Problem 1 - Advanced Water Mathematics

Question

A well has a nitrate level that exceeds the MCL of \(45 \mathrm{mg} / \mathrm{L}\). Over the last 3 sample results it has averaged \(52 \mathrm{mg} / \mathrm{L}\). A nearby well has a nitrate level of \(32 \mathrm{mg} / \mathrm{L}\). If both wells combined pump up to 2,275 gpm, how much flow is required from each well to achieve a nitrate level of \(40 \mathrm{mg} / \mathrm{L} ?\)

Step-by-Step Solution

Answer

910 gpm from the first well, 1,365 gpm from the second well.

Step 1 - Define the variables

Let the flow from the first well (the one with 52 mg/L) be denoted as \(Q_1\) and the flow from the second well (the one with 32 mg/L) be denoted as \(Q_2\). The total flow is given as 2,275 gpm.

Step 2 - Set up the equation for the nitrate level

The average nitrate level after mixing should be 40 mg/L. Use the formula for weighted average: \[ \text{Average nitrate level} = \frac{Q_1 \times 52 + Q_2 \times 32}{Q_1 + Q_2} \]We know that \(Q_1 + Q_2 = 2,275\).

Step 3 - Substitute the total flow equation

Since \(Q_1 + Q_2 = 2,275\), we can substitute \(Q_2 = 2,275 - Q_1\) into the nitrate level equation: \[ 40 = \frac{Q_1 \times 52 + (2275 - Q_1) \times 32}{2275} \]

Step 4 - Simplify and solve for \(Q_1\)

Multiply both sides by 2,275 to eliminate the denominator: \[ 40 \times 2,275 = Q_1 \times 52 + (2,275 - Q_1) \times 32 \]Simplify the equation: \[ 91,000 = 52Q_1 + 72,800 - 32Q_1 \]Combine like terms: \[ 91,000 = 20Q_1 + 72,800 \]Subtract 72,800 from both sides: \[ 18,200 = 20Q_1 \]Divide by 20: \[ Q_1 = 910 \]

Step 5 - Solve for \(Q_2\)

Using \(Q_1 = 910\) in the total flow equation \(Q_2 = 2,275 - Q_1\): \[ Q_2 = 2,275 - 910 = 1,365 \]

Step 6 - Verify the solution

Substitute \(Q_1 = 910\) and \(Q_2 = 1,365\) back into the weighted average formula to verify: \[ \text{Average nitrate level} = \frac{910 \times 52 + 1,365 \times 32}{2,275} = 40 \] The values satisfy the original condition.

Key Concepts

Weighted Average

To solve the problem, the concept of a weighted average is used. A weighted average takes into account both the values and their respective weights.
Here, the 'values' are the nitrate levels of each well, and the 'weights' are the flow rates.

• In our formula, \( \text{Average nitrate level} = \frac{Q_1 \times 52 + Q_2 \times 32}{Q_1 + Q_2} \), the numerator represents the total amount of nitrate ion from both wells.
• The denominator, \( Q_1 + Q_2 \, \), represents the total flow rate of the mixed water.

The objective is to achieve a desired nitrate level by combining flow rates in such a manner that the weighted average meets the regulation.
By solving a simplified equation, we can determine the necessary flow rates from both wells.

Fluid Dynamics

Understanding fluid dynamics helps to resolve how two water flows mix together.
Here are some basic principles:

• Fluid flow, in this case, is measured in gallons per minute (gpm).
• A higher flow rate from a well with high nitrate concentration will increase the overall nitrate level.
• Conversely, a higher flow rate from a well with low nitrate concentration will decrease the overall nitrate level.

In this problem, we are mixing two different flows to obtain a target nitrate concentration.
Combining 910 gpm from the first well with 1,365 gpm from the second well results in an overall flow that balances the nitrate levels to the desired 40 mg/L.
The underlying principle relies on the sum of products of each flow rate and its respective nitrate concentration, divided by the total flow.

Environmental Chemistry

Nitrate contamination in water sources is a major environmental issue.
High levels of nitrate in drinking water can be harmful. For this reason, regulatory bodies set Maximum Contaminant Levels (MCL) to ensure water safety.
Considerations in environmental chemistry for this problem include:

• The presence of nitrates, often from agricultural runoff or industrial processes, which can seep into groundwater sources.
• Human health risks associated with high nitrate levels, including conditions like methemoglobinemia or 'blue baby syndrome.'
• Regulating nitrate levels with mixing techniques to ensure safe drinking water without costly filtration processes.

The given exercise demonstrates a practical application where calculated mixing of two distinct water sources achieves a target nitrate concentration, ensuring environmental safety and regulatory compliance.
The solution employs chemistry principles, mathematical calculations, and an understanding of water flow dynamics for environmental protection.