# Filters calculation and selection

**Contents:**

## General information

Generally there are no chemical agents and compounds in pure form in the nature, but they are usually in compounding condition with other agents.

Muddy water, dusty air, fog – all of them is samples for multiphase medium. Agent compound doesn’t imply the forming of b links between its agents, that’s why the physical methods are usually enough for its separation.

In the early stages of its development, humankind generally used agents in that form as they were found in nature. So necessary condition for forming a large settlement was a source with drinking water, i.e. with low salt content and different kinds of impurities such as lower algae etc. However due to small population size and density there were no needs to look for water purifying methods for its consumption. The low human needs didn’t require to look for ways for separating of other kinds of found mixtures.

As in the case with many aspects of life, the development of culture and technology led to increased needs. In particular, the need in abstraction of pure agents from mixtures or, conversely, purification from impurities. These requirements could be determined by the urgent need, so to improve the quality of drinking water, about 2000 years BC it was started with additional purification, passing through a layer of sand or coal, which allowed to remove the smell and small impurities. In some cases, the task of agent mixture filtering could arise unexpectedly and require an urgent decision, which occurred on April 22, 1915 on the German-French front of the First World War, when Germany used chemical war gases for the first time in the history, what was a cause for the next invention and creation of a gas mask.

However, the greatest impetus to the study of filtration systems resulted from rapid development of science and technology in the 17th and next centuries. Many chemical reactions proceed in a liquid medium, and reaction products can already be solid formations. Requirements for many industrial processes do not allow using of ordinary water without additional purification stages, some of which concludes filtration, sedimentation, etc. The same applies to cleaning of the air drawn from the atmosphere or, conversely, discharged from the air unit.

The first filtration units, like the same sand filter, were extremely simple designed, made mostly of natural materials and did not require any serious calculation or research work. The increased need for filtration led to the development of filter equipment, which created a great variety of both in designing and in the choice of the physical or physico-chemical separation principle.

## Classification of mixtures. Formula for the mass fraction of the dispersed phase. Suspension viscosity calculation

A mixture is a physicochemical system, which contains at least two components. The mixture can be separated by physical methods into components, without any chemical transformation of the components. The components of the mixture can be in either one or in different aggregate states. According to this principle, two types of mixtures are to be extracted:

- Homogeneous (uniform)
- Heterogeneous (nonuniform)

In the general case, systems consisting of two or more non chemical interacting phases, in which one of them is spread in volume of the other, are called dispersed phases. In technological processes, dispersed systems are most common, in which a homogenous (continuous) phase is a liquid or a gas, i.e. different kinds of emulsions and suspensions. If we consider the easiest cases of dispersed system with two components, then the following types of phases are to be extracted:

- Dispersion (continuous)
- Dispersed (discontinuous)

Types of dispersed systems | Dispersed (discontinuous) medium | |||
---|---|---|---|---|

Gaseous phase | Liquid phase | Solid phase | ||

Dispersion (continuous) medium | Gaseous phase | Doesn’t form any dispersed systems | Fogs | Dusts Steams |

Liquid phase | Foams Gas emulsions |
Emulsions | Suspensions | |

Solid phase | Solid foams | Solid emulsions | Alloy Composite |

Despite the fact that the components included in the mixture don’t enter in chemical reactions with each other, its physical properties may vary from those of its components. Most often, a certain physical parameter of the mixture, such as density, will be between the values of the same parameter for its components. The main role here is played by the quantitative ratio of mixture components. Usually for dispersed systems the volume (Cv) or mass (Cm) concentration of the dispersed phase, expressed in fractions, is to be identified. So if we know the density of continuous and dispersed phase, as well as the volume fraction of dispersed phase, we can determine the density of formed system:

Ρ_{дс} = С_{v}·ρ_{д}+(1-С_{v})·ρ_{с}

ρ_{с} – homogenous phase density, kg/m³;

ρ_{д} – dispersed phase density, kg/m³;

ρ_{дс} – dispersed system density, kg/m³;

С_{v} – volume part of dispersed phase.

