Electrostatics of particles

Advanced Powder Technol., Vol. 14, No. 2, pp. 143– 166 (2003) Ó VSP and Society of Powder Technology, Japan 2003. Also available online - www.vsppub.com

Invited review paper Electrostatics of particles SHUJI MATSUSAKA ¤ and HIROAKI MASUDA Department of Chemical Engineering, Kyoto University, Kyoto 606-8501 Japan Received 2 October 2002; accepted 26 November 2002 Abstract—In powder handling, each particle collides with another particle or a wall, and consequently becomes charged up to a certain value. Such contact charging is experienced in various elds. In the present review, the basic concepts of contact charging are summarized; in particular, the effect of the contact potential difference and the initial charge on the charge transfer is described in detail. Furthermore, the variation of the particle charging caused by repeated impacts on a wall is formulated. This theory is extended to the particle charging in gas– solids pipe ow, where each particle has a different amount of charge; the distribution of the particle charge is also analyzed theoretically. In addition, the method of measuring important electrostatic properties, the technique of detecting particle charging and the application of particle charging are described. Keywords: Electrostatics; particle charging; contact potential difference; charge distribution; measurement.

NOMENCLATURE

a1 ; a2 b1 ; b2 C0 Cc c Di Dp d dl E; E1 ; E2 Ef ¤

constant in (36) and (37) (—) constant in (36) and (37) (C / kg) capacitance between bodies at critical separation (F) Cunningham slip correction factor (—) constant in (39) (kg/ A) inside diameter of pipe (m) particle diameter (m) constant in (39) (kg/ A) thickness of powder layer (m) Young’s modulus (Pa) strength of electric eld (V / m .D N/C))

To whom correspondence should be addressed. E-mail: [email protected]

144

e f; f1 ; f2 ; f3 fc fic fk g h I; I1 ; I2 k kN k0 k1 k2 k3 k4 ke m n nN n0 q qc qm qm0 ; qm1 qm1 q0 qr 1qc S t uN V V0 Vb Vc ; V1=2

S. Matsusaka and H. Masuda

elementary charge (C) probability density function of particle charge per unit mass (kg/ C) frequency of particle collision (1/s) probability density function of initial particle charge per unit mass (kg/ C) probability density function of impact charging factor (kg/ C) gravitational acceleration (m / s2 / Planck constant (J s) electric current (A) impact charging factor (D qm1 =n0 / (C /kg) mean value of impact charging factor (C /kg) constant in (6) (V / C) constant in (7) (charging efciency) (—) constant in (9) (1 / s) D Vb =q (V / C) constant in (35) (—) elasticity parameter (1 / Pa) mass ow ratio of particles to gas (—) number of particle collisions (—) mean number of particle collisions (—) relaxation number (—) net charge on particle (C) charge accumulated by contact on particle (C) particle charge per unit mass (charge-to-mass ratio) (C /kg) particle charge per unit mass at a reference point (C /kg) particle charge per unit mass at x D 1 (C /kg) initial charge (C) leakage charge (C) charge transferred by contact (C) contact area (m2 / elapsed time (s) average gas velocity (m / s) total potential difference (V) zero-point potential (V) potential difference arising from space charge (V) contact potential difference based on surface work function (V)

Electrostatics of particles

Ve VM=Au VP=Au vN ve vi Wp x 1x z0

145

potential difference arising from image charge (V) contact potential difference of metal against gold (V) contact potential difference of powder against gold (V) average particle velocity (m / s) electrical migration velocity of charged particle (m /s) impact velocity (m / s) mass ow rate of particles (kg/ s) distance from starting point (m / s) length of pipe (m) critical gap between contact bodies (m)

Greek ® 1; 11 ; 12 "0 "p ¹ º1 ; º2 ½ ½c ½p ¾0 ¾k Á; ÁI ; Á2 ; ÁM ; ÁI !

