IE010301A
Chapter 1
Derivation Of The Basic Airflow Force Equation
The airflow over wind generator blades and its effect on them is
amenable, to a certain degree, to mathematical treatment, something known
generally as theoretical aerodynamics. It is important at the outset to make
assumptions and establish limits for any analyses to be undertaken. Here the
approach will be two-dimensional. No attempt will be made to provide
three-dimensional treatment although equations handed down by pioneers in the
field are available for this more elaborate work. This does not, as we shall
see, reduce the quality or impact of this analysis significantly, for the
two-dimensional case is sufficiently accurate to do a quite acceptable job of
looking at wind machine airflow dynamics. It also is much more heuristic,
something that is needed within the context of decisionmaking by organizations
beset by many extraneous factors other than engineering
considerations.
Not only this but previous work has not been
characterized by a fully satisfactory set of working formulas for important
parameters even at this late stage of the technology. Before an analysis of any
greater detail is attempted, despite the availabilities of advanced electronic
processing, the foundation stones, so to speak, must be aligned and adjusted to
better support what other material may follow.
Two important features of
what is included in this treatment are these. The flow of air is understood in
terms of its mass, something previously given little credit for and incorporated
within only flow volume and density terms, and the airflow deflection produced
in the flow by the airfoil is finally given recognition for its being the
primary vehicle for the transfer of momentum to the blade.
The equation
considered to be the most basic in what is to follow is:
Short and not very fancy, its simplicity belies its importance and
usefulness. It can be described as a statement of conservation of momentum in
the limit for discrete points of a flow continuum and can be referred to as just
a particular case of Newton's Law, F = ma. Note also that it is a vector
equation.
Here is a derivation of it, making use of integral
calculus:
In previous analyses within the field of wind energy aerodynamics this equation appeared in a different form. It was seen often as:
and this is not in a form that can be quite so readily applied to
airflow over a blade. Rather it was applied to the larger dimension of airflow
impacting what was termed the "actuator disk", otherwise known as the blade
swept area. So just by the few steps necessary to reformulate the expression in
a more useful way, greater applicability is obtained and, of some importance
here, it can be made to provide results for the more detailed case.
This
new approach deserves consideration as being distinctly different. Not only are
the mass of the air and the airflow deflection given added emphasis as mentioned
earlier but also the equation has the powerful feature about it that it is
insensitive to details of flow during midway points, returning a net force from
information about only the start of a flow process at an entry point and the end
of it at the exit.
This feature even has application for the case of the
determination of aerodynamic lift for aircraft wings, something arrived at
normally by the suggested process of applying the Bernoulli equation to
calculate a pressure decrement at each point along the length of the airfoil
chord, not a method that is useful or meaningful in ordinary practice. The
description of such an aerodynamic lift force provided below, consistent with this approach,
was excerpted from material dating back to concepts originating with the Wright
Brothers. It remains the view seen in almost all references to it even until the
present day, something with which everyone is familiar.
While a point or two can be made in its favor, this treatment of this
important concept leaves a lot to be desired. For one thing, the Principle of
Conservation of Momentum is not observed. For another, if the velocity over the
top of the wing is so much faster than that below, then why is it that the the
air joins together with itself at the same place at the trailing edge, something
that can be visualized so as to conserve mass in the frame of reference in which the air is still and
the aircraft moving. And if a reduced pressure region is created above the wing
with no vertical momentum change in the air, then why doesn't it draw down air
from the open, unrestricted area farther up with the same rather substantial
force that is keeping the craft in flight? The basic force equation derived
above requires that a change must happen to the airflow and, upon careful observation, it can be seen in the above pictorial representation that no change has taken place, either in
its velocity or in its direction.
If these rather unsettling words on
standard beliefs so widely held on this subject are thought to be overly incisive and
something that does no one any good, seeing the safety and convenience of the
age of flight, then consider this. When these somewhat misguided principles are
applied to the aerodynamics of wind energy, they may very well make a difference
of greater consequence. So it may be worthwhile to take a look this far back at
this what almost the entire world has been taking for granted for so long.
For a better understanding of the aerodynamics of wind energy blades is theorized to be available
and it is theorized to be right there before us as the few printed characters
assembled together into the basic airflow force equation derived
above.