h1. Float Calculation
h2. !dosertank.JPG|align=center!
h5. Figure 1: Lever arm/float orientation
h3. Abstract
In the fall semester of 2009, the Non-linear Chemical Doser team developed a [Mathcad|Lever Arm Calculations 2009 NCDC TM.xmcd] file to help plant operators choose a float given a non-linear dosing system. This float will ensure that the angle of the lever arm will be kept at the the proper position to ensure the accurate dosing of alum as the plant flow rate is varied. For our prototype, we calculated that our float needs to have a diameter of 15.25 cm and 30.5 cm of height. Mass of the float will depend on the dosage that we want to calibrate our float on. The calculations for these numbers will be enumerated below.
h3. Method
The float design parameters can be determined using a moment balance around the pivot of the lever arm. As seen in Figure 2 below, the major forces acting on the lever arm are the center of masses of the lever arm on either side of the pivot, the weight of the dosing slide, the alum dosing tube, the tube connecting the lever arm with the rapid mix tube, and the components of the float. In this case, we are going to balance the lever, or calibrate it, with the dosage set to maximum dosage or the slider pushed all the way to the far end of the scale. A moment balance was performed in order to determine what mass of float would be required in order to balance the lever arm at the lever arm angle corresponding to maximum plant flow. The forces due to the masses of the lever arm to the left and right of the lever arm cancel out. All that remains then acting on the lever arm is the force due to the alum tubes, the weight of the sliding scale, and the tension caused by the float. The weight for the float is then changed until the moment about the pivot point becomes zero. The weight found makes the lever arm perfectly balanced at a maximum dosing rate of 100 mg/l at the maximum angle the doser arm will experience (max flow rate).
h2.!doser arm.JPG|align=center!
h5. Figure 2: Free Body Diagram of Lever Arm
The formula for the moment balance can be seen below.
{latex} \large $$ \sum {Moments_{pivot} = 0;} $$ {latex}
{latex}
\large
$$
T({L \over 2})\cos (\alpha ) = W_{alumtube1} ({L \over 4})\cos (\alpha ) + (W_{slide} + W_{alumtube2} )({L \over 2} - {{L_{slide} } \over 2})\cos (\alpha )
$$
{latex}
{latex}
\large
$$
F_{Buoy} = \gamma _{H2O} \times \pi \times {\textstyle{{D_{Float} ^2 } \over 4}} \times \left( {{\textstyle{{D_{Length} } \over 2}} - Y} \right)
$$
{latex}
*Where:*
L = Length of the lever arm
{latex}$$ \alpha $${latex} = The angle the lever arm is with the horizontal
T = Tension Force in the string
{latex}$$ W_{alumtube} $${latex} = The force caused by the weight of alum in the dosing tube
{latex}$$ W_{slide} $${latex} = The force caused by the weight of the slide
{latex}$$ W_{alumtube2} $${latex} = The force caused by the weight of alum in the tube which extends into the rapid mix unit.
The cosine terms cancel out of the equation and we can then substitute in the expression for the expression relating our float characteristics to the Tension force. This equation is shown below:
{latex} \large $$ T = W_{float}-F_b $$ {latex}
Archimedes principle, which predicts the upward force on the float cause by the displacement of water can be seen below.
{latex} \large $$ F_b = \gamma V $$ {latex}
*Where:*
{latex}$$ F_{b} $${latex} = The force of buoyancy (Newtons)
{latex}$$ \gamma $${latex} = Unit weight of water (1000 Kg/m^3)
V = Volume of displaced water
After plugging the formula for the tension in the rope into our moment balance equation, we can solve for our weight of the float required to cause our moments around the pivot point to be equal to 0. This approximation can be shown below.
