Calculating Discharge from a Circular Pipe

The question asks us to determine the discharge from a circular pipe given its diameter and the velocity of the flow within it. Discharge, often denoted by 'Q', is the volume of fluid passing through a cross-sectional area per unit time. It is a fundamental concept in fluid mechanics.

Understanding the Discharge Formula

The discharge (Q) is calculated using the simple formula:

\(Q = \text{Area} \times \text{Velocity}\)

Where:

• \(Q\) is the discharge (typically in m\({}^3\)/s).
• Area (\(A\)) is the cross-sectional area of the flow (typically in m\({}^2\)).
• Velocity (\(V\)) is the average velocity of the fluid flow (typically in m/s).

Calculating the Cross-sectional Area of the Pipe

The pipe is circular. The cross-sectional area (\(A\)) of a circle is given by the formula:

\(A = \pi r^2\)

Where \(r\) is the radius of the circle. We are given the diameter (\(D\)), and the radius is half the diameter:

\(r = \frac{D}{2}\)

So, the area can also be calculated as:

\(A = \pi \left(\frac{D}{2}\right)^2 = \frac{\pi D^2}{4}\)

Applying the Given Values

We are given:

• Diameter of the pipe (\(D\)) = 0.5 m
• Velocity of flow (\(V\)) = 10 m/s

First, let's calculate the radius:

\(r = \frac{0.5 \text{ m}}{2} = 0.25 \text{ m}\)

Next, let's calculate the cross-sectional area of the pipe:

\(A = \pi r^2 = \pi (0.25 \text{ m})^2\)

\(A = \pi \times 0.0625 \text{ m}^2\)

Using the value of \(\pi \approx 3.14159\):

\(A \approx 3.14159 \times 0.0625 \text{ m}^2 \approx 0.19635 \text{ m}^2\)

Now, we can calculate the discharge using the formula \(Q = A \times V\):

\(Q = 0.19635 \text{ m}^2 \times 10 \text{ m/s}\)

\(Q = 1.9635 \text{ m}^3\text{/s}\)

Comparing with Options

The calculated discharge is approximately 1.9635 m\({}^3\)/s. Let's look at the given options:

• 1.5 m\({}^3\)/s
• 1.96 m\({}^3\)/s
• 2.76 m\({}^3\)/s
• 3 m\({}^3\)/s

The calculated value \(1.9635 \text{ m}^3\text{/s}\) is closest to the option \(1.96 \text{ m}^3\text{/s}\).

Therefore, the discharge from the circular pipe is approximately 1.96 m\({}^3\)/s.

Revision Table: Key Concepts in Discharge Calculation

Additional Information: Understanding Discharge and Flow Rate

Discharge is also known as volume flow rate. It represents how much volume of fluid moves through a system over a specific period. The units of discharge are typically volume per unit time, such as m\({}^3\)/s, L/s, or ft\({}^3\)/s (cfs). Knowing the discharge is crucial in many engineering applications, including designing pipe networks, managing water resources, and analyzing fluid transport systems.

The continuity equation in fluid dynamics is based on the principle of mass conservation and often relates discharge in different sections of a pipe or channel:

\(Q_1 = Q_2\)

or

\(A_1 V_1 = A_2 V_2\)

This means that for an incompressible fluid in a pipe with varying cross-sectional area, the product of the area and velocity remains constant, implying that if the area decreases, the velocity must increase, and vice versa, to maintain the same flow rate.