D Vb =Ve .D k0 =k3 / (—) change of energy level caused by charge transfer (J) absolute permittivity of gas (F / m) absolute permittivity of powder layer (F /m) viscosity of gas (Pa s) Poisson’s ratio (—) gas density (kg/ m3 / charge density (C / m3 / particle density (kg/ m3 / standard deviation of initial charge per unit mass (C / kg) standard deviation of impact charging factor (C /kg) work function (J) frequency of radiation (1 / s)

1. INTRODUCTION

Electrostatic particle charging is an important phenomenon related to powder handling. It is commonly a nuisance [1 – 3] and the source of explosion hazards [4]. Various applications have, however, been developed, e.g. electrophotography [5, 6], dry powder coating [7 – 9], electrostatic precipitator [10], separation of powder [11, 12], electromechanical particulate operation [13, 14], powder ow measurement [15– 18], tomography [19] and many others. To improve the performance of these applications and to reduce the risk of dust explosion, a correct understanding of particle charging is required. Contact charging has been known for a old times [20]. In spite of the long history, there are still unsolved problems and inconsistent experimental results are also reported. This is because many factors, such as physical, chemical and electrical

146


characteristics, and environmental conditions, affect particle charging, and thus it is difcult to reproduce the same electrostatic phenomenon if there are some uncertain elements. Although complete theory applicable to any kind of particle charging is not established at present, there are several fundamentals to explain the particle charging and electrostatic phenomena [21– 23]. In the present review, we summarize the basic concepts of contact charging and then analyze impact charging of particles. We also describe several experimental techniques and show the actual particle charging in gas–solids pipe ow; furthermore, we mention the application of the electrostatic phenomenon.

2. BASIC CONCEPTS AND THEORY

2.1. Contact charging When two different materials are brought into contact and separated, an electric charge is usually transferred from one to the other. This phenomenon is often called ‘contact electrication’ or ‘contact charging’. When they are rubbed, it can be called ‘frictional electrication’ or ‘tribo-charging’; as for short contact, it can be called ‘impact charging’. Contact charging is also classied into three categories according to the contacting materials, i.e. metal – metal contacts, metal – insulator contacts and insulator– insulator contacts. 2.1.1. Metal– metal contacts. Charging of metals is usually unnoticeable because the charge transferred runs away from the contact point. In fact, charge transfer occurs even for metal – metal contact when the metals are isolated electrically before and after the contact. Figure 1 shows two metals with different work functions ÁI and Á2 in contact. Assuming that electrons transfer by tunneling so that thermodynamic equilibrium maintains, the contact potential difference (CPD) Vc is given by: Vc D V1=2 D ¡.Á1 ¡ Á2 /=e;

(1)

where V1=2 is the CPD of metal 1 against metal 2 and e is the elementary charge. The amount of transferred charge is equal to the product of the CPD and the capacitance between the two bodies. The capacitance depends on the state of the contacting part. Although the position of the electrons can vary as the metals are separated, the charge after the separation 1qc is approximated by the following simple equation: 1qc D C0 Vc ;

(2)

where C0 is the capacitance between the bodies at the critical separation distance. Figure 2 shows the theoretical and experimental results for the charge after contact [21]. Although the experimental results are somewhat less than the theoretical ones, the tendencies are in reasonable agreement. The difference is probably caused by the surface roughness, impurities, oxidized layer, separation speed, etc.


147

Figure 1. Electron potential energy at a metal– metal contact.

Figure 2. Charge on a chromium sphere, 4 mm in diameter, in contact with another metal sphere, 13 mm in diameter, as a function of the CPD of chromium against each metal [21].

