Cross-flow filtration: cakes with variable resistance and capture efficiency

O&E-2309,93 S&Ml + 0.00 0 1993 Pergamon Press Ltd

Chemical Engineering ScwnCe, Vol. 48, No. 16, pp. 2913-2918.1993. F-rintcd in Great Britain.

CROSS-FLOW FILTRATION: CAKES WITH VARIABLE RESISTANCE AND CAPTURE EFFICIENCY Department

N. E. SHERMAN of Chemical Engineering, University of Cambridge, Cambridge CB2 3RA, U.K. and

J. D. SHERWOOD’ Schlumberger Cambridge Research, PO Box 153, Cambridge CB3 OHG, U.K. (First received 14 August

1992;

accepted

in

revised form 8

January

1993)

Abstract-The cross-flow filtration model of Mackley and Sherman is reexamined with an alternative non-dimensionalization. The particle capture efficiency and cake resistance are assumed to depend on the

angle at which particles impact the cake. Predicted filtration rates for all values of the cross-flow velocity may be representedby a single master curve, in reasonable agreementwith the experimental results of Mackley and Sherman. The model ignores any concentrated boundary layer of uncaptured particles which flows across the surface of the cake when a final steady state is achieved. Estimates are obtained of the rate of growth of such a layer, and it is found to be negligible in two sets of laboratory experiments. 1. INTRODUCTION

In this paper we study in greater detail the cross-flow filtration experiments and theory of Mackley and Sherman (1992, henceforth denoted by MS). We adopt a different non-dimensionalisation to that used by MS. This enables us to test experiments against the MS theory, which predicts that after non-dimensionalisation the fluid loss rates measured in experiments performed at different shear rates should all fall on a single master curve. In Section 1 we set down the MS theory, and in Section 2 we show how results may be non-dimensionalised so as to fall on a master curve if the theory holds. The MS model does not explain the fate of particles which are brought by the filtration flow to the surface of the cake, but are not captured by the cake. Such particles will form a concentration boundary layer above the cake proper. In Section 3 we discuss how the magnitude of this boundary layer may be estimated, and show that the boundary layer was likely to be negfigible both in the experiments of MS and in those of Fordham and Ladva (1989). Finally, in Section 4 we compare the MS results against theory. We consider cross-flow filtration (Fig. 1) of a suspension of spheres of radius a flowing past a filter surface y = 0. The wall shear rate is $, and the filtration velocity is u. The tangential velocity one particle radius away from the upper surface of the cake is v = 4~. In the MS model it is assumed that the particle capture efficiency and cake resistance depend on the angle 4 = tan-’ (U/U) at which particles impact the cake. (Note that C#J as defined here is the complement of the angle used by MS.) The build-up of a concen-

‘Author

to

trated boundary layer of uncaptured particles is neglected. Sufficiently far downstream, this may become important, and it is considered in Section 3. We assume that the substrate on which the cake forms offers little resistance to flow. The filtration velocity I( is initially very large, and C#J = n/2, as in static filtration. The solids volume fraction within the suspension is co, and that within the cake, cCakc,is assumed constant, i.e. the cake is incompactible. = 1 - 8, where E is the porosity used by MS.) (c CaIEe The rate at which the cake height, h, grows is dh cOu = lu & = Ccatc- cg where 1 = c&,.,. - co). MS assumed that this growth rate is modified by a capture efficiencyf(#), so that dh z

=f(4) J.u

where f = 1 when 4 = ~12, and f decreases as 4 + 0. Iff(&,,) = 0, the cake will stop growing and the filtration rate will reach a limit u, = u tan (&,). Thus, the model predicts that the ultimate limiting filtration rate (if this exists) will be proportional to the crossflow velocity 0. The experiments of MS indicate that the permeability of the cake depends on the angle at which it was laid down. MS work with a cake resistance, R, rather than the permeability, and for a uniform cake Darcy’s law takes the form u = AP/Rh where AP is the pressure differential across the cake.

