Constant pressure filtration: The effect of a filter membrane

CONSTANT THE EFFECT

D. E. SMILES*, Division

of Environmental

PRESSURE FILTRATION: OF A FILTER MEMBRANE

P. A. C. RAATSt

Mechanics,

and J. H. KNIGHTS

CSIRO, P.O. Box 821, Canberra City, ACT 2601, Australia

(Received 25 May 1981; accepted 24 September 1981) Abstract-Theory developed to describe water movement and volume change in soils may be applied to many industrially important particulate liquid suspensions. The theory is used here to predict the important aspects of constant-pressure filtration where the filter membrane significantly impedes the escape of the liquid. The method requires measured liquid content-liquid potential and liquid content--liquid diffusivity relations of the suspensions, and the conductance of the filter membrane. Illustrative calculations for saturated bentonite slurry are presented. These calculatinns predict the evolution of the liquid and solid profiles in both material and physical space, and the cumulative volume of liquid expelled. as a function of time during filtration. Experiments using bentonite at two different pressures, and with a range of values of membrane conductance,

con&

integral predictions

of the model.

1. INTRODUCTION A useful approach to the one-dimensional processes of filtration and sedimentation of some classes of particulate suspensions is based on the so-called Richards equation[l] of soil water physics, but with the space coordinate replaced by a material coordinate defined in terms of the spatial distribution of the solid component of the suspension. The principal elements of this theory have been discussed elsewhere[2-5) but we reiterate its salient points below. We consider here the application of this approach to constant-pressure filtration flf a particulate suspension with an initially uniform liquid content 9,. This suspencell at the bottom of sion is contained in a pressure which is a membrane that permits passage of liquid but not solid. The membrane has a conductance (Y. The contents of the cell are subjected to a constant pressure p. relative to atmospheric pressure at a time f = 0, whereupon the liquid component of the suspension escapes through the membrane to a pool at atmospheric pressure, and the solid accumulates at the membrane. We treat the system as one-dimensional and assume that the liquid and solid phases of the two-phase system are incompressible. During flow the liquid phase is subject to three forces: a force associated with the gradient of the pressure head of the liquid, a gravitational force, and a drag force associated with the motion of the liquid relative to the solid phase. Darcy’s law expresses the balance between these forces. In addition, the pressure head will generally be the sum of two components: one (V) due to the local “Author to whom correspondence tpermanent

should be addressed. address: Institute for Soil Fertility, P.O. Box 30003,

HarenCr., The Netherlands. *Division of Mathematics and Statistics, CSIRO.

interaction between the liquid and the solid particles and their geometry, the other due to overburden[6]. In the first instance we neglect the influence of the gravitational force and the overburden component of the pressure head, asserting them to be of minor importance compared with the imposed pressure p. 1. THEORY The theory is phenomenological in the sense that it is based on macroscopic, measurable properties of the material[4,5]. The premises are: (a) The volumetric liquid content per unit volume of solid (4) is a well-defined function of the liquid potential (8). The S(Y) relationship may be measured, for example, by permitting a thin sample of the suspension to equilibrate with constant p in a filtration cell that permits escape of the liquid phase to a pool of free liquid at atmospheric pressure, and at the same height as the sample. At equilibrium q = -p; i.e. 9 is minus the compressive pressure on the solid phase. (h) The volumetric flux (0) of the Iiquid relative to the solid particles is described by Darcy’s law ” = -K(9)

awax = -D(9)

adax.

(1)

In eqn (1) K(fi) is the liquid content-liquid conductivity relation for the suspension, SIJlax is the space gradient of the potential of the liquid, and D(6) is a diffusion coefficient defined by D(b) = K(8)

dMd8.

