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Flowmeter: Gas properties

Flow through a tube can be described by the modified Bernoulli Equation, which states:

Po = P + 1/2 ρ*U^2

Where Po is a constant called the “stagnation pressure,” P is pressure, ρ is density, U and is equal to fluid velocity. Pressure does not always decrease in the direction of flow – if one assumes friction is negligible (and thus the total energy of the system remains constant), any change in the kinetic energy of a fluid (ex. a reduction as U^2 falls) must be accompanied by an oppositional change in potential energy increases (P rises)

Whether flow is laminar or turbulent is dependent on the Reynolds Number (Re), defined as:

Re = ρUL/u

Where ρ is density, U is equal to fluid velocity, L is the length of the tubing, and u is the viscosity of the fluid. Note that the modified Bernoulli Equation applies to laminar flow (ReD < 2300, parabolic flow profile), but not to turbulent flow (ReD > 4000, blunted flow profile)

Flowmeters based on a Venturi tube (circular tube with a gradual contraction and expansion in diameter, ex. those used in aircraft) maintain laminar flow at normal flow rates – since conservation of mass and energy is assumed to apply, restriction of the diameter leads to increased speed of gas flow and a corresponding decrease in the pressure on the walls

The floating bobbin rotameter that is familiar to anesthesiologists is known as the Thorpe tube – the sudden restriction caused by the bobbin leads to a proportional increase in velocity and the Bernoulli equation does not apply (flow is proportionate to the square root of the pressure drop). Because the pressure drop across the orifice is inversely proportional to orifice area squared, the pressure change across the bobbin will decrease as the bobbin rises (and the area of the tube increases). The bobbin reaches a steady state when the pressure drop exactly opposes the gravitational force on the bobbin – at this steady state the pressure drop is always the same (because gravitational forces are constant), however as flow rates increase, the bobbin will move higher because this critical pressure drop will occur over a larger cross-sectional area (since velocity is higher)

Gas Flow

  • Modified Bernoulli: Po = P + 1/2 ρ*U^2 (conservation of energy)
  • Reynolds Number (Re): Re = ρUL/u
  • Laminar (Re < 2300): parabolic flow profile, Bernoulli applies
  • Turbulent (Re > 4000): flat flow profile, Bernoulli does not apply