Problem 4 - Advanced Water Mathematics

Question

A water treatment plant processes a maximum of \(18.65\) MGD. The plant has 6 filters measuring \(20 \mathrm{ft}\) by \(22 \mathrm{ft}\) each. Assuming that each filter receives an equal amount of flow what is the filtration rate in gpm/ft \(^{2}\) ?

Step-by-Step Solution

Answer

4.91 gpm/ft²

Step 1 - Convert Maximum Daily Flow to Gallons per Minute (gpm)

First, convert the maximum daily flow from MGD (Million Gallons per Day) to gpm (gallons per minute). Utilize the conversion factor: 1 MGD = 1,000,000 gallons/day.18.65 MGD = 18,650,000 gallons/day.There are 1440 minutes in a day (24 hours * 60 minutes).Thus, \[ Q = \frac{18,650,000 \text{ gallons}}{1440 \text{ minutes}} \approx 12,951.39 \text{ gpm} \]

Step 2 - Determine Flow per Filter

The total flow is equally distributed among 6 filters. Therefore, the flow per filter is:\[ Q_{filter} = \frac{12,951.39 \text{ gpm}}{6} \approx 2158.57 \text{ gpm per filter} \]

Step 3 - Calculate Area of Each Filter

Calculate the area of each filter. Each filter measures 20 ft by 22 ft.\[ A_{filter} = 20 \text{ ft} \times 22 \text{ ft} = 440 \text{ ft}^{2} \]

Step 4 - Calculate Filtration Rate

The filtration rate is given by the flow per filter divided by the area of the filter. Therefore,\[ \text{Filtration Rate} = \frac{Q_{filter}}{A_{filter}} = \frac{2158.57 \text{ gpm}}{440 \text{ ft}^{2}} \approx 4.91 \text{ gpm/ft}^{2} \]

Key Concepts

Understanding Filtration Rate Calculations

The filtration rate is an essential concept in water treatment. It helps determine the efficiency of a filter system. To find the filtration rate, we must understand the flow of water through the filters and the area of each filter. By dividing the flow rate (in gallons per minute) by the area (in square feet), we get the filtration rate, measured in gpm/ft². This measure tells us how much water passes through each square foot of the filter surface per minute. This is critical for designing and operating water treatment plants efficiently.

Gallons Per Minute Conversion

In many water treatment problems, we need to convert between different units of flow. For example, here we converted Million Gallons per Day (MGD) to Gallons per Minute (gpm). The conversion is straightforward: 1 million gallons per day is equal to 1,000,000 gallons per day. Since there are 1,440 minutes in a day, we can use the conversion factor to find the flow rate in gpm. Using the formula:

\[ Q = \frac{18{,}650{,}000 \text{ gallons}}{1440 \text{ minutes}} \approx 12{,}951.39 \text{ gpm} \] We find that the flow rate is approximately 12,951.39 gallons per minute. This step is crucial because it allows us to use a consistent unit of measurement throughout our calculations.

Calculating the Area of Each Filter

To determine the filtration rate, we need to know the area of each filter. In this problem, each filter measures 20 feet by 22 feet. The area is found by multiplying these dimensions:

\[ A_{filter} = 20 \text{ ft} \times 22 \text{ ft} = 440 \text{ ft}^2 \] By knowing the area, we can then find out how much water each square foot of the filter is treating per minute. For each filter, this helps us evenly distribute the water flow and calculate the filtration rate accurately. Accurate area calculations ensure that each filter operates within its designed capacity, maintaining the overall efficiency of the water treatment process.