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Tube Length and Air Flow
Name: JR
Status: Educator
Grade: 6-8
Location: IL
Country: United States
Date: May 2007
Question:
What is the effect of tube length on air flow at a given
pressure. Example: If a tube of X length flows Y air at Z pressure.
What will the flow Y be if the tube length is 10 times X with the
same pressure.
Replies:
As with so many things, the relationship depends on how precise you want
your answer to be.
If you are looking for a precise mathematical relationship to calculate air
flow in a pipe, things get very complex very quickly. There is no one single
X, Y, Z relationship that can be used universally. There is a thing called
'turbulence'. At low enough velocity, fluids (such as air or water) flow
uniformly (called 'laminar' flow -- 'in layers'). As the fluids speed up
though, they start to flow chaotically, forming vortexes and eddies as they
move down the pipe ('turbulent'). The equations for laminar flow are quite
different than turbulent flow; the relationships for X, Y, and Z depend on
what X, Y, and Z are. And, the flow can transition from one regime to the
other during flow, so you might have to make three calculations: one for
each regime, and one for the transition phase. Several practical factors
also affect your answer. The geometry of your pipe, the roughness of the
sides, the purity of the air (is it oil-free?) all affect your answer. Thus,
you not only need X, Y, and Z, but also A, B, C, D, and E (so to speak).
Want more? Air (unlike water) is compressible. At high pressures, and high
pressure drops, compressibility affects flow. Also, as air flows, compresses
or decompresses, it can change temperature, which affects pressure and
therefore flow.
In short, a precise, predictive calculation is actually quite challenging.
Fortunately, there are tons of resources on-line that can perform
calculations for you. If you want to do it yourself, you can find several
different equations (they are different because they rely on different
assumptions -- for example, one might assume constant temperature). Google
'air flow in a pipe' or 'air flow in pipe calculator' and you'll have
several options.
If you're looking for a *very* rough estimate, you can assume pressure drop
to be linear across a straight pipe. In other words, if your inlet pressure
is 10 atmospheres at the start, and 9 atmospheres at X, then it would be 8
atmospheres at 2X, 7 atmospheres at 3X, etc. This is a rough estimate only
-- there are many factors that do not scale linearly. This is a much better
estimate for water than it is for air.
Hope this helps,
Burr Zimmerman
What you are asking is: "What is Poiseuille's Law?" If you do a Google (or other
search engine) search, you will find numerous sites, deriving and explaining P's
Law at any level of mathematical sophistication you care to use. Two points worth
making is that the resultant equation is different for "slow" and "fast" flow,
for turbulent flow, and for bends in the tube. So there is not a "universal"
equation. A couple of sites to get you started are:
http://en.wikipedia.org/wiki/Poiseuille's_law
http://en.wikipedia.org/wiki/Fluid_dynamics
One has to be careful about whether the fluid is compressible (air) or
incompressible (water), and whether viscosity is important. See: Compressible
vs incompressible flow
http://en.wikipedia.org/wiki/Fluid_dynamics
A fluid problem is called compressible if the pressure variation in the
flowfield are large enough to effect substantial changes in the density
of the fluid. Flows of liquids with pressure variations much smaller than
those required to cause phase change (cavitation), or flows of gases
involving speeds much lower than the isentropic sound speed are termed
incompressible.
For flow of gases, to determine whether to use compressible or incompressible
fluid dynamics, the Mach number of the problem is evaluated. As a rough guide,
compressible effects can be ignored at Mach numbers below approximately 0.3.
For liquids, whether the incompressible assumption is valid depends on the fluid
properties (specifically the critical pressure and temperature of the fluid) and
the flow conditions (how close to the critical pressure the actual flow pressure
becomes). Acoustic problems require allowing compressibility, since sound waves
can only be found from the fluid equations which include compressible effects.
The incompressible Navier-Stokes equations can be used to solve incompressible
problems. They are simplifications of the Navier-Stokes equations in which the
density has been assumed to be constant
Vince Calder
The formulas for fluid flow follow pretty much the same math as the formulas for
electrical current. Remember Ohm's law, V = IR, where V is voltage across a
component, I is the current through the component, and R is the resistance of
the component? Same for fluid flow, except use flow rate (say, gallons per
minute) instead of current and pressure instead of voltage. Resistance follows
the same rules; a 20-foot pipe has twice the resistance of a 10-foot pipe of the
same diameter.
So in your case, you are comparing pressure drop Z, flow rate Y, length X to
pressure drop Z, flow rate y, length 10X. We can say Z = (flow rate)
(resistance) in both cases, and we want to find the flow rate y. Let's
say the resistance in case 1 is R; the resistance of a 10-times longer
pipe will be 10R. So we set up for equal pressure drops:
Z = Z
YR = y (10R)
Now we need to solve for y:
y = YR/(10R) = Y/10
So not surprisingly the flow will be 1/10 as much for the same pressure
drop.
Richard Barrans
Department of Physics and Astronomy
University of Wyoming
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