Understanding the Time Required to Empty a Cistern: A Practical Example

When dealing with cisterns and outlet pipes, a common question that arises is: if an outlet pipe can empty a cistern in 8 hours, how long will it take to empty two-thirds of the cistern? At first glance, it may seem like the answer is straightforward, but the actual time required can vary based on several factors. This article provides a detailed analysis of the situation and explores the underlying principles of fluid dynamics.

Initial Misconception and Real-Life Differences

The initial instinct is often to assume that if an outlet pipe can empty a cistern in 8 hours, then it would take ( frac{2}{3} ) of 8 hours to empty ( frac{2}{3} ) of the cistern. This calculation, ( frac{2}{3} times 8 ) hours, results in approximately 5.33 hours. However, in reality, this assumption can be quite misleading due to the varying fluid dynamics throughout the cistern.

Consider a typical cistern, which is often vertical. The pressure at the outlet is highest when the water level is highest. This means that the water at the top third of the cistern will drain more quickly than the water at the middle and bottom thirds. This is because the fluid friction in the outlet pipe is higher for the lower levels due to the increase in water volume and pressure.

The Correct Approach: A Thorough Analysis

To accurately determine the time required to empty ( frac{2}{3} ) of a cistern, consider the following steps:

Define the Total Capacity: Let the total capacity of the cistern be ( 1 ) unit. Outlet Pipe Performance: If the outlet pipe can empty the cistern in 8 hours, it empties ( frac{1}{8} ) of the cistern in one hour. Time to Empty ( frac{2}{3} ) of the Cistern: To determine how long it takes to empty ( frac{2}{3} ) of the cistern, use the initial assumption that the time is directly proportional to the volume to be emptied. Dividing ( frac{2}{3} ) by ( frac{1}{8} ) gives ( frac{2}{3} times 8 5.33 ) hours. However, this is an overestimation due to varying fluid dynamics.

Given the complexities of fluid dynamics and the varying pressure levels, a more accurate approach involves breaking down the cistern into thirds and calculating the time for each third.

Breaking Down the Cistern into Thirds

Say the cistern is divided into three equal parts (top, middle, and bottom). The top third will drain out first, followed by the middle third, and then the bottom third. Assuming uniform flow rate and neglecting friction, each third would take the same time to drain. Therefore, the top and middle thirds each take 3 hours (as the cistern empties in 8 hours for the whole). This leaves the bottom third to be drained.

The bottom third takes 2 more hours to drain, as the remaining ( frac{2}{3} ) of the cistern is drained over the remaining 2 hours. Therefore, ( frac{2}{3} ) of the cistern is emptied in 2 hours, not 5.33 hours.

Conclusion

Understanding the dynamics of cistern emptying is crucial in various fields, including plumbing, engineering, and environmental science. The correct time to empty ( frac{2}{3} ) of a cistern is 2 hours, not 5.33 hours based on initial assumptions. This is due to the varying fluid pressure and friction within the cistern. The analysis demonstrates the importance of considering real-world factors when solving practical problems.

Related Keywords

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Conclusion

The time required to empty a cistern, especially when it comes to understanding the specifics of fluid dynamics, is a complex yet fascinating topic. Understanding these principles can help in various applications, from household plumbing to industrial processes. By breaking down the cistern into manageable parts and considering the real-world dynamics, we can arrive at accurate and practical solutions.