14.2. FILTRATION IN SOLID–LIQUID SEPARATION

14.2A. Introduction

In filtration, suspended solid particles in a fluid of liquid or gas are physically or mechanically removed by using a porous medium that retains the particles as a separate phase or cake and passes the clear filtrate. Commercial filtrations cover a very wide range of applications. The fluid can be a gas or a liquid. The suspended solid particles can be very fine (in the micrometer range) or much larger, very rigid or plastic particles, spherical or very irregular in shape, aggregates of particles or individual particles. The valuable product may be the clear filtrate from the filtration or the solid cake. In some cases, complete removal of the solid particles is required, in other cases only partial removal.

The feed or slurry solution may carry a heavy load of solid particles or a very small amount. When the concentration is very low, the filters can operate for very long periods of time before the filter needs cleaning. Because of the wide diversity of filtration problems, a multitude of types of filters have been developed.

Industrial filtration equipment differs from laboratory filtration equipment only in the amount of material handled and in the necessity for low-cost operation. A typical laboratory filtration apparatus is shown in Fig. 14.2-1, which is similar to a Büchner funnel. The liquid is made to flow through the filter cloth or paper by a vacuum on the exit end. The slurry consists of the liquid and the suspended particles. The passage of the particles is blocked by the small openings in the pores of the filter cloth. A support with relatively large holes is used to hold the filter cloth. The solid particles build up in the form of a porous filter cake as the filtration proceeds. This cake itself also acts as a filter for the suspended particles. As the cake builds up, resistance to flow also increases.

In section 14.2 the ordinary type of filtration will be considered, where a pressure difference is used to force the liquid through the filter cloth and the filter cake that builds up.

In Section 14.4E, centrifugal filtration will be discussed, where centrifugal force is used instead of a pressure difference. In many filtration applications, ordinary filters and centrifugal filters are often competitive and either type can be used.

14.2B. Types of Filtration Equipment

1. Classification of filters

There are a number of ways to classify types of filtration equipment; unfortunately, it is not possible to devise a single classification scheme that includes all types of filters. In one classification, filters are classified according to whether the desired product is the filter cake or the clarified filtrate or outlet liquid. In either case, the slurry can have a relatively large percentage of solids so that a cake is formed, or have just a trace of suspended particles.

Filters can also be classified by operating cycle. Filters can be operated as batch, where the cake is removed after a run, or continuous, where the cake is removed continuously. In another classification, filters can be of the gravity type, where the liquid simply flows by means of a hydrostatic head, or pressure or vacuum can be used to increase the flow rates. An important method of classification depends upon the mechanical arrangement of the filter media. The filter cloth can be in a series arrangement as flat plates in an enclosure, as individual leaves dipped in the slurry, or on rotating-type rolls in the slurry. In the following sections only the most important types of filters will be described. For more details, see references (B1, P1).

2. Bed filters

The simplest type of filter is the bed filter shown schematically in Fig. 14.2-2. This type is useful mainly in cases where relatively small amounts of solids are to be removed from large amounts of water in clarifying the liquid. Often the bottom layer is composed of coarse pieces of gravel resting on a perforated or slotted plate. Above the gravel is fine sand, which acts as the actual filter medium. Water is introduced at the top onto a baffle which spreads the water out. The clarified liquid is drawn out at the bottom.

The filtration continues until the precipitate of filtered particles has clogged the sand so that the flow rate drops. Then the flow is stopped, and water is introduced in the reverse direction so that it flows upward, backwashing the bed and carrying the precipitated solid away. This apparatus can only be used on precipitates that do not adhere strongly to the sand and can be easily removed by backwashing. Open-tank filters are used in filtering municipal water supplies.

