Chapter 8. FILTRATION PART II. Filtration variables, filtration mechanisms

8.1 Filtration variables – input of the filtration process I. Filtration variables are divided onto three groups: 1.Variables of filter material 2.Variables of filtered particles 3.Variables of filtration process

8.1.1 Variables of filter material: Filtration area Filter thickness Density and surface density of filter Uniformity of fibrous material Parameters of filter material surface interactions between the filter material and filtered particles electrical properties mechanical characteristics (tenacity, elongation...) resistance against surrounding factors (heat, solvents...) Parameters of fibers fiber diameter, fiber fineness shape of fiber cross-section fiber surface preparations Mechanical characteristics Filter structure filter density gradient fiber orientation

8.1.2 Variables of filtered particles Particle size Distribution of particle size Concentration of particles Shape and surface of particles Particle density Electrical properties Variable of filtration process Face velocity (speed of filtered particles in front of filter) Viscosity of the flow Temperature, pressure, humidity

8.2 Filtration mechanisms of deep filtration (the way how are particles captured) R fiber charge on the fiber surface diffusional deposition inertial impaction direct interception capture by electrostatic forces streamlines (air moving trajectory) Total filtration efficiency Ec is total efficiency, E r is efficiency of direct interception mechanism represented by parameter N r, E i is efficiency of inertial impaction represented by Stokes number Stk, E d is efficiency of diffusional deposition mechanism represented by Peclet number Pe and E e is efficiency of electrostatic mechanism represented by the parameter Nq. Mechanisms: direct interception inertial impaction diffusional deposition capture by electrostatic forces.

8.2.1 Direct interception Direct interception occurs when airborne particles behave in an entirely passive way with respect to the airflow. Airborne particles follow the streamline, which in steady state are independent of the air velocity. Particle will be captured when it is close to the fiber. This mechanism is independent of air velocity, air viscosity and density. Particle must be small, because inertial effects and external forces are neglected. This type of mechanism is common for simple respirators made from fibers of about 20  m, which operate in filration velocity about 0,04 m/sec. Furthermore interception acts along with other filtration processes. Parameter of direct interception: N r = dp/df (d p is particle diameter, d f is fiber diameter) dfdf fiber streamlines (air moving trajectory) dpdp Relation between parameter N r and efficiency of direct interceptiom mechanism: E R  N r 2 ; the simpliest relation is: E R =N R 2 / , more exactly: where  =-0,5.ln(c)-0,75 is hydrodynamic factor and m = 2/(3.(1-c))

8.2.2 Inertial impaction Any convergence, divergence or curvature of streamlines involves acceleration of the air, and under such conditions a particle may not be able to follow the airflow. What particle does depends upon its mass (inertia) and upon the Stokes drag exerted by the air. Stokes drag is defined as a force which acts on the moving sferical object inside of viscous liquid: F = 3. . .d p.v (where F is the force, d p is the particle diameter,  is the dynamic viscosity and v is the face velocity of the airflow). fiber inertial impaction streamlines (air moving trajectory) Intensity of the point particle inertia is determined by Stokes number: where d p is particle diameter,  is particle density, U is air face velocity,  is air viscosity and d f is fiber diameter. Efficiency of inertial impaction E i depends on the intensity of the point particle inertia. If inertia is negligible then E i will be zero, if the inertia is infinite then E i will be 100 %.

Relation between the Stokes number and efficiency of inertial impaction: For low Stokes number efficiency is lead by direct interception: E ir =E R +(2.  ) -2.J.St, where E R is efficiency of direct interception,  is hydrodynamic factor dependent on packing fraction c and J is constant dependent on c and parameter of direct interception Nr. For high Stokes number efficiency of inertial impaction is defined: E I =1-(  /St), where  is constant dependent on flow field.