2.1.2. Metal– insulator contacts. (i) Electron transfer. The concept of contact charging between metals can be extended to charging for metal – insulator contacts [24]; however, experimental data are often scattered widely [25] and inconsistent results are also reported [26, 27]. When a linear relationship between the transferred charge and the work function is obtained experimentally, it is very probable that the charging will occur by means of the electron transfer and the following relationship is applicable to the electrostatic characterization: 1qc / .ÁM ¡ ÁI /;

(3)

where ÁM is the work function of the metal and ÁI is the apparent work function of the insulator. The proportionality constant in this relationship depends on the characteristics of the insulator. Figure 3 shows the linear relationship obtained

148


Figure 3. Charge density of nylon 66 by contacting with various metals. The horizontal axis is the CPD of each metal against a gold reference VM=Au [24], where the CPD is dened as VAu=M .

experimentally by Davies [24]. Murata and Kittaka also produced evidence of electron transfer by means of the measurement of photoelectric emission [28]. (ii) Ion transfer. Several researchers have suggested that contact charging between a metal and an insulator may be due to the transfer of ions [20, 29, 30]. Insulators may contain ions in the body or on the surface. These ions can be transferred by diffusion, relative acidity, afnities and the kinetic effect based on shearing off. When ions exist in excess, ion transfer affects contact charging; however, when the ions are few, electron transfer will control the contact charging. (iii) Material transfer. The impact or friction between two bodies can result in the transfer of material from one to the other. When a metal slides over a polymer, large amounts of polymer may transfer to the metal, and the transferred polymer can carry charge and change the effective contact potential difference. Also, when brittle particles impact on a metal wall, elements of particles can be transferred on the metal [31]. (iv) Effect of the separation state on charging. Particle charging can be inuenced by the state of separation. If the charge transferred to the surface of the insulator ows back to the metal, the net charge will be reduced. This reduction depends on properties such as conductivity and the separation state such as the speed of sliding or rolling. In addition, a gaseous discharge may occur during the separation [32– 34].


149

Figure 4. Energy level diagram for insulator– insulator contact.

2.1.3. Insulator– insulator contacts. The tendency of the charging between insulators can be evaluated using a ‘triboelectric series’, in which insulators are ranked in an order such that a material higher up the series will always charge positive when touched or rubbed with a material lower down. However, it is not always correct and there is no theory in the triboelectric series. To analyze the contact charging between insulators in more detail, several models have been presented. Almost all of them are similar to those for metal – insulator contact, but the movement of electrons or ions in the body was more restricted. Figure 4 shows a contact charging model that includes the effect of an electric eld [35, 36]. When the insulators come in contact, charges ow from the lled surface state of the insulator 1 to the empty surface state of insulator 2. It is assumed that the system remains in equilibrium by tunneling until the surfaces are separated over the critical gap; nally, the expression for the energy level including the effect of the electric eld Ef is given as Á1 C 11 C eEf z0 D Á2 ¡ 12 ;

(4)

where 1 is the change of the energy level caused by the charge transfer. 2.2. Particle charging by repeated impacts on a metal wall 2.2.1. Charge transfer. When a particle impacts on a metal wall, each acquires an equal and opposite charge. The amount of charge transferred depends on the total potential difference V between the contact bodies, which is made up of two parts: Vc based on the surface work function and Ve arising from image charge, which is

150


induced in the wall by an external charge. These are related as follows [37– 39]: V D Vc ¡ Ve :

(5)

The potential difference Ve is approximated by: Ve D k0 q;

(6)

where k0 is a constant and q is the charge on the particle before impact. The charge transferred in a repeated impact process is analyzed by approximating the process to an equivalent rate process, i.e. the charge transfer is treated as a continuous quantity dqc =dn to obtain the enveloping curve of the remaining charge after a number of collisions: dqc (7) D k1 CV ; dn where n is the number of collisions, k1 is the charging efciency and C is the capacitance between the contact bodies, which is given by: CD

"0 S ; z0

(8)

where "0 is the absolute permittivity of the gas, S is the contact area and z0 is the critical gap including the surface roughness between the contact bodies. Charge relaxation with elapsed time dqr =dt is approximated by [40]: dqr D ¡k2 q; (9) dt where k2 is a constant. When the particle collides at regular intervals, i.e. the frequency is dened as fc , (9) is rewritten as: k2 dqr D ¡ q: fc dn

