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Chapter 11: Problem 69

A Venturi meter is a device that is used for measuring the speed of a fluid within a pipe. The drawing shows a gas flowing at speed \(v_{2}\) through a horizontal section of pipe whose cross-sectional area is \(A_{2}=0.0700 \mathrm{m}^{2}\) . The gas has a density of \(\rho=1.30 \mathrm{kg} / \mathrm{m}^{3} .\) The Venturi meter has a cross-sectional area of \(A_{1}=0.0500 \mathrm{m}^{2}\) and has been substituted for a section of the larger pipe. The pressure difference between the two sections is \(P_{2}-P_{1}=120 \mathrm{Pa}\) . Find \((\mathrm{a})\) the speed \(v_{2}\) of the gas in the larger, original pipe and \(\quad\) (b) the volume flow rate \(Q\) of the gas.

Short Answer

Expert verified
The speed of the gas, \(v_2\), is approximately 14.94 m/s, and the volume flow rate, \(Q\), is about 1.046 m鲁/s.

Step by step solution

01

Understand the Continuity Equation

The Venturi meter applies the principle of continuity, which states that the volume flow rate must be constant throughout the pipe. This leads to the equation:\[ A_1 \cdot v_1 = A_2 \cdot v_2 \]where \( A_1 \) and \( A_2 \) are the cross-sectional areas of the Venturi contraction and the larger pipe, respectively, and \( v_1 \) and \( v_2 \) are the gas velocities at these points.
02

Apply Bernoulli's Equation

Bernoulli's principle for incompressible flow provides the relation between speeds and pressures:\[ P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \]Given the values provided, rearrange for the velocity in the larger pipe, \( v_2 \), using the known pressure difference \( P_2 - P_1 = 120 \text{ Pa} \).\[ P_2 - P_1 = \frac{1}{2} \rho (v_2^2 - v_1^2) \Rightarrow v_2^2 - v_1^2 = \frac{2(P_2 - P_1)}{\rho} \]
03

Solve for Speed in the Venturi, \(v_1\)

Use the Continuity Equation to express \(v_1\) in terms of \(v_2\):\[ v_1 = \frac{A_2}{A_1} v_2 \]Substitute \(v_1\) into the equation from Bernoulli's principle:\[ v_2^2 - \left(\frac{A_2}{A_1} v_2\right)^2 = \frac{2 \times 120}{1.30} \]
04

Simplify and Solve the Quadratic Equation

This leads to a quadratic equation in \(v_2\):\[ v_2^2 - \left( \frac{A_2^2}{A_1^2} \right) v_2^2 = \frac{240}{1.30} \]\[ (1 - \frac{0.0700^2}{0.0500^2}) v_2^2 = \frac{240}{1.30} \]\[ v_2^2 \cdot \left(1 - \frac{0.0049}{0.0025}\right) = \frac{240}{1.30} \]Solve for \(v_2 \).
05

Calculate \(v_2\)

After solving the simplified equation:\[ 0.8272 v_2^2 = \frac{240}{1.30} \]\[ v_2^2 = \frac{240}{1.30 \times 0.8272} \]\[ v_2 = \sqrt{\frac{240}{1.07536}} = \sqrt{223.292} \]\[ v_2 \approx 14.94 \text{ m/s} \]
06

Calculate the Volume Flow Rate, \(Q\)

The volume flow rate \( Q \) is obtained by:\[ Q = A_2 \cdot v_2 \]Substitute for \( A_2 = 0.0700 \text{ m}^2 \) and \( v_2 \approx 14.94 \text{ m/s} \):\[ Q = 0.0700 \cdot 14.94 \]\[ Q \approx 1.046 \text{ m}^3/\text{s} \]

