Types of Industrial Filtration and Derivation equations

# Types of Industrial Filtration and Derivation equations

There are a variety of types of filtration in the chemical industry, each with its own benefits and drawbacks. Some common types of filtration are Activated carbon filtration, Carbon adsorption, High-pressure filtration, Magnetic filtration, and Pressure filtration.

There mainly are two types of Filtration (i) Constant Pressure filtration and (ii) Constant Rate Filtration.Â

## Difference between constant rate filtration and constant pressure filtration

### (i) Constant Pressure Filtration

Principle of constant pressure filtration: The method in which the pressure drop over the filter is held constant throughout a run so that the rate of filtration is maximum at the start of filtration and decreases continuously towards the end of the run is called Constant pressure Filtration.

If the outside pressure is constant, constant pressure filtration is carried out by applying a certain pressure at the inlet and maintaining it constantly throughout the run.

The constant pressure filtration, the application of high initial pressure results in a low rate of filtration as the first particles filtered will be compacted into a tight mass that largely fills the run whole cycle and is operated at less than maximum capacity.Â

#### Constant Pressure Filtration DerivationÂ

Filter medium resistance equation:- dt/ dV = u/Aâˆ†[e Î± V/A + Rm]. --------1.0

When âˆ†P is constant the only variables in the upper equation are V and t. When t=0, V=0 and âˆ†Pm henceÂ

uRm/ Aâˆ†P = ( dt/dV)â‚€ = 1/ qâ‚€ --------1.1

Equation (1.0) may therefore be written as:

dt/dV = 1/q = Kc V + 1/qâ‚€

dt/dV = u Î± c V/AÂ² âˆ†P + u Rm/A âˆ†PÂ

dt/dV = Kc V + 1/qâ‚€ -----1.2

Kc = u Î± c V/AÂ² âˆ†PÂ

Integrating equation 1.2 between the limits t =0 V= 0 and t = t' V= VÂ

à´½dt = à´½(Kc V + 1/qâ‚€)dVÂ

t = Kc VÂ²/2 + V/ qâ‚€ ------ 1.3

Rearranging equation (1.3), we get theÂ Constant pressure Filtration Equation.

t/V = Kc/2 V + 1/qâ‚€Â

Hence, a plot of t/V v/s V is will be linear with aa slogs equal to Kc/2 and an intercept of 1/qâ‚€. From the plot and equations (1.1) and (1.3) the values of Î± and Rm can be calculated.Â

### (ii) Constant Rate FiltrationÂ

The technique in which the pressure drops are varied usually from a minimum at the start of filtration to a maximum at the end of filtration so that the rate of filtration is throughout the run is called Constant rate filtration.

In constant rate filtration, nearly constant rate filtration is maintained by starting at low inlet pressure, and continuously increasing the pressure to overcome the resistance of the cake, until the maximum pressure is reached towards the end of the run.

#### Constant Rate Filtration DerivationÂ

When filtrate flows at a constant rate, the linear velocity u is constant

u = dV/dt/A = V/At -------- 1.4

We haveÂ

Î± = âˆ†Pc A/u Î¼ Mc -------1.5

and Mc = cV -------1.6Â

Substituting u from Equation (1.4) and Mc from Equation (1.6) into Equation (1.5) and rearranging, we getÂ

âˆ†Pc/Î± = Î¼c/t(V/A)Â² -------1.7

If Î± is known as a function of âˆ†Pc, and if âˆ†Pm can be estimated, then equation (1.7) can be used for relating the overall pressure drop to time when the flow rate of the filtrate is constant.

A more direct use of equation (1.7) can be made if the equation for cake resistance is Î± = Î±â‚€ (âˆ†P)^s is substituted in equation (1.7) and if âˆ†Pc is substituted by (âˆ†Pc - âˆ†Pm) then the result isÂ

(âˆ†Pc)Â¹â”€s = Î±â‚€ Î¼ct (V/At)Â² =Â  (âˆ†Pc - âˆ†Pm)Â¹â”€s ----1.8

The simplest method of correcting the overall pressure drop for the pressure drop through the medium is to assume the filter medium resistance constant during a given constant rate filtration. Then by equation âˆ†Pm/Rm = uÎ¼, âˆ†Pm is also constant in equation (1.8). As the only variables in equation (1.8) are âˆ†P and tÂ

(âˆ†Pc - âˆ†Pm)Â¹â”€s = Kr t -----1.9Â

Where

Kr = Î¼ uÂ² c Î±â‚€

Equation 1.9 is a constant rate filtration equation.Â

Take these Notes is, Orginal Sources:Â Unit Operations-II, KA Gavhane