Question

In order to properly back wash a certain filter a back wash rate of 20 inches per minute rise is needed. If the filter is \(20 \mathrm{ft}\) by \(25 \mathrm{ft}\), what is the backwash flow rate in gpm?

Short answer

The backwash flow rate is 6245.8 gpm.

Step-by-step solution

- Convert Rise Rate to Feet per Minute

First, convert the backwash rise rate from inches per minute to feet per minute. Given the rate is 20 inches per minute, we convert it by dividing by 12 (since there are 12 inches in a foot). So, 20 inches per minute / 12 = 1.67 feet per minute.

- Calculate Filter Area

Calculate the area of the filter. The filter dimensions are 20 feet by 25 feet. The area (A) is given by: A = length × widthA = 20 ft × 25 ft = 500 square feet.

- Calculate Volume of Backwash per Minute

Next, calculate the volume of water raised per minute. Volume (V) is given by the product of the rise rate (converted to feet per minute) and the area of the filter. Using the rise rate obtained in Step 1, V = rise rate × areaV = 1.67 feet/min × 500 sq ft = 835 cubic feet per minute.

- Convert Volume to Gallons per Minute

Convert the volume from cubic feet per minute to gallons per minute (gpm). Since 1 cubic foot of water is equal to approximately 7.48 gallons, use the conversion factor: 835 cubic feet per minute × 7.48 gallons/cubic foot = 6245.8 gpm.

Key concepts

Units Conversion

Units conversion is a crucial part of many mathematical and scientific problems. In this exercise, we need to convert different units to ensure we’re using consistent measurements.
For instance, the rise rate was given in inches per minute, but we needed it in feet per minute for subsequent calculations. To convert inches to feet, remember that 1 foot equals 12 inches.
So, if you have a rise rate of 20 inches per minute, dividing by 12 gives you the equivalent in feet per minute: \ \[ \text{Rise rate (feet/min)} = \frac{\text{Rise rate (in/min)}}{12} \ = \frac{20}{12} = 1.67 \text{ feet/min} \]
This converted unit is then used in the following steps for accurate area and volume calculations.

Area Calculation

Calculating the area is one of the fundamental skills required in geometry and real-life applications. For a rectangle, the area is found by multiplying its length by its width.
In this problem, our filter’s dimensions are given as 20 feet by 25 feet. So, to find the area, we use: \ \[ \text{Area (sq ft)} = \text{Length (ft)} \times \text{Width (ft)} \ = 20 \text{ ft} \times 25 \text{ ft} = 500 \text{ sq ft} \]
This calculated area will help us determine the volume of water raised during the backwash process in the next steps.

Volume Calculation

Volume calculation is essential to estimate the space occupied by an object or a fluid. In our case, we need to figure out the volume of water raised in the filter per minute.
Volume is calculated using the product of the area and the height (or rise rate). Hence, using the rise rate in feet per minute and the area obtained earlier: \ \[ \text{Volume (cubic ft/min)} = \text{Rise rate (ft/min)} \times \text{Area (sq ft)} \ = 1.67 \text{ ft/min} \times 500 \text{ sq ft} = 835 \text{ cubic ft/min} \]
This gives us the amount of water that the filter raises per minute in cubic feet.

Flow Rate Conversion

Flow rate conversion helps in understanding how much fluid passes through a point in a given time. We usually use gallons per minute (gpm) in many fluid flow applications instead of cubic feet per minute.
To convert cubic feet per minute to gallons per minute, we use the conversion factor where 1 cubic foot equals approximately 7.48 gallons. So, using the volume in cubic feet per minute obtained earlier: \ \[ \text{Flow rate (gpm)} = \text{Volume (cubic ft/min)} \times 7.48 \text{ gpm/cubic ft} \ = 835 \text{ cubic ft/min} \times 7.48 = 6245.8 \text{ gpm} \]
This gives us the final backwash flow rate in the desired units of gallons per minute.