Question

A liquid flows through a horizontal pipe whose inner radius is \(2.52 \mathrm{~cm}\). The pipe bends upward through a height of \(11.5 \mathrm{~m}\) where it widens and joins another horizontal pipe of inner radius \(6.14 \mathrm{~cm}\). What must the volume flux be if the pressure in the two horizontal pipes is the same?

Short answer

The volume flux (Q) can be calculated by substituting the found cross-sectional areas and the given height into the formula derived from Bernoulli's equation.

Step-by-step solution

Calculation of the cross-sectional area of the two pipes

We calculate the cross-sectional area using the formula: \( A = \pi r^2 \). Letting \( A_1 \) be the area of the first pipe, we have \( A_1 = \pi (0.0252 m)^2 \) We do the same for the second pipe, denoting the area as \( A_2 \): \( A_2 = \pi (0.0614 m)^2 \)

Set up Bernoulli's equation

We apply Bernoulli's equation, setting the height \( h_1 \) to 0 and the height \( h_2 \) at 11.5 m, and given the pressures at 1 and 2 are equal. Hence the Bernoulli’s equation becomes, \( \frac{1}{2} \rho v_1^2 = \frac{1}{2} \rho v_2^2 + \rho g h_2 \)

Substitute for velocities in terms of volume flux

We write down the velocity of the fluid in the two pipes in terms of the volume flux \( Q \) and their corresponding cross-sectional areas. Hence, \( v_1 = Q/A_1 \) and \( v_2 = Q/A_2 \). Substituting these into the equation from Step 2, we get \( \frac{1}{2} \rho (Q / A_1)^2 = \frac{1}{2} \rho (Q / A_2)^2 + \rho g h_2 \). This equation can be solved for Q.

Solve for Q

Surface area is a quadratic, we get \( (Q / A_1)^2 - (Q / A_2)^2 = 2 g h_2 \) By rearranging, we find: \( Q = \sqrt{\frac{A_1^2 A_2^2}{A_1^2 - A_2^2} 2 g h_2} \)

Substitute the calculated areas and given height into the formula

Substitute \( A_1 \), \( A_2 \) and \( h_2 \ = 11.5 m \) into the equation and calculate the values to find the volume flux \( Q \)

Key concepts

Bernoulli's Equation

Bernoulli's Equation is a fundamental concept in fluid dynamics that relates the speed of a fluid, its pressure, and its height or elevation in a fluid flow system. It is derived from the conservation of energy principle applicable to moving fluids. Imagine a pipe system where fluid travels through various sections. According to Bernoulli's equation, if a fluid is moving through a pipe with constant flow, the sum of its potential energy, kinetic energy, and pressure energy remains constant along the flow path.

The equation is expressed as:\[ P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant} \]where:

• \( P \) is the pressure of the fluid,
• \( \rho \) is the fluid density,
• \( v \) is the fluid velocity,
• \( g \) is the acceleration due to gravity,
• \( h \) is the height above a reference point.

In practical terms, if two points in a streamline have the same pressure but different heights, the fluid will adjust its speed to keep the equation balanced. This adjustment in speed due to height differences is key to solving many fluid dynamics problems.

Volume Flux

Volume flux, often denoted as \( Q \), is a measure of the volume of fluid that flows through a cross-sectional area per unit time in a fluid system. In simpler terms, it tells you how much fluid is passing through a pipe or channel. It's particularly important in engineering and physics because it gives insight into how fluids are transported from one place to another.

The formula for volume flux is:
\[ Q = A v \]where:

• \( Q \) is the volume flux,
• \( A \) is the cross-sectional area,
• \( v \) is the velocity of the fluid.

This equation shows that the volume flux is directly proportional to both the cross-sectional area through which the fluid moves and its velocity. Thus, if the pipe gets narrower (smaller \( A \)), either the speed must increase to maintain the same flow volume, or the flow decreases unless pressure or other conditions change.

Cross-Sectional Area

The cross-sectional area of a pipe or channel is crucial in determining fluid behavior, specifically how much fluid can pass through it at any time. It refers to the area of the section cut perpendicular to the flow direction. This area is typically circular for pipes but can be any shape in broader contexts.

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 pipe.

The larger the cross-sectional area, the more fluid can flow through the pipe, assuming that other factors like pressure and velocity are constant. It's a simple yet powerful relationship: larger pipes can transport more fluid, given the same flow conditions. This is why engineers often adjust pipe dimensions when designing systems to meet specific flow requirements efficiently. Changing the cross-sectional area affects not just the volume flux but also potentially alters velocity and pressure, intertwining closely with Bernoulli's Equation for comprehensive fluid management.