Understanding the Rate of Emptying for an Outlet Pipe

Have you ever wondered how an outlet pipe can drain a cistern in a specific amount of time, and what fraction of the cistern it can drain in just one minute? This question is a common problem in fluid dynamics and can be analyzed through basic mathematical models and physical principles. Let's explore how to approach and solve such problems.

Problem Statement

Given: An outlet pipe can empty a cistern in 15 minutes. What part of the cistern will it empty in one minute?

Basic Assumptions

Let V denote the total capacity of the cistern. Let F denote the fraction of the cistern that will be empty in one minute.

Calculating the Fraction Emptied in One Minute

Since the cistern is emptied in 15 minutes, the fraction of the cistern emptied in one minute is given by:

F V/15

Expressing this in numerical form:

F 100/15 6.67/100V

Thus, the pipe empties 6.67% of the total capacity of the cistern in one minute.

Example Calculation with a Specific Capacity

Let's consider a specific example where the cistern's capacity is 15 liters.

Since the pipe can empty the entire cistern in 15 minutes:

In 15 minutes, 15 liters are gone. So, in 1 minute, 1 liter will be gone. Therefore, 1/15th of the cistern will be emptied in one minute.

This simple ratio logic shows that the fraction emptied is 1/15. This is consistent with the general calculation done earlier.

General Case for a Different Cistern Size

Now, let's consider a different scenario: an outlet pipe can empty a cistern in 30 minutes. What part of the cistern will it empty in one minute?

Rationale:

Seemingly, one might initially think that the fraction emptied in one minute is:

100/30 3.33%

However, this approach is incorrect because the pipe's rate of flow depends on the height and pressure of the liquid in the cistern. The pressure and flow rate decrease as the liquid level in the cistern decreases. Therefore, the correct fraction emptied in one minute must also consider the shape and initial conditions of the cistern.

Mathematically:

Q(t) Q_0 e^{-kt}

Where Q(t) is the flow rate at time t, Q_0 is the initial flow rate, and k is a constant that depends on the cistern's shape.

Given that the cistern is fully emptied in 30 minutes, we can solve for the fraction emptied in one minute, but this will require knowledge of the initial conditions and the shape of the cistern.

Conclusion

Understanding the rate of flow of an outlet pipe is crucial in fluid dynamics. While the simple ratio method is useful for some scenarios, it is essential to consider the physical conditions of the cistern for more accurate calculations. The problem of determining the fraction of a cistern emptied by an outlet pipe in a single minute involves basic mathematical principles and a thorough understanding of fluid dynamics.