Solid particles undergoing separation from the mother suspension may be captured both on the surface of the filter medium and within the h e r 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 me&um. The network of pores within the filter medium can become blocked because of one or a combination of the following situations:
Pores may become blocked by the lodging of single particles in the pore passage.
Gradual blockage can occur due to the accumulation of many particles in pore passages.
Blockage may occur during intermediate-type filtration.
Proper filter medium selection is based on understanding these mechanisms and analyzing the impact each has on the filtration process.
174 WATER AND WASTEWATER =ATMEN" TECHNOLOGIES
In the case of single-particle blockage, we f i s t consider a 1 m2 surface of fdter medium containing N, number of pores. The average pore radius and length are r, and P,, 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:
Consequently, the initial filtration rate per unit area of filtration is:
Win = V'N,
Consider 1 m3 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 (N, - nq). Then the rate of filtration is:
W = V' ( N p - nq)
or
where
k' = V'n (43)
k' is a constant having units of see-'. 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:
WHAT PRESSURE AND CAKE FKTRATION ARE ALL ABOUT 175
Comparison with Equation 42 reveals:
dR - k' - - -
dq w 2
or
(45)
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 xo 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:
Win = BNprp 4
where
(47)
When the average pore radius decreases to r, the rate of filtration becomes:
W = BNpr4 (49)
176 WATER AND WASTEWATER TREATMENT TECHNOLOGIES
For a fink filtrate quantity, dq the amount of cake inside the pores is x,dq, and the cake thickness is dr. That is:
xodq = -NP2nrQpdr (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:
And from Equations 47 and 49, we may define the pore radii as follows:
r : = [ g ] 1 12
or simply:
r2 = [ % ) l n (53)
Substituting these quantities into Equation 51 and simplifying terms, we obtain:
w = [(w,J'" - Cq]2 (54)
where
It is convenient to define the following constant:
K=- 2c
WHAT PRESSURE AND CAKE FILTRATION ARE ALL ABOUT 177 From which Equation 54 may be restated as:
W = Wh(l - 1/2Kq)'
Since W = l / R , we may write:
R = 1
wi,( 1 - 1/2Kq)2
The derivative of this expression with respect to q is:
dR - k
I-
& y,(i - 1 / 2 ~ q ) 3
On some rearranging of terms, we obtain:
- dR = K(PV,,)'"R3"
dq
or
where
(57)
(59)
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 / = 2 c
178 WATER AND WASTEWATER TREATMENT TECHNOLOGIES
Substituting Equation 55 for C and using Equation 48 for B, the above expression becomes:
K“ = 2(W,,)’” -
[ N p L 1
Note that for constant 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.
Replacing W by dq/dt in Equation 57, we obtain:
dr = L( 1-5Kq) 1 -2 dq
Yn
Integration of this equation over the limits from 0 to ‘c for 0 to q we obtain:
and on simplification:
Equation 67 may be used to evaluate constants K (m-’) and 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:
Integration of this expression between the limits of 0 to q, from Rf to R gives:
WHAT PRESSURE AND CAKE FETRATION ARE ALL ABOUT 179
Substituting 1/W for R and 1/W, for Rf, the last expression becomes:
W.
W
- e r " q
Substituting dq/dz for Win Equation 71 and integrating over the limits of 0 to T between 0 and q we obtain:
Hence,
Accounting for Equation 70, the final form of this expression becomes:
To compare the different mechanisms of filtration, the governing equation of filtration must be rearranged. The starting expression is:
- - d v - 4
A & p [ r , x , ( V / A ) + R , ] (75)
180 WATER AND WASTEWATER TREATMENT TECHNOLOGIES
Replacing V by q, and denoting the actual filtration rate (dq/dt) as W, the governing filtration equation may be rewritten for a unit area of filtration as follows:
At the initial moment when q = 0, the filtration rate is
From Equations 76 and 77 we have:
W = W,
1 +. K"'Winq
where
The numerator of Equation 79 characterizes the cake resistance. The denominator contains information on the driving force of the operation. Constant K"' (sec/m2) 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:
The expression states that the intensity of increase in total resistance for cake filtration is constant with increasing filtrate volume. Replacing W by dq/dt in Equation 78 and integrating over the limits of 0 to q between 0 and t we obtain:
WHAT PRESSURE AND CAKE FILTRATION ARE ALL ABOUT 181
Note that tlus expression reduces to Equation 74 on substituting expressions for Win (Equation 77) and K"' (Equation 79).
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 cunstants 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 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(dq) 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 T vs d q
Intermediate 74 z vs 1/w
Cake 81 q vs z/q
All the mechanisms of filtration encountered in practice have the functional form:
where b typically varies between 0 and 2.
182 WATER AND WASTEWATER TREATMENT TECHNOLOGIES
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 and Qp, 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 cake-to-filtrate volumes. The influence of particle compressibility on the controlling mechdsmmay 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 (x,) 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 AP, for constant pressure drop AP we obtain:
W = B'hPi,,Npr;
where
(83)
The mass of particles deposited on the pore walls will be Gdq, and the thickness of this particle layer in each pore is dr. Hence
Integration over the limits of 0, q from rp to r yields
Radii rp and r are defined by Equations 83 and 85, respectively, from which we obtain the following expressions:
WHAT PRESSURE AND CAKE FILTRATION ARE ALL ABOUT 183
or
Since q = W t , Equation 88 may be stated in a reduced form as:
where
A plot of Equation 89 on the coordinate of T vs (l/Apin)’ - (l/Ap)lh 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 t = 0, Ap = Ap,,, which is in agreement with typical process observations.
The filtration time corresponding to total pore blockage, when A p W may be estimated from:
To express the relationship between AP and t more directly. Equation 89 is restated in the form:
Ap = 1
( A - C T ) 2
184 WATER AND WASTEWATER TREATMENT TECHNOLOGIES
where
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 accumulate onto a horizontal filter medium assists in understanding the influences that the particle settling velocity and particle concentration have on the controlling mechanisms. "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.
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.