3. Plate-and-frame filter presses

One of the important types of filters is the plate-and-frame filter press, which is shown diagrammatically in Fig. 14.2-3a. These filters consist of plates and frames assembled alternately with a filter cloth over each side of the plates. The plates have channels cut in them so that clear filtrate liquid can drain down along each plate. The feed slurry is pumped into the press and flows through the duct into each of the open frames so that slurry fills the frames. The filtrate flows through the filter cloth and the solids build up as a cake on the frame side of the cloth. The filtrate flows between the filter cloth and the face of the plate through the channels to the outlet.

The filtration proceeds until the frames are completely filled with solids. In Fig. 14.2-3a all the discharge outlets go to a common header. In many cases the filter press will have a separate discharge to the open for each frame. Then visual inspection can be made to see if the filtrate is running clear. If one is running cloudy because of a break in the filter cloth or other factors, it can be shut off separately. When the frames are completely full, the frames and plates are separated and the cake removed. Then the filter is reassembled and the cycle is repeated.

If the cake is to be washed, the cake is left in the plates and through-washing is performed, as shown in Fig. 14.2-3b. In this press a separate channel is provided for the wash-water inlet. The wash water enters the inlet, which has ports opening behind the filter cloths at every other plate of the filter press. The wash water then flows through the filter cloth, through the entire cake (not half the cake as in filtration), through the filter cloth at the other side of the frames, and out the discharge channel. It should be noted that there are two kinds of plates in Fig. 14.2-3b: those having ducts to admit wash water behind the filter cloth, alternating with plates without such ducts.

The plate-and-frame presses suffer from the disadvantages common to batch processes. The cost of labor for removing the cakes and reassembling plus the cost of fixed charges for downtime can be an appreciable part of the total operating cost. Some newer types of plate-and-frame presses have duplicate sets of frames mounted on a rotating shaft. Half of theframes are in use while the others are being cleaned, saving downtime and labor costs. Other advances in automation have been applied to these types of filters.

Filter presses are used in batch processes but cannot be employed for high-throughput processes. They are simple to operate, very versatile and flexible in operation, and can be used at high pressures, when necessary, if viscous solutions are being used or the filter cake has a high resistance.

4. Leaf filters

The filter press is useful for many purposes but is not economical for handling large quantities of sludge or for efficient washing with a small amount of wash water. The wash water often channels in the cake and large volumes of wash water may be needed. The leaf filter shown in Fig. 14.2-4 was developed for larger volumes of slurry and more efficient washing. Each leaf is a hollow wire framework covered by a sack of filter cloth.

A number of these leaves are hung in parallel in a closed tank. The slurry enters the tank and is forced under pressure through the filter cloth, where the cake deposits on the outside of the leaf. The filtrate flows inside the hollow framework and out a header. The wash liquid follows the same path as the slurry. Hence, the washing is more efficient than the through-washing in plate-and-frame filter presses. To remove the cake, the shell is opened. Sometimes air is blown in the reverse direction into the leaves to help in dislodging the cake. If the solids are not wanted, water jets can be used to simply wash away the cakes without opening the filter.

Leaf filters also suffer from the disadvantages of batch operation. They can be automated for the filtering, washing, and cleaning cycle. However, they are still cyclical and are used for batch processes and relatively modest throughput processes.

5. Continuous rotary filters

Plate-and-frame filters suffer from the disadvantages common to all batch processes and cannot be used for large-capacity processes. A number of continuous-type filters are available, as discussed below.

14.2C. Filter Media and Filter Aids

1. Filter media

The filter medium for industrial filtration must fulfill a number of requirements. First and foremost, it must remove the solids to be filtered from the slurry and give a clear filtrate. Also, the pores should not become plugged so that the rate of filtration becomes too slow. The filter medium must allow the filter cake to be removed easily and cleanly. Obviously, it must have sufficient strength not to tear and must be chemically resistant to the solutions used.

Some widely used filter media are twill or duckweave heavy cloth, other types of woven heavy cloth, woolen cloth, glass cloth, paper, felted pads of cellulose, metal cloth, nylon cloth, dacron cloth, and other synthetic cloths. The ragged fibers of natural materials are more effective in removing fine particles than the smooth plastic or metal fibers. Sometimes the filtrate may come through somewhat cloudy at first before the first layers of particles, which help filter the subsequent slurry, are deposited. This filtrate can be recycled for refiltration.

