3. The Science of Flight |
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The wings of an airplane lift the aircraft up by throwing a tremendous amount of air down. This basic fact about flying may not be obvious to everyone because air is invisible and so its movement usually goes unnoticed. Nevertheless, evidence of the air being displaced downward can be seen when an airplane flies over a cloud and in doing so it creates a trough of clear air cutting through the cloud. Likewise an airplane flying low over a runway or field will show the result of the downward breeze in the blowing dust and leaves as the airplane passes overhead. How the wings throw the air down is the key to understanding flight.
First thing to consider is whether wings generate lift by the bottom wing surface pushing the air down or is it the top wing surface generating the lift by pulling the air down. Conceptually it is easier to visualize the bottom wing surface lifting the plane up by pushing the air down. Essentially this is how a kite flies; the kite is lifted up by the bottom surface of the kite deflecting the wind down.
Many of the early aviation pioneers thought that this is how a airplane could fly. The wings of their aircraft were nothing more than large airtight fabrics that were intended to deflect the air downward as the propeller moved the plane forward. The idea that wing surface area equates to lift is a widely held belief even to this day. Aerodynamic engineers still calculate a plane’s wing loading – the weight of the plane divided by the area of the wing – as if the total surface area of the wings is all that mattered in determining the lift generated by the wings. But generating lift by deflecting air with a large surface is not how an airplane flies.
A major advancement in aviation came when a retired railroad engineer named Octave Chanute incorporated the same Pratt truss configuration used on railroad bridges to construct a long wingspan biplane glider. The long wingspan, in addition to the properly curved chambered profile of the wings, produced much more lift than the earlier wing designs that were based on maximizing wing area. The Wright brothers had the success of building the first controllable powered airplane by starting with Chanute’s wing as the central component of their design.
A properly cambered wing produces lift by the top wing surface pulling the air down. The wing is able to pull the air down because air has viscosity such that the air sticks to the surface of the wing and thus causes the air stream to follow the wing’s profile. This process of pulling the air down produces much more lift than the method of trying to generate lift by using the bottom surface to push the air down.
Because pushing down on the air produces so much drag, this is usually not a desirable way to produce lift unless the intent is for the airplane to lose air speed. Such is the case when a plane is landing and so planes could use this method for when they are landing if only they have extremely broad wings so that the pushing method could be effective. However most planes do not have extremely broad wings and so most planes do not use the bottom wing surface as a significant factor for generating their lift.
The exception to this rule is the supersonic aircraft that have been giving the extremely broad delta wings. Aviation engineers have discovered that it is easier to land supersonic aircraft by putting the airplane into a ‘power stall’ rather than try to land these fast jets in the conventional way. In a power stall the pilot puts the nose of the plane extremely high so that the huge surface of the broad delta wing is both pushing the air down to generate lift while it is pushing air forward so as to slow the aircraft down. Many birds will also land this way. Just before landing they take a more upright posture, flare their rear tail feathers, and allow their broad wings to push against the air for the last couple of seconds of their flight before they land. But other than these exceptions, the primary way that airplanes and birds generate lift is by the top wing surface pulling the air down.
For further evidence that it is the top wing surface that generates the lift please take notice that if an obstruction must be placed near the wing it is always place below the wing rather than above it. While obviously the landing gear needs to be placed below the wing it is conceivable that jet engines or military weaponry could be placed above the wing rather than below the wing. But these objects are not mounted above the wing because any object mounted this way would compromise that portion of the wing’s ability to generate lift.
It is because of a fluid’s tendency to stick to the surface that the wings are able to produce lift by pulling the air down. This phenomenon known as the Coanda effect is demonstrated by holding a smooth drinking glass horizontally so that it just touches the lightly flowing water falling from a faucet. Once the water comes in contact with the glass the stream of water follows the curvature of the glass instead of falling straight down.
Likewise the air stream adheres to the surface of a cambered wing. Just as the air flows over the thickest part of the wing it takes a downward turn so as to follow the profile of the wing. In taking this turn a force is being applied on the air pulling it down and likewise in accordance to Newton’s laws an equal and opposite force is being applied to the wing lifting it up.
