Cake filtarion research

Perspectives
Cake filtration research—a personal view
Chi Tien
Department of Chemical Engineering and Materials Science,Syracuse University,220Hinds Hall,Syracuse,NY 13244,USA
Received 13March 2002;accepted 15March 2002
Abstract
An outline of the directions and topics of future cake filtration research is presented.The proposed topics include the formulation of more accurate and efficient procedures of determining filter cake characteristics from experimental data,more complete analysis of cake formation and growth,the effect of accurate and non-intrusive measurements of the evolution of cake thickness histories.The selection is based on practical needs,intrinsic significance or both.The relationships between these proposed topics and some current and past research are also discussed.
D 2002Published by Elsevier Science B.V .
Keywords:Cake filtration;Solid –liquid separation process;Porosity;Solidosity
1.Introduction
Cake filtration,as a solid–liquid separation process,is widely used in the chemical and process industry.In the earlier part of the last century when the chemical engineer-ing profession was first established,it was also a popular topic of investigation.Among the earliest publications attributed of W.K.Lewis,an illustrious chemical engineer commonly credited for developing the chemical engineering profession,was an article on cake filtration [1].It is also interesting to note,as recalled by Gay [2],that one of the six publications which appeared in the first issue of the Trans-actions of the Institutions of Chemical Engineers dealt with cake filtration.The ample space devoted to cake filtration in the first and second editions of the classical text,‘‘Principles of Chemical Engineering’’by Walker,Lewis and McAdams was another example of the importance given to cake filtration by the profession during the earlier part of the 20th century.
This earlier interest and the effort devoted to cake filtration study,unfortunately,were not sustained and con-tinued.By the second half of the last century,the develop-ment of cake filtration technology was left mainly to equipment manufacturers which tended to be small scale operations with only modest technical capabilities.Cake filtration research,even in universities became a minor field
of endeavor.Among many academic engineers,there was the feeling that the field was well-studied and there remained few important problems worthy for further inves-tigations.
Resurgence of interest in separation technology in recent years has led to a revival of cake filtration studies.Problems arising from waste management and disposal,demand for more efficient mineral beneficiation and resources recovery,fabrication of new classes of materials and production of fine chemicals and pharmaceuticals can often be resolved with better filtration technology;the development of which requires a more insightful understanding and better infor-mation of the various aspects of the filtration process.As a field of study,cake filtration has begun to attract attention from investigators with various backgrounds different from those of chemical and process engineering.New perspec-tives about and different approaches to the solution of cake filtration have begun to emerge.
In light of the new and more promising situation,we offer in the following a broad outline of the need and opportunities of research in cake filtration.Generally speak-ing,cake filtration may consist of several stages of oper-ation.For batch operation,the process proceeds in the order of cake formation,cake consolidation and possibly cake washing.For operations using rotary and belt filters,the process may involve cake formation and dewatering by airflow.For cross-flow cases,cake formation may occur in the earlier stage of the process and media cleaning may or
0032-5910/02/$-see front matter D 2002Published by Elsevier Science B.V .PII:S 0032-5910(02)00063-3
E-mail address:ctien@ (C.Tien).
/locate/powtec
Powder Technology 127(2002)1–
8
may not follow.In all cases,cake formation and growth undoubtedly are the most important part of the process. Accordingly,it is this aspect of cake filtration study which we will address in the present article.It is well understood that any prognosis about the future is likely to be fraught with failures and what is stated below should be strictly regarded as the author’s personal opinion.
2.Better procedures of determining cake characteristics from experimental data
Designing and scaling-up filtration systems depend largely on the availability and accuracy of the relevant experimental data.One may therefore argue that a major part of the future effort in cake filtration research should be directed at improving the existing and developing new procedures of determining cake properties.The efforts should be aimed at both enhancing the accuracy of the results and reducing the time required to conduct the experi-ments.
