchange, and for the sake of simplicity, we consider that binary particle collisions are fully elastic. In other words we assume that, during binary collisions, the

H. Grosshans, M.V. Papalexandris / Powder Technology 315 (2017) 227–235

dissipative forces that are developed are negligible with respect to the collisional forces that result from momentum exchange. Since the powder considered in this study consists of particles of the same material, charge exchange between colliding particles takes place only if the particles carry different pre-charges, Q n . As proposed by Soo [39], the calculation of the charge exchange is based on an analogy to the charging/decharging of a capacitor and is described in the following. For particles of the same material (i.e. identical work functions and resistivities) the charge exchanges, DQ1 and DQ2 , during particle-particle contact time Dtp are given by the following relation, DQ1 =

C1 C2 C1 + C 2



Q1 Q2 − C2 C1

  1 − e−Dtp /tp = −DQ 2 .

(5)

where C is the electrical capacity and tpw the charge relaxation time. Vc denotes the contact potential between the particle and the wall. It is worth mentioning that the value of the contact potential might actually exhibit some spatial and temporal variations [2]. These can be attributed to a number of factors such as surface roughness of particles and pipe walls, and inhomogeneities in the chemical composition. However, in the literature there is currently no decisive information on how these factors actually influence the contact potential. For this reason, and following Kolniak and Kuczynski [24], in our study we assumed that the contact potential was constant and equal to Vc = 1 V. This assumption is deemed reasonable for the purposes of our study because it enables to assess directly the effect of varying a number of important electrical and mechanical parameters of the powder and the pipe walls. The electric capacity of two parallel plates is given by

The electric capacity Cn of a spherical particle is determined as follows: C= Cn = 4pe0 rp,n ,

C1 C2 rp,1 + rp,2 vp , C1 + C 2 A12

(7)

where vp denotes the resistivity of the particle in the above relation. The contact surface A12 is calculated according to the elastic theory of Hertz as A12 =

p rp,1 rp,2 a1 , rp,1 + rp,2

(8)

Apw = prp a2



2/5  rp,1 + rp,2 1 − mp2 5 pqp (1 + ke )|up, 12 |2 3 . 3 8 Ep rp,1 + rp,2

(9)

In the above relation, up,12 is the relative velocity of the two particles. Also, ke , mp and Ep denote the restitution ratio, the Poisson ratio and the Young’s modulus of the particle, respectively. The contact time Dtp is also calculated by the theory of Hertz according to 2.94 a1 . |up, 12 |

(10)

Reflections of the particles off the pipe walls are assumed to be imperfectly elastic. Thus, the velocity component of the particle normal to the wall changes its sign and reduces due to restitution to up,n = −ke up,n [19]. Further, during contact, the pipe wall and the particle exchange electric charge. The calculation of this exchange is based on the model by John et al. [23]. According to it, the total exchange DQpw consists of a dynamic charge transfer, DQc , and a charge transfer attributed to the particle pre-charge, DQ t , DQ pw = DQ c + DQ t

(11)

Since the radii of the particle and the pipe are large compared to the contact area, the dynamic charge transfer is modeled in a manner analogous to the charging of a parallel-plate capacitor. Thus, the charge exchange, DQ c , during the wall-particle contact time Dtpw is given by   DQc = CVc 1 − e−Dtpw /tpw ,

(12)

(14)

with

a2 = rp

a1 = rp,1 rp,2

(13)

where Apw is the plates surface area. Following the argumentation of John et al. [23] and Kolniak and Kuczynski [24], their distance h is chosen to be 10−9 m. This is assumed to be of the order of the range of repulsive molecular forces due to surface irregularities. The plates surface corresponds to the contact area between the particle and the wall. According to the elastic theory of Hertz it is calculated by



with

Dtp =

e0 Apw h

(6)

and the charge relaxation time tp as tp =

229

5 pqp (1 + ke )|up |2 8



1 − mp2 Ep

+

2 1 − mw Ew

2/5 (15)

where mw and Ew are the Poisson ratio and Young’s modulus of the wall, respectively. The contact time in Eq. (12) is also calculated by the theory of Hertz as follows, 2.94 Dtpw = a2 . up

