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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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class myclass
Figure 1: Nonlinear Dual Scale
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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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