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Our next step consists of developing the dual nonlinear scale and the two orifices for our two sets of target alum concentrations: 5-25 mg/L and 20-100 mg/L. Given a known maximum plant flow rate(
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{latex}\large$$Q_{P}$${latex} |
) and Alum Stock concentration(
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{latex}\large$$C_{C} $${latex} |
), we utilized the mass balance equation to determine alum flow rate required for each target alum concentration as shown below:
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{latex}
\large
$$
Q_{Alum} = {{Q_P \times C_T } \over {C_C }}
$$
{latex} |
Where
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{latex}\large$$Q_{Alum} $${latex} |
= Flow Rate of Alum Solution
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{latex}\large$$Q_{P} $${latex} |
= Plant Flow Rate
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{latex}\large$$C_{T} $${latex} |
= Target Alum Concentration
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{latex}\large$$C_{C} $${latex} |
= Alum Concentration in the Stock Tank
Because the orifice controls the flow of this alum solution, we again use the orifice equation. This time we use it to solve for the head loss necessary to achieve these different flow rates. These head losses, or the difference in height from the orifice to the water height in the constant head tank, are calculated as shown below:
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{latex}
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$$
h = {{\left( {{\textstyle{{Q_{Alum} } \over {K_{VC} \times {\textstyle{{D_{Orifice} ^2 } \over 4}} \times \pi }}}} \right)^2 } \over {C_C }}
$$
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Where
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{latex}\large$$Q_{Alum} $${latex} |
= Flow Rate of Alum Solution
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{latex}\large$$Q_{P} $${latex} |
= Plant Flow Rate
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{latex}\large$$h $${latex} |
= Head loss
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{latex}\large$$D_{Orifice} $${latex} |
= Diameter of the Orifice
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{latex}\large$$K_{VC} $${latex} |
= Orifice Constant
We then convert these head losses to points along our scale via simple geometry as shown below:
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{latex}
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$$
scale = {\textstyle{h \over {\sin (\theta _{Max} )}}}
$$
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Where
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{latex}\large$$scale $${latex} |
= distance from the pivot to a point on the scale
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{latex}\large$$h $${latex} |
= head loss from the previous paragraph
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{latex}\large$${\theta _{Max} }$${latex} |
= Maximum Angle Deflection
So that "scale" variable above corresponds to a specific head loss which corresponds to a specific alum flow rate, which corresponds to the target alum concentration that we want in our plant flow. Since we have nine target dosages, we utilized Mathcad to turn the nine target dosages into an array and apply the relationships shown above to produce arrays of corresponding alum flow rates, head losses, and scale points. The array of scale points is essentially the scale for our nonlinear scale. Since all above mentioned parameters are related to one another in a nonlinear relationship, the scale that is generated is nonlinear as shown below:
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| {center:class=myclass}h5.Figure 1: Nonlinear Dual Scale{center} |
Utilizing our Mathcad file, we varied the orifice diameter until we created a scale that maximized the total available length of the lever arm for the scale which for this lever arm is 0.4 m. We can also manipulate the alum stock concentration to affect orifice size. As the above mentioned equations show, lowering the stock alum concentration means more alum flow which means that we can use a greater orifice diameter while utilizing the same length of the scale part of the lever arm.
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