During a flight it is occasionally desirable for the wings to produce more lift than what they normally produce while cruising in level flight. To produce this additional lift the rear tail elevator is adjusted to lower the tail while pointing the nose of the plane up. In this new position the front of the wing is higher than the back of the wing. This increased angle of the tilt of the wing or Angle of Attack AOA effectively increases the camber of the wing. With this increases wing camber the air flowing over the wing is more sharply pulled down as it follows the profile of the top wing surface. By increasing the tilt of the wings above the usual approximately six percent AOA the wings can generate about twice as much lift as usual but this comes at the expense of creating greater drag on the aircraft.
Besides creating greater drag a high AOA increases the risk of causing the wings to stall. A stall is the aerodynamic name given for when the air stream breaks free of its attachment to the top wing surface. We can use our water flowing from facet to observe what is happening during a stall.
We place a wing profile next to the water stream so that the fluid sticks to the surface and in so doing it is diverted from flowing straight down. As we turn the wing profile we can increase how far we divert the water from flowing straight down. But if we turn our wing too far the water stream breaks free from our model wing surface. This is what happens during a stall except that it is air flowing horizontally rather than water falling vertically. The most likely time when the air stream may break from the wing is when the AOA is high and the plane is flying slowly. When a stall occurs the wings lose their lift.
When the air is sticking to the wing it follows the profile of the wing and in doing so it is thrown down. For the most part the air acts as if it is incompressible such that far above the wing the air streams continues to mimic the profile of the wing. Nevertheless as we go higher above the wing this following of the wing’s profile fades until we reach air that is unaffected by the passing of the airplane. Thus the air that is about a half wingspan above the wing is pulled down much harder than the air that is more than one wingspan above the airplane.
We need to know the effective amount of air being pulled down so as to calculate the lift provided by the wings. In their book Understanding Flight David F. Anderson and Scott Eberhardt introduced the visualization tool of there being a scoop above the wing diverting the air down. This simple conceptual model of having all the air within the scoop being thrown down the same way can be substituted for the more complicated reality of the downward pulling effect diminishing the farther we are above the wing. For the vast majority of aerodynamically correct wing profiles for a airplane in normal flight using a six degree AOA, the effective area A of the scoop is ½ b2 where b is the wingspan.
It is important to recognize that the wing’s surface area is not included in the equation above. This is a correction to many aerodynamic equations that incorrectly calculate lift based on the surface area of the wings. Obviously the cord of the wing needs to be long enough to create the profile that produces the Coanda effect, but beyond this increasing the wing’s surface area by increasing its cord length does not increase the lift produce by the wing. To give an example, if two planes have the same wing area yet one has short broad wings while the other has narrow wings that are twice as long as the first, the airplane with the longer wings will generate four times as much lift as the other. It is for this reason that planes such as sailplanes that wish to minimize weight while maximize lift will have long narrow wings.
This is not to say that the airplanes with broad wings are being wasteful. Broader wings enable a plane to safely fly in rougher weather or at a slower speed without stalling. This is because the broader wing gives the air stream a greater attachment area thus making it more difficult for the air stream to break free of the top wing surface so as to produce a stall. This ability to fly slower without stalling can be important to large birds or airplanes when they are taking off or landing. Take note during your next flight on an airliner to observe that the pilot will expand the wings at the early part of approaching the landing. The expansion of the wing lowers the stall speed of the wing and the downward turn of the expansion acts as a break in reducing the airplane's speed; both of these effects allow the airplane to make a safe landing.
With the correct understanding of the science of flight we can obtain tremendous insight into why the wings of birds or airplanes are shaped a particular way.
In this section the three key concepts to understanding flight are: 1) wings lift a plane up by throwing a tremendous amount of air down, 2) because air has viscosity it sticks to and follows the top wing surface, 3) in following the top wing surface the air is pulled down and as an equal and opposite reaction the wings are lifted up.
As a plane flies through the air the wings effectively grab the air to throw it down so as to lift the plane up. The act of throwing air down produces a reactionary force that lifts up on the wings. This upward force on the wings counterbalances the weight of the plane that is directed downward. Besides these two forces counterbalancing each other in the vertical direction, there is another pair of forces, the drag and the thrust, that counterbalance each other in the horizontal direction. While the plane is flying along at constant speed and level flight the vector summation of all of these forces is zero.
