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\large
$$
\eqalign{
& \sum
Unknown macro: {F = 0}
\cr
& F_b + F_d - mg\sin (\theta ) = 0{\rm{ (I)}} \cr
& - mg\sin \theta + \rho _w gV_
Unknown macro: {proj}
\sin \theta + \rho _w C_D A_
V^2 } \over 2}{\rm{ (II) \cr
& {\rm{Solving (II) for V }}... \cr
& {{V^2 } \over 2}\rho _w C_D A_
Unknown macro: {proj}
= \rho _p V_
g\sin \theta - \rho _w gV_
Unknown macro: {proj}
\sin \theta \cr
& V^2 = {2 \over {\rho _w C_D A_
}}(\rho _p V_
Unknown macro: {proj}
g\sin \theta - \rho _w gV_
\sin \theta ) \cr
& V^2 = {{2g\sin \theta V_
Unknown macro: {proj}
} \over {\rho _w C_D A_
}}(\rho _p - \rho _w ) \cr
& {\rm{This model initially assumes that the floc particles can be approximated as spheres}}. \cr
& {{V_
Unknown macro: {proj}
} \over {A_
}} = {{\pi d_p ^3 } \over 6 \over {{
Unknown macro: {pi d_p ^2 }
\over 4}}} =
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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 {{
Unknown macro: {24}
\over {{\mathop
Unknown macro: {rm Re}
\nolimits} }}}}
d_p {{(\rho _p - \rho _w )} \over {\rho _w }} \cr
& V = \sqrt {{4 \over {72}}g\sin (\theta )d_p {\mathop
Unknown macro: {rm Re}
\nolimits} {{(\rho _p - \rho _w )} \over
Unknown macro: {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
&
Unknown macro: {1 over r}
{\partial \over {\partial r}}(r{{\partial v_z } \over {\partial r}}) =
Unknown macro: {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_
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= 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 }}[(
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)^2 - 1] = {{2Q} \over {A_
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}}[(
)^2 - 1] \cr
& {\rm{Finally:}} \cr
& V_z (r) = 2V_
[(
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)^2 - 1] \cr
& {\rm{We can evaluate the velocity at the failure point, which for this approximtion is equal to (R}}_{{\rm
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}} {\rm{ - d}}_{\rm{p}} ) = P_f \cr
& V_z ({\rm{R}}_{{\rm
}} {\rm{ - d}}{\rm{p}} ) = 2V
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[({\rm{R_{{\rm
Unknown macro: {tube}
}} {\rm{ - d}}_{\rm{p}} } \over R})^2 - 1] \cr
& V_z (P_f ) = 2V_
[({\rm{R_{{\rm
Unknown macro: {tube}
}} - d_p } \over R})^2 ] = V_f \cr
& V_f = \sqrt {{4 \over {72}}g\sin (\theta )d_p {\mathop
\nolimits} {{(\rho _p - \rho _w )} \over
Unknown macro: {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_
Unknown macro: {av}
[({\rm{R_{{\rm
Unknown macro: {tube}
}} - d_p } \over R})^2 ] > \sqrt {{4 \over {72}}g\sin (\theta )d_p {\mathop
Unknown macro: {rm Re}
\nolimits} {{(\rho _p - \rho _w )} \over
}} \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 ) = (
Unknown macro: {1 over R}
)d_p ^3 - 4d_p ^2 + (4R^2 +
- 2)d_p - g\sin (\theta ){\mathop
Unknown macro: {rm Re}
\nolimits} {{(\rho _p - \rho _w )} \over {\rho _w }}{{A_
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^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}
$$