For a non-uniform cake, AP=u

whom correspondence should be addressed. 2913

h s0

R(4) dy

(2)

N.E.

2914

SHERMAN

and

where 4 = 4(y) is the impact angle at which the cake at y was created. If AP is held constant, differentiating eq. (2) leads to

J.D.

SHERWOOD

u*-tm’J(Z+a)

eqs (1) and (2) gives du -= dt

2.

- au3f (@)R @YAP.

(3)

NON-DIMENSIONALISATION

So far we have followed MS closely. We now scale the resistance R by Ro, where R. = R(7c/2) is the resistance of a cake grown in the absence of crossflow. Velocities are scaled by v, and lengths by AP/Rov. This gives us non-dimensional quantities u* =

u RL$ V’

t*=-.tR,v’ 0

du* -= dt*

_

s

au u _=-au v

u* - (3LR*(O) t*)-“3,

-u ’ 2f(ujv)R*(u/v)kR,v2 v AP

0

We assume that R* increases (as u* decreases) sufficiently rapidly thatf(u*)R* (u*) increases even though f decreases. Thus, initially the increase in cake resistance, R, dominates any effect due to changes in the particle capture efficiency factor, f: From eq. (6) we have

itR,u'

V2

u* = (2At*)-“z

2uz f(u/v)R*(u/v)

which is the result for static filtration. If we take f= sin Q, (equivalent to the example given by MS, which with their notation wasf = cos @), then we have the asymptote t’ + co

where we have assumed that R* + R*(O) -C 00 as u* 4 0. Increasing the resistance of the cake R(0) will decrease the filtration flux, as expected. We observe that the filtration velocity decreases as t- ‘j3, i.e. more slowly than in static filtration. More generally, if f --@ as 9 -+ 0, then

(6)

We consider how u(t) varies at a fixed time t when v changes. Differentiating eq. (6) with respect to u gives

which integrates to

For any given resistance R*(u*) and capture probability f(u*) it suffices to integrate eq. (4) to make predictions for all cross-flow velocities v. Iff= R* = 1, eq. (4) gives

Ql/fl+a

m du’ u,yu’Jtf(u’)R*(u’) = ‘tROvZ~Ap.

- Au*~~‘(u*) R* (u*)

du’ u’~~(IJ’) R* (u’) = It**

(5)

aj

In the above examples, the filtration flux u* decreases towards zero. If cake growth ceases due to the crossflow, and u* tends to a limit u: > 0, then f must tend to 0 as u* decreases to ~2. (We reject as unphysical the possibility that R* + 0.) The scaling adopted for t implies that events happen more rapidly at higher cross-flow velocities, and any variation of the capture probability or cake resistance will occur sooner. MS observed experimentally that R(#) increased as 4 decreases, and it is therefore possible that the cake resistance increases earlier at higher cross-flows. Thus, an increase in the cross-flow velocity can cause a decrease in the filtration rate, as was sometimes observed by MS. To investigate this, we rewrite eq. (4) as

AP

The impact angle 4 = tan-’ u*, and we may simply regard R and f as functions of u*. After non-dimensionalisation, eq. (3) becomes

t+

and a reciprocal rate plot of w 1 against the total volume of filtrate Q = 6 u dt would asymptote to u-1

Combining

as

u

-7iF-37

Hence,

%-CO

av

*

Thus, at fixed time t, an increase in cross-flow velocity, u, can cause a decrease in filtration velocity, u. However, if the particle capture efficiency f= 0 at some limiting flow rate ~2, then f(u*)R*(u*) -S 0 as u* + uz, and fR no longer increases. The above analysis fails, and, as discussed above, we expect the limiting filtration velocity urn= vu: to be proportional to the cross-flow velocity v.

v=ya Nuation velocity

K-

tangential ve’Ocity fdt r s ace

d

Fig. 1. The problem under consideration.