(2)

For a two-component system K(4) or D(6), and W(4) completely specify the properties of the suspension. In the development of the analysis it is convenient to use the solid phase as a reference frame to describe the

D. E.

708

SYLES

movement of the liquid. We therefore define a material coordinate m based on the cumulative volume of the solid phase standing above the filter, where we define x = 0. m(x) =

I

Ox(1 + a)-’ dr

(3)

et al.

tance a. Also, for constant t,, the potential PC, on the suspension-membrane interface and hence the liquid content e0 are unique functions of the reduced time T. We derive an approximate solution for the set of eqns (13-16) using the flux-concentration ratio F of Philip [71 defined by F(@, T) = V(@, T)/ V0( r)

In this coordinate the water is [3,4,5]

system

the continuity

equation

with 0 a normalized

(as/at), = -(adam),

(4)

and eqn (1) becomes u = -D’(a) with O’(6)

given

(17)

for

aslam

(3

+a)-‘.

(6)

liquid

@(l% T) = [6-

content

defined

by

&l/[t%o(~) - &I.

(18)

Since S, is a one-to-one function of T, we can use 7% as a time-like variable and write F(O, +). Introducing eqn (17) into eqn (14) and integrating at time T we obtain

by = D(9)(1

D’(6)

We seek a solution to the pair of eqns (4) and (5) for the case of an initially-uniform suspension in the pressure cell described above: i.e. the initial condition may be written 6=8_,,

m>O,

t=O.

(7)

The boundary condition, assuming that both the imposed pressure p. and the conductance a of the membrane are constant, may be written udt) = a[Y,

-Y&)1,

111= 0, t > 0

But from eqn (16), V,(T) = Y, -Y,(T), and S(Y) is a well-defined function, so eqn (19) represents a liquidcontent profile in M-space. This profile is characterized by 9,,(T) or Y,(T), and hence by the reduced time, T. We seek to define the family of these profiles for the range of & of concern and then to relate 9” (and PO) to T.

At time T, the variable I, representing in reduced form the cumulative volume of liquid (-i) filtered from the system, is found by integrating the liquid-content profile given by eqn (19). Thus,

(8) GO)

where u,(r) is the time-dependent flux of liquid escaping through the membrane, Y, = -p.. and Y,,(t) is the liquid pressure potential at the interface between the suspenThe subscript 0 refers to the sion and the membrane. suspension-membrane interface. In order to solve eqns (4) and (5) subject to eqns (7) and (8). it is convenient to introduce the reduced variables M=am

(9)

T=lr’t

(10)

v=

(11)

vln

I = ai

(12)

where -i is the cumulative outflow from the filter cell. Equations (4), (5), (7) and (8) may then be written

(wan,=-cav/aM), v=-o'(6) aslaM 9 = 79”,

Y=Y,,

V,(Tj=[Y’,-Y,,(T)],

Introducing eqn (19) into parts [4,1 I] then gives

I(&,) =

eqn

(20) and

integrating

by

[ Vd&)l-’ 1” I(8 - 9,)D’(9)/~(@, %)I d9. 8” (21)

Equation (21) calculates the reduced outflow I from C&,(T) [or Y,(T)], and provides the means for relating I to T: viz. V,(T)=dZ/dT.

(22)

But for given \Y,, V. and I are unique functions of YOU,, so eqn (22) may be integrated to yield the time T at which the potential on the interface is Y,. Thus

(13) (14)

M>O,

T=O

(15)

M-0,

T>O.

(16)

The solution of eqn (13) subject to eqns (151 and (16) represents an evolutionary family of liquid content profiles that does not depend explicitly on the conduc-

T(K) = 1’ [ v,(&t,tn,l-’ 0

dl.

(23)

The evaluation of eqn (23) is relatively straightforward. We calculate both I and V, for a particular value of YS and an appropriate range of values of & using eqns (21) and (16). [VJ’ can then be graphed as a function of 1 for each value of 4, and T evaluated graphically according to eqo (23).

