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[!Non Linear Doser Diagram.JPGjpg|width=300px, align=320px!|center!
h5. Illustration of Nonnon-linear Linearchemical Doser]doser
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h3. Methods


h4. Sizing the Orifice

The orifice between the rapid mix and floculation 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}$$
h_l 

Introduction

The non-linear dose controller was redesigned in order to reduce the amount of aeration caused as water traveled through the plant. For more information about the theory of the non-linear dose controller see the page for the original non-linear CDC design.

Methods

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:

$$
\Delta H = K_{DoseOrifice_{orifice} } {{V_{DoseTubejet} ^2 } \over {2g2*g}}
$${latex}

where

• Latex
  $$ \Delta H $$

  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:

$$V_{jet} = {Q \over {C_d *A_{orifice} }}
$$

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 :

$$
\theta = {{d_{orifice} } \over {V_{jet} }}
$$

where

• Latex
  $$ \theta $$

  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:

$$
\varepsilon = {{g*\Delta H} \over \theta }
$$

where

• Latex
  $$ \varepsilon $$

  is the energy dissipation rate
• g is gravity

• Latex
  $$ \Delta H $$

  is the head loss

• Latex
  $$ \theta $$

  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.

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.

$$
m_{doserful} = [({{.375in} \over 2})^2 *\pi *25cm + ({{D_{actual} } \over 2})^2 *\pi *1m]\rho _{water} + m_{doser}
$$

where

• D _actual is the difference between the given diameter of the dosing tube and the measured diameter of the dosing tube

• Latex
  $$ \rho _{water} $$

  is the density of water

• Latex
  $$ m_{doser} $$

  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.

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.

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!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
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Additionally we found 2 solutions with multiple orifices. We did this because more orifices at a smaller diameter will keep energy dissepation constant while increasing overall headloss across the system. The first solution gave us 8 orifices with a 4.5cm diameter and a minimum energy dissipation rate of .831.

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h5. Graph showing the non-linear relationship between the head loss vs. the flow rate through eight 4.5cm diameter orifices
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The second solution gave us 175 orifices with a 2 cm diameter and a minimum energy dissipation rate of .496.

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h5. Graph showing the non-linear relationship between the head loss vs. the flow rate through 175 2cm diameter orifices
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* h ~l~ the difference in head loss between the maximum CDC head loss and the actual head loss in the flexible dosing tube
* K ~DoseOrifice~ is the required minor loss coefficient through the orifice
* V ~DoseTube~ is the velocity in the dosing tube

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

V=

theta=


Once these values were determined, we were able to calculate the energy dissipation rate using the following equation.

We sought  to keep the energy dissipation between .5 and 1 W/kg.

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.

\]

where D.actual is the difference between the given diameter of the dosing tube and the measured diameter of the dosing tube

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.

h.3 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 20.3cm height difference.