(16)

The charge relaxation time in Eq. (12) is determined according to John et al. [23] as follows, tpw = e e0 vp

(17)

where e is the relative permittivity of the system and vp is the resistivity of the particle. For the case of an insulating particle hitting a conductive target, the literature is not conclusive as to which permittivity should enter the expression of the relaxation time. For example, John et al. [23] provided an expression for tpw in terms of the dielectric constant but without specifying if this constant relates to the particles or the surface material. Further, Kolniak and Kuczynski [24], who also dealt with dielectric particles and a conductive pipe, assumed that tpw depends on the permittivity of the pipe. On the contrary, Masuda et al. [28] used the permittivity of the particle in their expression for tpw . However, the polarization of the metal is overshadowed by its own conductivity [33]. Therefore, in this case, the charging time of the dielectric particle depends on its own permittivity, i.e. the re-orientation of dipoles. Thus, despite the afore-mentioned ambiguities in the literature, we assume e to correspond to the particle permittivity.

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Finally, the charge of the particle is assumed to be on its surface and the pre-charge on the contact area is transferred, 1 a2 Qn . 4 rp

(18)

3. Model validation The two-phase flow solver described in the previous section has been validated and employed for the simulation of for various types of flows. These include diesel sprays [11,17,18], gas turbines [10], electric powder charging [12–15] and even the electric charging of a helicopter while hovering in dusty atmosphere [16]. The Eulerian-Lagrangian flow solver (excluding electrostatic effects) employed in the present study has been validated previously by Grosshans and Papalexandris [13] with the experimental data provided by Tsuji et al. [43]. In those experiments, spherical polysterene particles of a diameter of 200 lm were pneumatically transported in a vertical pipe of an inner diameter of 30.5 mm. A particle-air mass flow ratio of 0.5 and Re = 23000 were considered. The grid resolution studies documented in Grosshans and Papalexandris [13] showed that simulations with sufficiently fine grid resolution (computational cell size equal to Dh = 1.0 mm) yield accurate predictions as regards both the average and the fluctuating velocity components. Also, in these simulations the solid-to-gas volume ratio per cell was maintained sufficiently low so that the powder could be modeled by DEM [17,37]. The model described in Section 2 accounts for the interaction of the gaseous phase with the particle dynamics as well as with the electrostatic charging of the particle and the electric field. This is visualized for a simple test case, consisting of only one particle, in Fig. 1. The particle advances from its initial position in the pipe center to its boundary. Between the second and the third snapshot, the particle collides with the pipe. Consequently, the particle gains electrical charge. This, in turn, causes the build-up of a spherically symmetric electrical field surrounding the particle. The isosurfaces of the field are visible in the third and fourth snapshot. After this qualitative test, we shall briefly summarize the previously undertaken quantitative comparison of the charge exchange during these impacts predicted by our model with experiments [14]. The experiments by Matsuyama and Yamamoto [32] dealt with the collision between single PTFE particles and a brass plate. The particles had a diameter of 3.2 mm, an impact velocity of 11.4 m/s and carried each a different amount of initial charge prior to the impact. Fig. 2 shows that our model predicts well the resulting impact charge. The data points follow precisely the linear charging line which was postulated by Masui and Murata [29] and Yamamoto and Scarlett [46] for uniformly pre-charged particles. However, the experiments exhibit a scatter around the charging line. This can be explained either by fluctuating impact conditions or, more likely, by a non-uniform imposition of the pre-charge. Nonetheless, the scope of the present paper is not to investigate the charging of a single particle but that of a complete powder. The specific charge of the powder after leaving the pipe

Impact charge (pC)

DQ t =

30 0

Matsuyama & Yamamoto (1995) Present model

-30 -60 -90 -120 -1200

-800

-400 0 Initial charge (pC)

400

800

Fig. 2. Results of the single particle charging experiment by Matsuyama and Yamamoto [32] and the present model. The figure shows that the applied model [13] predicts well the dependence of the impact charge on the initial particle charge. Note that for a better visualization not all data points are shown. Source: Modified from Grosshans and Papalexandris [14]; © 2016, Elsevier.

that was predicted by the model was compared by Grosshans and Papalexandris [13] to experimental data by Watano et al. [45]. The results suggested that the dependency of the charge on the conveying air velocity is accurately computed. However, the measured absolute powder charge was higher than the calculated one which was attributed to a non-zero initial particle charge in the experiments.