The same formula for the mass fraction of dispersed phase is as follows:

ρ_{дс} = [ρ_{д}·ρ_{с}] / [ρ_{д}-С_{м}·(ρ_{д}-ρ_{с})]

In case of suspensions, their viscosity is the result of viscosity change of the liquid phase under influence of solid particles of dispersed phase. Both the volume concentration of dispersed phase, and the size and shape of solid particles are important. If the volume fraction of disperse phase is less than 0.2, the viscosity calculation of suspension can be carried out using empirical formula:

Μ_{Sus} = μ_{ж}·(1+∑^{n}_{i=1}(a_{i}·C^{i}_{v}))

μ_{сус} – suspension dynamical viscosity, Pа·sec;

μ_{ж} – liquid dynamical viscosity (homogenous phase), Pa·sec;

С_{v} – volume fraction of dispersed phase;

i, n, a – empirical coefficients.

With a volume fraction of dispersed phase more than 0.2 suspensions already begin to behave like non-Newtonian fluids, i.e. their viscosity begins to depend on gradient of flow speed.

## Methods of mechanical separation of mixtures

Mixtures are separated due to differences in physical properties of its components. It is important to note, that most types of mixtures in one way or another are systems that are unstable and exposed to the breakdown process with time. However, natural breakdown as a rule is slow and takes a lot of time, which is unprofitable by providing technological processes. Therefore, in special units methods of separation processes intensifying are used. Mostly, this applies to heterogeneous mixtures.

In the case of homogeneous mixtures, the problem of their separation becomes much more complicated. Such systems often prove to be stable, i.e. they don’t disintegrate into components over time, and are not subject to natural separation. So familiar to us atmospheric air is a mixture of gases, mainly oxygen and nitrogen, and without additional manipulation it will be impossible to achieve its separation into components. Another example is the alloys of metals, which due to their structure are weakly subject to internal changes without additional external influence.

However, even when during separating of heterogeneous systems, one can meet with difficulties. True solutions and colloid systems are stable, since particles exposed to Brownian motion due to their small size are supported in a suspended condition and do not breakdown with time. Only coarsely dispersed systems are subject to breakdown. However in industry the most common are heterogeneous coarsely dispersed systems in which the dispersion medium is a liquid or a gas. The methods for their separation are described below.

**Settling. Specific surface of porous medium and solid phase. Calculation of hydraulic pore diameter**

Settling is one of the simplest separating methods of coarse-dispersed heterogeneous systems that doesn’t require complex equipment for execution. Driving force in this process is gravity force acting on solid particles (in suspensions, dusts and fumes) or drops (in fogs or emulsions). For emulsions whose dispersed phase is lighter than dispersion one, the force of ejection will act as driving force, making drops buoy to the surface of continuous phase.

This process is used mainly for primary rough separation of mixtures, since a small driving force allows efficient separation of only enough large solids or liquid particles. Primary separation is used to reduce the cost of total process, reducing the load on more complex and expensive next stages of fine cleaning. Sedimentation allows also to carry out densification of suspensions or their classification by solid particles. The most common devices, working on sedimentation principle, are settlers (purification of liquids) and dust-settling chambers (gas purification).

The mass of settled solids in suspensions forms a settlement. In the vast majority of cases, the structure of precipitation results in extremely difficult due to the different shape of solid particles and their chaotic conglomeration. It is characterized by such parameters as porosity (ε) representing the proportion of pore volume in residual volume, pore size and specific surface area (f_{sp}). In this case, the specific surface of the porous medium (f_{pm}) and the specific surface of the solid phase (f_{sp}) are to be extracted.

f_{пс} = F_{т}/V_{о}; f_{тф} = F_{т}/V_{т}; f_{тф} = f_{пс}/(1-ε); ε = (V_{о}-V_{т})/V_{о}

where:

V_{о} – settlement volume, m³;

F_{т} – total solid particles area in settlement volume V_{о}, m²;

V_{т} – total solid particles volume in settlement volume V_{о}, m³.

