h1. Orifice Size and the Dual Scale Design for the Nonlinear Alum Doser
h3. Abstract:
During the fall semester of 2009, the Nonlinear Chemical Dosing Team developed the dual scale, orifice-based doser in order to be able to deliver both turbulent and laminar alum flow. Like its linear predecessor, this doser must automatically increase or decrease the alum solution to maintain a target dosage set by the operator as the plant flow changes. As an additional feature, the two different scale provides the operator with additional precision through a low dosage (5-25 mg/L) and a high (20-100 mg/L) alum dosage range.
In order to meet our objectives above, we first researched and identified the nonlinear relationship between plant flow rate and alum dosage and the movement of the lever arm. We then utilized this relationship to develop the lever arm design to include the dual scale and the dual orifices. Attached is the [Mathcad File |^Lever Arm Calculations 2009 NCDC TM.xmcd]that contains the calculations for our dosing system.
Our current design consists of a 80cm long lever arm with equal lengths and two orifices of 3.175mm and 0.01587mm diameter, 9.525 mm PVC tubing, and other associated hydraulic components listed in our [component list.|^NCDC Component List.xlsx]
h3. Summary of the Design Process:
In order to meet our design objectives mentioned above, we must link plant flow to alum flow coming out of our doser.
We utilized Mathcad's vector calculation ability to help us in our calculations.
Our first and key step in developing this dosage system was the selection of the orifice to control the flow of alum. We increased the tubing size connecting the constant head tank to the orifice to 9.525mm which is wide and smooth enough to make the head loss from the tubing negligible compared to the head loss through each orifice, making the orifice the flow control component for the dosage system. The orifice equation, shown below, demostrates the nonlinear relationship between flow rate and the change in head loss.
{latex}
\large
$$
Q = K_{vc} A_{or} \sqrt {2gh}
$$
{latex}
Where
{latex}\large$$Q $${latex} = Flow Rate
{latex}\large$$h $${latex} = Head Loss
{latex}\large$$A_{or} $${latex} = Area of the Orifice
{latex}\large$$K_{vc} $${latex} = Orifice Constant
The plant itself is also controlled by the orifice. Head loss in the plant after the entrance tank occurs in the rapid mixer, the flocculation tank, and the launders. Those components are all controlled by orifices. The below table lists the major sources of headloss in the plant.
h5. Table 1: Head Loss Through the Plant
||Process||Head Loss||
|Rapid Mix Tube|10 cm|
|Flocculator|13.5 cm|
|Launder|5 cm|
|Weir|5 cm|
|*Total*|*33.5 cm*|
The only source of head loss not controlled by an orifice is the weir. Because the majority of the head loss is due to the orifice, we can state that the orifice equation dominates the relationship. Therefore, we can link the flow rate of the plant with the flow rate of alum required for the plant using the same square root relationship mentioned above. In other words, the rise and fall of the water height in the entrance tank caused by the change in flow rate, is nonlinearily proportional to the alum flow of our orifice based doser.
Consequently, the lever arm must be long enough to rise and fall with the minimum and maximum water height in the entrance tank. This range is equal to the total headloss in the plant, which is 33.5 cm as shown in Table 1 above. Therefore, we designed an equally balanced lever arm of .8 m that fits in our 1 m x 1 m entrance tank as well as respond to the 33.5 cm water height difference.
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({latex}\large$$Q_{P}$${latex}) and Alum Stock concentration({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:
{latex}
\large
$$
Q_{Alum} = {{Q_P \times C_T } \over {C_C }}
$$
{latex}
Where
{latex}\large$$Q_{Alum} $${latex} = Flow Rate of Alum Solution
{latex}\large$$Q_{P} $${latex} = Plant Flow Rate
{latex}\large$$C_{T} $${latex} = Target Alum Concentration
{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 headloss necessary to achieve these different flow rates. These headlosses, or the difference in height from the orifice to the water height in the constant head tank, are calculated as shown below:
{latex}
\large
$$
h = {{\left( {{\textstyle{{Q_{Alum} } \over {K_{VC} \times {\textstyle{{D_{Orifice} ^2 } \over 4}} \times \pi }}}} \right)^2 } \over {C_C }}
$$
{latex}
Where
{latex}\large$$Q_{Alum} $${latex} = Flow Rate of Alum Solution
{latex}\large$$Q_{P} $${latex} = Plant Flow Rate
{latex}\large$$h $${latex} = head loss
{latex}\large$$D_{Orifice} $${latex} = Diameter of the Orifice
{latex}\large$$K_{VC} $${latex} = Orifice Constant
We then convert these headlosses to points along our scale via simple geometry as shown below:
{latex}
\large
The $$
scale that is generated is nonlinear as shown below:
The Mathcad program's arrary function was utilized to tie these variables together in an orderly fashion. Each of the two alum concentration ranges have nine different alum dosage settings which can be captured via 1 x 9 array. The= {\textstyle{h \over {\sin (\theta _{Max} )}}}
$$
{latex}
Where
{latex}\large$$scale $${latex} = distance from the pivot to a point on the scale
{latex}\large$$h $${latex} = head loss from the previous paragraph
{latex}\large$${\theta _{Max} }$${latex} = Maximum Angle Deflection
So that point "scale" corresponds to a specific headloss which corresponds to a specific alum flow rate. That specific alum flow rate requiredcorresponds to meet these target concentration is generated as another 1 x 9 array. We just continue this process until we calculate anthe target alum concentration that we want in our plant flow.
Since we have nine target dosages, we utilized Mathcad array capability to package the nine target dosages as an array and apply the calculations above to produce an array of corresponding alum flow rates, headloss, and scales points. The array of scale points alongis essentially athe scale for asour shownnonlineary below. scale. Since all abovementioned parameters are non The scale that is generated is nonlinear as shown below:
Consequently,the the operator of this program can manipulate the diameter of the orifice and other design parameters until the dual scale utilizes maximum space on the lever arm (40 cm).
h3. Error Analysis
Although negligible, headloss via the tube is a source of error.
h3. Results and Discussions
Currently, our orifices are 3.175 mm for alum dosage of 20 to 100 mg/L offset by 10 mg/L and 1.587 mm for alum dosage of 5 to 25 mg/L offset by 2.5 mg/L. Our lever arm is 80 cm in length with equal lengths on each side. Tubing is made up of PVC and of 9.525 mm diameter which is wide and smooth enough to produce negligible head loss on the alum flow. For ease of operation, whenever this lever arm is used in the field, this lever arm can be delivered to the Aguaclara Plant with the dual scales already engraved on the arm. The operator simply has to calibrate the maximum dosage to the maximum flow rate and the lever arm will be ready for operation.
Our near goal is to build the [lever arm prototype|Prototype Design] and set up our hydraulic components and engrave our dual scale.
h2. Bibliography
* CEE4540 Flow Control Measurement Notes at https://confluence.cornell.edu/display/cee4540/Syllabus | 