Problem 9 - Applied Physics

Question

A liquid flows through a pipe with a diameter of \(0.50 \mathrm{~m}\) at a speed of \(4.20 \mathrm{~m} / \mathrm{s}\). What is the rate of flow in \(\mathrm{L} / \mathrm{min} ?\)

Step-by-Step Solution

Answer

The rate of flow is approximately 49,482 L/min.

Step 1: Calculate the Cross-Sectional Area of the Pipe

The cross-sectional area of the pipe is necessary to find the volume flow rate. The formula for the area of a circle, which describes the cross-section of the pipe, is \( A = \pi r^2 \), where \( r \) is the radius of the pipe. Given the diameter is 0.50 m, the radius \( r = 0.25 \) m. Using this, \( A = \pi \times (0.25)^2 \approx 0.19635 \mathrm{~m}^2 \).

Step 2: Calculate the Volume Flow Rate in Cubic Meters per Second

The volume flow rate can be determined using the formula \( Q = A \times v \), where \( A \) is the cross-sectional area and \( v \) is the velocity of the fluid. Here, \( v = 4.20 \mathrm{~m/s} \). Substituting the values, \( Q = 0.19635 \times 4.20 \approx 0.8247 \mathrm{~m}^3/ ext{s} \).

Step 3: Convert Cubic Meters per Second to Liters per Second

Since 1 cubic meter is equivalent to 1000 liters, the flow rate in liters per second is \( 0.8247 \mathrm{~m}^3/ ext{s} \times 1000 \approx 824.7 \text{ L/s} \).

Step 4: Convert Liters per Second to Liters per Minute

To convert from liters per second to liters per minute, multiply by 60 (the number of seconds in a minute). Thus, the rate of flow is \( 824.7 \mathrm{~L/s} \times 60 \approx 49482 \mathrm{~L/min} \).

Key Concepts

Understanding Volume Flow Rate

In fluid dynamics, the volume flow rate represents the amount of volume of fluid passing through a cross-section of a pipe or channel per unit of time. It's a crucial concept when working with liquids flowing in pipes. Volume flow rate is typically expressed by the symbol \( Q \) and can be calculated as \( Q = A \times v \), where \( A \) is the cross-sectional area and \( v \) is the velocity of the fluid. This calculation gives us the volume flow rate in cubic meters per second \( \mathrm{m}^3/\mathrm{s} \) when \( A \) is in square meters and \( v \) is in meters per second. It's essential to ensure that units are consistent when performing these calculations. Understanding the volume flow rate helps in many real-world applications, such as determining how fast a tank can fill or empty, designing irrigation channels, and managing fluid systems efficiently.

Calculating Cross-Sectional Area

The cross-sectional area of a pipe is fundamental for calculating the volume flow rate. For circular pipes, which are most common, the cross-sectional area can be found using the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle. In our example, the diameter of the pipe is given as \( 0.50 \mathrm{~m} \). To find the radius, simply halve the diameter: \( r = \frac{0.50}{2} = 0.25 \mathrm{~m} \). Plug this value into the area formula to find the area in square meters. In this case, \( A = \pi \times (0.25)^2 \approx 0.19635 \mathrm{~m}^2 \). The calculated area is then used to find the volume flow rate, linking the physical shape of the pipe to the fluid dynamics within it. Knowing the cross-sectional area is important for designing and evaluating systems that involve fluid movement.

Mastering Conversion of Units

Conversions are an indispensable skill in fluid dynamics, as measurements need to be consistent for accurate calculations. In our exercise, we started with the volume flow rate in cubic meters per second and needed to convert it to liters per minute to match common usage standards.

• Firstly, convert cubic meters to liters. Since \(1\, \mathrm{m}^3 = 1000\, \mathrm{L}\), multiply the flow rate by 1000. For example, \(0.8247\, \mathrm{m}^3/\mathrm{s}\) becomes \(824.7\, \mathrm{L/s}\).
• Next, convert liters per second to liters per minute by multiplying by 60, since there are 60 seconds in a minute. Therefore, \(824.7\, \mathrm{L/s} \times 60 = 49482\, \mathrm{L/min}\).

Converting units accurately ensures that the data can be compared and integrated effectively within systems, preventing costly errors and misinterpretations.