Problem 56 - Basic Technical Mathematics with Calculus

Question

Two pipes, each with a 25.0 -mm-diameter hole, lead into a single larger pipe (see Fig. 2.96). In order to ensure proper flow, the cross-sectional area of the hole of the larger pipe is designed to be equal to the sum of the cross- sectional areas of the two smaller pipes. Find the inside diameter of the larger pipe.

Step-by-Step Solution

Answer

The inside diameter of the larger pipe is approximately 35.4 mm.

Step 1: Find the Cross-sectional Area of One Smaller Pipe

The diameter of each smaller pipe is given as 25.0 mm. The radius will be half of that, which is 12.5 mm. The cross-sectional area (A) of a circle is given by the formula:\[ A = \pi r^2 \]Substituting the radius:\[ A = \pi \times (12.5)^2 \]

Step 2: Calculate the Total Cross-sectional Area of Two Smaller Pipes

Since there are two smaller pipes, the total cross-sectional area will be twice the area of one pipe:\[ A_{\text{total}} = 2 \times \pi \times (12.5)^2 \]

Step 3: Find the Required Cross-sectional Area of the Larger Pipe

According to the problem, the cross-sectional area of the larger pipe should be equal to the total cross-sectional area of the two smaller pipes:\[ A_{\text{larger}} = 2 \times \pi \times (12.5)^2 \]

Step 4: Solve for the Diameter of the Larger Pipe

Let's set the area of the larger pipe to equal the total area derived:\[ \pi \left(\frac{d}{2}\right)^2 = 2 \times \pi \times (12.5)^2 \]Divide both sides by \( \pi \) to simplify:\[ \left(\frac{d}{2}\right)^2 = 2 \times (12.5)^2 \]Take the square root of both sides:\[ \frac{d}{2} = \sqrt{2} \times 12.5 \]So,\[ d = 2 \times \sqrt{2} \times 12.5 \]

Step 5: Calculate and Provide the Final Diameter

Compute \( 2 \times \sqrt{2} \times 12.5 \):\[ \sqrt{2} \approx 1.414 \]\[ d = 2 \times 1.414 \times 12.5 \approx 35.355 \] mm.

Key Concepts

Pipe Diameter Calculation

Understanding how to calculate the diameter of a pipe is essential in ensuring fluid flow efficiency. Here, we look into determining the diameter of a larger pipe when smaller ones converge into it.
To determine this, we must first equate the cross-sectional areas of the smaller pipes to that of the larger one. This involves understanding basic circle geometry and the relationship between radius and diameter.

• The diameter of the larger pipe needs to be derived in such a way that its cross-sectional area is equivalent to two smaller pipes having a combined diameter.
• This means the equation is set up around balancing the areas, leading us to directly solve for the diameter of the larger pipe.

This calculation ensures that the flow characteristics, such as velocity and pressure, are maintained across transitions between different pipe sizes.

Cross-Sectional Area

The cross-sectional area of a pipe is essentially the area of the circular opening of the pipe. This is crucial to determine fluid dynamics as flow rate, pressure, and velocity are all functions of the cross-sectional area.

• For a circular pipe, the cross-sectional area (\(A\)) is calculated using the formula: \(A = \pi r^2\), where \(r\) is the radius of the circle.
• In the context of the exercise, each smaller pipe with a diameter of 25.0 mm has a radius of 12.5 mm, allowing us to compute its individual area.

Knowing this allows us to not only determine how flow will behave but also ensures that when different pipe sizes are involved, the transition doesn’t affect the overall efficiency of the system.

Circle Geometry

Circle geometry plays a crucial role in understanding and performing calculations related to pipes, as the cross-sections are circular.

• Key parameters include the radius and diameter, with the radius being half the diameter. This foundational knowledge is vital because the radius is used in calculating the area.
• The formula \(A = \pi r^2\) demonstrates the direct relationship between radius and area. When you understand this, you can solve for any related dimension when provided with certain variables, such as area or circumference.

These concepts are not only theoretical but are applied extensively in fields such as fluid mechanics, where understanding the behavior of fluids through channels with circular cross-sections is often needed.