NATO ASI Series Advanced Science Institutes Series
A Series presenting the results of activities sponsored by the NATO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities.
The Series is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division
A Life Sciences Plenum Publishing Corporation B Physics London and New York
C Mathematical and D. Reidel Publishing Company Physical Sciences Dordrecht and Boston
0 Behavioural and Martinus Nijhoff Publishers Social Sciences DordrechtlBoston/Lancaster
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G Ecological Sciences
Mathematical Models and Design Methods in Solid-Liquid Separation
edited by
A. Rushton Department of Chemical Engineering, UMIST Manchester, M60 10D UK
1985 Martinus Nijhoff Publishers Dordrecht / Boston / Lancaster
Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Study Institute on Mathematical Models and Design Methods in Solid-Liquid Separation, Lagos, Algarve, Portugal, January 4-15, 1982
Library of Congress cataloging in Publication Data
Main entry under title:
Mathematical models and design methods in solid-liquid separation.
(NATO AS! series. Series E, Applied sciences; no. 88) "Published in cooperation with NATO Scientific Affairs
Division. 1I
"Based on a series of lectures given at a Nato Advanced Study Institute held in Lagos, Portugal in January 1982"--Pref.
Includes bib! iographical references and index. 1. Separation (Technology) 2. Separation (Technology)-­
Mathematical models. I. Rushton, A. II. North Atlantic Treaty Organization. Scientific Affairs
Division. III. Series. TP156.S45M36 1985 660.2'842 84-29487
ISBN-13: 978-94-010-8751-3 e-ISBN-13: 978-94-009-5091-7 001: 10.1007/978-94-009-5091-7
Distributors for the United States and Canada: Kluwer Boston, Inc., 190 Old Derby Street, Hingham, MA 02043, USA
Distributors for the UK and Ireland: Kluwer Academic Publishers, MTP Press Ltd, Falcon House, Queen Square, Lancaster LA1 1RN, UK
Distributors for all other countries: Kluwer Academic Publishers Group, Distribution Center, P.O. Box 322, 3300 AH Dordrecht, The Netherlands
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publishers, Martinus Nijhoff Publishers, P.O. Box 163,3300 AD Dordrecht, The Netherlands
Copyright © 1985 by Martinus Nijhoff Publishers, Dordrecht Softcover repirnt of the hardcover 1 st edition 1985
Dedicated to the Memory of
Professor Dr. Ir. P.M. Heertjes University of Delft and Professor Lloyd A. Spielman University of Delaware
TABLE OF CONTENTS
A. Rushton Introduction
F.M. Tiller & J.R. Crump Recent Advances in Compressible Cake Filtration Theory
L.A. Spielman Flow Through Porous Media and Fluid-Particle Hydrodynamics
R. J. \-Iakeman Fi ltration Theory: Formation and Structure of
VII
3
25
K.J. I ves Deep Bed Filters 90
E.R. Baumann & C.S. Oulman Use of BDST Analysis Techniques for the Desi9n of Filtration Systems Using Coarse Media and Dual Media Filters in Series 150
A.S. "lard Pretreatment Processes 170
L.A. Spielman Hydrodynamic As'pects of Flocculation 207
R.V. Stephenson & E.R. Baumann Precoat Fi ltration Equations for Flat and Cylindrical Septa 233
VIII
C. Alt Centrifugal Separation 257
R.J. Wakeman Dewatering of Filter Cakes: Vacuum and Pressure Dewater i ng 286
J. Hermia Fi Iter Cake Washing 310
A. Rushton Filter Media: Woven & Non-Woven Cloths for Liquids 333
D. Leclerc & S. Reboui llat Dewatering by Compression
List of Participants and Authors
INDEX
356
393
397
PREFACE
The separation of finely-divided solids from liquids constitutes an important stage in many industrial processes. Separation of mixtures ranging from highly concentrated slurries to slightly turbid liquids must be effected in circumstances where the solids, liquid or both phases may have value.
Separations may be achieved by use of a membrane or filter medium which, positioned in the path of a flowing suspension, will allow passage of the fluid whilst retaining solids on the surface or within the medium. Alternatively the two phases may be separated by sedimentation processes involving gravitational or centrifugal force. In either mode, separation difficulties are sometimes experienced with the result that solid-liquid separation is often a bottleneck in commercial plants.
Operational difficulties and plant failures are associated with the random nature of the particles being separated; variations in size, shape, states of aggregation, compressibility, etc., produce a wide range of problems. Plugging of the filter medium or the collapse of the solids under applied stress lead to slow flowrates of liquid. The colloidal nature of some precipitates makes separation by settling virtually impossible without the use of chemical agents to enhance the size of basic units and to reduce repulsive surface forces. Unit operations such as filtration, comminution, etc., involve a seemingly bewildering array of machines which makes plant selection a difficult step and reflects the uncer­ tainties attaching to operations involving the solid )hase. Many types of pressure, vacuum and centrifugal filter are available. The older traditional units such as the plate-and-frame press, the rotary vacuum filter, the basket centrifuge and the deep sand-bed unit used in water clarification have all received modification in recent times. New machines have appeared, e.g., the variable chamber press, the cross-flow filter, the multi-layer sand bed, etc.
Many of these modifications and new designs have followed trends in the developing science of liquid-solid separation. The latter has, fortunately, attracted the attention of increasing numbers of
2
research scientists over the past decade and the large output of information made available in the literature had done much to transform the 'art' of filtration into a predictable operation.
This is not to claim, of course, that the picture is complete and that all separatiQn problems are easily solvable or may be avoided. This ideal situatiofi may never be fully obtained; new processes will probably always require an experimental basis for plant evalu­ ation, etc. Nevertheless, recent published work, much of which is reported in this text, points to correct modes of experimenta­ tion, results interpretation and application to plant design and operation.
The amount of information issuing from the research and development areas is quite enormous and it is fortunate, in view of the econo­ mic importance of filtration that several excellent texts have appeared recently (1) (2) (3) (4) (5) which taken together constitute a most valuable collection of information.
The chapters contained in this book are based on a series of lectures given at a Nato Advanced Study Institute held in Lagos, Portugal in January 1982. Lecturers from various engineering professions were invited to prepare notes which could form a bridge between the practical aspects of the subject and the recent, extensive theoretical developments available in the literature.
Each chapter contains the views of a recognised authority in the subject. No attempt has been made to alter the style of presen­ tation although in each chapter individual notation has been used in order to reduce the interdependence of the various subjects presented.
At the same time it is hoped that the material has been arranged in an order which, despite the individualistic character of the chapters, gives a measure of the entire set. Again it is hoped that this material presented forms a useful extension to the other Nato ASI publications in this field, e.g. the trilogy identified in the references (1) (5) (6).
References
(1) 'The Scientific Basis of Filtration', K.J. Ives Ed. Nato ASI Series E, Applied Science No.2, 1975
(2) Pruchas, D.B., 'Solid-Liquid Separation' Uplands Press Croydon (3) Wakeman, R.J. 'Advances in Liquid-Solid Separation', Elsevier (4) Svarovsky, L., 'Liquid-Solid Separation', Butterworths (5) 'The Scientific Basis of Flocculation', K.J. Ives Ed. Nato
ASI Series E, Applied Science No. 27, 1978 (6) 'The Scientific Basis of Flotation', K.J. Ives Ed. Nato
ASI Series E, Applied Science No. 75, 1984.
RECENT ADVANCES IN COMPRESSIBLE CAKE FILTRATION THEORY
Frank M. Tiller and Joseph R. Crump
Department of Chemical Engineering University of Houston, Houston, TX, U.S.A., 77004
CONTENTS
INTRODUCTION
BED STRUCTURE Constitutive Relations
OVERALL MATERIAL BALANCE
FLOW RESISTANCE RELATIONSHIPS
NOMENCLATURE
c
G
2 dimensionless variable, Rm /ca
unit vector in x-direction
o
J value of J when W=l wc w
J x
b "" 2 Darcy permea ll1ty, m
2 average value of K,m
cake thickness, m
unit normal to surface
dimensionless variable, R/R m
applied pressure, N/m2
pressure below which a and E are assumed constant, N/m2
pressure drop across cake, N/m2
hydraulic pressure, N/m2
pressure at cake-medium interface, N/m2
superficial flow rate at distance x, m3/Cm20s)
3 2 superficial flow rate at cake surface, m /Cm os)
3 2 filtration rate, dv/dt, m /Cm os)
3 2 average flow rate over entire cycle, v/t, m /Cm os)
R c
R m
total resistance, l/m
cake resistance, l/m
medium resistance, l/m
average mass fraction of solids in cake
time, s
3 2 v vol. of filtrate/unit area, m /m = m
w
total mass dry solids/unit area, kg/m2
ratio of w/w c
ratio of x/L
a average value of a, m/kg av
a. value of a below p., m/kg 1 1
a empirical constant, Eq. (5) m/kg o
5
6
Ei value of E below Pi
Eo empirical constant, Eq. (6)
Es volume fraction of solids, (I-E)
IT
ratio p /p s
INTRODUCTION
In compressible cake filtration, variables desired as a function of time are; filtrate volume/area, cake thickness, average cake porosity, and applied pressure.
In order to obtain relationships among these quantities, the following basic relationships must be developed;
1. Free-body analysis (neglecting momentum changes) relating liquid pressure to accumulative frictional drag on particulates.
2. Interaction of frictional drag and bed structure. 3. Law of continuity applied internally and to overall
system. 4. Rate equations relating variable permeability or
specific flow resistance to local conditions in (a) cake and (b) supporting medium.
