h1. Calculating Orifice Sizes and the Dual Scale for the Lever Arm.
!orifice.PNG|align=leftcentre,width=800,height=600!
h3. Abstract:
During the fall semester of 2009, the Non-linear Chemical Alum Doser Team developed the dual orifice scale system for the lever arm in order to deliver turbulent flow of alum dosing. 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 program that contains the calculations for our dosing system. Please see our [calculations|^Lever Arm Calculations 2009 NCDC TM.xmcd]
We have designed a 80cm long lever arm with equal lengths and two orifices of 0.122 inch and 0.061 inch, 3/8" pvc tubing, and other associated hydraulic components listed in our [hydraulic components list|^NCDC Component List.xlsx].
h3. Theory:
Our first step in developing a dosage system that can deliver both turbulent and laminar flow of alum was the selection of the orifice to control the flow of alum. We increased the tubing sizing connecting the constant head tank to the orifice to 3/8 inch which is wide and smooth enough to make the headloss from the tubing negligible. This action essentially makes 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 headloss. The flow rate is the result of the square root of the headloss times a constant.
{latex}
\large
$$
Q = K_{vc} A_{or} \sqrt {2gh}
$$
{latex}
We then learned that the orifice controls most of the other headloss occuring in the plant after the entrance tank since the flow through the rapid mixers, the flocculation tank, and the launder are all controlled by the orifice. The only source of headloss not controlled by an orifice is the weir. Because the majority of the headloss 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.
We then link the relationship between the lever and the plant flow rate through the following equations mentioned below.
{latex}
\large
$$
Q_C = K_C h_C^{n_C }
$$
$$
Q_P = K_P h_P^{n_P }
$$
{latex}
Above two equations show us that alum and plant flow rate are governed by the same nonlinear relationship as a result of the use of orifice.
{latex}
\large
$$
C_P = {{C_C Q_C } \over {Q_P }}
$$
{latex}
The mass balance equation above shows us how we calculate the flow of alum based on plant flow rate and the concentration of the alum chemical stock tank.
{latex}
\large
$$
C_P = {{C_C K_C h_C^{n_C } } \over {K_P h_P^{n_P } }}
$$
{latex}
We substitute the chemical 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, is nonlinearly related to the change in height of the scale. Because the relationship is defined by the orifice equation, square root of the change in height times a constant factor 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 nonlinear scale is shown below.
!scale.PNG|align=right!
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 water height in the entrance tank is determined by the sum of the the headloss from the flocculator, macro and micro mixer, weir, and launder orifices as shown below.
!HL Chart.PNG|align=centre,width=300,height=300!
This vertical distance establishes the upper and lower limit of flow that our lever arm must be designed to respond to.
In our calculations, we first define the scale of the alum dosage that we want the plant to operate around. We have two different orifices to handle two different levels of alum: high and low. They are offset to each other by a factor of 4. Consequently, as dictated by the area portion of the orifice equation relationship, the diameter of the larger orifice is twice the diameter of the smaller orifice.
!input.PNG|align=centre!
Diameter input is correct but this Mathcad program is designed so that we can find the right size diameter through a series of trial and error until we utilze the entire length of the scale which corresponds to the maximum and minimum height of the plant flow. The operator of this program will change the diameter of the program until the dual scale takes up as much of the scale(40 cm) as possible.
After we have inputed 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 headloss in the dosage system to produced the required flow rates.
Utilizing the sine function we convert the headloss to points along the lever scale arm.
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.
!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 diving the maximum dosage by the square root of the maximum length. 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 arm scale as shown below.
!output.PNG|align=centre!
Shown below is the scale that is generated from our calculations.
!Nonlinear Scale 2.PNG|align=centre!
h3. Results and Discussions
Currently, our orifices are 0.122 inch for alum dosage of 20 to 100 mg/L offset by 10 mg/L and 0.061 inch 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 inch diameter which is wide and smooth enough to produce negligible headloss 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 dicated by the orifice equation governs plant flow rate, the change in headloss, and the dual scale. We currently have purchased or acquired all hydraulic components needed for our lever arm.
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 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 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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