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الكلية كلية الهندسة     القسم  الهندسة البيئية     المرحلة 4
أستاذ المادة محمد عبد مسلم عبد الله الطفيلي       04/07/2018 07:05:21
GRAVITY SEPARATION THEORY

The removal of suspended and colloidal materials from wastewater by gravity separation is one of the most widely used unit operations in wastewater treatment. Sedimentation is the term applied to the separation of suspended particle that are heavier than water, by gravitational settling. The terms Sedimentation and settling are used interchangeably. A sedimentation basin may also be referred to as a sedimentation tank, clarifier, settling basin, or settling tank.
Particle Discrete Settling Theory (Type 1 settling)
The settling of discrete, non flocculating particles can be analyzed by means of the classic laws of sedimentation formed by Newton and Stokes. Newton’s law yields the terminal particle velocity by equating the gravitational force of the particle to the frictional resistance, or drag. The force balance is written so that the direction of gravitational force is positive. Therefore, a positive settling velocity means that the particle settles and a negative settling velocity means the particle rises. The gravitational and buoyant forces are given by F = ma, as follows:
F_g=ma= ?_p V_p g
F_b=ma= ?_(w ) V_p g
F_G=F_g-F_b= ?_p V_p g- ?_(w ) V_p g= ( ?_p- ?_(w ) )g V_p

Where:
m = mass, kg
a = acceleration, m/s2
Fg = gravitational force, (kg.m/s2 ) i.e. (N)
Fb = buoyant force, (kg.m/s2 ) i.e. (N)
FG = resultant gravitational force, (kg.m/s2 ) i.e. (N)
?p = density of particle, (kg/m3)
?w = density of water, (kg/m3)
g = acceleration due to gravity, (9.81 m/s2)
Vp = volume of particle, (m3)
The frictional drag force depends on the particle velocity, fluid density, fluid viscosity, particle diameter, and the drag coefficient Cd (dimensionless), and is given by:

F_d=(C_(d ) A_(p ) ?_(w ) ?v^2?_(p ))/2
Where:
Fd = frictional drag force, (kg. m/s2 )
Cd = drag coefficient, (unit less)
Ap = cross sectional or projected area of particles in direction of flow, (m2)
vp = particle settling velocity, (m/s)

Equating the resultant gravitational force to the frictional drag force for spherical particles yields Newton law:
v_(p(t))=?((4 g)/(3 C_(d ) ) (??_(p )- ??_(w )/?_w ) d_p ) = ?((4 g)/(3 C_(d ) ) (Sg_p-1) d_p )
Where:
v p(t) = terminal velocity of particle, (m/s)
dp = diameter of particle, (m)
SgP = specific gravity of the particle
The coefficient of drag Cd takes on different values depending on whether the flow regime surrounding the particle is laminar or turbulent. There are three distinct regions, depending on the Reynolds number NR:
laminar (NR < 1),
transition< (NR = 1 to 2000), and
turbulent (NR > 2000). Although

Although particle shape affects the value of the drag coefficient, for particles that are approximately spherical, the coefficient of drag Cd is approximated by the following equation:
C_d=24/N_R + 3/?(N_R )+0.34
The Reynolds number NR (or Re) for settling particles is defined as:

N_R=(v_(p ) d_(p ) ?_w)/?= (v_(p ) d_(p ))/?
where :
?? = dynamic viscosity, (N.s/m2)
? = kinematic viscosity, (m2/s)
Newton s Equation must be modified for non-spherical particles. An application that has been proposed is to rewrite the equation as follows :
v_(p(t))=?((4 g)/(3 C_(d ) ?) (??_(p )- ??_(w )/?_w ) d_p ) = ?((4 g)/(3 C_(d ) ?) (S g_p-1) d_p )
where ? is a shape factor. The value of the shape factor is 1.0 for spheres, 2.0 for sand grains, and up to and greater than 20 for fractal floc. The shape factor ? especially important in wastewater treatment where few, if any, particles are spherical.


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