Flow of a suspension of fine particles around a steel ball fixed in a cylindrical pipe

285

Powder Technology, 66 (1991) 285-292

Flow of a suspension cylindrical pipe

of fine particles

M. Benali*, K. Shakourzadeh-Bolouri Umversite’ de Technologre de Compi&ne, (Recemed

around

and J. F. Large Depamnent of Chemrcal Etgneenng,

May 30, 1989; m revised form December

a steel ball tied

BP

in a

649, 60206 ComprSgne (France)

12, 1990)

Abstract of a suspension of fine partlcles flowmg in a cylindrical pipe around a fixed steel ball has been studted An experimental technique, which is based on the dynamic equilibrium of the ball, is proposed for measuring the forces due to the gas and the dispersed phase. Expertmental results show a remarkable contribution of the forces acting on the ball as a function of the operating parameters: gas superficial velocity and mass flow rate of particles. A model for particle-ball colhstons is presented to interpret the phystcal origin of these forces The behaviour

Introduction

The earliest published studies on gas-solid suspensions show that the flow of a fluid or a suspension around bodies induce mteractive forces between the continuous (and/or dispersed) phase and the bodies. According to these studtes, rt is expected that the drag coefficient in a two-phase system may be different from that in a single-phase flow system. This study aims at identifying the force due to a fine suspension flowing in a cylindrical pipe around a fixed steel ball. It is related to the hydrodynamic analysis of many processes where a counter-current contacting of gas and sohds is required such as gas-solid or gas-solid-solid packed reactors which exhtbtt some very attractive properties: simpler geometry, stability of the flow, lower pressure drop along the packed zone, economical compared with shallow multistage fluidized beds [l]. Therefore, the understandmg and modelhng of direct contact mechanism between a gas and particulate phase IS required for predicting the hydrodynamic and thermal behavrour of these systems.

Literature

survey

The literature on flow around a body has been reviewed by Torobin and Gauvin [2, 31 and concerns *Author to whom correspondence should be addressed. Ecole Polytechmque, Department of Chemical Engineering, C.P. 6079 Succ. A, Montrtal, Que, H3C 3A7 (Canada).

0032-5910/91/$3 50

the roles of complex fluid dynamic phenomena which include boundary layer separation, vorticity transfer and those characterized by the Strouhal number (St). Barley [4], Foster et al. [5] and Roos and Willmarth [6] have studied the various experimental factors which affect the value of the drag force acting on a spherical body. According to these authors [3-61, the resistance to the flow is dependent on the Reynolds number (Ret,) related to body size, the ratio of sphere diameter to pipe diameter (d/D) and the radial oscillations of the sphere induced by high turbulence m the fluid. McCammon and Deutsch [7] have computed Stokes drag on a particle by solving a set of linear algebraic equations of the Kirkwood and Riseman type [8]. Subsequent developments along this line, by Trang-Cong and PhanThien [9], have given rise to the so-called ‘Boundary Element Method’ to compute the Stokes resistance of a multiparticle system whose size and shape are assumed to be arbitrary. Furthermore, Brenner [lo] has formulated a ‘Phenomenologtcal Scheme’ for calculating the quasistatic Stokes force on a rigid particle of any shape immersed in an arbttrary flow field. Using spectral methods to solve the stability problem of the flow past a sphere, Kim and Pearlstem [ll] show that the flow is subjected to disturbances occurring at critical Reynolds number and then experimental results revealed a strong influence of unstable flow on the drag undergone by the sphere. Whereas there are several studies related to the flow of a single fluid around an isolated spherical body, there are very few publications available for

0 Elsevier

Sequola/Prmted

m The Netherlands

286

the flow of two or more phases around an obstacle. Recently, Yokosawa et al. [12] have worked on this subject. They treated two-phase flow (gas-liquid) around a cylinder m a vertical pipe and noted the decrease in critical Reynolds number (Ret& compared to single-phase flow around an obstacle and also showed that the value of the drag coefficient depends on the turbulence in the liquid. Doig and Roper [13] studied the effect of dispersed phase on the suspension of spheres by a bounded gas-solids stream in 50-mm diameter glass tube. Four sizes of glass beads (99.5-l 285 pm) were chosen as the solids for the dispersed phase and four sizes of suspended spheres (3 050-l 1 853 pm) were chosen for suspension. In the absence of dispersed phase, Doig and Roper [13] found a mean value of the drag coefficient equal to 0.66, while in the presence of dispersed phase this value increased by a factor of 3.5. This amounts to demonstrating the contribution of dispersed solids to the suspension of the sphere. In [14], using the measurement of pressure drop due to solids flowing m a vertical pipe, it can be seen that the drag coefficients are greater than the standard values. The difference may be due to wall effects and fluid turbulence. More recently, Lee [15] has examined the flow properties of an upward solid particle-air two-phase turbulent suspension flow in a vertical pipe based on the local measurements of velocity both of the air and the particles. He proposed that longitudinal dynamic interaction between the phases, UZ., the drag forces may be described by a pseudo-Stokes law based on a turbulent particle Reynolds number in which, instead of the molecular viscosity, an equivalent turbulent viscosity of the fluid for the particles in the suspenston flow is used. This paper presents an experimental method to measure the forces acting on the ball and intends to examme the contribution of dispersed phase to forces acting on the ball.

