A fan (five blades, each blade = 225 mm)
diameter = 600 mm   
Speed = 2850 RPM 
Air flow = 4.5 m^3/s
Air temperature = 50 C
Pressure in inlet of the fan = 1 atmospheric Pressure
Operate at 100 m above sea level
 
I want to know the thrust (force) that this fan is making and how to calculate it.

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Request for Question Clarification by andrewxmp-ga on 29 Jun 2004 20:44 PDT

I definitely do not know how to go about this....but after researching
it for a while I did find out the following:

a) this is VERY complicated....not simple plug-and-chug from intro
physics class where you just multiply some variables together and get
some number for "force".  It requires a knowledge of fluid dynamics,
and some crazy mathematics to boot.  My own intuitions tell me that
even if you could theoretically calulate it, the value would not be
correct in practice, because there would be all kinds of weird thigns
like turbulence that would be nearly imossible to factor in.  I found
some simple computer programs that might be able to do it for you, but
they rely on other factors....which brings us to:

b) one of the most important things you will need is the shape of the
fan blades, and ultimately their areas (which could probably be found
from the shape and size).  You will also definitely need to know their
angle of attack (relative to the horizontal, or some other absolute
value).

c) there arae probably many other factors that are necessary as well. 
You said this is a "fan" so we can assume the blades are flat. 
However, many "fans" are like propellars in that the blades are
wing-shaped.  If that is the case here, then this becomes many times
more difficult, as an exact 3D representation of the blade shape would
probably be needed, along with some heavy-duty computer-based
processing to do the related calculations.

d) Am I looking too much into this?  Any engineers out there with some
experience along this topic?  It's pages like the following that make
me think it's way complicated.  (btw, the computer programs i was
referring to are those laced at the bottom of this page).
[ http://www.ae.su.oz.au/aero/propeller/prop1.html ]

Good luck....

-Andrewxmp

There are many computational, experimental, and analytical ways of
obtaining thrust from fans.  Take a deep breath - I'm going to quickly
give you ideas of all three.

It is important to know if the fan is in a duct or in a free-air environment.

1.  Blade element theory as given in the clarification was designed
for making new airfoil shapes.  Yes they have designed fan shapes
using this method.  However it is not accurate for determining
engineering data - I have tried designing airfoils with BET and it can
give approximate designs. Today noone uses BET for design unless its
in the classroom.
    The easiest way to figure out the thrust is with the Reynold's
transport thereom (analytical method).  You can go to any library,
pick up a book on Fluid Mechanics and use this theory.  You have all
the data you need to make a calulation.  It would be best if you knew
the AVERAGE inflow velocity and AVERAGE outflow velocity, The diameter
of the fan, and the density.  (you just don't have the inflow velocity
unless you assume it is zero upstream)

2.  You can use a computational method.  This would be a very
expensive simulation.  Depending on the details of the simulation you
can decide to use anythhing from a simple Euler calculation to a full
navier stokes simulation with turbulence modeling.  However your
simulation could only use what we call a zero equation turbulence
model... currently when we figure out the flow fields past aircraft we
use this class of turbulence models because of the lack of computer
power.  Its all very complicated... :(  I recommend checking out
www.cd-adapco.com or www.fluent.com for information on commercial
software packages that can perform these things.  Although they are
general codes they may perform a wide range of simulations (jack of
all trades syndrome).

3.  Experimental - You can mount the fan in a laboratory and put a
force transducer on the mount.  Turn on the fan and it will measure
the force.  That is the easiest way to measure the thrust!  The same
technique is done to measure the force of the new hybrid rocket
engines. :)
however if you don't have a force transducer (they aren't expensive)
then you can put the fan in front of a hanging board.  Hang the board
from the cieling.  Make sure that it swings freely.  Put the fan in
front of the board and turn it on.  You can then measure the average
angle that the board has turned away from the verticle (fan off)
position.  Using this angle and the Reynol'ds transport thereom you
can find the force to a pretty accurate level.

good luck.

