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Deposition and entrainment in pipe flow of a suspension
Powder Technology-Elsevier
Sequoia
SA., Lausanne-Printed
in the Netherlands
283
Deposition and Entrainment in Pipe Flow of a Suspension S. L. SO0 and S. K. TUNG* Department of Mechanical and Industrial Engineering, University of’lllinois at Urbana-Champaign, Urbana, Ill. 61801 (U.S.A.) (Received
February
7, 1972; in revised form March
7, 1972)
Summary Extending from previous studies of the fully developed flow of a suspension of particles in a turbulentfluid in gravitational and electricfields and a shearflowjeld, the effect of sedimentation is taken into account. Additional considerations from previous studies are diffusion and settling under field forces, the sticking probability of a particle at the wall and that to a bed of similar particles. The transient condition gives the rate of build-up of a bed of deposited particles. The method is applicable to pipes at any inclination to the direction of gravity.
1. INTRODUCTION
In our previous studies of pipe flow of a suspension, we have considered only cases offully developed motion where there is no net deposition of particles by field forces’ -4. It is well known that sedimentation due to gravity occurs in the hydrotransport of sands, ores, and sewage and that sedimentation due to electrostatic effect occurs in transfer lines of catalytic cracking system and chemical processing of powders in gaseous suspensions. Deposition by g field force may take place when the particle concentration at the pipe wall increases to its packed bed value or when particles start to adhere to the pipe wall. The former may result from a large concentration of particles, while the latter may occur even in a dilute suspension. In both cases, a layer of solid particles may build up to a point such that a sliding bed will proceed downstream and may actually reach a condition of steady flow. However, an alternative situation is unsteady flow with formation and blow away of dunes or unsteady * Present address : General Electric Company, Marine Turbine and Gear Department, 1100 Western Avenue, West Lynn, Mass. 01905. Powder Technol., 6
(1972)
flow with particles moving from one dune to the next undergoing deceleration or acceleration5. The idea of minimum transport velocity was suggested by Thomas6 in dealing with the transition from dilute phase transport to packed bed. He defined this minimum transport velocity as the mean stream velocity required to prevent accumulation of a layer of stationary or sliding particles at the bottom of a horizontal pipe. He showed that, for some solids, this velocity is directly proportional to the volume fraction of particles. Bagnold7 identified the ratio of densities of particles and fluid materials as the basic factor affecting minimum transport velocity at
S. L. SOO, S. K. TUNG
284 re-entrained gives CJ< 1. Another sticking probability cw concerns adhesion of particles at the immediate vicinity of the wall even when there is no drifting motion. Opposite to settling is the lifting of a particle in the shear flow field of a fluid. This leads to a redistribution of density of particle clouds and erosion of a bed of deposited particles. The fluid phase is taken to be turbulent in all cases, although the method is readily extended to a laminar flow field. Some experimental data are available for comparison to the results of the present study, but not in every case to be delineated here. The aim of this presentation is to develop further understanding and to identify important areas of study and needed measurements. 2. FLOW
SYSTEM
As in a previous studyj, the flow system consists of a circular pipe with its axis making an angle 0 with the direction of gravity of acceleration g as shown in Fig. 1. In Fig. 1, z, Y, and 4 are the axial, radial, and azimuthal coordinates; u, u, and w are
/u
the conjugate components of velocity of the fluid; and up, up, and w,, are those of the particle phase. We shall treat only the case of a single species of monodispersed suspension of spherical particles of radius a, although extension to include a distribution in particle size is readily accomplished17”. The material constituting the fluid and the particle phases are p and &, respectively, while those of the fluid phase and the particle cloud in the suspension are p and pP, respectively. The phases are further characterized by its inverse relaxation time for momentum transfer : F = F* [9 @/2a2)&]
Powder
Tech&.,
system and components
6 (1972)
of gravitational
accel-
(1)
including
the effect of apparent mass, and F*= accounts for the deviation (C,/~~)[~~(~IU-U,I)~CI] from Stokes’lawofdrag; othernotationsare :,Gis the viscosity of the fluid material, u, up are velocity vectors, C, is the drag coefficient for a sphere which is influence by both the Reynolds number of relative motion of particle to fluid, [2a(P 1u - u,l]/,ii, and the cloud density’7b ; F* = 1 in the Stokes’ law range. The non-sphericity of particles may be accounted for by modifying C,, but may also cause lift and moment depending on its orientation in the flow tie1d.A lift force is also exerted on a spherical particle in a shear layer of a fluid’,2,4. When dealing with particle diameters larger than the laminar sublayer of the fluid phase, the lift force due to shear may be neglected. Although the concepts presented here are applicable to laminar pipe flow also, turbulent fluid flow is used in illustrations in the following. The transfer of particles by the gradient in particle density is characterized by the diffusivity of particles D,. D, accounts for the random motion of particles in the flow field as induced by the diffusivity of the fluid whether laminar or turbulent, by the wake of the particles in their relative motion to the fluid, by the Brownian motion of particles, by wall interaction, and by perturbation of the flow field by the particles. For a dilute suspension, the transport of momentum of particles is given by its gradient or “stress” given by the tensor. Tp = pp D, [Vu, + vq
Fig. 1. Coordinate eration.
[ 1 + (P/2&)] - l
- $pp D, (V u,)U
(2)
where U is a unitary tensor. For a dilute suspension, particle-particle interaction can be neglected. The interaction length L, of particle-fluid interaction characterizes the response of a particle to the random (turbulent or molecular) motion of elements of the fluid. LP is defined as the
DEPOSITION
AND
ENTRAINMENT
IN PIPE FLOW
relative intensity of random motion of particles to fluid times the relaxation time of their momentum transfer. At a boundary (wall&w), the slip velocity tangential to the wall is given by (3)
“pw= &A,
where A,,, is the deformation tensor consisting of velocity components parallel to the wall. As was discussed earlier2, in a dilute suspension, that is, free path of particle-particle collision > L, > interparticle spacing >2a, the fluid motion is unaffected by the presence of the particles. This assumption is still valid when deposition of particles occurs if the thickness of the layer of deposit is much smaller than the pipe radius, so that the effective reduction in pipe radius is not enough to change appreciably the fluid velocity distribution. A stagnant layer having a thickness of 0.1 times the pipe radius changes the fluid velocity by a factor of 0.02. 3. GENERAL
RELATIONS
As in our earlier studies, we shall still restrict ourselves to cases where fluid motion is unaffected by the presence of particles. This means that the motion of the particle phase corresponds to viscous slip motion and particles are not correlated to each other even when the fluid is turbulent”. However, because of the deposition of particles, fully developed motion of the particle phase does not occur. The equation of motion of the particle phase should include the inertia forces or’ 7a d”, PP
dt
au, at
285
OF A SUSPENSION
The flux of particles due to field and fluid forces is now given by P,(“,-“)=(f,~,l~)+(~/~)V~~,
(6)
Where deposition may occur, unlike in fully developed motion, equality of fluxes due to diffusion and field forces’ cannot be specified. Rather, the conservation of mass has to be accounted for by the diffusion equation for a dilute suspension: (P ‘P,)
dp, _ - at
@P
dt
+ u.2
= -V[-DpVpp+pp
(up- ")I
(7)
Substitution of eqn. (6) gives, for a quasi-stationary state,
ap =
u .p
au
V~(DpVpp)-F-lV~ppfp
since substitution of eqn. (5) leads to V. Vx which gives zero and higher derivatives ofp, can be neglected. When applied to the pipe flow system, the components of field forces and the fluid force jr_ are (neglecting the $, z components of shear force due to shear flow in 4, z directions):
) (9)
d“,
=
PP
+
PP”P’F
=
Fpp(“-“P)+pPjp+V~t,
1
P
(4)
and v.z, = vx (ppDpVx “,)+2(v~&D,v)“, -$V L-P,D, (V. “,)I
V.E=p,@
(5)
where t is the time, P is the position vector, f, includes field force and fluid force per unit mass of the particle phase due to gravity and electrostatic effects. We shall treat the case where the axial gradient of the total particle flow is small such that we can assume a quasi-stationary state even for transient deposition; that is, a/&=0. Moreover, since particle velocity distribution over a pipe is not strongly influenced by different loading of similar particles3,17c, all the inertia terms can be neglected. PowderTechnol.,6 (1972)
where q is the charge, m is the mass of each particle, the E’s are components of the electric field E= - VV, V being the electric potential, and E is given by the Poisson equation
(10)
“0
where 8. is the permittivity of free space and E, is again small for gradual deposition. The boundary condition is given by the conservation of total flow: p,u,rdrd$
=
2a _
OR o
+
4v2~R~~dftlF)
‘pR&Rdd
-
“w 2R&‘,,
WF)
286
S. L. SOO. S. K. TUNG
wheref;, is the adhesive force per unit mass at the immediate vicinity of the wall. The flux of particles produced by the erosion from a bed is given by: (probability of lift) times (mass per unit area of surface layer of particles) times (frequency of lifting of particles). The mass per unit area of surface layer is given by : (4x/3)* @%a&, where Qb is the volume fraction particles in the bed. Since the frequency of lifting is given by the lift velocity (fJF) divided by this mean distance between successive particles lifted, the flux of particles produced by erosion is thus o&p,,JJF, where the probability ok accounts for the difference between bed density ppb and the above quantity of mass per unit area divided by the mean distance of lifted particles. Since we assume the motion of the fluid phase as fully developed (&@z=O), we have (E,-0) ;
(PpU,)= g CPp(Up-41+u
= p
-
O)ppR
+
dv
(12)
Ppt,] K./F)
(13)
which, for o=O, o,=O, &=O, reverts to the nondepositing boundary condition obtained before3. The relation offr to basic physical properties was dealt with previously 1,4. It is reconsidered in the
in the relations
(14) where U, is the local relative velocity of particles to gas and C, is the lift coefficient, which can be represented in the form:
The coefficient r1 and exponents m, n and p are given in Table 1’,2,19-21. In this general representation, the previous relation1,2 for.fr is now
(16) where the relative velocity is givenaby
Saffman19.,f~, (theoretical)
12.92/n
Eichorn’t (empirical)
7x lo4
Graf and AcarogluZ3.24
50
Kingsburyz6
Technol.,
6 (1972)
(17)
Au= l&/&la, and the shear response number Ns is now Ns=~i,~~~)“(~)“+p-m-l
(18)
correlating shear force of fluid to inertia force of
of lift force
Rejirence
Powder
When treating cases including turbulent flow over an erodible bed of solid particles whose settling is mainly due to gravity, .fL in eqn. (13) is the lift force per unit mass acting on a particle by fluid shear and
I
Constants
FLOW
u, = up-u+Au
F-' E,P,,-%P,,(j;/F)
TABLE
IN SHEAR
(15)
(1 -f)gcosO]~
[(I
4. LIFT FORCE
f$
Substitution of eqns. (12) and (8) into eqn. (11) gives, with ~~~-j~~lF: _ D, 2 = - (1 - 0) 1 - P S sin 8 cos f$ppR R PP 1 F ( +
next section in accordance with the above equation, (13), and experimental results.
