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Based on dimensional analysis, the velocity gradient G can be expressed as a function of the average energy dissipation rate (ε) and kinematic viscosity of the fluid (ν):

{latex}
\large
$$
G = \sqrt {{\varepsilon  \over \nu }}
$$
{latex}

(1.3)
Using conservation of energy, ε can be expressed as kinetic energy loss over a period of time:

{latex}
\large
$$
\varepsilon  = {{gh_L } \over \theta }
$$
{latex}

(1.4)
where: g is gravitational acceleration, hL is head loss and θ is average hydraulic residence time.
The head loss through a straight tube can, in turn, be defined as (Robertson et al, 1993):

{latex}
\large
$$
h_L  = f_s {L \over d}{{U^2 } \over {2g}}
$$
{latex}

(1.5)
where: L is the length of the flocculator and fs is the friction factor in a straight tube. For laminar flow, the friction factor fs = 64/Red, and Red is the Reynolds number as defined as:

{latex}
\large
$$
{\mathop{\rm Re}\nolimits} _d  = {{Ud} \over \nu }
$$
{latex}

(1.6)
where: U is the average axial velocity and d is the tube inner diameter.
The formulation for G derived by Gregory (1981) (see Equation 1.2) can also be derived from algebraic rearrangement of Equations 1.3-1.6. A correlation factor (Mishra & Gupta 1979) can be applied to Equation 1.7 to replace fs with fc and correct for the differences in head loss between straight and curved tubes.

{latex}
\large
$$
{{f_c } \over {f_s }} = 1 + 0.033\log \left( {De} \right)^4
$$
{latex}

(1.7)
where: De is the nondimensional Dean Number and characterizes the effect of curvature on fluid flow:

{latex}
\large
$$
De = \sqrt {{r \over {R_c }}} {\mathop{\rm Re}\nolimits} _d
$$
{latex}

(1.8)
where: r is the inner radius of the tube, Rc is the radius of curvature.
The average head loss measured as the pressure drop across the tube flocculator was within 2% of the head loss calculated using Equations 1.5 and 1.7 (Figure 1.4). The figure eight coil configuration used in this research was different from the flow regime modeled by Mishra and Gupta. The fact that our data agrees with their model suggests that the change in direction of the coil had only a small effect on total head loss. The following G value obtained from combining Equations 1.3-1.8 was used to design the experimental runs.

{latex}
\large
$$
G_c  = G_s \left( {1 + 0.033\log \left( {De} \right)^4 } \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/
{\vphantom {1 2}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{$2$}}}
$$
{latex}

(1.9)

We can also study the effect of increasing the residence time in the flocculator and holding shear constant by increasing the length of the flocculator while holding the flow rate constant; this will increase the amount of time water spends in the flocculator without changing the shear rate. Currently, setup can easily be modified to handle three different flocculator lengths, 27.96 m, 55.92 m, and 83.88 m. Process Controller can also be used to vary alum dosage, and set the desired influent turbidity for the raw water. This allows us complete control over what enters the flocculator, how long it spends in the flocculator, and how quickly it moves through the flocculator.

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