Experimental

apparatus

and operating

procedure

The experimental equipment is illustrated in Fig. 1. It consists of a glass transport line with internal diameter and overall height of 25 mm and 2 000 mm, respectively, and a precision balance to which a 4-mm diameter steel ball is hooked by a 30-pm diameter nylon wire (the wire’s weight is negligible compared with that of the ball). The ball is centred in the column by means of a stainless steel roller. Compressed air is used in all experiments. Before introducing air into the transport line, the compressed air is filtered to remove traces of oil and its pressure is regulated using two regulators placed in series

upstream of the gas flowmeter. The regulators are set to 4 and 2 bar, respectively. Solids are flowed through the installation in closed circuit. They are introduced into the hortzontal pipe injection system by gravity from the hopper feeder. Solids leaving the transport line are separated from the gas in a 265-mm x 200-mm Plexiglas disengaging chamber. An inclined deflector is placed 95 mm above the end of the transport line to break to gas-solids jet. At the bottom of the hopper, a sleeve valve is installed to control the solid flow rate which is measured volumetrically. The range of the operating conditions is as follows: Vr=4.17-21.50

m/s

W, = 2.67-16.27

g/s

The solids used are sand particles with a mean surface diameter of 205 pm and a density of 2 646 kg/m3.

Experimental

results

and discussion

Both of the forces due to single-phase gas flow (Fr) and the air-sand suspenston (F,) are deduced from the dynamic equilibrium of the steel ball (Fig. 2). Flow of a sqle

gas around

the jked

ball

Figure 3 shows the variation of the force due to single gas flow as a function of the gas superficial velocity: this means that the resistance force to flow around the fixed ball is more important for higher gas superficial velocity. Ftgure 4 represents the comparison between the measured force values and those calculated by Stokes law [12]. A remarkable aspect of the data in this figure is a deviation (2 to 25%) from the data given by the ‘Standard’ drag force (F,) at the same gas superficial velocity: this means that the gas flow around the ball is not very uniform. This is due to inhomogenous disturbances generated by small oscillations of the ball which appear clearly over the entire range of 8 to 15 m/s gas superficial velocity. Above V,= 15 m/s, these macroscopic instabilities become homogenous and tend to disappear. Therefore, the deviation of measured force from the 45” line decreases. Flow of suspension

of fine particles

around

the ball

The gas superficial velocity and mass flow rate of particles are used as independent variables. Typical variations for the measured forces due to the suspension (FrS) and those due the dispersed phase (FJ are illustrated m Figs. 5 and 6, respectively. As shown

287

1.

Compressor

2.

Gas pressure regulators

3.

Needle valve

4.

Rotmter

5.

Mercury mananeter

6.

Rotary plug flow

7.

Graduated CO~UM

8.

SLeeve valve

9.

Solid feed hopper

for measuring solid flourate

10. Precision balance 11. Converter 12. Graphical recording

16

13. Disengaging chamber 14. Plexiglass colwmtest

section

15. Steel ball 16. Pressure taps 17. Acceleration zone 18. Metal gird 19. Roller for ball centring

I 6

~18 19

Fig. 1. Schematic

view of the experimental

apparatus.