there ARE a lot more "dimensions" necessary for a more exact answer,
but if you want a ballpark figure, ThrustHP is a small piece of
freeware for modelers that will actually allow any diameter prop to be
input... not POSITIVE of the math behind it, some people say it's
accurate and a few don't... some basic inputs, ie 24in diameter, pitch
of 15(guess) 2 blades(and extrapolate to 5) rpm of 2850, and i use
master airscrew for the prop, yields an efflux velocity of 40.48 mph,
.823hp (*2.5 for 5 blades for 2.06 hp) and 7.63 lbs thrust (*1.9 for 5
blades for 14.5 lbs)... and know that an increase in pitch increases
efflux and hp linearly, and an increase in rpm increases; efflux by
(rpm2/rpm1), thrust by (rpm2/rpm1)^2, and hp by (rpm2/rpm1)*
2^(rpm2/rpm1), and an increase in diameter increases thrust and hp by
(dia2/dia1)^4... the diameter one i think is where the math might not
take into account an increase in chord as well as an increase in span,
so it isn't really working for ducted fans if that's your app (ie,
hovercraft/ helio) well, good luck, and if you find some software that
does it better, lemme know :D ps, there's also a sight for a guy who
converted a varieze plane to rotary engine and rear ducted fan, and he
has lots of sets of data on his sight, but it's more like a 37.5" fan,
http://www.bridgingworlds.com/DUCKT.HTM

We have the temperature, atmospheric pressure, and air flow, so do we
need to know anything about the blades? I don't know the formula, but
wouldn't the mass of air being blown gives us all we need to derive
thrust?

I agree with nikyzf-ga.  There appears to be enough information here
to solve the problem.

First, use the information given in the problem to establish that the
blades of the fan are subsonic.  From the information given, the tips
of the blades are moving at about 90 m/sec, so we establish that the
blades are moving much slower than the speed of sound (330 m/sec). 
Whatever dynamics are at work, we don't need to consider supersonic
effects, pressure waves, etc.

Ignoring the question of blade profiles (for which we don't have
enough information) for the moment, consider the inlet pressure.  If
the inlet pressure of 1 atm is accurately measured, we might try to
solve the problem this way:

From a table at http://www.usatoday.com/weather/wstdatmo.htm we can
get the standard atmospheric conditions for 15C and sea level:

Height Temperature	   Pressure	  Density
(m)	        (C)         (hPa)     (kg/m3)
0000       15.0         1013       1.2

1013 hPa is exactly one standard atmosphere.

Now I take a leap of faith and assume that atmospheric pressure should
obey the ideal gas law, and make a correction for the pressure we lose
by climbing 100m.

If those assumptions are valid, at 100m altitude and a temperature of
50C, the pressure should be something like:

(1*atm - (100m * 1.2 kg/m^3) * g) * 323/298 = 1.11 atm

where 
  1*atm is standard pressure
  100m is our altitude
  1.2 kg/m^3 is the approximate density of air at sea level
  323/298 is the difference in absolute temperature from 15C to 50C
  g is the gravitational constant for earth 9.8 m/sec^2

Then, since the problem states that the inlet pressure is only 1.0
atm, we can assume that the fan has created a pressure differential of
0.11 atm.  Multiplying this pressure by the area of the fan, we get:

0.11*atm * 0.283 m^2 = 700 lbf.

I haven't been very rigorous here, but then that's what you get for a
free answer.  Perhaps a researcher interested in grabbing the prize
could verify or discredit the above?  I must admit, I don't know
whether we should double the force above to account for both the
vacuum in front and increased pressure behind the fan, or if we should
only count it once.

Dividing the flow rate of 4.5 m^3/sec by the area of .283 m^2, we have
a linear velocity of about 16 m/s or 36mph.  Assuming we only count
the pressure once, for the given force of 700 pounds, you would need
to drive 70hp into the fan.  Otherwise, double the horsepower too.

Is there a physics professor out there with the answer?

78 Newtons or 17.5 lb.

All you need to know is A) the rate at which air is being moved, in
terms of mass per unit time and B) the velocity with which the air
leaves the fan.  Multiply these together, and you have your thrust. 
A) is given by 4.5 m^3 / s times the density of air at 50C and 100 m
above sea level, which is about 1.09 kg/m^3.  So air is blown at a
rate of about 4.9 kg/s.  To get the velocity, you divide the volume
rate of flow by the area of the fan, which is pi*(.3)^2 (the radius of
the fan is 0.3 m).  So the velocity is about 15.9 m/s.  4.9 kg/s *
15.9 m/s gives 78 kg m/s^2, or 78 N.

The way to understand this is to realize that force is just the rate
of change of momentum with time.  So if you take the momentum given to
air in a time interval, and divide the the length of that time
interval, you get the force.  The momentem given to the air is just
the mass of the air times the velocity of the air.