/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 A Venturi meter is a device that... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 63

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 (a) the speed \(v_{2}\) of the gas in the larger original pipe and (b) the volume flow rate \(Q\) of the gas.

Short Answer

Expert verified
The speed \( v_2 \) is approximately 20 m/s. The volume flow rate \( Q \) is approximately 1.4 m鲁/s.

Step by step solution

01

Understanding Bernoulli's Equation

Bernoulli's equation for fluid flow is given by \[ P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \] where \( P_1 \) and \( P_2 \) are pressures, \( v_1 \) and \( v_2 \) are fluid speeds, and \( \rho \) is fluid density. Use this equation to relate the pressure and the velocity at the two cross sections of the Venturi meter.
02

Using Continuity Equation

The continuity equation for incompressible flow states that \( A_1 v_1 = A_2 v_2 \). Since the areas \( A_1 \) and \( A_2 \) are known, you can express \( v_1 \) in terms of \( v_2 \) as \( v_1 = \frac{A_2}{A_1} v_2 \).
03

Substitute for v1 in Bernoulli's Equation

Replace \( v_1 \) in Bernoulli's equation using \( v_1 = \frac{A_2}{A_1} v_2 \). The equation becomes \[ P_1 + \frac{1}{2} \rho \left(\frac{A_2}{A_1} v_2\right)^2 = P_2 + \frac{1}{2} \rho v_2^2 \]. Simplify and solve for \( v_2 \) using the pressure difference \( P_2 - P_1 = 120 \text{ Pa} \).
04

Solve for v2

Simplifying the equation for \( v_2 \), you obtain \[ 120 = \frac{1}{2} \rho v_2^2 - \frac{1}{2} \rho \left(\frac{A_2}{A_1} v_2\right)^2 \]. Substitute \( \rho = 1.30 \text{ kg/m}^3 \), and solve the quadratic equation to find \( v_2 \).
05

Calculate Volume Flow Rate Q

Once you have \( v_2 \), compute the volume flow rate \( Q \) using the formula \( Q = A_2 v_2 \). Substitute the known values for \( A_2 \) and \( v_2 \) to find \( Q \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Venturi meter
A Venturi meter is an instrumental tool used in fluid dynamics to measure the flow rate or speed of a fluid traveling through a pipe. The device works based on the principle of pressure differential that occurs at different cross-sectional areas.
The Venturi meter narrows down a portion of the pipe which leads to a change in fluid velocity and pressure. By measuring the pressure difference, the speed of the fluid can be calculated. In simpler terms, a larger cross-sectional area results in a lower velocity and a smaller pressure drop, while a smaller cross-sectional area leads to higher velocity and a larger pressure drop. This unique property of a Venturi meter makes it highly effective for applications involving water, gas, or oil flow measurement in various industries.
Continuity equation
The continuity equation is a fundamental principle in fluid dynamics that expresses the conservation of mass in fluid flow. According to this equation, the mass flow rate must remain constant from one cross-section of a pipe to another, assuming the fluid is incompressible. This is effectively represented by the equation \[ A_1 v_1 = A_2 v_2 \] where \( A_1 \) and \( A_2 \) are the cross-sectional areas and \( v_1 \) and \( v_2 \) are the velocities at two different points along the pipe.
The equation is especially relevant when analyzing systems that involve fluid flow through various pipe diameters, such as a Venturi meter. The equation helps in simplifying the calculation of fluid speed and understanding how changes in pipe diameter affect flow conditions.
Fluid dynamics
Fluid dynamics is the study of the behavior and movement of fluids, including how they interact with their environment. This field is crucial for understanding the principles that govern everything from the flow of air over an airplane wing to the water through a Venturi meter.
The two main governing equations in fluid dynamics are Bernoulli's equation and the continuity equation.
Key considerations in fluid dynamics:
  • Fluid density - affects pressure and flow.
  • Pressure changes - related to velocity changes, as explained by Bernoulli's principle.
  • Fluid velocity - how fast the fluid particles are moving, influenced by factors like pipe diameter.
Understanding these factors helps predict the motion of fluids under various circumstances, assisting in design and analysis projects across engineering fields.
Volume flow rate
The volume flow rate, often denoted as \( Q \), quantifies how much fluid passes through a given surface or opening per unit of time. In the context of a Venturi meter, this measurement is crucial to determine how efficiently a fluid system operates.
The formula to calculate volume flow rate is given by \[ Q = A v \] where \( A \) represents the cross-sectional area through which the fluid is flowing, and \( v \) is the velocity of the fluid. This relationship ties together the size of the opening and the speed of the fluid, revealing the overall efficiency and capacity of the fluid flow system. Calculating the volume flow rate allows engineers and scientists to optimize systems for better performance and resource management.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Water is circulating through a closed system of pipes in a two-floor apartment. On the first floor, the water has a gauge pressure of \(3.4 \times 10^{5} \mathrm{~Pa}\) and a speed of \(2.1 \mathrm{~m} / \mathrm{s}\). However, on the second floor, which is \(4.0 \mathrm{~m}\) higher, the speed of the water is \(3.7 \mathrm{~m} / \mathrm{s}\). The speeds are different because the pipe diameters are different. What is the gauge pressure of the water on the second floor?