4. Material properties and numerical set-up In our study we focused on the influence of the material properties of the powder and the pipe. Accordingly, these properties were let to vary while the design parameters of the pneumatic system were kept constant. The constant parameters included the length and the diameter of the circular pipe, the solid mass loading, the (uniform) particle size, density and restitution ratio. Furthermore, the conveying gas was assumed to be air at ambient temperature. All numerical values are given in Table 1. Our study focused on a group of mechanical and electrical material properties. The mechanical properties were the pipe’s and particle’s Poisson ratios and Young’s moduli (mw , Ew , mp and Ep ). The electrical properties were the relative permittivity and the resistivity of the particle (e and vp ). Upon inspection of the governing equations, it becomes apparent that these properties are not independent of each other. The charge build-up of the powder relates solely to the particle-wall transfer process which is described by Eqs. (11)–(18). A closer look at these equations reveals that the charge exchange depends in fact only on two properties. The first one is a combination of the mechanical properties represented by the last term in Eq. (15). The second one is the product of the electrical properties appearing in Eq. (17).

Fig. 1. Sequence of snapshots of the simulation of trajectory of one particle. During the collision at t = 0.02 s with the pipe wall charge is exchanged. This results in a build-up of an electrostatic field which is visualized here by blue isosurfaces. The maximum field strength is |E| = 3 • 10 −5 V/m.

H. Grosshans, M.V. Papalexandris / Powder Technology 315 (2017) 227–235 Table 1 Above the separation line the constant parameters in all simulations are given. The values marked with an asterisk correspond to the base case whereas these values were varied by ± 50% for the other cases. Value

Pipe length Pipe diameter Solid mass loading Particle diameter Particle restitution ratio Particle density Average air velocity Density of air Kinematic viscosity of air Vacuum permittivity Effective separation Contact potential Relative permittivity∗ Particle resistivity∗ Particle elasticity parameter∗ Pipe elasticity parameter∗

L=1m di = 40 mm ˙ s = 10 g/s m dp = 300 lm ke = 0.95 qp = 2160 kg m −3 [19] 30 m/s qg = 1.1 kg m −3 mg = 1.46 • 10 −5 m2 s −1 e0 = 8.854 • 10 −12 F m −1 h = 10 −9 m [23] Vc = 1 V [24] e = 5.0 [42] vp = 8.0 • 108 Y m cp = 2.8 • 10 −10 m s2 /kg [5,19] cw = 9.2 • 10 −12 m s2 /kg [5]

However, the state of the powder electrification is determined not only by the total charge but also by the charge distribution. In turn, the charge distribution depends on the inter-particle charge transfer, as described by Eqs. (5)–(10). Herein, a combination of mp and Ep appears as an independent property in Eq. (9). Moreover, vp is independent of e in Eq. (7). As a consequence of the above considerations, the influence of four independent properties on the charging process was explored, namely, vp , e and the elasticity parameters cp and cw . These are defined as follows. cp =

1 − mp2 Ep

and

cw =

2 1 − mw . Ew

(19)