5. Boundary conditions at cake surface and at interface between supporting medium and cake.
Free-Body Force Balance
Conditions and nomenclature for a compressible filter cake are shown in Fig. 1. Flow is pictured from right to left origin­ ating in the slurry and exiting from the supporting medium. A maximum (E ) at the cake surface, the porosity decreases to a minimum, E~, at the medium. The hydraulic pressure follows a nonlinear path from the applied pump pressure p at x = L to PI at the medium where x = O. In Fig. lB, the cake is illustrated as being more compressed at the medium than at the surface. The portion of the cake between p and PL will be chosen for a free­ body force balance.
przl·o p~~;
o x L
L
The general nature of the frictional drag is depicted in Fig. 2A. Fluid flows through the interstices of the beds and exerts a drag force on each particle. Surface forces due to frictional flow result in internal stresses in each particle which are then communicated to other particles through an interconnecting network as shown. Hard particles are assumed to be in point contact. Manifestly there must be some area contact; or, otherwise, an infinite pressure would develop where forces are transmitted from one particle to another. The free­ body balance will be made on the assumption of universal point contact.
The line XX in Fig. IB is shown in Fig. 2B as it cuts through some of the particulates. Inasmuch as the forces are indeterminate over the portion intersected by the particles, we choose to construct a wavy membran~ which lies entirely in the fluid phase as indicated in Fig. 2B,C. The membrane intersects various dotted (Fig. 2B) constant pressure lines which are assumed to differ by no more than the variation over the thickness accomodating one particle. The x component
7
8
(A)
x
(C)
-+-+
pdAnoi = pA (1)
where p is the average pressure on the membrane and is assumed equal to the manometric pressure at position x. If point contact is accepted, the membrane area can be replaced by the cross-sectional area. We then assume that momentum changes are negligible and that the cake lying between x and L can be treated as a free body. A force balance yields:
APL + Fs = Ap (2)
where F represents the accumulated frictional drag over all particI~s. The increasing drag ,causes the structure to col­ lapse on a nearly irreversible basis leading to decreasing porosities as the medium is approached. The applied pressure, p, is a function of time but not of distance. Next, we divide (2) by A, and define "compressive drag pressure" as Ps Fs/A giving
Ps (x,t) + PL (x,t) = pet) (3)
The area A is not equal to the actual area of contact, but is simply the cross-section area of the filtration. Thus, p is a fictitious pressure which is used for convenience. It is analogous to the term "effective pressure" in soil mechanics. Next, taking differentials with respect to x at
constant t, we have
C:s ) + G:L) 0 (4)
Bed Structure
The nature of the interstices in a porous bed has been the subject of many investigations. Shape, size distribution, and arrangement of particulates determine pore geometry. In­ herent complexity and virtual impossibility of exact mathe­ matical representation of pore walls plus the continuous change which takes place in filter cakes requires various degrees of simplification. We shall assume that porosity is one essential component of the description of porous bed structure.
Cake structure is determined by the way in which particles are originally deposited and then collapsed into new positions as load in the form of fluid drag is applied. We shall assume that porosity can be described by the product of the initial unstressed value E and some function of effective pressure Ps' Two forms whigh have proved successful are
l-E = (l-E )(1 + P /p )S o s a
where Pa is an arbitrary parameter; and,
and
where
where
(5)
(6)
Eq. (5) possesses the advantage of providing a continuous func­ tion over the entire effective pressure range.
Continuity
As cake thickness grows, each layer is subjected to in­ creasing drag; and porosity at a fixed distance x from the medium decreases with time. In Fig. 3, void and solid volume fractions, E and E = l-E, are plotted against distance x at times t and t + dt: The porosity decreases at each point x while E increases. Solids are being squeezed into the voids as the take compresses. Solids are on the move, and must be
9
10
·5"1-.
x DISTANCE
Figure 3
-, ,-dL , , L
included in a continuity balance. The volume of liquid in the distance 0 - x is given by J x edx
o
Rate equations for solids and liquid yield (Tiller, 1981):
q(x,t) - ql (t) = ~t f: where ql = ~~, the filtration and liquids yields:
edx
rate.
(7)
(8)
(ii) ( ~~) = ~~ s (:: s ) x x
(9)
t
which is equivalent to the usual first order rartial differen­ tial equation of continuity.
Eqs. (7)-(9) clearly indicate that q is not constant throughout the cake (Tiller and Shirato, 1964). The nature
of the q distributions is illustrated in Fig. 4. For incom­ pressible materials and washing of beds with no volume change, qlql equals unity. With dilute slurries, squeezing has little effect on the local flow rate. The quantity of liquid which flows through the cake is large compared to the amount squeezed out. As slurry concentration increases, the effect increases as indicated by curve B. Highly compressible materials present a different picture in which most of the porosity reduction takes place in a skin close to the medium (Tiller and Green, 1973), a large portion of the cake is unconsolidated; and without a changing porosity, the local rate q does not change.
IMCOMPRESSIBLE CAKE
Figure 4 Overall Material Balance
A material balance over the cake, filtrate, and slurry yield
w c
(10)
Early researchers assumed c was constant, and textbook authors have followed the same path without investigating the effect of variation of s during filtration. With concentrated slurries having s7s about 0.5, c may vary as much as 20 percent during filt~ation. When sis < 0.1 variation in c is on the order of a few percent. c
II
12
Rate Equations
Flow rate equations are generally based on some form of Darcy's law. Inasmuch as solids are moving in compressible beds, the Shirato modification must be used in rigorous developments. The Darcy-Shirato equation takes two forms; first for spatial coordinates:
( dPL) = ~s (~_ ~) dX K £ 1 - £
t
(11)
where q/£ and q 1(1 - s) are respectively the average liquid and solid velocities in the pores and anti-pores. It is necessary to include £ as shown outside of the bracket in order that Eq. (11) reduce to the classical Darcy form when q = O. The superficial solid velocity, Eq. (8), and PL can b~ eliminated, resulting in
G:s) t
~(dPS ) ~ dX
t,x = 0
(12)
(13)
where R is defined as medium resistance and is assumed con­ stant. mThe medium resistance is equivalent to its thickness divided by its average permeability.
The second form of the Darcy-Shirato equation in material coordinates is
(14 )
W = PshX (1 - £)dx (15)
It is unfortunate that mass rather than volume of dry, inert solids was originally chosen for the material coordinate.
We shall not deviate from customary practice in this paper. Local permeability and specific flow resistance are related by K = lip (1 - £)0.. The spatial coordinate partial differ­ ential equ~tion which applies to compressible cake filtration is obtained by combining Eqs. (8), (9), and (14). The super­ ficial flow rate q as obtained from (12) is differentiated with respect to x at constant t and equated to Eq. (9) result­ ing in
(a2p S) _ ~q1 d£ (aps), + d1nK(1-£) (aps)2
a 2 K(l-£) dp ax dp ax x t s t s t
-~ dE (ap s ) K(l-£) dps at x (16)
This equation is equivalent to one used by Wakeman (1978). A similar equation for material coordinates results from first transforming the continuity equation into the form
(i;) =~~£(~!) + (17)
t t
o (18)
The coefficients of Eqs. (16) and (18) are functions of ps.
Flow Resistance Relationships
The basic equations for flow through porous media made use of the local porosity £ and local specific resistance a but do not indicate how they are to be related to p. The functional form of a and £ vs. p relationships dep~nds on empirical observation and is a k~y problem of filtration. Corresponding to Eq. (6), we use
a. = aps n where Ps ~ Pi
and n where Ps .::. Pi· (19) a. = o.. aPi 1
13
14
(20)
Although various simplifications are possible, they all lead to some contradictions. Few investigators have defined restrictions placed upon their theoretical developments. Wakeman (1978), Shirato, et. al. (1970) and Risbud (1974) have solved numerically the partial differential equations under different sets of boundary conditions.
Traditional developments generally neglect solids move­ ment and transform the Darcy partial into an ordinary differ­ ential equation. It is assumed that p = f(x/L, ~p ) or f(w/w , ~p ) and the continuity equati5n mayor mayCnot be used in th~ derivations. The following restrictions represent different approaches to filtration.
1. 2. 3. 4.
5.
Assume p = f(x/L, ~p ) Add the festriction tHat q = constant Add the restriction that £ = constant In constant pressure filtr~¥ion, assume that K and a are constants as well as £av av Negle~¥ medium resistance.
Also, it is assumed that a = a f (p ), 1-£ = (1-£ )f2(p ), and medium resistance R is coRstant. Migration o~ f1ne s (relative to pores) par~icles can lead to serious deviations from assumptions concerning a, £, and R. Minimum research attention has been focused on medium an~ cake clogging in spite of their industrial importance. Weber, et. al. (1981) modified traditional equations to take account of clogging in filtration of liquefied coal.
Quasi-Steady State Approximation
Quasi-steady state (QSS) is an unstated principle under­ lying derivation of conventional cake filtration formulas. Effective pressure and porosity profiles are assumed to adjust instantaneously to new value when the pressure drop across the cake ~p is changed. Fig. 5 illustrates the QSS approximation in whicH the £ vs. x curve is shown as a unique distribution for each value of ~p. As ~p increases, £ drops; and liquid must be expressed fr5m the ca~e leading to iVvariable q/qo.
o OX· x/L
NORMALIZED DISTANCE
Figure 5
Conventional theory requires that q/ql equal unity. The QSS approximation does not require that q/ql be constant but rather that it be a function of X = x/L. In order for the partial differential equation to collapse to an ordinary equation, the term on the RHS of (16) must be small compared to the terms of the LHS. We shall test the q/ql variation.