{latex} \large
$$
0 = - W_{float} + {{W_{alumtube1} } \over 2} + W_{slide} (1 - {{L_{slide} } \over L}) + W_{alumtube2} (1 - {{L_{slide} } \over L}) + {{\gamma _{H_2 O} \pi D^2 H} \over 4}
$$ {latex}
The solution for the mass of the float for calibration at maximum dosage was determined to be 2.9849 kg. This solution was obtained after inputing a reasonable value for the lenght and diameter of the float. The floats height was set to .305 m, which is reasonably long enough to provide stability to the provide. The diameter was set at a higher value of 15.2cm in order to reduce the amount of vertical movement of the float. We can also apply the above equations to derive the mass of the float needed to calibrate the lever arm at dosages different from the maximum. The values of calibration mass versus dosage are shown below:
||Dosage(mg/L)||Mass(kg)||
|20|2.783|
|30|2.786|
|40|2.791|
|50|2.797|
|60|2.805|
|70|2.814|
|80|2.824|
|90|2.836|
|100|2.849|
h3. Possible sources of Error
The largest source of error is due to the movement of slider assembly whenever the operator needs to adjust the doser to a dosage different from the dosage that the lever arm was initially calibrated at. In the example mentioned above, we calibrated the doser at maximum dosage. Consequently, error will increase as we decrease the dosage. Below diagram and equations will show how we calculated the error due to the movement of the slider assembly.
!Floaterror.PNG|align=center,width=500, height=300!
{latex}
\large
$$
\sum M = 0
$$
{latex}
{latex}
\large
$$
\cos (\theta ) \times {\textstyle{L \over 2}} \times \left( {W_{Float} - F_{Buoy} } \right) = \cos (\theta ) \times X \times W_{Assembly} + \cos (\theta ) \times {\textstyle{X \over 2}} \times W_{Tube}
$$
{latex}
{latex}
\large
$$
F_{Buoy} = \gamma _{H2O} \times \pi \times {\textstyle{{D_{Float} ^2 } \over 4}} \times \left( {{\textstyle{{D_{Length} } \over 2}} - \Delta Y} \right)
$$
{latex}
We rearrange the above equations to solve for the change in Y, or the error, that results from the movement of the slider assembly.
{latex}
\large
$$
\Delta Y = {\textstyle{{\left( {W_{Assembly} + .5W_{Tube} } \right) \times X \times 2/L - W_{Float} } \over {\gamma _{H2O} \times {\textstyle{{\pi \times D_{Float^2 } } \over 4}}}}}
$$
{latex}
The above equation demostrates several ways that we can reduce error. First of all, we can see that the diameter is inversely proportional to the error. Consequently we should increase the diameter as much as possible. Our entrance tank is 1 m by 1 m, giving us only .1 m clearance on each side of the lever. Consequently, a float diameter of 0.152m is the widest that we can make the float with some buffer room. The other observation is that the weight of the slider assembly and the alum tube is directly proportional to error. We can reduce our error by reducing our alum tube and slider assembly weight as much as possible. The most complete way to reduce error, or at least reduce the consequence of error is the selection of the dosage we calibrate our lever arm on. Since error is minimal in the dosage that we calibrate our lever arm on, we should calibrate our lever arm on the dosage that is most commonly used in the plant. Consequently, overall we would be achieving the most efficient dosing of alum.
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h3. Nonlinear Chemical Doser Calibration Procedure
Purpose: We need to make sure that our lever arm is able to adjust the alum dosage automatically for the entire flow range of the plant flow. Since alum dosage is ultimately set by the float moving up or down the entrance tank as plant flow changes, we need to make sure the lever arm is calibrated for minimum and maximum water height in the entrance tank, which corresponds to the minimum and maximum plant flow rate. Head loss in the plant determines the difference between minimum and maximum entrance tank water height. Greater the head loss in the tank, greater would be the difference betweenThe right side, therefore, has less head between the constant head tank and the dosing orifice, causing less flow of alum. Steps should be taken to ensure that the turbulence in the entrance tank is minimized so that there will be no additional forces on the float, which could cause errors in the alum dosage.