Cross-flow 3. THE BOUNDARY

LAYER OF UN CAPTURED

PARTICLES

The analysis in Section 2 neglects the flowing boundary layer of particles which must develop above the filter surface when capture is less than 100%. We now investigate the development of this boundary layer when a final steady state has been reached, with f=O. Our first analysis of the boundary layer is based on a convection4iffusion analysis of particle motion within the boundary layer. The surface of the filter cake occupies the plane y = 0, and the fluid velocity in the x-direction parallel to the surface is a shear flow u = Jay.Particles are convected by this cross-flow, and by the filtration velocity u = - u, towards the cake. There is also a flux of particles - DVc due to diffusion, where D is the coefficient of diffusion. The particle concentration, c, satisfies the convection-diffusion equation u-Vc = - VDVc.

filtration

2915

surface y = 0. The problem is very similar to that of heat transfer to/from a heated plate with a constantflux boundary condition. We look for a solution in which the particle concentration c has the form c=co+c1 Knco

=co+-

D

0D i’3

xi/3

(8)

&“)

7

where s=yx-‘/3

$

L’3.

0

Substituting eq. (8) into equation (7) we obtain d”+$j_+

the convection-diffusion

_umj-1/3D--2/3X1/3&

(9)

where the prime ’indicates differentiation with respect to s. The zero-flux boundary condition

We make the standard boundary layer approximation l&/axl Q l&z/ayl, and the governing equation becomes

w-D--O

ac ay

becomes d’+ In the experiments of MS, the total filtration flux increased linearly with the length of the filter, suggesting that, apart from entrance and exit regions, the filtration velocity, u, was independent of x. We shall also discuss the results of Fordham and Ladva (1989), who studied cross-flow filtration of bentonite suspensions, and similarly found that the final steady-state filtration flux was proportional to the length of the filter. We now investigate the fate of particles which are not captured by the cake. We shall assume that cake growth has stopped (so that no particles are captured) and that u = u, is uniform over the cake surface. We first obtain an equation governing the early stages of growth of a concentration boundary layer above the cake, and then estimate whether such a boundary layer should have been observed in the experiments. The particles which are not captured by the cake will flow downstream in a boundary layer adjacent to the cake surface. The concentration c of particles in the boundary layer will be higher than the conccntration c0 in the incoming slurry, and we set c = c,, + cl, We shall obtain an equation for c, which is valid only for cl 4 co, i.e. only for small perturbations to the concentration. The predictions of this equation will give an estimate for the distance downstream at which c1 E c,,, i.e. the distance at which a non-negligible change in particle concentration first occurs. Since we are assuming c1 @ co, any changes to the viscosity of the suspension will be negligible, and hence the shear rate 9 is constant. Similarly, although we expect that the particle diffusivity D will be a function of concentration, we may set D = D(q,). No particles are captured by the cake; we must therefore impose a zero-flux condition at the cake

1= _

,,$-1/3D-2/3X113d.

(10)

The terms on the right-hand sides of eqs (9) and (10) may be neglected as long as x 6 D21;ju:.

(11)

Under these conditions the similarity equation (9) with boundary condition (10) has the solution (e.g. Probstein, 1989, p. 75) (0 d(s) = 1.536 exp (- s3/9) - 0 texp ( - tj/9) dt ss

1

where the factor 1.536 ensures that the boundary condition d’ = - 1 is satisfied at s = 0, since 1

m

t

exp (-

t’/9) dt = 0.651 = l/1.536.

50 s

Thus, the concentration

at the wall is D

c(0) = co + 1.536y

0 -;

l/3

xw

Y

.

The boundary layer is negligible as long as IcyI < co, i.e. as long as eq. (11) holds. We may now make some order-of-magnitude estimates of the importance of the boundary layer. First we consider experiments performed on bentonite suspensions by Fordham and Ladva (1989). We assume that the shear rate at the wall is + = 100 s- ‘. Taking a typical particle radius Q = lo-’ m, the coefficient DB = kT/6nap corresponding to Brownian diffusion in fluid of viscosity p = lo- 3 Pa s is extremely small, and we shall instead use the estimate for the shearinduced self-diffusion coefficient D z 0.33azjcz

[l + 0.5 exp (8.8c)]

(12)

given by Leighton and Acrivos (1986) and used by

N. E.

2916

SHERMAN

and J. D.

