Calculation of Ratio of Settling Velocity to Particle Velocity
The calculation of this ratio is very important in order to begin modeling floc roll up. This ratio is the key to determining whether or not floc roll up will occur for a given set of conditions. In order to determine the critical velocity at which floc particles will begin to roll up the tube and into the effluent, we compare the settling velocity with the particle velocity experienced from the velocity gradient.
The settling velocity of a particle in a tube settler can be expressed as follows (Munson, 1998)
Unknown macro: {latex}
\large
$$
V_t = {{gd_0 ^{\left( {3 - D_
Unknown macro: {Fractal} } \right)} d^{\left( {D_
- 1} \right)} } \over {18\Phi \nu }}\left( {{{\rho _
Unknown macro: {Floc_0 }
} \over {\rho _
Unknown macro: {H_2 O}
}} - 1} \right)
$$
Where:
g = Gravity
Unknown macro: {latex} \large $$d_o $$
= size of the primary particles
Unknown macro: {latex}
\large $$D_
Unknown macro: {fractal}
$$
= fractal dimension of the floc particles
Unknown macro: {latex} \large $$\Phi $$
= shape factor for drag on flocs which is equal to
Unknown macro: {latex} \large $$\nu $$
= viscosity
Unknown macro: {latex}
\large $$\rho _
Unknown macro: {floc}
$$
= density of the floc particle
Unknown macro: {latex}
\large $$\rho _
Unknown macro: {H_2 O}
$$
= density of water
The particle velocity experienced as a result of the velocity gradient can be expressed as follows (Munson, 1998)
Unknown macro: {latex}
\large
$$
V_
Unknown macro: {particle}
= V_
Unknown macro: {ratio}
V_\alpha \left[ {1 - \left( {{{{{d_
Unknown macro: {tube}
} \over 2} - d_
Unknown macro: {Floc}
} \over {{{d_
Unknown macro: {Tube}
} \over 2}}}} \right)^2 } \right]
$$
Where
Unknown macro: {latex} \large $$V_\alpha $$
= directional velocity in the tube settler
Unknown macro: {latex}
\large $$d_
Unknown macro: {tube}
$$
= diameter of the tube settler
Unknown macro: {latex}
\large $$ d_
Unknown macro: {floc}
$$
= the diameter of floc particles
Vratio = the maximum velocity at the center of the tube- for a plate settler this value is 1.5 times the average velocity. For a tube settler this value is 2.
Therefore, the ratio between the settling velocity of the particle and the velocity experienced as a result of the velocity gradient can be expressed as by the below equation.
Unknown macro: {latex}
\large
$$
\Pi _V = {{{{g\sin (\alpha )d_0 ^2 } \over {18\Phi \nu }}{{\rho _
Unknown macro: {Floc_0 }
- \rho _
Unknown macro: {H_2 O} } \over {\rho _
}}\left( {{{d_
Unknown macro: {Floc}
} \over
Unknown macro: {d_0 }
}} \right)^{D_
Unknown macro: {Fractal}
- 1} } \over {V_
Unknown macro: {ratio}
{{V_
Unknown macro: {Up}
} \over {\sin (\alpha )}}\left[ {1 - \left( {{{{{d_
Unknown macro: {Tube}
} \over 2} - d_
} \over {{{d_
Unknown macro: {Tube}
} \over 2}}}} \right)^2 } \right]}}
$$
This ratio is a function of particle diameter, tube diameter, upflow velocity and the angle of the plate settler. When this ratio is greater than one (ie the settling velocity is greater than the velocity experienced by the floc particles in the tube), the flocs will fall back into floc blanket. When this ratio is equal to one, the particles will remain stationary in the tube settler. When the ratio is less than one, the velocity of the particles will exceed the settling velocity and the floc particles will roll up into the effluent, creating a highter turbidity.
References
Munson, B.,Young, D., Okiishi, T., (1998). Fundamentals of Fluid Mechanics (3rd ed.). New York, NY: John Wiley & Sons, Inc.