Fluid and Gas Flow Principles

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Figure above adapted from Reference 49

Figure adapted from Reference 46, Basic Physics and Measurement in Anaesthesia
  • An important clinical consideration has to do with a change in the relationship between flow and pressure as flow transitions from laminar to turbulent.  
    • In particular, for turbulence flow inside pathways that have rough internal edges, flow appears about proportional to the square root of pressure or a doubling of flow requires a quadrupling of pressure.
    • Furthermore, with turbulent flow, since pressure to flow relationships do not exhibit linearity, resistance will not be constant.  
    • In the turbulent flow case, resistance measurements must be specified in terms of the particular flow rate.
    • Physiologically, during breathing, airflow resistance will depend on the air flow rate assuming turbulence.
  • For turbulent flow, a nonlinear relationship exists between flow and pressure as shown below:

 

 

 

Occusion, Resistance and Pressure (courtesy of Dr. Rod Nave, Georgia State University) (http://hyperphysics.phys-astr.gsu.edu/hbase/hframe.html)
  • The  equation = Pd4 / (128 l ) (Hagen-Poiseulle equation) illustrates the important relationship between flow rate and pressure.  Note the direct proportionality-that is, if all other things are equal, to increased flow rate requires a directly proportional increase in pressure.
  • Now, using an example provided by Rod Nave, Ph.D., consider the case in which the areas the need for a significant increase in blood volume flow rate in a so-called "flight or flight" situation.  
    • If a 5-fold increase in blood flow were required in this could be accomplished only by increasing blood pressure, then the pressure would have to be increased from a nominal 120 mm Hg to 600 mm Hg, a clearly impossible physiological response.
  • By contrast, vasodilation of only 50% is sufficient to accomplish the fivefold increase in flow.  Note in the equation above the fourth-power dependency of diameter (or radius); so, going from 1r to 1.5r would provide (1.5)4 or 5.06 increase in volume flow rate.  Furthermore, a 50% vasodilation is physiologically reasonable.
  • Calculations

 

The  figures and Poiseuille's Law material below, courtesy of Dr. Rod Nave, Georgia State University

Calculations

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Bernoulli's theorem

  • "The Bernoulli Equation can be considered to be a statement of the conservation of energy principle appropriate for flowing fluids."



Bernoulli's Calculation
  • "The calculation of the "real world" pressure in a constriction of a tube is difficult to do because of viscous losses, turbulence, and the assumptions which must be made about the velocity profile (which affect the calculated kinetic energy). 
  • The model calculation here assumes laminar flow (no turbulence), assumes that the distance from the larger diameter to the smaller is short enough that viscous losses can be neglected, and assumes that the velocity profile follows that of theoretical laminar flow. 
  • Specifically, this involves assuming that the effective flow velocity is one half of the maximum velocity, and that the average kinetic energy density is given by one third of the maximum kinetic energy density.
  • Now if you can swallow all those assumptions, you can model* the flow in a tube where the volume flowrate is = cm^3/s and the fluid density is = gm/cm^3. For an inlet tube area = cm^2 (radius =cm), the geometry of flow leads to an effective fluid velocity of =cm/s. Since the Bernoulli equation includes the fluid potential energy as well, the height of the inlet tube is specified as = cm. If the area of the tube is constricted to =cm^2 (radius = cm), then without any further assumptions the effective fluid velocity in the constriction must be = cm/s. The height of the constricted tube is specified as = cm.

  • The kinetic energy densities at the two locations in the tube can now be calculated, and the Bernoulli equation applied to constrain the process to conserve energy, thus giving a value for the pressure in the constriction. First, specify a pressure in the inlet tube:
    Inlet pressure = = kPa = lb/in^2 = mmHg = atmos.
    The energy densities can now be calculated. The energy unit for the CGS units used is the erg.
Inlet tube energy densities
Kinetic energy density = erg/cm^3
Potential energy density = erg/cm^3
Pressure energy density = erg/cm^3
Constricted tube energy densities
Kinetic energy density = erg/cm^3
Potential energy density = erg/cm^3
Pressure energy density = erg/cm^3
  • "The pressure energy density in the constricted tube can now be finally converted into more conventional pressure units to see the effect of the constricted flow on the fluid pressure:

    Calculated pressure in constriction =
    = kPa = lb/in^2 = mmHg = atmos.

    This calculation can give some perspective on the energy involved in fluid flow, but it's accuracy is always suspect because of the assumption of laminar flow. For typical inlet conditions, the energy density associated with the pressure will be dominant on the input side; after all, we live at the bottom of an atmospheric sea which contributes a large amount of pressure energy. If a drastic enough reduction in radius is used to yield a pressure in the constriction which is less than atmospheric pressure, there is almost certainly some turbulence involved in the flow into that constriction. Nevertheless, the calculation can show why we can get a significant amount of suction (pressure less than atmospheric) with an "aspirator" on a high pressure faucet. These devices consist of a metal tube of reducing radius with a side tube into the region of constricted radius for suction."R. Nave

    "*Note: Some default values will be entered for some of the values as you start exploring the calculation. All of them can be changed as a part of your calculation." R. Nave


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