,n
n
P
Note
t
f
0
Very small particles and slow relative motionzO
2
2
2
Large particle in Poiseuille profile”
I
t
0
Present analysis of their results according to turbulent boundary layer theory”
1
I
0
Laminar bearing
motion
in slider
DEPOSITION
AND ENTRAINMENT
IN PIPE FLOW
OF
particles. We shall denote the Saffman expression as_L. In general,fi should be included over the whole shear layer in eqn. (9). jr_ is insignificant for both 2a <<6 and 2a 9 6,6 being the boundary layer thickness. The boundary condition in eqn. (13) for a horizontal pipe or 8 = n/2, rr/2 < 4 < 3x/2, and E, = 0, for density of bed ppb, D,
2 = -(l
-a)
R
+
_$0, c! 1- ;
$ppR
(I- 4PplJftRIF)+ 4vPpb(ftR ‘F)=
=-(l-a)R,-
287
A SUSPENSION
experiments has yet to be identified. A similar trend is seen in Wickszs. Comparison with eqns. (23) and (A12) shows that the range of experiments in Graf and Acaroglu23*24 is represented more closely by m= 1 and p =0 in eqn. (1.5)for the lift coefficient of the particles, while the value of n has no influence when the relative velocity is approximated by (&/&)a. Alternatively, it appears valid to visualize lifting of a particle as produced by film action as in a slider bearingz6. Based on the bearing theory, we get m= 1, n= 1, and p=O. With rn= 1, p=O. we get
- cr;] R, (24)
(19) In the above expression, the first term (R,) gives deposition by gravity and the second term (R2) gives re-entrainment by fluid shear. It is seen that, for a large pipe, R 9 L,, up N u, we have jLR = -N,(u,/R)“(alR)‘-“-“a
li?u/c3rl~-m
(20)
and (25)
A reasonable choice, to be substantiated in Section 8, is n=S. We have therefore, based on Graf and Acaroglu,
Further, for a turbulent fluid and a < y,. y, being the thickness of the laminar sublayerz2, or y,= 6O(u, R/C)-” R, u, = u. (y,/R)‘, we get
au
ar l-l
z-2
R
u
(21)
Ys
(26) 5. CORRELATION
In the following, we treat specific cases using these dimensionless quantities :
and, for c’, =(3)2n-m-360-(9)(2-m)C1 ,
r* =*rlR,
An interesting quantity is the ratio of the quantities R, and R, on the right-hand side of eqn. (19) at 4=x or
5 = (&-P)
R*=
&I
R2
=
$
($5)
(g-p-’
9 PPR _fiLR &b
f$)
pf]““ml-ci’
(23) which characterizes the sedimentation and transport and serves as a basis for comparison to correlations of experimental results in the Appendix. Equation (23) serves to substantiate the logic behind the dimensionless correlations in eqn. (A12) in the Appendix based on experimental results of Graf and Acaroglu 23,24.The diversity of the power of the Reynolds number (uoR/i) is believed to indicate the diversity of the state of relative motion of particles to fluid. The particle diffusivity in these Powder
Technul., 6 (1972)
PARAMETERS
z* = z/R,
pp*= pp/ppl = pp*(r*, z*, 4) 24:= up/u0 = $(r*,
u* = u/u0 = u*(r*),
E*= -V*I’*
V* = (q/m)V/D,F,
z*, 4)
(27)
where ue is the fluid velocity at the center of the pipe, ppl is the density of particles at the initial condition (z* = 0) of fully-developed motion, analogous to a thermal entrance condition. We also define the following parameters as before3 : 0:= (ppl/%)(dm)2
(R2/~,F)
/3= R’F/D, y = 2 [l - (p/p,)]R2g
q = 2[1-(p/p,)]Rg
cos B/D,&) sin f3/DpF
and the momentum transfer number N,,,and particle Knudsen number NKp N,,, = Q/RF &p = LpIR
S. L. SOO, S. K. TUNG
288
In this case, E,=O, deposition is by sticking at the wall alone. This was the case studied by Friedlander and Johnstonez7 and recently by Yoder and Silverman including effect of agglomeration2*. Here we are concerned with a very dilute suspension of, say, room dust. Equation (31) becomes
and an adhesive force parameter ;1= RfJFD, which has not been used before. f, acts only on particles at the immediate vicinity of the wall. In addition, the shear-lift parameterzv4 takes two possible forms :
(33) with u* =u* (r*) for turbulent boundary conditions are
according to the lift force of Saffman, and
r* = 0
6. ELECTROSTATIC
SEDIMENTARY
FLOW
&*
a - - r*p; r* ar*
+ flp;(u*-u,*)=o
(28)
(29)
the Poisson equation, i
a
r* ET = 4c(p,*
(30)
and the diffusion equation, -
I
a
r* &*
r*p,* E:
(31) The boundary conditions are r* = 0
r* = 1’
a6 ar*
a&Z_- -cQ”p;
1
’
&*
pp* = c
ck exp
[ -
k2
(z*lPKJlJ&r*)
k -
exp( - CJ,~~Z*/~/?N,,,)[~ -$a,h*2
+&a$E.2r*4...]
(35)
v,* = E,*/JN,,,
r* &*
-
and E,may include the Van der Waals force and the electrostatic force. Other than the dependence of u* = u*(r*), the solution of eqn. (33) is straightforward. For turbulent flow, we may take u* - 1 and we have
As an extension for the earlier study of turbulent flow of a charged suspension without gravity effect’, we treat the effect of particle charge on deposition. The above simplifications give the momentum equations, i
and the
%-0
’
according to the correlation of Graf and Acaroglu. r* =
motion,
’ u* = -N P
% Kp
i%*
av* -=
at-*
0
’
(32)
Note that eqns. (30)-(32) are satisfied by p: = [1-(0~/2)r*~]-~ when o=O, o,=O, thus reverting to the case without deposition. The following cases can be identified : e@t
Powder Technol.. 6 (1972)
BWnu*ap;_ ~ az*-
~ 4~r
-
&
[P,*Spp*d(r*2)]
(36)
For u*w 1, we have a similar case as in the electrohydrodynamic inflet flow, which has the solution, for 5 = z*(4cr/bN,,,),
-(1-0):&Z-0,i;
Case 1. Dij’hion
where J, is the zeroth order Bessel function of the first kind, ck is the Fourier coefficient for an eigenvalue k. The approximation is for small cr,$, and a,f,/4F is the deposition velocity defined by Friedlander and Johnstone27, now expressed in terms of material and surface properties. An example is the adhesion and deposition of quartz particles to glass”. This force amounts to 0.01 dyn/um of particle size. A deposition velocity of 10 cm/set in air suggests cw= 1.5 x 10m4. Case 2. Electrostatic force alone. When the electrostatic force due to self-field is significant, we can neglect both the effect of diffusion and 1. We get, from eqn. (31)
alone in a dilute suspension.
~p*=(l+[)-~+r*~(l+[)-~+...