It is obvious that the pores shape and size in settlement can be very different and practically are not direct measurable. To describe them, parameter such as the hydraulic pore diameter (d_{г}) is used. In the ideal case of a spherical solid particle diameter (d), the hydraulic pore diameter can be written as follows:

d_{г} = 2/3 · (ε·d)/(1-ε)

**Settlement humidity and saturation. Calculation**

The forming settlement also carries a part of liquid phase, and liquid content in settlement is characterized by a parameter called humidity (ω). Mass (ω_{m}) and volume humidity (ω_{v}) is to be singled out. The first shows the fluid mass per settlement mass unit, and the second - the volume of liquid per settlement volume unit. Two of these quantities can be related with the density of solid and liquid phases:

ω_{о} = [ω_{м}·ρ_{т}/ρ_{ж}] · [(1-ε)/(1-ω_{м})]

where:

ρ_{s} – solid phase density, kg/m³;

ρ_{l} – liquid phase density, kg/m³.

**Settling filters. The equation of the effective forces per particle when settling in a filter**

As already mentioned above, the main driving force of the settling process is the gravity acting on the particles of dispersed phase, and the particle settling speed of dispersed phase can be considered as main process characteristic. Let’s consider a spherical particle with mass (m_{t}) and diameter (d) moving in a viscous medium, which is impacted by series of forces: gravity (F_{a}), Archimedean buoyant force (F_{а}) and medium resistance force (F_{с}). According to this, we write the general equation of forces impacting on the particle:

F_{т} - F_{а} - F_{с} = m · (dw/dt)

where:

F_{т} = m_{т}·g = ρ_{т}·V·g = ρ_{т} · (π·d³)/6 · g;

F_{а} = m_{ж}·g = ρ_{ж}·V·g = ρ_{ж} · (π·d³)/6 · g;

F_{с} = ζ · S · (ρ_{ж}·w²)/2;

(ρ_{ж}·w²)/2 – particle motion energy

ζ – coefficient of resistance;

ρ_{т} – solid particle density, kg/m³;

ρ_{ж} – liquid density, kg/m³;

w – particle motion speed, m/sec;

S – The midlength section, i.e. the most section by the body of plane which is perpendicular to the motion direction (for a spherical particle S = (π·d²)/4), m².

As a rule, the acceleration time of particle is small, and it quickly goes to motion mode at constant velocity, so that we can lossless neglect the right side of the equation m dw / dt, taking it as 0. This implies:

F_{т}-F_{а}-F_{с} = 0

The settling mode also has a great impact on the determining of final settling speed. For each of modes in which the liquid flows around the particle, the resistance coefficient value is calculated in different ways, and therefore the formula for settling speed calculating also changes. This makes the speed calculation according to the obtained formula uncomfortable, since without knowing the settlement mode in advance, it is necessary to use the method of consequential iterations while calculating.

There is another method for calculating of settlement speed, based on the use of Archimedes number (Ar), which physical meaning is in the balance of gravity, viscosity, and Archimedean force. Like Reynolds number (Re), the Archimedes number has border values corresponding to the transition from one mode to another. Below is a table of settlement modes and respective values of Re and Ar, as well as formulas for calculating of resistance coefficient ζ.

Mode | Laminar | Transient | Turbulent |
---|---|---|---|

Value Re | Re < 2 | 2 < Re < 500 | Re > 500 |

Value Ar | Ar < 36 | 36 < Ar < 83000 | Ar > 83000 |

Resistance coefficient fromula (ζ) | ζ = 24/Re | ζ = 18,5/Re0,6 | ζ = 0,44 |

Re = (ω·d)/ν; Ar = [g·d³·(ρт-ρж)] / [ν²·ρж]; Re²·ζ = (4/3)·Ar |

The formulae mentioned above relate to the process of settling solids in liquid; they can be also applied for settling liquid droplets in gas. However it is recognized that the speed of settling a drop can be higher by half as the speed of settling a solid similar in size. This is because there is internal liquid circulation in the drop, which decreases by surface-active reagents or admixtures. The drops with low liquid circulation are called “hard” and their behavior can de described though the formulas used for solids. The increase of speed for non- contaminated drops has its limits also that conform to the critical drop diameter (d_{кр}). The drop diameter value is calculated as the sphere diameter with volume equal to its volume. The drops with the overcritical diameter often have some shape changes during settling; therefore they are called “oscillating”. The further increase of the oscillating drop causes the return slight reduction of the settling speed.