(10)

From (5)– (8) and (10), the following equation is derived: dqr dq dqc C D dn dn dn "0 S k2 .V c ¡ k0 q/ ¡ q: D k1 z0 fc

(11)

Solving (11) with initial conditions (n D 0; q D q0 /, the following equation is derived: ´ ´¼ n n C q1 1 ¡ exp ¡ ; q D q0 exp ¡ (12) n0 n0 where: q1 D

Vc ; k2 z0 k0 C k1 "0 Sfc

(13)


151

Figure 5. Charge on a rubber sphere accumulated by repeated impacts with a steel wall [39].

and: n0 D

1 : k2 k0 k1 "0 S C z0 fc

(14)

Figure 5 shows the variation in the charge on a rubber sphere by repeated impacts [39]. The transferred charge caused by an impact decreases with the number of collisions and the charge approaches a limiting value. The limiting value tends to decrease as the interval between collisions increases because the charge relaxation increases with elapsed time. The broken lines in Fig. 5 are the results calculated using (12). The experimental results are in agreement with the calculated values. 2.2.2. Effect of elasticity. When a particle impacts on a hard plate, the particle is deformed and a contact area is produced. If the particle is a sphere with a smooth surface and the contact deformation can be approximated by a Hertzian deformation pattern [41], the maximum contact area S during the impact is represented by: 4=5

S D 1:36ke2=5 ½p2=5 Dp2 vi ;

(15)

where ke is the elasticity parameter, ½p is the density of the sphere, Dp is the particle diameter and vi is the impact (incident) velocity. If the particle is not spherical, the particle shape should be taken into account [42]. The elasticity parameter ke is given by ke D

1 ¡ º22 1 ¡ º12 C ; E1 E2

(16)

where º is Poisson’s ratio, E is Young’s modulus, and subscripts 1 and 2 represent the sphere and the plate, respectively. Figure 6 shows the relationship between the transferred charge and the maximum contact area calculated using (15). The transferred charge is approximately

152


Figure 6. Relationship between transferred charge and maximum contact area during impact of a rubber sphere of 0.032 m in diameter on a steel wall [39].

proportional to the contact area and, therefore, it is deduced that the maximum contact area based on the elastic deformation controls the amount of the transferred charge. 2.3. Charging in gas–solids pipe ow In gas– solids pipe ow, particles are charged as a result of the collisions with the wall; the charged particles form an electric eld and the electric eld inuences the total potential difference V , i.e. [37, 38]: V D Vc ¡ Ve ¡ Vb

(17)

where Vb .D k3 q/ is the potential difference arising from the electric eld, which is called the ‘space charge effect’. When a length of metal pipe that is isolated electrically is grounded, the charge transferred from the particles to the wall ows to earth. The charge transferred per unit time is detected as electric current. The current I generated from the pipe of a length from x to x C 1x is expressed as: ´¼ ´¼ I n.1x/ n.x/ ; 1 ¡ exp ¡ (18) D fqm0 ¡ qm1 g exp ¡ Wp n0 n0 where Wp is the mass ow rate of particles, qm0 and qm1 are the charge-to-mass ratio at x D 0 and x D 1, respectively. n is the number of particle collisions and n0 is the relaxation number. When x D 0, (18) becomes: ´¼ n.1x/ I : (19) D fqm0 ¡ qm1 g 1 ¡ exp ¡ Wp n0


153

Furthermore, for n.1x/ ¿ n0 , (19) is simplied as: n.1x/ I ; D fqm0 ¡ qm1 g Wp n0

(20)

where: qm1 D

n

6Vc

¼½p Dp3 .k0 C k3 / C

k2 z0 o k1 "0 Sfc

;

(21)

and: n0 D

1 : k2 .k0 C k3 /k1 "0 S C z0 fc

(22)