2. Filter aids

Certain filter aids may be used to aid filtration. These are often incompressible diatomaceous earth or kieselguhr, which is composed primarily of silica. Also used are wood cellulose and other inert porous solids.

These filter aids can be used in a number of ways. They can be used as a precoat before the slurry is filtered. This will prevent gelatinous-type solids from plugging the filter medium and also give a clearer filtrate. They can also be added to the slurry before filtration. This increases the porosity of the cake and reduces resistance of the cake during filtration. In a rotary filter, the filter aid may be applied as a precoat; subsequently, thin slices of this layer are sliced off with the cake.

The use of filter aids is usually limited to cases where the cake is discarded or where the precipitate can be separated chemically from the filter aid.

14.2D. Basic Theory of Filtration

1. Pressure drop of fluid through filter cake

Figure 14.2-6 is a section through a filter cake and filter medium at a definite time t s from the start of the flow of filtrate. At this time the thickness of the cake is L m (ft). The filter cross-sectional area is A m² (ft²), and the linear velocity of the filtrate in the L direction is ν m/s (ft/s) based on the filter area A m².

The flow of the filtrate through the packed bed of cake can be described by an equation similar to Poiseuille's law, assuming laminar flow occurs in the filter channels. Equation (2.10-2) gives Poiseuille's equation for laminar flow in a straight tube, which can be written

Equation 14.2-1

where Δp is pressure drop in N/m² (lb_f/ft²), ν is open-tube velocity in m/s (ft/s), D is diameter in m (ft), L is length in m (ft), μ is viscosity in Pa · s or kg/m · s (lb_m/ft · s), and g_c is 32.174 lb_m · ft/lb_f · s².

For laminar flow in a packed bed of particles, the Carman–Kozeny relation, which is similar to Eq. (14.2-1) and to the Blake–Kozeny equation (3.1-17), has been shown to apply to filtration:

Equation 14.2-2

where k₁ is a constant and equals 4.17 for random particles of definite size and shape, μ is viscosity of filtrate in Pa · s (lb_m/ft · s), ν is linear velocity based on filter area in m/s (ft/s), ε is void fraction or porosity of cake, L is thickness of cake in m (ft), S₀ is specific surface area of particle in m² (ft²) of particle area per m³ (ft³) volume of solid particle, and Δp_c is pressure drop in the cake in N/m² (lb_f/ft²). For English units, the right-hand side of Eq. (14.2-2) is divided by g_c. The linear velocity is based on the empty cross-sectional area and is

Equation 14.2-3

where A is filter area in m² (ft²) and V is total m³ (ft³) of filtrate collected up to time t s. The cake thickness L may be related to the volume of filtrate V by a material balance. If c_S is kg solids/m³ (lb_m/ft³) of filtrate, a material balance gives

Equation 14.2-4

where ρ_p is density of solid particles in the cake in kg/m³ (lb_m/ft³) solid. The final term of Eq. (14.2-4) is the volume of filtrate held in the cake. This is usually small and will beneglected.

Substituting Eq. (14.2-3) into (14.2-2) and using Eq. (14.2-4) to eliminate L, we obtain the final equation as

Equation 14.2-5

where α is the specific cake resistance in m/kg(ft/lb_m), defined as

Equation 14.2-6

For the filter-medium resistance, we can write, by analogy with Eq. (14.2-5),

Equation 14.2-7

where R_m is the resistance of the filter medium to filtrate flow in m⁻¹ (ft⁻¹) and Δp_f is the pressure drop. When R_m is treated as an empirical constant, it includes the resistance to flow of the piping leads to and from the filter and the filter medium resistance.

