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 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. 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. In this case, we are going to balance the lever, or calibrate it, with the dosage set to maximum dosage. This means that the slider is 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).
Unable to render embedded object: File (doser arm.JPG) not found.
Figure 2: Free Body Diagram of Lever Arm
The formula for the moment balance can be seen below.
Unknown macro: {latex}
\large $$ \sum {Moments_
Unknown macro: {pivot}
= 0;} $$
Unknown macro: {latex}
\large
$$
T(
Unknown macro: {L over 2}
)\cos (\alpha ) = W_
Unknown macro: {alumtube1}
(
Unknown macro: {L over 4}
)\cos (\alpha ) + (W_
Unknown macro: {slide}
+ W_
Unknown macro: {alumtube2}
)(
- {{L_
Unknown macro: {slide}
} \over 2})\cos (\alpha )
$$
Unknown macro: {latex}
\large
$$
F_
Unknown macro: {Buoy}
= \gamma _
Unknown macro: {H2O}
\times \pi \times {\textstyle{{D_
Unknown macro: {Float}
^2 } \over 4}} \times \left( {{\textstyle{{D_
Unknown macro: {Length}
} \over 2}} - Y} \right)
$$
Where:
L = Length of the lever arm
Unknown macro: {latex}
$$ \alpha $$
= The angle the lever arm is with the horizontal
T = Tension Force in the string
Unknown macro: {latex}
$$ W_
Unknown macro: {alumtube}
$$
= The force caused by the weight of alum in the dosing tube
Unknown macro: {latex}
$$ W_
Unknown macro: {slide}
$$
= The force caused by the weight of the slide
Unknown macro: {latex}
$$ W_
Unknown macro: {alumtube2}
$$
= 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:
Unknown macro: {latex}
\large $$ T = W_
Unknown macro: {float}
-F_b $$
Archimedes principle, which predicts the upward force on the float cause by the displacement of water can be seen below.
Unknown macro: {latex}
\large $$ F_b = \gamma V $$
Where:
Unknown macro: {latex}
$$ F_
Unknown macro: {b}
$$
= The force of buoyancy (Newtons)
Unknown macro: {latex}
$$ \gamma $$
= 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.
Unknown macro: {latex}
\large
$$
0 = - W_
Unknown macro: {float}
+ {{W_
Unknown macro: {alumtube1}
} \over 2} + W_
Unknown macro: {slide}
(1 - {{L_
} \over L}) + W_
Unknown macro: {alumtube2}
(1 - {{L_
Unknown macro: {slide}
} \over L}) + {{\gamma _
Unknown macro: {H_2 O}
\pi D^2 H} \over 4}
$$
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 .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:
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 |
Possible sources of Error
The largest source of error is due to the movement of slider assembly whenever the operator needs to adjust the lever 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.
Figure 3: Error due to Slider Assembly Movement
Unknown macro: {latex}
\large
$$
\sum M = 0
$$
Unknown macro: {latex}
\large
$$
\cos (\theta ) \times {\textstyle{L \over 2}} \times \left( {W_
Unknown macro: {Float}
- F_
Unknown macro: {Buoy}
} \right) = \cos (\theta ) \times X \times W_
Unknown macro: {Assembly}
+ \cos (\theta ) \times {\textstyle{X \over 2}} \times W_
Unknown macro: {Tube}
$$
Unknown macro: {latex}
\large
$$
F_
Unknown macro: {Buoy}
= \gamma _
Unknown macro: {H2O}
\times \pi \times {\textstyle{{D_
Unknown macro: {Float}
^2 } \over 4}} \times \left( {{\textstyle{{D_
Unknown macro: {Length}
} \over 2}} - \Delta Y} \right)
$$
Where
Unknown macro: {latex}
\large$$W_
Unknown macro: {Float}
$$
=Float weight
Unknown macro: {latex}
\large$$F_
Unknown macro: {Buoy}
$$
=Buoyance force
Unknown macro: {latex}
\large$$W_
Unknown macro: {Assembly}
$$
=Weight of dosage slider and dosage tube
Unknown macro: {latex}
\large$$D_
Unknown macro: {Float}
$$
=Float diameter
Unknown macro: {latex}
\large$$D_
Unknown macro: {Length}
$$
=Float length
Unknown macro: {latex}
\large$$\Delta Y $$
=Change in y direction, the error
Unknown macro: {latex}
\large$$X $$
=Distance from pivot point to center of mass of the slider assembly
We rearrange the above equations to solve for the change in
Unknown macro: {latex}
\large$$\Delta Y $$
, or the error, that results from the movement of the slider assembly.
Unknown macro: {latex}
\large
$$
\Delta Y = {\textstyle{{\left( {W_
Unknown macro: {Assembly}
+ .5W_
Unknown macro: {Tube}
} \right) \times X \times 2/L - W_
Unknown macro: {Float}
} \over {\gamma _
Unknown macro: {H2O}
\times {\textstyle{{\pi \times D_
Unknown macro: {Float^2 }
} \over 4}}}}}
$$
The above equation demonstrates several ways that we can reduce error. First of all, we can see that the diameter of the float 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 effective way to reduce error, or at least mitigate the repercussions of error, is the selection of the dosage we calibrate our lever arm on. Since error is zero 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. By doing so, we reduce the overall alum dosage error in the plant.
Another way that we can reduce the error is to install a sliding counterweight to the float side of the lever that we can adjust as we change the dosage. The weight of the counterweight and the scale on the float side should be calculated to counteract the weight of the alum tube and dosing assembly and the actual dosage. For example, when the operator lowers the dosage or moves the slider assembly toward the pivot point, the operator would also move the counterweight slider assembly away from the pivot point to a point scaled specifically to counteract the movement of the slider to that specific dosage. This would mean that the lever arm needs to be calibrated with the counterweight and slider at specific starting points. Besides the additional calibration step, the other drawback is the need for another component for the doser arm.
There are other possible sources of error. 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 can respond to 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 a minor difference between calculated minimum and maximum water flow and the actual heights. Calibration will also minimize the sources of error mentioned above. These are the calibration steps:
Step 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. Find out what dosage is most commonly used. 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 that it is giving out approximately the correct flow rate of alum and that float is still submerged with a tight string connection to the lever.
Results and Discussions
We require a float with a diameter of 15.2 cm and 30.5 centimeter height for our lever arm prototype. The float must have a resealable cap that would allow us to vary the weight. Calibration is crucial in minimizing the error that is associated with the float. We should balance, or calibrate, the lever arm at the dosage that is most commonly used to reduce the overall alum dosage error in the plant.
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. Clear PVC tube is currently the best candidate. We will run the doser at different dosages and simulated water heights to confirm our above calculations. Further work will have to be done analyzing the errors in dosing mentioned above. The extent to which these errors will cause an improper amount of alum to be dosed is not fully known yet. The actual errors will be minimized before we order the final float for the dosing system for the EPA competition.
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