Assumption of QSS requires replacing the partial deriva­ tions in (12) and (14) with ordinary derivatives. Assuming p = f(X, 6p (t)), we find
s c
1.. ( ilp s) L ilX
6p c
(21)
( ilPS ') = (ilPs ) (.£!) + (ilPs ) (~(f'>Pc)) (22) ilt ilX ilt il(6pc) ilt
x 6p x X x c
Substituting 6p = P - ~ql R , we note that p and ql = dv/dt are functions of t. m
( :~s) x
6p c
( ilP )[ 2] s dp d v + il(6p) dt - ~Rm -2 (23)
c X dt
In this paper, we shall restrict ourselves to p = constant and R = 0; thus the last term in (23) will disappear. Substitut­ iWg (23) in the Darcy-Shirato spatial and material coordinate
15
16
(24 )
(25)
As the porosity is a function of X = x/L and ~p , equa­ tions equivalent to (24) and (25) can be developed iff which s replaces p. The resulting formulas can be substituted in the coptinuity equation (9) to yield
( aaxq ) - I dq _ (dS) _ x dL (as ) - L dX - at - - 2"" dt ax +
t x ~o
(a(~;c))x [* -~Rm ::~] (26)
Letting p be constant and Rm 0, (26) reduces to
~ - _ dL X ~ (27) dX - dt dX
We cancel the dX's, replace Xds = d(Xs) - sdX, and integrate over X(O,I) and q(ql,qo) to obtain
or
qo 0
o
Dividing both sides by ql dv/dt leads to
(28)
(29)
qo-qo dL -- = - (E -E ) ql dv 0 av (30)
The derivative dL/dv can be obtained from a volumetric balance over the slurry, wet cake, and filtrate. If ~s = volumetric fraction of solids in the slurry
L v
(31)
(32)
For the special cake under consideration in which ~p = P = constant, E is also constant. The ratio (ql-q )/q~ decreases as ~ incre~¥es. The maximum value of ~ leads ~o tfie largest q vafiation. We assume that the slurry toncentration cannot exceed the concentration of the unconsolidated cake and substi­ tute ~s = I-Eo as a limiting value of the slurry concentration. That YIelds q /ql < E where E generally ranges from 0.6-0.9. From a rheolo~ical-vigwpoint, ~ is limited to approximately 75% of (l-E ) if slurry viscosity is to be in a reasonable
o range.
We calculate the variation of (ql-q )/ql assuming ~ = 0.75 (l-E ) based upon data calculated by Til~er and Cooper (1962) withO~p = 689 kPa and list the results in Table 1.
c
Table 1 Variation in (ql-qo)/ql
E ~s = 0.75Cl-Eo) av
(ql-qo)/ql
0.029 0.035 0.09 0.16
The values in Table 1 indicate the maximum possible variation in q under conditions of constant pressure and negligible medium resistance.
17
18
[ 1 fX (q/ql-S) ] f6PC KdP
\1ql x X l-s dX = s
o Ps
(33)
The quantity in brackets on the LHS is termed J with the value J L when X = 1. Integrating over the entife cake yields
f Pc \1J Lql L = Kdps (34)
o
This equation provides a definition of average permeability. A similar analysis for the material coordinate form of the Darcy-Shirato equation leads to
[ 1 fW (q/ql-S) ] fi1Pc
Ps
(35)
The terms in brackets is called Jw. Integrating across the cake yields
l1J qlw =f"'Pc dp /a = i1p /a (36) wc c s c av o
It is this equation which serves as the basis for development of most theory. Substitution of (20) into (36) and integration produce
Pa [ -"(=-l--n""""")-a-o (1 + (37)
where i1IT = i1p /p. Eq. (37) emphasizes the QSS hypothesis in that theCrate &, aepends only on the cake mass w (or L) and i1p and is independent of past history. c
c
Noting that w = p (l-S )L, it is possible to eliminate w from (37) and oUtainsan e~~ation involving (qlL), the rate­ thickness product which is plotted in Fig. 6 as a function of i1IT at constant values of n + S. For incompressible solids, th~ flow rate is directly proportional to i1p /L. For
c
..J
-I g
Figure 6
moderately compressible materials, n + S would have a ~a5ue near 0.5, leading to a rate roughly proportional to ~p • IL. When the sum exceeds unity, the cake is highly compres~ible; and the rate-thickness product approaches a constant value. The rate is then proportional to IlL and is independent of ~p . It is possible to show that K approaches a form in which c it is inversely proportional t~ ~p so that K ~p = constant.
c av c
(38)
Integrating on the basis of Eqs. (5) and (20) while noting that K = lip a(l-£) produces s
(l+IT )l-n-S_ l Jx X = 1 _ s (39) J L (l+~ITc)(l-n-S)-l
19
20
where IT = P /p. Substitution of (1-£)/(1-£ ) for (l+ITs) using (5) le~dsato 0
1 -
The average porosity can be obtained directly by noting that
1 - £ av
= PsL = JJ~p = wc cKd o Ps
If JL/J = 1, Eq. (41) becomes wc
1-£ (l+~IT )l-n_ l av 1-n-8 ___ c--:-_-.".--_
1-£0 = ~ (l+~IT )1-n-8_ 1 c
J n K wc av av (41)
(42)
In Fig. 7, (1-£ )/(1-£) is plotted as a function of ~ at constant (n+8).avIncreaging ~p is always beneficial forcmoder­ ately compressible cakes. How~ver, for highly compressible beds, the porosity levels off with increasing ~p as illustrated for sums equal to 1.5 and 2.0. Increasing fi1tr~tion pressure does not cause £ to decrease indefinitely as ~p increases as might be expe~¥ed for highly compressible cake~. This adverse behavior mirrors the corresponding difficulty with response of q to ~pc when n + 8 is greater than unity.
1.6~=------1-----t---~~
Figure 7
Differential Equations for Cake Filtration
Assuming that ql is a function of time, (37) can be placed in the form (Tiller and Shirato, 1964)
_ dv _ P ql - dt - ~(J a w +R )
wc av c m (43)
Integration requires that w be replaced by cv. When R = 0 and p = constant, (43) can Be rigorously integrated int~ the form
2 ~J a cv /2 = pt wc av (44)
The parameters a ,J , and c are all constant when R = 0 and 11 av wc m Pc = p.
In the usual development of the more general filtration parabola when R is not zero, it is customary to assume J , a ,and c are ~onstant leading to wc
av
2 ~J a cv /2 + ~R v = pt wc av m
(45)
This equation is not valid although it can accurately approxi­ mate experimental data if a false medium resistance R f re­ places R. Tiller, Crump, and Ville (1980) showed th~t (45) can be improved for long filtrations (20-30 minutes) if R is replaced by its false value as given by m
R = (l-n) mf (l+ll)-(l+ll)n
(46)
As n becomes larger for highly compressible cake, Rmf approaches zero.
Correct integration of (43) when R is not zero requires that variation in the pseudo concentratTon c be accounted for. The average mass fraction s of solids in the cake is related to the average porosity by c
1 1 Eav 1 [1 ] -- = 1 + ------ = 1 + - -1 where 0 s 0 l-E 0 a K c av av av
where it has been assumed that JL/J = 1 in (41). ing (47) in (10) and the result lnt~c(43} yields
Ps P , (47)
Substitut-
21
22
dv [ 1 s 1 ] IIp c llpsV - = 1 - s (1 - -) - - --- --dt (J (J a K a av av av (48)
This equation provides an instaneous view of the cake. It basically yields a ql vs v relationship which is independent of time. To solve (48), a value of ql = dv/dt is chosen and ~p = p - llqR is calculated. With ~p known, a and K can beCobtained; Wnd then v can be calculated. Witha~ knownaXs a function of dv/dt, time is obtained as [dv/q.
Eq. (49) has been integrated numerically for a higher compressible material as illustrated in Fig. 8 where the follow­ ing dimensionless parameters are employed:
N = ---L- N R llqRm ' w
a w o c -R-
m pt/ll, G = R /ca . m 0
(49)
The NR vs. Nand Nt/GN vs. N curves correspond to the usual dt/dv vs. v ~nd t/v vs.wv plot~.
0.2 03
Figure 8
Traditional theory predicts that the plots will be linear as indicated by the asymptote lines shown on the graph. The largest deviation occurs at the beginning of the operation when w or N is zero and all the pressure drop is across the medium. The fal~e medium resistance is approximately 25% of the true value.
In actual practice, the portion of the curves with large curvature occurs during a very short period of time which may last from seconds to a few minutes. If data are missed during the first portion of the run, the investigator will draw a straight line through the remaining data and will obtain a false medium resistance or intercept.
If experimentation has a short duration, accurate values of R and low values of a result. For long runs, accurate valu~s of a can be obtafXed, but R will then be grossly in av m error.
Literature Cited
1. Risbud, H. M., "Mechanical expression, stresses at cake boundaries, and new compressibility-permeability cell," Ph.D Dissertation, University of Houston, Univ. Micro­ films, Ann Arbor, MI (1974).
2. Shirato, M., H. Kato, K. Kobyashi and H. Sakazaki, "Analysis of settling of thick slurries due to consoli­ dation," J. Chem. Eng., Japan, ~, 98-104 (1970).
3. Tiller, F. M. and H. Cooper, "The role of porosity in filtration, part 5, porosity variation in filter cakes," AIChE J., ~, 445-449 (1962).
4. Tiller, F. M., J. R. Crump and F. Ville, "Revised approach to the theory of cake filtration," Fine Particles Proces­ sing, Vol. 2, 1949-1980, Amer. Inst. Min. Met. Pet. Eng. (1980) .
5. Tiller, F. M. and Green, "The role of porosity in filtra­ tion, 9, skin effect with highly compressible materials," AIChE J., ~, 1266-1269 (1973).
6. Tiller, F. M. and M. Shirato, "The role of porosity in filtration, part 6, new definition of filtration resist­ ance, AIChE J., lQ, 61-67 (1964).