(37)
This relation is given in Fig. 10.16 of SOO”~. Case 3. Simultaneous action oj’dij’tision and electrostatic repulsion. Equations (28) and (31) have to be
DEPOSITION
AND ENTRAINMENT
IN PIPE FLOW
289
OF A SUSPENSION
solved numerically 3q4. Although any form of initial condition can be accounted for, the change from a fully developed condition at z =0 is sufficiently interesting. Figure 2 shows the change in particle density at the wall along the length of the pipe. The changes in density at the center of the pipe and density at the wall in the axial direction are readily seen. The particle velocity was determined from fluid velocity given by the Sth velocity law.
In this case, G is that for particle-particle sticking and f, is negligible. When the bed of deposits is stationary, the rate at which its thickness is built up is given by (40)
d&ldt = ~puvp,lpp,
At steady state, we may have a deposit layer at the wall of thickness 0, moving along the z-direction (41) where Qs is volume fraction particles in the moving bed, u, is the mean velocity of the deposit layer moving downstream. u, is given by the equalization of shear stress in the packed bed to that at the boundary of the dilute phase.
4
7. GRAVITY
EFFECT
Consider the gravity effect alone ; under the simplifying assumptions in Sections 2 and 3, the momentum equations take the same form as before2v3 :
3
l a (**~)+I&?&@) -SW,*+r*ar* r Pp
Ppw 2
+fip,*(u*-u,*)=O
1
(42)
and the diffusion equation now becomes ’
0
’
Fig. 2. Electrostatic 0=0.2. %=O.
Case4
’
2
’
’
4
Z/R sedimentation.
’
’
6
J
8
fl= 20, NKp = 0.2, N,,, = 2.0,
Attainment ofpacked
or moving bed density. A limiting condition is when CIis large enough so that the density of a packed or moving bed pps of volume fraction solid @, is reached at the wall ; that is, we have -2 PPR=
=“-=g
1-5 i
PPl
!
P Pl
(38) 1
when sedimentation occurs because of slowing down of a wall layer. For Qi,% G1, such a condition occurs when @, > 2/E
(39)
where _ a=G
Powder
PP -
4 0
2 R2
m
Technoi., 6 (1972)
The boundary conditions are 4 = 7112,3~12, ap;/ar*l,
= 0
and, from eqn. (13). $LZ = -(l-o)& &* 1
cos $hp,*-o,E.p,*
(44)
obviously for n/2 < 4 < 37c/2. Since the top cannot have particles falling into the suspension, the boundary condition becomes ap* 2 = -;yI cos 4p;-o,Ip,* ar* 1
(45)
for -n/2< $< ~12. Note that the reverse has to be specified when lip < p, such as in the case of buoyant particles or bubbles. The case where gravity effect is negligible obvi-
S. L. SOO, S. K. TUNG
ously reduces to Case 1 in the above section. We have the following additional cases : Case 5. Gravity flow alone, negligible I.. In this case, v-+ co, CJdoes not play a role. We introduce a coordinate 5’ such that
1
sin edz* (46) and, taking u*w 1 for turbulent flow, eqn. (43) becomes (47) In this case, pp*remains constant except that, starting from fully developed motion, the top boundary of the particle phase falls according to the case of batch settling; that is, the top surface of the suspension is at y=2R-Rt’
Z/R :
(48)
with y=O at 4=x, r=R, for 0fO (0=0 for vertical pipe) sedimentation following Fig. 9.1 of SOO”~, Fig. 4. Deposition N,,, = 2.0. i. = 0.
rate at the wall. fl= 10, I’= 1, q=2,
N,,=O.2,
with a packed bed of Qsbuilding up from the bottom of the pipe. Case 6. Simultaneous action of diffusion and gravity effect. Solution can be obtained only
0.5
1.0
1.5
L
2.5
2.0
3.0
PP
0.
0
:
0.5
=
numerically. With fully-developed flow as initial condition and Sth turbulent velocity flow from the fluid phase, computations were carried out for A= 0. Figure 3 shows the change in density distribution and Fig. 4 shows the deposition flux at the wall. Case 7. Attainment of packed bed density at the bottom ofthe pipe. This case is analogous to Case 4.
1.0
This may lead to sliding bed motion in both the axial and peripheral directions.
-
8. EFFECT 1.0 ’
0.5
1.0
1.5
*
2.0
2.5
3.0
PP
Fig. 3. Density distribution when deposition occurs due to gravity and diffusion. p= 10, y= 1, q=2, N,,=O.2, N,,, =2.0, 1= 0.
Powder Technol., 6 (1972)
OF LIFT IN SHEAR
FORCE
The influence of lift of particles by shear motion of a turbulent fluid is seen to be important in many cases of liquid-solid suspensions, including settling and erodible beds. However, most of the experimental data are on overall transport relationsz4.
DEPOSITION
AND ENTRAINMENT
IN PIPE FLOW
Detailed measurements on density distributions are available only in the case of non-settling pipe flows. When there is no settling, the distribution in particle density under the influence of gravity and shear flow is given byze4
aln PP*_ ar*
TOP 1.0
r*
0
-+q cos c#)-L*
a lnp* ~ = $j sin f$~ r* a4
(50) 1.0
where L* takes the form
I I
(51)
based on Saffman’s lift, or
f (u,*-u*+Au*)
L* =i
2
based on Graf and Acaroglu. Computations were made with both relations for comparison to results of the experiments performed on a suspension with particles of neutral buoyancy in a fluid in laminar motion by Segrk and Silberbergz9 and on suspensions of sand and perspex in water in turbulent motion by Newitt, Richardson and Shook3’. Case 8. In the horizontal flow of liquid suspension of sands and perspex, computations with parameters based on properties of materials given in experiments of Newitt, Richardson and Shook3’ yielded mass flows as shown in Fig. 5 for coarse sand and Fig. 6 for perspex. The related material properties and parameters are given in Table 2. It is
I -
0.5
,
I
,
7’10, @50,[,=84or --
7’8,
0.6
0.8
1.0
1.2
1.4
Pp”$
Fig. 6. Mass flow distribution in vertical plane of perspex in water (y=O, nlR=l/l3.5, N,,=O.l, NRc=104) as compared to experimental results of Newitt et al. (mean volume fraction solid : A 0.0983, 17 0.0295).
TABLE
I
2
Properties
and parameters
in the experiments
of Newitt et al.
Ptrrticles
Perspev
Cmme
20, in.
0.73 13.5 186 0.173 1.17 2.5 1200 0.1 0.6 100 10-S
0.0275 35.8 103 0.313 2.66 10 50 0.2 5 250 1o-6
Rla
F, set-
’
g/F, fps P,!P 4 B N, &VI il D,, in.‘/sec v=[O]
,
0.4
(52)
[up*-u*+Au*l+
I
0.2
Bo~+om
du* + dr*
,?I =[,(u,*-u*+Au*)
Top 10
291
OF A SUSPENSION
10-6m2/sec,R=0.0125m,N,,=[0]
sand
104,y=0,NKp=0.1.
,
5,=21
p=500, cl=250 or c2=62
IX 0
0.5
noted that their measurements3’ yielded mass flow instead of density. Figure 6 shows that the lift force according to eqns. (26) and (52) is closer to the experimental facts. In Fig. 5, however, corresponding values of i, and i2 gave similar curves. From these two comparisons, for D, given by cl, we can determine c;. We get c; -0.5, and c1 w 50, in eqns. (15) and (24).
10 10-z Bottom
10-l
Fig. 5. Mass water (y=O, experimental A 0.0550, 0
Powder
1
10
102
PZUP” flow distribution in vertical plane of coarse sand in n/r=1/35.8, N,,=O.l, NRe=104) as compared to results of Newitt et nl. (mean volume fraction solid : 0.0648).
Technol.,
6 (1972)
9. DISCUSSION
It is seen that sedimentation in pipe flow may occur for a variety of reasons, although their results are often equally undesirable. The cases treated in
292
S. L. SOO, S. K. TUNG
the above show how sedimentation may get started. Once started, three more cases call for a detailed treatment of the time dependence: Case 9. Transient deposition may continue until the pipe is completely plugged. Case 10. Transient deposition together with bed movement. Case 11. Formation of dunes and repeated piling up of deposits until a large pressure difference in the fluid is built up to blow it away. This causes a pulsat-
ing motion in the pipe. This wave motion is produced by a variation in fluid velocity due to flow restriction by the dunes. We have now identified the parameters LX, ,$ y, v], i, A, N,,, and A&, for treating the problem of sedimentation. They permit the prediction of density distribution and conditions for deposition in design. When dealing with gas-particle suspension, deposition in the form of dipolesr3*14 or space charges can be equally problematic unless the interest is in collection of particles. Precharged particles are deposited in pipes by space charge in the above or by instability under the image force even when there is only a single particle, because diffusion prevents a particle from staying in the middle of the pipe. By stretching a high voltage wire along the axis of a grounded pipe conveying a suspension, we can charge the particles by a corona discharge and deposit them by the field ; this constitutes an electrostatic precipitator3’. In liquid slurry lines, or other high density conveyance systems, deposition occurs primarily by the formation of packed bed. The computer programs for Cases 3,4,6.7 and 8 are given in a thesis by Tung3’.
