Problem 54 - Fundamentals of Physics

Question

The water flowing through a \(1.9 \mathrm{~cm}\) (inside diameter) pipe flows out through three \(1.3 \mathrm{~cm}\) pipes. (a) If the flow rates in the three smaller pipes are \(26,19,\) and 11 L. \(/\) min, what is the flow rate in the \(1.9 \mathrm{~cm}\) pipe? (b) What is the ratio of the speed in the \(1.9 \mathrm{~cm}\) pipe to that in the pipe carrying \(26 \mathrm{~L} / \mathrm{min} ?\)

Step-by-Step Solution

Answer

(a) The flow rate in the 1.9 cm pipe is 56 L/min. (b) The speed ratio is approximately 1.0086.

Step 1: Understand the Flow Rate Relationship

According to the principle of conservation of mass, the total flow rate in the larger pipe must equal the sum of the flow rates in the smaller pipes. This principle means that the flow rate through the large pipe should be the total flow rate of the three smaller pipes added together.

Step 2: Calculate the Total Flow Rate of Smaller Pipes

Add the flow rates of the three smaller pipes:\[26 + 19 + 11 = 56 \text{ L/min}\]

Step 3: Determine the Flow Rate in the Larger Pipe

Since the flow rate in the larger pipe must match the total flow rate of the smaller pipes, the flow rate in the larger pipe is:\[56 \text{ L/min}\]

Step 4: Calculate the Cross-sectional Area of Both Pipes

To find the flow speed ratio, we need the cross-sectional areas. The formula for the area of a circle is \(A = \pi (d/2)^2\).1. For the 1.9 cm pipe: \[ A_{large} = \pi (1.9/2)^2 = \pi (0.95)^2 \approx 2.835 \text{ cm}^2 \]2. For the 1.3 cm pipe: \[ A_{small} = \pi (1.3/2)^2 = \pi (0.65)^2 \approx 1.327 \text{ cm}^2 \]

Step 5: Calculate the Flow Speed for Both Pipes

Use the flow speed equation \(v = Q/A\):1. For the 1.9 cm pipe: \[ v_{large} = \frac{56 \times 1000}{2.835} \, \text{\frac{cm}{min}} \] Simplifying: \[ v_{large} \approx 19761.9 \, \text{cm/min} \]2. For the 1.3 cm pipe carrying 26 L/min: \[ v_{small} = \frac{26 \times 1000}{1.327} \, \text{\frac{cm}{min}} \] Simplifying: \[ v_{small} \approx 19592.8 \, \text{cm/min} \]

Step 6: Calculate the Speed Ratio

To find the ratio of the speed in the larger pipe to the pipe with 26 L/min flow rate:\[\text{Speed Ratio} = \frac{v_{large}}{v_{small}} = \frac{19761.9}{19592.8} \approx 1.0086\]

Key Concepts

Flow Rate Calculation

Flow rate calculation is a fundamental concept in fluid dynamics, often needed in engineering applications. It tells us how much fluid passes through a section of a pipe in a given time. This is usually expressed in liters per minute (L/min) or cubic meters per second (m³/s).

For the exercise at hand, we have a larger pipe branching into three smaller ones. To find the flow rate in the larger pipe, add up the flow rates from the three smaller pipes. This means if the smaller pipes have flow rates of 26, 19, and 11 L/min, you simply add these values to get the total flow rate in the larger pipe:

• 26 + 19 + 11 = 56 L/min

This shows that the larger pipe has a flow rate of 56 L/min.

Understanding how to sum these values is key to predicting how much fluid is in motion, ensuring systems function efficiently.

Conservation of Mass

Conservation of mass is a critical principle in fluid dynamics. It states that mass cannot be created or destroyed in an isolated system. Therefore, the mass of a fluid entering a pipe must equal the mass leaving it, assuming there is no leakage.

In pipe systems, this principle translates to the flow rate being consistent throughout. This means the total flow rate entering a series of branches from a larger pipe must equal the total flow rate exiting those branches.

In our example, the total output from the three smaller pipes must equal the input to the larger pipe. By understanding the conservation of mass, we ensure that the flow rate calculations are correct and all sections of the piping system are accounted for properly. This principle is essential for designing efficient piping networks.

Cross-Sectional Area

The cross-sectional area of a pipe is crucial when studying fluid flow. It's defined as the area of a slice through the pipe perpendicular to its length. For a circular pipe, you calculate it using the formula:

• \( A = \pi (d/2)^2 \)

where \(d\) is the diameter of the pipe.

By knowing the diameter, you can determine how much fluid the pipe can carry at any given time. In our exercise, the larger pipe has a diameter of 1.9 cm, giving it a calculated area of approximately 2.835 cm². Meanwhile, a smaller pipe of 1.3 cm diameter has an area of about 1.327 cm².

Knowing these areas allows us to relate flow rates to the speed of flow inside the pipe, using the relationship between flow speed and cross-sectional area.

Pipe Flow Speed

Flow speed within a pipe is determined by how quickly the fluid moves. This can be calculated using the flow rate and the cross-sectional area of the pipe. The relationship is given by:

• \( v = Q/A \)

where \(v\) is the flow speed, \(Q\) is the flow rate, and \(A\) is the cross-sectional area.

For example, in the exercise, the larger pipe has a flow rate of 56 L/min and a cross-sectional area of 2.835 cm². Using these values, the speed of flow can be calculated as approximately 19761.9 cm/min.

Understanding pipe flow speed is vital, as it helps in designing efficient systems by predicting how quickly transport through pipes can occur, affecting pressure and system stability. Calculating the speed ratio, like in the example where the speed in the large pipe is compared to one of the smaller pipes, informs decisions about pipe sizing and flow efficiency within a network.