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Cake Filtration and Filter Media Filtration
3 CAKE F I L T R A T I O N AND F I L T E R M E D I A FILTRATION Introduction
Cake filtration is the most common form of filtering employed by the chemical and process industries. This manifests into handling the permeability of a bed of porous material, the schematic of which is shown in Figure 1. With high-solids-concentration suspensions, even relatively small particles (in comparison to the pore size) will not pass through the medium, but tend to remain on the filter surface, forming "bridges" over individual openings in the filter material. Filtrate flows through the filter medium and cake because of an applied pressure, the magnitude of which is proportional to the filtration resistance. This resistance results from the frictional drag on the liquid as it passes through the filter and cake. Hydrostatic pressure varies from a maxinmm at the point where liquid enters the cake, to zero where liquid is expelled from the medium; consequently, at any point in the cake the two are complementary. That is, the sum of the hydrostatic and compression pressures on the solids always equals the total hydrostatic pressure at the face of the cake. Thus, the compression pressure acting on the solids varies from zero at the face of the cake to a maximum at the filter medium. When solid particles undergo separation from the mother suspension, they are captured both on the surface of the filter medium and within the inner pore passages. The penetration of solid particles into the filter medium increases the flow resistance until the filtration cycle can no longer continue at economical throughput rates, at which time the medium itself must be replaced. This chapter provides a summary of standard calculation methods for assessing cake formation, behavior, and the overall efficiency of the filter-medium filtration process. 59
60
Liquid Filtration
Dynamics of Cake Filtration When the space above the suspension is subjected to a source of compressed gas (e.g., air) or the space under the filter plate is connected to a vacuum source, filtration is accomplished under a constant pressure differential (the pressure in the receivers is constant). In this case, the rate of the process decreases due to an increase in the cake thickness and, consequently, flow resistance. A similar filtration process results from a pressure difference due to the hydrostatic pressure of a suspension layer of constant thickness located over the filter medium. If the suspension is fed to the filter with a reciprocating pump at constant capacity, filtration is performed under constant flowrate. In this case, the pressure differential increases due to an increase in the cake resistance. If the suspension is fed by a centrifugal pump, its capacity decreases with an increase in cake resistance, and filtration is performed at variable pressure differentials and flowrates. The most favorable filtration operation with cake formation is a process whereby no clogging of the filter medium occurs. Such a process is observed at sufficiently high concentrations of solid particles in suspension. From a practical standpoint this concentration may conditionally be assumed to be in excess of 1% by volume.
Figure 1. Operating scheme of a filtration process: I-filter; 2-filter medium; 3-suspension; 4-filtrate; 5-cake. To prevent pore clogging in the filter medium when handling relatively low solids concentrations (e.g., O. 1-1% by volume), general practice is to increase the solids concentration in thickeners before the suspension is fed to the filter. Filtration is frequently accompanied by hindered or free gravitational settling of solid particles. The relative directions of action between gravity force and filtrate motion
Cake Filtration and Filter Media Filtration
61
may be concurrent, countercurrent or crosscurrent, depending on the orientation of the filter plate, as well as the sludge location above or below the filter plate. The different orientations of gravity force and filtrate motion with their corresponding distribution of cake, suspension, filtrate and clear liquid are illustrated in Figure 2. Particle sedimentation complicates the filtration process and influences the controlling mechanisms. Furthermore, these influences vary depending on the relative directions of gravity force and filtrate motion. If the suspension is above the filter medium (Figure 2A), particle settling leads to more rapid cake formation with a clear filtrate, which can be evacuated from the filter by decanting. If the suspension is under the filter medium (Figure 2B), particle settling will prevent cake formation, and it is necessary to mix the suspension to maintain homogeneity.
Figure 2. Direction of gravity force action and filtrate motion in filters: A-cocurrent; Bcountercurrent; C-crosscurrent; solid arrow-direction of gravity force action; dashed arrowdirection of filtrate motion; 1-filter plate; 2-cake; 3-sludge; 4-filtrate; 5-clear liquid. When the cake structure is composed of particles that are readily deformed or become rearranged under pressure, the resulting cake is characterized as being compressible. Those that are not readily deformed are referred to as semicompressible, and those that deform only slightly are considered incompressible. Porosity (defined as the ratio of pore volume to the volume of cake) does not decrease with increasing pressure drop. The porosity of a compressible cake decreases under pressure, and its hydraulic resistance to the flow of the liquid phase increases with an increase in the pressure differential across the filter media. Cakes containing particles of inorganic substances with sizes in excess of 100 ~m may be considered incompressible, for all practical purposes. Examples of incompressible cake-forming materials are sand and crystals of carbonates of calcium and sodium. The cakes containing particles of metal hydroxides, such as ferric hydroxide, cupric hydroxide, aluminum hydroxide, and sediments consisting of easy deforming aggregates, which are formed from primary fine crystals, are usually compressible.
62
Liquid Filtration
At the completion of cake formation, treatment of the cake depends on the specific filtration objectives. For example, the cake itself may have no value, whereas the filtrate may. Depending on the disposal method and the properties of the particulates, the cake may be discarded in a dry form, or as a slurry. In both cases, the cake is usually subjected to washing, either immediately after its formation, or after a period of drying. In some cases, a second washing is required, followed by a drying period where all possible filtrate must be removed from the cake; or where wet discharge is followed by disposal: or where repulping and a second filtration occurs; or where dry cake disposal is preferable. Similar treatment options are employed in cases where the cake is valuable and all contaminating liquors must be removed, or where both cake and filtrate are valuable. In the latter, cake-forming filtration is employed, without washing, to dewater cakes where a valueless, noncontaminating liquor forms the residual suspension in the cake. To understand the dynamics of the filtration process, a conceptual analysis is applied in two parts. The first half considers the mechanism of flow within the cake, while the second examines the external conditions imposed on the cake and pumping system, which brings the results of the analysis of internal flow in accordance with the externally imposed conditions throughout. The characteristics of the pump relate the applied pressure on the cake to the flowrate at the exit face of the filter medium. The cake resistance determines the pressure drop. During filtration, liquid flows through the porous filter cake in the direction of decreasing hydraulic pressure gradient. The porosity (e) is at a minimum at the point of contact between the cake and filter plate (i.e., where x = 0) and at a maximum at the cake surface (x = L) where sludge enters. A schematic definition of this system is illustrated in Figure 3.
Figure 3. Important parameters in cake formation.
The drag that is imposed on each particle is transmitted to adjacent particles. Therefore, the net solid compressive pressure increases as the filter plate is approached, resulting in a decrease in porosity. Referring to Figure 4A, it may be
Cake Filtration and Filter Media Filtration
63
assumed that particles are in contact at one point only on their surface, and that liquid completely surrounds each particle. Hence, the liquid pressure acts uniformly in a direction along a plane perpendicular to the direction of flow. As the liquid flows past each particle, the integral of the normal component of force leads to form drag, and the integration of the tangential components results in frictional drag. If the particles are non-spherical, we may still assume single-point contacts between adjacent particles as shown in Figure 4B. Now consider flow through a cake (Figure 4C) with the membrane located at a distance x from the filter plate. Neglecting all forces in the cake other than those created by drag and hydraulic pressure, a force balance from x to L gives:
F +ApL : Ap
(1)
The applied pressure p is a function of time but not of distance x. F s is the cumulative drag on the particles, increasing in the direction from x = L to x = 0. Since single point contact is assumed, the hydraulic pressure PL is effectively over the entire cross section (A) of the cake; for example, against the fictitious membrane shown in Figure 4B. Dividing Equation 1 by A and denoting the compressive drag pressure by ps = F/A, we obtain: (2)
Figure 4. Frictional drag on particles in compressible cakes.
The term Ps is a fictitious pressure, because the cross-sectional area A is not equal to either the surface area of the particles nor the actual contact areas In actual cakes, there is a small area of contact Ac whereby the pressure exerted on the solids may be defined as Fs/Ac. Taking differentials with respect to x, in the interior of the cake, we obtain:
dPs + dPc = 0
(3)
64
Liquid Filtration
This expression implies that drag pressure increases and hydraulic pressure decreases as fluid moves from the cake's outer surface toward the filter plate. From Darcy's law, the hydraulic pressure gradient is linear through the cake if the porosity (c) and specific resistance (tO are constant. The cake may then be considered incompressible. This is illustrated by the straight line obtained from a plot of flowrate per unit filter area versus pressure drop shown in Figure 5. The variations in porosity and specific resistance are accompanied by varying degrees of compressibility, also shown in Figure 5. As noted in Chapter 1, filtration is primarily an application of fluid mechanics; that is, filtrate flow is induced through a porous filter cake and filter medium. The rate of the filtration process is directly proportional to the driving force and inversely proportional to the resistance.
Figure 5.
Flowrate/area versus pressure drop across the cake.
Because pore sizes in the cake and filter medium are small, and the liquid velocity through the pores is low, the filtrate flow may be considered laminar: hence, Poiseuille's law is applicable. Filtration rate is directly proportional to the difference in pressure and inversely proportional to the fluid viscosity and to the hydraulic resistance of the cake and filter medium. Because the pressure and hydraulic resistances of the cake and filter medium change with time, the variable rate of filtration may be expressed as: u -
dV Adz
(4)
Cake Filtration and Filter Media Filtration
where
65
V = volume of filtrate (m 3) A = filtration area (m 2) 1: = time of filtration (sec)
Assuming laminar flow through the filter channels, the basic equation of filtration as obtained from a force balance is:
1 dV u
where
-
A d~.
