In the case of single-particle blockage, we first consider aim surface of filter medium containing Np number of pores. The average pore radius and length are rp and { , 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:

Consider 1 m 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:

k' is a constant having units of sec"1. 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:

Comparison with Equation 42 reveals:

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 x0 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:

When the average pore radius decreases to r, the rate of filtration becomes

where

7tAp

7tAp

From which Equation 54 may be restated as:

The derivative of this expression with respect to q is:

On some rearranging of terms, we obtain:

where

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:

Substituting Equation 55 for C and using Equation 48 for B, the above expression becomes:

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:

Integration of this equation over the limits from 0 to t for 0 to q we obtain:

WJ2-Kq)

and on simplification:

Equation 67 may be used to evaluate constants K (mi1) 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:

Replacing V by q, and denoting the actual filtration rate (dq/dx) 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:

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/dx in Equation 78 and integrating over the limits of 0 to q between 0 and x we obtain:

Note that this 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 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 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(x/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.

Type of Filtration |
Equation |
Coordinates |

With Total Pore Blocking |
42 |
q vs W |

With Gradual Pore Blocking |
67 |
T vs t/q |

Intermediate |
74 |
T vs 1/W |

Cake |
81 |
q vs t/q |

All the mechanisms of filtration encountered in practice have the functional form:

where b typically yaries 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 5p, 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 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 (x0) 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:

The mass of particles deposited on the pore walls will be xGdq, and the thickness of this particle layer in each pore is dr. Hence

where

x0dq = -Np2itrtpdz

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:

where

It is important to note that pore blocking occurs when suspensions have the

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

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

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.

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