Question

1. a water plant is treating 2,000,000 gallons per day. a test for free chlorine residual at the plant indicates a level of 0.2 mg/l. based on experience, the operator knows they must bring the free chlorine level up to 1.7 mg/l at the plant in order to maintain a level of 1.0 mg/l free residual in the distribution system. if the rotameter on the chlorinator is set at 35, what must the new setting be?

Explanation:

Step1: Understand the formula for chlorine dosage adjustment

The key concept here is that the required chlorine dosage is proportional to the difference in chlorine levels. The formula for the new rotameter setting (\(S_{new}\)) can be derived from the proportionality of the chlorine demand change and the original setting. The formula is based on the fact that the amount of chlorine needed is proportional to the volume of water and the difference in chlorine concentration.

First, we need to find the change in chlorine concentration. The initial chlorine level (\(C_{initial}\)) is 0.2 mg/L, and the target chlorine level at the plant (\(C_{target}\)) is 1.7 mg/L. The difference in concentration (\(\Delta C\)) is \(C_{target}-C_{initial}\).

The original rotameter setting (\(S_{original}\)) is 35. The relationship between the rotameter setting and the chlorine dosage is proportional, so we can set up a proportion:

\(\frac{S_{new}}{S_{original}}=\frac{C_{target}-C_{initial}}{C_{initial, old}-C_{initial, old?}}\) Wait, actually, more accurately, the amount of chlorine added is proportional to the concentration difference and the flow rate. Since the flow rate (2,000,000 gallons per day) is constant, the rotameter setting (which controls the chlorine dosage) is proportional to the required chlorine concentration difference.

So, the initial chlorine concentration at the plant is 0.2 mg/L, and we need to bring it up to 1.7 mg/L. So the required increase in concentration is \(1.7 - 0.2=1.5\) mg/L. Wait, but actually, maybe the original setting was for a certain concentration, but let's think again.

Wait, the problem says "the rotameter on the chlorinator is set at 35" for the initial situation. We need to find the new setting when the required chlorine concentration at the plant is 1.7 mg/L (from 0.2 mg/L). So the formula for the rotameter setting (which is related to the chlorine dosage) is proportional to the chlorine concentration needed. So:

\(S_{new}=S_{original}\times\frac{C_{target}-C_{initial}}{C_{initial, old}-C_{initial, old?}}\) No, actually, the initial chlorine level is 0.2 mg/L, and with the original setting (35), that's the current dosage. Now we need to have a dosage that brings the level to 1.7 mg/L. So the ratio of the new concentration difference to the old concentration difference (but wait, maybe the old concentration difference was such that with setting 35, the concentration was 0.2 mg/L. Wait, maybe the initial setting of 35 gives a chlorine dosage that results in 0.2 mg/L. Now we need a dosage that results in 1.7 mg/L. So the ratio of the new dosage to the old dosage is equal to the ratio of the new concentration to the old concentration (since dosage is proportional to concentration when flow is constant).

So, \( \frac{S_{new}}{35}=\frac{1.7}{0.2} \)? Wait, no, that can't be, because 1.7/0.2 is 8.5, and 35*8.5 is 297.5, which seems too high. Wait, maybe I misinterpret the problem. Wait, the operator knows they must bring the free chlorine level up to 1.7 mg/L at the plant in order to maintain 1.0 mg/L in the distribution system. Wait, maybe the initial chlorine level at the plant was such that in the distribution system it was... Wait, no, the problem says "a test for free chlorine residual at the plant indicates a level of 0.2 mg/L. Based on experience, the operator knows they must bring the free chlorine level up to 1.7 mg/L at the plant in order to maintain a level of 1.0 mg/L free residual in the distribution system." So the initial plant level is 0.2 mg/L, and they need plant level 1.7 mg/L to get distribution level 1.0 mg/L. So the current setting (35) is for plant level 0.2 mg/L. Now they need plant level 1.7 mg/L. So the required chlorine dosage is proportional to the plant level (since the flow rate is constant). So the ratio of the new setting to the old setting is equal to the ratio of the new plant level to the old plant level.

So:

\( S_{new}=35\times\frac{1.7}{0.2} \)

Wait, let's calculate that:

\( \frac{1.7}{0.2}=8.5 \)

\( 35\times8.5 = 297.5 \)? That seems too high. Wait, maybe I made a mistake. Wait, maybe the initial chlorine level at the plant is 0.2 mg/L, and with that, the distribution level is some value, but now they need plant level 1.7 mg/L to get distribution level 1.0 mg/L. Wait, maybe the problem is that the current setting (35) gives a plant level of 0.2 mg/L, and we need to find the setting that gives a plant level of 1.7 mg/L. So the rotameter setting is proportional to the plant chlorine level (since the flow rate is constant). So:

\( S_{new}=S_{original}\times\frac{C_{target}}{C_{initial}} \)

Wait, \( C_{initial}=0.2 \) mg/L, \( C_{target}=1.7 \) mg/L, \( S_{original}=35 \)

So \( S_{new}=35\times\frac{1.7}{0.2} = 35\times8.5 = 297.5 \)? That seems very high, but maybe that's correct? Wait, no, maybe the difference is (1.7 - 0.2) instead of 1.7/0.2. Wait, let's re-examine the problem.

Wait, the plant is treating 2,000,000 gallons per day. The initial free chlorine residual at the plant is 0.2 mg/L. They need to bring it up to 1.7 mg/L at the plant to have 1.0 mg/L in the distribution. So the chlorine dosage (controlled by the rotameter) is what's adding chlorine to the water. So the amount of chlorine added per day is proportional to the rotameter setting. The amount of chlorine in the water is (chlorine added) / (volume of water). So:

Chlorine added (\(Cl_{added}\)) = rotameter setting (\(S\)) * some constant (\(k\))

Chlorine concentration (\(C\)) = \(Cl_{added}\) / (volume of water, \(V\))

So \( C = \frac{S\times k}{V} \)

Since \(V\) and \(k\) are constant, \(C\) is proportional to \(S\). So \( C \propto S \), so \( \frac{S_{new}}{S_{original}}=\frac{C_{new}}{C_{original}} \)

Wait, that makes sense. Because if you double the rotameter setting, you double the chlorine added, hence double the concentration (since volume is constant). So in the initial case, \( C_{original}=0.2 \) mg/L, \( S_{original}=35 \). In the new case, \( C_{new}=1.7 \) mg/L, so:

\( S_{new}=S_{original}\times\frac{C_{new}}{C_{original}} \)

So plugging in the numbers:

\( S_{new}=35\times\frac{1.7}{0.2} \)

Calculate \( \frac{1.7}{0.2}=8.5 \)

Then \( 35\times8.5 = 297.5 \)

Wait, but that seems very high. Maybe I misread the problem. Wait, the problem says "the rotameter on the chlorinator is set at 35" for the initial situation (0.2 mg/L). Now we need to set it so that the plant level is 1.7 mg/L. So according to the proportionality, since concentration is proportional to rotameter setting (because more setting means more chlorine added, hence higher concentration), then the calculation is correct.

Alternatively, maybe the initial chlorine level was such that with setting 35, the concentration was 0.2 mg/L, and now we need a concentration of 1.7 mg/L, so the ratio is 1.7/0.2, so the new setting is 35*(1.7/0.2)=297.5.

Answer:

The new rotameter setting must be \(\boxed{297.5}\) (or if we consider rounding, maybe 298, but the calculation gives 297.5).