Davis and Leighton (1987). A detailed discussion of shear-induced diffusion has been given by Leighton and Acrivos (1987) and by Koch (1989). The result in eq. (12) is for suspensions of sphericai particles. The clay particles used by Fordham and Ladva were plate-like. Eckstein et al. (1977) measured shear-induced diffusion in suspensions of both spheres and discs. The experimental scatter in their results was such that there was no clear dependence upon particle shape. Taking c = 5 x lo-‘, eq. (12) predicts D N lo-” m* s- ’ The filtration velocity was typically 2 x 10e7 m s-l. The boundary layer concentration is predicted by eq. (11) to be negligible as long as x Q 1 m. Thus, the experiments by Fordham and Ladva in cells of length 0.1 m would not have produced a noticeable boundary layer. In the experiments of MS, the particle radius was typically cr = 8 x lo- 5 m, and the concentration c = 5 x lo-‘. In these experiments the filtration flux ~1, was typically 10m3 m s-l (much larger than in the experiments of Fordham and Ladva), and we again estimate the shear rate as $ = 100 s-l. These values lead to a diffusivity D u 8 x 10-‘4m2 s-l. The estimate in eq. (11) for the downstream distance at which the boundary layer becomes important is now 6 x lo-l6 m. However, the analysis in this case predicts a highly concentrated boundary layer which is much thinner than the size of a typical particle. A simpler analysis is therefore appropriate. The volume flux of particles (per unit area) brought to the surface is u,c. These particles are convected downstream and at a distance x downstream the volume flux of particles in the boundary layer (per unit length in the z-direction) is xu,c. The particles are assumed to move with velocity a+, and if the boundary layer is precisely one particle high, the volume fraction within the boundary layer is cb = xu,cf2az3. This monolayer of rp-titles will become close packed when cb = 0.6, i.e. 1 the basis of the estimates given above) at a distar D x = 1.5m along the filter. The filtration cell used L, MS was only 0.1 m long, and the single flowing layer of particles would not become close-packed. The initial estimate of the boundary layer thickness based on continuum analysis was sufficiently small (compared to the particle size) that we have chosen to base our estimates of the MS experiments on a discrete, particulate model. However, it is possible that the flowing boundary layer, far downstream, is sufficiently thick that a continuum analysis is appropriate. The particle concentration will no longer be uniform, and changes in the diffusivity, D, and slurry viscosity may no longer be neglected. Davis and Leighton (1987) showed that the maximum cake thickness under these conditions was $ = p0 u,y/a”T = 0.08, where pcois the filtrate viscosity, and 7 the wall shear stress. Taking 7/p,, = Jo = 100 s-l as before, we find that the maximum thickness of the flowing layer is y = O.&la

SHERWOOD

for the filtration rate u, = tom3 m s-’ and particle size a = 8 x lop5 m used in the earlier estimates. We conclude that the flowing layer cannot be modelled by a continuum analysis under these conditions. Studies of cross-flow filtration frequently find that the filtration velocity, u, decreases as x increases (Blatt et al., 1970). This behaviour can be predicted by convection4iffusion models in which particles are free to enter or leave the surface of the filter cake. The appropriate boundary condition is usually thought to be constant solids concentration on the surface of the filter, with c equal to the maximum packing value, or to the value at which the slurry can no longer flow. If the resistance of the filter membrane is negligible (compared to that of the cake), there are similarity solutions in which the filtration velocity u, cc x- li3, as discussed by Pearson and Sherwood (1988) and by Davis and Sherwood (1990). Fordham and Ladva concluded that such a choice of boundary conditions was inappropriate for their bentonite particles, which were irreversibly attached to the surface of the cake at filtration velocities u > u,. MS, using optical observation, similarly failed to see any re-suspension of particles. Nevertheless, simple considerations of mass conservation imply that, once particles are no longer captured by the cake, an appreciable boundary layer of uncaptured particles will build up at large distances downstream. This bed of flowing particles will impede the flow of filtrate towards the filter cake. Eventually, the resistance of this mobile layer will become large compared to that of the fixed bed of particles. The analyses of Pearson and Sherwood, and of Davis and Sherwood, in which filtration is controlled solely by the mobile layer of particles, then become appropriate. At intermediate distances downstream, the resistance of the fixed bed (and of the filter itself) cannot be neglected, and full numerical models, such as those of Zydney and Colton (1986) or Romero and Davis (1988), are required. The model of MS assumes that particle deposition depends only on the angle of impact of particles arriving at the cake surface. A change in the pressure differential AP will modify the thickness of the cake and the rate at which the cake grows, but will not change the ultimate steady filtration velocity, u,. However, in the experiments of both MS and of Fordham and Ladva, care was taken to ensure that the pressure drop along the length of the channel was sufficiently small that changes in AP were negligible. Modifications to the analysis of the model will also be required if the loss of filtrate is such that u decreases along the length of the filter. 4 THE EXPERIMENTS

