Question

A \(15 \mathrm{ft}\) by \(17 \mathrm{ft}\) filter needs to be back washed at a rate of \(17 \mathrm{gpm} / \mathrm{ft}^{2}\) for a minimum of 20 minutes. How many gallons are used during the backwashing process?

Short answer

86700 gallons

Step-by-step solution

- Calculate the Area of the Filter

To find the area of the filter, multiply the length by the width. The dimensions given are 15 feet and 17 feet. So, the area is: \[ 15 \, \text{ft} \times 17 \, \text{ft} = 255 \, \text{ft}^2 \]

- Calculate the Total Backwash Flow Rate

The flow rate is given as 17 gallons per minute per square foot. Multiply this by the area calculated in Step 1 to find the total flow rate: \[ 255 \, \text{ft}^2 \times 17 \, \text{gpm/ft}^2 = 4335 \, \text{gpm} \]

- Calculate the Total Gallons Used

To find the total number of gallons used, multiply the flow rate by the duration of the backwash (in minutes). The duration is given as 20 minutes. So, \[ 4335 \, \text{gpm} \times 20 \, \text{min} = 86700 \, \text{gal} \]

Key concepts

Area Calculation

When you're dealing with filters, understanding area calculation is crucial. To find the area of a rectangular filter, you simply multiply its length by its width. For instance, with a filter of 15 feet in length and 17 feet in width, you would calculate the area as follows:
\[ 15 \text{ ft} \times 17 \text{ ft} = 255 \text{ ft}^2 \]
In this example, the filter has an area of 255 square feet. This measurement is essential, as it will be used in further calculations like determining the total flow rate required.

Flow Rate

Flow rate is a key concept in fluid dynamics and is typically expressed in units such as gallons per minute (gpm). It tells you how much liquid flows through a given area in a specific time. In this exercise, you're given a flow rate of 17 gallons per minute per square foot. To find the total flow rate, you multiply this rate by the filter's area:
\[ 255 \text{ ft}^2 \times 17 \text{ gpm/ft}^2 = 4335 \text{ gpm} \]
This means that the filter needs to handle 4335 gallons of water per minute during the backwashing process.

Volume Calculation

Understanding how to calculate volume is indispensable when dealing with tasks like backwashing filters. In our example, after calculating the total flow rate, you'll want to find out the total volume of water used in the backwashing process. First, recall that flow rate is 4335 gallons per minute. Then multiply this by the duration of the process in minutes. If the backwash lasts 20 minutes:
\[ 4335 \text{ gpm} \times 20 \text{ min} = 86700 \text{ gal} \]
Therefore, the total volume of water used in this backwashing process is 86,700 gallons.

Time Duration

The duration of the backwashing process is another important factor. Time duration is the total time that the process requires to complete. In this exercise, it's given as 20 minutes. To find out how much water is used during this time, you multiply the flow rate by the time duration. Without knowing the duration, you couldn't accurately calculate volume or total flow.

Unit Conversion

Unit conversion is critical in ensuring all your measurements align, making calculations coherent and accurate. In this problem, you’ll mainly deal with minutes and gallons per minute. Ensure the units of your measurements match:
- Flow rate: gallons per minute (gpm)
- Time duration: minutes (min)
After verifying units are consistent, you can perform the necessary calculations without any conversion issues. Proper unit conversion ensures your results are in the correct units and easy to interpret.