23
24
7. Tiller, F. M., "Revision of kynch sedimentation theory," accepted by AIChE J. (1981).
8. Wakeman, R. J., "Numerical integration of the differential equations describing the formation of and flow in compres­ sible filter cakes," Trans. I. Chern. E., 56, 258-265 (1978) .
9. Webber, Wm., O. Davies, R. Chow and F. M. Tiller, "Clog­ ging phenomena in filtration of liquefied coal," accepted by Chern. Engr. Progr. (1981).
FLOW THROUGH POROUS MEDIA AND FLUID-PARTICLE HYDRODYNAMICS
Lloyd A. Spielman
Departments of Civil and Chemical Engineering University of Delaware, Newark, Delaware
CONTENTS
Introduction
Packed Bed Hydrodynamics Relationship of Stokes Flow to Darcy's Law Kozeny-Carman Theory Flow through Assemblages of Spheres
Particle-Collector Interactions
cell radius
F3(H) interactions (dimensionless)
dimensionless separation
unit vector
fluid velocity
axisymmetric stagnation velocity field normal to collector
superficial velocity (unless otherwise stated)
-1 kg m
-1 kg m
thickness of section of porous bed
bed solidity (dimensionless
bed porosity (dimensionless)
angular position of particle (radians)
permeabili ty
3 -1 m s
Two important factors affecting the economics of water filtration are effectiveness of particle removal and hydrodyna­ mic resistance to flow. These factors are in turn closely associated with the mechanics of pore flow. Particle removal is in large part governed by the interaction of various physical and chemical forces, such as Brownian motion, gravity settling and colloid chemical attraction and repulsion, with hydrodynamic forces in the heighborhood of the collecting surfaces. The hydrodynamic resistance of deep filter beds results from the integrated action of hydrodynamic stresses on the stationary bed matrix, In fact, much about both aspects of deep bed filtration can be understood using theoretical models which focus on the detailed flow of water-borne particles around a single character­ istic bed grain.
It is important to recognise that because most of the filtered particles are in the range of submicrons to tens of
27
28
.. 4 microns (1 micron • 10 em). while the bed grains are usually no SlIIaller than a few tenths of a milUmeter, that the particles are much saaller than the pores or grains and so can penetrate to considerable bed depths before encountering surface and being captured. 'Io deal iD aD effective way with both bed resistance and the lIIOVemat of particles in porous media. requires con­ sideration of the fundamental equations governing particle-fluid hydrodynamiCS, especially those describing small scale flows.
stOKES EQUA:tIQlIJS
'Ihe equations generally goveruing isothermal incompressible fluid flow are the full Navier-Stokes equations,
II01HIlt UIIl : ~ a~ + + 2+ PDt • p(at + u • Vu) • -Vp + lIV u (1)
contiDuity: v • ~ • 0 (2)
+ Here u i. the fluid velocity field. p the dynamic pressure field. p the constant fluid density and lI. the constant fluid viscosity.
Beeau .. Eq. (1) is DGnlinear, its solution can be obtained only under suitable approximations or ideal circumstances. If one considers the flow within a small scale geometry. the magni­ tudes of the inertial acceleration terns can be small compared with the viscous teras in Eq. (1). Because the ratio of inertial to viscous terms is estimated to order of magnitude by the Reynolds number. one obtains
el)
Here 1 is a characteristic length and V a characteristic velocity. For pore flow 1 is on the order of the grain or pore dimension and V the pore velocity. When the inequality (3) is satisfied. tbe left hand terms of Eq. el) may be neglected. in which case Eqs. (1) and (2) simplify to Stokes equations of creeping flow:
Vp • Ilvl: (4)
(2)
It should be stressed that for porous media flow the smallness of
Re in (3) results mainly from the smallness of the characteristic length t, in contrast with those of larger scale flows which occur in geophysical, atmospheric or most hydraulic situations. Furthermore, whereas the inequality (3) is on occasion only marginally satisfied or even violated for pore flow, it is almost always satisfied for the disturbance flows of the water­ borne particles, whose micron dimensions assure correspondingly smaller Reynolds numbers, with fluid inertia playing an even more minor role.
Linearity and Superposition
The applicability of the Stokes Eqs. (4) and (2), rather than the full Navier-Stokes Eqs. (1) and (2). implies major simplifications in both the mathematics of equation solving as well as the developing of scaling criteria by dimensional ana­ lysis. Because Eqs. (4) and (2) are linear, they are often re­ sponsive to the variety of existing methods suited to linear partial differential equations, such as separation of variables. One of the most powerful tools used to solve problems described by Eqs. (4) and (2) is the method of linear superposition of ~olutions. +Clearly, if each of the velocity and pressure fields ul. PI ~nd u2' P2 satisfy ~qs.+(4) !nd (2), respectively, then their l~near combinations u • ul + u2. P • PI + P2 also give a solution. Using this property one can construct complicated flows describing the hydrodynamics of particles near one another or near collectors, from component solutions governing relatively simpler flo~s. An example of this is construction of the flow fiell! describing simult·aneous translation and rotation of a particle, ;"y linear superposition of the flow fields governing its translation alone and its rotation alone. Another example is describing the translation of a particle toward or away from a solid planar surface at some oblique angle, by superposition of the flo~s governing its movements respectively perpendicular and tangential to the surface; if, in addition, the particle is rotating, then its isolated rotational flow field can be super­ imposed too, and so forth. Of course, great care must be taken to ~ke certain that all the boundary conditions add up properly so the resultant flow field is precisely the one desired.
Forces and Flow Reversal Symmetry
To compute the hydrodynamic forces acting on an object immersed in a given flow field, one usually integrates the local hydrodynamic stresses acting over its surface. The local hydro­ dynamic stresses are straightforwardly related to the velocity and pressure fields given by the solution of the governing equations under appropriate boundary conditions. In Cartesian
29
30
coordinates the local stresses for incompressible viscous flow are given by
au TXX • P - 21.1 --!
Cly (11)
au au T
• T • -lJ(--! + ~) (iv) xy yx 3y ax
3u au T - T • -lJ(~+ _z) (v) yz zy Clz ay
3u au T • T • -lJ (.-...!. + -!) (vi) xz zx ax Clz
in which P is the thermodynamic pressure. The stresses given by !qs. (i) through (vi) are linearly related to the viscosity and velocity gradients. This follows for the normal stresses given by Eqs. (i) through (iii) because the pressure also is seen to be proportioaa! to II and velocity gradients via Eq. (4). If a given flow field satisfies Eqs. (4) and (2). then so must that cor­ responding to reversal of the velocities and pressure gradient. Eqa. (i) through (vi) then imply that all drag forces exerted by the flow field are Simply reversed also.
Quasistatic Property
ADother convenient property of Eqs. (4) and (2). which is not in general possessed by Eqs. (l) and (2). is that the former are quasistatic. That is, their time varying flows may be viewed as a smooth sequence of instantaneous steady state flows. To see how this property can be used. let us consider the well-known Stokes resistance formula for the drag under steady movement of a sphere with radius a. moving with constant velocity U through an unbounded stationary fluid having viscosity Il. wbich says the particle experiences a steady dra~ force of magnitude
!' • 611'L1aU (5)
If the particle is permitted to accelerate such that its velocity
1s aD arbitrary specified function of time, U(t),then Eqs. (4) and (2) imply immediate extension of ~q. (5) to describe the time varying drag force as
F(t) .. 6lT:JaU(t) (6)
This extension results because the condition (3) assures us that fluid inertia is effectively absent so the fluid responds and adjusts to time variations instantaneously. Of course, if the particle is accelerated too suddenly, or brought to such large velocities that the inequality (3) is violated, then Eqs. (4) and (2) won't apply and one must resort to the nonlinear Eqs. (1) and (2) for a realistic description.
Dimensional Analysis and Scaling
We notice that because the inertia terms are absent, the fluid density does not appear explicitly in Eqs. (4) and (2). This greatly simplifies the development of scaling criteria through dimensional analysis because the fluid density does not usually have to be included in our list of parameters. For instance, let us assume we did not know the Stokes formula, Eq. (5), but wished to obtain as ~uch information as possible about the relationship of drag force to the other parameters, without undertaking solving Eqs. (4) and (2), as Stokes did. Careful consideration of the relevant parameters appearing in Eqs. (4) and (2) and their boundary conditions, tells us our list should include F, U, a, and u. but not p. From these parameters, only one dimensionless group can be formed, thus we obtain
F ~aU = constant (7)
Dimensional analysis gives nearly the entire formula (5) and the detailed !Qlution of Eqs. (4) and (2) merely gives the value of the dimensionless constant = 611. Had we been less perceptive in inspecting Eqs. (4) and (2), and conservatively included the fluid density p in our list, we would have obtained instead
~aup r = \laU (-) 1.1
(8)
in which :r is an unknown functiQn of the Reynolds number and which conveys far greater ambiguity concerning the desired rela­ tionship aeong the parameters than Eq. (7). It is a character­ istic of Stokes flows, that drag forces and stresses are directly proportional to viscosity and velocity, with the coefficient of proportionality depending on geometry. For a fuller discussion of
31
32
PACKED BED h~DRODYN~~ICS
Relationship of Stokes Flow to Darcy's Law
By using the simple ideas concerning dimensional analysis outlined above with some further plausible arguments, we can ~erive Darcy's law for flow through porous media. In its simplest form Darcy's law for one dimensional flow may be stated as
IC aP U .. - ~ ax (9)
In Eq. (9), U is the superficial velocity = volume flow rate/cross sectional area of bed, ~ the viscosity, llP is the dynamic pressure difference across thickness 6x of porous medium, and ~ is the hydraulic permeability, which is experimentally found to be a property of the porous solid. In wnat follows, the porolls solid is taken to be macroscopically uniform in the x-direction, but not having any special microscopic geometry. That is, the solid is not in particular assumed to be a bundle of tubes or a regular array of spheres, but can be of any degree of microscopic corn­ plexity so long as its overall bulk character is uniform.
