Page History

Why Drag Analysis is Necessary

Observation of the tube settlers yielded interest in floc buildup and floc flow in tubes. As the flocs began to build up, some started to roll up the tubes and flow out into the effluent instead of falling back into the floc blanket. It was determined that a drag force was possibly preventing the flocs from settling out. The question remained, however, as to why smaller tubes at higher flow rates experienced the rolling flocs but not the larger tubes.
It was determined that velocity gradients vary with the tube diameter, and that the drag force was related to the velocity gradient at the tube wall. As the diameter decreased at the same

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!Plate Settler Spacing Research Fall 2008^Uniformflow.png|width=200px|height=250px!
h5. Figure 1: Development of fully developed flow
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Velocity Gradients

The Reynolds number and entrance region length were calculated to determine whether the flow through the tubes was transient or laminar. With Reynolds numbers below 100 the length of the entrance region was determined by the following equation:

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$l_e  = 0.06{\mathop{\rm Re}\nolimits}  \cdot d$

It was then determined that the flow though the tubes became fully developed very quickly.

Figure 1 shows the evolution from uniform to fully developed. The parabola represents the fully developed velocity profile through the tubes. It is clear from this image that flocs experience higher velocity gradients in the entrance region.

The next calculation involved the Navier Stokes equation for laminar flow through a cylindrical tube, as seen below.

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$\frac{{\partial v}}{{\partial r}} = \frac{1}{\mu }\left( {\frac{{\partial p}}{{\partial z}}} \right)R + c_1 $

Final equation:

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$\frac{{\partial v}}{{\partial r}} = \frac{{4 \cdot V}}{R}$

This equation was evaluated at R, the radius of the tube, to find the maximum velocity gradient at the tube walls. Table 1 lists the velocity gradient values for each tube at the test critical velocities.

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*Table 1. Results: The Velocity Gradients for the Tested Critical Velocities*
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In order to determine a minimum plate spacing for the tanks in AguaClara plants, a Navier Stokes equation for laminar flow between two flat plates was used.

$\frac{{\partial u}}{{\partial y}} = \frac{1}{\mu }\left( {\frac{{\partial p}}{{\partial x}}} \right)y + c_1 $

Final equation:

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$\frac{{\partial u}}{{\partial y}} = \frac{{3V}}{{2h}}$

The minimum spacing for plate settlers can be determined using the above equations.

The velocity gradients found in each tube over the range of critical velocities can be found in Table 1. By comparing the velocity gradients in table 1 and the results table from the flow rate experiment it can be determined that once the velocity gradient in the tube reaches a certain value, failure occurs. From the data it appears that failure occurs around 2.4 1/s, as velocity gradients beyond this value correspond with failure in the two smallest tube sizes tested.

Drag on a floc

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!Drag on floc.png|width=90px|height=90px!
*Figure 2: Force Balance on a floc*
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Figure 2 shows the force balance on a floc. As stated before, it is believe that when the drag force on a floc exceeds force due to gravity, the floc beings to roll up the tube.