Substituting k0 D 2z0 =.¼ "0 Dp2 / (for the image charge effect), k2 D 0 (for low 3 N N (for the space charge effect) electric relaxation) and k3 D 3z0 m½Di u=.2¼" 0 ½p D p v/ into (21) and (22), gives the following equations, respectively [38, 43]: qm1 D

3"0 Vc ; ½p Dp z0 .1 C ®/

(23)

and: n0 D

¼Dp2 2k1 S.1 C ®/

;

(24)

in which ® is the ratio of the space charge to image charge effect, i.e.: ®D

3 ½Di uN m : 4 ½p Dp vN

(25)

where m is the mass ow ratio of particles to gas, ½ is the gas density, ½p is the particle density, Di is the inside diameter of the pipe, uN is the average gas velocity and vN is the average particle velocity. 2.4. Charge distribution of particles in gas– solids pipe ow Although particle charging depends on various factors, the main factors are considered to be the number of particle collisions, initial charge on the particles and the state of the impact charging. To derive the charge distribution of particles, we introduce the probability density functions of these factors [44]. 2.4.1. Probability density function of particle collision with the wall inside. Assuming that the probability density function of particle collision is expressed as a normal distribution, which is derived from a binomial distribution, and also using the relationship between the number of particle collision and the particle charge, the


154

probability density function of the charge to mass ratio f1 .qm / is represented by: 2 n ± o2 3 qm ¡ qm0 ² q C nk ¡N m1 ln 1 7 6 qm1 1 qm1 ¡ qm 7 ; (26) p f1 .qm / D ¡ exp 6 5 4 2 qm1 ¡ qm 2¼ nk 2nk N N where nN is the mean number of particle collisions and k is the impact charging factor .D qm1 =n 0 /. When qm0 D 0 and jqm j ¿ jqm1 j in the early stages of the particle charging, (26) is simplied as: ¼ .qm ¡ nk/ 1 N 2 : f1 .qm / ¼ p (27) exp ¡ 2nk N 2 2¼ nk N 2.4.2. Probability density function of initial charge on particles. Using a probability density function of initial charge fic .qm0 /, which is based on the form of (26), the probability density function f2 .qm / after traveling through the pipe is represented by the following equation: Z f2 .qm / D D

qm1 ¡1

f1 .qm /fic .qm0 / dqm0

qm1 1 q qm1 ¡ qm 2¼.nk N 2 C ¾02 / 2 n ± ² o2 3 qm C q nk ln 1 ¡ N m1 6 7 qm1 ¡ qm 7; £ exp 6 ¡ 2 4 5 2 2.nk N C ¾0 /

(28)

where ¾0 is the standard deviation of the initial charge. When jqm j ¿ jqm1 j, (28) is simplied as: ( ) .qm ¡ nk/ 1 N 2 : f2 .qm / ¼ q (29) exp ¡ 2.nk N 2 C ¾02 / 2¼.nk N 2 C ¾ 2/ 0

Substituting ¾0 D 0 in (29) gives the same form as (27). Figure 7 shows the charge distributions calculated using (28). In this calculation, it is assumed that the charge transferred from the particles to the wall is positive. As a result, the amount of the particle charge increases negatively with the mean number of particle collisions n.D N 0; 2; 4; 8; 16; 32; 64/ and approaches an equilibrium value (qm1 D ¡6 mC /kg). Since the transferred charge depends both on the particle charge and the number of collisions, the variation of the charge distribution is somewhat complicated.


155

Figure 7. Effect of the mean number of particle collisions nN on the charge distribution (qm1 D ¡6 mC/ kg, qm0 D 0 mC/ kg, ¾0 D 0:3 mC/ kg, k D ¡0:2 mC/ kg).