Straightforward reasoning shows that the overall pressure difference ~P should be directly proportional to the bed depth Ax. This is because llP times the bed cross sectional area measures the net force on the opposite faces of the bed and must be equal to the total drag force exerted over all the micro­ scopic surface inside the bed, since it is equal and opposite to the total force necessary to hold the bed fixed. It therefore follows that doubling the bed depth, 6x, will double the pressure drop, because it doubles the aJIlOunt of porous solid over which the internal drag force is exerted, hence 6P must be directly proportional to 6x.
AP and Ax should therefore enter the final expression only as their ratio, (AP/Ax). If we now require that the small pore Reynolds number condition (3) be satisfied so the microscopic flow is in the Stokes regime and Eqs. (4) and (2) govern, then in accord with the previous discussion the fluid density p should not appear explicitly in the end result. The quantities which do appear should therefore include only (llP/6x), U and U, as well as a potentially long list of independent geometric parameters. These geometric parameters would formally appear in a very COQ­
plica ted expression describing the internal solid surface at which the no-slip boundary condition for Eqs. (4) and (2) would be applied (we don't really have to be able to write down all
these geometric parameters or the equation of the surface, but only recognize that such an expression applies). In the list of geometric parameters, there must be at least one characteristic length, say 1, since even the simplest imaginable pore shapes require one parameter to describe them (e.g. circular). The list of quantities appearing in the final expression would then look like:
AP ai- U, Il, t, .It', 1", ••• , other lengths, angles, etc.
Dimensional analysis then gives
IlU l' 1" ~~=----- f(.-. ~, ••• , other geometric ratios) (10) .lt2 (I1P/l1x) ..
in which the left hand side of Eq. (10) is a dimensionless group and the right hand side a dimensionless function of geometric ratios.
Hut Eq. (10) can be rewritten as Eq. (9) if we interpret the permeability in the latter as
(11)
Eq. (11) shows the permeability has dimensions of length squared and depends only on the ~eometry of the porous solid. It also follows from the flow reversal principle that reversing the direction of flow through a porous solid cannot alter its per­ meability.
The foregoing derivation of Darcy's law clearly shows the key assumption which underlies it. Namely, that because the pores are small, the pore Reynolds number is usually small so fluid inertia effectively plays no role in the dynamics of flow. If not, Darcy's law doesn't apply. A common misconception about porous media says that as flow rate increases Darcy's law first breaks down upon the onset of turbulence in the pores. In fact, pore Reynolds numbers rarely become large enough for turbulent flow in the pores. Breakdown of Darcy's law really uarks the onset of inertial forces in laminar flow which occurs at pore Reynolds numbers on the order of 1-10. 2,1 For very complicated soli~ geometries, especially consolidated porous solids, the best route to determining the permeability is by direct measure­ ment, using Eq. (9). Usually U is plotted against 6P to get a straight line and K calculated from the measured slope.
33
34
Over the years there have been many attempts to relate per­ meabilities to the geometry of the porous solid by using special models. One of the most widely used of these theories is the Kozeny-Canaan development which is outlined below.
The average velocity for laminar flow through a straight circular tube is given by the well-known formula:
a .9. a _ R2AP Uavs A 8~L (12)
where Q is the volume flow rate. A the tube cross sectional area, R. tbe tube radius and L the tube length. For noncircular tubes, Eq. (l2) baa been generalized to
~AP Vavi a_ kolAL (13)
In Eq. (13). ~ is the hydraulic radius defined as (flow cross sectional areaTwetted perimeter). For a circular tube k • 2 and Rh a R/2. For straight tubes with noncircular crossosections. such as rectangles. ellipses, etc •• whose aspect ratios are not very different from unity, the coefficient ko is Eq. (13) varies from ~bout 2.0 to 2.5 ana so may be considered as roughly inde­ pendent of shape. Assuming the porous solid to be a bundle of irregularly shaped, straight channels, the above definition of bydraulic radius gives
~ - ~/s (14)
where ~ 1s the voids fraction and s is the internal surface area per unit volume of bed. It is then argued that the effective pore length is really somewhat greater tl-,an the bed depth because the fluid travels a tortuous path, thus the apparent length L in Eq. (13) should be replaced by Le where the tortuosity factor is Le/L > 1. The interst1tial velocity U' is related to the super­ f~cial velocity U by
v' • U~.~ (15)
However, it is then argued that because the tortuous fluid path is longer than L by tbe factor LeIL, the velocity along the tortuous path must be correspondingly greater than that for travel straight through which is given by Eq. (15), hence the proper velOCity to use in Eq. (13) is
L U La U" • V' (....!) • -(-) L £ L (16)
Setting U avg U" and 1
R.2 t-P U" -11 .. -~
Now substituting Eqs. (14) and (16) into Eq. (17) gives,
&3 l1P
;--z~ all!>
L ~",'2 L ", .. ,
gives
u
(17)
(18)
(19)
(20)
The numerical factor in Eq. (20) is in fair accord with experi­ ment for unconsolidated granular beds in which the pores do not vary too greatly in size. One of the most important uses of Eq. (20) is in determining the internal surface areas of porous materials from permeability data. It also indicates how changes in packing density should affect permeability.
The Kozeny-Carman theory has received much criticism, large­ ly undeserved since it correlates bed resistance data for a wider class of porous media than any other permeability theory. On the other hand~ to develop theories of particle removal by granular beds requires a more detailed picture of the flow field near the collecting surfaces within the bed than the Kozeny model affords. Progress to overcome this has been made adopting Happel's cell model to say more about the microscopic flow field. Instead of viewing a packed bed as a bundle of tortuous channels as the Kozeny theory does, the cell models view the bed ~rains as an assemblage of interacting, but essentially individual spheres, with the flow field about an average sphere being described in detail. ~~reover, by summing up the drag forces acting on the individual bed grains, the cell model also permits self-consistent prediction of bed permeabilities, which agree with data for un­ clogged media at least as well as the Kozeny equation. It also yields predictions of bed expansion in backflow as well as hindered settling of suspensions. To analyze the cell model, however, requires a closer look at solutions of Eqs. (4) and (2) which describe the fluid mechanics of particles in general.
35
36
Plow TIu:ou&h As ..... l.... of Spher.s
Lamb' s general solution. Lamb 4 gave a general solution to Eqs. (4) ad (2), suited to treating boundary value problems in which velocities are prescribed on spherical surfaces. Lamb's solutioG takas the form -
p - t p n-- n -+ [+ (n+3) 2
u - n=-- Vx(rXa) + Vtn + 2~(n+I)(2n+3) r VPn
n + ~(n+l)(2n+3) rpn]
in which P ,x and t are each solid spherical harmonics which are dete~ednfrom tee specified boundary conditions. Happel and BreDDer illustrate the adaptation of Lamb's general solution to treat a vadety of boUDdary value problems. Although Lamb's solution provides a general approach to such problems, many axisymmetric flow problema involving spheres can be solved usin~ the simpler, though mare restricted method which follows.
Stokes solution for a single sphere. A number of important boundary value problems in axisymmetric Stokes flow may be treated using a simple general solution of Eqs. (4) and (2), first ob­ tained by Stokes. Among these are uniform flow past an isolated ~111d sphere and the cell models mentioned previously, as well as circulating droplets and particles mavi_. by electrophoresis. I •5
£qa. (4) and (2) may be simplified by introducing a stream function '/I such that
1 a." 2 -
u -----e r sine 3r (21)
In £qs. (21), rand e are spherical coordinates. Eq. (2) is then automatically satisfied and eliminating the pressure between the r- and a-components of Eq. (4) gives
[L + ~ 1- 1 II ]2 _ 0 " 2 2 ae (sine as) '/I "r r
For solutions of the form,
." - fer) sin2e
A 2 4 f(r) = - + Br + Cr + Dr r
(24)
~n which A, B, C. D are integration constants to be determined from boundary conditions.
For an isolated sphere with no-slip at its surface and uniform flow at infinity, the boundary conditions are
Ur • 0, ue • 0 at r - a
or, equally, from Eq. (21),
l!. _ l!. .. 0 at a6 Clr
and ur + -U.COS 6, as r + ...
or, using Eq. (21) we have equivalently,
1 2 2 '" + '2 U .. r sin 6 as r + CD
The constants in Eq. (24) are then determined as
A • ! U .. 3 4 ..
Tbis gives the velocity field,
ur ( 3 a 1 a 3 if" '"' - 1 - 2'(-;) + 2'(-;) leose .. Us 3 a 1 a 3 if" - [1 - 4(-;) - 4(-;) )sin6 ..
and the pressure distribution,
(25)
(26)
(27)
(28)
The total normal stress is given in spherical coordinates by
au 1 .. P - 211 --E rr Clr
(29)
37
38
(30)
Integration of Eqs. (29) and (30) over the entire sphere surface gives the drag force. Eq. (5). Equating the drag force to the weight minus buoyancy gives the well-known Stokes law for the terminal settling velocity of an isolated particle.
2 u .!~ s 9 \l
(31)
whete A~ is the density difference between particle and fluid and g is the acceleration due to gravity.
