Float Calculation
Figure 1: Lever arm/float orientation
Abstract
In the fall semester of 2009, the Non-linear Chemical Doser team developed a [Mathcad] 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 mass of 2.9 Kg,a diameter of 15.25 cm, and 30.5 cm of height. The calculations for these numbers will be enumerated below.
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. 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).
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Figure 2: Free Body Diagram of Lever Arm
The formula for the moment balance can be seen below.
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\large $$ \sum {Moments_
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= 0;} $$
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\large
$$
T(
Unknown macro: {L over 2}
)\cos (\alpha ) = W_
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(
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)\cos (\alpha ) + (W_
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+ W_
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)(
- {{L_
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} \over 2})\cos (\alpha )
$$
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\large
$$
F_
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= \gamma _
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\times \pi \times {\textstyle{{D_
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^2 } \over 4}} \times \left( {{\textstyle{{D_
Unknown macro: {Length}
} \over 2}} - Y} \right)
$$
Where:
L = Length of the lever arm
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$$ \alpha $$
= The angle the lever arm is with the horizontal
T = Tension Force in the string
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$$ W_
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$$
= The force caused by the weight of alum in the dosing tube
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$$ W_
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$$
= The force caused by the weight of the slide
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$$ W_
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$$
= 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:
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\large $$ T = W_
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-F_b $$
Archimedes principle, which predicts the upward force on the float cause by the displacement of water can be seen below.
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\large $$ F_b = \gamma V $$
Where:
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$$ F_
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$$
= The force of buoyancy (Newtons)
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$$ \gamma $$
= 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.
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\large
$$
0 = - W_
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+ {{W_
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} \over 2} + W_
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(1 - {{L_
} \over L}) + W_
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(1 - {{L_
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} \over L}) + {{\gamma _
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\pi D^2 H} \over 4}
$$
The solution for the mass of the float was determined to be 2.9 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. Any change in the position of water on the float will introduce errors in the amount of alum being dosed. So it is essential to minimize the amount of change in H, shown in Figure 2. So we tried to make the diameter of the float as wide as possible without hitting the side of the entrance tank, while minimizing the height of the float. Given the amount of clearance on either side of the lever arm (.1 meters), the diameter of the float was chosen to be .152 m; this diameter should give adequate amount of clearance between the float and the entrance tank wall.
Possible sources of Error
An assumption was made in this analysis which will cause error in our final calculation. In order to calculate the tension force in the line connecting the lever arm and the float, an assumption for the amount of displaced water caused by the float had to be made. It was assumed that the float was positioned halfway up in the water. This assumption for the position of the float allowed for the force of buoyancy and thus the tension force to be calculated. However, as the sliding arm is moved up and down the lever arm, this will change the amount of tension applied to the float, causing the float to not be halfway up in the water.
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\large
$$
\sum M = 0
$$
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\large
$$
\cos (\theta ) \times {\textstyle{L \over 2}} \times \left( {W_
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- F_
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} \right) = \cos (\theta ) \times X \times W_
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+ \cos (\theta ) \times {\textstyle{X \over 2}} \times W_
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$$
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\large
$$
F_
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= \gamma _
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\times \pi \times {\textstyle{{D_
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^2 } \over 4}} \times \left( {{\textstyle{{D_
Unknown macro: {Length}
} \over 2}} - Y} \right)
$$
Additionally, the position for the center of mass of the dosing alum arm was assumed to be directly in the center of the right side of the lever arm. This is obviously an approximation and will introduce error. The same can be said for the position of the center of mass of the rigid tube which carries alum to the rapid mix tube. The force of this tube was assumed to be acting directly at the location of the slide's center of mass.
Maintaining tension on the rope connecting the float and the lever arm is essential for the accurate dosing of alum into the rapid mix tube. Upon inspection, this tension should be maintained since the right side of our lever arm is heavier than the left side of our lever arm due to the added components. As the float drops, the tension force will cause the left side of the lever arm to drop, but the weight of the components on the right will not allow there to be "slack" in the tension line. As the float drops it maintains tension in the rope and drops the left side of the lever, raising the right side. The 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.
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 between minimum and maximum water height given any plant flow rate. Even though we would design the lever arm based on analytically calculated head loss, there is bound to be minor difference between calculated minimum and maximum water flow and the 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 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 in the plant. Mark the location of the maximum water height.
2. Install the nonlinear chemical doser: After checking or adjusting the water height range in the entrance tank, set up 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 center of the float length to the maximum water height mark and ensure that the string connecting the float to the lever arm is tight.
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 the correct flow rate of alum flow rate.
5. Adjust the float connection: Tie the float closer or farther to the pivot point until the maximum alum float rate is correct.
Conclusion and Future Work
We require a float with a mass of 1.24 Kg, diameter of 12.7 cm, and 5.08 centimeter height for our lever arm prototype.
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.
Bibliography
- CEE4540 Flow Control Measurement Notes at https://confluence.cornell.edu/display/cee4540/Syllabus
Deliverables
- Final Float Design Parameters and Calculations
- Float protype for March 2010 EPA Competition