Problem 5 - Advanced Water Mathematics

Question

1 has a tota… # Two wells need to achieve a daily flow of \(3.24 \mathrm{MG}\) and a total hardness level of \(90 \mathrm{mg} / \mathrm{L}\) as calcium carbonate (CaCOs.) Well #1 has a total hardness level of \(315 \mathrm{mg} / \mathrm{L}\) as \(\mathrm{CaCO}_{3}\) and Well #2 has a level of \(58 \mathrm{mg} / \mathrm{L}\) as \(\mathrm{CaCO}_{3} .\) What is the gpm that each well must pump?

Step-by-Step Solution

Answer

Q₁ ≈ 0.403 MG/day, Q₂ ≈ 2.837 MG/day.

Step 1: Define variables

Let:1. Q₁ be the daily flow rate of Well #1 (in million gallons per day, MG).2. Q₂ be the daily flow rate of Well #2 (in million gallons per day, MG).3. H₁ = 315 mg/L (hardness level from Well #1).4. H₂ = 58 mg/L (hardness level from Well #2).5. Qₜ = 3.24 MG (total daily flow rate).6. Hₜ = 90 mg/L (total hardness level for the mixed water).

Step 2: Set up the total daily flow equation

Since the total daily flow is the sum of the flows from both wells, write the equation:\[Q₁ + Q₂ = Qₜ\]\[Q₁ + Q₂ = 3.24\]

Step 3: Set up the total hardness equation

The total hardness of the mixed water is the flow-weighted average of the hardness levels from each well. Write the equation:\[Q₁ H₁ + Q₂ H₂ = Qₜ Hₜ\]Substituting the known values:\[Q₁(315) + Q₂(58) = 3.24(90)\]\[315Q₁ + 58Q₂ = 291.6\]

Step 4: Solve the system of equations

Solve the system of linear equations formed in the previous steps:1. \[Q₁ + Q₂ = 3.24\] (Equation 1)2. \[315Q₁ + 58Q₂ = 291.6\] (Equation 2)From Equation 1, express Q₂ in terms of Q₁:\[Q₂ = 3.24 - Q₁\]Substitute this into Equation 2:\[315Q₁ + 58(3.24 - Q₁) = 291.6\]Expand and simplify:\[315Q₁ + 187.92 - 58Q₁ = 291.6\]Combine like terms:\[257Q₁ = 103.68\]Solve for Q₁:\[Q₁ = \frac{103.68}{257} \approx 0.403\] million gallons per day.

Step 5: Find Q₂

Using the value of Q₁:\[Q₂ = 3.24 - Q₁\]\[Q₂ = 3.24 - 0.403\]\[Q₂ \approx 2.837\] million gallons per day.

Key Concepts

water hardness

Water hardness refers to the concentration of certain minerals, primarily calcium and magnesium, in water. This is usually measured in milligrams per liter (mg/L) as calcium carbonate (CaCO₃). Hard water can cause scaling in pipes and reduce the effectiveness of soaps and detergents. There are two main types of water hardness:

• Temporary hardness - caused by the presence of dissolved bicarbonate minerals (calcium bicarbonate and magnesium bicarbonate) which can be removed by boiling the water.
• Permanent hardness - caused by the presence of sulfate or chloride compounds of calcium and magnesium, which cannot be removed by boiling.

linear equations

Linear equations are mathematical statements that describe a straight line when plotted on a graph. They generally take the form:

• a continuous and constant rate of change
• such as y = mx + b

where y is the dependent variable, m is the slope (rate of change), x is the independent variable, and b is the y-intercept. In the context of water flow and hardness calculations, we use linear equations to represent the relationships between different quantities. For this problem, we set up systems of linear equations to solve for the flow rates of each well, expressed as:
1. \(Q₁ + Q₂ = 3.24\)
2. \(315Q₁ + 58Q₂ = 291.6\)

mixed water quality

Mixed water quality refers to the combined characteristics of water from two or more different sources. In this exercise, we are considering the combined hardness of water from two wells. When mixing waters from different sources, the overall quality is often a weighted average based on the proportion of water contributed by each source. This means the total hardness of the mixed water (Hₜ) depends on the flow rates and individual hardness levels (H₁ and H₂) of the contributing wells. The formula used is:
\(Q₁H₁ + Q₂H₂ = QₜHₜ\)
where

• \(Q₁\) is the daily flow rate from Well #1
• \(H₁\) is the hardness level of Well #1
• \(Q₂\) is the daily flow rate from Well #2
• \(H₂\) is the hardness level of Well #2
• \(Qₜ\) is the total daily flow rate of the mixed water
• \(Hₜ\) is the hardness level of the mixed water