1 6 !fappel's cell model. Happel' treated the problem of flow through an assemblage of spheres by assuming a typical sphere to be enclosed within a spherical envelope of radius b, whose volume corresponds to the voids ratio in the overall assemblage. i.e.,
4 1/3 1/3 b • Y • ~ - (1 - ~) (32)
where ~ is the solidity (volume fraction spheres) and E the poro­ sity. He then used the general form given by Eqs. (23) and (24), ret3iniD& the DO-slip surface conditions (25). but instead of the isolated flow condit.:.on (26). used boundary conditions at the envelope, r • b. to fix all the constants in Eq. (24). thus he takes
Ur • -U cose, Tre • 0 at r· b (33)
with T 9 ~iv~n by Eq. (30), The first of conditions (33) sets the rd1aJ. component of velocity equal to that corresponding to the superficial velocity U (or, equivalently, to the velocity U of the assemblage as it moves through the fluid). the second of conditions (33) assumes the envelope at r • b to b~ a free surface, which CaD be justified in some sense by arguing that a free surface of a different shape, but equivalent volume, must exist for regular arrays of equal spheres. This determines all the constats and gives the flow field near the sphere. Tbe force on the sphere can then be evaluated as previously for the isolated sphere,
the apparent arbitrariness of the free surface assumption is made cl~r by considering an alternate condition used by Kuwabara in his cell model. Instead of the vanishing shear
condition (33) Kuwabara assumed vanishing vorticity, i.e.,
(34)
Whether the vanishing shear or the vanishing vorticity assumption is more correct cannot convincingly be answered on theoretical grounds, but is better judged by comparison with experiment.
Happel's model gives for the drag on each sphere,
F = 5 41fuaU(3+2y )
5 6 (2-3y+3y -2y ) (35)
where y d~pends on voids fraction through Eq. (32). In the limit that the voids fraction & tends to unity, Y ~ 0, and Eq. (35) appropYleL~y reduces to Eq. (5) for the isolated sphere.
One ~an now use Eq. (35) to predict pressure drop through a packed bed of equal spheres. Equating the force difference due to pressure on the opposite faces of a thickness of bed 6x, to the sum of the drag forces on all the spheres in the thickness, gives
AP - - ~ F (36) 4 3 3"a
Substituting from Eqs. (35) and (32) and rearranging, gives
256 U __ (~~(3-9Y/Z+9r /2-3y )] ~
9 y3 3+2y5 u6x (37)
The bracketed term in Eq. (37) is a function of a and y = (1 - &)~ only and corresponds to the Darcy permeability, K, defined by Eq. {9}. Eq. (37) is found to be in good agreement with experi­ ment, closely agreeing with the Kozeny equation. (20) in the porosity range. 0.4 < & < 0.7. At higher porosities Happel's model is superior to Eq. (20) because the former reduces to an assemblage of isolated spheres whereas Eq. (ZO) does not. On the other hand Kuwabara's model leads to a stronger dependence on 1 porosity, giving somewhat higher pressure drops than observed. This supports Happel's model as giving the more realistic flow field near a typical grain.
Rappel's model may also be applied to assemblage settling, giving
39
40
(38)
in which U is th_ .ldered settling velocity and Uo that given by Eq. (31). H~ __ too agreement with experiment appears to be good.
1 Happel and Brenner discuss the use of the cell model to describe fluidized bed behavior during the expanded bed phase.
PARTICLE-COLLECTOR INTERACTIONS
Here we consider theoretical aspects of particle motion near a muc.h larger collec.ting grain of radius a_. This discu~;sion is not intended to deal in a complete manner ~~th collection mechanisms, but to outline how fluid mechanical effects enter the particle capture process. Early treatments of particle capture assume the particles move with the undisturbed fluid velocity except for the action of external forces such as van der t~aals attraction or gravity. However, recent treatoents8,9 consider the exact Stokes disturbance flow field created by the particle in proximity to the collector. The particle is taken to be pro­ pelled by the undisturbed flow near the collector rather than artificially superimposed upon it. The entrained particle freely translates and rotates as it &hould according to its equations of motion under the hydrodynamic and external forces which act upon it. The particle thus creates a locally confined hydrodynamic disturbance which is governed by Stokes Eqs. (4) and (2). The boundary conditions are taken to be the undisturbed flow field far frOID the particle with no slip at both spherical particle and collector surfaces and no net force or torque acting on the particle (all inertia is neglected). Because the curvature of the collec~or is so mpch smaller than that of the particle, the former 1s approximated as a plane surface in the neighborhood of the particle. Also, external field forces such as van der Waals attraction, double layer repulsion, and gravity, can be included in the overall force balance. In this way, both external and hydrodynamic interaetions are simultaneously taken into account in a rigorous manner. Neglect of inertia is justified by the smallness of the particle and its Reynolds number.
The flow field very near the spherical collector can be obtained by expanding Eq. (23) in Taylor series about the surface. This gives, to lowest order,
(39)
which 1s restricted to small (r - a )/a , where a is the grain radius. In Eq. (39), A is a dimen:ion!ess param~ter character­ izing the flow model. for an isolated sphere in Stokes flow with a uniform velocity U at infinity, A = 1. For a spherical grain within a packed bed, A is a known function of bed porosity. l~ppel's model for flo~ around a characteristic grain gives
A -s 2(1_y5)
5 6 2-Jy+Jy -2y (40)
Let us now define a system of local cylindrical coordinates wand z whose origin is on the collector surface at r = a • e - e~; ep is the an~le corresponding to the center posit!on of an entral-ned particle. The origin of the coordinate system thus GI~nges position as the entrained particle moves around the col­ lector (Fig. 1). By straightforward transformations, the undis­ turbed flow field can be expressed as
(41)
where
and
.... .. Here i y • iw. and i z ' are unit vectors in the y. w, and z direc­ tions respectively. The above expression for the undisturbed field is correct through terms of lowest order in wand z. In Eq. (41) the undisturbed flow field near the collector has been decomposed into two additive fields, each of which satisfies
Fig. 1. Local coordinates as particle moves around collector.
41
42
Eqa. (4) aDd (2) separately. The field U t is axisymmetric about the Z'-axis and has a stagnation point at ~ .. 0, Z .. 0; ush is a uni.form shear field directed parallel to the collector surface. Letting h be the minimum separation between the entrained particle and the locally flat collector surface, the particle center is located at ~- 0, z - z - a + h, e = Op' For Eq. (41) to give the boundary condilion gn the disturbance field. the parti~le must be so saaall compared with tile collector, that within separa­ tions where the particle deviates appreciably from an undisturbed streamline. the collector can be approximated as a planar wall (except inasmuch as its geometry determines the undisturbed flow). Thus a «a and Eq. (41) applies only near the moving origin and oulside the region of the disturbance.
the aovement of the entrained particle and its corresponding disturbance flow field are now decomposed into the fields cor­ respondiaa to its normal and tangential motions separately. This is permitted because the governing Eqs. (4) and (~) are linear and all the velocity boundary conditions <at the particle sur­ face, obstacle surface and far from the particle) are arranged to be additive. Tbe method of superposition of solutions discussed previously is used to construct the solution for the particle freely moving near the collector. Also, because the creeping flow equations (4) and (2) are quasistatic, they apply at any instant a8 the entrained particle proceeds along its trajectory.
the disturbance flow corresponding to the z-directed particle motion may further be decocposed into two additive flows. These are summarized in Table I. In one such flow, the particle moves in the z--direction under the influence of an instantaneously applied normal force Fn , which, for the present may be viewed as unspecified, with the velocity field taken to vanish far from the particle and no-slip at both the particle and the effectively planar collector surfaces. The particle motion in this Stokes flow i8 liven by
(42)
the dimensionless function F (hi a ) - F (H) is known for all H f.rom the exact solution of S!okesP equations given by Brenner .10 The function F (H) i8 shown graphically in Fig~re 2. In a second flow contributtng to the z-directed motion of the~article, the particle is taken to be held~fixed in a field which becomes the axisymmetric velocity field ust ' given by Eq. (41), far from the particle. again with no-slip at both the particle and'plaoar col­ lector surfaces. Because of the axisymmetry of this flow, the particle experiences a purely z-directed force,
T hl
.ll .E
m ni
ar y
o f
S u p ~ r i m p o s e d
F lo
w F
ie ld
s G
iv in
g R
e su
2 2 a
s
(43)
The dimensionless function FZ(II) is known for all 11 '" h/a from the exact solution of Stokes equations p,iven by Gorenll agd Goren and O'Neill12 and also is shown graphically in Fi~ure 2. Eq. (43) r,ivlng J:'st is subsequently made use of in the force balance '.Jith Fn which occurs in reconstituting the original flow field. Let Fext(H) be an external field force, which for simpli­ city, "\lill be taken to have a z-component only. That is, .it is directed perpendicular to the collector and depends only on the distance of the particle from the surface, for instance, surface forces of colloidal origin. The resultant z-directed motion of the particle and its accompanying fluid motion can now b~ obtained by linear superposition of the separate flows described above. i~eglect:ing inertia, the motion of the entrained particle is outained by combining the flows such that the net force on the particle is zero. This requires that the applied force of Eq. (42) equals the sum of the hydrodynamic force of Eq. (43) and the external force, i.e.,
F • F + F n st ext (44)
Substituting Eqs. (42) and (43) into Eq. (44) and rearrangin~, gives the motion of particle perpendicular to the collector sur­ face at any instant:
2 6!1~a dii -----2. _ .. Fl(H) dt
-6rr~a3A U cosOp 3 2 --,P:...;::;---.;;.. 2(11 + 1) F 2 (1I) + F ext (H)
a s
(45)
The motion of the particle tangential to the collector is obtained by considering the part of the flow field which describes motion of the particle parallel to the effectively planar col­ lector surface. In this flow, the particle undergoes free rota­ tion ~nd translation (experiences zero net torque and force)+as it cre3tes a confined disturbance in the uniform shear flow ush given by Eq. (41), which is recovered away from the disturbance. The solution to this Stokes boundary value problem was ohtained by Goldman, Cox and Brenner,l3 and the resulting expression for the induced a-directed particle velocity is
dOp _
(46)
45
46
This induced velocity is purely O-directed (Fig. 1). The dimensionless function F3 (1l) is shown in Figure 2. The deviations of FI , F~, and FJ from unity in Fig. 2, reflect the strengths of the partl.cle-collector hydrodynamic interactions.
