h1. Orifice Size and the Dual Scale Design for the Nonlineary Alum Doser
h3. Abstract:
During the fall semester of 2009, the 2009 Nonlinear Chemical Dosing Team developed the dual scaled orifice-based doser in order to deliver turbulent alum flow. This doser, once set for a specific dosage by the operator, must automatically increase or decrease the alum solution to maintain that target dosage as the plant flow changes. TheIn addition, 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.
We have currently researched and identified the nonlinear relationship between plant flow rate and alum dosage and the movement of the lever arm. We utilized this relationship to develop the lever arm design to include the dual scale and the 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 diameer, 9.525 mm pvc tubing, and other associated hydraulic components listed in our [component list.|^NCDC Component List.xlsx]
h3. TheorySummary of 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 design.
Our first and key step in developing a dosage system that can automatically deliver both turbulent and laminar flow of alum 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 flowrate and the change in head loss. The flow rate(Q) is the result of the square root of the head loss(h), the area of the orifice(Aor), and the orifice constant,(Kvc).
{latex}
\large
$$
Q = K_{vc} A_{or} \sqrt {2gh}
$$
{latex}
Where
Q = Flow Rate
h = Headloss
A_or = Area of the Orifice
Kvc = Orifice Constant
The plant itself is controlled by the orifice. Head loss occuring in the plant after the entrance tank including rapid mix, the flocculation tank, and the launders are all controlled by orifices. The only source of head loss not controlled by an orifice is the weir. Because the majority of the head loss is dominated by the orifice 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.
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 leveronly armsource dosingof flowhead rateloss and plant flow rate are both governed by the same nonlinear relationship as a result of the mutual use of the orifice.
Now, we would demonstrate how the movementnot controlled by an orifice is the weir. Because the majority of the leverhead armloss andis thedominated rise and fall of plant flow rate are related.
{latex}
\large
$$
Q_C = K_C h_C^{n_C }
$$
$$
Q_P = K_P h_P^{n_P }
$$
{latex}
where
Q_C = dosing flow rate
K_C = orifice coefficient
Q_P = the plant flow rate
K_P = the orifice coefficient.
The mass balance equation calculates the flow of alum needed based on plant flow rate and the concentration of the alum chemical stock tank.
{latex}
\large
$$
C_P = {{C_C Q_C } \over {Q_P }}
$$
{latex}
{latex}
\large
$$
C_P = {{C_C K_C h_C^{n_C } } \over {K_P h_P^{n_P } }}
$$
{latex}
C_P=target alum concentration for our plant
C_C=the alum solution concentration
Q_C=the alum flow rate
We substitute the dosing flow rate equation in and link the two heights with a lever and cancel out any duplicate variables.
{latex}
\large
$$
h_C = K_L h_P
$$
$$
C_P = {{C_C K_C K_L^{n_C } h_P^{n_C } } \over {K_P h_P^{n_P } }}
$$
$$
C_P = {{C_C K_C K_L^{n_C } h_P^{n_C } } \over {K_P h_P^{n_P } }}
$$
$$
C_P \propto {{K_L^{n_C } h_P^{n_C } } \over {h_P^{n_P } }}
$$
$$
C_P = {{C_C K_C K_L^{n_C } h_P^{n_C } } \over {K_P h_P^{n_P } }}
$$
$$
C_P = \sqrt {K_L^{} } {{C_C K_C } \over {K_P }}
$$
$$
\sqrt {K_L } \propto C_P
$$
{latex}
The height of the water, or the flow rate to the plant via the entrance tank, is non-linearly related to the change in height of the scale. Because the relationship is defined by the orifice equation,the change in height gives us the change in flowrate required. Linking the different sets of heights to the scale, we generate a scale that an operator can use to adjust the dosage. Because the orifice equation that controls this relationship is nonlinear, the scale is nonlinear. An example of a nonlinear scale is shown below.
+(Your scale is a good visual representation. I would include a table as well of these values and perhaps a sample calculation for what you described above.)+
!scale.PNG|align=centre!
h3. Method:
We utilized [Mathcad|^Lever Arm Calculations 2009 NCDC TM.xmcd] to calculate our design parameters. The rest of this section illustrates our method in designing the orifice diameter and the dual scale for the plant.