Ap -
I~(R c + Rf)
(5)
= pressure difference (N/m z) = viscosity of filtrate (N-sec/m 2) = filter cake resistance (m -1) = initial filter resistance (resistance of filter plate and filter channels) (m -1) = filtration rate (m/sec), i.e., filtrate flow through cake and filter plate filtration rate (m3/sec), i.e., filtrate flow rate dV/dq: = Ap /x Rc Rf
Filter cake resistance (Rc) is the resistance to filtrate flow per unit area of filtration. Rc increases with increasing cake thickness during filtration. At any instant, Rc depends on the mass of solids deposited on the filter plate as a result of the passage of V (m 3) filtrate. Rf may be assumed a constant. To determine the relationship between volume and residence time "c, Equation 5 must be integrated, which means that R c must be expressed in terms of V. We denote the ratio of cake volume to filtrate volume as x0. Hence, the cake volume is x0V. An alternative expression for the cake volume is hcA; where hc is the cake height in meters. Consequently
x ov
= hcA
(6)
Hence, the thickness of the cake, uniformly distributed over the filter plate, is" V
h c = x o -A
(7)
The filter cake resistance may be expressed as: V
R c = r o X o -~
where r 0-- specific volumetric cake resistance (m-2).
(8)
66
Liquid Filtration
As follows from Equation 8, ro characterizes the resistance to liquid flow by a cake having a thickness of 1 m. Substituting for Rc from Equation 8 into Equation 5, we obtain:
1 dV
Ap --
U
A dz
=
#[ roXo ( V/A ) + Rf]
(9)
Filtrate volume, Xo, can be expressed in terms of the ratio of the mass of solid particles settled on the filter plate to the filtrate volume (Xw) and instead of ro, a specific mass cake resistance r w is used. That is, r w represents the resistance to flow created by a uniformly distributed cake, in the amount of 1 kg/m 2. Replacing units of volume by mass, the term ro Xo in Equation 9 changes to rwXw. Neglecting filter plate resistance (Rf -" 0 ) , and taking into account Equation 7, we obtain from Equation 3 the following expression"
@ ro-
tzhc u
(10)
At/z = 1 N-sec/m 2, hc = 1 m and u = 1 m/sec, ro = Ap. Thus, the specific cake resistance equals the pressure difference required by the liquid phase (with a viscosity of 1 N-sec/m 2) to be filtered at a linear velocity of 1 m/sec through a cake 1 m thick. This hypothetical pressure difference, however, is beyond a practical range. For highly compressible cakes, r0 can exceed 1012m 2. Assuming V = 0 (at the start of filtration) where there is no cake over the filter plate. Equation 9 becomes: Rf-
@
(11)
#u
At /z = 1 N-sec/m 2 and u = 1 m/sec, Rf = Ap. This means that the filter plate resistance is equal to the pressure difference necessary for the liquid phase (with viscosity of 1 N-sec/m 2) to pass through the filter plate at a rate of 1 m/sec. For many filter plates R e is typically 10 I~ m -~ . For a constant pressure drop and temperature filtration process all the parameters in Equation 9, except V and "r, are constant. Integrating Equation 9 over the limits of 0 to V, from 0 to ~:, we obtain v
f 0
or
v
#( roXo ~ + Rf ) d V :
ApAdr, 0
(12a)
Cake Filtration and Filter Media Filtration
67
g 2
tXoroXo ~
(12b)
+ ~ R f V = A p A r.
Dividing both sides by/xroXo/2A gives:
V 2+2
roX~
V =
2
. I~ roX o
-c
(13)
Equation 13 is the relationship between filtration time and filtrate volume. The expression is applicable to either incompressible or compressible cakes, since at constant Ap, r0 and Xo are constant. If we assume a definite filtering apparatus and set up a constant temperature and filtration pressure, then the values of Rf, ro,/x and Ap will be constant. The terms in parentheses in Equation 13 are known as the "filtration constants", and are often lumped together as parameters K and C; where: 2ApA 2 K
__
l~ roX o
(14)
RA
C-
f roxo
(15)
Hence, a simplified expression may be written to describe the filtration process as follows: V 2 + 2 V C = KI:
(16)
Filtration constants K and C can be experimentally determined, from which the volume of filtrate obtained over a specified time interval (for a certain filter, at the same pressure and temperature) can be computed. If process parameters are changed, new constants K and C can be estimated from Equations 14 and 15. Equation 16 may be further simplified by denoting ~:0 as a constant that depends on K and C:
68
Liquid Filtration
C 2
a:0
-
K
(17)
Substituting t0 into Equation 16, the equation of filtration under constant pressure conditions is: (V+C)
2 =
K(I: + t o )
(18)
Equation 18 defines a parabolic relationship between filtrate volume and time. The expression is valid for any type of cake (i.e., compressible and incompressible). From a plot of V + C versus (z + %), the filtration process may be represented by a parabola with its apex at the origin as illustrated in Figure 6. Moving the axes to distances C and % provides the characteristic filtration curve for the system in terms of volume versus time. Because the parabola's apex is not located at the origin of this new system, it is clear why the filtration rate at the beginning of the process will have a finite value, which corresponds to actual practice.
Figure 6. Typical filtration curve.
Constants C and z o in Equation 18 have physical interpretations. They are basically equivalent to a fictitious layer of cake having equal resistance. The formation of this fictitious cake follows the same parabolic relationship, where t o denotes the time required for the formation of this fictitious mass, and C is the volume of filtrate required. Differentiating Equation 16 gives:
69
Cake Filtration and Filter Media Filtration
dV
K
dl:
2( V+ C)
(19)
And rearranging in the form of a reciprocal relationship: d'c
--
dV
2V
+
K
2C
(20)
K
This form of the equation provides a linear relation as shown by the plot in Figure 7. The expression is that of a straight line having slope 2/K, with intercept C. The experimental determination of d~:/dV is made simple by the functional form of this expression. Filtrate volumes Vl and V2 should be measured for time intervals q:, and 1:2. Then, according to Equation 16:
172- 171
1
V2 - V 2
Vz - Vl
V2- Vl
K
2 c(g 2 -
VI)
K (21)
Vl+V2) 2
2 k
+ 2C K
In examining the right side of this expression, we note that the quotient is equal to the inverse value of the rate at the moment of obtaining the filtrate volume, which is equal to the mean arithmetic value of volumes V1 and V2:
172-'I71 = { dg / V2_V1 ~ Vl+V2 2
(22)
Filtration constants C and K can be determined on the basis of several measurements of filtrate volumes for different time intervals. As follows from Equations 14 and 15, values of C and K depend on r0 (specific volumetric cake resistance), which in turn depends on the pressure drop across the cake. This Ap, especially during the initial stages of filtration, undergoes changes in the cake. When the cake is very thin, the main portion of the total pressure drop is exerted on the filter medium. As the cake becomes thicker, the pressure drop through the cake increases rapidly but then levels off to a constant value. Isobaric filtration shows insignificant deviation from Equation 16. For approximate calculations, it is
70
Liquid Filtration
Figure 7. Plot o f Equation 20.
possible to neglect the resistance of the filter plate, provided the cake is not too thin. Then the filter plate resistance Rf = 0 in Equation 15, C = 0 (Equation 15) and % = 0 (Equation 17). Therefore, the simplified equation of filtration takes the following form:
V 2 = K~
(23)
For thick cakes, Equation 23 gives results close to that of Equation 16.
Constant-Rate Filtration When sludge is fed to a filter by a positive-displacement ptunp, the rate of filtration is nearly constant (i.e., dV/dz = constant). During constant-rate filtration, the pressure increases wit1 an increase in cake thickness. Therefore, the principal variables are pressure and filtrate volume, or pressure and filtration time. Equation 9 is the principal design relation, which may be integrated for a constant-rate process. The derivative, dV/dl:, may be replaced simply by V/T: Ap
/troXo A 2z
(24)
The ratios in parentheses express the constant volume rate per unit filter area. Hence, Equation 24 is the relationship between time 1: and pressure drop Ap. For incompressible cakes, ro is constant and independent of pressure. For compressible cakes, the relationship between time and pressure at constant-rate filtration is:
Cake Filtration and Filter Media Filtration
A p = i~ axo AP s
(v)2 -~
71
(25)
"r. + I~
Filtration experiments are typically conducted in pilot scale equipment and generally tests are conducted either at constant pressure or constant rate to determine ax0, as well as s and R t-, for a given sludge and filter medium. Such tests provide empirical information that will enable the time required tor the pressure drop to reach the desired level for a specified set of operating conditions to be determined. In the initial stages of filtration, the filter medium has no cake. Furthermore, Ap is not zero, but has a value that is a function of the resistance of the medium for a given flowrate. This initial condition can be stated as: V
(26)
For an incompressible cake (where s = 0), Equation 25 takes the form:
ap- axo
(27)
As noted earlier, for thick cakes, the resistance of the filter medium may be neglected. Hence, for Rt-= 0, Equation 25 simplifies to: Apl-S = I~ax o
(v/2
~:
(28)
An increase in pressure influences not only coefficient r0, but the cake's porosity as well. Since the cake on the filter plate is compressed, residual liquid is squeezed out. Thus, for constant feed, the flowrate through the medium will not be stable, but will fluctuate with time. The weight of dry solids in a cake is: W = x0 V
(29)
where Xo = weight of solids in the cake per unit filtrate volume. The concentration of solids in the feed sludge is expressed by weight fraction c. It is also possible to evaluate experimentally the weight ratio of wet cake to its dry content m. Hence, a unit weight of sludge contains mc of wet cake. We denote y as the
72
Liquid Filtration
specific weight of feed sludge. This quantity contains c amount of solids; hence, the ratio of the mass of solids in the cake to the filtrate volume is: cy x~ =
1 - me
(30)
Thus, from the sludge concentration c and the weight of wet cake per kg of dry cake solids m, Xo can be computed. If the suspension is dilute, then c is small; hence, product mc is small. This means that Xo will be approximately equal to c. According to Equations 29 and 30, the weight ratio of wet to dry cake will vary. Equation 30 shows also that because Xo depends on the product mc, at relatively moderate suspension concentrations this effect will not be great and can, therefore, be neglected. However, when filtering concentrated sludges the above will play some role; that is, at constant feed, the filtrate changes with time. Variable-Rate and -Pressure Filtration
The dynamics of variable-rate and -pressure filtrations can be illustrated by pressure profiles that exist across the filter medium. Figure 8 shows the graphical representation of those profiles. According to this plot, the compressed force in the cake section is: P = Pl-Pst
where
Pl ps
(31)
= pressure exerted on the sludge over the entire cake thickness = static pressure over the same section of cake
p corresponds to the local specific cake resistance (rw),. At the sludge-cake interface Pst= Pt and p = 0; and for the interface between the cake and filter plate Pst - Pst and P - Pl -P's,. P's, corresponds to the resistance of filter plate pf, and is expressed by: Apf
= I~Rf W
(32)
where W = rate of filtration (m3/m2-sec). Note that Apt. is constant during the operation. Pressure p is also the driving force of the process. Therefore, starting from the governing filtration equations, the general expression for an infinitesimal increment of solid particle weight in a cake of unit of area is xwdq (q - filtrate volume obtained from 1 m 2 filtering area, m3/m:). The responding increment dp may be expressed as:
C a k e Filtration and Filter Media Filtration
73
Figure 8. Distribution o f static pressure Pst in liquid and p along the cake thickness and filter plate: I, H-boundaries between the cake and sludge at r " and r ; III, IV-boundaries between cake layers or cake and filter plate at r Mand r ; V- boundary line between the cake and filter plate or free surface o f filter plate; 1,3-curves Pst =f(hoc) and p = f ( h oc) at ~ ; 2, 4-curves Pst=f(hoc) and p=f(hoc ) at r':
dq xwdP
- tx ( r w ),. W
(33)
Xw is not sensitive to changes in p. In practice, an average value for Xw can be assumed. Note that W is constant for any cross section of the cake. Hence, Equation 33 may he integrated over the cake thickness between the limits of p = 0 and P = P l - P ' s t , from q = 0 to q = q:
P l -P.~ir
q -
(34) Ixx wW
0
( r w ),-
Parameters q and W are variables when filtration conditions change. Coefficient (rw)x is a function of pressure: (rw)x = f ( P )
(35)
74
Liquid Filtration
Figure 9. Compression-permeability cell.
