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Fluid dynamics, Pipe Flow, Hagen Poiseuille, Darcy's equation

  1. Feb 3, 2014 #1
    1. The problem statement, all variables and given/known data
    A water injection line is made from smooth capillary tubing with inside
    diameter D = 25.0 mm . If the length of the pipe is 0.75 m and assuming
    laminar flow is present up to Re = 2000, find

    (i) the maximum average velocity at which the flow is laminar
    (ii) the pressure drop required to deliver this maximum velocity.

    [Answer: (i) u = 8 m/s; (ii) ∆p = 3 072 MPa ]


    2. Relevant equations
    Everything related to laminar flows in pipes used the hagen poiseuille equation and Darcy's equations.

    u=-1/4μ*(dP/dx)(R^2-r^2)


    3. The attempt at a solution

    Just some random substitutions, like setting -(dP/dx)=Δp/L and then equating that to Darcy's 4f/d*ρU^2/2

    But the issue is the viscosity. I can't get rid of it, and I can't find it. Though I can understand why it is not given, as it would make things way too easy.

    Anything about viscosity?
     
  2. jcsd
  3. Feb 3, 2014 #2
    Look up the viscosity of water at room temperature. At 20C, it's about 1 centipoise.

    Are you sure about that diameter? 25 mm sounds like an awfully big capillary. Did you mean 25μm?

    If you know the viscosity, the density, the diameter, and the Reynolds number, you have enough info to calculate the average velocity.

    The equation you wrote is not the Hagen Poiseuille equation. That equation expresses the pressure drop in terms of either the volumetric throughput rate, or equivalently, the average velocity. What is that equation?

    Chet
     
  4. Feb 3, 2014 #3
    You are right about the diameter. Its 0.25mm.

    Non the less, if viscosity was given it would be awfully easy to get the velocity. Another thing is, no temperature is given, which it is a bit iffy to make an assumption about what viscosity to use. Is there no way to get it out of there?
     
  5. Feb 3, 2014 #4

    pasmith

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    Homework Helper

    What's the context of this exercise? What values for the viscosity of water have you been given in your notes/textbook? In short, is there anything which suggests that you should do otherwise than assume a temperature of 20 degrees? Working backwards from the given answer suggests that you are supposed to make that assumption, so there must be something in the context to tell you that or the exercise is poorly designed.
     
  6. Feb 3, 2014 #5
    You know that, in the real world, you are not going to be spoon fed all the data. If you knew that the problem statement was missing the viscosity, you should have looked it up. After all, the problem statement did say room temperature. Would it have mattered much to the answer if you had used 22C or 23C? All you needed to do was to google viscosity of water. PAsmith confirmed that this would have been the right thing to do.

    Chet
     
  7. Feb 3, 2014 #6
    I assumed that there must be a way to circumvent viscosity because the emphasis of the topic was the equation u=-1/4μ*(dP/dx)*(R^2-r^2) (for laminar flows). Knowing viscosity means you can find the velocity through reynolds number Re=ρud/μ which is kinda trivial. Only thing that's left is finding the pressure difference which is not difficult either.

    All that was given was what I wrote. Something like that can pop up on the exam and I don't get to check water viscosity on the internet.

    It does work out in the end, yeah. I'm asking for clarity though, is there no way to circumvent needing viscosity there?
     
  8. Feb 3, 2014 #7
    Definitely not.

    Chet
     
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