Since the resistances of the cake and the filter medium are in series, Eqs. (14.2-5) and (14.2-7) can be combined, and become

Equation 14.2-8

where Δp = Δp_c + Δp_f. Sometimes Eq. (14.2-8) is modified as follows:

Equation 14.2-9

where V_e is a volume of filtrate necessary to build up a fictitious filter cake whose resistance is equal to R_m.

The volume of filtrate V can also be related to W, the kg of accumulated dry cake solids, as follows:

Equation 14.2-10

where c_x is mass fraction of solids in the slurry, m is mass ratio of wet cake to dry cake, and ρ is density of filtrate in kg/m³ (lb_m/ft³).

2. Specific cake resistance

From Eq. (14.2-6) we see that the specific cake resistance is a function of void fraction ε and S₀. It is also a function of pressure, since pressure can affect ε. By conducting constant-pressure experiments at various pressure drops, the variation of α with −Δp can be determined.

Alternatively, compression–permeability experiments can be performed. A filter cake at a low pressure drop and atm pressure is built up by gravity filtering in a cylinder with a porous bottom. A piston is loaded on top and the cake compressed to a given pressure. Then filtrate is fed to the cake and α is determined by a differential form of Eq. (14.2-9). This is then repeated for other compression pressures (G1).

If α is independent of −Δp, the sludge is incompressible. Usually, α increases with −Δp, since most cakes are somewhat compressible. An empirical equation often used is

Equation 14.2-11

where α₀ and s are empirical constants. The compressibility constant s is zero for incompressible sludges or cakes. The constant s usually falls between 0.1 to 0.8. Sometimes the following is used:

Equation 14.2-12

where , β,and s' are empirical constants. Experimental data for various sludges are given by Grace (G1).

The data obtained from filtration experiments often do not have a high degree of reproducibility. The state of agglomeration of the particles in the slurry can vary and have an effect on the specific cake resistance.

14.2E. Filtration Equations for Constant-Pressure Filtration

1. Basic equations for filtration rate in batch process

Often a filtration is done under conditions of constant pressure. Equation (14.2-8) can be inverted and rearranged to give

Equation 14.2-13

where K_p is in s/m⁶ (s/ft⁶) and B in s/m³ (s/ft³):

Equation 14.2-14

Equation 14.2-15

For constant pressure, constant α, and incompressible cake, V and t are the only variables in Eq. (14.2-13). Integrating to obtain the time of filtration in t s,

Equation 14.2-16

Equation 14.2-17

Dividing by V

Equation 14.2-18

where V is total volume of filtrate in m³ (ft³) collected to t s.

To evaluate Eq. (14.2-17) it is necessary to know α and R_m. This can be done by using Eq. (14.2-18). Data for V collected at different times t are obtained. Then the experimental data are plotted as t/V versus V, as in Fig. 14.2-7 Often, the first point on the graph does not fall on the line and is omitted. The slope of the line is K_p/2 and the intercept B. Then, using Eqs. (14.2-14) and (14.2-15), values of α and R_m can be determined.

EXAMPLE 14.2-1. Evaluation of Filtration Constants for Constant-Pressure Filtration Data for the laboratory filtration of CaCO3 slurry in water at 298.2 K (25°C are reported as follows at a constant pressure (−Δp) of 338 kN/m2 (7060 lbf/ft2) (R1, R2, M1). The filter area of the plate-and-frame press was A = 0.0439 m2 (0.473 ft2) and the slurry concentration was cs = 23.47 kg/m3 (1.465 lbm/ft3). Calculate the constants α and Rm from the experimental data given, where t is time in s and V is filtrate volume collected in m3. t V t V t V 4.4 0.498 × 10−3 34.7 2.498 × 10−3 73.6 4.004 × 10−3 9.5 1.000 × 10−3 46.1 3.002 × 10−3 89.4 4.502 × 10−3 16.3 1.501 × 10−3 59.0 3.506 × 10−3 107.3 5.009 × 10−3 24.6 2.000 × 10−3         Solution: First, the data are calculated as t/V and tabulated in Table 14.2-1. The data are plotted as t/V versus V in Fig. 14.2-8 and the intercept is determined as B = 6400 s/m3 (181 s/ft3) and the slope as Kp/2 = 3.00 × 106 s/m6. Hence, Kp = 6.00 × 106 s/m6 (4820 s/ft6). Table 14.2-1. Values of t/V for Example 14.2-1 (t =s, V = m3) t V ×σ103 (t/V) ×10−3 0 0   4.4 0.498 8.84 9.5 1.000 9.50 16.3 1.501 10.86 24.6 2.000 12.30 34.7 2.498 13.89 46.1 3.002 15.36 59.0 3.506 16.83 73.6 4.004 18.38 89.4 4.502 19.86 107.3 5.009 21.42 Figure 14.2-8. Determination of constants for Example 14.2-1. At 298.2 K the viscosity of water is 8.937 × 10−4 Pa ∊· s = 8.937 × 10−4 kg/ms. Substituting known values into Eq. (14.2-14) and solving, Substituting into Eq. (14.2-15) and solving,