OF

MACKLEY

NON-DIMENSIONAL

AND SHERMAN:

RESULTS

MS report experiments performed on polyethylene particles with diameters 2a in the range 125-180 pm, suspended in a water-methylated spirit mixture at a solids volume fraction 5 x 10-O. Filter cakes created

Cross-flow filtration

in the absence of cross-flow had a mean resistance R,-, = 1.8 x lO’Nsm_‘. Cross-flow filtration experiments were performed over a filtration surface of length 0.1 m in the direction of flow, and width 0.007 m in a channel of total width 0.03 m. The height of the channel above the filter surface was h = 0.01 m. The pressure differential AP was 5 lcPa, and the mean flow rate, r?, varied over the range 0.06-0.24 ms-‘. The filtration velocity, u (measured as an average over the surface of the filter), was shown to be independent of the filter length. We non-dimensionalise velocities by the mean velocity, i?, within the channel, rather than by the velocity +a = 6fia/h one particle radius away from the wall. Figure 2 shows the filtration velocity u* = u,& as a function of the non-dimensional time t* = tR,iJ*/AP. Results are shown for a series of different cross-flow velocities B. MS gave data at velocities 0 = 0.06,0.10,0.17 and 0.24 m s- 1 up to times t* = 4 x 105, 106, 2 x lo6 and 8 x 106. We see that there is little evidence to suggest that a limiting filtration velocity has been achieved at these times. Further experiments were therefore performed in order to investigate the behaviour of the filtration velocity, u*, at values oft* up to 2 x lOa, an order of magnitude longer than in the earlier work. If the capture efficiency f is reduced to zero, the filtration velocity, u* , should reach a limiting value, u:. Alternatively, iff0 as 4 4 0, we expect that u* will continue to decrease, albeit slowly, as t+ co, as predicted by eq. (5) for the case f- 9”. There is some evidence in Fig. 2 that the filtration rate curves are beginning to flatten out, but the experimental scatter is such that no definite conclusions may be drawn. The results at early times appear to follow a master curve, as predicted by the MS theory. Although the scatter at small times Seems to depend on the crossflow velocity O, at later times the curves diverge, and the results do not depend on B in any systematic way. This suggests that the capture efficiency, J is not

solely a function of the angle of impact, or perhaps indicates experimental difficulties at the lowest filtration rates. For static filtration, the filtration velocity u a t- ‘I’, and hence a log-log plot of u against t should have slope -0.5. Cross-flow will tend to reduce the capture efficiency,f, and this will increase the slope towards zero. The curves in Fig. 2 have slope approximately -0.7, indicating that the velocity is decreasing more rapidly than for static filtration. This is strong evidence to support the MS conclusion that the filter cake permeability decreases as the angle of impact 4 became smaller. All that is necessary is that R increases more rapidly than f decreases, so that fR is initially an increasing function of u*. At low impact angles only the deeper hollows on the bed surface will be able to capture incoming particles, leading to lower porosities and permeabilities. Acknowledgements-We thank Dr M. Mackley (University of Cambridge) and Dr G. Meeten (Schlumberger) for helpful discussions. This work was funded by the SERC and by

