Laminar suspension flow with diffusive, gravitational deposition in the entrance of a converging and diverging channel

Powder Technology, @ Elsevier Sequoia

23 (1979)

S.A.,

47 - 53

Lausanne -

Printed in the Netherlands

47

Suspension Flow with Diffusive, Gravitational Deposition in the Entrance of a Converging and Diverging Channel

Laminar

T. A_ KORJACK* New Jersey Institute (Received

and R. Y. CHEN of Technology,

June 8,197s;

Newark.

N-J_

in revised form December

07102

(U.S.A.)

5,197s)

SUMMARY

The simultaneous development of the fluid and particle phases was solved numerically to effect deposition rate results in converging and diverging straight wall channels with significant gravity effects. The flow was laminac and two-dimensional with non-reacting dilu’te suspensions in an incompressible carrier. Deposition rate was found to be higher in a favorable pressure field (i.e. converging channel) than that in an adverse pressure field (i.e. diverging channel) when diffusion was the predominant force involved. However, in a gravity field, bottom deposition rates were found to increase in adverse pressure fields, whereas favorable pressure fields effected less deposition rates. Moreover, top deposition rates were found to decrease with both increasing convergence and divergence angles. Finally, the bottom deposition rates were found to increase to very high proportions (near separation points) for high No in diverging channels due to the large magnitude of the particle density on the bottom wall.

INTRODUCTION

Many investigators [l - 41 have undertaken studies dealing with deposition in either parallel-plate channels or circular tubes. Only recently Eldighidy [S] studied the symmetrical laminar flow of suspensions with deposition in the entrance region of a diffuser with diffusive and electrostatic effects, exclusive of unsymmetrical gravitational influences. The main objective of this article was to investigate the deposition of suspensions in *Present address: Clarkson College, Postdam,

NY.

laminar flow in the entrance region of a converging and also a diverging channel with diffusive and gravitational effects. Since experimental work in most cases utilizes particles for which the gravity effect is very significant, this analysis will serve as a practical mode? for experimental data analysis. Thus, a mathematical model for the entrance solution of suspensions in laminar flow with significant gravity effects in the entrance region of a nozzle (converging channel) and diffuser (diverging channel) has been developed for the deposition process on both the t?p and bottom walls of the channels. The characteristics of the rate of deposition for different flow parameters with gravitational forces was also considered, along with the effect of the angle of divergence and convergence with gravity on the rate of deposition of the solid particles.

ANALYSIS

The particle cloud was treated as a continuum whereby the flow of suspensions was regarded as a mixture of two interpenetrating continuous fluids. The particulate concentration was assumed low enough such that the particles have negligible effect upon the fluid phase. The fluid phase was assumed to be viscous and incompressible, and both the fluid phase and particulate phase were assumed to be in two-dimensional steady flow. A Cartesian coordinate system was selected in such a way that the X-axis was the centerline of the channels, and the Y-axis was normal to the flow. According to the usual boundary layer assumptions [ 53 , the governing equations are c31 (A) Fluid phase

48 au au -+-=(I ax ay

u

(9)

au

au

1 dp

a2u

ax

ay

P

a~

-+v-=--+v2

*

(2)

a vp av, UlJ- ax +v,----ay

_ -(V-V,) 1

--N+

N,

B

m

(B) Particulate phase

au*_

au,

UP -cup--

ax

ay (3)

au,

-+

ax

UP

ah

uax+v-=-

iz =

z=

VP

ay

CUV,P,

+

(41

-g

aPP -+pPg

ay

(5)

F

(6)

where F = 9~//2a2~p for spherical particles in the Stokes’ law range. Note that & represents the total deposition rate. Introducing the following dimensionless quantities: X=x&

Y = y/h,, Y/(l+Xtane)

v = v/u,

U,

v,

P=@

= uoho/v

K

UP

ilk*

=

R

=

N,

Nx,

= uoh,lD,

=

ax

aY

v=v,=o

uniform

R=l P=O

(at the lower wall)

AtY=-H* jy=TT=O

aR (-)aY

~pl~p,, = h,,g/D,F

no slip condition

y=-_H’

+I!&,

(14)

y=-_n* forX<

N,

forX>

N,

(15)

= -R[aNBVp

-N_q

+N,-,]y=-_H*

(16) At Y = H* (at the upper wall) U = V = 0 U,

= a,(1

no slip condition -X/N,)

(17)