The relevant quantities characterizing cake characteristics are the cake porosity,e(or solidosity,e s=1Àe),and cake permeability,k.From the relevant continuity equations and based on certain assumptions(i.e.negligible solid velocity, ignoring the moving boundary effect of the cake-suspension interface,and e(or e s)and k dependent solely on the cake compressive stress which is related to the liquid pressure by the relationship d p S+d p s=0for the one-dimensional recti-
linear case),the filtration rate per unit filter area,q S
m can be
expressed as
q‘
m ¼
d V
d t
¼
P o
l q s
1Àms
ða avÞP
s m
þR m
h ið1Þ
where V is the cumulative filtrate volume(per unit filter area)and t is the time.P o is the operating pressure.m is the wet to dry cake mass ratio.R m,s,l and q are medium resistance,solid mass fraction of the feed suspension,filtrate viscosity and density,respectively.a av is the average spe-
cific cake resistance over p s ranging from0to p s
m .With
p s
m =D p c on account of the p s–p S relationship stated above,
[a av]p
sm =D p c
is
½a av p
s m ¼D p c
¼
D p c
Z D p c
ðaÞÀ1d p‘
ð2Þ
where
a¼ðk e s q sÞÀ1ð3ÞFor a cake in formation,a is a local and time-dependent quantity.[a av]can be viewed as the stress-averaged value of the specific resistance of a cake across which the liquid pressure drop is D p c such that the cake compressive stress varies from zero at the cake-suspension interface to p s
m
=D p c at the cake-medium interface.
Further integration of Eq.(1)under the condition P o=constant and assuming R m being constant(no medium clogging)yields
P o t¼
lq s
v
av p
s m
¼D p c
1Àms
2
4
3
5V2þl R m Vð4Þ
where
ða avÞ
s m c
ð1ÀmsÞ
2
4
3
5¼
2
Z V
ða avÞ
s m c
ð1ÀmsÞ
d V
V2
ð5Þ
It is clear that both(a av)p
sm
=D p c and m vary with time in a filtration operation.Eq.(4),therefore,does not give t as a parabolic function of V.However,if the following approx-imations are introduced on account that the medium resist-ance is negligible
ða avÞp
s m
¼D p c
¼ða aÞp
s m
¼P o
ð6aÞand
ð1ÀmsÞ¼
q
q p
!
s
e s
i
ð6bÞ
where e s
o
is the solid volume fraction of the feed suspension.
Eq.(4),upon rearrangement,becomes
t
V
¼
l
P o
q p e s
o
v
ða avÞp
s m
¼P o
VþR m
h i
ð7Þ
Namely,a plot of t/V vs.V yields a linear relationship,the slope of which can be related to(a av)at p s
m
=P o.This behavior is clearly shown in Fig.1.The results shown in Fig.1were obtained from filtration experiments of2% CaCO3suspensions with media of various resistance[3].It is clear that the linearity of t/V vs.V is seen only when the value of V(or time)is sufficiently large.The initial part of the data was clearly nonlinear.
The commonly used procedure of plotting t/V vs.V enables the determination of(a av)(corresponding to p s
m
=P o)utilizing only part of the data when the cake thickness is sufficiently large.In order to obtain the values
C.Tien/Powder Technology127(2002)1–8 2
of (a av )over a range of p s ,a number of filtration experi-ments under different values of P o must be carried out.On the other hand,the pressure drop across a cake in a filtration experiment,D p c ,is actually a monotonically increasing function of time with P o as its asymptotic value.In princi-ple,it therefore is possible to evaluate (a )av corresponding to the range of D p c values experienced by the cake.Such a procedure certainly would be more efficient than that based on the t /V vs.V plot.
Fathi-Najafi and Theliander [5]experimentally deter-mined the evolution of the liquid pressure profile across a forming cake and calculated (a )av based on the pressure data.The inherent difficulty associated with the procedure they used made the pressure drop data not always reliable as the authors stated themselves.Recently,Usher et al.[4]conducted filtration experiments in which the operating pressure was increased incrementally and determined a av corresponding to each of the applied pressure.However,to insure that the approximations of Eqs.(6a)and (6b)hold and to minimize the disturbances caused by the sudden change in operating pressures,after each pressure increase,filtration must proceed over a sufficient period of time.In a sense,there is no difference between the method of Usher et al.[4]and the conventional one since either procedure uses only part of the data collected.