Also by hindered settling, the non-uniformity of speeds occurs as per the device height that results from the dispersed phase rising flow coming from the bottom and being displaced by the mass of settling particles of the dispersed phase, and as result it leads to retardation in the bottom layer. Moreover, the larger particles settle down faster in spite of partial grading of the settling speeds. This leads to formatting some settling zones. We can see that there is an almost completely free settling zone in the upper part of the device, a hindered settling zone in the middle of the device and a sediment layer on the bottom.

**Speed calculation for hindered settling in a settler-filter**

It is evident that hindered settling speed differs from free settling speed. As a rule, the different empirical and semiempirical formulae are sued for this purpose. One method for hindered settling speed calculation (w_{ст}) is based on that w_{ст} is a function of free settling speed (w_{св}) and a volume ratio of the dispersed phase (ε). Two calculation formulae are applied depended on ε:

1) w_{ст} = w_{св} · ε² · 10^{-1,82·(1-}^{ε}^{)} (at ε>0,7)

2) w_{ст} = w_{св} · 0,123 · ε³ · [1/(1-ε)] (at ε≤0,7)

The universal formula suitable for all settlings modes (laminar, transient, turbulent) can be used also:

Re_{ст} = [Ar·ε^{4,74}] / [18+0,6·√(Ar·e^{4,75})]

where:

*Re _{ст} = (ρ_{ж}·w_{ст}·d)/μ_{ж}* – Reynolds criterion for hindered settling;

*Ar = [ρ*– Archimedes criterion;

_{ж}·g·d³·(ρ_{т}-ρ_{ж})]/μ²ρ

_{т}– solid density, kg/m3;

ρ

_{ж}– liquid density, kg/m3;

μ – dynamic liquid viscosity, cPs;

d – diameter of dispersed phase solids, m.

The value d can be used only when spherical particles with the same size are involved in settling. When non- spherical particles settle, the diameter of sphere as value d will be considered, which is equal to the mass of the settling particle:

d = ((6·V_{ч})/π)^{1/3}

where:

V_{ч} – particle volume, m3.

The correction index (ψ<1) is used to factor in deviations of shape and size, by which to multiply the received settling speed value:

ψ = 4,836·(V_{ч}^{2/3})/S_{ч}

**Settling in centrifugal force field. Centrifugal force acts on the particle in a filter**

A serious disadvantage of standard settling process is its limited moving force – the gravitational force. The process is carried out in the centrifugal force field for its intensification, which can be created specially and reach high values compared with the gravitational field of the earth.

Usually the centrifugal force field is created with one of two methods: a separating medium is fed into a rotating aggregate where the medium accepts rotating movements (centrifuge process); or the flow itself receives the rotating movement while the aggregate is fixed (cyclone process). As the name implies, settling centrifuges are used for the first method and cyclones (hydro cyclones) are used for the second one.

The centrifugal force as a process moving force can de indicated according to the formula:

F_{ц} = (m·w_{r}²)/r

where:

Fц – centrifugal force acts on the particle, N;

m – particle mass,kg;

r – circle radius of particle rotation, m;

w_{r} – linear speed of particle rotation, m/sec.

To estimate the efficiency by centrifugal separation compared with standard one, the value as separation coefficient (K_{p}) equal to ratio of centrifugal force and gravitational force acting on the same particle:

K_{p} = F_{ц}/F_{т} = [(m·w_{r}²)/r] / [m·g] = [w_{r}²] / [r·g]

where:

F_{т} = m·g – gravitational force acting on the particle by mass (m).

