Filtration of new processes and new technologies - basic principles of filtration and calculations (1)

(1) Filtration Overview Filtration is a process of separating solid particles from a fluid. The basic principle is to feed a mixture of liquid-solid two phases to the surface of a porous medium (ie, filter medium, generally with a filter cloth, etc.) at a pressure difference. Under the action of the liquid, the liquid passes through the medium, and the solid particles remain on the medium, which is called a filter cake; the liquid passes through the cake layer and the medium layer to become a clear filtrate. A filter that uses a filtrate as a product is generally easier to handle than a filter that obtains a filter cake as a product. For the further dehydration of the finely divided material after preliminary dehydration, the most common method currently used is filtration. Filtration energy consumption is lower compared to other separation methods.
A Filtration method The method of applying filtration in the industry can be divided into four types according to the different filtering power.
(1) Gravity filtration. This type of filtration is a deep bed filtration (i.e., a thick filter layer) characterized by the deposition of solid particles within the thicker granular media bed. The particle diameter in the suspension is less than the diameter of the bed channel and adheres to the filter media as it passes through the tortuous channels in the bed. This filtration is suitable for applications where the particles in the suspension are very small and the content is very small. For example, in the waterworks, a quartz sand layer is used as a filter medium to purify the water.
(2) Vacuum filtration. The vacuum pump causes a certain pressure difference between the two sides of the filter medium. Under the driving force, the liquid of the suspension passes through the filter cloth, and the solid particles are deposited in the form of a cake layer on the upstream side of the filter cloth. This method is generally suitable for treating suspensions having a relatively small liquid solid and a relatively fine solid particle.
(3) Pressure filtration. Use high pressure air 785 kPa (8 kg / cm ² ) or high pressure water 883 ~ 1569 kPa (9 ~ 16 kg / cm ² ) to fill the diaphragm mounted on one side or both sides of the filter chamber, by means of expansion By uniformly pressing the filter cake, a filter cake having a very low water content can be obtained. It is generally suitable for the treatment of fine-grained materials that are difficult to filter. In recent years, pressurized-vacuum combined filtration has been developed.
(4) Centrifugal filtration. The centrifugal force acts to cause the liquid in the suspension to be scooped out, and the particles are trapped on the surface of the filter cloth. The centrifugal force field can provide a stronger filtering driving force than the gravity field, and the separation speed is high and the effect is good. Suitable for processing slurries containing tiny solid particles.
B Filter media. There are many types of commonly used filter media, and the main ones can be divided into three categories:
(1) Granular medium. Such as fine sand, gravel, glass slag, charcoal, bone charcoal, acid chalk and so on. Such media, the particles are hard and can be deposited into layers, and the fine pores between the particles are sufficient to trap suspended solids while allowing only liquid to pass. For example, sand filters in urban and factory water supply facilities are constructed using such media.
(2) Fabric medium (or filter cloth medium). It is a filter cloth woven from natural or artificial fibers. Materials used are cotton, hemp, rayon and wool as well as various metal wire. Such media most widely used, especially in cotton canvas, nylon, rayon, wool woolen cloth used in the beneficiation plant is most prevalent.
(3) Porous ceramic or plastic media. Such media are used for sand filters in the laboratory and for special filter tanks for drinking water. The purpose of the filtration is to obtain a filtrate containing a lower water content or a filtrate containing no solids. According to the difference in properties, the filter cake can be divided into two types, namely, an incompressible filter cake and a compressible filter cake. The former consists of non-deformable particles, and mineral crystals fall into this category. The latter consists of amorphous particles, mainly colloidal filter residues such as lead hydroxide and various hydration precipitates. When the incompressible filter residue accumulates on the filter medium to form a filter cake, the mutual arrangement of the individual particles and the pores between the particles do not change greatly due to the increase of the pressure. However, when the compressible filter residue is filtered, the pores between the particles and the particles become significantly smaller as the pressure increases, and thus the flow of the filtrate is hindered.
