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Chapter 4: Problem 4

At a particular section in a 100 -mm-diameter pipe, the velocity is measured as \(1200 \mathrm{~m} / \mathrm{s}\). Downstream of this section, the flow is expanded to a \(300-\mathrm{mm}\) pipe in which the velocity and density of the airflow is measured as \(700 \mathrm{~m} / \mathrm{s}\) and \(1.1 \mathrm{~kg} / \mathrm{m}^{3},\) respectively. If flow conditions are steady, estimate the density of the air at the upstream (100mm-diameter) section.

Short Answer

Expert verified
The upstream air density is approximately 8.67 kg/m鲁.

Step by step solution

01

Understand the Problem

We need to determine the density of air at the upstream section of a pipe system given certain conditions downstream. The known factors include the diameters, velocities, and downstream density.
02

Apply Continuity Equation

The continuity equation for steady incompressible flow states that mass flow rate must be conserved between two sections of a pipe. Mathematically, this is expressed as \( \rho_1 A_1 V_1 = \rho_2 A_2 V_2 \), where \( \rho \) is density, \( A \) is the cross-sectional area, and \( V \) is velocity at each section.
03

Calculate Cross-sectional Areas

Compute the cross-sectional areas for both the upstream and downstream sections. For a circular pipe, \( A = \frac{\pi d^2}{4} \). Thus,\[ A_1 = \frac{\pi (0.1)^2}{4} = 7.85 \times 10^{-3} \text{ m}^2 \]\[ A_2 = \frac{\pi (0.3)^2}{4} = 7.07 \times 10^{-2} \text{ m}^2 \]
04

Substitute Known Values into Continuity Equation

Plug the calculated areas and given velocities into the continuity equation. We know that \( V_1 = 1200 \text{ m/s}, \rho_2 = 1.1 \text{ kg/m}^3, \) and \( V_2 = 700 \text{ m/s} \). Substitute these values:\[ \rho_1 \times 7.85 \times 10^{-3} \times 1200 = 1.1 \times 7.07 \times 10^{-2} \times 700 \]
05

Solve for Density \( \rho_1 \)

Rearrange the equation to solve for \( \rho_1 \):\[ \rho_1 = \frac{1.1 \times 7.07 \times 10^{-2} \times 700}{7.85 \times 10^{-3} \times 1200} \]Calculate \( \rho_1 \):\[ \rho_1 = \frac{1.1 \times 7.07 \times 700}{7.85 \times 1200} \approx 8.67 \text{ kg/m}^3 \]
06

Final Solution

The density of air at the upstream section of the pipe is approximately \( 8.67 \text{ kg/m}^3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fluid Mechanics
Fluid Mechanics is the branch of physics that studies how fluids (liquids and gases) behave both when at rest and when in motion. It involves analyzing forces, pressures, and fluid flow to understand their interactions and effects.
In the world of Fluid Mechanics, the behavior of fluids in pipes is of great significance, especially in engineering applications. This is where concepts like the Continuity Equation come into play, which helps ensure the conservation of mass or volume within a closed system.
  • Fluids can be either compressible or incompressible. Incompressible fluids maintain constant density, while compressible fluids do not. In our exercise, we're dealing with air, which generally can compress but is treated as incompressible for steady flow conditions in this context.
  • Understanding Fluid Mechanics helps engineers design systems such as pipelines, hydraulic machinery, and even airplanes.
Pipe Flow
In fluid mechanics, pipe flow refers to the movement of a fluid through a closed conduit, typically circular in cross-section. Pipe flow is influenced by many factors, including pressure differences, friction, and changes in diameter, which often occur in actual systems. This exercise considers flow through a pipe system with varying diameters.
Different diameters cause changes in the velocity of the fluid as it attempts to maintain constant mass flow rate throughout its journey.
  • Smaller diameters result in higher velocities if the mass flow is constant.
  • Larger diameters allow the fluid to travel slower, as seen in our problem where the flow slows down after entering the 300-mm diameter pipe.
Pipe flow is typically turbulent or laminar, with the exercise focusing on velocity changes due to pipe diameter changes rather than flow regime.
Density Calculation
Density Calculation is a critical step in solving problems related to fluid flow, especially when using the Continuity Equation. Density represents the mass per unit volume of a fluid and is crucial for determining other flow parameters.
In the given exercise, the challenge is determining the upstream density using downstream measured values and the Continuity Equation.
  • The calculation uses the known values of velocity and density downstream to infer the unknown upstream density.
  • The formula for density calculation derived from the Continuity Equation is \( ho_1 = \frac{\rho_2 A_2 V_2}{A_1 V_1} \), demonstrating that when area and velocity are known, density can be calculated.
This principle shows how interconnected fluid properties can be used to solve real-world fluid behavior questions.
Mass Flow Rate
Mass Flow Rate measures the amount of fluid mass passing through a given cross-sectional area of a pipe per unit time. It combines fluid density, velocity, and pipe area to describe the overall transfer of mass through a system.
Mass Flow Rate is essential for designing efficient fluid systems, like pipelines or air ducts, to ensure they're transporting enough fluid to meet system demands without excess capacity.
  • Expressed as \( \dot{m} = \rho A V \), where \( \rho \) is fluid density, \( A \) is cross-sectional area, and \( V \) is velocity.
  • In steady flows, mass flow rate remains constant across different sections of the pipe, explaining the use of the Continuity Equation.
Understanding mass flow rate helps ensure systems are neither underbuilt nor overbuilt, optimizing operational efficiency.
Steady Flow
Steady Flow describes a fluid flow condition where the fluid properties at a point in the system do not change over time. This concept simplifies analysis by focusing on a single, unchanging set of conditions.
It means that all variables (velocity, pressure, density, etc.) remain constant over time at every point in the fluid system. This doesn't imply they are the same at different points, but rather unchanging with time.
  • Steady Flow allows engineers to create models and calculations without accounting for changes over time.
  • In the exercise, assuming steady flow makes it reasonable to use the Continuity Equation without accounting for time-dependent changes.
By considering steady conditions, we can better predict the behavior of the fluid and design systems accordingly.