LIST OF SYMBOLS
a
radius of a particle Fourier coefficient cr, c; constants as defined drag coefficient for a sphere Cn lift coefficient for a sphere in shear flow field Cr. particle diffusivity D* E electric field force vector per unit mass of a particle :% radial, tangential, and axial force per unit ck
2: 1 F
Powder
mass of a particle inverse of relaxation time or relaxation time constant for momentum transfer Technol.,
6 (1972)
F*
coefficient for deviation from Stokes’ law gravitational acceleration Bessel function of zero order ;0 particle-fluid interaction length -& m mass of a particle m, n, p constants in exponents NKP Knudsen number of the particle phase momentum transfer number N, shear response number N, electric charge per particle 4 radial coordinate r position vector pipe radius k time t u V U’ I UP
VP
axial, radial, and tangential component velocity of the fluid
of
axial, radial, and tangential component velocity of the particle
of
WP I uo
U DW
maximum or core velocity of the fluid phase velocity of particles at the wall unitary tensor vectorial velocity of the fluid vectorial velocity of the particle cloud electric potential axial coordinate
dimensionless groups as defined deformation tensor permittivity of free space dimensionless axial coordinate angle of inclination of the pipe with the direction of gravity viscosity of the material constituting the fluid phase density of the fluid phase density of the material constituting the fluid phase density of the particle cloud density of the material constituting the particles sticking probabilities as defined shear stress tensor of the particle cloud azimuthal angle volume fraction particles Superscript * dimensionless quantities as defined
transposed tensor
DEPOSITION
AND ENTRAINMENT
IN PIPE FLOW
Subscript 1 initial condition cases as indicated I.2 for particle phase P s packed bed condition R at radius R w for wall condition
REFERENCES 1 S. L. Soo, Pipe flow of suspensions, Appl. Sci. Res., 21 (1969) 68884. of states of hydraulic transport of 2 S. L. Soo, Correlation solids in pipes, Proc. Hydrotrnnsporr Conf., Univ. o/‘Worwick, England, BHRA, A-l, 1970. 3 S. L. Soo and S. K. Tung, Pipe flow of suspensions in turbulent fluids--electrostatic and gravity effects, Appl. Sci. Res., 24 (1971) 84-97. 4 S. L. Soo and S. K. Tung, Gravity effect in pipe flow of suspensions, Paper No. l-5-164, Symposium o/’ Flow, Pittsburgh, Pa., May 9-14, 1971. 5 J. F. Kennedy, The formation of sediment ripples, dunes and antiduores, Ann. Rev. Fhid Mech., 1 (1969) 147-168. characteristics of suspensions. II. 6 D. G. Thomas, Transport Minimum transport velocity for floculated suspensions in horizontal pipes, Am. Inst. Chem. Engrs. J., 7 (3) (1961) 43340. 1 R. A. Bagnold, The movement of a cohesionless granular bed by fluid flowing over it, Brit. J. Appt. Phys., 2 (2) (1951) 29934. 8 G. W. Seman and G. W. Penny, Photographic records of particle trajectories during electrostatic precipitation, Power App. Systems, 86(3) (1965) 365-368. 9 S. L. Soo and L. W. Rodgers, Further studies on the electroaerodynamic precipitator, Powder Technol., 5 (1971) 43-50. 10 M. Corn, The adhesion of solid particles to solid surfaces, J. Air Pollution Control Assoc., 11(l) (1961) 523-528; ll(12) (1961) 566575, 584. 11 D. D. Eley, Adhesion, Oxford Univ. Press, London, 1961, p. 119. 12 W. Pietsch, E. Hoffman and H. Rumpf, Tensile strength of moist agglomerates, Ind. Eng. Chem. Prod. Res. Develop., 8 (1) (1969) 5862. 13 G. W. Penny and E. H. Klingler, Constant potential and the adhesion of dust, Commun. Electron., 81(l) (1962) 20&205. 14 J. M. Niedra and G. W. Penny, Orientation and adhesion of particles, Ind. Electron. Control Instrumentation, 12(2) (1965) 4650. 15 W. Kottler, H. Krupp and H. Rabenhorst. Adhesion of electrically charged particles, Z. Angew. Phys.. 24(4) (1968) 219-223. 16 H. Czichos. Uber den Zusammenhang zwischen Adhiision und Elektronenstruktur von Metallen bei der Rollreiburg im elestischen Bereich, 2. Angew. Physik, 27(l) (1969) 4@46. 17 S. L. Soo. Fluid Dynamics of’ Multiphase Sysrems, Blaisdell, Waltham, Mass., 1967. (a) Ch. 6, (b) Ch. 5, (c)p. 169, (d) p. 469, (e) p. 368. 18 S. L. Soo, Think two-phase, in I. Zandi (ed.), Adoances in Solid-Liquid Flow in Pipes and Its Application, Pergamon, Oxford. 1971, pp. 35-38.
Powder Technot., 6 (1972)
293
OF A SUSPENSION
19 P. G. Saffman, The lift on a small sphere in a slow shear flow, J. Fluid Mech., 22(2) (1965) 385-400; On the motion of small spheroidal particles in a viscous liquid, ibid., I (1956) 540-553. 20 M. T. Lawler and P. C. Lu, The role of lift in the radial migration of particles in a pipe flow, in I. Zandi (ed.), Adoances in Solid-Liquid Flow in Pipes and Its Application, Pergamon, Oxford, 1971. 21 R. Eichorn and S. Small, Spheres suspended in Poiseuille flow, .I. Fluid Mech., 20(3) (1964) 513-527. 22 H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1960, p. 508. 23 W. A. Graf and E. R. Acaroglu, Sediment transport in conveyance systems. Part I. A physical model for sediment transport in conveyance systems, Bull. Intern. Assoc. Sci. Hydrol., 23 (2) (1968) 20-39 : Part II. The modes of sediment transport and their related bed forms in conveyance systems, ibid., 23(3) (1968) 123-135. 24 W. H. Graf. Hydraulics of Sediment Transport, McGraw-Hill, New York, 1971, p. 442. in horizon25 M. Wicks, Transport of solids at low concentration tal pipes, in I. Zandi (ed.), Aduances in Solid-Liquid Flow in Pipes and Its Application, Pergamon, Oxford, 1971, pp. 101-124. New York, 1942, 26 A. E. Norton, Lubrication, McGraw-Hill, pp. 73, 77. 27 S. K. Friedlander and H. F. Johnstone, Deposition of suspended particles from turbulent gas streams, Znd. Eng. Chem., 49 (1957) 1151-1156. on 28 J. D. Yoder and L. Silverman, Influence of turbulence aerosol agglomeration and deposition in a pipe, Paper No. 67-33. 60th Ann. APCA Mfg., Cleveland, June 13-17, 1967. 29 D. M. Newitt, J. F. Richardson and C. A. Shook, Hydraulic conveying of solids in horizontal pipes. Part II. Distribution of particles and slip velocities, Proc. Symp. Interaction between Fluids and Particles. London, June 1962, Inst. Chem. Engrs., pp. 87-100. 30 H. J. White, Industrial Electrostatic Precipitation, AddisonWesley, Reading, Mass., 1963, p. 35. 31 S. K. Tung, Flow of suspensions under field forces (August 1971) (microfilm copy available from University Microfilms, Inc., Ann Arbor, Mich.).
APPENDIX CORRELATION OF EXPERIMENTAL OF GRAF AND ACAROGLU
RESULT
The correlations of Graf and Acaroglu23*24 on deposition and erosion of particles appear to be most general and straightforward. They give a “shear intensity parameter” Y=
=
critical shear stress to lift available shear stress
(Al)
294
S.
where s is the slope of energy grade line and R, is the hydraulic radius. s is given by v’, = r,,/p = gR,s
(-42)
or
They also gave a “transport parameter” (A4)
U being the mean velocity U=O.Su for pipe flow or pP,U is the mean mass flow. A large number of test data from various sources were correlated to give Q,= 10.39(Y)-2.52
(A5)
making possible estimations of transport capacity. Depending on the mode of transport, Y is given by the shear motion and the whole range given by Graf and Acaroglu may be correlated according to Y Rc$.*~ = 9360(2~,,)‘.*~
PowderTechnol.,6 (1972)
TUNG
Equation (A6) was delineated over the following ranges : critical condition (start motion) Y=8 Re,520 ripples and dunes (beginning of ripple) Y=4 Re, 58 (disappearance of ripple) Y =0.85 Re, I plane bed (bed load and suspension) Y=O.2 Re,,<2.6 sliding bed or anti-dunes (predominantly suspension) Y = 0.06 Re, 5 1.4 Experimental results are often correlated in terms of fluid shear velocity v.+or r: = %lP
(A8)
where r,, is the shear stress at the bed of particles. For pipe flowz5 : ~1,= O.l5u, (T/Ru,,)*
(A9)
Substitution into eqn. (19) gives
(A6)
where a mmdenotes a in mm and Re, is the Reynolds number based on shear velocity and particle diameter or Re, = 2a(v,/V)
L. SOO.S. K.
(A7)
For flow through circular pipes, we have Re, = 0.30(a/R)(uo R/C?