A possible procedure for evaluating (a )av utilizing the entire body of data collected in a single constant pressure filtration experiment may be stated as follows:assuming that the results of V vs.t and L vs.t are available,they can then be used to estimate d V /d t and m at various
times.From the filtration rate at t =0,(d V /d t )o ,the medium resistance,R m ,can be estimated and D p c ,the pressure drop across the cake,can be readily determined as
R m ¼P o =
d V
d t
o
;D p m ¼d V d t
R m ¼
ðd V =d t Þ
ðd V =d t Þo P o and D p c ¼P o ÀD p m ¼P o 1Àðd V =d t Þ
ðd V =d t Þo !
ð8Þ
with (d V /d t ),m and D p c known at a given time,the value of (a av )corresponding to p s m =D p c can be readily eval-uated from Eq.(1).Thus,from a single filtration experi-ment,one can obtain (a av )vs.p s with 0<p s <P o instead of a single value of (a av )at p s =P o .While the principle of the procedure is simple,its implementation may not be straightforward.To obtain (d V /d t )from V vs.t by differ-entiation will inevitably introduce errors.This,in turn,implies that the data of V vs.t used must have a high degree of accuracy.The accuracy of D p c depends upon the initial value of (d V /d t ),(d V /d t )o and is based on the assumption that R m is constant.A number of studies have demonstrated the difficulties of accurately determining R m [6–8].The accuracy of estimating D p c depends upon the accuracies of (d V /d t )and (d V /d t )o (see Eq.(8))and obtaining (d V /d t )o requires both differentiating and extrap-olating experimental data.It is obvious that these prob-lems can only be resolved by careful and detailed treatment of accurate data.
3.Independent determination of cake solidosity and permeability
The physical parameter which characterizes cake forma-tion is the specific cake resistance which is assumed to be a function of the compressive stress.Although the stress-averaged a ,(a )av ,can be obtained from filtration data,the physical significance of a becomes more certain if it can be determined from independent measurement so that a could not be regarded as merely a fitting parameter.Furthermore,if the independent measurement can be carried out in a less time-intensive manner,it would have an added advantage over the t /V vs.V procedure.
The so-called compression–permeability cell (C–P cell)measurement offers a conceptually simple,independent method of determining a (p s ).The C–P cell method was first proposed by Ruth [9]and further developed by Grace [10].In essence,a C–P cell measurement consists of preparing a homogeneous (or nearly homogeneous)cake from a known solid–liquid suspension sample under a specified compression load.Upon equilibrium,the cake solidosity,e s ,is determined from the cake height and
the
Fig.1.Plot of t /V vs.t .Data were obtained from filtration of 2%CaCO 3–H 2O suspension with P o =1barg and media consisting of 3,4,10and 18sheets of Whatman #1filter paper.The relationship between t /V vs.t was clearly not linear initially [3].
C.Tien /Powder Technology 127(2002)1–83
mass of the solid in the sample used.The cake permeability, k,is determined from the pressure drop–flow rate measure-ment and the specific cake resistance is found to be (e s q s k)À1.By forming cakes with different load(or p s), the constitutive relationships,e s=e s(q s)and a(p s)can be established.
Generally speaking,C–P measurement is relatively sim-ple to conduct.Independent determination of a as stated before also adds credence to the physical significance of a. However,in spite of its simplicity in operation,C–P measurement does have its problems.The presence of the wall friction along the cell surface makes it impossible to prepare a truly homogeneous cake although it was found possible to minimize the wall friction effect as shown in a more recent study[11].Further,Lu et al.[12]have devel-oped procedures to compensate the wall friction effect in interpreting C–P cell data.
The purpose of the C–P cell measurements is,of course,to obtain cake solidosity and specific resistance data(as a function of the compressive stress)to be used in design and simulation.In cake filtration,compressive stress arises from the cumulative effect of fluid drag acting upon the particles constituting a cake.A relation-ship between the pressure of the pore liquid,p S,and the cake compressive stress,p s,therefore can be expected. However what is this relationship?Intuitively,one may assume that d p S+d p s=0.For the one-dimensional recti-linear case,this assumption has been used in most filtra-tion studies.