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Most popular questions from this chapter

A jet engine is mounted on an aircraft that is cruising at an altitude of \(10 \mathrm{~km}\) in a standard atmosphere, and the speed of the aircraft is \(300 \mathrm{~m} / \mathrm{s}\). The intake area of the engine is \(2 \mathrm{~m}^{2},\) the fuel consumption rate is \(30 \mathrm{~kg} / \mathrm{s},\) and the exhaust gas exits at a speed of \(250 \mathrm{~m} / \mathrm{s}\) relative to the moving aircraft. The pressure of the exhaust gas is approximately equal to the ambient atmospheric pressure. Under these conditions, what thrust is expected from the engine?

A 400 -mm-diameter pipe divides into two smaller pipes, each of diameter \(200 \mathrm{~mm}\). If the flow divides equally between the two smaller pipes and the velocity in the \(400-\mathrm{mm}\) pipe is \(2 \mathrm{~m} / \mathrm{s},\) calculate the velocity and flow rate in each of the smaller pipes.

Water flows at a rate of \(0.025 \mathrm{~m}^{3} / \mathrm{s}\) in a horizontal pipe whose diameter increases from 6 to \(11 \mathrm{~cm}\) by an enlargement section. If the head loss across the enlargement section is \(0.45 \mathrm{~m}\) and the kinetic energy correction factor at both the inlet and the outlet is \(1.05,\) determine the pressure change.

A storage tank with a volume of \(0.2 \mathrm{~m}^{3}\) contains compressed air. Air is released from the tank by opening a valve connected to a 12 -mm- diameter tube. At the instant the valve is opened, air exits at a velocity of \(150 \mathrm{~m} / \mathrm{s}\). Find the density and the rate of change of density of the air in the tank the instant the valve is opened under the following conditions: (a) The density of the air exiting the tank is \(6 \mathrm{~kg} / \mathrm{m}^{3}\) and (b) the air exiting the tank has a temperature of \(-10^{\circ} \mathrm{C}\) and an absolute pressure of \(400 \mathrm{kPa}\).

Water at \(20^{\circ} \mathrm{C}\) in a \(10-\mathrm{m}\) -diameter, \(2-\mathrm{m}\) -deep aboveground swimming pool is to be emptied by unplugging a \(3-\mathrm{cm}\) -diameter, \(25-\mathrm{m}\) -long horizontal plastic drain pipe attached to the bottom of the pool. Determine the time it will take to empty the swimming pool completely, assuming that the entrance to the drain pipe is well-rounded with negligible loss. Assume the friction factor of the pipe to be \(0.022 .\)
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