2.4.3. Probability density function of impact charging factor. The impact charging factor k is not always constant, i.e. it depends on electrostatic properties, particle impact velocity, particle diameter, surface roughness, angle of incidence, etc. In the same manner as (28), the probability density function of particle charge f3 .qm / is represented by: Z 1 f3 .qm / D f2 .qm /fk .k/ dk; (30) ¡1

where fk .k/ is the probability density function of the impact charging factor. Assuming that fk .k/ is represented by a normal distribution, (30) becomes: Z f3 .qm / D

1

qm1 1 q ¡1 qm1 ¡ qm 2¼.nk N 2 C ¾02 / 2 n ± ² o2 3 qm C q nk ln 1 ¡ N m1 7 6 qm1 ¡ qm 7 £ exp 6 ¡ 4 5 2.nk N 2 C ¾02 /

¼ N 2 .k ¡ k/ exp ¡ dk £p 2¾k2 2¼ ¾k 1

156


Z 1 qm1 1 1 q D qm1 ¡ qm 2¼ ¾k ¡1 nk N 2 C ¾02 2 n 3 ± ² o2 qm C q nk ln 1 ¡ N m1 N 27 6 .k ¡ k/ qm1 ¡ qm 7 dk; (31) £ exp 6 ¡ ¡ 4 2.nk N 2 C ¾02 / 2¾k2 5

Figure 8. Effect of the standard deviation of the impact charging factor ¾k on charge distribution (¾ 0 D 0:3 mC/ kg, nN D 3:0; kN D ¡0:2 mC / kg).

Figure 9. Comparison between experimental data and theoretical curves (y ash: Dp50 D 12¹m; (E) uN D 77 m/ s, 1x D 0:5 m; (n) uN D 58 m/ s, 1x D 2:0 m [44].


157

where kN and ¾k are the mean value and the standard deviation of the impact charging factor, respectively. When jqm j ¿ jqm1 j, (31) is simplied as: ( ) Z 1 N 2 .qm ¡ nk/ .k ¡ k/ 1 1 N 2 q f3 .qm / ¼ exp ¡ ¡ dk: (32) 2¼ ¾k ¡1 .nk 2.nk N 2 C ¾02 / 2¾k2 N 2 C ¾02 / Figure 8 shows the effect of the standard deviation of the impact charging factor ¾k on the charge distribution. The charge distributions for ¾k D 0 and for ¾k 6D 0 are calculated using (29) and (32), respectively. The shape of the distribution curve becomes skewed as the value of ¾k increases, i.e. the tail on the left-hand side of the distribution curve becomes larger. Figure 9 shows typical examples of the results for the charge distribution of particles. The values of particle charge are widely distributed, including positive charge as well as negative charge, and the distributions are skewed, i.e. there is a longer tail on the left-hand side. Calculated lines are also added to Fig. 9. The experimental data agree well with the theoretical curves taking into account the distribution of the impact charging factor.

3. CHARACTERIZATION AND MEASUREMENT

3.1. Electrostatic properties 3.1.1. Work function. When materials are exposed to electromagnetic radiation, electrons are librated. The number of electrons emitted depends on the intensity of the radiation. The kinetic energy of the electrons emitted depends on the frequency of the radiation. The radiation is regarded as a stream of photons, each having an energy h!, where h is the Planck constant and ! is the frequency of the radiation. A photon can only eject an electron if the photon energy exceeds the work function Á of the solid, i.e. there is the minimum frequency (or threshold frequency) at which ejection occurs. For many solids, the photoelectric effect occurs at ultraviolet frequencies or above, but for some materials (having low work functions) it occurs with light. The maximum kinetic energy of the photoelectron depends on the energy of a photon and the work function. This is applied to a technique for determining the work function or analyzing the surface properties of various materials [28, 45]. 3.1.2. CPD. The CPD directly controls the contact charging as mentioned in Section 2.1. In fact, the surface of materials is not pure and is usually covered with an oxide lm, and thus the CPD of the materials used may differ from the values shown in literature. To measure the CPD between a powder and a wall, a measuring system based on the Kelvin – Zisman method was developed [40, 46, 47]. The measuring system is shown in Fig. 10. This system has an electric circuit in which there is a capacitor made up of two electrodes, a DC supply and an electrometer in series. Powder is lled in the concavity of the lower electrode and upper electrode

158


Figure 10. Measurement of the CPD based on the Kelvin– Zisman method.

