To find the solids loading rate applied to the gravity thickener in pounds per square foot per day, we can break the problem down into the following steps:

Step 1: Convert the flow rate to gallons per day

We are given the flow rate of 80 gallons per minute. To convert this to gallons per day:

$80 \, \text{gallons per minute} \times 60 \, \text{minutes per hour} \times 24 \, \text{hours per day} = 115,200 \, \text{gallons per day}$

Step 2: Convert the concentration to a decimal

The concentration of solids in the sludge is given as 4%. To work with this, we need to convert it to decimal form:

$4\% = \frac{4}{100} = 0.04$

Step 3: Calculate the amount of solids in pounds

The amount of solids in the sludge is 4% of the total flow rate. First, we need to convert the gallons of sludge to pounds of sludge. The weight of water is approximately 8.34 pounds per gallon. Thus, the total pounds of sludge per day is:

$115,200 \, \text{gallons per day} \times 0.04 \, \text{(concentration of solids)} \times 8.34 \, \text{pounds per gallon} = 38,438.4 \, \text{pounds of solids per day}$

Step 4: Calculate the surface area of the thickener

The thickener has a diameter of 40 feet, so its radius is:

$\frac{40}{2} = 20 \, \text{feet}$

The surface area $A$ of the thickener is the area of a circle, which can be calculated using the formula:

$A = \pi r^2 = \pi (20)^2 = 1,256.64 \, \text{square feet}$

Step 5: Calculate the solids loading rate

Now, we can calculate the solids loading rate by dividing the total pounds of solids by the surface area of the thickener:

$\text{Solids loading rate} = \frac{38,438.4 \, \text{pounds per day}}{1,256.64 \, \text{square feet}} = 30.6 \, \text{pounds per square foot per day}$

Final Answer:

The solids loading rate applied to the gravity thickener is approximately 31 pounds per square foot per day (rounded to the nearest whole number).

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Would you like any additional clarification on the steps involved?

Here are 5 related questions to expand on this topic:

1. How does the initial sludge concentration affect the solids loading rate?
2. What would happen if the diameter of the thickener was reduced to 30 feet?
3. How would increasing the sludge flow rate affect the performance of the thickener?
4. What role does gravity play in the thickening process?
5. How can the efficiency of a gravity thickener be improved?

Tip: Always ensure that units are consistent throughout your calculations to avoid errors.