A more careful examination reveals that the p S–p s relationship can be obtained from the volume-averaged equations of motion of the liquid and particle phases. Depending upon the assumptions used in carrying out volume averaging,in addition to the relationship
d p‘þd p s¼0ð9aÞther
e are other possibilities including[13]
ð1Àe sÞd p‘þd p s¼0ð9bÞð1Àe sÞd p‘þe s d p s¼0ð9cÞd½ð1Àe sÞp‘ þd½e s p s ¼0ð9dÞComparisons between the specific cake resistance obtained from the filtration experimental results and those from the C–P cell measurements using the various p S–p s relationships were made in a recent investigation[14].The p S–p s relationship which yields the best agreement was found to be system specific and varies with cake compres-sibility.Rational rules which can be used to select the proper p S–p s relationships remain to be established.This is one area of study which should be pursued by future investi-gators.
4.Determination of cake thickness history
In the chemical and process industries,cake filtration is commonly applied for separating solid from fluid or the recovery of filtrate.Filtration performance therefore can be described by the relationship between the cumulative filtrate volume,V,with time,t.On the other hand, information concerning cake formation,including the increase in cake thickness with time is also of practical importance if the recovered solid is the primary product of separation.
A fundamental question concerning cake formation and growth is cake definition.Put it in another way,is there a threshold value of e s,e s o which distinguishes the cake formed from the feed suspension?If so,what is this threshold value and how can it be determined?Exact answers to these questions are not available but it is often assumed that a cake is formed if the particle volume fraction (or solidosity)reaches a certain value which is given by the constitutive relationship e s=e s(p s)at p s=0or e s=e s o.By implicit assumptions,this threshold value is taken to be an average value of e s over a sufficiently large cake surface area and the cake/suspension interface is assumed to be smooth.
Determining L vs.t experimentally is a rather involved undertaking for the following reasons:The value of a s o is not well defined,and microscopically,cake surface is not smooth.Because of these reasons,earlier attempts of determining cake thickness were based on indirect infer-ence.Murase et al.[15]showed that by placing an interchangeable plate with a small opening(as compared with the cell cross section)at a predetermined height in their experimental filtration cell,a sharp change(decrease) in filtration rate was observed when the growth of the filter cake reached the plate.Thus,by carrying out experiments with the plate placed at different heights, the results of L vs.t can be established.The disadvantage of this procedure is that it is time-consuming.A more recent study also revealed that the placing of such a plate may cause errors in the data of V vs.t as shown in Fig.2
[11].
A different approach of experimentally determining L vs. t was suggested by Fathi-Najafi and Theliander[5].The procedure was based on the fact that the pressure within the suspension phase in cake formation remains relatively con-stant while the pressure change is significant within the cake.Thus,by placing pressure sensors along the surface of an experimental cell at various heights and recording the respective pressure histories,a history of L vs.t was obtained(see Fig.3).The difficulty of this method arose for the same reasons mentioned earlier,namely,the cake/
C.Tien/Powder Technology127(2002)1–8 4
suspension interface is not smooth.The results obtained at a particular point may not be representative of the entire surface.
In addition to the studies mentioned above,a number of methods of observing cake growth have been devel-oped principally for cross-flow membrane filtration.Taka-hashi et al.[16]applied an ultrasound method for on-line measurement of cake thickness.However,it is doubtful whether their method can be directly applied to cakes with significant compression.For cross-flow filtration,direct observations of cake growth using optical or video cameras were made by Mackley and Sherman [17]and Wakeman [18]and the use of laser sensors was also explored [19].More recently,Tung et al.[20]presented cake thickness results based on measurements using photointerrupt sensors.With the exception of the work of Tung et al.[20],all the other methods give results at a given point.Significant progress in the study of cake growth can be made if these methods can be extended to cover the entire cake surface and arrangement made to adapt these methods to dead-end filtration as well.
The information of L vs.t is also important to the determination of the constitutive relationship a =a (p s )as described in the preceding section.Referring to Eq.(1),the value of [a av ]p sm can be evaluated from the instantaneous filtration rate if the value of m is known.By definition,m is given as
m ¼1þ
ð1Àe s Þq e s q s
ð10
Þ
Fig.3.Liquid pressure histories obtained in a filtrate experiment by Fathi-Najafi and Theliander [5].The pressure sensors 1–7were placed at 3.9,6.8,9.7,13.0,16.0,19.0and 22.0mm from the filter medium.The response of sensor no.6was earlier than expected and there was no response of sensor no.7;suggesting uneven cake
formation.
