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The settling velocity of a particle in a tube settler can be expressed as follows (Munson, 1998)

{latex}
\large
$$
V_t = {{gd_0 ^{\left( {3 - D_{Fractal} } \right)} d^{\left( {D_{Fractal} - 1} \right)} } \over {18\Phi \nu }}\left( {{{\rho _{Floc_0 } } \over {\rho _{H_2 O} }} - 1} \right)
$$
{latex}

Where:

g = Gravity

{latex}\large $$d_o $$ {latex}

= size of the primary particles

{latex}\large $$D_{fractal} $$ {latex}

= fractal dimension of the floc particles

{latex}\large $$\Phi $$ {latex}

= shape factor for drag on flocs which is equal to

{latex}\large $$\nu $$ {latex}

= viscosity

{latex}\large $$\rho _{floc}$$ {latex}

= density of the floc particle

{latex}\large $$\rho _{H_2 O} $$ {latex}

= density of water
The particle velocity experienced as a result of the velocity gradient can be expressed as follows (Munson, 1998)

{latex}
\large
$$
V_{particle} = V_{ratio} V_\alpha \left[ {1 - \left( {{{{{d_{tube} } \over 2} - d_{Floc} } \over {{{d_{Tube} } \over 2}}}} \right)^2 } \right]
$$
{latex}

Where

{latex}\large $$V_\alpha $$ {latex}

= directional velocity in the tube settler

{latex}\large $$d_{tube}$$ {latex}

= diameter of the tube settler

{latex}\large $$ d_{floc} $$ {latex}

= 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.

{latex}
\large
$$
\Pi _V = {{{{g\sin (\alpha )d_0 ^2 } \over {18\Phi \nu }}{{\rho _{Floc_0 } - \rho _{H_2 O} } \over {\rho _{H_2 O} }}\left( {{{d_{Floc} } \over {d_0 }}} \right)^{D_{Fractal} - 1} } \over {V_{ratio} {{V_{Up} } \over {\sin (\alpha )}}\left[ {1 - \left( {{{{{d_{Tube} } \over 2} - d_{Floc} } \over {{{d_{Tube} } \over 2}}}} \right)^2 } \right]}}
$$
{latex}

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.

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