drew
|
|
Wind propulsion calculations
| 
Aug 11 20:41 UTC 2000 |
[This item has some diagrams and graphs, which I hesitate to upload for
policy reasons, along with a couple of Lotus 123 worksheets. I'll send
the whole .ZIP file (about 73K) on request. Anyways, here's the main
text file:]
In this item I will attempt to analyse the expected performance of
both a windmill-rotor and an airfoil-sail on an arbitrary vehicle. It
is customary in the aeronautical sciences to work in the frame of reference
in which the vehicle is stationary, so we will use that convention here.
Most of the calculations will be in the vehicle FOR, with the ground (or
water as the case may be) and the air in motion - hopefully with different
vectors.
Variables used:
alpha Angle of attack of airfoil
beta Angle between airfoil force and vehicle FOR wind vector
Cl,Cd,Cf,Ct Dimensionless coefficients of L,D,F, and T
CdA Effective area of hull exclusive of wind components
D Drag of airfoil or rotor
F Total force of airfoil or rotor
k Ratio of vehicle speed to wind speed
L Airfoil lift (perpendicular to vehicle FOR wind vector)
phi Wind direction from forward, ground FOR
Ro Air density
S Area of airfoil or rotor
T Thrust (net forward airfoil force component)
theta Wind direction from forward, vehicle FOR
u Wind speed, vehicle FOR
ue Wind speed exiting a rotor, vehicle FOR
v Vehicle ground speed
w Wind speed, ground FOR
W' Power (net, gross, reqd, etc.)
Z Rotor power-to-area-speed-cubed ratio
The first thing that must be realized is that the wind direction (and
speed) will be different in the vehicle and ground FORs. The vehicle's
ground velocity and that of the wind add vectorially (wvectors.gif), with
the result that the wind direction swings toward the bow with increasing
ground speed. (Note 1.) The wind vector in the vehicle FOR is
w [ (k + cos(phi)), sin(phi) ].
The absolute speed and direction can easily be converted:
u = w ( k^2 + 2 k cos(phi) + 1 )^0.5
theta = atan2( y = sin(phi), x = k + cos(phi) ). (Note 2.)
wspeed.gif Vehicle FOR wind speed versus ground FOR wind direction
wdir.gif Ground FOR versus vehicle FOR wind direction
Now we will consider a rotor-equipped vehicle (wrotor.gif). Let there
be a vane on the rotor so that it tracks with the vehicle FOR wind direction.
The power produced by such a machine is proportional to intake area and
the cube of the wind speed:
W' = 0.5 Ro u^3 S Z.
In producing this power, the rotor can be expected to encounter a force
along its axis due to the change in momentum of the air flow. The air
in providing its energy slows down to some speed ue:
W' = 0.5 Ro S ( u^3 - ue^3 ) = 0.5 Ro u^3 S Z.
u^3 - ue^3 = Z u^3.
ue^3 = u^3 ( 1 - Z ).
F = (Ro S u) (u - ue) = Ro S u^2 ( 1 - ( 1 - Z )^(1/3) ).
The sideways component of this force will presumably be resisted by
the wheels (or keel); while the forward component will contribute to the
resistance to motion over and above the hull drag.
D = F cos(theta).
This force will contribute to the power required to keep the vehicle
moving, reducing the net power accordingly. The drain is proportional
to the fore-aft component and also to the ground speed, since the rotor
would be supplying its total power to wheels/screws.
W'net = W' - D v.
The power required to push the vehicle itself, exclusive of the rotor,
is likewise proportional to ground speed (for the same reason), and also
to the fore-aft component of its own drag.
W'req = 0.5 CdA Ro u^2 cos(theta)^2 v.
We want our final equations in terms of the wind speed; so we will
substitute some terms:
W' / w^3 = 0.5 Ro S Z ( k^2 + 2 k cos(phi) + 1 )^(3/2).
D / w^2 = Ro S cos(theta) (k^2 + 2 k cos(phi) + 1) (1 - (1 - Z)^(1/3)).
W'net / w^3 = W'/ w^3 - k D / w^2.
W'req / w^3 = 0.5 CdA Ro cos(theta)^2 k (k^2 + 2 k cos(phi) + 1)
It is evident now that all power parameters are directly proportioal to
windspeed cubed (and all force parameters proportional to windspeed squared).
We can now write our spreadsheet.
rotor.wk1: Wind rotor calcs.
1. Set course in degrees from windward, and if desired, wind speed.
2. Execute the macro ADJUST or \A. This will adjust k for zero net power.
3. Adjust K manually to calculate additional power required or surplus.
4. Macro MAKETABLE or \T generates a table of speed versus course.
Choices of values:
Z: The windmill power constant was calculated from a wind power table in
Mother Earth News _Handbook Of Homemade Power_ (p 141) by averaging all the
values of W'/(0.25 Ro pi D^2 V^3). A copy of this table with the averaging code
is provided in windtabl.wk1.
CDA_HULL: I used a value corresponding to 100 watts at 7 meters per
second (15.6583 MPH), as I'm mostly curious about applications on otherwise
unpowered craft; and this seems to be the most reasonable estimate I've
been able to come up with for bicycle performance. I would expect other
pedal powered vehicles to show similar performance. (Note 3.)