If the concentrate filter cake of the ore dressing plant does not contain ingredients with special recycling value, it is generally not washed. However, in the hydrometallurgical (such as electrolytic copper, manganese, etc.) the factory, to the filter cake after washing, in order to fully recover the filtrate, and the filter cake was purer, but also to ensure product quality can enhance the recovery of useful components.
For compressible filter cakes, the pores between the particles narrow as the filtration pressure increases, and sometimes the channels are clogged due to the particles being too fine. In this case, some coarse-grained materials can be mixed into the suspension to improve the properties of the slurry, forming a loose filter cake and improving the filtration efficiency. These mixed substances may be the same ingredients or other materials or agents, collectively referred to as filter aids. [next]
(two) filtering theory
Theoretical studies on the filtration of liquid-phase materials consisting of liquid-solid two-phase materials through pores began in the late 19th century. The theoretical study of vacuum filtration in industrial production began only in the early part of this century. With the development of industrial production and technology, the requirements for accurately calculating and selecting filtration equipment are more urgent. In recent years, there have been some developments in filtration theory, but the development is still very slow.
The problems involved in the study of filtration theory are more complicated. For example, only the resistance of the filter is related not only to the weaving method of the filter medium (filter cloth), the shape, size and density of the pores, the surface roughness of the filter cloth, the expansion ratio and the breakage rate, but also The extent also depends on the resistance of the filter cake layer on the surface of the filter cloth, which in turn depends on the nature of the slurry, the temperature of the filtrate, the looseness of the filter cake and the internal structure, such as the size, shape, and Factors such as the mutual position within the cake, the porosity of the filter cake, the pore size and the curvature of the pores. However, most of the factors that determine the characteristics of the filter cake are related to the pressure applied to the filter. Therefore, the measurement of the filtration resistance is difficult to find the exact theoretical formula. In order to facilitate research problems, people have to resort to certain assumptions about the relationship between changes in filtration resistance and other factors when conducting filtration theory studies. This makes the filtering formula extremely complicated and its practical value is greatly reduced. Thus, filtration theory is not yet available to people to more accurately calculate the full range of filtering equipment. However, it can help explain some of the general conditions and influencing factors in the filtration process. So far, the pre-calculation and rational operation of the vacuum filter for industrial applications is decisive, and it is still the empirical data obtained from the correct simulation test and actual production.
Regardless of the type of filter media and the source of the driving force of the filter, the productivity of the filter is determined by the rate at which the filtrate passes through the filter cake and the filter media. Through multiple filtration experiments, it can be determined that when the filtered liquid flows through the pores of the filter slag and the filter medium, the fluid is in a laminar flow state. Accordingly, according to the law of laminar motion of liquid in the capillary channel, the filtration rate expression can be derived (ie, the flow rate of the filtrate passing through the filtration area of ​​1 m ² per unit time, m ³ / (m ² · s), but due to the filter cake and The number, radius and bending degree of the capillary in the filter cloth are difficult to measure and have no practical value. Later, further analysis, and experimentally proved that under certain operating conditions, the relevant parameters of the capillary, the filtration area and the viscosity of the liquid are all It is a constant. The filtration speed at this time varies only with the applied driving force and the thickness of the filter cake. On this basis, according to the concept of driving force and resistance, a new understanding of the filtration speed is proposed, namely the filtration speed and The filtered thrust is proportional to the resistance of the filter. In addition, the volume of the filtrate ( ^m3 / sec) obtained per unit time during the filtration process is called the filtration rate. After reasonable assumptions and derivation, the filtration rate can be established. The general relationship between the relevant factors is:

Where V - the actual filtrate volume, m ³ ;
p———the pressure drop across the capillary, which can be used to replace the vacuum gauge pressure before the filter cake, pa;
t———Filter time, s;
——the viscosity of the liquid, pa · s;
ρ———the resistance of the unit thickness of the incompressible filter cake, ie the specific resistance, (m ² ) ^-1 ;
W———The volume of the filter cake contained in the unit volume filtrate, no dimension or m ³ /m ³ ;
A———Filter area, m ² ;
V ₀ — The volume of filtrate obtained from the start of filtration to the filter cake equal to the resistance of the medium, referred to as the equivalent filtrate volume of the filter medium, or the virtual filtrate volume, m ³ .[next]
When the filter cake is compressible, the change in resistance is:
ρ=ρ'p ^s (2)
Where S——— is the compressibility factor of the filter cake, determined by the test. S = 0 of the incompressible filter cake;
ρ′———The specific resistance of the filter cake when the pressure is 98.1 kPa (1 kg/cm ² ), 1/m ² .
Therefore, for the compressible filter cake, according to formula (1) and (2):

The 3 formula is called the basic equation of filtration, which indicates the relationship between the filtration rate at any instant in the filtration process and various related factors, and is the basic basis for filtering calculation. This formula is suitable for compressible filter cakes and incompressible filter cakes.
When applying (3) for the calculation of the filter, it is necessary to integrate the formula for the specific way in which the process is performed. At the time of integration, one of the three independent variables in the equation, namely, the gauge pressure p, the filtrate volume V, and the filtration time t, must be maintained. In fact, the filtering operation has three modes: constant pressure, constant speed, and constant speed after constant speed. The concentrator filtration operation works more at constant pressure, and constant velocity filtration is less common. At the beginning of the filtration, because there is no filter cake on the surface of the medium, the filtration resistance is the smallest. If the maximum pressure is suddenly applied, the fine particles will be washed through the medium pores, causing the filtrate to turbid or block the pores.
The filtration performed on the filter is constant pressure filtration, and only the theoretical calculation of constant pressure filtration is now discussed.
In the constant pressure filtration, the filter cake is continuously thickened so that the filtration resistance is continuously increased, but the driving force (pressure) of the filtration is constant, and thus the filtration rate is gradually reduced. Therefore, except for V and t are variables in (3), all other parameters are constant. If order:

The integral form of equation (3) can be found as:
∫(V+V ⁰ )dV=kA ² p ^1-s ∫dt
When the filtration conditions change, the filtration time is from 0→t ₀ ; then from t ₀ →t+t ₀ . The filtrate volume is from 0→V ₀ and then V ₀ →V+V ₀ .
The filtration time referred to herein refers to the sum of the virtual filtration time t ₀ and the actual filtration time t; the filtrate volume refers to the sum of the virtual filtrate volume V ₀ and the actual filtrate volume V. Then two integrals obtained under the above two conditions of change:
∫ ^V ₀ (V+V ₀ )d(V+V ₀ )=kA ² p ^1-s ∫ ^t ₀ d(t+t ₀ ) (5)
And ∫ _V0 ^V+V0 (V+V ₀ )d(V+V ₀ )=kA ² p ^1-s ∫ _t0 ^t+t0 d(t+t ₀ ) (6)
Integral on the second formula and order: K=2kp ^1-s (7)
The following two formulas are available: V ² ₀ = KA ² t ₀ (8)
And V ² +2V ₀ V=KA ² t (9)
From this, the filter equation for the filter cake formation time can be obtained:
(V+V ₀ ) ² =KA ² (t+t ₀ ) (10)[next]
The formula (10) is called the constant pressure filtration equation. It shows that the relationship between filtrate volume and filtration time during constant pressure filtration is a parabolic equation, as shown in Figure 1. The Ob segment of the curve in the figure represents the relationship between the actual filtration time t and the actual filtrate volume V; and O ₀ O represents the relationship between the virtual time t ₀ corresponding to the medium resistance and the virtual filtrate volume V ₀ .

When the filter medium resistance is negligible, ie V ₀ =0, t ₀ =0, the formula can be simplified as:
V ² =KA ² t (11)
During the filtration process of the concentrator, since the resistance of the filter cake is much larger than the resistance of the medium, only the resistance of the filter cake can be considered in the calculation, and the resistance of the medium is not calculated. The calculation of the filtration process in the water treatment system is just the opposite. If order:

Then (8) to () 10 can be written as:
q ² ₀ =Kt ₀ (14)
q ² ₀ +2q ₀ q=Kt (15)
(q+q ₀ ) ² =K(t+t ₀ ) (16)

The formula (16) is also called a constant pressure filtration equation.
K (7) in the constant pressure filtration equation is a constant determined by the material characteristics and the difference between the filtration pressures. It is called the filter cake constant, and its unit is m ² / sec. t ₀ and q ₀ reflect the resistance of the filter medium. The constants are called dielectric constants, and their units are respectively seconds and meters ³ / m ^{2, the} total filter constant, which can be measured by the filtration test. After obtaining the above test parameters, the filter area A required to process a certain amount of slurry per unit time can be obtained by substituting the formula (11) and substituting K and k values:

Where μ—the viscosity of the liquid, Pa · s;
p———Filtered vacuum gauge pressure, Pa;
ρ, W, V, S——— Same as (1) and (2).

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