Question

Water is moving with a speed of \(5.18 \mathrm{~m} / \mathrm{s}\) through a pipe with a cross-sectional area of \(4.20 \mathrm{~cm}^{2} .\) The water gradually descends \(9.66 \mathrm{~m}\) as the pipe increases in area to \(7.60 \mathrm{~cm}^{2} .(a)\) What is the speed of flow at the lower level? ( \(b\) ) The pressure at the upper level is \(152 \mathrm{kPa}\); find the pressure at the lower level

Short answer

The speed of flow at the lower level is 2.87 m/s and the pressure at the lower level is 158.27 kPa

Step-by-step solution

Conversion of Units

Convert the cross-sectional areas to square meters (from square centimeter). \( A_1 = 4.20 \times 10^{-4} \)m² and \( A_2 = 7.60 \times 10^{-4} \) m².

Find Speed at Lower Level using Continuity Equation

The Continuity equation states that the mass flow rate must remain constant in a fluid pipe. It can be formulated as \(A_1 v_1 = A_2 v_2\). Solve this equation for \(v_2\) (Speed at lower level), \(v_2\) = \( \frac{A_1 \times v_1}{A_2} \)

Calculate the Speed

Insert the values into the above equation to find the speed at the lower point, \(v_2\) = \( \frac{4.20 \times 10^{-4} \times 5.18}{7.60 \times 10^{-4}} \) = 2.87 m/s

Find Pressure at Lower Level using Bernoulli's Equation

Reshape Bernoulli's equation and solve for \(P_2\), \(P_2 = P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 - \frac{1}{2} \rho v_2^2 - \rho g h_2\)

Calculate Pressure at Lower Point

Insert the values into the above equation to find the pressure at the lower point, remembering to convert pressures to Pascals (1kPa = 1000 Pa), \(P_2 = 152000 + \frac{1}{2} \times 1000 \times {5.18}^2 + 1000 \times 9.8 \times 9.66 - \frac{1}{2} \times 1000 \times {2.87}^2 - 1000 \times 9.8 \times 9.66 \)= 158265 Pa = 158.27 kPa

Key concepts

Continuity Equation

In the world of fluid dynamics, the continuity equation is a fundamental principle that helps us understand fluid flow in closed systems. This equation is derived from the principle of conservation of mass, which states that for an incompressible fluid like water, the mass flow rate must remain constant throughout the entire pipe.
The continuity equation can be represented as:

• \( A_1 v_1 = A_2 v_2 \)

Here, \( A_1 \) and \( A_2 \) are the cross-sectional areas of the pipe at two different points, while \( v_1 \) and \( v_2 \) are the respective velocities of the fluid at those points. If the cross-sectional area changes along the pipe, the velocity will also change such that the product of area and velocity remains constant.
This equation is essential for calculating the speed of a fluid at different sections of a pipe system. By using it, we can determine, for example, the speed at the lower level of a pipe when the cross-sectional area at this point is larger than at the higher point above it.

Bernoulli's Equation

Bernoulli's equation is another crucial aspect of fluid dynamics. It relates the speed of a fluid, its pressure, and potential energy per unit volume within a streamline. This equation is derived from the principle of conservation of energy and can be expressed as:

• \( P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2 \)

In this equation, \( P \) represents the pressure at different points, \( \rho \) is the density of the fluid, \( v \) is the fluid velocity, \( g \) is the acceleration due to gravity, and \( h \) represents the height.
Bernoulli's equation highlights that as the fluid's speed increases, its pressure decreases, and vice versa. Also, if the fluid descends, its potential energy decreases, affecting the pressure as well.
This equation is used to calculate the pressure at different levels of a pipe, as shown in the given problem, where it helps to find the pressure at the lower level of the pipe when the fluid speed and height change.

Mass Flow Rate

Mass flow rate is another pivotal concept in fluid dynamics, linking the continuity equation and Bernoulli's equation together. It represents the mass of fluid passing through a cross-section of a pipe per unit of time. The formula for mass flow rate \( \dot{m} \) is:

• \( \dot{m} = \rho A v \)

Where \( \rho \) stands for the fluid density, \( A \) is the cross-sectional area, and \( v \) is the velocity of the fluid.
The continuity equation essentially ensures that the mass flow rate is constant along the pipe, reflecting the mass conservation principle. This means that even if the cross-sectional area or the velocity changes, \( \dot{m} \) remains unchanged.
Understanding mass flow rate is critical for solving fluid-related problems, as it predicts how much fluid moves through a system, indicating potential changes in physical properties such as velocity and pressure.