Tank Filling Problem: A Comprehensive Analysis

Understanding the process of filling a tank with a specified flow rate and additional valve can be a complex yet fascinating problem to solve. In this article, we will analyze the scenario where a tank is filled by a single inlet valve with a flow rate of 12 liters per second (Lps). We will then extend this scenario to include a second inlet valve, which is energized 30 minutes after the first one. Our goal is to determine the total time required to fill the tank under these conditions.

Initial Scenario: Filling with a Single Valve

To begin, let's consider the situation where a tank is filled by a single inlet valve with a flow rate of 12 liters per second. To find the total volume of the tank, we first convert the flow rate to liters per minute (Lpm). Given that 12 Lps is equivalent to 720 Lpm, we can calculate the tank volume by multiplying the flow rate by the total active time.

The total time to fill the tank with a single inlet valve is specified as 72 minutes. Therefore, the total volume of the tank can be calculated as:

Total Tank Volume Flow Rate × Total Time

Total Tank Volume 720 Lpm × 72 mins 51840 liters

Extending the Scenario with a Second Valve

In the second part of the problem, we need to determine how long it takes to fill the tank if a second inlet valve is opened 30 minutes after the first one. For this part, let's break down the problem step by step.

Step 1: Determine the Volume Filled by the First Valve

Let's first consider the time period during which only the first valve is active. This is for the initial 30 minutes of the filling process.

V1 Flow Rate × Time1

V1 720 Lpm × 30 mins 21600 liters

Step 2: Determine the Remaining Volume

The total volume of the tank is 51840 liters. After the first 30 minutes, the remaining volume to be filled is:

Remaining Volume Total Volume - Volume Filled by First Valve

Remaining Volume 51840 - 21600 30240 liters

Step 3: Calculate the Time for the Remaining Volume

When the second valve is opened, the combined flow rate of both valves is 1440 liters per minute (720 Lpm 720 Lpm). To find the time required to fill the remaining volume, we use the formula:

T2 Remaining Volume / Combined Flow Rate

T2 30240 / 1440 21 minutes

Step 4: Determine the Total Filling Time

The total time to fill the tank is the sum of the time the first valve is active and the time for the second valve to finish the process:

Total Time Time with First Valve Time with Both Valves

Total Time 30 mins 21 mins 51 minutes

Conclusion

Based on the analysis, the total time required to fill the tank is 51 minutes. This is consistent across all calculations and confirms that the flow rate is not a limiting factor as long as the conditions of the problem are adhered to.

It is important to note that this solution method is applicable regardless of the specific flow rate, as long as the time to fill the tank with a single valve is known. The key factors are the initial volume filled by the single valve and the combined flow rate when both valves are active.