As rotary speed is often used instead for rotary aggregates of the linear one, it is required to make some modifications to receive the separation coefficient through rotation frequency. Linear speed and frequency are related to the following formula:

w_{r} = 2·π·r·n

where:

n – rotation particle frequency (aggregate), с^{-1}.

Substituting the received result into the formula for the separation coefficient we receive as follows:

K_{p} = (2·π·r·n)²/(r·g) = (4·π²·n²·r)/g

It is clear from the received equation that essential increase of the separation coefficient is reached by increasing the rotation frequency but not the centrifuge/cyclone diameter. The separation coefficient value can significantly differ among aggregates that depend on their usage and application. On the whole, the value K_{р} for cyclones is calculated in hundreds and this figure for centrifuges is calculated in thousands as rotary speed is higher there.

**Filtration. Filtration speed calculation**

Filtration is a process of separating a dispersed medium through a porous diaphragm. Pores are made to pass through one phase free and lock the other one. In fact, the process of separating is carried out due to holding some components on the diaphragm. If a suspension is filtered, the liquid passed through the diaphragm is called “filtrate” and the solids remain in the filter - “settlement”.

Indeed the filtration process is more difficult as mainly the settlement layer forming on the diaphragm also has the essential impact on the process; it involves in filtration and becomes an additional porous wall. To be noted that hydraulic resistance of the filtration diaphragm is usually the same, more or less, within the process (except for when small particles hold inside pores reducing their passed size), while hydraulic resistance of the settlement rises with increase its thickness. It is clear that hydraulic resistance of the settlement is equal to zero at the beginning of the filtration process as there is not one. Other important factor for the settlement, impacting on the final hydraulic resistance value, is the ability to change or not change the porosity by increasing pressure. So the settlements are classified into compressible and non-compressible.

The filtration process can be under different conditions, therefore some modes are identified:

- Filtration at continuous differential pressure (at compressed air over the filtration diaphragm or discharging under the diaphragm);
- Filtration at continuous speed (suspension is fed by piston pump);
- Filtration at variable-pressure and speed (suspension is fed by centrifugal pump).

Filtration speed can be considered as filtrate’s volume passing per unit of time through unit of filtering surface:

w = dV/(S·dτ)

where:

w – filtration speed, m/sec;

V – filtrate volume, m3;

S – filtration area, m2;

τ – filtration time, sec.

Also experience has shown that filtration speed is directly proportional to differential pressure of filter and inversely proportional to liquid viscosity as well as hydraulic resistance created by diaphragm and settlement layer:

w = ∆p / [μ·(R_{фп}+R_{со})]

where:

μ – dynamic liquid viscosity, cPs;

R_{фп} – hydraulic resistance of filtration diaphragm, м^{-1};

R_{со} – hydraulic resistance of settlement layer, м^{-1}.

**Filtration in centrifugal force field **

The filtration process like the settling process can be intensified by its carrying out in centrifugal force field. The centrifuges with a slightly different structure compared with settling centrifuges are used for this purpose. Such centrifuge has a drum with screen surface and operates as a porous wall in the filter. Generally there are three process stages: settlement formation, settlement consolidation and then its mechanical drying.

The filtration process in filters and filtrating centrifuges as well as the calculation methods differ essentially. One of the differences is the irregularity in the distribution of the main moving forces. For example, centrifugal force in a filtrating centrifuge is nonuniform and rises by increasing a radius. Moreover, a round centrifuge causes the change in settlement area because of its layer growth.

However, the most important is the possibility to create the significant field of centrifugal forces. It can leads to that the particles contacting to the filtration wall will be distorted and clog up all passes and this will decrease the filtration speed essentially. The significant efforts acting on the settlement can also cause an excessive decrease of the settlement porosity at its bigger compressibility. Sometimes it is better to carry out the process in filters than in centrifuges though the centrifuge can reach higher pressures in liquid.