Schlumberger Cambridge Research. NOTATION

a C Cc.ke

co Cl

44 g, R Ro R* s t t* u %I

Ll’

-1 _ 10 :

0

ii X

Y Id

I07

16

108

t* Fig. 2. Plot of non-dimensionalised filtration velocity u* = u/t! as a function of non-dimensional time t* = tR,t?/AP, at six different cross-flow velocities: (0) ~=0.t_h5ms-1;(0)~=0.10ms-‘;(0)V=0.15ms-’;(+)

particle radius solids volume fraction solids volume fraction in cake solids volume fraction in incoming slurry perturbation to solids volume fraction

( = c - co) cb

U* **

2917

volume fraction in a boundary layer one particle high scaled perturbation c1 in boundary layer particle diffusivity capture probability function cake height specific cake resistance cake resistance, R, in absence of cross-flow non-dimensional cake resistance ( = R/R,) similarity variable [ = yx- ‘/3(+j/D)“3] time non-dimensional time ( = tRooZ/AP) filtration velocity limiting filtration velocity at which cake growth ceases non-dimensional filtration velocity (= u/u) dummy variable of integration tangential velocity one particle radius from cake surface ( = j~a) mean tangential velocity within channel coordinate paraIle1 to the filter surface coordinate normal to the filter surface

Greek letters wall shear rate at cake surface LP pressure drop across cake cake growth parameter [ = c~/(c~.~. - co)] 1 filtrate viscosity PO wall shear stress at the cake surface z impact angle of particles arriving on cake 4

N. E.

2918 limiting ceases

impact

SHERMAN

and J. D.

angle at which cake growth

REFERENCES

Biatt, W. F., David, A., Michaels, A. S. and Nelsen, L., 1970, Solute polarization and cake formation in membrane ultrafiltration: causes, consequences, and control techniques, in Membrane Science and Technology (Edited by J. E. Flinn), pp. 47-97, Plenum Press, New York. Davis, R. H. and Leighton, D. T., 1987, Shear-induced transport of a particle layer along a porous wall. Chem. Engng Sci. 42, 275-281. Davis, R. H. and Sherwood, J. D., 1990, A similarity solution for steady-state crossflow rnicrofihration. Chem. Engng Sci. 45.3203-3209. Eckstein,? E. C., Bailey, D. G. and Shapiro, A. H., 1977, Self-diffusion of particles in shear flow of a suspension. J. Fluid Mech. 79,191-208. Fordham, E. J. and Ladva, H. K. J., 1989, Cross-flow filtration of bentonite suspensions. PhysicoChem. Hydrodyn. 11,411+39.

SHERWOOD

Koch, D. L., 1989, On hydrodynamic diffusion and drift in sheared susnensions. Phvs. Fluids Al. 1742-1745. Leighton, D. -and Acrivos; A., 1986, V&coas resuspension. Chem. Engng Sci. 41, 1377-1384. Leighton, D r&d AC&OS, A., 1987, The shear-induced migration of particles in concentrated suspensions. J. *Fluid Mech. 181,415439. Mackley, M. R. and Sherman, N. E., 1992, Cross-flow cake filtration mechanisms and kinetics. Chem Engng Sci. 47, 3067-3084. Pearson, J. R. A. and Sherwood, J. D., 1988, Continuum modelling of cross-flow filtration. PhysicoChem. Hydrodyn. lQ647-661. Probstein, R. F., 1989, Physicochemical Hydrodynamics. Butterworth, Boston. Romero, C. A. and Davis, R. H., 1988, Global model of crossflow micro6ltration based on hydrodynamic particle diffusion. J. Membrane Sci. 39, 157-185. Zydney, A. L. and Colton, C. K., 1986, A concentration polarization model for the filtrate flux in cross-flow microChem. Engng filtration of particulate suspensions. Commun. 47, l-21.