288

!-

Roller

(a)

far

ball

centrnlg

(b)

Fig. 2. Dynamic equihbrium of the ball (a), Flow of single gas around the ball; (b) flow of dilute suspension air-sand around the ball.

v,

(m/s)

Fig. 3. Varlatlon of the force due to a smgle gas phase as a function of the gas superficial velocity. In Fig. 5, the force Ffsincreases with the gas superficial velocity. In the range 8 4.65 g/s). The reproducibihty of runs carried out continuously under these conditions for a week (Fig. 7) confirms this observation and shows the absence of the resonance effect which could be generated by low radial fluc-

Fig. 4. Comparison of the values of the force due to a single gas phase and those calculated by the ‘Standard’ drag force law.

vi

(m/s)

Fig. 5. Variation of the force due to the air-sand suspension as a function of the gas superficial velocity, at constant mass flow rate of the solids. tuations of the ball (radial displacement of ball is of the order of the magnitude of its radius). Consequently, this ‘discontinuous phenomenon’ could correspond to the change in the flow pattern around the ball. In Fig. 6, it appears that the measured force due to the presence of particles increases with mass flow

10

0

20

Fig 6. Variation of the force due to the dispersed phase as a function of mass flow rate of the solids, at constant gas superficial velocity. rate of sand particles. Otherwise, the extension of these results in terms of dispersed contribution to the forces acting on the ball (FrJFr) is summarized in Fig. 8. It is subsequently demonstrated that for a given gas superficial velocity, this ratto increases with mass flow rate of solids. This implies negligible particle-particle interaction due to the use of dilute suspensions. The interaction can inhibit the action of dispersed phase, viz., the force F, remains on the increase.

Theoretical interaction

approach to the particle-ball

Figure 9 illustrates the impact of a sand particle with the stationary ball (V, = 0). As the particles are not rotating about their axis either before or after collisions, the vector velocity ?s defines the kinematic state of the particle-ball system before collisions. When the sand particle comes near the fixed ball with a velocity i;;, their centres are at a distance apart of 6=

&+ds 2

(If they are not disturbed). Moreover, in order that a given sand particle (centre 0, and velocity FJ collides with the ball, it must be inside the useful

(b) Fig. 7. Variation air-sand

suspension

m the time of the force due to the (a), W,=4 65 g/s; (b) W,= 8.795 g/s

volume surrounding the centre by the fictive cylinder with sectton&. The random number on the ball with an average time interval dt is given as ‘Ps=N

z (d +db)‘p

“4”

s

dt

Ob which is an effective of particles velocity of

described colhsion ?Psacting ps in the

(2)

Equation (2) is based on the simple hypothesis that the volumetric solid concentration and the veloc~ty profiles are uniform across the pipe section. Because the suspension is dilute, it is assumed that the presence of the ball in the axis of the pipe does not perturb the average value of the volumetric solid concentration. Since the collisions are considered to be instantaneous (the particle-ball contact time 1s infinitesimally short), the random number (lys) of incident particles will be less than unity and equal to the collision number. Thus, the collision frequency per unit volume is given by us,,b =

N,

;

(db + d,)'1;:

(3)

290

As the average solid concentration is proportional to the mass flow rate of solids and inversely proportional to the particle veloctty, eqn. (4) may be expressed as 6 W, d,+d, * v&,= -3 rpsd ( D )

(5)

When a collision occurs, the force per unit volume exerted by the dispersed phase on the ball is given by the product (I,v~,,,) where 1, represents the axial momentum loss of the particle colliding with the ball; I, is proportional to the difference between the incident and the reflecting velocities: Is = n&

- rn&

The projection as:

(6) offs on the OX-axis may be written

I,=m,V,-m,(-V:cos(O+@)

; (g/s) Fig. 8. Contrlbutlon acting on the ball.

of the dispersed

phase to the forces

(7)

The reflecting velocity in the axial direction is related to the Incident veloctty by a tangential coefficient of restitution:

(8)

ps = kti, Substituting becomes

ps by Its expresston

Z,=m,[l +k COS(~+/~)]~~

in eqn. (7), Z,

(9)

Assuming that the sand particle has no spin motion (0= p), eqn. (9) becomes I,=m,(l

+k cos ZO)~s

(10)

As shown by Fig. 10, the prolection of collision surface on the horizontal plane is written as cos

@sin ede

(11)

Integration of eqn. (10) weighted by the d&(0) over the entire range O< 8<~/2 gives the value of the momentum loss of the particle:

FIN 9. Schematrc

diagram of the partxle-ball

collision.

The mean value of particle number per unit volume of suspension (Ns) is proportional to the dispersed phase concentration divided by the spherical sand parttcle volume. Therefore, eqn. (3) becomes (4)

90

I’

Fig. 10 Effective

scctlon of colhsion.

s d2

m,(l +k cos 2@‘s2~

I,=

sin ecos

BdB

O

T2r(Frsin

0cos

0d0

(12) Rearranging

s

of eqn. (12) yields

d2

m,vJl

I, =

+k cos 20) sin 20 d0

0

(13)

lrL?

s0

sin 28 de

I, =m,Fs

(14)

