Yesterday Mr. Toto suggested that the Hagen-Poiseuille law could be
used to calculate the relative flows between player piano bleeds of
differing sizes.

This law can be reduced to the conclusion that the ratio of flow
between two different flow conduits is proportional to the ratio of the
fourth power of their respective radii.  In other words, increasing the
size of a hole by 10% increases the flow through it by 46%.

However, this won't work for holes in a thin plate, like a bleed.  For
the Hagen-Poiseuille law to apply, the flow must be 'fully laminar'.
This requires a hole that is quite long compared with its diameter,
having a very small flow through it, and a smooth transition from the
chamber supplying the fluid into the channel, undisturbed by any nearby
bends or sharp edges.

Player bleeds (indeed anything in a player, including even the valves),
always have 'turbulent flow', for which completely different calcu-
lation methods apply.

It is not possible here to go into the methods for determining that
such flows are turbulent; indeed such methods don't even exist for
anything much more complex than a simple, straight channel.  But one
can apply this practical test: "If you can hear it, it ain't laminar."

The problem is that textbooks often spend a lot of time on 'laminar
flow' examples, because they are 'elegant', and the relationships that
govern them can be derived strictly from theory.  In real life, laminar
flow, except on surfaces like airplane wings or bearings, is extremely
rare.  Empirical flow calculations, 'back-derived' from measurements
made on real systems, almost always have to be used in industrial
practice.

One of the basic conclusions one can draw from all of this practical
work is the 'square-root law' which governs almost any flow through
any restriction:

Flow = some constant K, times the square of the hole's dimension,
times the square-root of the pressure differential across that hole.

But 'K' is different for each different physical situation, and has
to be determined experimentally.  In other words, the ratio of flow
through two otherwise similar orifices is proportional to the ratio
of the squares of the two diameters.

Quoting "Crane's Handbook of Flow Through Pipes, Valves, and Fittings",
the relationship for calculating flow through a flat-plate orifice is:

  q' = 678 ( Y * do^2 * C ) * (( DP * P ) / ( T * Sg ))^0.5

Where

  q'm = flow in 'cubic feet per minute',
  Y = expansion factor, above 0.98, say 1 at the moderate
      absolute pressure ratios in a player piano,
  do = hole diameter, in 'inches',
  C = Orifice Flow Coefficient, about 0.53, when the hole diameter
      is less than 0.2 of the channel diameter, which is usually
      the case for a bleed,
  DP = pressure differential across the orifice in PSI; in a player,
      the stack vacuum in IN-WC / 27.29; say 0.366 for 10"WC vacuum,
  P = upstream (atmospheric pressure) in PSI-absolute, 14.7,
  T = Absolute temperature, about 530 for a 70 deg. room,
  Sg = specific gravity of the fluid, 1.0 for air.

This reduces to:

    q'm = 359 * do^2 * Sqrt( 0.00102 * Stack-Vac )

in Cubic Feet per Minute, or, for a more useful result,

    q"s = 10339 * do^2 * Sqrt( 0.00102 * Stack-Vac )

in Cubic Inches per Second.

However, it must be noted that this formula must be used with caution.
The experimentally determined "C" can vary all over the place, from 0.1
to 0.5, depending on the finish and concentricity of the hole, and the
physical character of the channel in which the orifice is placed.
However, when comparing two orifices which are alike in everything but
the hole diameter, the comparative result should be reliable.

Richard Vance

 [ Editor's note:
 [
 [ MMDer Philip Dayson did much experimental work with the typical
 [ flat plate bleed which occurs in the pneumatic musical instruments.
 [ I believe he concluded that the flow was turbulent even at the
 [ low pressures and velocities encountered in pianos and organs.
 [
 [ In contrast, modern air flow measuring instruments are available
 [ which have a laminar-flow restriction ("bleed").  This complex
 [ "cluster of tubes" creates a pressure differential in linear propor-
 [ tion to the air flow; the pressure drop is easily and precisely
 [ measured with a manometer.
 [
 [ MMDer Vicki Webb reported testing new organ pipe designs at the
 [ "smoke table" which is used to locate turbulence problems on scale
 [ models of new automobiles.  I heard a couple of the pipes: sweet-
 [ sounding, indeed!
 [
 [ I sure hope we hear more from all you folks about music and air
 [ flow, and how the traditional air-flow theory applies to practical
 [ pneumatic valves and organ pipes.  :)
 [
 [ -- Robbie