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{latex}
\large

$$
\eqalign{
  & \sum {F = 0}   \cr
  & F_b  + F_d  - mg\sin (\theta ) = 0{\rm{   (I)}}  \cr
  &  - mg\sin \theta  + \rho \_w gV_{proj} \sin \theta  + \rho \_w C_D A_{proj} {{V^2 } \over 2}{\rm{  (II)}}  \cr
  & {\rm{Solving (II) for V }}...  \cr
  & {{V^2 } \over 2}\rho \_w C_D A_{proj}  = \rho \_p V_{proj} g\sin \theta  - \rho \_w gV_{proj} \sin \theta   \cr
  & V^2  = {2 \over {\rho \_w C_D A_{proj} }}(\rho \_p V_{proj} g\sin \theta  - \rho \_w gV_{proj} \sin \theta )  \cr
  & V^2  = {{2g\sin \theta V_{proj} } \over {\rho \_w C_D A_{proj} }}(\rho \_p  - \rho \_w )  \cr
  & {\rm{This model initially assumes that the floc particles can be approximated as spheres}}.  \cr
  & {{V_{proj} } \over {A_{proj} }} = {{{{\pi d_p \^3 } \over 6}} \over {{{\pi d_p \^2 } \over 4}}} = {2 \over 3}d_p   \cr
  & {\rm{Assuming that the flow in the tube settlers is creeping, we can approximate the drag coefficient as 24/Re}}{\rm{. Thus:}}  \cr
  & V^2  = {{2g\sin \theta } \over {{{24} \over {{\mathop{\rm Re}\nolimits} }}}}{2 \over 3}d_p {{(\rho \_p  - \rho \_w )} \over {\rho \_w }}  \cr
  & V = \sqrt {{4 \over {72}}g\sin (\theta )d_p {\mathop{\rm Re}\nolimits} {{(\rho \_p  - \rho \_w )} \over {\rho \_w }}}   \cr
  & {\rm{At this velocity, a floc shouldn't move}}{\rm{.  We should call this something like the critical velocity, since any velocity value (at the edge of the floc) larger than this value will produce floc roll - up}}{\rm{.}}  \cr
  &   \cr
  & {\rm{Analysis of the velocity gradient}}  \cr
  & {1 \over r}{\partial  \over {\partial r}}(r{{\partial v_z } \over {\partial r}}) = {1 \over \mu }({{\partial P} \over {\partial z}})r  \cr
  & {\rm{After integration }}  \cr
  & r{{\partial v_z } \over {\partial r}} = {1 \over {2\mu }}({{\partial P} \over {\partial z}})r^2  + c_1   \cr
  & {\rm{c}}_{\rm{1}} {\rm{ = 0, since v}}_{\rm{z}} {\rm{ must be finite at the center of the tube}}{\rm{.  Dividing through by r \-  - }}  \cr
  & {{\partial v_z } \over {\partial r}} = {1 \over {2\mu }}({{\partial P} \over {\partial z}})r  \cr
  & {{\partial P} \over {\partial z}} = {{\Delta P} \over L}  \cr
  & V_{av}  = {{R^2 \Delta P} \over {8\mu L}} = {Q \over {\pi R^2 }}  \cr
  & {{\Delta P} \over L} = {{Q8\mu } \over {\pi R^4 }}  \cr
  & {{\partial v_z } \over {\partial r}} = {1 \over {2\mu }}{{Q8\mu } \over {\pi R^4 }}r  \cr
  & {{\partial v_z } \over {\partial r}} = {{4Q} \over {\pi R^4 }}r  \cr
  & {\rm{Integrate to obtain an expression for the velocity profile with respect to z}}{\rm{.}}  \cr
  & V_z (r) = {{2Q} \over {\pi R^4 }}r^2  + c_1   \cr
  & {\rm{Now apply the no - slip condition; }}V_z  = 0{\rm{ when r = R}}  \cr
  & c_1  = {{ - 2Q} \over {\pi R^2 }}  \cr
  & V_z (r) = {{2Q} \over {\pi R^2 }}\[({r \over R})^2  - 1\] = {{2Q} \over {A_{tube} }}\[({r \over R})^2  - 1\]  \cr
  & {\rm{Finally:}}  \cr
  & V_z (r) = 2V_{av} \[({r \over R})^2  - 1\]  \cr
  & {\rm{We can evaluate the velocity at the failure point, which for this approximtion is equal to (R}}_{{\rm{tube}}} {\rm{ \-  d}}_{\rm{p}} ) = P_f   \cr
  & V_z ({\rm{R}}_{{\rm{tube}}} {\rm{ \-  d}}_{\rm{p}} ) = 2V_{av} \[({{{\rm{R}}_{{\rm{tube}}} {\rm{ \-  d}}_{\rm{p}} } \over R})^2  - 1\]  \cr
  & V_z (P_f ) = 2V_{av} \[({{{\rm{R}}_{{\rm{tube}}}  - d_p } \over R})^2 \] = V_f   \cr
  & V_f  = \sqrt {{4 \over {72}}g\sin (\theta )d_p {\mathop{\rm Re}\nolimits} {{(\rho \_p  - \rho \_w )} \over {\rho \_w }}}   \cr
  & {\rm{When the velocity at the failure point exceeds the critical velocity, flocs will roll up}}{\rm{. }}{\rm{This suggests an inequality}}{\rm{.}}  \cr
  & 2V_{av} \[({{{\rm{R}}_{{\rm{tube}}}  - d_p } \over R})^2 \] > \sqrt {{4 \over {72}}g\sin (\theta )d_p {\mathop{\rm Re}\nolimits} {{(\rho \_p  - \rho \_w )} \over {\rho \_w }}}   \cr
  & {\rm{If we solve this expression for }}d_p {\rm{ we can obtain a function to evaluate the critical velocity as a function of the floc diameter}}{\rm{.  }}  \cr
  & {\rm{The function is cubic and has the form:}}  \cr
  & {\rm{(}}d_p ) = c_1 d_p \^3  - c_2 d_p \^2  + c_3 d_p  - \omega   \cr
  & {\rm{(}}d_p ) = ({1 \over R})d_p \^3  - 4d_p \^2  + (4R^2  + {1 \over R} - 2)d_p  - g\sin (\theta ){\mathop{\rm Re}\nolimits} {{(\rho \_p  - \rho \_w )} \over {\rho \_w }}{{A_{tube} \^2 } \over {Q^2 }}  \cr
  & {\rm{Theoretically, when:}}  \cr
  &   \cr
  & {\rm{                                  }}{\rm{(}}d_p ) < 0,{\rm{ no roll - up occurs}}  \cr
  & {\rm{                                  }}{\rm{(}}d_p ) = 0,{\rm{ critical diameter}}  \cr
  & {\rm{                                  }}{\rm{(}}d_p ) > 0,{\rm{ roll - up occurs}}{\rm{.}} \cr}
$$
{latex}