The exact relationship can be derived from experiments in a device called a compression-permeability cell which is illustrated in Figure 9. Once this relationship is defined, the integral of the right side of Equation 34 may be evaluated analytically (or if the relationship is in the form of a curve, the evaluation may be made graphically). The interrelation between W and P~ is established by the pump characteristics, which define q = f(W) in Equation 34. Filtration time may then be determined from the following definition:
dq _ W
(36)
r. = I d q W o
(37)
da:
Hence,
75
Cake Filtration and Filter Media Filtration
Constant-Pressure and -Rate Filtration This mode of operation is achieved when a pure liquid is filtered through a cake of constant thickness at a constant pressure difference. Cake washing by displacement when the washing liquid is located over the cake may be considered to be filtration of washing liquid through a constant cake thickness at constant pressure and flowrate. The rate of washing is related to the rate of filtration during the last stages and may be expressed by Equation 9, where Ap is the pressure at the final moment and V is the filtrate volume obtained during filtration, regardless of the filtration method used (i.e., constant-pressure or constant-rate operation). In the final stages, filtration usually is performed under constant pressure. Then, the rate of this process may be calculated from Equation 19. From filtration constants C and K, at constant pressure for a given system, the filtration rate for the last period is determined. If the washing liquid passes through the filter in the same pore paths as the sludge and filtrate, then the difference between the washing rate and filtration rate for this last period will be mostly due to a difference in the viscosities of the wash liquor and filtrate. Therefore, Equation 19 is applicable using the viscosity of the washing liquid, ~w. Denoting the rate of filtration in the last period as (dV/dz), the washing rate is:
i,v)
(38)
Filter-Medium Filtration Formulas Solid particles undergoing separation from the mother suspension may be captured both on the surface of the filter medimn and within the inner pore passages. This phenomenon is typical in the separation of low-concentration suspensions, where the suspension consists of viscous liquids such as sugar liquors, textile solutions or transformer oils, with fine particles dispersed throughout. The penetration of solid particles into the filter medium increases the flow resistance until eventually the filtration cycle can no longer proceed at practical throughputs and the medium must be replaced. In this section standard filtration formulas are provided along with discussions aimed at providing a working knowledge of the filter-medium filtration process.
Constant-Pressure-Drop Filtration Constant-pressure drop filtration can result in saturation or blockage of the filter medium. The network of pores within the filter medium can become blocked because of one or a combination of the following situations: 1. Poreslnay become blocked by the lodging of single particles in the pore passage. 2. Gradualblockage can occur due to the accumulation of many particles in pore passages. 3. Blockagemay occur during intermediate-typefiltration.
76
Liquid Filtration
Proper filter medium selection is based on understanding these mechanisms and analyzing the impact each has on the filtration process. In the case of single-particle blockage, we first consider a 1 m 2 surface of filter medium containing Np number of pores. The average pore radius and length are rp and Qp, respectively. For laminar flow, the Hagen-Poiseuille equation may be applied to calculate the volume of filtrate V' passing through a pore in a unit of time"
8/~Op
(39)
Consequently, the initial filtration rate per unit area of filtration is"
Wi, ' : V ' N
(40)
Consider lm 3 of suspension containing n number of suspended particles. If the suspension concentration is low, we may assume the volume of suspension and filtrate to be the same. Hence, after recovering a volume q of filtrate, the number of blocked pores will be nq, and the number unblocked will be (Np - nq). Then the rate of filtration is" W
= V' ( N p - n q )
(41)
or
W = Wi, ' - k ' q
(42)
where k'=
V'n
(43)
k' is a constant having units of sec -l. It characterizes the decrease in intensity of the filtration rate as a function of the filtrate volume. For constant V', this decrease depends only on the particle number n per unit volume of suspension. The total resistance R may be characterized by the reciprocal of the filtration rate. Thus, W in Equation 42 may be replaced by 1/R (sec/m). Taking the derivative of the modified version of Equation 42 with respect to q, we obtain:
Cake Filtration and Filter Media Filtration
dR
k
dq
( W.,, - k ' q ) 2
77
l
(44)
Comparison with Equation 42 reveals: e W2
(45)
2
(46)
l
dR
m
dq
or
dR
aq
-
k'R
Equation 46 states that when complete pore blockage occurs, the intensity of the increase in the total resistance with increasing filtrate volume is proportional to the square of the flow resistance. In the case of multiparticle blockage, as the suspension flows through the medium, the capillary walls of the pores are gradually covered by a uniform layer of particles. This particle layer continues to build up due to mechanical impaction, particle interception and physical adsorption of particles. As the process continues, the available flow area of the pores decreases. Denoting x o as the ratio of accumulated cake on the inside pore walls to the volume of filtrate recovered, and applying the Hagen-Poiseuille equation, the rate of filtration (per unit area of filter medium) at the start of the process is: WI.,, = B N r p
4
(47)
where
B-
~Ap
8~
(48)
When the average pore radius decreases to r, the rate of filtration becomes: W - BN
p
r4
(49)
For a finite filtrate quantity, dq the amount of cake inside the pores is xodq, and the cake thickness is dr. That is:
78
Liquid Filtration
xo dq = -Np 2 = rQpdr
(50)
Note that the negative sign indicates that as q increases, the pore radius r decreases. Integrating this expression over the limits of 0 to q, for rp to r we obtain:
q
NprCQp -
Xo
2 (rp
-
r 2)
(51)
And from Equations 47 and 49, we may define the pore radii as follows: 1/2
(52)
or simply:
-
(53)
Substituting these quantities into Equation 51 and simplifying terms, we obtain: W = [(W/,,)1/2 _ Cq ]2
(54)
where
X~ c
=
1/2
rCQp
(55)
It is convenient to define the following constant: K=
2C (w/,,) 1/2
(56)
From which Equation 54 may be restated as:
W = Win ( 1 - 1/2Kq)2
(57)
Cake Filtration and Filter Media Filtration
79
Since W = l/R, we may write: R Win ( 1 - 1 / 2 K q ) 2
(58)
The derivative of this expression with respect to q is: dR
k
dq
Win ( 1 - 1 / 2 K q ) 3
(59)
On some rearranging of terms, we obtain: dR
aq
- K(
Win)l/2R 3/2
(60)
K"R
(61)
or
dR
-
3/2
@
where K " = K(Wi,,) 1/2
(62)
Equation 61 states that the intensity of increase in total resistance with increasing filtrate amount is proportional to resistance to the 3/2 power. In this case, the total resistance increases less sensitively than in the case of total pore blockage. As follows from Equations 56 and 62: K" = 2C
(63)
Substituting Equation 55 for C and using Equation 48 for B, the above expression becomes:
x":e(wi")"~I ~r~X~
(64)
Note that for c o n s t a n t Win , parameter K " is proportional to the ratio of the settled volume of cake in the pores to the filtrate volume obtained, and is inversely proportional to total pore volume for a unit area of filter medium.
80
Liquid Filtration
Replacing W by dq/dt in Equation 57, we obtain" 1 (
1)
-2
aq
d'r. = Wi,, 1 - - K q 2
(65)
Integration of this equation over the limits from 0 to z for 0 to q we obtain" 2q W~,,(2 - K q )
=
(66)
and on simplification: K
1:
1
2
q
W/,I
Equation 67 may be used to evaluate constants K (m-1) and
(67)
Win.