-KR,, Y=EP

(7)

(8)

au*_ i_Jpa~+vp-aY

(13)

L,lho

Hence, all upper case letters denote non-dimensional quantities. The governing equations were expressed in nondimensional form as

av

(Inlet)

lJ=U,=l

Ns = o,hofwlFD,

au -+-=o

Y
=uo/izo F

NR

m/pPouo

AtX=Ofor-1I

= v&o

N,

-Po)Pd

(12)

aV,)lu=-n*

-

The boundary conditions expressed in nondimensional form were:

Up =a,(l-X/N,)

H* = h(x)/h,

u = u/u, = up/u0

+

UIlifOlXXl

u,f,Pp/Fly=h(r,

+ Ca,f,p,/F--v,p,l,=--h(r)

y*=

+Ns/Np)]ycH*

CR(Ns/Np F(V --up)

a ay

ah

ni* = [R(aV,

forX<

N,

forX>

N,

(18)

Y=Hf

=-R[uNBVP Y=IP

+Ns

+N,-,]

(19)

49

The slip velocity of the particles at the wall, given by eqns. (17) and (20) for X < Nnl , took into consideration the distance the solid particles travel before they are decelerated to the slip velocity given in these equations for X > N,. A factoru, (12 (T,b 0) was multiplied to this additional slip velocso that the particles on the ity, 1 --X/N,. wall did not have a velocity higher than the particles adjacent to the wall. The governing equations and respective boundary stipulations were expressed in implicit finite difference forms. Discretization was implemented by the superimposition of a trapezoidal mesh internal to the nozzle and diffuser fiow fields. An accuracy of four significant figures was realized in the computations_ The implicit finite difference technique applied to the particulate phase equations was envisioned by consideration of the Eulerian reference frame. Hence, the momentum and diffusion equations were expressed as

+NqAyi+~ M*

El=

-Ri+l.K-1

Ri+l.K+l

(22)

2Np
C&+l.N

+2(aVPi+1.N+2

+NdNp)l

+ C&+1. zW&‘J,T --oVPi+,.2)1

(23)

Finite difference expressions for the fluid phase followed similarly. Twenty-one mesh points in the Y-direction and up to 141 mesh points in the X-direction were used. For the converging and diverging channel flow, AX E {O.Ool,

0.01,

0.1,0.2)

while A Y varied according to the converging or diverging angle (20 )Different values for 0 were chosen, in particular, 0 E {2”, 4”) 7.5”) 10”)

for diffuser flow

and 0 E {-

2”,-44”,--7.5”,-10”)

fornozzleflow

Considering all variations of parameters and mesh sizes utilized in this analysis, the general condition of stability [S] given by -

UPi+l.K+l VPi.KAYi+l

=

AYi~l

(AX/NR,)I(AY)~

-

K -

ui+l

-’

UPi*l.K

N, -2UPi+l.K

+

UPi+l.K--I

Np(AYit~12

Ri.K+I

+ Ayi+,

-Ri,K-i

Upi+1_K+l

2IVoAyi+l

-UP~+I.EC-I

2A Yi+lRi.K (20)

UPi.K

< 0.5

was satisfied and convergence was realized. It must be noted that the flow field was divided into either diverging or converging grids. The finite difference expressions (20) (23) took into consideration the variation of the vertical grid, A Y, with axial change.

2AYi+l

UPi+l.K+l

+AYi+-1

uPi+l.K-l

A&+,

AYi+l

&+l.K

A&VPi.K

q+l.K

-

VPii-l.K

-

A Yi+lNq

A yi +I% +I,K AX

R- x+l.K+l

(21)

NaN,

NUl

A YiRi A

%+=.x+1

K

-Ri+l.x-I_

2A Yi +I -2Ritl.K

No(Aui+l)2

+Ri+l.K-1

-

AND

DISCUSSICN

computations were performed for c = 0.5, N, = 2 and Na = 103. The special case of complete absorption at the boundary for both pIug and Poiseuilie flows at No = 100, Nv = I,8 = O” was performed for the purposes of comparison with Ingham [43 Excellent agreement was found with at most a discrepancy of 0.1% in ah comparable cakulations. Figure 1 depicts the axial distribution of the particle cloud with surface adhesion effect in a converging channel of 0 = - 4”. As can be seen, with low adhesion (Ns = O-l), the solid particles descended towards the bottom wall in the streamwise direction, but when _ Ns = 10, more particles adhered to the botAll

u,

=

AX

-I- Vi+l.xAYi+l = AYiel

-

VPi+l.K

RESULTS

=

Fig. 1. Axial distribution of particle concentration with surface adhesion effect in a convergent channel (8=-4°,N~=5,Np=40,N,=2,N~=1000,~= 0.5, us = 0.5).