First, a base case with properties that correspond to those of a steel pipe and PMMA powder was simulated (see Table 1). Afterward, we performed simulations with different values of these parameters; more specifically, each parameter was varied by ±50%. The results are discussed in the following section. Concerning the numerical set-up, the Eulerian equations were discretized on a grid consisting of uniform cells each of the size 2.5mm • 1.0mm • 1.0mm where the first value relates to the streamwise direction. Through this choice the gaseous phase is well resolved (cf. [13]) and at the same time the volume per cell occupied by the solid phase can be considered negligible. At the inlet a velocity profile was imposed which reflects the average profile of a turbulent flow [36,47] whereas convective outflow condition were considered at the outlet. 5. Results and discussion The particle distribution and their absolute charge are visualized in Fig. 3 for a given instance of time of the base case. It can be observed that the particles at the bottom of the pipe, which are the

3 base case ε - 50 % ε + 50 % γp - 50 % γp + 50 % γw - 50 % γw + 50 % ϕp - 50 % ϕp + 50 %

2.5 qav (μC/kg)

Parameter

231

2 1.5 1 0.5 0 0

5

10 15 20 cluster of 1000 particles

25

Fig. 4. Specific absolute powder charge (qav ) for all simulated cases over time. In order to present statistically representative quantities, we plotted the average charge of clusters of each 1000 particles leaving the pipe.

ones that collide most often with the wall because of gravity, gain the highest charge. Further, the charge of the particles at the exit of the pipe is higher; this is directly related to their longer residence times compared to the particles close to the inlet. In order to evaluate the influence of the particle and pipe properties on the powder charge, nine different cases were computed as discussed in Section 4. For each case, the resulting average particle charge, qav , leaving the pipe is plotted in Fig. 4. It is noted that since the average particle charge is given as specific quantity it is equivalent to the powder charge. Due to the uncertainty concerning the polarity of the surface potential it was decided to give qav in terms of an absolute value. In order to evaluate a statistically representative quantity, the average charge of each 1000 exiting particles is accounted for. For all considered cases, a low charge of the powder output at the beginning of the conveying process can be observed. However, after approximately 10 000 particles, the powder charge stabilizes and becomes statistically stationary. This was observed previously by Grosshans and Papalexandris [12] who attributed it to the solid mass loading inside the pipe. In the beginning, the loading is low and particles can cross the pipe while experiencing no or only few collisions. Later the loading is higher and, consequently, the inter-particle collision frequency increases. As a result, the particle trajectories get deflected which leads to more frequent particle-wall collisions. Since charge is transferred during those collisions, the powder electrification is intensified at later stages. In the present study we are particularly interested in the characteristics of the specific powder charge in the statistically stationary regime. In that respect and from Fig. 4 it can be concluded that cw does not significantly influence the charging process. In particular,an increase or decrease of its value by 50% does not change the powder charge when compared to the base case. Further, the small differences that can be observed in Fig. 4 relate to fluctuations that are inherent in turbulent flows. We can therefore conclude that, at least for the ranges of values considered herein, the mechanical properties of the pipe used for pneumatic conveying does not appear to be a efficient measure for the control of the powder charge.

Q n (pC) 0.1 0.05 0 0m

0.5 m

1m

Fig. 3. Snapshot of the pneumatic transport of PMMA powder through a steel pipe corresponding to the base case at t = 0.33 s. The color of the particles corresponds to their individual charges gained through collisions with the pipe. For better visualization the radial coordinate is scaled by the factor three and only 20% of the particles currently present in the pipe are displayed.

