h3. Theoretical Analysis of the Velocity Gradient
When an incompressible fluid flows through a cylindrical tube its velocity relative to the walls changes as a function of the tube radius. In general, this velocity distribution is parabolic: the greatest velocities are achieved at the center of the tube (where R=0) eventually tapering off to 0 at the walls. The parabolic nature of the distribution arises from cylindrical symmetry as well as the fact that the fluid does not move at the walls (the "no-slip" condition).
This gradient in the velocity profile contibutes to the force that a floc experiencing roll-up feels. Flocs actually begin to roll up when the velocity at their edge exposed to the flow exceeds some critical value, which is highly dependent on the floc's diameter, its density, and the capture velocity of the system, among other things.
!Floc Roll up picture.png|width=250!
{latex}
\large
$$
Applying a force balance to this (where F_b = force of buoyancy, F_d = force of drag, F_g = force of gravity, and \theta = 60 degrees):
$$
\sum {F = 0}
$$
{latex}
{color:white}.{color}
{latex}
\large
$$
F_b + F_d - mg\sin (\theta ) = 0
$$
{latex}
{color:white}.{color}
{latex}
\large
$$
- mg\sin \theta + \rho _w gV_{proj} \sin \theta + \rho _w C_D A_{proj} {{V^2 } \over 2}
$$
Solving this equation for the velocity:
{latex}
{color:white}.{color}
{latex}
\large
$$
{{V^2 } \over 2}\rho _w C_D A_{proj} = \rho _p V_{proj} g\sin \theta - \rho _w gV_{proj} \sin \theta
$$
{latex}
{color:white}.{color}
{latex}
\large
$$
V^2 = {{2g\sin \theta V_{proj} } \over {\rho _w C_D A_{proj} }}(\rho _p - \rho _w )
$$
For the purposes of this approximation, it is assumed the floc's diameter can be modeled as a sphere.
{latex}
{color:white}.{color}
{latex}
\large
$$
{{V_{proj} } \over {A_{proj} }} = {{{{\pi d_p ^3 } \over 6}} \over {{{\pi d_p ^2 } \over 4}}} = {2 \over 3}d_p
$$
Assuming that the flow through the tube settlers is creeping flow (Re <<1), we can approximate the drag coefficient as 24/Re:
{latex}
{color:white}.{color}
{latex}
\large
$$
V = \sqrt {{4 \over {72}}g\sin (\theta )d_p {\mathop{\rm Re}\nolimits} {{(\rho _p - \rho _w )} \over {\rho _w }}}
$$
At this velocity a floc that is sitting on the bottom wall of the tube settler should not move. We should call this something like the critical or failure velocity, since velocity values greater than this quantity (at the edge of the floc exposed to the flow) will cause roll-up.
{latex}
{color:white}.{color}
h5. VELOCITY GRADIENT ANALYSIS
{latex}
\large
Navier-Stokes for a cylindrical system
$$
{1 \over r}{\partial \over {\partial r}}(r{{\partial v_z } \over {\partial r}}) = {1 \over \mu }({{\partial P} \over {\partial z}})r
$$
{latex}
{color:white}.{color}
{latex}
\large
Integrate again
$$
r{{\partial v_z } \over {\partial r}} = {1 \over {2\mu }}({{\partial P} \over {\partial z}})r^2 + c_1
$$
This constant of integration should be zero, since no velocity gradient exists at the point of maximum velocity, achieved at the center of the tube where R=0.
