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 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. The 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.
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. The purpose of calculating the float weight is to ensure the stability of the float in the entrance tank. As in any stable hull, the force of buoyancy needs to be below the center of gravity. This concept which will result in a stable float is shown in the figure below:
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
$$
T(
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)\cos (\alpha ) = W_
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(
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)\cos (\alpha ) + (W_
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+ W_
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)(
- {{L_
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} \over 2})\cos (\alpha )
$$
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\large
$$
F_
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= \gamma _
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\times \pi \times {\textstyle{{D_
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^2 } \over 4}} \times \left( {{\textstyle{{D_
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} \over 2}} - Y} \right)
$$
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 the pvc apparatus which delivers the alum from the orifice to the rapid mix tube.
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 substituting 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 for calibration at maximum dosage was determined to be 2.849 kg. This solution was obtained after inputing a reasonable value for the height and diameter of the float. The float's height was set to 0.305 m, which is reasonably tall enough to provide stability. The diameter was set at a high 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:
Table 1: Float Calibration Mass
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 |
An error analysis of the float calculations was done to determine the sensitivity of the calculated results.