Figure 11. Characterizationof the coated particles by the Kelvin– Zisman method [47].

made of gold oscillates vertically. When the voltage applied on the upper electrode is equal to the potential difference between the powder and the upper electrode, the induced currents detected by the electrometer become zero. Hereafter, the applied voltage is called a zero-point potential V0 , which is expressed by the following equation: V0 D VP=Au C

½c dl2 ; 2"p

(33)

where VP=Au is the CPD between the powder and gold reference, ½c is the charge density, dl is the thickness of the powder layer, and "p is the absolute permittivity of the powder layer. When the charge of the powder layer is sufciently small .½c ¼ 0/; V0 is equal to VP=Au. Figure 11 shows an example of the measurements


159

for analyzing the CPD of coated particles. The value of the CPD varies according to the thickness of the coat, i.e. the value approaches that of the coating material as the thickness increases. In addition, various studies such as evaluation of toners [40] and particles with different functional groups [48], and analysis of the effect of temperature and humidity [49] have also been carried out using the method. 3.2. Particle charge 3.2.1. Charge-to-mass ratio. In general, the charge on particles is measured with a ‘Faraday cage’ because the method is simple and reliable. A number of charged particles are put into a metal enclosure that is isolated electrically and the induced charge is measured. The charge-to-mass ratio is obtained by dividing the charge by the mass. In gas– solids pipe ow, the charged particles are collected on the lter in the Faraday cage. 3.2.2. Distribution of particle charges. Mazumder et al. [50] developed the electrical-single particle aerodynamic relaxation time (E-SPART) analyzer, which is based on the dynamics of the aerosol particles in an electrostatic and acoustic eld. The motion of the particles can be analyzed with a laser Doppler velocimeter and particle diameter Dp is determined from the phase lag of the particle motion relative to the motion of the gas. The charge of each particle can be determined from the electrical migration velocity Ve . When the motion is governed by Stokes’ low, the charge is given by: qD

3¼¹Dp ve ; Ef Cc

(34)

where ¹ is the viscosity of the gas, Ef is the strength of the electric eld and Cc is the Cunningham slip correction factor. Masuda et al. developed a simple method to directly determine the charge-to-mass ratio of each aerosol particle qm [51]. The value can be calculated from the two-dimensional trajectory of the particle moving in an electrostatic and gravitational eld: g q m D k4 ; (35) Ef where k4 is a constant, which is the ratio of the vertical velocity to the horizontal velocity controlled by electrostatic force, and g is the gravitational acceleration. Since these methods enable us to collect data efciently, the distribution of the data can be obtained easily.

4. PROCESS

In pneumatic transport, particles repeatedly collide with an internal wall, and particles and the wall are charged. When a metal pipe is grounded, the charge

160


Figure 12. Electric current generated from a detecting pipe as a function of powder ow rate (quartz sand: D p50 D 320 ¹m [43].

transferred from the particles to the wall ows to the ground, and can be detected as electric currents. The currents are caused by the particle charging in the pipe and thus the value of the current should be proportional to the particle ow rate. As shown in Fig. 12, a linear relationship is obtained experimentally [43]. This relationship is also explained by (18), where the right-hand side term must be a constant. In general, this is satised in dilute and high-speed conveying systems. As for dense-phase conveying systems, the surrounding particles prevent the free contacts between the particles and the wall; as a result, the currents are less than those expected under ideal conditions. Using smaller particles, the adhesiveness also prevents free contacts. In addition, the initial charge on the particles affects the charge transfer (see (18)). In fact, the particles collide with many different walls before arriving at the pipe, e.g. hopper, feeder, chute, disperser, etc., and hence the polarity and the amount of the particle charge change according to the operating condition. To estimate the charge transferred from the particles to the wall, the initial charge has to be known beforehand.