EXAMPLE 14.2-2. Time Required to Perform a Filtration The same slurry used in Example 14.2-1 is to be filtered in a plate-and-frame press having 20 frames and 0.873 m2 (9.4 ft2) area per frame. The same pressure will be used in constant-pressure filtration. Assuming the same filter-cake properties and filter cloth, calculate the time to recover 3.37 m3 (119 ft3) filtrate. Solution: In Example 14.2-1, the area A = 0.0439 m2, Kp = 6.00 × 106 s/m6, and B = 6400 s/m3. Since the α and Rm will be the same as before, Kp can be corrected. From Eq. (14.2-14), Kp is proportional to 1/A2. The new area is A = 0.873(20) = 17.46 m2 (188 ft2). The new Kp is The new B is proportional to 1/A from Eq. (14.2-15): Substituting into Eq. (14.2-17), Using English units,

2. Equations for washing of filter cakes and total cycle time

The washing of a cake after the filtration cycle has been completed takes place by displacement of the filtrate and by diffusion. The amount of wash liquid should be sufficient to give the desired washing effect. To calculate washing rates, it is assumed that the conditions during washing are the same as those that existed at the end of the filtration. It is assumed that the cake structure is not affected when wash liquid replaces the slurry liquid in the cake.

In filters where the wash liquid follows a flow path similar to that during filtration, as in leaf filters, the final filtering rate gives the predicted washing rate. For constant-pressure filtration, using the same pressure in washing as in filtering, the final filtering rate is the reciprocal of Eq. (14.2-13):

Equation 14.2-19

where (dV/dt)_f = rate of washing in m³/s (ft³/s) and V_f is the total volume of filtrate for the entire period at the end of filtration in m³ (ft³).

For plate-and-frame filter presses, the wash liquid travels through a cake twice as thick and an area only half as large as in filtering, so the predicted washing rate is one-fourth of the final filtration rate:

Equation 14.2-20

In actual experience the washing rate may be less than predicted because of cake consolidation, channeling, and formation of cracks. Washing rates in a small plate-and-frame filter were found to be from 70 to 92% of that predicted (M1).

After washing is completed, additional time is needed to remove the cake, clean the filter, and reassemble the filter. The total filter-cycle time is the sum of the filtration time, plus the washing time, plus the cleaning time.

EXAMPLE 14.2-3. Rate of Washing and Total Filter-Cycle Time At the end of the filtration cycle in Example 14.2-2, a total filtrate volume of 3.37 m3 is collected in a total time of 269.7 s. The cake is to be washed by through-washing in the plate-and-frame press using a volume of wash water equal to 10% of the filtrate volume. Calculate the time of washing and the total filter-cycle time if cleaning the filter takes 20 min. Solution: For this filter, Eq. (14.2-20) holds. Substituting Kp = 37.93 s/m6, B = 16.10 s/m3, and Vf = 3.37 m3, the washing rate is as follows: The time of washing is then as follows for 0.10(3.37), or 0.337 m3 of wash water: The total filtration cycle is

3. Equations for continuous filtration

In a filter that is continuous in operation, such as a rotary-drum vacuum type, the feed, filtrate, and cake move at steady, continuous rates. In a rotary drum the pressure drop is held constant for the filtration. The cake formation involves a continual change in conditions. In continuous filtration, the resistance of the filter medium is generally negligible compared with the cake resistance. So in Eq. (14.2-13), B = 0.

