What determines the dominant pressure-flow relationship for a gas across a flow restriction?If one measures the pressure drop across any gas flow restriction you can generally fit the relationship to Δ P = K 2 Q 2 + K 1 Qwhere Δ P is the pressure drop and Q is the volumetric flowand what I've observed is that if the restriction is orifice-like, K 2 &gt;&gt; K 1 and if the restriction is somewhat more of a complex, tortuous path, K 1 &gt;&gt; K 2 and K 2 tends towards zero.I get that the Bernoulli equation will dominate when velocities are large and so the square relationship component. But what's determining the K 1 component behavior? Is this due to viscosity effrects becoming dominant? Does the Pouiselle relationship become dominant?

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What determines the dominant pressure-flow relationship for a gas across a flow restriction?If one measures the pressure drop across any gas flow restriction you can generally fit the relationship to  Δ  P  =      K    2        Q    2    +      K    1    Qwhere   Δ  P is the pressure drop and   Q is the volumetric flowand what I&#039;ve observed is that if the restriction is orifice-like,       K    2    &amp;gt;&amp;gt;      K    1   and if the restriction is somewhat more of a complex, tortuous path,       K    1    &amp;gt;&amp;gt;      K    2   and       K    2   tends towards zero.I get that the Bernoulli equation will dominate when velocities are large and so the square relationship component. But what&#039;s determining the       K    1   component behavior? Is this due to viscosity effrects becoming dominant? Does the Pouiselle relationship become dominant?

Darion Sexton · Accepted Answer

Solutions that depend upon   Δ  P  ∝      Q          2       are related to the normal aerodynamic drag, which just depends upon the ram pressure,   ρ         U          2            /    2, a drag coefficient,       C          d      , and the cross-sectional area affected,   A, or:                    (1)                              F                      d                          =                  1          2                         ρ                           U                      2                                             C                      d                                   A            Solutions that depend upon   Δ  P  ∝  Q are dominated by Stokes drag, where the drag force is given by:                    (2)                              F                      s                          =        6                 π                 η                 r                 U            where   η is the dynamic viscosity,   r is an effective radius or scale-size of the object, and   U is the bulk flow velocity relative to the object.&quot;I get that the Bernoulli equation will dominate when velocities are large and so the square relationship component. But what&#039;s determining the       K          1       component behavior? Is this due to viscosity effrects becoming dominant? Does the Pouiselle relationship become dominant?&quot;Yes, it&#039;s an effective viscous effect. When the restriction or obstacle are a single shape and flow is relatively steady, laminar, then Equation 1 above dominates. If the flow path has multiple turns or the fluid is highly viscous or the flow is turbulent, then Equation 2 above dominates.Stokes drag (Equation 2) arises from the strain tensor in the Navier-Stokes equations while the aerodynamic drag (Equation 1) arises from an approximation for the pressure tensor from Bernoulli&#039;s equation.The separation between the two is approximated by the Reynolds number given by:                    (3)                    R        e        =                              ρ                         U                         L                    η                    where   ρ is the mass density of the fluid and   L is the characteristic scale size. Turbulent flow onsets for high   R  e, but the onset depends on whether the fluid flows around an obstacle (e.g.,   R  e  &amp;gt;      10          5      ) or through a pipe (e.g.,   R  e  &amp;gt;      10          3      ).There simple limits/examples to consider when Equations 1 and 2 dominate. If you try to drag a stick/rod through a thick, viscous fluid like honey then obviously Equation 2 is the relevant drag force. If you drag the same stick/rod through the air as fast as your arm can move, then Equation 1 will dominate.If you have a fluid that falls in between, then both equations can play significant roles. For instance, air is generally not tremendously viscous but if you force it through a small pipe with multiple bends at high enough speeds, it can behave like a viscous fluid. Unfortunately, the Reynolds number is not an exact parameter, in that it does not state that at the value   R  e  =  X then the flow is exactly laminar.

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