Constant

pressure

If we knew D’(B), 8(Y) and F(0, T), then the problem would be solved since eqn (23) describes the evolution of the interfacial liquid potential, Y,, with time T, and Yuo(r) then permits calculation of I( ir’) using eqn (21) and %(A4, T) using eqn (19). The material properties are relatively easily measured[4,7,8] and in our calculations we use the experimentally derived Y(%) and D’(B) relations for bentonite given as Figs. 2 and 3 by Smiles [4]. The specification of F(0, 7) presents a problem, however, and in general one cannot define it a priori. Physically reasonable bounds may be placed upon it, however, and practical estimates within these bounds appear useful. The arguments are based on the following: (1) During the very early stages of the process described here, F(B. T’) should approximate that appropriate to constant-rate filtration of a “linear” material (in which D is constant). As Pbilip[7] and White[lO] show, this F(0) should provide a lower bound to the filtration behaviour of materials in which D’ is dependent on 6 as shown in Fig. 3 of [4]. This lower bound is defined exactly [7] by F(8)

= erfc [invierfc (0/q/n)]

(24)

and is shown as the lower bound of the shaded area in Fig. 1 (2) As filtration proceeds, the solid accumulates at the membrane and the membrane conductance becomes less significant in the flow process, which then tends to resemble constant-pressure filtration with a relatively permeable membrane. For this case F(0, 7) tends to that appropriate to the constant-concentration boundary condition. This F-curve necessarily[7] lies above the constant-flux relation (24) but for a “non-linear” material

1.0

O-6 F 0’4

filtration

709

(where 1y is %-dependent)

F(e) = exp[-inverfc

0

02

o-4

O-6

08

10

8

Fig. I. The flux ratio-water content relation. The points represent the empirical fit F = I - (1 - 63)‘.” used in the calculations. The shading defines the area within which F(B) will tend to lie for all D(S) relations 171.

0)*]_

(25)

Equation (25) gives the flux ratio for the constantconcentration boundary condition in a “linear” material and defines the upper bound of the shaded area of Fig. 1. (3) For many situations[4, P] the F-function appears not to be strongIy time dependent and, furthermore, differs very little from that appropriate to the constantconcentration boundary condition. This particular F(0) may be determined by an iterative procedure [ll]. For our clay this relation is approximated by F = 1 - (1 - 0)‘.75,

(26)

and is shown as the dashed line in Fig. 1. It lies within the bounds defined by eqn (24) and eqn (25). implying that the shaded area probably represents the region within which all such relations will lie. We use eqn (26) in the illustrative calculations that follow. In conclusion, however, it must be admitted that the use of any F(0) for conditions other than the constant-concentration boundary condition will depend on insight and must be justified by experiments.

J. lLLU.WRATlVE CALCULATIONS The main results of interest are: (a) the successive liquid content profiles in m- and x-space; (b) the corresponding families of solid content profiles; and (c) the cumulative volume of liquid expelled from the system as a function of time. These calculations depend centrally on the evaluation of eqns (19) and (21), and then on their manipulation according to eqn (23) to establish the time-dependence of the process. We calculate the profiles and then examine how they evolve. Figure 2 shows liquid-content profiles in the material variable -MV,(T). evaluated using eqn (19) for a number of values of &. If we treat as a family the profiles that evolve for a particular imposed Y=, then this figure represents, in a concise way, the evolution of all such families. Because each member of a family is identified with a particular value of %,, (and by implication, VI,), each profile may be presented in terms of the variable M by scaling it according to its particular value of V,(n = [Ys -Y,,(T)]. Thus, if Y= = -0.14 m, and we consider the profile identified by 9, = 16, then the reduced flux at M = 0 is, from eqn (16) V,(T)

0

below F(B) given by

= [-14-Y,,(T)]

= -12.8m

(27)

because \Iro(T) corresponding to So = 16, in Fig. 2 of [41, is -1.2 m. Tbe M-coordinate of 6 = 30 (for example) on this curve then becomes 2.77 X IO-‘” m SC’, and for (I = 1.2 = IO-’ s-l, m = 2.3 X lo-’ m. We chose this value of a because it corresponds approximately with that used in the experimental section below. We note, in passing, that V,,(Z’) is negative because it represents outflow from the system. Similarly, the profile identified by %,, = 10 (Yu, =

710

D.

