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      First, begin with the description of the velocity profile for laminar flow in a cylindrical tube set at a vertical angle alpha:
    

{latex}
     \large
    $$
v_z = {1 \over {4\mu \sin \alpha }}({{\partial p} \over {\partial z}})(r^2 - R^2 )
$$

     {latex}

     Here, mu is the kinematic viscosity of the fluid flowing through the tube (in this case, water), dp/dz represents the pressure difference across the tube, R is the tube radius, and small r represents the distance between the tube center and the position of itnerest. The key is to determine a relationship that eliminates this pressure term in the equation.
     We know that the flow rate can be defined as:
   
   

{latex}

     \large
     $$
     Q = {{ - \pi R^4 } \over {8\mu }}\left( {{{\partial p} \over {\partial z}}} \right)
$$
     {latex}

     Solving this equation for the pressure differential and inserting the term v ratio (equal to 2 for tubes) gives:
   
   

{latex}
    
\large
$$
({{\partial p} \over {\partial z}}) = {{ - 4v_{ratio} \mu Q} \over {\pi R^4 }}
$$

     {latex}

     This can then be plugged into the velocity profile equation to yield, after simplification:
   
   

{latex}
    
\large
$$
v_z = {{v_{ratio} Q} \over {\pi R^4 \sin \alpha }}(R^2 - r^2 )
$$
{latex}

     The differential of this velocity with respect to the position within the tube gives the velocity gradient:
    
   

{latex}
     \large

$$
{{\partial v_z } \over {\partial r}} = {{ - 2v_{ratio} Q} \over {\pi R^4 \sin \alpha }}r
$$



    
{latex}

     This term can be simplified by substituting V alpha, which represents the average velocity through the cylindrical tube.
    
   

{latex}
\large

$$
{{dv_z } \over {dr}} = {{ - 2v_{ratio}V_\alpha } \over {R^2 }}r
$$
{latex}

      It is important to note that the profile only depends on the tube radius, the flow rate, and r, the distance from the tube wall, which corresponds to the tube radius minus the diameter of the floc particle of interest.The velocity gradient can either be analyzed in terms of the specific radial position of a particle in the tube, or in terms of the velocity gradient that all particles experience at the tube wall. This term corresponds to the previous equation evaluated at the tube wall:
    
   

{latex}
\large

$$
{{dv_z } \over {dr}} = {{ - 2v_{ratio} V_\alpha } \over R}\,;\quad r = R
$$

{latex}

Results and Discussion

     As mentioned in the experimental methods, five velocity gradients (6, 9, 12, 15, 18 1/s) were tested for two tube diameters; 6.35 mm and 15.3 mm. For each of these tubes, two placements of the tube settler were also tested; 1.3 cm and 2.7 cm above the floc blanket. These placements were designated as "low" and "high" heights, respectively, in the following analysis. The team only had a certain amount of flexibility in terms of where the entrance to the tube settler could be placed in the sedimentation column, and thus chose only to look for qualitative differences between the results obtained for experiments at these two different heights.

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