Filtration of incompressible materials

Filtration involves the production of a porous bed of particles through which the suspending medium flows. It therefore involves flow through a porous medium whose thickness (and other properties) change with throughput. As particles forming the bed (or cake) are usually small and flow velocities low, the regime is usually streamline. The superficial flow velocity (uc) is therefore well represented by application of the Kozeny's equation:-

Where A is the total cross sectional area of the cake V is the volume of filtrate passed in time t e is the voidage of the cake S the specific surface of the particles therein l the thickness of the cake i?? the viscosity of the filtrate ?P the applied pressure difference.

If the particles are rigid and so non-compressible and e is constant then the quantity e3/[5(1? e)2S2] [2] is a constant property of the particles in the form of the cake (specific hydraulic conductance) whose reciprocal is usually termed the specific resistance r, so that 1 dV ?? P uc = ? ? = ?? [3] A dt ri?? l r may depend upon rates of deposition, nature of particles and forces between them, it has the dimensions L? 2.

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The variables V and l are connected in the sense that as a unit of suspension passes through the system a unit of cake is deposited. The relation may be deduced by a material balance for solids transferred from slurry to filter cake. As the cake is permeable it also contains some of the liquid from the slurry. For solids of density ? s, the mass in a filter cake Ms will be Ms P) ? = ??? [11] dt ri?? vV Equ.

11 is the basic filtration equation arising from Kozeny's analysis. However, in filtration the total resistance to flow increases as cake is deposited. Filtration at constant flowratepractice, operation at constant ?

P is usual. However, there is usually some initial period during which ? P rises to its steady operating value. If this takes time t1 (from t = 0) during which volume V1 of filtrate passes then on integration of Eqn.  Period of constant pressure filtration is t? t1 for passage of V? V1 filtrate. Note (t? t1)/( V? V1) is ? V? V1. Flow through the filter medium or cloth Experiments have shown that filtration operations are affected by the manner in which the first few layers of particles are collected on (and often in) the filter medium or cloth (this may be critical)

The support geometry for the cloth is arranged Usual practice Consider the cloth plus the first few layers of particles to be equivalent in thickness to a thickness L of cake deposited at some later stage. The total equivalent cake thickness is then L + l. Including L in eqn 3 gives:- 1 dV ?? P uc = ? ? = ??? [19] A dt ri?? (l + L) and Eqn 9 may then be used to give l in terms of For an initial period of constant rate filtration integrate between t = 0 and V = 0 to t = t1 and V = V1 and then from t= t1, V = V1 to t = t and V = V for subsequent filtration at constant pressure.

For constant rate:-P) A(?? P) The same linear relation is obtained as for equ. 18. The slope is proportional to the specific resistance r (just as for flow through the cake alone), but the line does not pass through the origin. In principle the intercept on (t ? t1)/(V? V1) when V? V1 = 0 allows L to be found, but in practice it is so sensitive to initial filtration conditions that precise determination is difficult. Why might the linear relationship break down? (Why might r increase ? ) Filtration Example A slurry with volumetric solids concentration (C) of 0. 04 is to be filtered on a rotary drum filter which is 1 m in diameter and 1.

5 m long. The drum operates with 30% of its surface immersed in the slurry under a constant pressure difference of 70 kN m? 2. Filtration tests using a laboratory filter of area 0. 025 m2 in which the slurry was fed with a slurry pump to give a constant rate of filtration of 15 ? 10? 6 m3 s? 1 resulted in a pressure difference of 14 kN m? 2 after 250 s and 29 kN m? 2 after 800s. If the drum rotates at 1/3 rpm determine the thickness of cake formed on the drum and the mass rate of production of solids. Assume that the cake is incompressible and that the resistance of the cloth is the same in both filters. Data Cake voidage 0.

4 Solids density 2000 kg m? 3 Solution Properties of slurry: Basis 1 m3 of slurry. Volume m3 Total solids (in cake) 1 ? 0. 04 = 0. 04 Total volume of liquid 0715 = 0. 0051 m3 s? 1. Filtration Practice The Filter Cloth Serves to support the cake. The initial layers of cake provide the true filter. Types: Woven cloths/fabrics (ease of cake removal) Porous solids (sintered materials)

Granular solids Loss of permeability The initial cake formation is critical to the overall filtration performance. Particles may block or shield pores in the filter medium. Concentration: Dilute suspensions ? impermeable cakes (Particles follow streamlines into pores) Concentrated suspensions ? permeable cakes (Particles spread evenly over the medium and bridge over pores) Effect of Sedimentation

a) Downward flow (filter receives settling large particles) assisting initial low r b) Upward flow (finer particles filtered) high r Delayed Cake Filtration Cake is removed by a scraper or blade (a short distance above the filter cloth) and returned to the feed slurry. Solids concentration in the feed increases until its almost cake. Fluid flow through slurry falls. As ?? P/dl falls (at constant ? P) the forces both retaining particle and forcing them into pores decrease and l reaches an equilibrium value (which may be below the height of the blade). At steady state the (volume) rate of feed must balance the (volume) rate of filtrate.