0

0.2

E. SMILESet al.

-10”

M V,(T)/(

10

O-8

0.6

0.4

d

12

s-‘)

Family of liquid-content profiles expressed in material coordinates; these protiles are calculated using eqn (19) and the material properties shown in Figs. 2 and 3 of Smilesl4). The numbers on the curve.s are appropriate values of &. Fig. 2.

-3.7 m) must scale according to V&T)= -10.3 m for T, = -14 m. Profiles corresponding to Fig. 2, showing the distribution of solid, appear in Fig. 3. I-Iere the volumetric solid content #, is graphed as a function of --MV,(T). This is in fact a phase diagram which also shows the distribution of the volumetric liquid content 0, because ‘e,+e,=1. As with Fig. according to The data presented in

(28)

2 these profiles must be individually scaled their appropriate V,,(T) value. of Fig. 2 (and of Fig. 3) may readily be terms of the spatial coordinate x because X-ax=

M(l+fi)dM

(29)

Finally, it is of interest to calculate the cumulative volume of solid and its distribution away from the membrane. This relation is simply determined if we recall that nr is exactly this quantity and is defined formally by eqn (3). Figure 4 shows a family of such distributions in the reduced coordinates M and X The lowest curve in this distribution represents the initial solid distribution. It is linear because the initial liquid content (and by inference solid content) is constant throughout the system. All the profiles presented so far are perfectly general with respect to conductance o. The appropriate profiles for a particular value of a (determined independently) are easily calculated when we recall from the definitions of M, T, V and X in eqns (9). (lo), (11) and (26) that a is simply a constant scaling factor.

Initial 0 0

I

I

I

I

I

I

0.2

0.4

0.6

0.6

1.0

1.2

-10”

Fig. 3. Volumetric

IF& = O-027

solid-content

_

MVo(T)/(m2s-‘)

protiles corresponding to the liquid-content

profiles

shown in

Fig.

2

Constant

pressure

filtration

711

Table I. Representative data from Fig. 2 of [4] and Fig. 5 used to demonstrate the calculation of I(T) (and by implication B,(T)) for t, = - 14 m using eqn (23)

0.75

16

3.836

7.55

28.95

1.20

16

5.255

7.81

41.05

3.05

2.00

14

7.043

8.33

58.69

4.47

5.30

2.15

0 0

10

20

30

-1d”XV,(T),/(in*s-‘)

Fig. 4. The relationships between the material coordinate M and coordinate X for some of the profiles shown in Fig. 2. the spatial Abe lowest curve represents the initial condition, for which M = (1 + &-'X. The numbers on the curves are appropriate values of &. We turn now to the cumulative expression I of liquid from the system and the time dependence of the process. Figure 5 shows IV,,(T) as a function of So. calculated using eqn (21) [or using eqn (20) and Fig. 21. This diagram relates the areas between each member of the family of profiles shown in Fig. 2 and the initial distribution given by eqn (15). It provides the basis for calculating the evolution of all profiles in time. This calculation is conveniently demonstrated for q’, = -14 m. Using Fig. 2 of Smiles[41 and eqn (16) we calculate the appropriate value of V;;’ for each value of &, and, using Fig. 5, infer the corresponding value of I.