Eqs. (45) and (46) are the differential equations which des­ cribe the normal and tangential translation of the entrained particle in the vicinity of the collector surface. The equation describing the particle trajectories is obtained by eliminating the time t between Eqs. (45) and (46). The resulting equation is
(47)
The numerical solution of Eq. (47) has been reported to predict collection by London-van der Waals attraction and gravi­ tational external forcesS,9 and compared with experiment. lli Its solution for capture by combined London attraction and electrical double layer repulsion has been recently reported. ls
REFERENCES
1. Happel, J ~ and Brenner, H., .!:!£.~_~eY_l!.0ld_s_;I~ber~.!odynamics, Prentice-llall, Englewood Cliffs, N.J., 1%5.
2. Bird, R.B., Stewart, W.E., Lightfoot, E.N., ~~~port Phenomena, John Wiley and Sons Inc., New York, 1960, chap.6.
3. Scheidegger, A.E., The Physics of Flo~~~~o~~Porous Media, University of Toronto Press, 1960.
4. Lamb, H., Hydrodynamics. Cambridge University Press, 1932; reprint Dover Publications, New York, 1945, 594.
5. Levich, V.G., !!!l'.sicochemica1 Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J., 1962.
6. Happel, J., AIChE~. 4, 197, 1958.
7. Kuwabara, S., J. Phys. Soc. Jap., 14, 527. 1959.
S. Spielman, L.A. and Goren, S.L., Cn~iro_l!.~Sci._ ]:'~chnol., 4, 134, 1970; 5, 254, 1971.
9. Spielman, L.A. and fitzPatrick, J .A., ~C_c:!l.E..ii. Interface, 42. 607. 1973.
10. Brenner, H., Chem. Eng. Sci., 16, 242, 1961-
11. Goren, S.L., J. F1~i~clech., 41, 619, 1970.
12. Goren, S.L. and O'NeiU, H.E., Chern. Eng. Sci., 26,325, 1971.
13. Goldman, A.J., Cox, R.G., and Brenner, H., Chern. Eng. Sci., 22, 637, 653, 1967.
14. FitzPatrick, J.A. and Spielman, L.A., J. Colloid Interface ~., Hay 1973.
15. Spielman, L.A. and Cukor, P.!1., J. Colloid Interface £cL, 43, 51, 1973.
47
48
Richard J. Wakeman
Department of Chemical Engineering University of Exeter, Exeter, Devon, U.K.
CONTENTS
INTRODUCTION
COMPRESSION-PERMEABILITY CELL DATA
CONCLUDING REMARKS
A Filtration area (m2)
E Compressibility coefficient (m2 s-l)
ftc) Function of c, to account for significance of buoyancy
Fd Interfacial drag force per unit volume of solids (N m- 3)
g Acceleration due to gravity (m s-2)
i Integer
j Number of time increments into which total filtration time is divided
k Permeability (m2 )
L' Defined in equation (25)
n Constant in equation (81), or, number of layers into which cake is divided
p Pressure based on total area (N m-2)
p Dynamic pressure (N m-2)
Constant in equations (81) to (83)
Medium resistance (m- I )
t Time (s)
v Absolute flux relative to a stationary containe~, e.g. pore velocity of a liquid (m 3m-2 s-I)
V Volume of filtrate (m 3)
w
Mass of dry solids (kg)
Coordinate distance (m)
Specific cake resistance (m kg-I)
Value of C! at which solids stresses are just starting to exist (m kg-I)
Constant in equation (82)
49
£ o
e Porosity at which solids stresses are just starting to exist
Dimensionless time, see equation (34)
A
)l
p
= b/ Va , a similarity variable, or, constant in equation (83)
Viscosity (Pa s)
1 of liquid
50
A bar sign (-) over a variable indicates an average value
INTRODUCTION
Recent analyses of cake filtration have been aimed at providing more detailed descriptions of the fluid motion through the cake due to the hydraulic pressure gradient. This gradient causes an inter­ facial momentum transfer in the form of viscous drag at the particle­ fluid interfaces. If the shape (or the physical strength) of the solids is such that the packing arrangement in the bed can sustain this drag force without further movement, then the cake is regarded as incompressible. However, some particle rearrangement generally occurs to yield a compressible cake. During compression the poro­ sity decreases with time at any given distance from the filter cloth, and simultaneously a porosity distribution is obtained throughout the depth of the cake. This distribution often ranges from a minimum at the cake/cloth interface to a maximum at the growing cake surface, but instances of a minimum porosity some distance from the filter cloth have been reported 1-3 when charac­ teristics of the solid/liquid system are such that the cake collapses after deposition of a critical amount of solids. The compressive action itself causes the interstitial flow rate of liquid to increase towards the medium.
The development of filtration theory has been based upon dif­ ferential equations involving local flow resistances and variable flow rates 4-8. Attempts have been made to correlate data from actual filtration tests with values from compression-permeability cells introduced into filtration studies by Carman 9 and Ruth 10. Tiller et al 4-7 have elucidated the internal flow mechanisms, developing equations for the variable fluid flow rate at discrete points within the cake, but in doing so the validity of the compression-permeability cell was accepted. As a result, more unusual E versus x/L distributions such as those found by Rietema 1 and Baird and Perry 2 were not revealed, and for short filtration times the instantaneous porosity-pressure equilibrium inherent in the theories may not be realised, However, the work of Tiller et al led to improved definitions of average filtration resistance which recognised movement of the solid particles as well as the liquid,
A fundamental approach to the modelling of constant pressure filtration was attempted by Smiles 11. Attention is drawn to two alternative non-steady-state analyses of cake filtration which both
51
recognise that Darcy's law describes fluid flow relative to the solid particles rather than to fixed space, and both take account of mass flow of the solid component. These factors were, however, previously acknowledged by Tiller 4 and Shirato 5-7, but the transient nature of the problem dictates that it must properly be described by a non-linear partial differential equation. A similar approach was later adopted by Atsumi and Akiyama 12 but, unlike Smiles 11, they identified a moving boundary condition at the growing cake surface. Atsumi and Akiyama 12 used the constant pressure filtration data of Okamura and Shirato 13 combined with relevant compression-permeability cell data 14 to test their theory. In doing so it was presumed that compression-permeability cell information could be used to interpret filtration results, a major assumption being that local porosity and specific filtration re­ sistances (in a dynamic state in a growing filter cake) are equi­ valent to the corresponding values in a compressed cake (in static equilibrium in a compression-permeability cell).
variable liquid flows in the cake are important in short time filtrations of concentrated slurries, when errors ranging from 5% to 25% may result from their neglect 7. In general, for filtration the slurry must contain less solids than the cake surface (where the fraction of solids is a minimum), but the transition from cake to slurry is difficult to identify by experiment or by reasoning through the packing .characteristics of randomly sized and shaped particles.
CAKE RESISTANCES FROM FILTER MACHINES
Filtration theory has evolved from the classical law govern­ ing fluid flow through porous media, Darcy's law. It has necessa­ rily been assumed that particles do not pass into or through the interstices of the porous membrane or filter medium, but that a machanism of surface deposition {cake law filtration) holds and that the filter medium characteristics are unchanged during the process.
Under these conditions, and on the assumption that the cake, once formed, has time independent permeability and porosity char­ acteristics which are also uniform throughout the entire depth of the cake, a general filtration expression can be derived between the filtrate volume V collected in process time t:
(1)
The average resistance of the filter cake a is related to the averaged permeability K and the mean fraction of cake occupied by the solids (I-E) by:
52
ps(l-E)k
It has long been recognised that the filter cake resistance is pressure dependent, and that it generally varies less with pres­ sure than does the cake permeability.
For the analysis of data from filter machines to yield cake resistances equation (1) (or an integrated form) is the most useful relationship presently available. The cake resistances
(2)
and porosities so obtained must be interpreted as average values. Extensive experiments have been carried out 15 using "constant pressure filtration" in a plate and frame press using calcium carbonate, Hyflo-Supercel, mixtures of the two, and calcium sul­ phate. All the results indicated broadly similar trends. Figures lea) and l(b) show the variation of cake resistances with slurry concentration and filtration pressure for a 75% CaC03/25% Hyflo­ Supercel mixture. At filtration pressures below about 70 kN m-2 the specific resistance increases with pressure at all slurry con­ centrations. At a critical pressure the cake resistance passes through a maximum, this pressure depending on the slurry concen­ tration; the peakedness of the curve becomes less pronounced as the concentration increases, and no maximum occurs at the highest concentration. A critical concentration apparently exists above which medium blocking is less likely. An alternative explanation is that at lower concentrations the liquid velocity is greater at any filtration pressure, producing a flow stabilised cake. As the solids concentration increases the liquid velocity decreases, causing consolidation with formation or the cake to "collapse" in the extreme. That is, retarded packing may be important below the critical concentration. Above the critical concentration the flow velocity is sufficiently low to produce inherently more stable cakes, and although retarded packing may still occur its import­ ance is secondary to the compression" mechanism and so the conven­ tionally accepted a vs. 6p relation results.