The idea that an object with zero net force acting on it will continue moving in a straight line is Newton’s first law, while the action and reaction forces acting on the wing is an example of Newton’s third law that all forces are equal and opposite. What is next needed is the application of Newton’s second law of motion for equating the weight of the plane to how much air the wings need to throw down for the purpose of lifting the plane up.
Most scholars of physical science are familiar with Newton’s second law as being F = m a; when a net force acts on an object its acceleration in the same direction will be inversely proportional to its mass. But while this equation is useful in solving the vast majority of physics problems involving objects, it is not really helpful when we are generating a force by constantly throwing down a fluid. Fortunately Newton actually stated his second law as force being equal to a change in momentum with a change in time: F = Δ(v m) / Δt. By starting with this more general statement we can write Newton’s second law applicable to a fluid as F = v dm / dt.
This is the same equation that we would use to describe rocket propulsion. It tells us that we generate the greatest lifting force by maximizing the rate of fluid mass per unit time that we thrown down and we throw that fluid down at the highest speed possible. For our airplane this force is the lifting force acting on the wings that in turn is equal in magnitude to the weight of the plane so that
W = v (dm / dt)
W is the weight of the plane or flying animal, v is the downward speed of the air, and dm / dt is the mass of the air being thrown down per unit time.
This flow of mass per unit time is produced by the wings grabbing the air and throwing it down. Earlier it was stated that the cross sectional area of air A grabbed by the wings is about ½ b2. Thus
(dm / dt) = D dV / dt
(dm / dt) = D A (dx / dt)
(dm / dt) = D A v
(dm / dt) = D (½ b2) v
Substituting this into our main lift equation gives:
W = v D (½ b2) v
Where W is the weight of the airplane, v is the speed of the air being thrown down, D is the density of the air, v is the relative speed of the plane through the air, and ½ b2 is the effective area A surrounding the plane affected by the wings as they cut through the air.
This equation tells us that for a greater weight of our plane and its cargo we can either 1) throw the air down at a faster speed, 2) fly at lower altitude where the air’s density is greater, 3) fly faster, or 4) build a plane with longer wings that are capable of influencing more of the air surrounding the airplane.
| The Problem with Hardheaded Know-It-Alls |
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After finally conceding that the Equal Transit Time explanation of lift is incorrect, some old school ‘experts’ still insist that even though they cannot give a conceptual explanation of lift, that nevertheless their use of elaborate equations and software to calculate lift is proof that they know how airplanes fly. There are two flaws to their argument: first, hand-waving mathematics is not a replacement for conceptual understanding, and second, their claim of competency lacks credibility when their primary equation for calculating lift is incorrect.
By substituting lift FL for weight W
and the more common
Correct Equation for Lift
Aviation Industry’s Equation
The most serious problem with the aviation industry’s lift |
As the plane flies through the air, the wings are converting some of the airplanes forward thrust to throwing air down so as to lift the airplane up. This process of throwing the air down requires energy; energy that the airplane gives to the air. The kinetic energy of a moving object is calculate as
E = ½ m v2.
Since the airplane is continuously giving this energy to the air it is more appropriate to discuss power, the use of energy per unit time.
P = dE /dt
Substituting the energy equation into the power equation gives:
PL = d(½ m v2) /dt
This simplifies to:
PL = ½ (dm / dt) v2.
The mass flow rate (dm / dt) was calculated earlier as
(dm / dt) = D (½ b2) v
and we can use the final equation of the last section to solve for the speed that the air is thrown down as:
v = 2 W / D b2 v.
Making these substitutions gives the power needed for lift based on known values:
PL = W2 / (D b2 v).
Where PL is the power needed for lift, W is the weight of the plane, D is the density of the air, b is the wingspan, and v is the speed of the airplane. This equation gives us great insight into the fundamental requirements for flight, yet we are still not finished since we must also account for power required to overcome the fluid drag.
In addition to the power being used by the wings to lift the plane up, power is also required to overcome the parasitic drag. For most large high speed objects moving through a fluid the parasitic drag can be accurately modeled as:
FP = ½ A C D v2.
Where FP is the parasitic drag, A is the front facing or cross sectional area, C is the aerodynamic drag coefficient, D is the density of the fluid, and v is the object’s relative speed through the fluid.
The general equation for power is
P = F v.
So the power needed to overcome the parasitic drag is:
P = ½ A C D v3.