Integrating Eq. (14.2-13), with B = 0,

Equation 14.2-21

Equation 14.2-22

where t is the time required for formation of the cake. In a rotary-drum filter, the filter time t is less than the total cycle time t_c by

Equation 14.2-23

where f is the fraction of the cycle used for cake formation. In the rotary drum, f is the fraction submergence of the drum surface in the slurry.

Next, substituting Eq. (14.2-14) and Eq. (14.2-23) into (14.2-22) and rearranging,

Equation 14.2-24

If the specific cake resistance varies with pressure, the constants in Eq. (14.2-11) are needed to predict the value of α to be used in Eq. (14.2-24). Experimental verification of Eq. (14.2-24) shows that the flow rate varies inversely with the square root of the viscosity and the cycle time (N1).

When short cycle times are used in continuous filtration and/or the filter medium resistance is relatively large, the filter resistance term B must be included, and Eq. (14.2-13) becomes

Equation 14.2-25

Then Eq. (14.2-25) becomes

Equation 14.2-26

EXAMPLE 14.2-4. Filtration in a Continuous Rotary-Drum Filter A rotary-vacuum-drum filter having a 33% submergence of the drum in the slurry is to be used to filter a CaCO3 slurry as given in Example 14.2-1 using a pressure drop of 67.0 kPa. The solids concentration in the slurry is cx = 0.191 kg solid/kg slurry and the filter cake is such that the kg wet cake/kg dry cake = m = 2.0. The density and viscosity of the filtrate can be assumed as those of water at 298.2 K. Calculate the filter area needed to filter 0.778 kg slurry/s. The filter-cycle time is 250 s. The specific cake resistance can be represented by α = (4.37 × 109) (−Δp)0.3, where −Δp is in Pa and α in m/kg. Solution: From Appendix A.2 for water, ρ = 996.9 kg/m3, μ = 0.8937 × 10−3 Pa · s. From Eq. (14.2-10), Solving for α, α = (4.37 × 109) (67.0 × 103)0.3 = 1.225 × 1011 m/kg. To calculate the flow rate of the filtrate, Substituting into Eq. (14.2-24), neglecting and setting B = 0, and solving, Hence, A = 6.60 m2.

14.2F. Filtration Equations for Constant-Rate Filtration

In some cases filtration runs are made under conditions of constant rate rather than constant pressure. This occurs if the slurry is fed to the filter by a positive-displacement pump. Equation (14.2-8) can be rearranged to give the following for a constant rate (dV/dt) m³/s:

Equation 14.2-27

where

Equation 14.2-28

Equation 14.2-29

K_V is in N/m⁵ (lb_f/ft⁵) and C is in N/m² (lb_f/ft²).

Assuming that the cake is incompressible, K_V and C are constants characteristic of the slurry, cake, rate of filtrate flow, and so on. Hence, a plot of pressure, −Δp, versus the total volume of filtrate collected, V, gives a straight line for a constant rate dV/dt. The slope of the line is K_V and the intercept is C. The pressure increases as the cake thickness increases and the volume of filtrate collected increases.

The equations can also be rearranged in terms of −Δp and time t as variables. At any moment during the filtration, the total volume V is related to the rate and total time t as follows:

Equation 14.2-30

Substituting Eq. (14.2-30) into Eq. (14.2-27),

Equation 14.2-31

For the case where the specific cake resistance α is not constant but varies as in Eq. (14.2-11), this can be substituted for α in Eq. (14.2-27) to obtain a final equation.