h3. Nonlinear Chemical Doser Calibration Procedure
Purpose: We need to make sure that our lever arm is able to adjust the alum dosage automatically for the entire flow range of the plant flow. Since alum dosage is ultimately set by the float moving up or down the entrance tank as plant flow changes, we need to make sure the lever arm is calibrated for minimum and maximum water height givenin anythe plantentrance flowtank, rate.which corresponds Evento thoughthe weminimum wouldand designmaximum theplant lever arm based on analytically calculated head loss, there is bound to be minorflow rate. Head loss in the plant determines the difference between calculated minimum and maximum entrance tank water flowheight. andGreater the actualhead heightsloss therein willthe betank, othergreater sourceswould ofbe errorthe thatdifference canbetween beminimum minimizedand bymaximum calibratingwater theheight levergiven armany toplant theflow plantrate. Even though we would design the
1.lever Confirmarm and/orbased adjuston theanalytically totalcalculated head loss, inthere theis plant:bound to Eitherbe throughminor inspectiondifference ofbetween thecalculated plantminimum orand dialoguemaximum withwater theflow experts onand the ground, we need to confirm the minimum and maximum water height possible in the plant. If there is too much difference between minimum and maximum water height and the lever arm would not be long enough to respond to the change, we must decrease the total head loss in the plant. If needed, we can make adjustments to the rapid mixer to increase or decrease the head loss actual heights there will be other sources of error that can be minimized by calibrating the lever arm to the plant.
1. Confirm and/or adjust the total head loss in the plant: Either through inspection of the plant or dialogue with the experts on the ground, we need to confirm the minimum and maximum water height possible in the plant. If Markthere theis location of thetoo much difference between minimum and maximum water height.
2. Installand the lever nonlineararm chemicalwould doser:not be Afterlong enough checkingto orrespond adjustingto the waterchange, heightwe rangemust indecrease the entrancetotal tank,head setloss upin the non-linear doser.
3. Balance the lever: The float is weighted so that when at equilibrium with water it is halfway submerged. Manually bring the centerplant. If needed, we can make adjustments to the rapid mixer to increase or decrease the head loss in the plant. Mark the location of the float length tomaximum water height.
2. Install the maximum water height mark and ensure that the string connecting the float to the lever arm is tight. nonlinear chemical doser: After checking or adjusting the water height range in the entrance tank, set up the non-linear doser.
3. Calibrate the float weight on the dosage that is most commonly used. Use the chart below:
||Dosage(mg/L)||Mass(kg)||
|20|2.783|
|30|2.786|
|40|2.791|
|50|2.797|
|60|2.805|
|70|2.814|
|80|2.824|
|90|2.836|
|100|2.849|
4. Check the doser at maximum alum flow rate: Push the slider to the maximum alum dosage and measure the alum flow to ensure it is giving out approximately the correct flow rate of alum flow rate.
5. Adjust the and float connection: Tie the float closer or farther to the pivot point until the maximum alum float rate is correct.
h3. Conclusion and Future Workis still submerged with a tight string connection to the lever.
h3. Results and Discussions
We require a float with a mass of 1.24 Kg, diameter of 1215.72 cm, and 30.5.08 centimeter height for our lever arm prototype. The float must have a sealable cap that would allow us to vary the weight.
Further work will have to be done analyzing the errors in dosing caused by the assumptions made in the calculation of the float size. The extent to which these errors will cause an improper amount of alum to be dosed is not fully known yet.
Upon construction of the lever arm, we will utilize a makeshift float in which we can adjust the weight of the float easily to verify our calculations here. This will be done before we order the final float for the dosing system.
h2. Bibliography
* CEE4540 Flow Control Measurement Notes at https://confluence.cornell.edu/display/cee4540/Syllabus
h2. Deliverables
* Final Float Design Parameters and Calculations
* Float protype for March 2010 EPA Competition
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