Question

Solve the following chemical dosage problems. Be sure to account for the differences in chemical percent concentrations. A water softening treatment process uses \(25 \% \mathrm{NaOH}\) during \(20 \%\) of the year and \(50 \%\) \(\mathrm{NaOH}\) for \(80 \%\) of the year. Assuming a constant flow rate of 1,100 gpm and a dosage of \(70 \mathrm{mg} / \mathrm{L}\), what is the annual budget if the \(25 \% \mathrm{NaOH}(\mathrm{SG}=1.18)\) costs \(\$ 0.95\) per gallon and the \(50 \% \mathrm{NaOH}(S G=1.53)\) costs \(\$ 1.70\) per gallon?

Short answer

The annual budget is $42,552.02.

Step-by-step solution

- Determine Annual Flow Volume

Convert the flow rate from gallons per minute (gpm) to gallons per year. There are 525,600 minutes in a year.\[ \text{Annual Flow (gallons)} = 1,100 \text{ gpm} \times 525,600 \text{ minutes} = 578,160,000 \text{ gallons/year} \]

- Convert Dosage to Pounds per Year

Convert the dosage from mg/L to pounds. Using the conversion factor: 1 mg/L = 8.34 lbs/million gallons (MG).\[ 70 \text{ mg/L} = 70 \text{ mg/L} \times 8.34 \frac{\text{lbs}}{\text{MG}} = 583.8 \text{ lbs/MG} \]Calculate the total pounds per year using the annual flow volume.\[ \text{Total lbs/year} = 583.8 \text{ lbs/MG} \times 578.16 \text{ MG} = 337,458.048 \text{ lbs/year} \]

- Calculate Pounds of Each NaOH Concentration Used

Determine the amount used from each concentration. First, for 25% NaOH used for 20% of the year:\[ \text{Pounds of 25% NaOH} = 337,458.048 \text{ lbs/year} \times 0.20 = 67,491.6096 \text{ lbs/year} \]Next, for 50% NaOH used for 80% of the year:\[ \text{Pounds of 50% NaOH} = 337,458.048 \text{ lbs/year} \times 0.80 = 269,966.4384 \text{ lbs/year} \]

- Convert Pounds of NaOH to Gallons

Use the specific gravity (SG) to convert pounds to gallons. For 25% NaOH (SG=1.18):\[ \text{Gallons of 25% NaOH} = \frac{67,491.6096 \text{ lbs/year}}{1.18 \times 8.34 \frac{\text{lbs}}{\text{gallon}}} = 6,790.89 \text{ gallons/year} \]For 50% NaOH (SG=1.53):\[ \text{Gallons of 50% NaOH} = \frac{269,966.4384 \text{ lbs/year}}{1.53 \times 8.34 \frac{\text{lbs}}{\text{gallon}}} = 21,235.69 \text{ gallons/year} \]

- Calculate Annual Cost for Each NaOH Concentration

Calculate the cost for each concentration. For 25% NaOH:\[ \text{Cost for 25% NaOH} = 6,790.89 \text{ gallons/year} \times 0.95 \text{ \/ gallon} = 6,451.35 \text{ dollars/year} \]For 50% NaOH:\[ \text{Cost for 50% NaOH} = 21,235.69 \text{ gallons/year} \times 1.70 \text{ \/ gallon} = 36,100.67 \text{ dollars/year} \]

- Calculate Total Annual Budget

Add the cost for each concentration to get the total annual budget.\[ \text{Total Annual Budget} = 6,451.35 \text{ dollars/year} + 36,100.67 \text{ dollars/year} = 42,552.02 \text{ dollars/year} \]

Key concepts

Water Treatment Processes

Water treatment processes involve multiple techniques to ensure safe and clean water. One important process is water softening. This method removes minerals like calcium and magnesium that cause hard water. In our exercise, the water softening process uses Sodium Hydroxide (NaOH) as a chemical reagent.
A constant flow rate of 1,100 gallons per minute (gpm) is maintained. The effectiveness of this process is measured in terms of dosage, given in milligrams per liter (mg/L). Dosage determines how much of the chemical is needed to treat the water. Calculations are essential to ensure optimal chemical use and cost-efficiency.

Sodium Hydroxide Concentration

Sodium Hydroxide (NaOH) is a common chemical used in water treatment. It comes in various concentrations, such as 25% and 50% in our case. Understanding these concentrations is crucial for accurate chemical dosage calculation.
Percent concentration refers to the proportion of NaOH in a solution. For example, 25% NaOH means 25 grams of NaOH in every 100 grams of solution. This concept is important when determining how much of each concentration is used based on the treatment time. Specific gravity (SG) is also factored in to convert the weight of NaOH to volume, as it represents the density of the NaOH solution compared to water.

Annual Chemical Budget Calculation

Calculating the annual chemical budget involves several steps. First, determine the annual flow volume by converting the gpm flow rate to gallons per year. Next, convert the dosage from mg/L to pounds per year. This tells you the total amount of NaOH needed annually.
Then, determine the pounds of each NaOH concentration used based on the percentage of time each concentration is applied. Convert these pounds to gallons using specific gravity. Finally, calculate the annual cost by multiplying the gallons used by the cost per gallon. Adding these costs gives the total annual budget, essential for financial planning and efficiency in water treatment operations.