Finally, for the case of intermediate filtration, the intensity of increase 'in total resistance with increasing filtrate volume is less than that occurring in the case of gradual pore blocking, but greater than that occurring with cake filtration. It may be assumed that the intensity of increase in total resistance is directly proportional to this resistance: dR
aq
-
K'"R
(68)
Integration of this expression between the limits of 0 to q, from R,. to R gives"
e
RI
-
e
K'"q
(69)
Substituting 1/W for R and 1/Win for Rf, the last expression becomes
Wilt W
_ eK'"q
(70)
or W
= Wine -K'"q
(71)
Cake Filtration and Filter Media Filtration
81
Substituting dq/dz for W in Equation 71 and integrating over the limits of 0 to between 0 and q we obtain" 1: -
1
e K'''q- 1
W/.
K"'
(72)
Hence, K ' "'c -
e x'''q W.,,
1 Wi, '
(73)
Accounting for Equation 70, the final form of this expression becomes: 1
1
-
W
+ K'"'r
Wi, ,
(74)
Filtration Mechanisms
To compare the different mechanisms of filtration, the governing equation of filtration must be rearranged. The starting expression is: dV
_
Ap
A cl~
tx [ roX o ( V/A ) + R r ]
(75)
Replacing V by q, and denoting the actual filtration rate (dq/d'c) as W, the governing filtration equation may be rewritten for a unit area of filtration as follows: @ W
__
I~(roXoq + Rf)
(76)
At the initial moment when q = 0, the filtration rate is Wi n _
Ap
(77)
From Equations 76 and 77 we have"
Win W
.._.
1 + K"'Win q
(78)
82
Liquid Filtration
where
#roXo Ap
K....
(79)
The numerator of Equation 79 characterizes the cake resistance. The denominator contains information on the driving force of the operation. Constant K " ' (sec/m 2) characterizes tile intensity at which the filtration rate decreases as a function of increasing filtrate volume. Substituting 1/R for W in Equation 78 and taking the derivative with respect to q, we obtain:
dR
- K"'
dq
(80)
The expression states that the intensity of increase in total resistance for cake filtration is constant with increasing filtrate volume. Replacing W by dq/dz in Equation 78 and integrating over the limits of 0 to q between 0 and ~: we obtain: K'"
__i_
q
-
1: q
_
1 Wi n
(81)
Note that this expression reduces to Equation 74 on substituting expressions for (Equation 77) and K ' " (Equation 79).
Win
Examination of Equations 46, 61, 68 and 80 reveals that the intensity of increase in total resistance with increasing filtrate volume decreases as the filtration process proceeds from total to gradual pore blocking, to intermediate type filtration and finally to cake filtration. Total resistance consists of a portion contributed by the filter medium plus any additional resistance. The source of the additional resistance is established by the type of filtration. For total pore blockage filtration, it is established by solids plugging the pores; during gradual pore blockage filtration, by solid particles retained in pores; and during cake filtration, by particles retained on the surface of the filter medium. The governing equations (Equations 42, 67, 74 and 81) describing the filtration mechanisms are expressed as linear relationships with parameters conveniently grouped into constants that are functions of the specific operating conditions. The exact form of the linear functional relationships depends on the filtration mechanism. Table 1 lists the coordinate systems that will provide linear plots of filtration data depending on the controlling mechanism. In evaluating the process mechanism (assuming that one dominates) filtration data may be massaged graphically to ascertain the most appropriate linear fit and, hence, the
Cake Filtration and Filter Media Filtration
83
type of filtration mechanism controlling the process, according to Table 1. If, for example, a linear regression of the filtration data shows that q = f(z/q) is the best linear correlation, then cake filtration is the controlling mechanism. The four basic equations are by no means the only relationships that describe the filtration mechanisms. Table 1. Coordinates for representing linear filtration relationships.
Type of Filtration
Equation
Coordinates
With Total Pore Blocking
42
q vs W
With Gradual Pore Blocking
67
1: vs l:/q
Intermediate
74
1: vs 1/W
Cake
81
q vs z/q
All the mechanisms of filtration encountered in practice have the functional form: dR
- KR ~
clq
(82)
where b typically varies between 0 and 2. Constant Rate Filtration
Filtration with gradual pore blocking is most frequently encountered in industrial practice. This process is typically studied under the operating mode of constant rate. We shall assume a unit area of medium which has Np pores, whose average radius and length are rp mad ~?p,respectively. The pore walls have a uniform layer of particles that build up with time and decrease the pore passage flow area. Filtration must be performed in this case with an increasing pressure difference to compensate for the rise in flow resistance due to pore blockage. If the pores are blocked by a compressible cake, a gradual decrease in porosity occurs, accompanied by an increase in the specific resistance of the deposited particles and a decrease in the ratio of caketo-filtrate volumes. The influence of particle compressibility on the controlling mechanism may be neglected. The reason for this is that the liquid phase primarily flows through the available flow area in the pores, bypassing deposited solids. Thus, the ratio of cake volume to filtrate volume (Xo) is not sensitive to the pressure difference even for highly compressible cakes. From the Hagen-Poiseuille relation (Equation 39) replacing Win in Equation 40 with constant filtration rate W and substituting APin for constant pressure drop AP we obtain: ,
4
W = B meinNprp
(83)
84
Liquid Filtration
where B' -
71;
(84)
8~[~p
The mass of particles deposited on the pore walls will be xod q, and the thickness of this particle layer in each pore is dr. Hence
Xo d q = -Np 2r~rQpdX
(85)
Integration over the limits of 0, q from rp to r yields
q
NprtOp -
2 (rp
2) -
(86)
r
xO
Radii rp and r are defined by Equations 83 and 85, respectively, from which we obtain the following expressions:
q
-/w)
m
xo
B' APinN p
B' ApNp
(87)
or
q
._.
(88)
.
xo
B'
Since q = Wz, Equation 88 may be stated in a reduced form as" (89)
where C =
xo
n Qp
Np
(90)
A plot of Equation 89 on the coordinate of z vs (1/Apin) '/2 - (1/Apt/2 results in a straight line, passing through the origin, with a slope equal to C. Thus, if experimental data correlate using such coordinates, the process is gradual pore blocking. Note that at z = 0, Ap = Apin, which is in agreement with typical process observations.
85
Cake Filtration and Filter Media Filtration
The filtration time corresponding to total pore blockage, when zXp--+oo may be estimated from:
1 ( 1 )1/2
z = --
(91)
To express the relationship between AP and 1: more directly. Equation 89 is restated in the form: Ap =
(a
1 -
Cq;) 2
(92)
where 1
A--
/ 1/2 (93)
It is important to note that pore blocking occurs when suspensions have the following characteristics" 1. relatively small particles" 2. high viscosity; and 3. low solids concentrations.
Both particle size and the liquid viscosity affect the rate of particle settling. The rate of settling due to gravitational force decreases with decreasing particle size and increasing viscosity. The process mechanisms are sensitive to the relative rates of filtration and gravity sedimentation. Examination of the manner in which particles accunmlate onto a horizontal filter medium assists in understanding the influences that the particle settling velocity and particle concentration have on the controlling mechanisms. The separation process through a cross section of filter medium is illustrated in Figure 10. "Dead zones" exist on the filter medium surface between adjacent pores. In these zones, particle settling onto the medium surface prevails. After sufficient particle accumulation, solids begin to move under the influence of fluid jets in the direction of pore entrances. This leads to favorable conditions for bridging. The conditions for bridge formation become more favorable as the ratio of particle settling to filtration rate increases. An increase in the suspension's particle concentration also enhances accumulation in "dead zones" with subsequent bridging. Hence, both high particle settling velocity increases and higher solids concentrations create favorable conditions for cake filtration. In contrast, low settling velocity and concentration results in favorable conditions for gradual pore blocking.
86
Liquid Filtration
The transition from pore-blocked filtration to more favorable cake filtration can therefore be achieved with a suspension of low settling particles by initially feeding it to the filter medium at a low rate for a time period sufficient to allow surface accumulation. This is essentially the practice that is performed with filter aids.
Figure 10. Suspension flow downward onto a filter medium. An initial accumulation of solids occurs around the pott entrance followed by particle bridging.
Suggested Readings 1. Cheremisinoff, P.N., Wastewater Treatment Pocket Handbook, Pudvan Publishing, Northbrook, IL, 1987 2. Cheremisinoff, P.N., Pocket Handbook for Solid-Liquid Separations, Gulf Publishing Co., Houston, TX, 1984 3. Noyes, R., Unit Operations in Environmental Engineering, Noyes Publishers, NJ, 1994 4. Kirkpatrick, J., Mathematics for Water and Wastewater Treatment Plant Operators, Ann Arbor Science Pub., Ann Arbor, MI, 1976
C a k e Filtration and Filter Media Filtration
,
87
Environmental Law Institute, Clean Water Deskbook, Environmental Law Reports, Washington, DC, 1988.
Nomenclature A
B,B' C
= =
C Fs hc
=
area (m 2) empirical parameters filtration parameter concentration (kg/m 3) force (N) cake height (m)
K,K" ,K'" L
=
Qp
=
n
=
Np
=
p q
= =
r
~-
ro rp
=
rw
--
R Rc,Rf =
U
--
V W
=
X0
filtration constants cake thickness (m) pore length (m) number of suspended particles number of pores pressure (N/m 2) filtrate volumer per unit area of filter (mB/m3) or volume (m 3) specific resistance (m -l) specific volumetric cake resistance (kg/m 2) pore radius (m) specific mass cake resistance (kg/m 2) resistance (m/sec) cake and filter resistances, respectively (m -~) average velocity (m/sec) filtrate volume (m 3) mass of dry solids (kg), or rate of filtration (m3/m2-sec) ratio of cake to filtrate volume.
