Question

A water treatment plant is in the process of redesigning their sedimentation basin. The plant treats \(4.5 \mathrm{MGD}\) with an average detention time of \(1.85\) hours. Portable storage tanks will be used when the basin is under construction. The portable storage tanks are \(25 \mathrm{ft}\) tall and \(20 \mathrm{ft}\) in diameter. How many tanks will be needed?

Short answer

6 tanks are needed.

Step-by-step solution

Convert Flow Rate to Cubic Feet per Hour

First, convert the flow rate in million gallons per day (MGD) to cubic feet per hour (ft³/hr). There are 7.48 gallons in a cubic foot and 24 hours in a day. Thus, the flow rate in ft³/hr is: \ \( 4.5 \text{ MGD} \times \frac{10^6 \text{ gallons}}{1 \text{ MGD}} \times \frac{1 \text{ ft}³}{7.48 \text{ gallons}} \times \frac{1 \text{ day}}{24 \text{ hours}} \)}

Calculate the Number of Storage Tanks Required

Divide the total volume of water required by the volume of one storage tank to find the number of tanks needed: \( \frac{46379.92 \text{ ft}³}{7853.98 \text{ ft}³/\text{tank}} \approx 5.91 \text{ tanks} \). Since you cannot have a fraction of a tank, round up to the next whole number: 6 tanks.

Key concepts

Flow Rate Conversion

To solve problems related to water treatment, it's essential to convert the flow rate to a useful unit of measurement. In this exercise, we convert the flow rate from million gallons per day (MGD) to cubic feet per hour (ft³/hr). Understanding these conversions ensures accuracy in subsequent calculations.

You begin by noting that 1 million gallons is equivalent to 1,000,000 gallons and that there are 7.48 gallons in one cubic foot. Additionally, there are 24 hours in a day.

To convert from MGD to ft³/hr:

• Multiply the MGD value by 1,000,000 to get the number of gallons per day.
• Divide by 7.48 to convert the total gallons to cubic feet.
• Finally, divide by 24 to convert the flow from cubic feet per day to cubic feet per hour.

Using the exercise's information, the calculation would be:
\[ \text{4.5 MGD} \times \frac{10^6 \text{ gallons}}{1 \text{ MGD}} \times \frac{1 \text{ ft}³}{7.48 \text{ gallons}} \times \frac{1 \text{ day}}{24 \text{ hours}} \]
Therefore, the flow rate in cubic feet per hour (ft³/hr) is calculated to ensure consistent results in further calculations.

Detention Time

Detention time is the period water stays in a treatment process, allowing sufficient time for sediment to settle. In this problem, the detention time is given as 1.85 hours.

The formula to calculate the total water volume needed in the basin, considering the flow rate and the detention time, is:
\[ \text{Volume} = \text{Flow Rate} \times \text{Detention Time} \]

Using the flow rate calculated earlier, let's plug in the values:

• Flow rate is the value we got from converting 4.5 MGD to ft³/hr.
• Detention time is given as 1.85 hours.

Hence, the total volume of water required in cubic feet is calculated as:
\[ 20610 \times 1.85 = 38128.5 \text{ ft}³ \]
Understanding detention time ensures that the system provides adequate treatment by allowing particles to settle out effectively.

Volume Calculation

Volume calculation is crucial to determine how much space is needed to store or treat the water. In our scenario, the storage tanks have specific dimensions that are used to calculate their volume.

The formula for the volume of a cylindrical tank is:
\[ \text{Volume} = \text{π} \times \text{radius}^2 \times \text{height} \]

Given the diameter of the tank is 20 ft, the radius is half of that: 10 ft. The height of the tank is given as 25 ft.

• Calculate the area of the base using π times the radius squared.
• Multiply the area by the height to find the volume.

So, the volume of one storage tank is:
\[ \text{π} \times (10 \text{ ft})^2 \times 25 \text{ ft} = 7853.98 \text{ ft}³ \]
This volume is essential for determining the number of tanks needed to hold a certain amount of water.

Storage Tank Capacity

Calculating storage tank capacity is the final step to ensure that the total water volume can be adequately stored. With the volume of one tank known, we can determine the total number of tanks needed.

First, calculate the total volume of water required (as determined in the detention time section) and divide by the volume of a single tank.

• Total water volume required is calculated as 38128.5 ft³.
• Volume of one tank is 7853.98 ft³.

Then, the number of tanks required is:
\[ \frac{38128.5 \text{ ft}³}{7853.98 \text{ ft}³/\text{tank}} \]

This results in approximately 4.85 tanks. Since we can't have a fraction of a tank, we round up to the nearest whole number:
\[ \text{5 tanks} \]
Calculating the tank capacity ensures you always have adequate storage without risking overflow.