Theoretical Fluid Mechanics: Venturi Meter

A Venturi Meter is a device that allows flow rates through pipes to be calculated by measuring the difference in pressure created by a contraction in a pipe.  When the flow goes through the contraction it must speed up, and so the pressure must drop.  By measuring the two pressures, engineers can directly calculate the velocity of the fluid.  Knowing the pipe diameter, this velocity can be converted into a flow rate.

Venturi Meters work based on the principles found in Bernoulli's equation.  Here, Bernoulli's equation is set so each term is in units of length, allowing an engineer to calculate the "head" on a pipe.  Remember that for a pipe without energy losses, H is constant.

Because H is constant, we can compare Bernoulli's equation at a point before and during the contraction:

If we can measure the elevation of each pipe segment (or if it does not change), and the pressure head using a Piezometer (A small tube with an opening flush with the wall of the pipe), than we have one equation with two unknowns.

If the pipe diameters are known, conservation of mass law, will give us a second equation allowing us to solve for velocities (or flow rates):

I made this little Mathematica demo (You might need a special plug-in) which allows you to play with a pipe to see how this might work.  The yellow parts show a fluid element in the pipe.  Notice when the pipe contracts, the fluid element must stretch out (accelerate) in order to fit through the pipe.  The demo also shows the Energy Grade line (EGL) and the Hydraulic Grade Line (HGL). Think of  the EGL being the total energy  at a pont in the pipe.  If you stuck a pitot tube into the pipe, the water would rise to this line.  The HGL is how high the flow would rise in a piezo tube, and considers just the first two terms of Bernoulli's equation.  If the Pressure drops too low, Cavitation might occur!

Feel free to play around with the sliders, and have fun.

For more information on how Venturi Meters work, I'd recommend looking at this youtube video:

If you have any specific questions about Venturi Meters, How they work, or Fluid Mechanics in general, please feel free to ask below.  I'd be happy to help.

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  • jim rohr

    Mike - wondering if you have a good way to explain the following:

    Consider steady, fully developed flow - no acceleration - no inertial forces;
    yet Reynolds number is critical to characterize - yet Re = inertial forces / viscus forces.

    Jim

    • mrsoltys

      Jim,
      Thanks for your question! I work very little with flows that have negligible inertial forces, such as a creeping flow or stokes flow; however, it does make me think of a coral sperm (very relevant to my research) trying to make its way through a seemingly incredibly viscous fluid. What level students are you trying to explain the above problem to? I hope my answer here isn't too technical (or wrong).

      My short answer is there is an important distinction between a very small number and zero. For flows with negligible inertial forces, the Re=inertial forces / viscous forces « 1. Here, it is important note that a very small number is much different than 0. Just like we can have super high Re flows (such as a volcano erupting Re~10^8) we can also have super low Re flows (such as a bacterium swimming through water Re~10^-4).

      The important question, and what we mean when we say that inertial forces are 0, is what assumptions can we make when Reynolds number « 1? More specifically, what assumptions do Low reynolds numbers allow us to make in the Navier Stokes equations? If

      then Re « 1 happens when the velocity (u) and the length scale (L) approach zero while the viscosity gets very large.

      Looking at the incompressible Navier-Stokes Equation, and labeling what each term means:

      By my labeling we can assume that if inertial forces are negligible, we can drop the left side of the equation, but how can we prove this? Scaling arguments! Each term scales by:

      The clearest way to show how these scales relate to Reynolds number is to multiply through by [L^2 /U ν]. The relevant scales for each term become:

      it is clear that if Re « 1 (aka the inertial forces are negligible) we can drop the Left hand side of the equation and we're left with what is commonly called the "Stoke's Equations" for creeping flow.

      A good place I visited to get the gears turning was: http://www.hitech-projects.com/euprojects/artic/index/Low%20Reynolds%20number%20flows.pdf

  • sameer

    it is helpful but Nre ? actual discharg? inertia? velocity?