This equation shows that the power requirements go up dramatically with an increase in speed. Therefore if the intend is to travel fast then all effort should be made to streamline the object, reduce its thickest cross sectional area, and if possible travel through a low density fluid. The reason fast cars, trains, and jets look fast even while stationary is due to the necessary of streamlining the form so as to reduce the power needed to reach the fastest speeds.
Flying is the exception from most forms of travel in that a jet airliner has a means of reducing the density of fluid that it is traveling through. The density of air decreases with altitude. A primary reason that jet airliners fly at high altitude is so that they are flying through air that has a density that is about one fourth the density of air at sea level. By flying through lower density air the airliners reduce their power requirements and achieve savings by getting better fuel mileage. To make a comparison, despite a submarine’s extremely good streamline form it can not travel nearly as fast as a jet airliner because the density of sea water is two or three thousand times greater than the density of high altitude air.
Flying is also different from other forms of travel in regards to the various speeds that they can travel. While other forms of travel can move as slow as they like, an airplane is restricted to flying at speeds above its stall speed. Not only is flying well above the stall speed necessary for safe flying it is also best to always fly fast so as to produce the best fuel mileage. Eventually though because the power used to overcome drag is a function of the speed cubed, even for planes there is a limit to how fast they can fly and still be fuel efficient. For airplanes there is a sweet spot between flying too slow and flying too fast where the plane gets the best fuel efficiency.
Starting on the left, the power required to lift the plane goes down as the plane flies faster. In fact an airplane can only fly so slowly before it will stall; this part of the graph is shown with a dash line. The center graph shows power output required to overcome the parasitic drag increases with the cube of the speed. The addition of the power required in overcoming drag and the power required for lift gives the total power needed to fly. This is shown by the graph on the right. Number 1 is the minimum power required for flight. Airplanes and flying animals actually need to fly faster than this speed, to the right of point 1. Number 2 marks the cruise speed.
We can calculate the minimum power speed by summing together the two power requirements and then taking the derivative of the total power requirement with respect to speed. The total power is:
PT = PP + PL
PT = ½ A C D v3 + W2 / (D b2 v)
Taking the derivative of the total power equation with respect to speed then setting the result equal to zero, we determine the speed for minimum power:
![v_min = [(2 W^2) / (3 A C b^2 D^2)]^1/4](flight_eq21.gif)
The minimum power can now be determined by substituting this minimum power speed back into our total power equation. Yet when we do this we notice that at this speed the minimum power for lift is ¾ of the total power. Thus we can simplify our minimum power equation to be:
![P_T-min = 4/3 [ W^2 / (b^2 D v_min)]](flight_eq22.gif)
Esker’s Equations of Flight are relatively simple equations that give the minimum take-off speed and minimum power requirements to achieve flight based on the atmospheric density and known data of any given airplane. These equations are general to all types of airplanes and flying vertebrates.
If we continued building on the concepts presented so far we can derive a general relationship for how the power requirements increase with size. The general equation is
Thus we see that as the size of our planes increase our power requirements increase dramatically. As an example of using this power equation we can compare a Cessna 172 to a 747 jetliner. A Cessna 172 is 8.25 m long and has a 120 kW engine, so in applying our equation to the 41 m 747 we get a ballpark power requirement of 42 MW. The actual power of a 747 is about 50 MW.
To produce accurate answers when applying this equation our flying objects should be of approximately the same shape and overall density. Yet as the example demonstrates the equation can produce reasonably accurate ballpark values for the require power even when our flyers are not that similar. For example, based on the power requirements of the Cessna 172 we could also use this equation to calculate the approximate power requirements of a sparrow or a housefly.
Yet regardless of our fondness for theoretical work, it is all worthless if it is not verified by the evidence. To test the equations the author has selected an assortment of airplanes and flying animals and then used the known data regarding these flyers weight and form as input to calculate their flight performance. When tested these equations not only prove themselves to be accurate, but in addition the equations are able to provide insightful answers to some historical aerodynamic controversies. The reader is encouraged to select their own airplanes or animals to do their own testing to validate Esker's Equations of Flight.
