The flow of any liquid through any porous medium offers a resistance to its flow. The rate of filtration in such cases is expressed as:
Filtration Rate= Driving force/Resistance by filter medium
Rate is expressed as the volume of filtrate per unit time (dv/dt) and gets affected by various factors,
It can be explained by,
Kozeny-Carman Equation.
Darcy's Equation.
Poisulle’s Equation.
Poisulle’s Equation:
Poiseullie considered that filtration is similar to the streamline flow of liquid under pressure through capillaries.
Poiseullie's Equation is
Where,
V = rate of flow, m³/s (l/s)
AP= Pressure difference across the filter, Pa
r = radius of capillary in the filter bed, m
L = thickness of filter cake (capillary length), m
n = viscosity of filtrate, Pa.s
If the cake is composed of a bulky mass of particles and the liquid flows through the interstice, then flow of liquids through these may be expressed by this equation.
Darcy’s Equation:
Poiseullie's law assumes that the capillaries found in the filter are highly irregular and non-uniform.
Therefore, if the length of capillary is taken as the thickness of bed, a correction factor for radius is applied so that the rate is closely approximated and simplified.
The factors influencing the rate of filtration has been incorporated into an equation by Darcy, which is:
Where, K = permeability coefficient of cake, m²
A= surface area of porous bed (filter medium), m²
Other terms are same as the previous equation.
K depends on characteristics of cake, such as porosity, specific surface area and compressibility.
Other terms are same as previous equation
Permeability may be defined quantitatively as the flow rate of a liquid of unit viscosity across a unit area of cake having unit thickness under a pressure gradient of unity.
This equation is valid for liquids flowing through sand, glass beds and various porous media.
This model is applied to filter beds or cakes and other types of depth filter.
This equation is further modified by including characteristics of K by Kozeny-Carman.
Kozeny-Carman Equation:
The Kozeny-Carman equation is widely used for filtration.
Where,
ε = porosity of cake (bed)
S = specific surface area of particles comprising the cake m²/m³
K = Kozeny constant (usually taken as 5) Other terms are same as previous equations
Limitations:
It does not consider the fact that depth of granular bed is lesser than the actual path traversed by the fluid.
The actual path is not the same throughout the bed, but it is sinuous or tortuous.