8.2.3 Diffussional deposition The trajektories of individual small particles do not coincide with the streamlines of the fluid because of Brownian motion. With decreasing particle size the intensity of Brownian motion increases and, as a consequence, so does the intensity of diffusion deposition [Pich J,1964]. However the air flow effects on the particles motion too. Thus the real motion of small particles depends on Brownian motion and air flow. Brownian motion is determined by diffusion coefficient D defined by the Einstein equation: where k B is Boltzmann constant, K is Kelvin temperature,  is air viscosity, d p is particle diameter and Cn is the Cunningham correction, which involve aerodynamic slip flow of particles: where  is mean free path of molecules (at NTP it is 0,065  m) and A, B, Q are constants (A=1,246; B=0,87; Q=0,42) [Brown RC, 1993]. diffusional deposition streamlines (air moving trajectory) fiber

Coefficient of diffusional deposition: Capture of particles by a diffusional deposition will depend on the relation between the diffusional motion and the convective motion of the air past the fiber. Dimensionless coefficient of diffusional deposition N D is defined: where d f is fiber diameter, U is air flow velocity and P e is named „Peclet number“. Diffusional capture efficiency: According to Fokker-Planck equation was aproximated relation between the N D (or 1/P e ) and diffusional capture efficiency E D = 2,9.  -1/3. Pe -2/3 where  is hydrodynamic factor (  = -0,5. ln(c)-0,75 by Kuwabara) [Brown RC, 1993]. Previous equation was verified by experiments with model filters with the some  and observed functional dependance was the some with little different numerical coefficient: E D = 2,7. Pe -2/3 When we calculate with the slip flow (see chapter 9) the resulting capture efficiency is bigger.

8.2.4 Electrostatic forces: Both the particles and the fibers in the filter may carry electric charges. Deposition of particles on the fibers may take place because of the forces acting between charges or induced forces. [Pich J, 1964]. The capture of oppositely charged particles is given by coulomb forces. The capture of neutral particles comes about by the action of polarisation forces. We can define three cases of interaction between particle and fiber. Used equations were derived from Coulomb´s law. fiber charge on the fiber surface capture by electrostatic forces streamlines (air moving trajectory) 1. Charged particle, charged fiber where q is the particle charge, Q is fiber charge per unit lenght of fiber and x is the distance between fiber and particle. 2. Charged fiber, neutral particles where D 1 is the dielectric constant of the particle and d p is particle diameter. 3. Charged particles, neutral fiber where D 2 is dielectric constant of the fiber and d f is fiber diameter.

Coefficient of electrostatic mechanism, efficiency of electrostatic mechanism We can interpret this parameter as a ratio of electrostatic forces to drag forces. From this parameter were derived equations for efficiency [Pich J, 1964]. B is mechanical mobility of the particle, U 0 is the velocity far form the fiber, d f is fiber diameter, d p is particle diameter and  is viscosity Coefficient of electrostatic mechanism Efficiency of electrostatic mechanism Charged fiber and charged particle Charged fiber and neutral particle Carged paricle and neutral fiber

8.3 Filtration variables vs.capture efficiency of filtration mechanisms Efficiency of each filtration mechanisms Relations how some filtration variables increase or decrease or not affect the efficiency of each filtration mechanisms filter density fiber diameter particle diameter particle mass face velocity viscosity of air relative charge direct interception -  ---- inertial impaction ??  - diffusional deposition  -  - electrostatic deposition -  - 

Efficiency of each filtration mechanisms Numeric relations between the filter variables and capture efficiency of each mechanisms filter density c fiber diameter d f particle diameter d p particle mass  face velocity U viscosity of air  relative charge q, Q direct interception - 1/d f 2 dp2dp inertial impaction 1/(ln c) 2 1/d f or 1 – k.d f d p 2 or 1-1/d p 2  or 1-k/  U or 1-k/U 1/  - diffusional deposition 1/(ln c) 1/3 1/d f 2/3 -1/U 2/3 1/  2/3 - electrostatic deposition -1/d f 1/d p or d p 2/3 or 1/d p 1/2 - 1/U or 1/U 1/3 or 1/U 1/2 1/  q.Q or Q 2/3 or q

8.4 Filtration mechanism of flat filtration – „Sieve effect“ E s = 1 for d p  d pore; ; E s = 0 for d p < d pore, where E s is efficiency of sieve effect and d pore is pore diameter. Relation between fiber and pore diameter according to Neckar [Neckar B., 2003]: () where q is fiber shape factor (zero for cylindrical fibers), c is packing factor, a and k are constats related to filter structure (usually a is ½). For cylindrical fibers with hexagonal structure is k = 2 -1/2.