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Water enters the top of the tube through the large-scale mixing orifice, where it is dosed with the aluminum sulfate and begins the rapid mix process. This orifice is in place to create large scale mixing in the first section of the tube. The design of this orifice size is based on the exit loss coefficient through the orifice, K. The target K value for this orifice is 2, which provides the best mixing in the first section of the tube for large-scale rapid mix. To calculate the necessary area and diameter of the large scale mixing orifice, the following equation was used:

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$${K_{ex}}\left( {{A_{out}},{A_{in}}} \right) = {\left( {{{{A_{out}}} \over {{A_{in}}}} - 1} \right)^2}$$
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Solving for A.in, which will be the area of the stream of water entering the tube through the orifice, we obtain:

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$${A_{in}}\left( {{A_{RapidMixOrifice}},P{i_{VC}}} \right) = {A_{RapidMixOrifice}}*P{i_{VC}}$$
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In this equation, A.in is taken to be the area of contracted flow through the orifice, which is the area of the large-scale mixing orifice multiplied by the vena contracta coefficient, which accounts for the contraction of flow through an orifice. The equation the describes this is as follows:

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$${A_{RMOrificeLS}}\left( {{A_{out}},{K_{ex}},P{i_{VC}}} \right) = {{{A_{out}}} \over {P{i_{VC}}*\left( {1 + \sqrt {{K_{ex}}} } \right)}}$$
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Figure 4 illustrates the effect of the water contraction flowing through an orifice.

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After water flows into the rapid mix tube and through the top section of the pipe to achieve large-scale mixing, it reaches the small-scale mixing orifice. The area of the small-scale mixing orifice is designed to achieve a target head loss, providing a mechanism to measure the level of water in the plant to assist in the correct dosing of the plant's raw water source with aluminum sulfate. The equation used to calculate the area of the small-scale mixing orifice is as follows:

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$${A_{RMOrificeSS}}\left( {Q,\Delta h} \right) = {Q \over {{K_{vc}}\sqrt {2g\Delta h} }}$$
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In this equation, Q signifies the flow rate of water through the plant, K.vc is the vena contracta coefficient as discussed above in the large-scale orifice design section, and Δh is the target head loss for which the plant is being designed; this design value can be varied based on the desired characteristics of each plant. K.vc is used here again because the flow of water through the small-scale mixing orifice is also a flow contraction, and the area of the water stream entering the pipe following is a fraction of the area of the orifice.

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Each of these designs is preferred in different cases depending on the plant flow rate, pipe diameter, desired head loss, and desired energy dissipation rate. Tentatively, the Agalteca plant will feature a small-scale orifice featuring the multiple round orifices, which will best serve this plant in evenly mixing the aluminum sulfate dosed to the raw waste, as well as achieving the desired energy dissipation rate through the orifice. To calculate the dimensions of the round orifices that will occur in the small-scale mixing orifice, the following equation in used:

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$$Width\left( {\Delta h,\varepsilon } \right) = {{{{\left( {2g\Delta h} \right)}^{1.5}}} \over {20\varepsilon }}$$
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Here, ε is the value of the maximum energy dissipation rate for the plant, and the orifice is thus designed to achieve this value. Δh is the same target value for the head loss from the small-scale orifice design equation. This equation thus calculates the maximum minimum dimension of a rectangular orifice. This dimension can be adapted to the proposed Agalteca design with multiple small orifices, however, because of the presence of many small orifices in entire small-scale mixing orifice. The dimension calculated in this equation will then be used as the diameter of the multiple orifices that must be put into the small-scale mixing orifice. Figure 6 provides a schematic of the rapid mix tube as well as the placement of the two orifices and a detail of the multiple-orifice small scale mixing orifice that will likely be used in the Agalteca plant.

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Headloss Calculations and Significance
The total headloss through the system is comprised of minor losses, caused by water flow through the orifices and through pipe fittings such as elbows, and major losses due to friction on the pipe walls. The equation used to calculate total headloss through the system is:

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$$H{L_{system}} = H{L_{L\arg eScaleMixing}} + H{L_{SmallScaleMixing}} + H{L_{PipeFittings}} + H{L_{PipeFriction}}$$
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Here, HL.LargeScaleMixing and HL.SmallScaleMixing refer to the minor losses through the large and small scale mixing orifices, respectively. HL.PipeFittings and HL.PipeFriction are caused by minor losses through the pipe fittings and major losses sue to shear along the pipe walls.

Flow through both the large and small scale orifices is modeled as an expansion; as the water flows through the orifices, the flow is contracted and then expands on the other side of the orifice. Thus, head loss can be calculated using the equation for head loss in a flow expansion:

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$${h_{ex}}\left( {{Q_{plant}},{A_{pipe}},{K_{ex}}} \right) = {{{Q_{plant}}} \over {2g{{\left( {{A_{pipe}}} \right)}^2}}}*{K_{ex}}$$
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The calculation of the headloss through the system is important because the headloss through the rapid mix tube system partially determines the change in height of the water in the entrance tank when the plant flow changes. The nonlinear chemical doser is designed to change the flow rate of alum with the flow rate of the plant, and it is thus very important to calculate the headlosses through the rapid mix tube system and the sections of the plant following it in order to correctly design the chemical doser to maintain a constant concentration of alum in the raw water source despite changing flow rates.

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