Russell McMahon wrote:

> There are two identical ducts in series with an essentially lossless =

> section in between. .
> =

> Per duct:
> =

> Duct is a very wide and shallow rectangle (a slot in cross section )
> =

> - Duct length is (only) 400mm
> - Duct width is 350mm.

I think most formulas for duct resistance will fail here. If I understand
fluid dynamics correctly, this is a length/width ratio that creates a flow
situation that's quite different from the one that the normal resistance
formulas assume (they assume much longer than wide). Normal formulas mostly
assume a laminar flow profile, which can only be expected to start after a
length of about 10 diameters.

> - Air flow is about 10l / second

Here is a formula for round pipes:

(1) delta p =3D lambda * (l/d) * (rho/2) * w^2

  delta p: pressure difference [N/m^2]
  lambda: friction coefficient [1]
  l, d: length, diameter [m]
  rho: specific mass [kg/m^3]
  w: flow speed [m/s]

For rectangular ducts, an equivalent diameter can be calculated:

(2) d equ =3D 2 * a * b / (a + b)

  d equ: equivalent diameter [m]
  a, b: sides of the rectangle [m]

You now can express d in (1) in terms of a and b from (2), where a is
constant and b is what you want. You have delta p and l given. rho is known
(at least the range, depending on your temperature range). w can be
expressed as a function of the volume flow (given) and cross section
(composed of the given a and the result b). That leaves us lambda as only
unknown.

lambda (probably something like "friction coefficient" in English) depends
on the flow profile (laminar, turbulent -- in your case the latter, given
your short length), the smoothness of the inner surface and the Reynolds
number.

For turbulent flow and ideally smooth surfaces is

(3) lambda =3D 0.3164 * Re^(-0.25)

(If you want to go into more detail here and possibly take the surface
smoothness into account, I have other formulas for this. But the principle
remains the same; you'd only have one more variable in here and much more
complex formulas.)

The Reynolds number is

(4) Re =3D w * d / nu

  w: flow speed [m/s]
  d: (equivalent) diameter [m]
  nu: "kinematische Z=E3higkeit" (don't know the English name) [m^2/s]

Now we can insert lambda from (3) in (1) and substitute Re with (4). In
(4), we can substitute w and d like before (with a, b and the given volume
flow), and nu is ~13 at 0=B0C, ~15 at 20=B0C, ~17 at 40=B0C (for air at 1 b=
ar, in
10^-6 m^2/s).

I didn't do all the substitutions, but you'll end up with a formula that
you can solve for b (height of the duct), dependent on only known values.
You'll get even a notion how much the various parameters influence the
result; this could potentially be more interesting than the actual result
itself.

Gerhard

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