(All)
6412)
Sequoia
SA., Lausanne-Printed
in the Netherlands
283
Deposition and Entrainment in Pipe Flow of a Suspension S. L. SO0 and S. K. TUNG* Department of Mechanical and Industrial Engineering, University of’lllinois at Urbana-Champaign, Urbana, Ill. 61801 (U.S.A.) (Received
February
7, 1972; in revised form March
7, 1972)
Summary Extending from previous studies of the fully developed flow of a suspension of particles in a turbulentfluid in gravitational and electricfields and a shearflowjeld, the effect of sedimentation is taken into account. Additional considerations from previous studies are diffusion and settling under field forces, the sticking probability of a particle at the wall and that to a bed of similar particles. The transient condition gives the rate of build-up of a bed of deposited particles. The method is applicable to pipes at any inclination to the direction of gravity.
1. INTRODUCTION
In our previous studies of pipe flow of a suspension, we have considered only cases offully developed motion where there is no net deposition of particles by field forces’ -4. It is well known that sedimentation due to gravity occurs in the hydrotransport of sands, ores, and sewage and that sedimentation due to electrostatic effect occurs in transfer lines of catalytic cracking system and chemical processing of powders in gaseous suspensions. Deposition by g field force may take place when the particle concentration at the pipe wall increases to its packed bed value or when particles start to adhere to the pipe wall. The former may result from a large concentration of particles, while the latter may occur even in a dilute suspension. In both cases, a layer of solid particles may build up to a point such that a sliding bed will proceed downstream and may actually reach a condition of steady flow. However, an alternative situation is unsteady flow with formation and blow away of dunes or unsteady * Present address : General Electric Company, Marine Turbine and Gear Department, 1100 Western Avenue, West Lynn, Mass. 01905. Powder Technol., 6
(1972)
flow with particles moving from one dune to the next undergoing deceleration or acceleration5. The idea of minimum transport velocity was suggested by Thomas6 in dealing with the transition from dilute phase transport to packed bed. He defined this minimum transport velocity as the mean stream velocity required to prevent accumulation of a layer of stationary or sliding particles at the bottom of a horizontal pipe. He showed that, for some solids, this velocity is directly proportional to the volume fraction of particles. Bagnold7 identified the ratio of densities of particles and fluid materials as the basic factor affecting minimum transport velocity at
S. L. SOO, S. K. TUNG
284 re-entrained gives CJ< 1. Another sticking probability cw concerns adhesion of particles at the immediate vicinity of the wall even when there is no drifting motion. Opposite to settling is the lifting of a particle in the shear flow field of a fluid. This leads to a redistribution of density of particle clouds and erosion of a bed of deposited particles. The fluid phase is taken to be turbulent in all cases, although the method is readily extended to a laminar flow field. Some experimental data are available for comparison to the results of the present study, but not in every case to be delineated here. The aim of this presentation is to develop further understanding and to identify important areas of study and needed measurements. 2. FLOW
SYSTEM
As in a previous studyj, the flow system consists of a circular pipe with its axis making an angle 0 with the direction of gravity of acceleration g as shown in Fig. 1. In Fig. 1, z, Y, and 4 are the axial, radial, and azimuthal coordinates; u, u, and w are
/u
the conjugate components of velocity of the fluid; and up, up, and w,, are those of the particle phase. We shall treat only the case of a single species of monodispersed suspension of spherical particles of radius a, although extension to include a distribution in particle size is readily accomplished17”. The material constituting the fluid and the particle phases are p and &, respectively, while those of the fluid phase and the particle cloud in the suspension are p and pP, respectively. The phases are further characterized by its inverse relaxation time for momentum transfer : F = F* [9 @/2a2)&]
Powder
Tech&.,
system and components
6 (1972)
of gravitational
accel-
(1)
including
the effect of apparent mass, and F*= accounts for the deviation (C,/~~)[~~(~IU-U,I)~CI] from Stokes’lawofdrag; othernotationsare :,Gis the viscosity of the fluid material, u, up are velocity vectors, C, is the drag coefficient for a sphere which is influence by both the Reynolds number of relative motion of particle to fluid, [2a(P 1u - u,l]/,ii, and the cloud density’7b ; F* = 1 in the Stokes’ law range. The non-sphericity of particles may be accounted for by modifying C,, but may also cause lift and moment depending on its orientation in the flow tie1d.A lift force is also exerted on a spherical particle in a shear layer of a fluid’,2,4. When dealing with particle diameters larger than the laminar sublayer of the fluid phase, the lift force due to shear may be neglected. Although the concepts presented here are applicable to laminar pipe flow also, turbulent fluid flow is used in illustrations in the following. The transfer of particles by the gradient in particle density is characterized by the diffusivity of particles D,. D, accounts for the random motion of particles in the flow field as induced by the diffusivity of the fluid whether laminar or turbulent, by the wake of the particles in their relative motion to the fluid, by the Brownian motion of particles, by wall interaction, and by perturbation of the flow field by the particles. For a dilute suspension, the transport of momentum of particles is given by its gradient or “stress” given by the tensor. Tp = pp D, [Vu, + vq
Fig. 1. Coordinate eration.
[ 1 + (P/2&)] - l
- $pp D, (V u,)U
(2)
where U is a unitary tensor. For a dilute suspension, particle-particle interaction can be neglected. The interaction length L, of particle-fluid interaction characterizes the response of a particle to the random (turbulent or molecular) motion of elements of the fluid. LP is defined as the
DEPOSITION
AND
ENTRAINMENT
IN PIPE FLOW
relative intensity of random motion of particles to fluid times the relaxation time of their momentum transfer. At a boundary (wall&w), the slip velocity tangential to the wall is given by (3)
“pw= &A,
where A,,, is the deformation tensor consisting of velocity components parallel to the wall. As was discussed earlier2, in a dilute suspension, that is, free path of particle-particle collision > L, > interparticle spacing >2a, the fluid motion is unaffected by the presence of the particles. This assumption is still valid when deposition of particles occurs if the thickness of the layer of deposit is much smaller than the pipe radius, so that the effective reduction in pipe radius is not enough to change appreciably the fluid velocity distribution. A stagnant layer having a thickness of 0.1 times the pipe radius changes the fluid velocity by a factor of 0.02. 3. GENERAL
RELATIONS
As in our earlier studies, we shall still restrict ourselves to cases where fluid motion is unaffected by the presence of particles. This means that the motion of the particle phase corresponds to viscous slip motion and particles are not correlated to each other even when the fluid is turbulent”. However, because of the deposition of particles, fully developed motion of the particle phase does not occur. The equation of motion of the particle phase should include the inertia forces or’ 7a d”, PP
dt
au, at
285
OF A SUSPENSION
The flux of particles due to field and fluid forces is now given by P,(“,-“)=(f,~,l~)+(~/~)V~~,
(6)
Where deposition may occur, unlike in fully developed motion, equality of fluxes due to diffusion and field forces’ cannot be specified. Rather, the conservation of mass has to be accounted for by the diffusion equation for a dilute suspension: (P ‘P,)
dp, _ - at
@P
dt
+ u.2
= -V[-DpVpp+pp
(up- ")I
(7)
Substitution of eqn. (6) gives, for a quasi-stationary state,
ap =
u .p
au
V~(DpVpp)-F-lV~ppfp
since substitution of eqn. (5) leads to V. Vx which gives zero and higher derivatives ofp, can be neglected. When applied to the pipe flow system, the components of field forces and the fluid force jr_ are (neglecting the $, z components of shear force due to shear flow in 4, z directions):
) (9)
d“,
=
PP
+
PP”P’F
=
Fpp(“-“P)+pPjp+V~t,
1
P
(4)
and v.z, = vx (ppDpVx “,)+2(v~&D,v)“, -$V L-P,D, (V. “,)I
V.E=p,@
(5)
where t is the time, P is the position vector, f, includes field force and fluid force per unit mass of the particle phase due to gravity and electrostatic effects. We shall treat the case where the axial gradient of the total particle flow is small such that we can assume a quasi-stationary state even for transient deposition; that is, a/&=0. Moreover, since particle velocity distribution over a pipe is not strongly influenced by different loading of similar particles3,17c, all the inertia terms can be neglected. PowderTechnol.,6 (1972)
where q is the charge, m is the mass of each particle, the E’s are components of the electric field E= - VV, V being the electric potential, and E is given by the Poisson equation
(10)
“0
where 8. is the permittivity of free space and E, is again small for gradual deposition. The boundary condition is given by the conservation of total flow: p,u,rdrd$
=
2a _
OR o
+
4v2~R~~dftlF)
‘pR&Rdd
-
“w 2R&‘,,
WF)
286
S. L. SOO. S. K. TUNG
wheref;, is the adhesive force per unit mass at the immediate vicinity of the wall. The flux of particles produced by the erosion from a bed is given by: (probability of lift) times (mass per unit area of surface layer of particles) times (frequency of lifting of particles). The mass per unit area of surface layer is given by : (4x/3)* @%a&, where Qb is the volume fraction particles in the bed. Since the frequency of lifting is given by the lift velocity (fJF) divided by this mean distance between successive particles lifted, the flux of particles produced by erosion is thus o&p,,JJF, where the probability ok accounts for the difference between bed density ppb and the above quantity of mass per unit area divided by the mean distance of lifted particles. Since we assume the motion of the fluid phase as fully developed (&@z=O), we have (E,-0) ;
(PpU,)= g CPp(Up-41+u
= p
-
O)ppR
+
dv
(12)
Ppt,] K./F)
(13)
which, for o=O, o,=O, &=O, reverts to the nondepositing boundary condition obtained before3. The relation offr to basic physical properties was dealt with previously 1,4. It is reconsidered in the
in the relations
(14) where U, is the local relative velocity of particles to gas and C, is the lift coefficient, which can be represented in the form:
The coefficient r1 and exponents m, n and p are given in Table 1’,2,19-21. In this general representation, the previous relation1,2 for.fr is now
(16) where the relative velocity is givenaby
Saffman19.,f~, (theoretical)
12.92/n
Eichorn’t (empirical)
7x lo4
Graf and AcarogluZ3.24
50
Kingsburyz6
Technol.,
6 (1972)
(17)
Au= l&/&la, and the shear response number Ns is now Ns=~i,~~~)“(~)“+p-m-l
(18)
correlating shear force of fluid to inertia force of
of lift force
Rejirence
Powder
When treating cases including turbulent flow over an erodible bed of solid particles whose settling is mainly due to gravity, .fL in eqn. (13) is the lift force per unit mass acting on a particle by fluid shear and
I
Constants
FLOW
u, = up-u+Au
F-' E,P,,-%P,,(j;/F)
TABLE
IN SHEAR
(15)
(1 -f)gcosO]~
[(I
4. LIFT FORCE
f$
Substitution of eqns. (12) and (8) into eqn. (11) gives, with ~~~-j~~lF: _ D, 2 = - (1 - 0) 1 - P S sin 8 cos f$ppR R PP 1 F ( +
next section in accordance with the above equation, (13), and experimental results.