Fig.2.Filtration data of t /V vs.t demonstrating sudden change in filtration rate as cake approached the interchangeable plate placed at various height in the filtration cell corresponding to cake thickness of 10mm (o ),20mm (5),30mm (+),40mm (w ),and 50mm (D ).
C.Tien /Powder Technology 127(2002)1–85
where¯e s is the average cake solidosity defined as
e s¼Z L
e s d x
L
ð11Þ
By definition,the particle mass fraction of the feed suspension,s,can be expressed as
s¼
L e s q s
L e s q sþLð1Àe sÞqþV q
ð12Þ
From which,one has
e s¼
ðV=LÞþ1
1þðq s=qÞ1
s
À1
ÀÁð13Þ
Accordingly,from the data of V vs.t and L vs.t,the value of m at different times can be found using Eqs.(13) and(10),which,in turn,can be applied to calculate(a av)p
sm from Eq.(1).
5.Effect of particle–particle interaction on cake structure and properties
It has long been recognized that cakes formed from the same suspension but with different pH values may differ significantly in their solidosities and permeabilities[10]. In industrial practice,using polymer additives or filter aids to facilitate filtration and dewatering is widespread. These differences and improvement in performance are believed to be due to the particle–particle interactions’effect.On the other hand,analyses delineating the effect of particle–particle interactions on cake performance are incomplete and there is also a lack of experimental data which systematically demonstrate the effect of solution variables(e.g.pH,electrolyte concentrations,etc.)on cake characteristics.
A rigorous approach to the study of the particle–particle interaction effect in cake filtration can be made by first analyzing the fluid–particle and particle–particle interac-tions on the particle level.The result of this analysis can then be averaged to obtain a macroscopic description.The two key steps involved are the identification and quantifi-cation of the interactions and the construction of an appro-priate averaging process.Questions regarding the latter have been mentioned before[see the various relationships between p s and p S of Eqs.(9a)–(9d)].For the first step, the particle–particle interactions are due to mainly the double layer force,the London van der Waals force and possibly the hydration force.Bowen and Jenner[21] assumed that the sum of the above three,or the disjoining pressure is the same as the compressive stress.Koenders and Wakeman[22,23],on the other hand,assumed that com-pressive stress is due to the double layer force ing either assumption,it is possible,in principle,to determine, for a given system,the compressive stress as a function of the average particle–particle separation which can be related to the solidosity(or porosity);thus establishing the constitutive relationship e s=e s(p s).
Explicit expression of the interaction forces between two particles mentioned above are readily available.To obtain the compressive stress,the sum of a large number of such pair interactions must be considered.Expressing the force expression in continuum variable,however,is not a simple undertaking.The efforts of Koenders and Wakeman have only yielded expression of limiting situations with undeter-mined coefficients.The practical utility of their work is therefore rather limited.
To be practically useful,the knowledge and capability of directly relating the constitutive relationships with the solution and electrokinetic variables of suspension is desired.In the following sections,we will briefly explore the possibility of acquiring such a capability based on the work of Bowen and Jenner[21].
As stated before,Bowen and Jenner assumed the com-pressive stress may be expressed as
p s¼
ffiffiffi
6
p
A h
½fðDÞþF AðDÞþF hyðDÞ ð14Þ
where D is the average particle–particle separation.For particles of radius,a,the relationship between e s,a and D is
e s¼
4
ffiffiffi
2
p
3
p
a3
ð2aþDÞ
ð15Þ
and A h is the effective area occupied by one particle and equal to2
ffiffiffi
3
p
aþD
ÀÁ2
.The configurational electrostatic force,the London van der Waals force,and the hydration force are f,F A and F hy,respectively.Explicit expressions of F A and F hy are available.The configurational electrostatic force expression was determined by Bowen and Jenner using a Wigner–Seitz cell and its value can be obtained from the numerical solution of the relevant Poisson–Boltz-mann equation.