{latex}
{color:white}.{color}
{latex}
\large
$$
{{\partial v_z } \over {\partial r}} = {1 \over {2\mu }}({{\partial P} \over {\partial z}})r
$$
{latex}
{color:white}.{color}
{latex}
\large
$$
- {{\partial P} \over {\partial z}} = {{\Delta P} \over L}
$$
{latex}
{color:white}.{color}
{latex}
\large
$$
V_{av} = {{R^2 \Delta P} \over {8\mu L}} = {Q \over {\pi R^2 }}
$$
{latex}
{color:white}.{color}
{latex}
\large
$$
{{\Delta P} \over L} = {{Q8\mu } \over {\pi R^4 }}
$$
{latex}
{color:white}.{color}
{latex}
\large
$$
{{\partial v_z } \over {\partial r}} = {1 \over {2\mu }}{{Q8\mu } \over {\pi R^4 }}r
$$
{latex}
{color:white}.{color}
{latex}
\large
$$
{{\partial v_z } \over {\partial r}} = - {{4Q} \over {\pi R^4 }}r
$$
{latex}
{color:white}.{color}
{latex}
\large
$$
V_z (r) = {{2Q} \over {\pi R^2 }}[1 - ({r \over R})^2 ] = {{2Q} \over {A_{tube} }}[1 - ({r \over R})^2 ]
$$
{latex}
{color:white}.{color}
{latex}
\large
Finally:
$$
V_z (r) = 2V_{\max } [1 - ({r \over R})^2 ]
$$
{latex}
{color:white}.{color}
latex}
\large
$$
V_z ({\rm{R}}_{{{}_\rm{_}{}}}{{
{tube}}} {\rm{ - d}}_{\rm{p}} ) = V_{\max } [(1 - {{{\rm{R}}_{{\rm{tube}
}}
{{}}
{\rm{ - d}}_{\rm{p}} } \over R})^2 \]
$$
{latex}
{color:white}.{color}
{latex}
\large
$$
V_z (P\_
{failure}
) = V\_
{\max } [(1 - {{{\rm{R}}_{{\rm{tube}}} - d_p } \over R})^2 ] = V_{critical}
$$
{latex}
{color:white}.{color}
{latex}
\large
$$
V_f = \sqrt {{4 \over {72}}g\sin (\theta )d_p {\mathop{\rm Re}\nolimits} {{(\rho _p - \rho _w )} \over {\rho _w }}}
$$
{latex}
{color:white}.{color}
{latex}
\large
When the velocity at the critical point exceeds the failure velocity, roll-up occurs. This suggests an inequality of the form:
$$
V_{\max }
} [1 -
{tube}}} - d_p } \over R})^2 "linktype="raw" linktext="1 - ({{{\rm{R}}_{{\rm
{tube}
}} - d_p } \over R})^2 " class="linkerror">1 - ({{{\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 }}}
$$
{latex}
{color:white}.{color}
{latex}
\large
Solving this inequality for the diameter of the particle allows us to generate a function that will allow us to analyze roll-up in terms of a critical floc diameter. This equation actually happens to be cubic in the diameter of the floc and has the following form:
$$
{\rm{F(}}d_p ) = c_1 d_p ^3 - c_2 d_p ^2 + c_3 d_p - \omega
$$
{latex}
{color:white}.{color}
{latex}
\large
$$
{\rm{F(}}d_p ) = ({1 \over {R^4 }})d_p ^3 - ({2 \over {R^3 }})d_p ^2 + ({1 \over {R^2 }})d_p - g\sin (\theta ){\mathop{\rm Re}
}\nolimits} {{(\rho \_p - \rho \_w )} \over {\rho \_w }}{1 \over {V\_
{\max }
\^2 }}
$$
{latex}
{color:white}.{color}
{latex}
\large
The coefficients in this function are only functions of the radius of the tube, and an analysis of how they change with the radius could possibly give some physical insight into the phenomenon of floc roll-up as it relates to the geometry of the tube settlers.
$$
{\rm{F(}}d_p ) < 0
$$
No roll-up.
{latex}
{color:white}.{color}
{latex}
\large
$$
{\rm{F(}}d_p ) = 0
$$
This is the critical diameter at which a floc will not move.
{latex}
{color:white}.{color}
{latex}
\large
$$
{\rm{F(}}d_p ) > 0
$$
Roll-up occurs.
{latex}
{color:white}.{color}
During Spring 2009 the Plate Settler Spacing team first identified the roll-up phenomenon, and their subsequent analysis is centered around a critical velocity gradient occuring at the walls of the tube settler. While this analysis provides reasonable data, the fundamental force balance has steered the Summer 2009 team in a direction that is more focused on the physical properties of the floc as they relate to roll-up. This is beneficial because a model based entirely on the velocity gradient may lend itself to difficulty in calculations, since this gradient could very well depend on the influent water turbidity and chemistry.
It might be useful to analyze the final cubic function's behavior around its inflection points -- these may indicate critical values or "states" the system moves through as the capture velocity changes. | 