## The main principles of calculating and selecting the filters used for settling and filtration

**Settling filter**

Calculating and selecting the settling filters are based on the principle that the smallest particles of dispersed phase shall be separated from a medium to be cleaned, which are in the zone most difficult for settling. If this condition is ruled, then obviously the particles with big size will be settled also.

The zone most difficult for settling is the suspension surface where the particle going from the bottom travels over the biggest distance and it means it requires more time. Let us denote the time of settling the particles the most distant from the bottom by τ_{ос}.To provide settling of the particles of dispersed phase fully, their total residence time in an aggregate (τ) shall not be less τ_{ос}. If τ≥τ_{ос}_{, }it means, that a part of the settler will be out of the settling process; if τ≤τ_{ос}_{, }not all particles will settle by passing through the settler, it means, the settling process will be not complete.

As a simple example, let us consider the rectangular settler with length (l) and width (b), where a suspension with speed (v) flows, and height of liquid layer is (h). When residence time of the separate particle in the settler is follows:

τ = l/v

Because consumption of the cleaned liquid can be considered as the cross sectional area multiplied by flow speed (Q_{оч} = v·h·b)), the residence time of the particle in the aggregate can be expressed in the consumption value:

τ = (l·h·b)/Q_{оч} = (h·F)/Q_{оч}

where:

F – settling area of the settler, m2.

In turn, let us denote the settling speed of the particles of dispersed phase as (w_{ос}), then the settling time (τ_{ос}) of the particles the most distant from the bottom will be equal to:

τ_{ос}= h/ω_{ос}

The condition for the complete particles settling is the equation τ = τ_{ос}_{. }Using the formula received before we transform this equation as follows:

(h·F)/Q_{оч} = h/ω_{ос}

We express the value of settler settling area from the received equation:

F = Q_{оч}/ω_{ос}

As is obvious, the value F does not depend on height and width of suspension speed directly, therefore the values h and b can be selected based on practical condition. It is only essential to provide the laminar mode for liquid flow to create the condition the most optimal for settling.

**Calculating and selecting the industrial filters **

When the filtration process is carried out as periodic, usually there are three consecutive stages: filtration, settlement flushing, secondary operations. Each stage requires the certain time, and their sum determines the total filtration cycle time.

T = τ_{ф} + τ_{пр} + τ_{вс}

where:

T – total filtration cycle time , sec;

τ_{ф} – time for carrying out filtration, sec;

τ_{пр} – time for carrying out settlement flushing, sec;

τ_{вс} – time for carrying out secondary operations, sec.

Time for carrying out the filtration stage can be determined as per the formula:

τ_{ф} = [(r·V_{ос}·q²)/(2·∆p)] + [R_{фп}·q)/∆p]

where:

r – specific resistivity of settlement, m^{-2};

V_{ос} – settlement volume for filtrate volume unit;

q – specific capacity of filter, m³/m²;

Δp – pressure drop at filtration, Pa;

R_{фп} – hydraulic resistance of filtrating wall, m^{-1}.

Expressing q from the formula mentioned above we receive the design formula for specific capacity:

q = √([R_{фп}/(r·V_{ос})]² + [(2·∆p)/(r·V_{ос})]·τ_{ф}) - [R_{фп}/(r·V_{ос})]

Time to be required for flushing is calculated as follows:

τ_{пр} = [Q_{пв}·V_{ос}·q·(r_{пр}·V_{ос}·q+R_{фп})] /∆p_{пр}

where:

Q_{пв} – flow rate of flushed water related to settlement volume unit;

Δp_{пр} – pressure drop at flushing, Pa.

Time for carrying out the secondary operations is selected based on the condition that capacity of the periodic filter is maximal by execution of the equation:

τ_{вс} = τ_{ф}+τ_{пр}

Filter area is related to filtration cycle time through the following formula:

F = (Q_{ф}·T)/q

where:

F – filtration area, m²;

Q_{ф} – filter capacity for filtrate, m³/sec.

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