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Key Concepts

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

Continuity equation
The Continuity Equation is an essential concept in fluid dynamics, especially when analyzing systems like Venturi meters. It is based on the principle of the conservation of mass. In a flow system, the mass flow rate must remain consistent, which means the amount of fluid entering a section must equal the amount exiting. This principle leads us to the equation:\[ A_1 \cdot v_1 = A_2 \cdot v_2 \]Here:
  • \( A_1 \) is the cross-sectional area of the first segment of the pipe.
  • \( v_1 \) is the velocity of the fluid in that segment.
  • \( A_2 \) is the cross-sectional area of the second segment.
  • \( v_2 \) is the velocity of the fluid in the second segment.
This equation shows that if the cross-sectional area decreases, the velocity must increase to maintain a constant volume flow rate. It highlights why the fluid speeds up in the narrowing section of a Venturi meter.
Bernoulli's equation
Bernoulli's Equation provides a way to relate the pressure, velocity, and height of a moving fluid. It applies the conservation of energy principle to fluid flow, assuming incompressible and frictionless movement. The equation can be written as:\[ P_1 + \frac{1}{2} \rho v_1^2 + \rho gh = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh \]Where:
  • \( P_1 \) and \( P_2 \) are the pressures at two different points.
  • \( \rho \) is the fluid density.
  • \( v_1 \) and \( v_2 \) are velocities at these points.
  • \( g \) is the acceleration due to gravity, and \( h \) is the height above a reference level.
In a horizontal flow like a Venturi meter, the height terms cancel out, simplifying the equation to relate only pressure and velocity changes. This relationship helps us determine how pressure decreases when velocity increases through a constriction.
Volume flow rate
The Volume Flow Rate, often denoted as \( Q \), is a measure of the volume of fluid that passes a point in a system per unit of time. It's a crucial quantity in fluid dynamics because it provides insight into the efficiency and capacity of fluid transport systems. The formula for calculating the volume flow rate is:\[ Q = A \cdot v \]Where:
  • \( A \) is the cross-sectional area through which the fluid flows.
  • \( v \) is the velocity of the fluid across that area.
In our Venturi meter context, we compute \( Q \) to understand how much gas moves in the larger pipe, using the area of the pipe and the speed of the gas calculated from previous equations. The continuity of flow rate throughout the pipe ensures we can consistently apply this calculation.
Fluid dynamics
Fluid Dynamics is the broad study of fluids (liquids and gases) in motion. This field tackles complex behaviors of fluid flow through various mediums and is crucial in engineering, meteorology, oceanography, and even medicine. Fluid dynamics involves different equations and laws, including both the Continuity Equation and Bernoulli's Equation. Key Principles in Fluid Dynamics:
  • Conservation of mass, which the Continuity Equation represents.
  • Conservation of energy, represented by Bernoulli's Equation.
  • Understanding forces acting on and within fluids, which affect movement and interaction.
By utilizing Venturi meters, engineers apply these principles to measure flow speed and pressure changes in pipes. Such understanding enables the design of systems ranging from water supply networks to engines, optimizing the handling and transport of fluid substances.

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Most popular questions from this chapter

A paperweight, when weighed in air, has a weight of \(W=6.9 \mathrm{N}\) . When completely immersed in water, however, it has a weight of \(W_{\text { in water }}=4.3 \mathrm{N}\) . Find the volume of the paperweight.

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When an object moves through a fluid, the fluid exerts a viscous force \(\overrightarrow{\mathbf{F}}\) on the object that tends to slow it down. For a small sphere of radius \(R\) moving slowly with a speed \(v\) , the magnitude of the viscous force is given by Stokes' law, \(F=6 \pi \eta R v,\) where \(\eta\) is the viscosity of the fluid. (a) What is the viscous force on a sphere of radius \(R=5.0 \times 10^{-4} \mathrm{m}\) that is falling through water \(\left(\eta=1.00 \times 10^{-3} \mathrm{Pa} \cdot \mathrm{s}\right)\) when the sphere has a speed of 3.0 \(\mathrm{m} / \mathrm{s} ? \quad\) (b) The speed of the falling sphere increases until the viscous force balances the weight of the sphere. Thereafter, no net force acts on the sphere, and it falls with a constant speed called the "terminal speed If the sphere has a mass of \(1.0 \times 10^{-5} \mathrm{kg}\) , what is its terminal speed?

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