,n
n
P
Note
t
f
0
Very small particles and slow relative motionzO
2
2
2
Large particle in Poiseuille profile”
I
t
0
Present analysis of their results according to turbulent boundary layer theory”
1
I
0
Laminar bearing
motion
in slider
DEPOSITION
AND ENTRAINMENT
IN PIPE FLOW
OF
particles. We shall denote the Saffman expression as_L. In general,fi should be included over the whole shear layer in eqn. (9). jr_ is insignificant for both 2a <<6 and 2a 9 6,6 being the boundary layer thickness. The boundary condition in eqn. (13) for a horizontal pipe or 8 = n/2, rr/2 < 4 < 3x/2, and E, = 0, for density of bed ppb, D,
2 = -(l
-a)
R
+
_$0, c! 1- ;
$ppR
(I- 4PplJftRIF)+ 4vPpb(ftR ‘F)=
=-(l-a)R,-
287
A SUSPENSION
experiments has yet to be identified. A similar trend is seen in Wickszs. Comparison with eqns. (23) and (A12) shows that the range of experiments in Graf and Acaroglu23*24 is represented more closely by m= 1 and p =0 in eqn. (1.5)for the lift coefficient of the particles, while the value of n has no influence when the relative velocity is approximated by (&/&)a. Alternatively, it appears valid to visualize lifting of a particle as produced by film action as in a slider bearingz6. Based on the bearing theory, we get m= 1, n= 1, and p=O. With rn= 1, p=O. we get
- cr;] R, (24)
(19) In the above expression, the first term (R,) gives deposition by gravity and the second term (R2) gives re-entrainment by fluid shear. It is seen that, for a large pipe, R 9 L,, up N u, we have jLR = -N,(u,/R)“(alR)‘-“-“a
li?u/c3rl~-m
(20)
and (25)
A reasonable choice, to be substantiated in Section 8, is n=S. We have therefore, based on Graf and Acaroglu,
Further, for a turbulent fluid and a < y,. y, being the thickness of the laminar sublayerz2, or y,= 6O(u, R/C)-” R, u, = u. (y,/R)‘, we get
au
ar l-l
z-2
R
u
(21)
Ys
(26) 5. CORRELATION
In the following, we treat specific cases using these dimensionless quantities :
and, for c’, =(3)2n-m-360-(9)(2-m)C1 ,
r* =*rlR,
An interesting quantity is the ratio of the quantities R, and R, on the right-hand side of eqn. (19) at 4=x or
5 = (&-P)
R*=
&I
R2
=
$
($5)
(g-p-’
9 PPR _fiLR &b
f$)
pf]““ml-ci’
(23) which characterizes the sedimentation and transport and serves as a basis for comparison to correlations of experimental results in the Appendix. Equation (23) serves to substantiate the logic behind the dimensionless correlations in eqn. (A12) in the Appendix based on experimental results of Graf and Acaroglu 23,24.The diversity of the power of the Reynolds number (uoR/i) is believed to indicate the diversity of the state of relative motion of particles to fluid. The particle diffusivity in these Powder
Technul., 6 (1972)
PARAMETERS
z* = z/R,
pp*= pp/ppl = pp*(r*, z*, 4) 24:= up/u0 = $(r*,
u* = u/u0 = u*(r*),
E*= -V*I’*
V* = (q/m)V/D,F,
z*, 4)
(27)
where ue is the fluid velocity at the center of the pipe, ppl is the density of particles at the initial condition (z* = 0) of fully-developed motion, analogous to a thermal entrance condition. We also define the following parameters as before3 : 0:= (ppl/%)(dm)2
(R2/~,F)
/3= R’F/D, y = 2 [l - (p/p,)]R2g
q = 2[1-(p/p,)]Rg
cos B/D,&) sin f3/DpF
and the momentum transfer number N,,,and particle Knudsen number NKp N,,, = Q/RF &p = LpIR
S. L. SOO, S. K. TUNG
288
In this case, E,=O, deposition is by sticking at the wall alone. This was the case studied by Friedlander and Johnstonez7 and recently by Yoder and Silverman including effect of agglomeration2*. Here we are concerned with a very dilute suspension of, say, room dust. Equation (31) becomes
and an adhesive force parameter ;1= RfJFD, which has not been used before. f, acts only on particles at the immediate vicinity of the wall. In addition, the shear-lift parameterzv4 takes two possible forms :
(33) with u* =u* (r*) for turbulent boundary conditions are
according to the lift force of Saffman, and
r* = 0
6. ELECTROSTATIC
SEDIMENTARY
FLOW
&*
a - - r*p; r* ar*
+ flp;(u*-u,*)=o
(28)
(29)
the Poisson equation, i
a
r* ET = 4c(p,*
(30)
and the diffusion equation, -
I
a
r* &*
r*p,* E:
(31) The boundary conditions are r* = 0
r* = 1’
a6 ar*
a&Z_- -cQ”p;
1
’
&*
pp* = c
ck exp
[ -
k2
(z*lPKJlJ&r*)
k -
exp( - CJ,~~Z*/~/?N,,,)[~ -$a,h*2
+&a$E.2r*4...]
(35)
v,* = E,*/JN,,,
r* &*
-
and E,may include the Van der Waals force and the electrostatic force. Other than the dependence of u* = u*(r*), the solution of eqn. (33) is straightforward. For turbulent flow, we may take u* - 1 and we have
As an extension for the earlier study of turbulent flow of a charged suspension without gravity effect’, we treat the effect of particle charge on deposition. The above simplifications give the momentum equations, i
and the
%-0
’
according to the correlation of Graf and Acaroglu. r* =
motion,
’ u* = -N P
% Kp
i%*
av* -=
at-*
0
’
(32)
Note that eqns. (30)-(32) are satisfied by p: = [1-(0~/2)r*~]-~ when o=O, o,=O, thus reverting to the case without deposition. The following cases can be identified : e@t
Powder Technol.. 6 (1972)
BWnu*ap;_ ~ az*-
~ 4~r
-
&
[P,*Spp*d(r*2)]
(36)
For u*w 1, we have a similar case as in the electrohydrodynamic inflet flow, which has the solution, for 5 = z*(4cr/bN,,,),
-(1-0):&Z-0,i;
Case 1. Dij’hion
where J, is the zeroth order Bessel function of the first kind, ck is the Fourier coefficient for an eigenvalue k. The approximation is for small cr,$, and a,f,/4F is the deposition velocity defined by Friedlander and Johnstone27, now expressed in terms of material and surface properties. An example is the adhesion and deposition of quartz particles to glass”. This force amounts to 0.01 dyn/um of particle size. A deposition velocity of 10 cm/set in air suggests cw= 1.5 x 10m4. Case 2. Electrostatic force alone. When the electrostatic force due to self-field is significant, we can neglect both the effect of diffusion and 1. We get, from eqn. (31)
alone in a dilute suspension.
~p*=(l+[)-~+r*~(l+[)-~+...
