Constant Pressure Differential Filtration
When the space above the suspension is subjected to compressed gas or the space under the filter plate is under a vacuum, filtration proceeds under a constant pressure differential (the pressure in the receivers is constant). The rate of filtration 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. Filtration is frequently accompanied by hindered or free gravitational settling of solid particles. The relative directions of action between gravity force and filtrate motion 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 1. 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 1A), 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 IB), particle settling will prevent cake formation, and it is necessary to mix the suspension to maintain homogeneity.
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Figure 1. Direction of gravity force action and filtrate motion in filters: Acocurrent; Bcountercurrent; Ccrosscurrent; solid arrowdirection of gravity force action; dashed arrow direction of filtrate motion; 1filter plate; 2cake; 3sludge; 4filtrate; 5clear 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 semcompressible, 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 pm may be considered incompressible. Examples of incompressible cakeforming 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. 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, cakeforming 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 2.
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Figure 2. Important parameters in cake formation.
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Figure 2. 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 3A, it may be 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 nonspherical, we may still assume singlepoint contacts between adjacent particles as shown in Figure 3B.
Consider flow through a cake (Figure 3C) 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:
The applied pressure p is a function of time but not of distance x. Fs 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 pLis effectively over the entire cross section (A) of the cake; for example, against the fictitious membrane shown in Figure 3B. Dividing Equation 1 by A and denoting the compressive drag pressure by ps= F/A, we obtain:
Figure 3. Frictional drag on particles in compressible cakes.
Figure 3. Frictional drag on particles in compressible cakes.
The terra ps is a fictitious pressure, because the crosssectional 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:
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 (e) and specific resistance (a) 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 4. The variations in porosity and specific resistance are accompanied by varying degrees of compressibility, also shown in Figure 4.
The rate of the filtration process is directly proportional to the driving force and inversely proportional to the resistance.
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:
dV Adx
where V = volume of filtrate (m3) A = filtration area (m2) t = time of filtration (sec)
Assuming laminar flow through the filter channels, the basic equation of filtration as obtained from a force balance is:
A dT
where Ap =
pressure difference (N/m2) viscosity of filtrate (Nsec/m2) filter cake resistance (m"1) initial filter resistance (resistance of filter plate and filter channels) (m"1) u = filtration rate (m/sec), i.e., filtrate flow through cake and filter plate dV/dx = filtration rate (m3/sec), i.e., filtrate flow rate
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 (m3) filtrate. Rf may be assumed a constant. To determine the relationship between volume and residence time t, Equation 5 must be integrated, which means that Rc must be expressed in terms of V.
We denote the ratio of cake volume to filtrate volume as x^. Hence, the cake volume is x^V. An alternative expression for the cake volume is hcA; where hc is die cake height in meters. Consequently:
Hence, the thickness of the cake, uniformly distributed over the filter plate, is:
The filter cake resistance may be expressed as:
where r0= specific volumetric cake resistance (m"2).
As follows from Equation 8, r0 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:
Filtrate volume, Xq 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 r0, a specific mass cake resistance rw is used. That is, r w represents the resistance to flow created by a uniformly distributed cake, in the amount of 1 kg/m2. Replacing units of volume by mass, the term r0 Xq 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:
At fn= 1 Nsec/m2, hc = 1 m and u = 1 m/sec, r 0 = Ap. Thus, the specific cake resistance equals the pressure difference required by the liquid phase (with a viscosity of 1 Nsec/m2) 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 1012m2. Assuming V = 0 (at the start of filtration) where there is no cake over the filter plate. Equation 9 becomes:
At ^ = 1 Nsec/m2 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 Nsec/m2) to pass through the filter plate at a rate of 1 m/sec. For many filter plates Rf is typically 1010 m"1 .
For a constant pressure drop and temperature filtration process all the parameters in Equation 9, except V and t, are constant. Integrating Equation 9 over the limits of 0 to V, from 0 to t, we obtain:
Dividing both sides by ¡jlï^IK gives:
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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 Xq are constant. If we assume a definite filtering apparatus and set up a constant temperature and filtration pressure, then the values of Rf, r0, fi 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: Hence, a simplified expression may be written to describe the filtration process as follows: 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 x0 as a constant that depends on K and C: Substituting Tq into Equation 16, the equation of filtration under constant pressure conditions is: 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 (t+t0), the filtration process may be represented by a parabola with its apex at the origin as illustrated in Figure 5. Moving the axes to distances C and t0 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 5. Typical filtration curve. Constants C and x0 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 x0 denotes the time required for the formation of this fictitious mass, and C is the volume of filtrate required. Differentiating Equation 16 gives: And rearranging in the form of a reciprocal relationship: This form of the equation provides a linear relation like the plot in Figure 6. The expression is that of a straight line having slope 2/K, with intercept C. The experimental determination of dx/dV is made simple by the functional form of this expression. Filtrate volumes V, and V2 should be measured for time intervals t, and t2. Then, according to Equation 16: T2"T1 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 V, and V2: Filtration constants C and K can be determined on the basis of several measurements of filtrate volumes for different time intervals. dr dv Figure 6. Plot of Equation 20. Figure 6. Plot of Equation 20. 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 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 t0 = 0 (Equation 17). Therefore, the simplified equation of filtration takes the following form: For thick cakes, Equation 23 gives results close to that of Equation 16. 
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