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.