Greek Symbols C
/x II
= =
T
=
TO
--
porosity viscosity (P) ratio of filtration rate to gravity setting time (sec) time constant (sec)
filtrate
Cake filtration is the most common form of filtering employed by the chemical and process industries. This manifests into handling the permeability of a bed of porous material, the schematic of which is shown in Figure 1. With high-solids-concentration suspensions, even relatively small particles (in comparison to the pore size) will not pass through the medium, but tend to remain on the filter surface, forming "bridges" over individual openings in the filter material. Filtrate flows through the filter medium and cake because of an applied pressure, the magnitude of which is proportional to the filtration resistance. This resistance results from the frictional drag on the liquid as it passes through the filter and cake. Hydrostatic pressure varies from a maxinmm at the point where liquid enters the cake, to zero where liquid is expelled from the medium; consequently, at any point in the cake the two are complementary. That is, the sum of the hydrostatic and compression pressures on the solids always equals the total hydrostatic pressure at the face of the cake. Thus, the compression pressure acting on the solids varies from zero at the face of the cake to a maximum at the filter medium. When solid particles undergo separation from the mother suspension, they are captured both on the surface of the filter medium and within the inner pore passages. The penetration of solid particles into the filter medium increases the flow resistance until the filtration cycle can no longer continue at economical throughput rates, at which time the medium itself must be replaced. This chapter provides a summary of standard calculation methods for assessing cake formation, behavior, and the overall efficiency of the filter-medium filtration process. 59
60
Liquid Filtration
Dynamics of Cake Filtration When the space above the suspension is subjected to a source of compressed gas (e.g., air) or the space under the filter plate is connected to a vacuum source, filtration is accomplished under a constant pressure differential (the pressure in the receivers is constant). In this case, the rate of the process decreases due to an increase in the cake thickness and, consequently, flow resistance. A similar filtration process results from a pressure difference due to the hydrostatic pressure of a suspension layer of constant thickness located over the filter medium. If the suspension is fed to the filter with a reciprocating pump at constant capacity, filtration is performed under constant flowrate. In this case, the pressure differential increases due to an increase in the cake resistance. If the suspension is fed by a centrifugal pump, its capacity decreases with an increase in cake resistance, and filtration is performed at variable pressure differentials and flowrates. The most favorable filtration operation with cake formation is a process whereby no clogging of the filter medium occurs. Such a process is observed at sufficiently high concentrations of solid particles in suspension. From a practical standpoint this concentration may conditionally be assumed to be in excess of 1% by volume.
Figure 1. Operating scheme of a filtration process: I-filter; 2-filter medium; 3-suspension; 4-filtrate; 5-cake. To prevent pore clogging in the filter medium when handling relatively low solids concentrations (e.g., O. 1-1% by volume), general practice is to increase the solids concentration in thickeners before the suspension is fed to the filter. Filtration is frequently accompanied by hindered or free gravitational settling of solid particles. The relative directions of action between gravity force and filtrate motion
Cake Filtration and Filter Media Filtration
61
may be concurrent, countercurrent or crosscurrent, depending on the orientation of the filter plate, as well as the sludge location above or below the filter plate. The different orientations of gravity force and filtrate motion with their corresponding distribution of cake, suspension, filtrate and clear liquid are illustrated in Figure 2. Particle sedimentation complicates the filtration process and influences the controlling mechanisms. Furthermore, these influences vary depending on the relative directions of gravity force and filtrate motion. If the suspension is above the filter medium (Figure 2A), particle settling leads to more rapid cake formation with a clear filtrate, which can be evacuated from the filter by decanting. If the suspension is under the filter medium (Figure 2B), particle settling will prevent cake formation, and it is necessary to mix the suspension to maintain homogeneity.
Figure 2. Direction of gravity force action and filtrate motion in filters: A-cocurrent; Bcountercurrent; C-crosscurrent; solid arrow-direction of gravity force action; dashed arrowdirection of filtrate motion; 1-filter plate; 2-cake; 3-sludge; 4-filtrate; 5-clear liquid. When the cake structure is composed of particles that are readily deformed or become rearranged under pressure, the resulting cake is characterized as being compressible. Those that are not readily deformed are referred to as semicompressible, and those that deform only slightly are considered incompressible. Porosity (defined as the ratio of pore volume to the volume of cake) does not decrease with increasing pressure drop. The porosity of a compressible cake decreases under pressure, and its hydraulic resistance to the flow of the liquid phase increases with an increase in the pressure differential across the filter media. Cakes containing particles of inorganic substances with sizes in excess of 100 ~m may be considered incompressible, for all practical purposes. Examples of incompressible cake-forming materials are sand and crystals of carbonates of calcium and sodium. The cakes containing particles of metal hydroxides, such as ferric hydroxide, cupric hydroxide, aluminum hydroxide, and sediments consisting of easy deforming aggregates, which are formed from primary fine crystals, are usually compressible.
62
Liquid Filtration
At the completion of cake formation, treatment of the cake depends on the specific filtration objectives. For example, the cake itself may have no value, whereas the filtrate may. Depending on the disposal method and the properties of the particulates, the cake may be discarded in a dry form, or as a slurry. In both cases, the cake is usually subjected to washing, either immediately after its formation, or after a period of drying. In some cases, a second washing is required, followed by a drying period where all possible filtrate must be removed from the cake; or where wet discharge is followed by disposal: or where repulping and a second filtration occurs; or where dry cake disposal is preferable. Similar treatment options are employed in cases where the cake is valuable and all contaminating liquors must be removed, or where both cake and filtrate are valuable. In the latter, cake-forming filtration is employed, without washing, to dewater cakes where a valueless, noncontaminating liquor forms the residual suspension in the cake. To understand the dynamics of the filtration process, a conceptual analysis is applied in two parts. The first half considers the mechanism of flow within the cake, while the second examines the external conditions imposed on the cake and pumping system, which brings the results of the analysis of internal flow in accordance with the externally imposed conditions throughout. The characteristics of the pump relate the applied pressure on the cake to the flowrate at the exit face of the filter medium. The cake resistance determines the pressure drop. During filtration, liquid flows through the porous filter cake in the direction of decreasing hydraulic pressure gradient. The porosity (e) is at a minimum at the point of contact between the cake and filter plate (i.e., where x = 0) and at a maximum at the cake surface (x = L) where sludge enters. A schematic definition of this system is illustrated in Figure 3.
Figure 3. Important parameters in cake formation.
The drag that is imposed on each particle is transmitted to adjacent particles. Therefore, the net solid compressive pressure increases as the filter plate is approached, resulting in a decrease in porosity. Referring to Figure 4A, it may be
Cake Filtration and Filter Media Filtration
63
assumed that particles are in contact at one point only on their surface, and that liquid completely surrounds each particle. Hence, the liquid pressure acts uniformly in a direction along a plane perpendicular to the direction of flow. As the liquid flows past each particle, the integral of the normal component of force leads to form drag, and the integration of the tangential components results in frictional drag. If the particles are non-spherical, we may still assume single-point contacts between adjacent particles as shown in Figure 4B. Now consider flow through a cake (Figure 4C) with the membrane located at a distance x from the filter plate. Neglecting all forces in the cake other than those created by drag and hydraulic pressure, a force balance from x to L gives:
F +ApL : Ap
(1)
The applied pressure p is a function of time but not of distance x. F s is the cumulative drag on the particles, increasing in the direction from x = L to x = 0. Since single point contact is assumed, the hydraulic pressure PL is effectively over the entire cross section (A) of the cake; for example, against the fictitious membrane shown in Figure 4B. Dividing Equation 1 by A and denoting the compressive drag pressure by ps = F/A, we obtain: (2)
Figure 4. Frictional drag on particles in compressible cakes.
The term Ps is a fictitious pressure, because the cross-sectional area A is not equal to either the surface area of the particles nor the actual contact areas In actual cakes, there is a small area of contact Ac whereby the pressure exerted on the solids may be defined as Fs/Ac. Taking differentials with respect to x, in the interior of the cake, we obtain:
dPs + dPc = 0
(3)
64
Liquid Filtration
This expression implies that drag pressure increases and hydraulic pressure decreases as fluid moves from the cake's outer surface toward the filter plate. From Darcy's law, the hydraulic pressure gradient is linear through the cake if the porosity (c) and specific resistance (tO are constant. The cake may then be considered incompressible. This is illustrated by the straight line obtained from a plot of flowrate per unit filter area versus pressure drop shown in Figure 5. The variations in porosity and specific resistance are accompanied by varying degrees of compressibility, also shown in Figure 5. As noted in Chapter 1, filtration is primarily an application of fluid mechanics; that is, filtrate flow is induced through a porous filter cake and filter medium. The rate of the filtration process is directly proportional to the driving force and inversely proportional to the resistance.
Figure 5.
Flowrate/area versus pressure drop across the cake.
Because pore sizes in the cake and filter medium are small, and the liquid velocity through the pores is low, the filtrate flow may be considered laminar: hence, Poiseuille's law is applicable. Filtration rate is directly proportional to the difference in pressure and inversely proportional to the fluid viscosity and to the hydraulic resistance of the cake and filter medium. Because the pressure and hydraulic resistances of the cake and filter medium change with time, the variable rate of filtration may be expressed as: u -
dV Adz
(4)
Cake Filtration and Filter Media Filtration
where
65
V = volume of filtrate (m 3) A = filtration area (m 2) 1: = time of filtration (sec)
Assuming laminar flow through the filter channels, the basic equation of filtration as obtained from a force balance is:
1 dV u
where
-
A d~.