Experiments using the same particles were also carried out on a rotary vacuum filter. Integration of equation (1) and suitable rearrangement of the result permits calculation of the specific cake resistance if the medium resistance is assumed negligible and the dry solids yield is measured. If two experiments are performed on the same slurry at only slightly different vacua two sets of data become available. These can be used to calculate both cake and medium resistances at the mean filtration vacuum of the two experiments. Sedimentation in the slurry trough, and drop-off of the outer layers of the cake, reduce the solids recovered. Figure 2(a) shows the dry solids yield from a calcium carbonate slurry, one yield being based on the volume of filtrate collected and the
75% Calcium carbonate - 250/0 Hyfto-Supercel
Shxry concentratIon
+~~.-I . .~"
o.n8
FILTRATION PRESSURE. />p kNlm2 lo7------A~~5~O---------7,100~----~~~1~~~~~~2~OO~~~~~2~50~----j
Figure 1 (a)
Figure 1 (b)
Filtraiton pressure,AP kNlm2 j
SLURRY CONCEN1RATION. s
53
54
other being the actual measured yield. The measured yield is always lower due to sedimentation and partial cake drop-off effects. In Figure 2(b) the specific cake resistances calculated from the different data of Figure 2(a) is shown, and it is readily seen that sedimentation and/or drop-off significantly increases the cake resistance. Resistances based on the measured yield are of the same order as those found in a plate and frame press for the same slurry and driving force. As with the pressure filtration results, and for the same reasons, cake resistances can decrease with filtration pressure. These data point to the care which must be exercised when cake resistances are being obtained for scale-up purposes.
COMPRESSION-PERMEABILITY CELL DATA
Due to the important role played by compression-permeability (C-P) cells in the past in the prediction and interpretation of filtration data and the fact that they are still being recommended as a means of obtaining data for thickeners 16, and to the doubt which may be cast over their application to the problem, it is pertinent here to collate existing knowledge and present a critical review of the validity of their use. The methods used for C-P cell tests have significant effects on the reproducibility of test data. A marked decrease in permeability occurs when liquid is allowed to flow through the cake for a long period of time 17, but Hameed 18 claimed that this problem could be substantially elimi­ nated by using distilled, filtered water. Experiments carried out in connection with this study substantiate the work of Lu et al 17,
Tiller 19 and Okamura and Shirato 13 present data illustrating the time dependence of porosity after a load has been applied to the cake, but subsequent theoretical analyses fail to take this into account and simply uqilise an equilibrium porosity obtained after a long time period, Whilst it may be argued that this assumption was reasonable for the analysis of their experiments, for short time filtrations this cannot be acceptable 20. Okamura and Shirato 13, 14 measured the liquid pressure distribution in an actual filter cake and interpreted the results using C-P cell data1 the fact that agreement between theory and experiment was quite good must be largely attributable to their filtering only a mode­ rately compressible solid, and one which behaves in similar manner in both filtration and C-P tests.
Rushton et al 21, 22 have compared extensive data from various types of filters with that obtained from C-P cells, Large differ­ ences obtain between specific filtration resistances (al calculated from the two methods. The errors in porosity values are less serious. C-P cell data do not show any effect of slurry concen­ tration, whereas this does affect a and E values in an actual
t'f' E "-co .~
30 ,- <II in w a: w ?i 20 u
~ \!, u w • 10 <II
CalcIum camonate-, 5= CLOt.
ODO\!0--~--~40~-L--~5~0--~~~~~--~7~0~-L--~60· FIL TRATION PRESSURE, Il p kN/m2
The effect of sedimentation of solids yield from an upward filtration
CalcIum carbonate. S = 0.0.1.
" 45 5U 55 60 &5 70 75
FILTRATION PRESSURE,6p kN/m2
Figure 2(b) - The effect of sedimentation on a in an upward filtration
55
56
filtration. Although concentration sensitive data might be pre­ sented empirically 21, 22, Tiller's theoretical developments 4-6 did predict some concentration dependence.
Willis et al 23 examined two basic assumptions necessary to obtain a unique correspondence between C-P and filtration data. The two assumptions are that the specific filtration resistance is a function solely of cumulative drag stress, and that this cumulative drag stress is equal to the hydraulic pressure drop across the cake. They validate the former assumption but found that the cumulative drag stress is equal to the hydra41ic pressure drop multiplied by the average cake porosity. However, only dilute suspensions (the mass fraction of solids in the slurry being between 1.64% and 4.67%) were used in these experiments, with two experiments of similar solids concentration being carried out on each slurry. Any con­ centration dependence of a did not, therefore, show itself.
To summarise, previous agreements between C-P and filtration data could be explained by the methodology 17, wall effects in the cell 24, and the LID dependence 17. Further inaccuracies in C-P cell testing result from side-wall friction, the time lag required to reach an equilibrium porosity, the change of cake characteristics with time, and the inability of a C-P test to reveal concentration effects. A unique one-to-one correspondence between C-P and fil­ tration data is not possible without a priori knowledge of the cake porosity. Although C-P cells have been useful as a research tool to simulate filtration data, filtration times predicted from C-P data are likely to be considerably in error and unacceptable to engineering practice. For the above reasons it is desirable to develop experimental techniques and theories which obviate the need for C-P, and give a further insight into the formation and struct­ ure of filter cakes. Although local porosity and local filtration resistances are not predictable from C-P measurements such cells may, however, still find application when attempts are being made to classify the behaviour of different kinds of materials under stress25 •
THE THEORY OF COMPRESSIBLE CAKE FILTRATION
Before deposition has started the particle arrival rate at the septum. is believed to influence the specific resistance of the cake immediately adjacent to the cloth 22, 26. This was predicted by Heertjes 27 who argued that surface deposition would only be obtained with higher slurry concentrations, when larger numbers of particles are arriving at the filter medium surface in unit time. If dilute suspensions are filtered, as in clarification processes, the particle is more likely to follow a fluid streamline and be directed towards a pore in the medium. This could explain the variations of a with filtration pressure and slurry concentration.
57
At the surface of a growing cake formed by upward deposition, the stabilising force created by fluid flow into the interstices of the cake must be greater than those forces which promote parti­ cle movement away from the cake. These may be body forces due to gravity, or surface forces resulting from fluid shear at the grow­ ing surface. In the absence of forces which act to hinder cake growth, such as cake formation on an upward facing surface, the arrival of solid particles is determined by the instantaneous filtration velocity at the cake surface. The structure of the cake determines the fluid velocity. Once cake has been deposited, its characteristics are possibly time dependent and affected by a number of factors. These are primarily:
(i) re-alignment of the particles; this generally causes overall compression of the cake and hence a reduction of the overall porosity (this does not preclude local porosity increases as a result of caverns being formed by particles moving out of a volume and not being replaced by others moving in),
(ii) deformation of particles under high pressures,
(iii) a gradual increase in the medium resistance throughout the filtration as particles tend to penetrate the cloth pores,
(iv) (i) and (ii) together with migration of the finest particles within the cake in the direction of flow leading to an increase in the specific cake resistance.
As the cake grows, so the pressure loss across it increases and less is available to cause particle deposition. This may, how­ ever, be offset by a reducing pressure loss over the filter cloth as the filtrate rate decreases. The filtration velocity will inevitably decrease and may become smaller than the settling velo­ city of the larger particles in the suspension. Under free settling conditions the larger particles will be lost from the filtration zone, when deposition is on a downward facing surface, whence the average particle size forming the deposit will decrease simultane­ ously with the concentration of the particles available for fil­ tration.
The complexity of the sequence of events described in the aforegoing is further complicated if a tangential velocity compo­ nent of the liquid at the cake surface has the effect of stripping particles from the formed cake. This is believed to happen on, for example, rotary vacuum filters 15, 20, 22. Perry and Dobson 28 have described a model vacuum filtration cell designed for cake formation on a downward facing surface. The filtrate flow rate, cake thickness and resistivity were monitored continuously, and it was found in several experiments that gas bubble formation (presu­ mably dissolved air coming out of solution) within the growing cake was an important factor in controlling the course of the fil­ tration. It was concluded that, in order to adequately model the
58
filtration process, the appropriate pressure gradients on the liquid, solid and, where appropriate, gas phases should be repro­ duced in the model. The measurement of local porosity using resistivity data from growing filter cakes was further investigated by Baird and Perry 2 and Shirato and Aragaki 29 who concluded that the accuracy of the electrical method was good except in the region immediately adjacent to the growing cake surface, where the electric field has the tendency to bend towards the direction of larger local porosity.
Compressible Cake Filtration Equations
Considering Figure 3, a mass balance on the particles in either the suspension or cake yields:
ac a (cvs )
at ax (3)
where c l-€ is the volume fraction of solids in the mixture.
';vU1 -\0""'-- Flow rat. of liquid
-vso ---- Migration rat. of solids
, , IEm :€ I Ei Eo Porosity , ., I ' : I ,
Ip I , , , Hydraulic :Pl I PLi pressure Lm , " , ,
I: .Itl J , v------'
The liquid material balance can also be written as:
a€ at
a (€vR,)
ax
cake
(4)
where € is the volume fraction of liquid (porosity) in the mixture. Vs and vR, are the true velocities of the solid and liquid respect­ ively, written relative to the chamber in which the solid-liquid mixture is contained. As both liquid and solids move towards the septum then both vR, and Vs have negative values.
The force-momentum balance on the solids when their concen­ tration is greater than the critical value (at the critical concen-
tration the so-called solids stress is just beginning to be felt through the particulate structure, and so some strength is just starting to develop in the filter cake) is obtained by equating
59
the net force on the particles, the gravitational force minus the sum of the buoyancy and drag forces and the solids stress gradient, to the net rate of convection of momentum of the particles and the rate of change of particle momentum in a differential layer of the solid liquid mixture of thickness dx. This is indicated in Figure 4, and can be written formally as:
dP a (cv 2)
Drag force
~x-------~--~----
dx ",.
force
(5)
Figure 4 - Forces acting on the solids in an element of thickne