The maximum movement of the by the orifice 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 nonlinearly 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. isThis determinedrange byis theequal sum ofto the thetotal headloss fromin the flocculatorplant, macrowhich andis micro mixer, weir, and launder orifices as shown in Table 1.
!HL Chart.PNG|align=centre,width=300,height=300!
This vertical distance of 33.5 cm establishes the upper and lower limit of flow to which our lever arm must be designed to respond.
In our calculations, we first define the applicable alum dose range of plant operation. Two different orifices handle two different scales of alum dosage: high (20-100 mg/L) and low (5-25 mg/L). They are offset to each other by a factor of four. Consequently, as dictated by the area portion of the orifice equation, the diameter of the larger orifice is twice the diameter of the smaller orifice.
+(Please label and describe your mathcad screenshot)+
!input.PNG|align=centre!
The operator of this program will change the diameter of the program until the dual scale utilizes maximum space on the lever arm (40 cm). This Mathcad program is designed to find the right size diameter through a series of trial and error iterations until we utilze the entire length of the scale which corresponds to the maximum and minimum height of the plant flow.
After we have input the tentative diameter for the larger orifice and the dual series of doses we conduct the calculations mentioned in the theory section to produce the nonlinear dual scale. We convert the dosage to the alum flow rate required by utilizing the mass balance equation.
!angle flow.PNG|align=centre!
The above snap shot also shows how the maximum headloss dicatates the maximum angle that our lever arm will be operating around.
!h scale.PNG|align=centre!
We then calculate the necessary head loss in the dosage system to produced the required flow rates.
Utilizing the sine function we convert the head loss to points along the lever scale arm. +(Show the equation if possible)+
We can also calculate the dual scale through a quicker method by utilizing the orifice equation relationship governing this whole process by linking the dosage directly to the scale. The resulting scale is shown below with the rest of the results for comparison. +(Did you compare your two methods here? I didn't see this.)+
!k.PNG|align=centre!
We know that the governing equation for this system dictates that the flow is equal to square root of distance times a constant K. We solve for the K by dividing the maximum dosage by the square root of the maximum length +(can you refer to an equation here?)+. We then utilize the K to solve for the dual scale.
The snapshot below shows us our final results. We manipulated the larger orifice size until we fully utilize the lever arm33.5 cm in this case. Therefore, we selected 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 as shown below:
Our next step consists of setting up two sets of target alum concentrations: 5-25 mg/L and 20-100 mg/L. Given
a known maximum plant flow rate(Qp) and Alum Stock concentration(Cc), we utilized the mass balance equation to
determine alum flow rate required for each target alum concentration as shown below:
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:
We then convert these headlosses to points along our scale via simple geometry as shown below:
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 alum flow rate required to meet these target concentration is generated as another 1 x 9 array. We just continue this process until we calculate an array of points along a scale as shown below.
!output.PNG|align=centre!
Shown below is the scale that is generated from our calculations.
+(Please connectConsequently,the the calculationsoperator youof performedthis inprogram MathCADwill tochange some sort of diagramthe diameter of the systemorifice anduntil referthe todual it.scale Also,utilizes makemaximum surespace thaton yourthe mathcadlever screenshotsarm and graphs are separate and that you have a caption for both figures and that you refer to your caption in the text.)+
!Scale 2.PNG|align=centre!
(40 cm).
h3. Error Analysis
h3. Results and Discussions
Currently, our orifices are 03.310175 cmmm for alum dosage of 20 to 100 mg/L offset by 10 mg/L and 01.0155587 cmmm 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 3/8 inch9.525 mm diameter which is wide and smooth enough to produce negligible head loss on the alum flow. The scale developed by linking the dosage to the scale is very similiar to our dual scale developed via our longer Mathcad programs, validating that the nonlinear relationship dictated by the orifice equation governs plant flow rate, the change in head loss, and the dual scale.
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. We also plan on conducting experiments to test our lever arm and find a solution to the [clogging problem|Clogging Experiment] currently besetting actual alum dosers in Honduras.
h2. Bibliography
* CEE4540 Flow Control Measurement Notes at https://confluence.cornell.edu/display/cee4540/Syllabus
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