Calculating Orifice Sizes and the Dual Scale for the Lever Arm.
Abstract:
During the fall semester of 2009, the Non-linear Chemical Alum Doser Team is developing the dual orifice scale system for the lever arm in order to handle turbulent flow chemical dosing used in conjunction with the newly designed Rapid Mix Tube. We have currently researched and identified the relationship between plant flow rate and alum dosage and lever arm components to include the dual scale and developed a Mathcad program to help us pick out components for our nonlinear dosing system. Mathcad file.
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 in order to make the headloss from the tubing negligible and 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.
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We then recognized that the orifice control most of the headloss occuring in the plant after the entrance tank since the flow through the macro and micro mixer, 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 but 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.
We then link the relationship between the lever and the plant flow rate.
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Q_P = K_P h_P^
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Both alum and plant flow rate are controlled by the orifice equation and share the same relationship as shown above.
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C_P = {{C_C Q_C } \over {Q_P }}
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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.
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C_P = {{C_C K_C h_C^
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We substitute the chemical flowrate equation in and link the two heights with a lever and cancel out necessary variables.
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h_C = K_L h_P
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C_P = {{C_C K_C K_L^
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h_P^
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$$
C_P = {{C_C K_C K_L^
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$$
C_P \propto {{K_L^
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C_P = {{C_C K_C K_L^
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C_P = \sqrt {K_L^{} } {{C_C K_C } \over {K_P }}
$$
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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 as shown below is nonlinear.
Because the plant flow rate and
This relationship justifies
Our method consisted of 1)establishing the relationship between plant flow rate, alum dosage, delta h (elevation difference or head loss?) between the constant head tank, and the dual scale (lever arm) 2)utilization of Mathcad Array function to link these variables, and` 3) a trial and error process to determine orifice size and the dual arm scale.
We first linked the plant flow to the target alum flow rate via the mass balance equation. (Show the mass balance equation, and specify your variables. You can copy the mass balance equation on the front page) We then utilized the orifice equation to link the alum float flow rate to the different delta h's. Delta h divided by the sin(angle deflection) gives us the points along the lever which is our dual scale. (I understand what you are saying, but a visualization how to determine your dual scale would be more explanatory than just the words here.)
We utilized the array calculation ability of Mathcad to link the plant flow rate to the dual scale. The Mathcad file first asks the user to input all the necessary plant parameters to include target alum dosage scale as an array. It then develops another arrary of alum flow rate necessary to meet those target alum dosage given the plant's flow rate and the alum concentration of the stock tank. We then created an array of delta H for each alum flowrate utilizing the orifice equation. The Mathcad then created another array indicating points along the scale that would correspond to each delta H. (I think it could be useful to show these arrays as screenshots here, but you do not have to.)
Given a set maximum angle deflection and other specified plant parameters, we would vary the orifice diameter through a trial and error process until we are able to both achieve the desired array of alum dosage and the corresponding array of dual scale that fully utilized the entire length of the scale arm. (The process seems to be very well thought out. I still think some visualization or screenshots of your Mathcad calculations would enhance your clever method).
Results/Discussion
The two orifice sizes that we calculated were 0.082 inch and 0.044 inch. Because the dosage tube is so wide(3/8 inch), these two orifices control the flow of alum from the constant head tank to the entrance tank. Given these two diameters and other plant parameters we are able to utilize the entire length of the scale arm on our lever as a scale.
Conclusion and Future Work
Plant flow rate, the alum flow rate, orifice size, delta h, and the distance along the lever from the pivot point(dual scale) are related to one another and the Lever Arm mathcad file links these different parameters and is designed to serve as a tool to enable the user to develop both the best orifice sizes and the dual scale by enabling the user to change the orifice sizes given other constant plant and lever arm parameters to meet the dosage needs of the plant as well as fully utilize the entire length of the scale of the lever arm(0.5m in length). (A future challenge for you (a task on the backburner if you ever have free time) or the design team could be to figure out a way to automate your array calculation method so that the design tool could be used to also calculate these parameters. Is this possible or too complicated?)
Our next step is conduct experiments to find a solution to the clogging problem currently occurring with actual alum dosers in the fields of Honduras.
Bibliography
- CEE4540 Flow Control Measurement Notes at https://confluence.cornell.edu/display/cee4540/Syllabus
Deliverables
*Final hydraulic component list
*Final orifice and dual scale calculations-2009 NCDC Lever Arm Calculations
*Hydraulic components for the lever arm prototype
*Dual scale engraved on the lever arm prototype