Ap -
I~(R c + Rf)
(5)
= pressure difference (N/m z) = viscosity of filtrate (N-sec/m 2) = filter cake resistance (m -1) = initial filter resistance (resistance of filter plate and filter channels) (m -1) = filtration rate (m/sec), i.e., filtrate flow through cake and filter plate filtration rate (m3/sec), i.e., filtrate flow rate dV/dq: = Ap /x Rc Rf
Filter cake resistance (Rc) is the resistance to filtrate flow per unit area of filtration. Rc increases with increasing cake thickness during filtration. At any instant, Rc depends on the mass of solids deposited on the filter plate as a result of the passage of V (m 3) filtrate. Rf may be assumed a constant. To determine the relationship between volume and residence time "c, Equation 5 must be integrated, which means that R c must be expressed in terms of V. We denote the ratio of cake volume to filtrate volume as x0. Hence, the cake volume is x0V. An alternative expression for the cake volume is hcA; where hc is the cake height in meters. Consequently
x ov
= hcA
(6)
Hence, the thickness of the cake, uniformly distributed over the filter plate, is" V
h c = x o -A
(7)
The filter cake resistance may be expressed as: V
R c = r o X o -~
where r 0-- specific volumetric cake resistance (m-2).
(8)
66
Liquid Filtration
As follows from Equation 8, ro characterizes the resistance to liquid flow by a cake having a thickness of 1 m. Substituting for Rc from Equation 8 into Equation 5, we obtain:
1 dV
Ap --
U
A dz
=
#[ roXo ( V/A ) + Rf]
(9)
Filtrate volume, Xo, can be expressed in terms of the ratio of the mass of solid particles settled on the filter plate to the filtrate volume (Xw) and instead of ro, a specific mass cake resistance r w is used. That is, r w represents the resistance to flow created by a uniformly distributed cake, in the amount of 1 kg/m 2. Replacing units of volume by mass, the term ro Xo in Equation 9 changes to rwXw. Neglecting filter plate resistance (Rf -" 0 ) , and taking into account Equation 7, we obtain from Equation 3 the following expression"
@ ro-
tzhc u
(10)
At/z = 1 N-sec/m 2, hc = 1 m and u = 1 m/sec, ro = Ap. Thus, the specific cake resistance equals the pressure difference required by the liquid phase (with a viscosity of 1 N-sec/m 2) to be filtered at a linear velocity of 1 m/sec through a cake 1 m thick. This hypothetical pressure difference, however, is beyond a practical range. For highly compressible cakes, r0 can exceed 1012m 2. Assuming V = 0 (at the start of filtration) where there is no cake over the filter plate. Equation 9 becomes: Rf-
@
(11)
#u
At /z = 1 N-sec/m 2 and u = 1 m/sec, Rf = Ap. This means that the filter plate resistance is equal to the pressure difference necessary for the liquid phase (with viscosity of 1 N-sec/m 2) to pass through the filter plate at a rate of 1 m/sec. For many filter plates R e is typically 10 I~ m -~ . For a constant pressure drop and temperature filtration process all the parameters in Equation 9, except V and "r, are constant. Integrating Equation 9 over the limits of 0 to V, from 0 to ~:, we obtain v
f 0
or
v
#( roXo ~ + Rf ) d V :
ApAdr, 0
(12a)
Cake Filtration and Filter Media Filtration
67
g 2
tXoroXo ~
(12b)
+ ~ R f V = A p A r.
Dividing both sides by/xroXo/2A gives:
V 2+2
roX~
V =
2
. I~ roX o
-c
(13)
Equation 13 is the relationship between filtration time and filtrate volume. The expression is applicable to either incompressible or compressible cakes, since at constant Ap, r0 and Xo are constant. If we assume a definite filtering apparatus and set up a constant temperature and filtration pressure, then the values of Rf, ro,/x and Ap will be constant. The terms in parentheses in Equation 13 are known as the "filtration constants", and are often lumped together as parameters K and C; where: 2ApA 2 K
__
l~ roX o
(14)
RA
C-
f roxo
(15)
Hence, a simplified expression may be written to describe the filtration process as follows: V 2 + 2 V C = KI:
(16)
Filtration constants K and C can be experimentally determined, from which the volume of filtrate obtained over a specified time interval (for a certain filter, at the same pressure and temperature) can be computed. If process parameters are changed, new constants K and C can be estimated from Equations 14 and 15. Equation 16 may be further simplified by denoting ~:0 as a constant that depends on K and C:
68
Liquid Filtration
C 2
a:0
-
K
(17)
Substituting t0 into Equation 16, the equation of filtration under constant pressure conditions is: (V+C)
2 =
K(I: + t o )
(18)
Equation 18 defines a parabolic relationship between filtrate volume and time. The expression is valid for any type of cake (i.e., compressible and incompressible). From a plot of V + C versus (z + %), the filtration process may be represented by a parabola with its apex at the origin as illustrated in Figure 6. Moving the axes to distances C and % provides the characteristic filtration curve for the system in terms of volume versus time. Because the parabola's apex is not located at the origin of this new system, it is clear why the filtration rate at the beginning of the process will have a finite value, which corresponds to actual practice.
Figure 6. Typical filtration curve.
Constants C and z o in Equation 18 have physical interpretations. They are basically equivalent to a fictitious layer of cake having equal resistance. The formation of this fictitious cake follows the same parabolic relationship, where t o denotes the time required for the formation of this fictitious mass, and C is the volume of filtrate required. Differentiating Equation 16 gives:
69
Cake Filtration and Filter Media Filtration
dV
K
dl:
2( V+ C)
(19)
And rearranging in the form of a reciprocal relationship: d'c
--
dV
2V
+
K
2C
(20)
K
This form of the equation provides a linear relation as shown by the plot in Figure 7. The expression is that of a straight line having slope 2/K, with intercept C. The experimental determination of d~:/dV is made simple by the functional form of this expression. Filtrate volumes Vl and V2 should be measured for time intervals q:, and 1:2. Then, according to Equation 16:
172- 171
1
V2 - V 2
Vz - Vl
V2- Vl
K
2 c(g 2 -
VI)
K (21)
Vl+V2) 2
2 k
+ 2C K
In examining the right side of this expression, we note that the quotient is equal to the inverse value of the rate at the moment of obtaining the filtrate volume, which is equal to the mean arithmetic value of volumes V1 and V2:
172-'I71 = { dg / V2_V1 ~ Vl+V2 2
(22)
Filtration constants C and K can be determined on the basis of several measurements of filtrate volumes for different time intervals. As follows from Equations 14 and 15, values of C and K depend on r0 (specific volumetric cake resistance), which in turn depends on the pressure drop across the cake. This Ap, especially during the initial stages of filtration, undergoes changes in the cake. When the cake is very thin, the main portion of the total pressure drop is exerted on the filter medium. As the cake becomes thicker, the pressure drop through the cake increases rapidly but then levels off to a constant value. Isobaric filtration shows insignificant deviation from Equation 16. For approximate calculations, it is
70
Liquid Filtration
Figure 7. Plot o f Equation 20.
possible to neglect the resistance of the filter plate, provided the cake is not too thin. Then the filter plate resistance Rf = 0 in Equation 15, C = 0 (Equation 15) and % = 0 (Equation 17). Therefore, the simplified equation of filtration takes the following form:
V 2 = K~
(23)
For thick cakes, Equation 23 gives results close to that of Equation 16.
Constant-Rate Filtration When sludge is fed to a filter by a positive-displacement ptunp, the rate of filtration is nearly constant (i.e., dV/dz = constant). During constant-rate filtration, the pressure increases wit1 an increase in cake thickness. Therefore, the principal variables are pressure and filtrate volume, or pressure and filtration time. Equation 9 is the principal design relation, which may be integrated for a constant-rate process. The derivative, dV/dl:, may be replaced simply by V/T: Ap
/troXo A 2z
(24)
The ratios in parentheses express the constant volume rate per unit filter area. Hence, Equation 24 is the relationship between time 1: and pressure drop Ap. For incompressible cakes, ro is constant and independent of pressure. For compressible cakes, the relationship between time and pressure at constant-rate filtration is:
Cake Filtration and Filter Media Filtration
A p = i~ axo AP s
(v)2 -~
71
(25)
"r. + I~
Filtration experiments are typically conducted in pilot scale equipment and generally tests are conducted either at constant pressure or constant rate to determine ax0, as well as s and R t-, for a given sludge and filter medium. Such tests provide empirical information that will enable the time required tor the pressure drop to reach the desired level for a specified set of operating conditions to be determined. In the initial stages of filtration, the filter medium has no cake. Furthermore, Ap is not zero, but has a value that is a function of the resistance of the medium for a given flowrate. This initial condition can be stated as: V
(26)
For an incompressible cake (where s = 0), Equation 25 takes the form:
ap- axo
(27)
As noted earlier, for thick cakes, the resistance of the filter medium may be neglected. Hence, for Rt-= 0, Equation 25 simplifies to: Apl-S = I~ax o
(v/2
~:
(28)
An increase in pressure influences not only coefficient r0, but the cake's porosity as well. Since the cake on the filter plate is compressed, residual liquid is squeezed out. Thus, for constant feed, the flowrate through the medium will not be stable, but will fluctuate with time. The weight of dry solids in a cake is: W = x0 V
(29)
where Xo = weight of solids in the cake per unit filtrate volume. The concentration of solids in the feed sludge is expressed by weight fraction c. It is also possible to evaluate experimentally the weight ratio of wet cake to its dry content m. Hence, a unit weight of sludge contains mc of wet cake. We denote y as the
72
Liquid Filtration
specific weight of feed sludge. This quantity contains c amount of solids; hence, the ratio of the mass of solids in the cake to the filtrate volume is: cy x~ =
1 - me
(30)
Thus, from the sludge concentration c and the weight of wet cake per kg of dry cake solids m, Xo can be computed. If the suspension is dilute, then c is small; hence, product mc is small. This means that Xo will be approximately equal to c. According to Equations 29 and 30, the weight ratio of wet to dry cake will vary. Equation 30 shows also that because Xo depends on the product mc, at relatively moderate suspension concentrations this effect will not be great and can, therefore, be neglected. However, when filtering concentrated sludges the above will play some role; that is, at constant feed, the filtrate changes with time. Variable-Rate and -Pressure Filtration
The dynamics of variable-rate and -pressure filtrations can be illustrated by pressure profiles that exist across the filter medium. Figure 8 shows the graphical representation of those profiles. According to this plot, the compressed force in the cake section is: P = Pl-Pst
where
Pl ps
(31)
= pressure exerted on the sludge over the entire cake thickness = static pressure over the same section of cake
p corresponds to the local specific cake resistance (rw),. At the sludge-cake interface Pst= Pt and p = 0; and for the interface between the cake and filter plate Pst - Pst and P - Pl -P's,. P's, corresponds to the resistance of filter plate pf, and is expressed by: Apf
= I~Rf W
(32)
where W = rate of filtration (m3/m2-sec). Note that Apt. is constant during the operation. Pressure p is also the driving force of the process. Therefore, starting from the governing filtration equations, the general expression for an infinitesimal increment of solid particle weight in a cake of unit of area is xwdq (q - filtrate volume obtained from 1 m 2 filtering area, m3/m:). The responding increment dp may be expressed as:
C a k e Filtration and Filter Media Filtration
73
Figure 8. Distribution o f static pressure Pst in liquid and p along the cake thickness and filter plate: I, H-boundaries between the cake and sludge at r " and r ; III, IV-boundaries between cake layers or cake and filter plate at r Mand r ; V- boundary line between the cake and filter plate or free surface o f filter plate; 1,3-curves Pst =f(hoc) and p = f ( h oc) at ~ ; 2, 4-curves Pst=f(hoc) and p=f(hoc ) at r':
dq xwdP
- tx ( r w ),. W
(33)
Xw is not sensitive to changes in p. In practice, an average value for Xw can be assumed. Note that W is constant for any cross section of the cake. Hence, Equation 33 may he integrated over the cake thickness between the limits of p = 0 and P = P l - P ' s t , from q = 0 to q = q:
P l -P.~ir
q -
(34) Ixx wW
0
( r w ),-
Parameters q and W are variables when filtration conditions change. Coefficient (rw)x is a function of pressure: (rw)x = f ( P )
(35)
74
Liquid Filtration
Figure 9. Compression-permeability cell.
