A liquid flows through a horizontal pipe whose inner radius is 5.35cm. The pipe bends upward through a height of through a height of 9.5m, where it widens and joins another horizontal pipe of inner radius of 6.8cm. What must the volume flux be, if the pressure in the two horizontal pipes is same?

To determine the volume flux of the liquid flowing through the pipes, we can utilize the principle of conservation of mass, specifically the continuity equation, along with some basic fluid dynamics concepts. Since the pressure in both horizontal sections of the pipe is the same, we can infer that the flow rate must remain constant throughout the system, despite the changes in pipe diameter and elevation.

Understanding the Problem

We have two sections of a pipe: the first section has an inner radius of 5.35 cm, and the second section has an inner radius of 6.8 cm. The liquid flows from the first section, bends upward by 9.5 m, and then enters the second section. Since the pressure is the same in both horizontal pipes, we can focus on the relationship between the cross-sectional areas and the velocities of the liquid in each section.

Applying the Continuity Equation

The continuity equation states that the mass flow rate must remain constant in a closed system. This can be expressed mathematically as:

• A₁v₁ = A₂v₂

Where:

• A₁ = cross-sectional area of the first pipe
• A₂ = cross-sectional area of the second pipe
• v₁ = velocity of the liquid in the first pipe
• v₂ = velocity of the liquid in the second pipe

Calculating Cross-Sectional Areas

First, we need to calculate the cross-sectional areas of both pipes using the formula for the area of a circle, A = πr².

• For the first pipe (r₁ = 5.35 cm = 0.0535 m):
• A₁ = π(0.0535 m)² ≈ 0.00899 m²
• For the second pipe (r₂ = 6.8 cm = 0.068 m):
• A₂ = π(0.068 m)² ≈ 0.01454 m²

Relating Velocities

Since the pressure is the same in both horizontal sections, we can set up the equation:

• 0.00899 m² * v₁ = 0.01454 m² * v₂

From this, we can express v₁ in terms of v₂:

• v₁ = (0.01454 m² / 0.00899 m²) * v₂ ≈ 1.617 * v₂

Volume Flux Calculation

The volume flux (Q) can be defined as:

• Q = A * v

For the first pipe, we have:

• Q₁ = A₁ * v₁ = 0.00899 m² * v₁

Substituting v₁:

• Q₁ = 0.00899 m² * (1.617 * v₂) = 0.01454 m² * v₂ = Q₂

This shows that the volume flux is constant throughout both sections of the pipe. To find the actual volume flux, we need to know either the velocity in one of the sections or the flow rate. If we assume a certain velocity in the second pipe, we can calculate the volume flux accordingly.

Final Thoughts

In summary, the volume flux remains constant between the two sections of the pipe due to the conservation of mass. The relationship between the diameters and velocities allows us to understand how the liquid behaves as it flows through different pipe sections. If you have a specific velocity or flow rate in mind, we can plug that into our equations to find the exact volume flux.