The influx of solids, at a concentration (1? eo), is (1? eo)dV/dt At any time (t) the volume of solids in the filtration vessel (of volume V) is V(1? et) a solids balance gives:- (1? eo)dV/dt = d/dt{V(1(1? et)} or d/dt(1? et) = (1/V)(1? eo)dV/dt when dV/dt = constant the fractional solids hold-up (1? et) increases linearly with time until resistance to flow in the slurry becomes significant - then dV/dt falls rapidly towards zero. Filtration Of Materials Forming Compressible Cakes In this kind of process the usual relationships such as Kozeny-Carman equations are not applicable.

Experiments to assess whether or not the resistance to flow is a function of ? P are required for a wide range of ? P/dl. If resistance is dependent on ? P then large ? Ps may not lead to significant improvements in throughput as the resistance will be highest at the highest pressure. Compressible cakes occur in the paper and biotechnology industries. Sewage sludge is a good example. Deformation of cakes is usually a consolidation process with little elastic recovery. Flow through a compressible cake Voidage (e) varies through the cake. Impressive forces vary from a maximum at the base of the cake to a minimum at the free liquid surface.

Compressibility will be a function of cake structure and the nature of interparticle contacts. An empirical or heuristic approach is commonly used to facilitate progress in this subject. The compressive force = f (Pl ? Pz). This will be a function of the pressure between Pl (at the surface of the cake) and that at depth z in the cake (at pressure Pz). Since e at a point is a function of compressive force at that point, ez = f(Pl ? Pz ) In general e decreases from the surface to the base of the cake and the corresponding resistance increases down the cake reaching a maximum at the filter medium.

Definition Diagram P1 Pz P2 bulk flow z dz filter medium cake P1 = pressure at interface of cake/slurry Pz = is the pressure at a thin slice of cake dz P2 = pressure at free surface of the medium As r is a f(z) the basic equation must be used in differential form.  dz where S is assumed constant. The passage of a small volume of filtrate (dV) results in the formation of a small quantity of cake of thickness dz.

The volume of cake deposited or associated with passage of a unit volume of filtrate is now not constant but the volume of solids deposited (v? ) will be almost independent of the conditions of cake formation. A material balance provides:-note a comparison of the basic equations here v? rz = vr, i. e. rz = r v/ v? Next useful dodge to facilitate integration:-

At any instant in a constant pressure drop filtration we have on integration through the depth of the entire cake the following:- At any instant dV/dt is constant throughout the cake therefore Now rz has been shown to be a function of ? P not absolute pressure. Suppose (a power law relation with r tending to high values at high ? P). and (which if n = 1, is physically untenable) hence (where r? = (1? n)r? ) where is the mean cake resistance i. e. = r"(?? P)n Take a standard pressure drop filtration experiment and evaluate r then use another value of ? P and see if r changes.

For low compressibility n = 0.01 to 0. 15 whilst highly compressible cakes may have values of ~ 0. 9. Typical plots of log r as flog(? P)) suggest the use of an optimum pressure drop. Increasing ? P merely causes an increase in resistance. Pretreatment & Conditioning Dilute suspensions 1) If particles settle at significant rate then thicken slurries prior to filtration to reduce volume. 2) If particles don't settle at significant rate then use a coagulant or adjust liquid conditions (pH, I) to increase the effective particle size. 3) Filter aids This involves addition of particles which pack to form cakes of high e.

They contaminate the cake, may render it compressible and introduce an additional cost and so require evaluation. Filtration Equipment The Plate and Frame Press (Batch Filtration) Consists of a rack into which rectangular frames are held between plates, which are stacked and clamped together. The area for filtration is controlled by the number of frames and the maximum cake thickness (per unit area) by the frame thickness. Cake (which occupies the internal space of each of the frames) is retained by a filter cloth which rests on the adjacent plates. Feed enters the frames via ports located in designated corners of plates and frames.

The feed is distributed to the cloth at both sides and so the when a frame is full, the resistance to flow is provided by a cake of half the thickness of the frame. Two types of plates may be used  filter plates allow filtrate to enter via the cloth and to leave via an outlet port at the base.  wash plates allow wash liquor to pass from the downstream side of the filter cloth on one side of a frame, across the entire thickness of cake within the frame and then out via the cloth on the opposite side. Frames of various thickness are available. When frames are full of cake the feed is stopped.

If required, the cake is firstly washed, the assembly is then dismantled (to recover the cake), cleaned and reassembled. Advantages Disadvantages flexible labour intensive low maintenance cost wears out cloths high A is small space leakage easily detected high (?? P) operation suited to solid or liquid product Washing of Cakes a) Simple washing Wash liquor follows the feed path. Consequently The cake may be disrupted and flow channels formed, washing is likely to be uneven. If feed liquid held in the cake needs to be recovered then this method is not recommended.

For given P liquor flowrate = final filtrate flowrate c) Through or thorough washing Wash liquor enters the cake via the cloth passes across the entire frame thickness of cake and leaves via the cloth on the opposite side. As flow area halved and cake thickness and doubled For given ?? P liquor flowrate = (final filtrate flowrate)/4 Optimum cycle time for batch operation of a filter press As a consequence of having to dismantle and reassemble a filter press the equipment is out of service for the period required (t? say). This period is effectively independent of the thickness of frames used in the press.

What thickness of frame is required to produce the maximum average rate of production of filtrate (W) from a period of filtration t and the downtime t?? Overall filtrate flowrate W = V/(t + t? ) Now at constant pressure operation (throughout):-from which L may be determined by material (solids volume- pore volume) balance and the number and area of frames.