h3. Possible sources of Error
The largest source of error is due to the movement of the 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.
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h5. Figure 3: Error due to Slider Assembly Movement
{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}
Where
{latex}\large$$W_{Float} $${latex}=Float weight
{latex}\large$$F_{Buoy} $${latex}=Buoyance force
{latex}\large$$W_{Assembly} $${latex}=Weight of dosage slider and dosage tube
{latex}\large$$D_{Float} $${latex}=Float diameter
{latex}\large$$D_{Length} $${latex}=Float length
{latex}\large$$\Delta Y $${latex}=Change in y direction, the error
{latex}\large$$X $${latex}=Distance from pivot point to center of mass of the slider assembly
We rearrange the above equations to solve for the change in {latex}\large$$\Delta Y $${latex}, 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 demonstrates several ways that we can reduce error. First of all, we can see that the diameter of the float, {latex}\large$$D_{Float} $${latex}, 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, {latex}\large$$W_{Assembly} $${latex}, 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 toward the pivot point to counteract the change in moment. 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.
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