In the table below a power ratio of 1.0 is the minimum requirement for a airplane or animal to takeoff in ideal weather conditions. While a power ratio of around 2.2 and above allows a airplane or animal to fly at its best efficiency per distance along with giving it a safety margin for flying in foul weather.
| Flyer | Weight (N) |
Front Area Estimate (m2) |
Drag Coefficient Estimate (Front Area) |
Wingspan (m) |
Speed for least Power (m/s) |
Minimum Power (kW) |
Available Power (kW) |
Power Ratio |
|---|---|---|---|---|---|---|---|---|
| Cessna 172 | 10900 | 4.0 | 0.17 | 11 | 28 | 37 | 84 | 2.3 |
| Sopwith Camel | 6500 | 2.8 | 0.50 | 8.5 | 21 | 30 | 68 | 2.3 |
| Wright Flyer | 3340 | 1.7 | 0.50 | 12.3 | 14 | 5.7 | 6.3 | 1.1 |
| Spruce Goose | 2300000 | 110 | 0.17 | 97.5 | 60 | 9900 | 13000 | 1.3 |
| Human Flight | 980-610 | 1.7 | 0.12 | 23.2 | 7.8 | 0.24 | 0.29 | 1.2 |
| Starling | 0.74 | 0.0080 | 0.20 | 0.36 | 5.8 | 0.00077 | 0.0051 | 6.6 |
| Giant Bat | 11.8 | 0.070 | 0.50 | 1.5 | 5.2 | 0.013 | 0.021 | 1.6 |
| Condor | 102 | 0.19 | 0.20 | 2.9 | 11 | 0.12 | 0.13 | 1.1 |
| Argentavis | 1400 | 1.1 | 0.20 | 7 | 17 | 2.5 | 0.77 | 0.3 |
| Quetzalcoatlus | 7000 | 2.5 | 0.50 | 12 | 18 | 20 | 1.8 | 0.09 |
Cessna 172 and Sopwith Camel- Other than the Cessna having more power and consequently greater speed, the Cessna 172 and the Sopwith Camel are similar flying aircraft. The modern day Cessna is only slightly heavier, faster, more aerodynamic, and more powerful than the world war one Sopwith Camel. These first two flyers are supplied with enough power to be safe fliers even in foul weather.
Wright Flyer - The bi-wing of the World War One Sopwith Camel is not much different from that of the Wright Flyer. Yet the performance characteristics are dramatically different because the Wright Flyer is underpowered: it had only a 12 hp engine.
Calculations based on the historical data show that the overland speed of the Wright Flyer was only about 4 or 5 m/s even though the least power speed of the Wright Flyer is 14 m/s. The explanation for the difference is that the Wright brothers flew their airplane into a strong headwind as all airplanes do when they takeoff. The Power for Flight equations predict that at least a 10 m/s headwind was required to be added to the Wright brothers’ 4 m/s overland speed to give the Wright Flyer the 14 m/s relative speed it needed to fly, and sure enough a check of the historical weather data states that the top wind speed on December 17, 1903 was 22 mph or 10 m/s. The brothers from Dayton, Ohio specifically chose to do their flying at Kitty Hawk, North Carolina because of its record of steady, strong winds.
Spruce Goose – How much does it actually weigh is the key to answering the “Can it really fly” question. Depending on the source its empty weight is either 300,000 or 400,000 lbs (1.3 to 1.8 MN). It seems odd that this key data is not precisely known. It is also odd that the stated cargo weight of either 100,000 or 130,000 lbs does not correspond with the calculated cargo weight. Seven hundred and fifty men at 200 lbs each gives us 150,000 lbs for the cargo. But if we are going to fly over the Atlantic then the plane also needs 12,500 gallons or 70,000 lbs of fuel for a total of 220,000 lbs. So depending on which empty weight is correct, its full weight is either 520,000 lbs or 620,000 lbs. Expressed in metrics its full fuel and cargo weight is 2,300,000 N to 2,800,000 N.
Loaded with cargo and fuel its power ratio is only 1.1 to 1.3 thus indicating that the Spruce Goose had only half as much power as what it needed. During the design of the Spruce Goose, Howard Hughes and at least one consultant disagreed on if the Spruce Goose exceeded the limitation of how large a propeller driven airplane could be. But at that time there were no simple equations for determining the power to weight limitation or if a size limitation even existed; so Howard was undeterred in his goal of building an exceptionally large airplane.