The exact relationship can be derived from experiments in a device called a compression-permeability cell which is illustrated in Figure 9. Once this relationship is defined, the integral of the right side of Equation 34 may be evaluated analytically (or if the relationship is in the form of a curve, the evaluation may be made graphically). The interrelation between W and P~ is established by the pump characteristics, which define q = f(W) in Equation 34. Filtration time may then be determined from the following definition:
dq _ W
(36)
r. = I d q W o
(37)
da:
Hence,
75
Cake Filtration and Filter Media Filtration
Constant-Pressure and -Rate Filtration This mode of operation is achieved when a pure liquid is filtered through a cake of constant thickness at a constant pressure difference. Cake washing by displacement when the washing liquid is located over the cake may be considered to be filtration of washing liquid through a constant cake thickness at constant pressure and flowrate. The rate of washing is related to the rate of filtration during the last stages and may be expressed by Equation 9, where Ap is the pressure at the final moment and V is the filtrate volume obtained during filtration, regardless of the filtration method used (i.e., constant-pressure or constant-rate operation). In the final stages, filtration usually is performed under constant pressure. Then, the rate of this process may be calculated from Equation 19. From filtration constants C and K, at constant pressure for a given system, the filtration rate for the last period is determined. If the washing liquid passes through the filter in the same pore paths as the sludge and filtrate, then the difference between the washing rate and filtration rate for this last period will be mostly due to a difference in the viscosities of the wash liquor and filtrate. Therefore, Equation 19 is applicable using the viscosity of the washing liquid, ~w. Denoting the rate of filtration in the last period as (dV/dz), the washing rate is:
i,v)
(38)
Filter-Medium Filtration Formulas Solid particles undergoing separation from the mother suspension may be captured both on the surface of the filter medimn and within the inner pore passages. This phenomenon is typical in the separation of low-concentration suspensions, where the suspension consists of viscous liquids such as sugar liquors, textile solutions or transformer oils, with fine particles dispersed throughout. The penetration of solid particles into the filter medium increases the flow resistance until eventually the filtration cycle can no longer proceed at practical throughputs and the medium must be replaced. In this section standard filtration formulas are provided along with discussions aimed at providing a working knowledge of the filter-medium filtration process.
Constant-Pressure-Drop Filtration Constant-pressure drop filtration can result in saturation or blockage of the filter medium. The network of pores within the filter medium can become blocked because of one or a combination of the following situations: 1. Poreslnay become blocked by the lodging of single particles in the pore passage. 2. Gradualblockage can occur due to the accumulation of many particles in pore passages. 3. Blockagemay occur during intermediate-typefiltration.
76
Liquid Filtration
Proper filter medium selection is based on understanding these mechanisms and analyzing the impact each has on the filtration process. In the case of single-particle blockage, we first consider a 1 m 2 surface of filter medium containing Np number of pores. The average pore radius and length are rp and Qp, respectively. For laminar flow, the Hagen-Poiseuille equation may be applied to calculate the volume of filtrate V' passing through a pore in a unit of time"
8/~Op
(39)
Consequently, the initial filtration rate per unit area of filtration is"
Wi, ' : V ' N
(40)
Consider lm 3 of suspension containing n number of suspended particles. If the suspension concentration is low, we may assume the volume of suspension and filtrate to be the same. Hence, after recovering a volume q of filtrate, the number of blocked pores will be nq, and the number unblocked will be (Np - nq). Then the rate of filtration is" W
= V' ( N p - n q )
(41)
or
W = Wi, ' - k ' q
(42)
where k'=
V'n
(43)
k' is a constant having units of sec -l. It characterizes the decrease in intensity of the filtration rate as a function of the filtrate volume. For constant V', this decrease depends only on the particle number n per unit volume of suspension. The total resistance R may be characterized by the reciprocal of the filtration rate. Thus, W in Equation 42 may be replaced by 1/R (sec/m). Taking the derivative of the modified version of Equation 42 with respect to q, we obtain:
Cake Filtration and Filter Media Filtration
dR
k
dq
( W.,, - k ' q ) 2
77
l
(44)
Comparison with Equation 42 reveals: e W2
(45)
2
(46)
l
dR
m
dq
or
dR
aq
-
k'R
Equation 46 states that when complete pore blockage occurs, the intensity of the increase in the total resistance with increasing filtrate volume is proportional to the square of the flow resistance. In the case of multiparticle blockage, as the suspension flows through the medium, the capillary walls of the pores are gradually covered by a uniform layer of particles. This particle layer continues to build up due to mechanical impaction, particle interception and physical adsorption of particles. As the process continues, the available flow area of the pores decreases. Denoting x o as the ratio of accumulated cake on the inside pore walls to the volume of filtrate recovered, and applying the Hagen-Poiseuille equation, the rate of filtration (per unit area of filter medium) at the start of the process is: WI.,, = B N r p
4
(47)
where
B-
~Ap
8~
(48)
When the average pore radius decreases to r, the rate of filtration becomes: W - BN
p
r4
(49)
For a finite filtrate quantity, dq the amount of cake inside the pores is xodq, and the cake thickness is dr. That is:
78
Liquid Filtration
xo dq = -Np 2 = rQpdr
(50)
Note that the negative sign indicates that as q increases, the pore radius r decreases. Integrating this expression over the limits of 0 to q, for rp to r we obtain:
q
NprCQp -
Xo
2 (rp
-
r 2)
(51)
And from Equations 47 and 49, we may define the pore radii as follows: 1/2
(52)
or simply:
-
(53)
Substituting these quantities into Equation 51 and simplifying terms, we obtain: W = [(W/,,)1/2 _ Cq ]2
(54)
where
X~ c
=
1/2
rCQp
(55)
It is convenient to define the following constant: K=
2C (w/,,) 1/2
(56)
From which Equation 54 may be restated as:
W = Win ( 1 - 1/2Kq)2
(57)
Cake Filtration and Filter Media Filtration
79
Since W = l/R, we may write: R Win ( 1 - 1 / 2 K q ) 2
(58)
The derivative of this expression with respect to q is: dR
k
dq
Win ( 1 - 1 / 2 K q ) 3
(59)
On some rearranging of terms, we obtain: dR
aq
- K(
Win)l/2R 3/2
(60)
K"R
(61)
or
dR
-
3/2
@
where K " = K(Wi,,) 1/2
(62)
Equation 61 states that the intensity of increase in total resistance with increasing filtrate amount is proportional to resistance to the 3/2 power. In this case, the total resistance increases less sensitively than in the case of total pore blockage. As follows from Equations 56 and 62: K" = 2C
(63)
Substituting Equation 55 for C and using Equation 48 for B, the above expression becomes:
x":e(wi")"~I ~r~X~
(64)
Note that for c o n s t a n t Win , parameter K " is proportional to the ratio of the settled volume of cake in the pores to the filtrate volume obtained, and is inversely proportional to total pore volume for a unit area of filter medium.
80
Liquid Filtration
Replacing W by dq/dt in Equation 57, we obtain" 1 (
1)
-2
aq
d'r. = Wi,, 1 - - K q 2
(65)
Integration of this equation over the limits from 0 to z for 0 to q we obtain" 2q W~,,(2 - K q )
=
(66)
and on simplification: K
1:
1
2
q
W/,I
Equation 67 may be used to evaluate constants K (m-1) and
(67)
Win.