Eq.(14)provides the basis of predicting the relationship of e s=e s(p s)from fundamental quantities characterizing the nature of particle–particle interactions.Because of the large number of assumptions used in its derivation,it is unlikely that Eq.(14)can be a truly predictive tool.The results obtained from it,however,can be helpful in assessing the relative importance of the various electrokinetic parameters in cake filtration and in developing empirical correlations
C.Tien/Powder Technology127(2002)1–8 6
from experimental data.In particular,it can be applied to estimate,at least,the approximate magnitude of e s o,the value of e s at p s=0,which is the threshold value of e distinguishing cake from suspension.No experimental procedure is available presently for its determination.
6.Analysis and simulation of cake formation and growth
In more recent years,analysis and simulation of cake formation and growth have been a popular subject of study as evidenced by the relatively large number of publications of this type in the literature.It is not our purpose here to offer a detailed review and critique of these studies.Rather, the discussions given below will be restricted to a brief summary as background to the suggested future studies.
A review of the earlier investigations indicate that all the analyses were based on the solutions of the volume-aver-aged equations of continuities of the fluid and particle phases.Their major differences reside mainly in the follow-ing areas.
(1)The continuity equations may be given in either physical(spatial)coordinate or material coordinate.When the latter is used,the governing equation can be reduced to a form similar to the diffusion equation.
(2)Different parameters were used to characterize cake properties.However,there is correspondence between these different sets of quantities;thus making it moot to claim superiority of one set of quantities over others.
(3)Different physical domains were used in the analyses depending upon the mode of operation;continuous or batch. As a result,cake compaction and/or the presence of a suspension phase may or may not be included.These differences also led to the use of different boundary con-ditions.
(4)The question whether there is a discontinuity in solidosity across the cake/suspension interface is unsettled, and formulation of the moving boundary condition due to this discontinuity was not always done properly.
(5)Most of the analysis were made numerically.Some investigators have attempted analytical solutions but with only limited success.
That there are a relatively large number of studies dealing with the same subject but using seemingly differ-ent notations and often with spurious claims have caused confusion especially to potential users of the results. Studies aimed at reconciling these results and their comparisons(in terms of user-friendliness and accuracy) are therefore needed.Other studies which are worthwhile include the incorporation of the simulation as part of a parameter search algorithm which can be used to evaluate the parameters of the constitutive relationships from experimental data.Because of the relatively large number of parameters to be determined,a two-stage search which first determines some approximate values followed by refinement may be required.
Another problem which may warrant attention is the development of approximate analytical solutions which yield results more accurate than those of the conventional theory but do not require excessive competition.For exam-ple,Landman et al.[24]gave a similar solution by assuming D p c being constant.If an approximate correction can be made to account for the medium resistance,a new procedure of evaluating cake property parameters with advantage over the conventional t/V vs.V plot could possibly be developed leading to the establishment of more accurate constitutive relationship.
As a more fundamental approach to the analysis of cake formation and growth,the use of computer simu-lation based on particle dynamics consideration offers significant possibilities.In principle,computer simulation allows the monitoring of the movement of individual particle and the determination of liquid flow field within a forming cake,taking into account all the forces involved including the particle–fluid,particle–particle interactions.It is therefore possible to obtain results of filtration performance as well as the evolution of the cake structure from simulation.Lu and Hwang[25,26]pre-sented results on cake structure from their simulation studies,which did not consider the presence of par-ticle–particle interactions and gave no results on liquid flow.Extending their studies to include some or all these variables would be a worthwhile future investigation. 7.Concluding remarks
The brief presentation given above is to demonstrate that cake filtration is indeed a fertile field of research in spite of its being a subject of study and engineering practice of long standing.Its rigorous study based on micromechanical approach requires interfacing knowledge and information of several disciplines.It is therefore not surprising that investigators active in the field nowadays have various educational and professional backgrounds. An important factor to be recognized is that regardless of the specific topics of research one may undertake,the study must have consequences in practice.Thus,it is important that analysis of cake filtration should yield results not only in predicting filtrate performance,but also provide basis for data interpretation and process control.There should also be close interaction between analysis and experiments.In this regard,establishing a database of cake filtration work accessible to the public should be highly desirable.
Cake filtration research offers opportunity of not only obtaining results useful and significant in themselves but providing experiences which are essential to engineering graduate education.In this context,one can only hope that in the not too distant future,more interest in its study will be developed in universities and educational institutions.
C.Tien/Powder Technology127(2002)1–87。