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Continuity Equation and the Bernoulli's Equation

  1. Jul 31, 2011 #1
    1. The problem statement, all variables and given/known data

    Skill Level II Problem
    Use the Continuity equation to explain how jet engines provide a forward thrust for an airplane.

    Skill Level Problem III
    The Contintuity Equation is related to a powerful equation from fluid dynamics called Bernoulli's Equation. Do the research and answer the questions listed below.
    • Do research necessary to find Bernoulli's Equation (DONE)
    • Identify what each term in Bernoulli's Equation represents. (DONE)
    • Reread the problem given in Skill Level I of this thought Project. What assumption is built into the wording of the problem that becomes unnecessary if you use Bernoulli's?
    • Discuss how the Continuity Equation and Bernoulli's Equation might relate to one another.

    Skill Level I Problem
    One evening, a husband and wife are working in their garden. While watering the grass, the wife holds the hose horizontally and the water flows from the end of the hose. She twists the nozzle of the hose to partially close it and suddenly the water shoots farther from the end of the hose. This seems odd to her husband, who wonders, "Why does the stream of the water travel farther, since it appears that less water is flowing from the end of the hose?"
    Your challenge is to give a scientific explanation for this seemingly unnatural phenomenon.



    2. Relevant equations
    The Continuity Equation according to my textbook, A1v1=A2v2 A1 and A2 are the cross-sectional areas at points 1 and 2 v1 and v2 are the velocity of the fluid at point 1 and 2.

    The Bernoulli's Equation according to my textbook, P1+pgy1+1/2pv1=P2+pgy2+1/2pv2
    I understand P1 and P2 are the pressures on the fluid at points 1 and 2
    p is the density of the fliud
    g is the acceleration due to gravity, 9.8m/s sq.
    y1 and y2 are the respective heights of the pipe at points 1 and 2
    v1 and v2 are the velocities of the fluid at points 1 and 2.

    3. The attempt at a solution

    In regard to Skill Level I: With my written expression learning disability problem, I have a hard time in math, only when answers to the question are essay style. In truth, we are learning how to use George Polya Four-Step Process to solve problems.
    Understand the Problem
    "Why does the stream of the water travel farther, since it appears that less water is flowing from the end of the hose?" Draw a picture to help visualize.
    Devise a Plan
    Discuss possible problem solving strategies for solving this problem. Which strategy (physical experimentation, information research, or mathematics) would be most likely to generate a successful solution? Why? Why do you think that the strategies that you eliminated would not be viable problem solving options for this situation?​

    Carry out the Plan
    Does the introduction of the Continuity Equation modify your interpretation of the original problem, or your strategy for solving it? If so how?

    The equation give to use was A1v1=A2v2 A1 and A2 are the cross-sectional areas at points 1 and 2 v1 and v2 are the velocity of the fluid at point 1 and 2.

    Choose some real numbers and show how this equation provides an answer to the husband's apparent problem.

    I used made up numbers like 2 meter for A1, 1 meter for A2, and 10m/s for v1 and we solved for v2. therefore v2 is 20m/s. ​

    What does this equation suggest about the relationship between velocity and area?

    Write out a detailed explanation of your solution to this thought project. Suggest some other situations in which Continuity Equation would come into play. ​

    Look Back
    We had nothing in regards to look back.

    In regards to Skill Level II: I found this information in my research trying to do this on my own but I still not sure how to answer anything correctly. I print a copy of this article on Jet engines from About.com Inventors. Which I will turn in with my assignments that are due Monday. http://inventors.about.com/library/inventors/blhowajetengineworks.htm".

    I also found this follow information which might be of help to explain but I not sure of the right words. I have a hard time putting thing into my own words. Or, I have no one to talk to and hear me out and state if I am right and should right that down.

    Identify Examples of Newton’s Three Laws of Motion.
    NEWTON’S FIRST LAW – THE LAW OF EQUILIBRIUM
    ○ An airplane in straight and level flight at a constant velocity is acted upon by four forces: thrust, drag, lift and weight
    ○ When thrust is equal to drag, and lift is equal to weight, the airplane is in equilibrium
    NEWTON’S SECOND LAW – THE LAW OF ACCELERATION
    ○ When an airplane’s thrust is greater than its drag (in level flight), the excess thrust will accelerate the airplane until drag increases equal to thrust
    NEWTON’S THIRD LAW – THE LAW OF INTERACTION
    ○ The thrust from a jet engine consists of hot gases exhausted rearward producing a thrust acting forward
    Describe the relationship between airflow velocity and cross-sectional area within a streamtube using the continuity equation.
    • Velocity and area in a streamtube are inversely related
    • The Continuity Equation: A1V1 = A2V2
    • Mass flow is equal to the density times the area time the volume
    • Mass flow = (rho)AV
    • The density times the volume times the area are equal to the same for each cross section of the streamtube.
    • (rho)1A1V1 = (rho)2A2V2​
    Describe the relationship between total pressure, static pressure, and dynamic pressure within a streamtube using Bernoulli’s equation.
    • Total pressure (PT), also called head pressure (HT), is the sum of static and dynamic pressure
    • Static pressure (PS) is a measure of potential energy per unit volume
    • Dynamic pressure (q) is the pressure of a fluid resulting from its motion, and is equal to 1/2(rho)V*2
    • Equation is PT = PS + q
    • Static pressure and dynamic pressure are inversely related
    Marine Gouge. Aerodynamics 1. MediaWiki. 28 January 2011
    <http://marinegouge.com/mediawiki-1.13.3/?title=Aerodynamics_1> [Broken]
    Date Access: July 31, 2011

    I post help in the wrong area my second night on this forum. And now it will be too late for anyone to help. My assignment in due Monday night. August 2, 2011 I know it is against the rules but I need help more than a normal person for I suffer from a Written Expression Learning Disability.
     
    Last edited by a moderator: May 5, 2017
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