lo*

1 V,

10

12.00

11.49

1.38.0

12.50

8.5

8

15.40

18.18

280.1

33.20

11.0

I

17.32

33.33

577.4

I.10

14.0

6

19.57

_

_

_

T(I) is then calculated using eqn (23); Table 1 shows appropriate values involved. The time-dependence of 19~ which emerges is shown in Fig. 6. In particular, note that &only very slowly approaches the equilibrium value of 6 corresponding to 8,, = -14 m because of the continuing, but decreasing, potential drop across the filter membrane. Figure 6 also shows predicted &,( 7’) curves for q’. = -11 m and \y, = -5.75 m. With the calculation of S,(T), all features of the filtration process are resolved, since Fig. 6 permits us to retate the profiles shown in Fig. 2 to T (for Ur. = -14 m)

(T)/(rnz

S-‘)

Fig. 5. The general relation between surface liquid content & and the “reduced” cumulative volume of desorbed The manipulation of these data, for particular values of Y,,. permits calculation of the timeliquid, V&n. dependence of the filtration process.

D. E. SMU.ESet al.

-13

-12

-11 ‘o9,o

-10

-9

[T/b-‘,]

Fig. 6. The calculated timedeper.dence of the surface watercontent& for threeimposed pressures. These cnrves .become asymptotic to the equilibrium value of 9 appropriate to the particular imposed pressure. rather than to 17~. Tn addition, the cumutative volume of liquid IiItered from the system is defined in Table 1. Figure 7 shows 1 as a function of fi for q’, = -ldm. These data are taken from Table 1 and are graphed in this way to reveal some basic features of the filtration process. In particular, we note that three stages in the desorption process are identifiable: (a) in the early stages I is a linear function of T and the membrane controls the flow. There is then a (b) transitional stage leading to a (c) continuing stage in which I( T”2) is linear. In this fmal stage the material properties of the suspension (particularly close to the membrane) control the flow and the system behaves as if the boundary conditions were w = ?d=V!=),

m = 0.

(30)

The cumulative outtlow i(=#a) is then linear with respect to VT[ = */a]. It is evident that if D’(8), .19(q) and F(E), T) are well-defined, then one can predict the course of constant-

0

2

pressure filtration of a particulate slurry through a membrane whose transfer coefficient is constant. The effect of a given membrane may be treated simply by scaling the reduced variables, I, M, T, V and Xaccording to the transfer coefficient U.

4. FXPERMENTS

The experimental test of some details of the analysis presents difficulties so we examine here only the integral predictions embodied in Fig. 7. This is, however, still a searching test because prediction of I( T”*) from q’(6) and D’(8) requires not only that we accurately calculate the integrals of the profiles shown in Fig. 2 but also the relation between these integrals and So, shown in Fig. 5. The experiments were performed in a pressure filtration cell identical to that described in [d], using “Western bentonite” provided by the National Lead Company, Houston, Texas. A large volume of clay was prepared by mixing this bentonite with an appropriate volume of

6

4 Id

T $,/

Fig. 7. The cumuIative outflow I(=P~) as a function of T’n(=&z)

8

Kl

a-h caIculated according

to Table I for

YO = -14 m

Constant pressure filtration: the effect of a filter membrane

0

Fig. 8. Experimental

2

4

6

8

and predicted cumulative outflow during constant pressure impermeable membranes for two imposed pressures.

distilled water; all experiments were performed on subsamples of this stock. In order to obtain a sufficiently small value of a, it was necessary to use one of more “Pellicon” ultra-filtration membranes. Each of these membranes has a conductance a = 1.2 x 10e7 s-’ when measured at steady state using deaied distilled water. For these experiments the pressure cell was filled with water, a constant pressure applied and the outflow measured on a top-weighing balance reading to 53 x 10eb kg_ This procedure was adopted for each set of membranes used. To commence an experiment, the excess water from the a-measurement was removed from the cell and 10 or 20 ml of clay suspension added. A constant air pressure was then applied and the outflow measured on the balance. Two sets of experiments were performed at imposed pressures of 5.75 m and 11 m. In the process it was found that (Y measured with distilled water was about 60% greater than that given by the early stages of outflow when i vs I was approximately linear. We assume that the effect arises because of blinding of pores in the membrane by the clay. There seems to he no simple method for scaling the value of (2, measured in the absence of clay, to account for this interaction effect, so we have used (1’ based on the early stages of outflow to calculate I and T from the experimental data according to eqns (IO) and (12). Ivs T”’ curves, so calculated, are superimposed on predicted curves and shown in Fig. 8. The important features of each experiment are tabulated in this figure. Observe that the use of reduced coordinates I and T consolidates all i(t) data identified with a particular value of Y,, regardless of the value of 0’. Furthermore, the correspondence between prediction and observation at small T is excellent; as I( T”‘) becomes linear, however, prediction tends to overestimate the observed outflow