(37)
This relation is given in Fig. 10.16 of SOO”~. Case 3. Simultaneous action oj’dij’tision and electrostatic repulsion. Equations (28) and (31) have to be
DEPOSITION
AND ENTRAINMENT
IN PIPE FLOW
289
OF A SUSPENSION
solved numerically 3q4. Although any form of initial condition can be accounted for, the change from a fully developed condition at z =0 is sufficiently interesting. Figure 2 shows the change in particle density at the wall along the length of the pipe. The changes in density at the center of the pipe and density at the wall in the axial direction are readily seen. The particle velocity was determined from fluid velocity given by the Sth velocity law.
In this case, G is that for particle-particle sticking and f, is negligible. When the bed of deposits is stationary, the rate at which its thickness is built up is given by (40)
d&ldt = ~puvp,lpp,
At steady state, we may have a deposit layer at the wall of thickness 0, moving along the z-direction (41) where Qs is volume fraction particles in the moving bed, u, is the mean velocity of the deposit layer moving downstream. u, is given by the equalization of shear stress in the packed bed to that at the boundary of the dilute phase.
4
7. GRAVITY
EFFECT
Consider the gravity effect alone ; under the simplifying assumptions in Sections 2 and 3, the momentum equations take the same form as before2v3 :
3
l a (**~)+I&?&@) -SW,*+r*ar* r Pp
Ppw 2
+fip,*(u*-u,*)=O
1
(42)
and the diffusion equation now becomes ’
0
’
Fig. 2. Electrostatic 0=0.2. %=O.
Case4
’
2
’
’
4
Z/R sedimentation.
’
’
6
J
8
fl= 20, NKp = 0.2, N,,, = 2.0,
Attainment ofpacked
or moving bed density. A limiting condition is when CIis large enough so that the density of a packed or moving bed pps of volume fraction solid @, is reached at the wall ; that is, we have -2 PPR=
=“-=g
1-5 i
PPl
!
P Pl
(38) 1
when sedimentation occurs because of slowing down of a wall layer. For Qi,% G1, such a condition occurs when @, > 2/E
(39)
where _ a=G
Powder
PP -
4 0
2 R2
m
Technoi., 6 (1972)
The boundary conditions are 4 = 7112,3~12, ap;/ar*l,
= 0
and, from eqn. (13). $LZ = -(l-o)& &* 1
cos $hp,*-o,E.p,*
(44)
obviously for n/2 < 4 < 37c/2. Since the top cannot have particles falling into the suspension, the boundary condition becomes ap* 2 = -;yI cos 4p;-o,Ip,* ar* 1
(45)
for -n/2< $< ~12. Note that the reverse has to be specified when lip < p, such as in the case of buoyant particles or bubbles. The case where gravity effect is negligible obvi-
S. L. SOO, S. K. TUNG
ously reduces to Case 1 in the above section. We have the following additional cases : Case 5. Gravity flow alone, negligible I.. In this case, v-+ co, CJdoes not play a role. We introduce a coordinate 5’ such that
1
sin edz* (46) and, taking u*w 1 for turbulent flow, eqn. (43) becomes (47) In this case, pp*remains constant except that, starting from fully developed motion, the top boundary of the particle phase falls according to the case of batch settling; that is, the top surface of the suspension is at y=2R-Rt’
Z/R :
(48)
with y=O at 4=x, r=R, for 0fO (0=0 for vertical pipe) sedimentation following Fig. 9.1 of SOO”~, Fig. 4. Deposition N,,, = 2.0. i. = 0.
rate at the wall. fl= 10, I’= 1, q=2,
N,,=O.2,
with a packed bed of Qsbuilding up from the bottom of the pipe. Case 6. Simultaneous action of diffusion and gravity effect. Solution can be obtained only
0.5
1.0
1.5
L
2.5
2.0
3.0
PP
0.
0
:
0.5
=
numerically. With fully-developed flow as initial condition and Sth turbulent velocity flow from the fluid phase, computations were carried out for A= 0. Figure 3 shows the change in density distribution and Fig. 4 shows the deposition flux at the wall. Case 7. Attainment of packed bed density at the bottom ofthe pipe. This case is analogous to Case 4.
1.0
This may lead to sliding bed motion in both the axial and peripheral directions.
-
8. EFFECT 1.0 ’
0.5
1.0
1.5
*
2.0
2.5
3.0
PP
Fig. 3. Density distribution when deposition occurs due to gravity and diffusion. p= 10, y= 1, q=2, N,,=O.2, N,,, =2.0, 1= 0.
Powder Technol., 6 (1972)
OF LIFT IN SHEAR
FORCE
The influence of lift of particles by shear motion of a turbulent fluid is seen to be important in many cases of liquid-solid suspensions, including settling and erodible beds. However, most of the experimental data are on overall transport relationsz4.
DEPOSITION
AND ENTRAINMENT
IN PIPE FLOW
Detailed measurements on density distributions are available only in the case of non-settling pipe flows. When there is no settling, the distribution in particle density under the influence of gravity and shear flow is given byze4
aln PP*_ ar*
TOP 1.0
r*
0
-+q cos c#)-L*
a lnp* ~ = $j sin f$~ r* a4
(50) 1.0
where L* takes the form
I I
(51)
based on Saffman’s lift, or
f (u,*-u*+Au*)
L* =i
2
based on Graf and Acaroglu. Computations were made with both relations for comparison to results of the experiments performed on a suspension with particles of neutral buoyancy in a fluid in laminar motion by Segrk and Silberbergz9 and on suspensions of sand and perspex in water in turbulent motion by Newitt, Richardson and Shook3’. Case 8. In the horizontal flow of liquid suspension of sands and perspex, computations with parameters based on properties of materials given in experiments of Newitt, Richardson and Shook3’ yielded mass flows as shown in Fig. 5 for coarse sand and Fig. 6 for perspex. The related material properties and parameters are given in Table 2. It is
I -
0.5
,
I
,
7’10, @50,[,=84or --
7’8,
0.6
0.8
1.0
1.2
1.4
Pp”$
Fig. 6. Mass flow distribution in vertical plane of perspex in water (y=O, nlR=l/l3.5, N,,=O.l, NRc=104) as compared to experimental results of Newitt et al. (mean volume fraction solid : A 0.0983, 17 0.0295).
TABLE
I
2
Properties
and parameters
in the experiments
of Newitt et al.
Ptrrticles
Perspev
Cmme
20, in.
0.73 13.5 186 0.173 1.17 2.5 1200 0.1 0.6 100 10-S
0.0275 35.8 103 0.313 2.66 10 50 0.2 5 250 1o-6
Rla
F, set-
’
g/F, fps P,!P 4 B N, &VI il D,, in.‘/sec v=[O]
,
0.4
(52)
[up*-u*+Au*l+
I
0.2
Bo~+om
du* + dr*
,?I =[,(u,*-u*+Au*)
Top 10
291
OF A SUSPENSION
10-6m2/sec,R=0.0125m,N,,=[0]
sand
104,y=0,NKp=0.1.
,
5,=21
p=500, cl=250 or c2=62
IX 0
0.5
noted that their measurements3’ yielded mass flow instead of density. Figure 6 shows that the lift force according to eqns. (26) and (52) is closer to the experimental facts. In Fig. 5, however, corresponding values of i, and i2 gave similar curves. From these two comparisons, for D, given by cl, we can determine c;. We get c; -0.5, and c1 w 50, in eqns. (15) and (24).
10 10-z Bottom
10-l
Fig. 5. Mass water (y=O, experimental A 0.0550, 0
Powder
1
10
102
PZUP” flow distribution in vertical plane of coarse sand in n/r=1/35.8, N,,=O.l, NRe=104) as compared to results of Newitt et nl. (mean volume fraction solid : 0.0648).
Technol.,
6 (1972)
9. DISCUSSION
It is seen that sedimentation in pipe flow may occur for a variety of reasons, although their results are often equally undesirable. The cases treated in
292
S. L. SOO, S. K. TUNG
the above show how sedimentation may get started. Once started, three more cases call for a detailed treatment of the time dependence: Case 9. Transient deposition may continue until the pipe is completely plugged. Case 10. Transient deposition together with bed movement. Case 11. Formation of dunes and repeated piling up of deposits until a large pressure difference in the fluid is built up to blow it away. This causes a pulsat-
ing motion in the pipe. This wave motion is produced by a variation in fluid velocity due to flow restriction by the dunes. We have now identified the parameters LX, ,$ y, v], i, A, N,,, and A&, for treating the problem of sedimentation. They permit the prediction of density distribution and conditions for deposition in design. When dealing with gas-particle suspension, deposition in the form of dipolesr3*14 or space charges can be equally problematic unless the interest is in collection of particles. Precharged particles are deposited in pipes by space charge in the above or by instability under the image force even when there is only a single particle, because diffusion prevents a particle from staying in the middle of the pipe. By stretching a high voltage wire along the axis of a grounded pipe conveying a suspension, we can charge the particles by a corona discharge and deposit them by the field ; this constitutes an electrostatic precipitator3’. In liquid slurry lines, or other high density conveyance systems, deposition occurs primarily by the formation of packed bed. The computer programs for Cases 3,4,6.7 and 8 are given in a thesis by Tung3’.