Finally, for the case of intermediate filtration, the intensity of increase 'in total resistance with increasing filtrate volume is less than that occurring in the case of gradual pore blocking, but greater than that occurring with cake filtration. It may be assumed that the intensity of increase in total resistance is directly proportional to this resistance: dR
aq
-
K'"R
(68)
Integration of this expression between the limits of 0 to q, from R,. to R gives"
e
RI
-
e
K'"q
(69)
Substituting 1/W for R and 1/Win for Rf, the last expression becomes
Wilt W
_ eK'"q
(70)
or W
= Wine -K'"q
(71)
Cake Filtration and Filter Media Filtration
81
Substituting dq/dz for W in Equation 71 and integrating over the limits of 0 to between 0 and q we obtain" 1: -
1
e K'''q- 1
W/.
K"'
(72)
Hence, K ' "'c -
e x'''q W.,,
1 Wi, '
(73)
Accounting for Equation 70, the final form of this expression becomes: 1
1
-
W
+ K'"'r
Wi, ,
(74)
Filtration Mechanisms
To compare the different mechanisms of filtration, the governing equation of filtration must be rearranged. The starting expression is: dV
_
Ap
A cl~
tx [ roX o ( V/A ) + R r ]
(75)
Replacing V by q, and denoting the actual filtration rate (dq/d'c) as W, the governing filtration equation may be rewritten for a unit area of filtration as follows: @ W
__
I~(roXoq + Rf)
(76)
At the initial moment when q = 0, the filtration rate is Wi n _
Ap
(77)
From Equations 76 and 77 we have"
Win W
.._.
1 + K"'Win q
(78)
82
Liquid Filtration
where
#roXo Ap
K....
(79)
The numerator of Equation 79 characterizes the cake resistance. The denominator contains information on the driving force of the operation. Constant K " ' (sec/m 2) characterizes tile intensity at which the filtration rate decreases as a function of increasing filtrate volume. Substituting 1/R for W in Equation 78 and taking the derivative with respect to q, we obtain:
dR
- K"'
dq
(80)
The expression states that the intensity of increase in total resistance for cake filtration is constant with increasing filtrate volume. Replacing W by dq/dz in Equation 78 and integrating over the limits of 0 to q between 0 and ~: we obtain: K'"
__i_
q
-
1: q
_
1 Wi n
(81)
Note that this expression reduces to Equation 74 on substituting expressions for (Equation 77) and K ' " (Equation 79).
Win
Examination of Equations 46, 61, 68 and 80 reveals that the intensity of increase in total resistance with increasing filtrate volume decreases as the filtration process proceeds from total to gradual pore blocking, to intermediate type filtration and finally to cake filtration. Total resistance consists of a portion contributed by the filter medium plus any additional resistance. The source of the additional resistance is established by the type of filtration. For total pore blockage filtration, it is established by solids plugging the pores; during gradual pore blockage filtration, by solid particles retained in pores; and during cake filtration, by particles retained on the surface of the filter medium. The governing equations (Equations 42, 67, 74 and 81) describing the filtration mechanisms are expressed as linear relationships with parameters conveniently grouped into constants that are functions of the specific operating conditions. The exact form of the linear functional relationships depends on the filtration mechanism. Table 1 lists the coordinate systems that will provide linear plots of filtration data depending on the controlling mechanism. In evaluating the process mechanism (assuming that one dominates) filtration data may be massaged graphically to ascertain the most appropriate linear fit and, hence, the
Cake Filtration and Filter Media Filtration
83
type of filtration mechanism controlling the process, according to Table 1. If, for example, a linear regression of the filtration data shows that q = f(z/q) is the best linear correlation, then cake filtration is the controlling mechanism. The four basic equations are by no means the only relationships that describe the filtration mechanisms. Table 1. Coordinates for representing linear filtration relationships.
Type of Filtration
Equation
Coordinates
With Total Pore Blocking
42
q vs W
With Gradual Pore Blocking
67
1: vs l:/q
Intermediate
74
1: vs 1/W
Cake
81
q vs z/q
All the mechanisms of filtration encountered in practice have the functional form: dR
- KR ~
clq
(82)
where b typically varies between 0 and 2. Constant Rate Filtration
Filtration with gradual pore blocking is most frequently encountered in industrial practice. This process is typically studied under the operating mode of constant rate. We shall assume a unit area of medium which has Np pores, whose average radius and length are rp mad ~?p,respectively. The pore walls have a uniform layer of particles that build up with time and decrease the pore passage flow area. Filtration must be performed in this case with an increasing pressure difference to compensate for the rise in flow resistance due to pore blockage. If the pores are blocked by a compressible cake, a gradual decrease in porosity occurs, accompanied by an increase in the specific resistance of the deposited particles and a decrease in the ratio of caketo-filtrate volumes. The influence of particle compressibility on the controlling mechanism may be neglected. The reason for this is that the liquid phase primarily flows through the available flow area in the pores, bypassing deposited solids. Thus, the ratio of cake volume to filtrate volume (Xo) is not sensitive to the pressure difference even for highly compressible cakes. From the Hagen-Poiseuille relation (Equation 39) replacing Win in Equation 40 with constant filtration rate W and substituting APin for constant pressure drop AP we obtain: ,
4
W = B meinNprp
(83)
84
Liquid Filtration
where B' -
71;
(84)
8~[~p
The mass of particles deposited on the pore walls will be xod q, and the thickness of this particle layer in each pore is dr. Hence
Xo d q = -Np 2r~rQpdX
(85)
Integration over the limits of 0, q from rp to r yields
q
NprtOp -
2 (rp
2) -
(86)
r
xO
Radii rp and r are defined by Equations 83 and 85, respectively, from which we obtain the following expressions:
q
-/w)
m
xo
B' APinN p
B' ApNp
(87)
or
q
._.
(88)
.
xo
B'
Since q = Wz, Equation 88 may be stated in a reduced form as" (89)
where C =
xo
n Qp
Np
(90)
A plot of Equation 89 on the coordinate of z vs (1/Apin) '/2 - (1/Apt/2 results in a straight line, passing through the origin, with a slope equal to C. Thus, if experimental data correlate using such coordinates, the process is gradual pore blocking. Note that at z = 0, Ap = Apin, which is in agreement with typical process observations.
85
Cake Filtration and Filter Media Filtration
The filtration time corresponding to total pore blockage, when zXp--+oo may be estimated from:
1 ( 1 )1/2
z = --
(91)
To express the relationship between AP and 1: more directly. Equation 89 is restated in the form: Ap =
(a
1 -
Cq;) 2
(92)
where 1
A--
/ 1/2 (93)
It is important to note that pore blocking occurs when suspensions have the following characteristics" 1. relatively small particles" 2. high viscosity; and 3. low solids concentrations.
Both particle size and the liquid viscosity affect the rate of particle settling. The rate of settling due to gravitational force decreases with decreasing particle size and increasing viscosity. The process mechanisms are sensitive to the relative rates of filtration and gravity sedimentation. Examination of the manner in which particles accunmlate onto a horizontal filter medium assists in understanding the influences that the particle settling velocity and particle concentration have on the controlling mechanisms. The separation process through a cross section of filter medium is illustrated in Figure 10. "Dead zones" exist on the filter medium surface between adjacent pores. In these zones, particle settling onto the medium surface prevails. After sufficient particle accumulation, solids begin to move under the influence of fluid jets in the direction of pore entrances. This leads to favorable conditions for bridging. The conditions for bridge formation become more favorable as the ratio of particle settling to filtration rate increases. An increase in the suspension's particle concentration also enhances accumulation in "dead zones" with subsequent bridging. Hence, both high particle settling velocity increases and higher solids concentrations create favorable conditions for cake filtration. In contrast, low settling velocity and concentration results in favorable conditions for gradual pore blocking.
86
Liquid Filtration
The transition from pore-blocked filtration to more favorable cake filtration can therefore be achieved with a suspension of low settling particles by initially feeding it to the filter medium at a low rate for a time period sufficient to allow surface accumulation. This is essentially the practice that is performed with filter aids.
Figure 10. Suspension flow downward onto a filter medium. An initial accumulation of solids occurs around the pott entrance followed by particle bridging.
Suggested Readings 1. Cheremisinoff, P.N., Wastewater Treatment Pocket Handbook, Pudvan Publishing, Northbrook, IL, 1987 2. Cheremisinoff, P.N., Pocket Handbook for Solid-Liquid Separations, Gulf Publishing Co., Houston, TX, 1984 3. Noyes, R., Unit Operations in Environmental Engineering, Noyes Publishers, NJ, 1994 4. Kirkpatrick, J., Mathematics for Water and Wastewater Treatment Plant Operators, Ann Arbor Science Pub., Ann Arbor, MI, 1976
C a k e Filtration and Filter Media Filtration
,
87
Environmental Law Institute, Clean Water Deskbook, Environmental Law Reports, Washington, DC, 1988.
Nomenclature A
B,B' C
= =
C Fs hc
=
area (m 2) empirical parameters filtration parameter concentration (kg/m 3) force (N) cake height (m)
K,K" ,K'" L
=
Qp
=
n
=
Np
=
p q
= =
r
~-
ro rp
=
rw
--
R Rc,Rf =
U
--
V W
=
X0
filtration constants cake thickness (m) pore length (m) number of suspended particles number of pores pressure (N/m 2) filtrate volumer per unit area of filter (mB/m3) or volume (m 3) specific resistance (m -l) specific volumetric cake resistance (kg/m 2) pore radius (m) specific mass cake resistance (kg/m 2) resistance (m/sec) cake and filter resistances, respectively (m -~) average velocity (m/sec) filtrate volume (m 3) mass of dry solids (kg), or rate of filtration (m3/m2-sec) ratio of cake to filtrate volume.
Greek Symbols C
/x II
= =
T
=
TO
--
porosity viscosity (P) ratio of filtration rate to gravity setting time (sec) time constant (sec)
filtrate