713

10

filtration through relatively

rate although the rate is still within the range to be expected if D’(4) lay close to the lower bound shown in Fig. 3 of [4]. The difference is also consistent with a small loss of outflow due to evaporation. Because it is difficult to reliably measure D’(8) to better than about t15% using any particular method of determination[l2], we do not believe that there is any advantage in pursuing the definition of D’(9) further. We therefore conclude that the approach provides a valid analysis of the process of constant-pressure filtration through a relatively impermeable membrane provided the material properties D’(9) and 6(Y) are well defined. In this respect we emphasize the great generality of the approach; these macroscopic properties are readily established directly from routine macroscopic measurements and they may be of quite arbitrary functional form. In addition, we are not limited to any simplifying assumptions about internal geometry of the medium, nor do we depend on arbitrary simple models of (for example) capillary flow. The analysis is totally free of irrelevant abstractions and simplifications relating to scales of discourse other than the macroscopic one of immediate concern, und we demand only that D’(B) (or K(6)) and 9(6) exist and arc measureable. Finally we note that the procedure described above is not restricted to situations where L+ is constant: we do not pursue the issue here, but the analysis may also be presented for cz an arbitrary function of time, and for (I dependent on YO(t). We note also that the analysis may be performed for situations where gravitational effects cannot be neglected.

NOTATION i

m

cumulative liquid flux, m material coordinate, m

D. E. SMILBSet al.

714 P t v

; D’ F I K M T V X

imposed pressure (relative to atmospheric pressure), m time, s

volume flux density of liquid relative to the solid particles, m s-’ space coordinate, m liquid diffusion coefficient, m2 s-’ liquid diffusion coeficient in material coordinates, m2 s-l flux ratio cumulative reduced flux, m s-’ liquid conductivity defined by Darcy’s law, m s-’ reduced material variable, m s-’ reduced time variable, s-’ reduced flux variable, m reduced space variable, m s-’

Greek symbols 6 volumetric liquid content per unit volume of solid ? liquid potential component of the total potential of the iiquid, m

u 0 0, 0,

liquid conductance of the membrane, s-’ normalized liquid content volume fraction of solid volume fraction of liquid

REFERENCES

[ll Richards L. A.. Physics 1931 1 318. 121 Shirato M.. Sambuichi M., Kate H. and Aragaki T.. Am. Inst. Chem Eng. J. I%9 15 405. I31 Smiles D. E., Chem Engng Sci I%9 ZS 985. I41 Smiles D. E., Chem. Engng Sci 1978 33 1355. IS1 Smiles D. E., Knight I. H. and Nguyen-Hoan T. X. T., Sqor. Sci TechnoL 1979 14 175. [4l Philip J. R., Aust. 1 Soif Res. 1%9 ‘I 99. VI Philip J. R., Soil Sci 1973 116 328.

I81 Smiles D. E. and Rosenthal M. J.. Aust. J. Soi1 Res. 1968 6 237. 191 Smiles D. E. and Harvey A. G., Soil Sci 1973 116 391. 1101 White 1.. Soil Sci SM. Am. 1. 1979 43 1074. il Ii Knight ;. H. and Philip I. R., ,902 SCL 1973 116 407. I121 Smiles D. E., Soil Sci Sm. Am. J. 1978 42 11.