S: I sized the rotor for 2 meters in diameter, in order to fit it
comfortably into a single traffic lane. There should also be ample clearance
under most bridges.
W: Winds in the 10-20 MPH range seem to be the most common these days.
So 7 m/sec (15.6583 MPH) is a reasonable assumption. Since my intended
output is in multiples of windspeed, however, and since the windspeed
terms cancel out of the equations, it doesn't really matter what number is
entered here.
rotorpf.gif: Wind rotor performance, wind speed multiples versus direction.
----
Airfoil performance:
As in the case of the rotor, the effective wind vector shifts to
forward with increasing speed. The principal forces are the lift and
the wing drag, which are really components of a single force vector.
The useable thrust is the fore-and-aft component of this force (wwing.gif).
F = ( L^2 + D^2 )^0.5
therefore
Cf = ( Cl^2 + Cd^2 )^0.5
and the angle between this force and the air stream is given by
beta = atan( Cl / Cd ).
It is also evident that
T = 0.5 Ro u^2 S Ct.
T = 0.5 Ro u^2 S Cf cos(pi - theta - beta).
T = 0.5 Ro w^2 S Cf (k^2 + 2 k cos(phi) + 1) cos(pi - theta - beta).
T / w^2 = 0.5 Ro S Cf (k^2 + 2 k cos(phi) + 1) cos(pi - theta - beta).
This propulsive force is compared with the hull drag force:
Dhull = 0.5 Ro u^2 cos(theta) CdA.
Dhull = 0.5 Ro w^2 cos(theta) (k^2 + 2 k cos(phi) + 1) CdA.
Dhull/w^2 = 0.5 Ro cos(theta) (k^2 + 2 k cos(phi) + 1) CdA.
Again, the prominent forces can be expressed in terms of the amount
per unit of windspeed.
(T - D) / w^2 = 0.5 Ro ( S Ct - CdA cos(theta) ) (k^2 + 2 k cos(phi) + 1).
The net power surplus can now be directly calculated:
W'surplus = (T - D) w (k^2 + 2 k cos(phi) + 1)^0.5.
As to figuring out Cl and Cd, I cheated and downloaded SNACK
(www.dreesecode.com) which can calculate coefficients for an airfoil of
arbitrary NACA number (though it balked at zero thickness). I ended up
trying out five different airfoil sections:
NACA-0001 As close to a flat plate as the software could work with.
This one was, surprisingly, the most resistant to stall.
Even at 20 degrees angle of attack, the maximum that
SNACK can handle, this section still produced a lift
coefficient of over 1.9.
NACA-0010 A standard symmetric wing. It was more prone to stall,
but did considerably better at providing net sailing
thrust than the flat wing.
NACA-9301 Here I sacrificed symmetry in the hope of getting a bit more
thrust. It helped a little. The 1% chord thickness, again
the smallest that the software could handle, is to enable
using a sheet, which can be reversed on the opposite tack,
in place of a rigid airfoil. The sheet would be given its
profile by a pair of restraining frames or booms.
Joukowsky .300-.010
Again an ultra-thin airfoil to enable flexibility. I
decided here to depart from the NACA series as these wings
didn't provide any more lift than the Cl=2 range of the
others. This wing is extremely sensitive to angle of attack,
stalling a couple of degrees above its best L/D.
Joukowsky .300-.010 with flap
Just for the hell of it I turned on the Flap Bender. I got
a wing that was both virtually immune to stall and provided
grossly high lift coefficients. I decided to include it as
it gives the best performance of them all, allowing close
hauling as low as 10 degrees off the wind (barely moving)
and reaching as fast as six times the wind speed. It does
the wind's speed on a 20 degree tack. (Note 4.)
wings.wk1: Airfoil sail performance.
Works much the same as above; adjust K for any given course and note
power surplus (or deficiency). The same macros are available for solving
for zero power/net drag and generating a table.
Choices of values:
CDA_HULL and W: Same as for rotor.
S: I wanted a comparison of sails/wings to the rotor, so it seemed
appropriate to use the same area.
wingpf.gif: Airfoil performance, multiples of windspeed versus direction.
Comparison:
The figure of 20 degrees offwind is a bit small an estimate for where
sails begin to outperform the rotor. Even the Joukowsky-with-flap airfoil
only does the wind's speed on this course, while the rotor does almost 1.37
times the wind. Nevertheless, all of the airfoils are far superior to the
rotor on most other courses.
The rotor, in fact, performs miserably enough overall so as not to be
worth building. Its maximum speed barely gets above 1.39 times windspeed,
and is closer to 1X most of the time. It is, however, almost uniformly
consistent in performance in all directions.
Notes:
1. My guess is that this is the reason why the wind powered electric car
I asked about has to get up some speed before the airfoils can provide
enough thrust to keep the car moving.
2. The ATAN2 function takes the x and y argument in different orders depending
on the software. Lotus-123 expects the x co-ordinate first, while Matlab
expects the y term first. Refer to the documentation for your software.
3. I actually had in mind equipping one of Greenspeed's trikes with some
sort of windcatcher.
4. This performance does sound way too good to be true. It may be wondered
just where the energy comes from to provide that much of a push. The
airstream would, at least, have to be tapped a considerable distance
from the vehicle itself. The results for this airfoil should, in any case,
be treated with some skepticism.
|