LIST OF SYMBOLS
a
radius of a particle Fourier coefficient cr, c; constants as defined drag coefficient for a sphere Cn lift coefficient for a sphere in shear flow field Cr. particle diffusivity D* E electric field force vector per unit mass of a particle :% radial, tangential, and axial force per unit ck
2: 1 F
Powder
mass of a particle inverse of relaxation time or relaxation time constant for momentum transfer Technol.,
6 (1972)
F*
coefficient for deviation from Stokes’ law gravitational acceleration Bessel function of zero order ;0 particle-fluid interaction length -& m mass of a particle m, n, p constants in exponents NKP Knudsen number of the particle phase momentum transfer number N, shear response number N, electric charge per particle 4 radial coordinate r position vector pipe radius k time t u V U’ I UP
VP
axial, radial, and tangential component velocity of the fluid
of
axial, radial, and tangential component velocity of the particle
of
WP I uo
U DW
maximum or core velocity of the fluid phase velocity of particles at the wall unitary tensor vectorial velocity of the fluid vectorial velocity of the particle cloud electric potential axial coordinate
dimensionless groups as defined deformation tensor permittivity of free space dimensionless axial coordinate angle of inclination of the pipe with the direction of gravity viscosity of the material constituting the fluid phase density of the fluid phase density of the material constituting the fluid phase density of the particle cloud density of the material constituting the particles sticking probabilities as defined shear stress tensor of the particle cloud azimuthal angle volume fraction particles Superscript * dimensionless quantities as defined
transposed tensor
DEPOSITION
AND ENTRAINMENT
IN PIPE FLOW
Subscript 1 initial condition cases as indicated I.2 for particle phase P s packed bed condition R at radius R w for wall condition
REFERENCES 1 S. L. Soo, Pipe flow of suspensions, Appl. Sci. Res., 21 (1969) 68884. of states of hydraulic transport of 2 S. L. Soo, Correlation solids in pipes, Proc. Hydrotrnnsporr Conf., Univ. o/‘Worwick, England, BHRA, A-l, 1970. 3 S. L. Soo and S. K. Tung, Pipe flow of suspensions in turbulent fluids--electrostatic and gravity effects, Appl. Sci. Res., 24 (1971) 84-97. 4 S. L. Soo and S. K. Tung, Gravity effect in pipe flow of suspensions, Paper No. l-5-164, Symposium o/’ Flow, Pittsburgh, Pa., May 9-14, 1971. 5 J. F. Kennedy, The formation of sediment ripples, dunes and antiduores, Ann. Rev. Fhid Mech., 1 (1969) 147-168. characteristics of suspensions. II. 6 D. G. Thomas, Transport Minimum transport velocity for floculated suspensions in horizontal pipes, Am. Inst. Chem. Engrs. J., 7 (3) (1961) 43340. 1 R. A. Bagnold, The movement of a cohesionless granular bed by fluid flowing over it, Brit. J. Appt. Phys., 2 (2) (1951) 29934. 8 G. W. Seman and G. W. Penny, Photographic records of particle trajectories during electrostatic precipitation, Power App. Systems, 86(3) (1965) 365-368. 9 S. L. Soo and L. W. Rodgers, Further studies on the electroaerodynamic precipitator, Powder Technol., 5 (1971) 43-50. 10 M. Corn, The adhesion of solid particles to solid surfaces, J. Air Pollution Control Assoc., 11(l) (1961) 523-528; ll(12) (1961) 566575, 584. 11 D. D. Eley, Adhesion, Oxford Univ. Press, London, 1961, p. 119. 12 W. Pietsch, E. Hoffman and H. Rumpf, Tensile strength of moist agglomerates, Ind. Eng. Chem. Prod. Res. Develop., 8 (1) (1969) 5862. 13 G. W. Penny and E. H. Klingler, Constant potential and the adhesion of dust, Commun. Electron., 81(l) (1962) 20&205. 14 J. M. Niedra and G. W. Penny, Orientation and adhesion of particles, Ind. Electron. Control Instrumentation, 12(2) (1965) 4650. 15 W. Kottler, H. Krupp and H. Rabenhorst. Adhesion of electrically charged particles, Z. Angew. Phys.. 24(4) (1968) 219-223. 16 H. Czichos. Uber den Zusammenhang zwischen Adhiision und Elektronenstruktur von Metallen bei der Rollreiburg im elestischen Bereich, 2. Angew. Physik, 27(l) (1969) 4@46. 17 S. L. Soo. Fluid Dynamics of’ Multiphase Sysrems, Blaisdell, Waltham, Mass., 1967. (a) Ch. 6, (b) Ch. 5, (c)p. 169, (d) p. 469, (e) p. 368. 18 S. L. Soo, Think two-phase, in I. Zandi (ed.), Adoances in Solid-Liquid Flow in Pipes and Its Application, Pergamon, Oxford. 1971, pp. 35-38.
Powder Technot., 6 (1972)
293
OF A SUSPENSION
19 P. G. Saffman, The lift on a small sphere in a slow shear flow, J. Fluid Mech., 22(2) (1965) 385-400; On the motion of small spheroidal particles in a viscous liquid, ibid., I (1956) 540-553. 20 M. T. Lawler and P. C. Lu, The role of lift in the radial migration of particles in a pipe flow, in I. Zandi (ed.), Adoances in Solid-Liquid Flow in Pipes and Its Application, Pergamon, Oxford, 1971. 21 R. Eichorn and S. Small, Spheres suspended in Poiseuille flow, .I. Fluid Mech., 20(3) (1964) 513-527. 22 H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1960, p. 508. 23 W. A. Graf and E. R. Acaroglu, Sediment transport in conveyance systems. Part I. A physical model for sediment transport in conveyance systems, Bull. Intern. Assoc. Sci. Hydrol., 23 (2) (1968) 20-39 : Part II. The modes of sediment transport and their related bed forms in conveyance systems, ibid., 23(3) (1968) 123-135. 24 W. H. Graf. Hydraulics of Sediment Transport, McGraw-Hill, New York, 1971, p. 442. in horizon25 M. Wicks, Transport of solids at low concentration tal pipes, in I. Zandi (ed.), Aduances in Solid-Liquid Flow in Pipes and Its Application, Pergamon, Oxford, 1971, pp. 101-124. New York, 1942, 26 A. E. Norton, Lubrication, McGraw-Hill, pp. 73, 77. 27 S. K. Friedlander and H. F. Johnstone, Deposition of suspended particles from turbulent gas streams, Znd. Eng. Chem., 49 (1957) 1151-1156. on 28 J. D. Yoder and L. Silverman, Influence of turbulence aerosol agglomeration and deposition in a pipe, Paper No. 67-33. 60th Ann. APCA Mfg., Cleveland, June 13-17, 1967. 29 D. M. Newitt, J. F. Richardson and C. A. Shook, Hydraulic conveying of solids in horizontal pipes. Part II. Distribution of particles and slip velocities, Proc. Symp. Interaction between Fluids and Particles. London, June 1962, Inst. Chem. Engrs., pp. 87-100. 30 H. J. White, Industrial Electrostatic Precipitation, AddisonWesley, Reading, Mass., 1963, p. 35. 31 S. K. Tung, Flow of suspensions under field forces (August 1971) (microfilm copy available from University Microfilms, Inc., Ann Arbor, Mich.).
APPENDIX CORRELATION OF EXPERIMENTAL OF GRAF AND ACAROGLU
RESULT
The correlations of Graf and Acaroglu23*24 on deposition and erosion of particles appear to be most general and straightforward. They give a “shear intensity parameter” Y=
=
critical shear stress to lift available shear stress
(Al)
294
S.
where s is the slope of energy grade line and R, is the hydraulic radius. s is given by v’, = r,,/p = gR,s
(-42)
or
They also gave a “transport parameter” (A4)
U being the mean velocity U=O.Su for pipe flow or pP,U is the mean mass flow. A large number of test data from various sources were correlated to give Q,= 10.39(Y)-2.52
(A5)
making possible estimations of transport capacity. Depending on the mode of transport, Y is given by the shear motion and the whole range given by Graf and Acaroglu may be correlated according to Y Rc$.*~ = 9360(2~,,)‘.*~
PowderTechnol.,6 (1972)
TUNG
Equation (A6) was delineated over the following ranges : critical condition (start motion) Y=8 Re,520 ripples and dunes (beginning of ripple) Y=4 Re, 58 (disappearance of ripple) Y =0.85 Re, I plane bed (bed load and suspension) Y=O.2 Re,,<2.6 sliding bed or anti-dunes (predominantly suspension) Y = 0.06 Re, 5 1.4 Experimental results are often correlated in terms of fluid shear velocity v.+or r: = %lP
(A8)
where r,, is the shear stress at the bed of particles. For pipe flowz5 : ~1,= O.l5u, (T/Ru,,)*
(A9)
Substitution into eqn. (19) gives
(A6)
where a mmdenotes a in mm and Re, is the Reynolds number based on shear velocity and particle diameter or Re, = 2a(v,/V)
L. SOO.S. K.
(A7)
For flow through circular pipes, we have Re, = 0.30(a/R)(uo R/C?
(All)
6412)