5. APPLICATIONS

5.1. Online measurement of particle ow rate in the dilute phase The electric current detected from a pneumatic transport pipeline depends on the particle ow rate and the initial charge per unit mass of particles. If the current and the initial charge are known, the particle ow rate can be determined using (19). The continuous measurement of the current is very easy, but that of the initial


161

Figure 13. A novel method for measuring the particle ow rate.

charge is not so easy. If the particle ow rate is determined from only the electric currents, it is very convenient for the measurement. Figure 13 shows a method for measuring the particle ow rate [52, 53]. The system has two different detecting pipes connected in series with electrical isolation. The charge balances for the rst and the second detecting pipes are given by the following equations, respectively: I1 D a1 qm0 C b1 ; Wp

(36)

I2 D a2 qm1 C b2 ; Wp

(37)

and:

where I is the electric current, Wp is the particle ow rate, qm is the charge per unit mass of particles at the inlet of each detecting pipe, a and b are constants, and subscripts 1 and 2 refer to the rst and second detecting pipes, respectively. Since the charge at the outlet of the rst pipe is equal to that at the inlet of the second pipe, the charge balance is expressed as: qm0 ¡ qm1 D

I1 : Wp

(38)

From (36)– (38), the powder ow rate Wp is given by the following equation: Wp D cI1 C dI2 ;

(39)

162


here: cD

.a1 ¡ 1/a2 ; a1 b2 ¡ a2 b1

(40)

dD

a1 : a1 b2 ¡ a2 b1

(41)

and:

In (40) and (41), the denominator of the right-hand side must not be zero. Therefore, the electrostatic property of the rst detecting pipe must differ from that of the second, i.e.: b1 b2 6D : a1 a2

(42)

Figure 14 shows the electric currents generated from the two detecting pipes. The polarity of the currents is determined by the relative electrostatic property between particles and the wall. Since particles are fed continuously into the pipe, the currents almost keep constant. The responsiveness in this system is so high that a small uctuation of the feed rate can be detected as the variation of the current. Figure 15 shows the results on the measurement of the particle ow rate. The relative error is less than 10%. This system can be applied to even polymer particles over wide range [54].

Figure 14. Electric currents generated from the two detecting pipes [52].


163

Figure 15. Comparison between the online method and weighing method [52].

Figure 16. Comparison between the online method and Faraday cage method [52].

5.2. Online measurements of particle charge The above system can be applied to the measurement of particle charge. From (36)– (38), the following equation is derived: qm0 D

.b2 ¡ a2 b1 /I1 ¡ b1 I2 : a2 .a1 ¡ 1/I1 C a1 I2

(43)

164


Figure 16 shows the comparison between the values obtained by this online method and those obtained by the Faraday cage method.

6. CONCLUSION

Contact charging is usually classied into three categories according to the contacting materials. For metal – metal contacts, the charge transferred can be explained by the CPD. As for metal – insulator contacts and insulator– insulator contacts, many other factors affect the charge transfer. When a linear relationship between the transferred charge and the work function is obtained experimentally, it is very probable that the charging will occur by means of electron transfer and the CPD plays an important role. When a particle impacts on a metal wall, the charge transfer depends on the total potential difference, including the effect of the initial charge. The charge transferred is approximately proportional to the contact area. When the particle deforms elastically, the charge can be estimated theoretically. The particle charging caused by repeated impacts on a metal wall can be expressed using an equation with exponential functions. This theoretical analysis can be extended to the particle charging in gas– solids pipe ow, where the charge transferred per unit time, i.e. electric current, is an important evaluation factor. Since many particles ow in the pipe, the charge on particles is widely scattered. The charge distribution can be explained by introducing the probability density functions of (i) the number of particle collision, (ii) the initial charge on the particles and (iii) the amount of charge transferred by an impact. In addition to the theoretical study, extensive experimental studies have been carried out. Various measuring techniques to characterize the electrostatic properties and charge transferred have been developed, and some of them are used widely in the laboratory and industry.

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