Fig. 3. Axial distribution of deposition rate with converging and diverging angle effect (N,, = 5, Np = 40, 1000, NS = 1, (T= 0.5, a, = 0.5). NIU =2,Nx=

0.3

0

Fig. 2. Axial distribution of the bottom rate of deposition with converging and diverging angle effect (N,.,=10,NP=40,N,=2,NR=1000,NS=1,0= 0.5,os = 0.5).

tom wall rather than being carried downstream. Figure 2 shows the effect of the angle of convergence on the bottom deposition rate in the presence of a large gravitational field (shown NV = 10). As can be cIea.rIy discerned, the increase of convergence, i.e. as B = - 4”) - 7.5” and lo”, caused a marked decrease in deposition rate because of the increase of the fluid velocity resulting from the decrease of the fIuid static pressure: however, when the angle became divergent, as in the case for 6 = 4”, the deposition rate clearly increased in magnitude due to the decrease in axial fluid velocity near the boundary resulting from the adverse pressure gradient. Hence the deposi-

Fig. 4. Effect of surface adhesion parameter on axial distribution of particle concentration in a divergent channeI (6 = 4”. N,, = 5, Ns = 40, N, = 2, NR = 100, u = 0.5, as = 0.5).

tion rate for 0 = 4” reached a maximum at X = 0.9 of 2.1 (not shown). The surface adhesive parameter on the rate of deposition in a convergent channel was investigated in detail for 0 = - 4O, NV = 5, ND = =103 ando=oS=0.5.1t 40, N, =2,N, was found that for values of Ns above 3, the maximum penetration rate was a maximum at the inlet. For Ns Lessthan 3, the maximum penetration occurred at a finite value of X which, for example at Ns = 0.1, occurred at X= N,. Figure 3 again shows the effect of the convergence and divergence angle on the rate of deposition on both the bottom and top walls when the gravity field was maintained at Nq = 5. As also witnessed from Fig. 2, the increase in convergence caused a decrease in bottom

51

0.5

0

1

PJrU,

Fig. 5. Axial distribution of mass flux distribution cf =olid particles with gravity effect (8 = 2O. NP = 40, N, = 2, Ns = O-1, NR = 1000, (i = 0.5, os = 0.5).

0 0;

i -II *,

-.

---___

109

0

------__________

I 5

IO

Fig. 6. Effect of surface adhesion on deposition rate in a divergent channel (8 = 4”. NV = 5, No = 40, N,.,, = 2, NR = 1000, u = 0.5, a, = 0.5).

deposition rate while an increase towards divergence effected an increase in bottom deposition rate. Moreover, the top deposition rate decreased for increases in divergence as well as increases in convergence. Figure 4 illustrates the surface adhesion effect on the particle concentration in a diffuser of 8 = 4” _ The effect was similar to that in a nozzle as shown in Fig. 1. However, the concentration was higher near the top wall for flow in a nozzle than that in a diffuser. This was an indication of greater gravity effect in a diffuser in comparison to a nozzle. Figure 5 shows the axial distribution of the mass flux of the solid particles in a diffuser of 2” with gravity effect. As gravity was increased, more particles were of course depos-

I

in a divergent channel

ited much sooner upon inlet, thereby decreasing the number of particles farther downstream. Figure 6 illustrates the effect of the surface adhesion parameter on the rate of deposition in a 4O diffuser. Since the angle of divergence was not large, typical deposition rates results were obtained similar to the convergent channel but with increase of magnitude for the bottom deposition rate for Ns = 0.1. Figure 7 ikstrates the divergence angle effect when NP = 10’. The increase in divergence clearly showed how the bottom deposition rate increased and top deposition rate decreased. As the point of separation was approached, the bottom deposition rate approached very high values due to the rapid increase of particle concentration at the bottom wall. For example, when Np = 40 and 0 = 4”) the particle concentration at the bottom wall attained values of 2.17, 1.7,1.5 and 1.3 for axial locations of 1, 5,7 and 9, respectively. However, when NP = 10’ and 0 = 4”, the particle concentration at the bottom wall attained values of 1.4, 2.17, 2.5 and 3.9 for the same respective axial locations. Hence, as Np increased, the deposition increased to very high proportions near the point of separation. Figure 8 depicts the situation when only the diffusive and surface adhesive forces were present -.vith the converging-diverging angle influence affecting the deposition rate. Deposition was higher in a favorable pressure field (i.e. when pressure was decreasing in the

52

S- PO17i

OF SEPAOATIO:.

