Flow measurement of helium

How Fluid Machines Work
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A pitot tube placed in a flow of helium (rho = .166 kg/m3) at 20C and 1 atm shows a reading of 11 mm H2o. What is the speed of the helium

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This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:

  1. PURPOSE OF THE EXERCISE:
    Connect the use of a Pitot tube to the Bernoulli-equation analysis that is used to interpret it. In a lesser sense, remind a student to reflect on how a manometer works and how to read one.

  2. ENGINEERING CONTEXT:
    Pitot tubes are used extensively to infer flow speeds on aerodynamic vehicles. The approach here demonstrates how to design or calibrate flow measurement instruments.

  3. PHYSICAL CONTEXT:
    The dominant effect is the inviscid conversion of kinetic energy (i.e., dynamic pressure) to static pressure, as captured by Bernoulli’s equation applied along a streamline that terminates at the stagnation point at the tip of the pitot tube.

  4. ASSUMPTIONS:
    Assume steady, inviscid incompressible flow in the region upstream of the Pitot tube; this is reasonable because the measured pressure is small and the Mach number is low.

  5. SOLUTION TECHNIQUE:
    A Pitot-static tube measures the dynamic pressure

\Delta p = p_\mathrm{stag} - p_\infty = \tfrac{1}{2}\rho_\mathrm{He} U^2\,.

Because the measurement is given as an 11 mm H$_2$O head, convert this to Pascal using

\Delta p = \rho_\mathrm{w}\,g\,h\,.

With \rho_\mathrm{w}=1000\,\mathrm{kg/m^3}, g=9.81\,\mathrm{m/s^2}, and h=0.011\,\mathrm{m},

\Delta p = 1000\,\mathrm{kg/m^3}\times9.81\,\mathrm{m/s^2}\times0.011\,\mathrm{m}\approx107.9\,\mathrm{Pa}\,.

Then, solving for the velocity via

U=\sqrt{\frac{2\Delta p}{\rho_\mathrm{He}}}\,,

and substituting \rho_\mathrm{He}=0.166\,\mathrm{kg/m^3},

U=\sqrt{\frac{2(107.9\,\mathrm{Pa})}{0.166\,\mathrm{kg/m^3}}}\approx\sqrt{1299.4\,\mathrm{m^2/s^2}}\approx36.0\,\mathrm{m/s}\,.

A student might benefit from sketching the streamline that terminates at the front of the Pitot tube.

  1. REFLECTION:
    The calculation shows the sensitivity – the Helium is moving pretty fast and the manometer is only 11mm. One can’t really measure a manomter more precisely than about a millimeter, so this setup will measure Helium speed, but at these speeds it will only measure it to a precision of about 3 m/s.

In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 5 How Stuff Works: How a Pitot–static tube works (HIGHLY RELEVANT)
  2. Chapter 9.3: Why We Use Irrotational Flow Analysis Part 5 (HIGHLY RELEVANT)
  3. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 2 (HIGHLY RELEVANT)
  4. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 5 Part 1 (HIGHLY RELEVANT)
  5. Chapter 3 Engineering Models: Hydrostatic Equation (HIGHLY RELEVANT)

Disclaimer:

See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.