As a cargo plane completed too late to fulfill its war-time mission and, more important, lacking the power to carry a cargo the Spruce Goose was not a useful or safe flyer. Yet the Spruce Goose is a historical icon and the numerous engineering challenges that were solved during the design and construction of this extremely large airplane did advance aviation technology.
Human Powered Flight – Two values are given for the weight; the 980 N is the total weight of both man and airplane while the 610 N is the weight of just the man. The total weight is used for calculating the airplane’s performance while the weight of the man is use to calculate the power available.
The performance numbers show that human flight aircraft are clearly built around the limitation of a top athlete being capable of producing only about 290 watts. With only a 1.2 power ratio it is probably safe to state that ground effect is a significant factor in enabling these airplanes to fly.
Human powered flight can be achieved only by keeping the weight of both the athlete and the airplane to a minimum while extending the wingspan out as far as possible. Minimize the front facing area and minimize the drag coefficient is helpful yet of secondary importance to the weight, power, and wingspan criteria.
Starling – With an 6.6 power ratio flying is a not a challenge for these and other small and medium size birds. With such an abundance of relative power it is no wonder that small and medium size birds often appear playful as they fly.
Giant Bat – Giant bats have a power ratio of only 1.6. This marginal power ratio explains why the largest bats are rarely ever seen flying in the rain.
California Condor – This is another extremely marginal flyer that can only fly when the weather conditions are favorable. A 1.1 power ratio is barely enough to allow these large birds to lift off after feasting on a carcass. But once they lift off they glide themselves over the rising air thermals that act like a rising elevator to lift them higher. Once they are soaring, these large birds require only enough power to guide their flight over the rising air thermals.
Argentavis – Six million years ago there existed an extremely large bird known as the Argentavis. It flew in South America like the modern large flyer the Andean condor. However the wingspan of the Argentavis was two and a half to three times that of the Andean condor. Its mass would be proportional of the wingspan cubed and so its mass would have been 16 to 27 times greater than the Andean condor. This places it far beyond the modern crowded field of bird species competing for the title of being the largest flying birds. Consequently many scientists investigating this matter are baffled in their attempts to explain how the Argentavis flew.
It is probable safe to assume that these ancient relatives of the California condor utilized the warm rising air thermals in same way as the large soaring birds of today. Thus the power ratio of 1.1 for the California condor should be seen as the cutoff value for birds using thermal assisted flying. Yet these birds have a power ratio of only 0.3. So even with the assistance of rising air thermals this animal could not have flown in today’s atmospheric conditions.
Quetzalcoatlus – Unlike the Argentavis there are no living relatives of the Quetzalcoatlus and this makes it difficult to estimate the Quetzalcoatlus’ mass. The author produce values that fell in a range between 500 kg to two tons; thus arriving at a rough estimate of 700 kg. With a 12 m wingspan and a chest cavity larger than that of a horse there is no getting around the fact that this was a huge animal.
However some paleontologists tell a different story in
estimating the Quetzalcoatlus mass to be between 90 and 250 kg. Thus they
are claiming that the Quetzalcoatlus had a body density that was about
seven times less than any animal presently flying. These paleontologists
need to do some explaining as to how the muscle, bone and other bodily
parts of a Quetzalcoatlus could have a density seven times less than what
is found in present day birds.
Some paleontologists have suggested that the pterosaurs may have been warm-blooded. While warm-bloodedness is unusual for reptiles this is nevertheless a feasible hypothesis and so the available power for the Quetzalcoatlus is calculated using the power-to-weight equation for a warm-blooded mammal. Even so, the power ratio of the Quetzalcoatlus is still only nine percent of the minimum requirement for flight.
So how did the Argentavis and Quetzalcoatlus fly? The answer is in the equations:
If the answer is still not clear then just read on to the
next chapter, the chapter that searches and finds the solution to the
paradox of gigantic birds, pterosaurs, and dinosaurs.
*If at the beginning of this chapter on flight you passed over the
link to the discussion of animal power, this would now be a good time to
read this section to fully understand the relationship between animal
power and the power required for flight.
PERMISSION IS GRANTED FOR TEACHERS AND OTHER EDUCATORS TO COPY AND DISTRIBUTE THIS MATERIAL, UNALTERED OR CHANGED IN ANY WAY, TO STUDENTS IN BOTH PUBLIC AND PRIVATE EDUCATIONAL INSTITUTIONS.