0 0

2

Ii

6

8

10

Fig_ ‘i. Effect of angle of divergence on deposition N, = 2, NR = 1000, u = 0.5, Ns = 1, os = 0.5).

12

I4

16

Y

rate with high diffusive Peclet number (NV = 5, Np = 107,

CONCLUSION

Fig. 8. Axial distribution of rate of deposition with converging-diverging angle effect in absence of gravity field (NV = 0, Np = 40, NS = 1, N, = 2. hk = 1000, u = o-5,0, = 0.5).

downstream axial direction) than that in a adverse pressure field when gravity was neglected and the diffusive force was the predominant deposition influence_ On the contrary, the inclusion of gravity affected the deposition rate in a complete reverse trend as seen by referring to Figs. 2 and 3. For all situations investigated, the particle Knudsen number, K, , maintained the constant vahre of O.OOOl~irnplying that the fluid-particle interaction length, i.e. the free path ower which a particle changes its direction because of particle diffusivity, was almost of negligible consequences.

The angle of convergence and divergence had a considerable effect on the deposition rate in a gravity flow field. Increasing the convergence angle caused a decrease in both top and bottom deposition rate in a gravity flow field, whereas increasing the divergence caused a larger bottom deposition rate and a decrease in top deposition rate when gravity was considered. However, when the predominant force affecting deposition was diffusion, i.e. neglecting gravity, the deposition rate increased with increasing divergence angle and decreased with increasing divergence angie. When NP became very high in diffusers in a gravity field, the bottom deposition rate increased as much as 10 times that upon inlet as the point of separation was approached. Furthermore, it was determined that higher us values (0 < CT~G 1) gave higher particle velocities near the inlet; smaller N, values promoted shorter distances of particle velocity slip and thus increased the deposition rate near the inlet.

ACKNOWLEDGEMENT

This study was supported Army Research Office.

by the U.S.

53 LIST OF SYMBOLS

F ho

Kn, L.P m

NUZ NR P

PO R

u, v UP,

VP

uo

u, v UP,

VP

x, Y

x y Y*

radius of a particle particle diffusivity adhesive force per unit mass of particles at the immediate vicinity of the wall inverse of relaxation time half the channel width at inlet particle Knudsen number fluid-particle interaction length total rate of mass flow at the solid particles deposited on the walls momentum transfer number Reynolds number static pressure of the fluid static pressure at inlet dimensionless density of the particle cloud axial and vertical component of the fluid velocity axial and vertical component of the particle velocity inlet velocity (uniform) dimensionless axial and vertical component of fluid velocity dimensionless axial and vertical component of particle velocity axial and vertical coordinates, respectively dimensionless axial and vertical coordinates dimensionless vertical component with respect to angle 8, Y* = Y/ (1 +xtan@)

P

3 PP 0 PPO

V a

density of the fluid phase density of the particle cloud density of the material constituting the particle phase half the converging or diverging angle inlet density of the particle cloud kinematic viscosity of the fluid phase sticking probability accounting for viscous forces sticking probability accounting for adhesive forces at the wall velocity impedance factor

REFERENCES R. Y. Chen, Diffusive deposition of particles in a short channel, Powder Technol., 16 (1977) 131 135. C. N. Davies, Diffusion and sedimentation of aerosol particles from Poiseuille flow in pipes, J. Aerosol Sci., 4 (1973) 317. S. M. Eidighidy, Deposition of suspensions in laminar flow in the entrance region of a channel and diffuser, Dr. Eng. Sci. Thesis, New Jersey Inst. of Technology, 1975. D. B. Ingham, Diffusion and sedimentation of aerosol particles from Poiseuihe flow in rectangular tubes, J. Aerosol Sci., 7 (1976) 13. H. Schlichting, Boundary Layer Theory, McGrawHill, New York, 6th edn., 1968. A_ Quarmby, A finite difference analysis of developing slip flow, Appl. Sci. Res., 19 (1978) i8 - 33.