On the other hand, if turbulence effects in the vertical pipe are neglected, the slip velocity between the gas and the particles would be equal to the free fall terminal velocity (U,) of the particle. Therefore, the total force due to the presence of sand particles (FJ may be expressed as F,=W,

(

de

1

‘(&-l/l)

(15)

It ensues from eqn. (15) that the force F, is independent of the coefficient of restitution, viz., the physical nature of the collisions does not affect the action of the particles on the ball. Comparison of the values of the measured force and those of the calculated force, using eqn. (IS), is summarized in Fig. 11. At lower gas superficial velocities, an accurate agreement appears between experimental and calculated values of force F,. So, when Vr> 13.22 m/s and W,> 8.67 g/s, a remarkable difference between these values (10 to 20%) is observed. This signifies that the assumption made in the theoretical approach (U,,= U,) becomes very sensitive under these conditions, viz., in dilute phase pneumatic conveying the mean slip velocity is not necessarily equal to the free fall terminal velocity at high gas superficial velocities. This could give rise to an important relative motion between the gas and the particles before the shocks undergone by the ball and could affect the calculated force.

50

Fig. 11. Comparison of the values of the measured and those of the calculated force.

An experimental device has been used to directly measure the interactive forces between flowing particles and a fixed body. It is shown that the proposed method may be applied to dilute gas-solid transport for the measurement of gas-particle and parti-

force

cle-particle interactive forces. Moreover, it is found that the force due to the dispersed phase acting on the ball is proportional to the gas superficial velocity and the mass flow rate of solids. Finally, a model which is based on the particle-ball collision is proposed to predict this force as a function of the operating variables.

Acknowledgements The authors would like to thank Dr. Yuri Molodtsof for many helpful discussions.

List of symbols

Ab AC D

& 4 dt

Conclusion

100

FD Fr F, FS t

steel ball section, m2 effective section of collision, m2 internal pipe diameter, m mean diameter of steel ball, m surface mean diameter of sand particle, m time interval of collision, s ‘Standard’ drag force, N force due to a single gas, N force due to suspension of fine particles, N force due to dispersed phase, N gravitational acceleration, m/s2 vector-impulse loss of the incident particle, kg. m/s

292 k 111, N, 0, 0, r Ret, (R‘%)w t us, u, Vf K W,

coefficient of restitution mass of sand particle, kg particle number per unit of volume, part./ m3 centre of steel ball centre of sand particle radial position, m Reynolds number related to body size critical Reynolds number time, s slip velocity, m/s free fall terminal velocity, m/s gas superficial velocity, m/s sand particle velocity, m/s mass flow rate of sand particles, kg/s

Greek Jymbols a;

P e vsb PS

%

volumetric solids concentration, reflecting angle of sand particle, ’ incident angle of sand particle, ’ collision frequency per unit volume density of sand particles, kg/m3 random number of particles colliding with ball

Subscripts

D

ball collision drag

f fs sb

gas gas-solid particle-ball

b C

slip sand particle terminal

Sl S

t

References Benali, Docforaf fhesrr, UniversitB de Technologte de Compiegne (1989). 2 L. B. Torobin and W. H. Gauvm, Can. J. Chem. Eng., (1959) 224. 3 L. B Torobin and W. H. Gauvm, Can J. Chem Eng, (1959) 167. 4 A B. Barley, J Flurd. Mech, 65 (1974) 401. 5 B. P. Foster, A. R. Han and I. D. Doig, Chem. Eng. J., 9 (1975) 241. 6 F. W. Roos and W. W. Willmarth, AL4A J, 9 (1971) 285. 7 .I. A. McCammon and J M Deutch, Blopolym, IS (1976) 1397. 8 J G. Kirkwood and J. M. Rtseman, J Chem. Phys., 16 (1948) 565. 9 T. Trang-Gong and N. Phan-Thien, Phys Fluids A, 1 (1989) 453. 10 H. Brenner, Chem. Eng. Ser., 19 (1964) 703. 11 I. Ktm and A. J. Pearlstein, J. Fluid Mech , 211 (1990) 1 M.

73.

12 M. Yokosawa, Y. Kosawa, A. Inoue and S. Aoki, Int. J. Multrphase Flow, 12 (1986) 169. 13 I. D. Doig and G. H. Roper, Ind Eng. Chem Fundam, 7 (1968) 459. 14 S. Matsumoto, H. Harakawa, M. Suzuki and S. Ohtani, Int J Multiphase Flow, I2 (1986) 445. 15 S. L. Lee, Int J. Multiphase Flow, 13 (1987) 247. 16 H. Schhchting, Boundary Layer Theov, McGraw-Hrll, New York, 4th edn., 1960.