Problem 6 - Advanced Water Mathematics

Question

The State Health Department has requested a blending plan to lower levels of sulfate from a small water utility well. The well has a constant sulfate level of \(525 \mathrm{mg} / \mathrm{L}\). The utility needs to purchase the water to blend with the well. The purchased water has a sulfate level of \(135 \mathrm{mg} / \mathrm{L}\). They need to bring the sulfate levels down to \(265 \mathrm{mg} / \mathrm{L}\) and supply a demand of \(1.15 \mathrm{MGD}\). The purchased water costs \(\$ 550 / \mathrm{AF}\). How much will the purchased water cost for the entire year?

Step-by-Step Solution

Answer

The cost for the purchased water for the entire year is approximately $536,008.

Step 1: Define Variables

Let the volume of well water needed per day be denoted by variable W (in MGD). Let the volume of purchased water be denoted by variable P (in MGD).

Step 2: Set Up the Equation

The final sulfate concentration after blending is given by:\[(W \times 525 + P \times 135) / (W + P) = 265\]

Step 3: Express Daily Water Demand

The total daily water demand must satisfy:\[W + P = 1.15\]

Step 4: Solve for P

Substitute W in terms of P using the total daily water demand equation:\[W = 1.15 - P\]Plug this into the sulfate concentration equation:\[((1.15 - P) \times 525 + P \times 135) / 1.15 = 265\]

Step 5: Simplify

Simplify the equation to get a single variable equation in P:\[603.75 - 525P + 135P = 304.75P\]Combine like terms:\[603.75 - 390P = 304.75P\]

Step 6: Solve for P

Rearrange and solve for P:\[603.75 = 304.75P + 390P\]\[603.75 = 694.75P\]\[P = 603.75 / 694.75 \]\[P \approx 0.87 \text{ MGD}\]

Step 7: Calculate Total Volume of Purchased Water

Multiply the daily purchased water by the number of days in a year:\[P_{year} = 0.87 \text{ MGD} \times 365 \text{ days} = 317.55 \text{ MG}\]

Step 8: Convert Volume to Acre-Feet

Convert the total volume to acre-feet, knowing 1 MG = 3.068 acre-feet:\[Volume = 317.55 \text{ MG} \times 3.068 = 974.56 \text{ acre-feet}\]

Step 9: Calculate Total Cost

Multiply by the cost per acre-foot:\[Cost = 974.56 \times 550 \approx 536,008 \text{ dollars}\]

Key Concepts

sulfate concentration

Understanding how to adjust sulfate concentrations in water is crucial for meeting health standards. In this exercise, the sulfate concentration of the existing well water is 525 mg/L. To safely reduce it to acceptable levels, water with a lower sulfate concentration of 135 mg/L is blended in. This balance ensures that the final mixture meets the demand of 265 mg/L. The equation used to calculate the required blend is offered significant insight into how different sources with varying concentrations should be managed to meet regulatory standards.

When blending, consider these factors:

• Sulfate levels in each water source
• The desired final concentration
• The proportion of each water source

This systematic approach ensures that the final sulfate concentration is within acceptable limits.

water demand calculation

Meeting water demand accurately involves detailed calculations. Here, the utility's daily water demand is 1.15 million gallons per day (MGD). To find out how much water is needed from each source, we define:

• W: Volume of well water in MGD
• P: Volume of purchased water in MGD

Using the equation, we deduce that the required proportion of each water source ensures the total demanded is met.

The steps involve:

• Solving the blending equation to determine the amount of purchased water
• Re-calculating based on daily requirements to ensure consistency
• Multiplying the daily water needs by the number of days in the year to get the annual water demand

Ascertaining these volumes guarantees an adequate supply while maintaining the required sulfate concentration.

cost calculation

Determining the financial implications of water blending is essential for optimal resource management. In this example, the cost of purchased water is \(550 per acre-foot. To calculate the annual expense:

• Calculating the total volume of purchased water per year in million gallons (MG)
• Converting the annual volume from MG to acre-feet (1 MG = 3.068 acre-feet)
• Multiplying the total volume in acre-feet by the cost per acre-foot

For instance, in the solution:


Purchased water volume for the year, P_year, is 0.87 MGD \times 365 days = 317.55 MG.
Converting this to acre-feet is 317.55 MG \times 3.068 = 974.56 acre-feet.
Finally, the cost calculation is 974.56 acre-feet \times \)550 = $536,008.

This method helps utilities plan and budget effectively, ensuring they meet legal standards without incurring excessive costs.