H. Grosshans, M.V. Papalexandris / Powder Technology 315 (2017) 227–235

This can also be inferred upon examination of Eq. (15) in conjunction with Eq. (19) where cw enters the powder charging process. Therein, the elasticity parameters of both the pipe and the particle play a role. However, if we compare the values of both we note that cp is an order of magnitude higher than cw and, therefore, its influence is much higher. In other words, the pipe considered in our study is sufficiently stiff so that its deformation during a contact with a particle does not influence the pipe-particle contact area. Instead, the contact area is determined by the elastic deformation of the particle. On the other hand, our simulations predicted that the other properties have a significant effect. While in the basic case the specific powder charge was equal to qav = 1.32 lC/kg, a decrease (in absolute terms) of cp by 50% causes a decrease to qav = 0.78 lC/kg. On the other hand, an increase by 50%, results in qav = 1.85 lC/kg. The effects of vp and e were similar. This is to be expected because these two parameters contribute equally to the charging process, as can be inferred upon insertion of Eqs. (17)–(12). According to our simulations, an increase by 50% leads for both properties to a decrease of qav to 0.88 lC/kg. The decrease of vp or e by 50% shows the strongest effect of all parameter variations investigated herein and leads to a powder charge of 2.65 lC/kg. Therefore, it can be concluded that varying cp , vp and e may provide a convenient means to control the powder charge in a desired way. Furthermore, the effect of these parameters can be clearly identified in Fig. 5 in which we display not only the charge of the complete powder, but also the distribution of charge on the individual particles for each parameter variation. As expected, the parameters vp and e are the ones that influence the distribution the most. Further, as mentioned above, they exert the same influence on the powder charging. But vp additionally appears in the equations for the inter particle charge exchange, cf. Eq. (7). Thus, one may hastily assume

0.4

0.8 base case ε - 50 % ε + 50 % γp - 50 % γp + 50 % ϕp - 50 % ϕp + 50 %

0.6 0.4 0.2 0 0

5

20

base case γp - 50 % γp + 50 %

pdf

0.3 0.2 0.1

0

0 0

1

2

3 4 5 q (μC/kg)

6

7

8

0

1

2

(a) 0.4

3 4 5 q (μC/kg)

6

7

8

7

8

(b) 0.4

base case γw - 50 % γw + 50 %

base case ϕp - 50 % ϕp + 50 %

0.3 pdf

0.3 0.2 0.1

0.2 0.1

0

0 0

1

2

3 4 5 q (μC/kg)

(c)

6

7

8

25

that a variation of vp affects the distribution of the total charge on the individual particles. However, no significance difference can be observed when comparing Fig. 5 (a) and (d), thus the influence of a variation of vp is minor. Further, we remark that all distributions presented in Fig. 5 exhibit a peak at q = 0 lC/kg. This implies that a substantial fraction of particles exits the pipe uncharged. Nonetheless, it is also interesting to note that most distribution reveal a second peak between 1 lC/kg and 2 lC/kg. Since this second peak is observed for all cases

0.4

0.2

15

Fig. 6. Number of wall collisions (nwcoll ) per particle for all cases considered in our study. The average number of wall collisions per particle is independent of cp , vp and e.

0.1

pdf

10

cluster of 1000 particles

base case ε - 50 % ε + 50 %

0.3 pdf

1

n wcoll

232

0

1

2

3 4 5 q (μC/kg)

6

(d)

Fig. 5. Probability density function (pdf) of the specific particle charge (q) of the individual particles for each parameter variation.

H. Grosshans, M.V. Papalexandris / Powder Technology 315 (2017) 227–235

0.2

0.15

0.3

base case γp - 50 % γp + 50 %

0.25

pdf

0.2 pdf

0.3

base case ε - 50 % ε + 50 %

0.2

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

0

0 0

0.02 0.04 0.06 ΔQ pw (pC)

0.08

base case ϕp - 50 % ϕp + 50 %

0.25

pdf

0.3 0.25

233

0 0

0.02 0.04 0.06 ΔQ pw (pC)

(a)

0.08

0

0.02 0.04 0.06 ΔQ pw (pC)

(b)

0.08

(c)

Fig. 7. Probability density function (pdf) of the charge transfer (DQ pw ) during each particle-wall collision. An increase of cp , respectively a decrease of vp leads to a significant shift of the distributions to higher values.