A 1.3-m length of horizontal pipe has a radius of \(6.4 \times 10^{-3} \mathrm{~m}\). Water within the pipe flows with a volume flow rate of \(9.0 \times 10^{-3} \mathrm{~m}^{3} / \mathrm{s}\) out of the right end of the pipe and into the air. What is the pressure in the flowing water at the left end of the pipe if the water behaves as (a) an ideal fluid and (b) a viscous fluid \(\left(\eta=1.00 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}\right)\) ?

A solid cylinder (radius \(=0.150 \mathrm{~m},\) height \(=0.120 \mathrm{~m}\) ) has a mass of \(7.00 \mathrm{~kg}\). This cylinder is floating in water. Then oil \(\left(\rho=725 \mathrm{~kg} / \mathrm{m}^{3}\right)\) is poured on top of the water until the situation shown in the drawing results. How much of the height of the cylinder is in the oil?

A meat baster consists of a squeeze bulb attached to a plastic tube. When the bulb is squeezed and released, with the open end of the tube under the surface of the basting sauce, the sauce rises in the tube to a distance \(h\), as the drawing shows. It can then be squirted over the meat. (a) Is the absolute pressure in the bulb in the drawing greater than or less than atmospheric pressure? (b) In a second trial, the distance \(h\) is somewhat less than it is in the drawing. Is the absolute pressure in the bulb in the second trial greater or smaller than in the case shown in the drawing? Explain your answers. Using \(1.013 \times 10^{5} \mathrm{~Pa}\) for the atmospheric pressure and \(1200 \mathrm{~kg} / \mathrm{m}^{3}\) for the density of the sauce, find the absolute pressure in the bulb when the distance \(h\) is (a) \(0.15\) \(\mathrm{m}\) and (b) \(0.10 \mathrm{~m}\). Verify that your answers are consistent with your answers to the Concept Questions.

A ship is floating on a lake. Its hold is the interior space beneath its deck and is open to the atmosphere. The hull has a hole in it, which is below the water line, so water leaks into the hold. (a) How is the amount of water per second (in \(\mathrm{m}^{3} / \mathrm{s}\) ) entering the hold related to the speed of the entering water and the area of the hole? (b) Approximately how fast is the water at the surface of the lake moving? Justify your answer. (c) What causes the water to accelerate as it moves from the surface of the lake into the hole that is beneath the water line? Explain.
See all solutions

Recommended explanations on Physics Textbooks

Radiation

Read Explanation

Thermodynamics

Read Explanation

Modern Physics

Read Explanation

Dynamics

Read Explanation

Energy Physics

Read Explanation

Fluids

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.