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|Nonlinear Chemical Dose Controller old].

h3. Methods


h4. Sizing the Orifice

The orifice between the rapid mix and flocculation tanks is designed to produce a difference in water level high that can then be sensed by a float which would then change the flow rate of aluminum sulfate:
{latex}$$
\Delta H = K_{_{orifice} } {{V_{jet} ^2 } \over {2*g}}
$$
{latex}
where
* {latex}$$
\Delta H
$${latex}
is the difference in head loss between the rapid mix and flocculation tank
* K ~orifice~ is the required minor loss coefficient through the orifice
* V ~jet~ is the velocity in the dosing tube

This head loss was then used to determine the velocity of the water through the orifice and the residence time. Using the following equations:

Velocity of Jets:
{latex}
$$V_{jet} = {Q \over {C_d *A_{orifice} }}
$${latex}
where
* V ~jet~ is the velocity of the jet
* Q is the flow rate through the system
* C ~d~ is the vena contracta coefficient for exit condition in orifice
* A ~orifice~ is the area of the orifice

Residence time :
{latex}$$
\theta = {{d_{orifice} } \over {V_{jet} }}
$${latex}
where
* {latex} $$
\theta
$$ {latex}
is the residence time

* d ~orifice~ is the diameter of the orifice
* V ~jet~ is the velocity of the jet

Once these values were determined, we were able to calculate the energy dissipation rate using the following equation:
{latex}$$
\varepsilon = {{g*\Delta H} \over \theta }
$${latex}
where
* {latex}$$
\varepsilon
$${latex}
is the energy dissipation rate
* g is gravity
* {latex}$$
\Delta H
$${latex}
is the head loss
* {latex}$$
\theta
$${latex}
is the residence time

We sought to keep the energy dissipation rate between .5 and 1 W/kg so that molecular scale diffusion works and in order for small scale turbulent mixing to be effective.

h4. Lever Arm and Float

We first must determine the size of the counterweight on the doser arm in order to ensure that the dosage will only be a function of the difference in water height in the flocculation and rapid mix tanks. The mass of the weight is calculated by determining the mass of the doser when full.
{latex}$$
m_{doserful} = [({{.375in} \over 2})^2 *\pi *25cm + ({{D_{actual} } \over 2})^2 *\pi *1m]\rho _{water} + m_{doser}
$${latex}
where
* D ~actual~ is the difference between the given diameter of the dosing tube and the measured diameter of the dosing tube
* {latex}$$
\rho _{water}
$${latex}
is the density of water
* {latex}$$
m_{doser}
$${latex}
is the mass of the doser empty

The size of the float can be determined using a moment balance around the pivot of the lever arm. This is to ensure that a change in head in the entrance tank will cause a similar change in the relative height of the float. The float was sized using the same float sizing algorithm used by the linear CDC. Based on this we found that a float of 13.3 inches would theoretically be able to measure a .25cm height difference.

h3. Conclusion

Based on our calculations, we found that an orifice of 8cm would give us an acceptable energy dissipation rate of .927 W/kg and would require a 13.3in float. This float would have a .25cm sensitivity over a 15.2cm height difference.
{float:left|border=2px solid black|width=300px}
!head loss vs flow rate.jpg|width=300px, align=center!
h5. Graph showing the non-linear relationship between the head loss vs. the flow rate through a 8cm diameter orifice
{float}

{float:left|border=2px solid black|width=300px}
!8 orifices.jpg|width=300px, align=center!
h5. Graph showing the non-linear relationship between the head loss vs. the flow rate through eight 4.5cm diameter orifices
{float}

{float:left|border=2px solid black|width=300px}
!175 orifices.jpg|width=300px, align=center!
h5. Graph showing the non-linear relationship between the head loss vs. the flow rate through 175 2cm diameter orifices
{float}