with the wall. However, inter-particle contacts do not alter the net charge of the powder (cf. Eq. (5)). For this reason, we now proceed to analyze further the charge transferred during each particle-wall collision. The distributions in Fig. 7 show for the lower value of cp and the higher value of vp and e a peak at around DQ pw = 0.01 pC, i.e. this is the charge transferred during most of the particle-wall collisions. However, if cp is increased, respectively if vp or e is decreased, significantly more charge is exchanged during collisions. The resulting shift in the distributions can explain the appearance of a second peak in the related powder charge curves, as discussed above and as shown in Fig. 5. In order to understand the reason for this increase in charge transfer, the parameters related to the collisions, namely the particle-wall contact area and time, are further examined. Their distributions are plotted in Figs. 8 and 9, respectively. From these plots, we readily observe that both the contact area and the contact time increase for higher values of cp . According to the definition of cp , cf. Eq. (19), this relates either to a decrease in the particle’s Poisson ratio or Young’s modulus. This is to be expected because a lower Poisson ratio means that a particle, while contracted in one direction during collision, expands more in the directions orthogonal to the direction of the contraction. On the other hand, a lower Young’s modulus results in a stronger elastic deformation during particle-wall contact. Actually, as it can inferred from Eqs. (14) and (15), reducing either of these two parameters leads to a larger contact area and, consequently, to a higher contact time. Therefore, they both result in an increase of the charge exchange during each collision. On the contrary, Figs. 8 and 9 reveal that a change of vp or e does not influence either the contact area or the contact time. Their

that result in a relatively high powder charge, we expect that this feature relates to the charge transfer during individual particle wall collisions. In these cases, most particles gain a charge of at least 1 lC/kg during the first impact. Thus, only few particles whose charge lies between q = 0 lC/kg and 1 lC/kg were found. On the contrary, the charge transfer during each impact of the cases which represent an increase in e or vp or a decrease in cp is lower. Consequently, the related distributions are also smooth between q = 0 lC/kg and 1 lC/kg. This relation is further investigated below. Thus far, we have identified the mechanical and electrical parameters that influence the process of powder charging the most. However, the issue regarding the identification of the underlying physical mechanisms remains open. We now proceed to elaborate on this issue. It becomes clear that the total charge of the powder is determined by two factors. The first one is the number of times each particle collides with the wall, i.e. the fluid dynamics of the flow. The second one is the amount of charge which is transferred between the pipe and the particle during each collision. In order to assess their role, we plotted the average number of collisions against the powder output in Fig. 6. It can be observed that the number of collisions, which is known to depend on the dynamics of the flow, does not depend on the electrical and mechanical parameters examined in our study, at least for the range of values considered herein. In other words, the differences in the powder charge between the various cases of our study (cf. Fig. 4), cannot be attributed to the number of collisions. In turn, this implies that these differences should be attributed to the characteristics of each collision. It is further noted, that during the stationary stage of the conveying process particles collide approximately 3.2 times more frequent with each other than

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Fig. 8. Probability density function (pdf) of the particle wall contact area during each collision (Apw ).

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H. Grosshans, M.V. Papalexandris / Powder Technology 315 (2017) 227–235

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Fig. 9. Probability density function (pdf) of the particle wall contact time during each collision (Dtpw ).

effect can be evidenced upon examination of the equation Eq. (17) for the particle-wall charge relaxation time. A decrease of each of these parameters leads to a decrease of tpw . Thus, even if the contact time and area is constant, more charge is exchanged during each collision. 6. Conclusions In our study we investigated the effect of the particle and pipe bulk material properties on powder electrification during pneumatic transport, via numerical simulations and parametric studies. Our simulations predict that altering some of those properties represents a promising method to control the powder electrification in a desired way. In particular, an increase of the particle’s Young’s modulus, Poisson ratio or electrical resistivity causes a significant reduction of the powder charge. The same turned to be true regarding the permittivity of the material of the particle. The results of our investigations can contribute to mitigate undesired electrostatic effects in industrial applications. This can be realized by an appropriate choice of the pipe material or the addition of agents to the powder in order to modify its characteristics. However, it is worth mentioning that the list of parameters examined in our study is not exhaustive. In particular, triboelectric charging can be affected by other parameters such as variations in the contact potential of the materials involved and ambient humidity. Acknowledgments The first author gratefully acknowledges the financial support of the National Research Fund of Belgium (FNRS) under the GRANMIX Projet de Recherche grant. The 3-d figures contained in this paper are created using VisIt [3].

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