Aeronautical Calculations: What’s with All Those Coefficients Anyway?

Aeronautical Calculations: What’s with All Those Coefficients Anyway? 

         When aircraft designers decide on the design specifications for a new aircraft, they develop a model that is tested in a wind tunnel. The data collected from wind tunnel tests are used to develop a comparison to how the aircraft will fly at full scale. They make these and other comparisons by using proven aeronautical equations using dimensionless coefficients. 

In the previous sections you saw that thrust, lift, weight, and drag are forces and are usually expressed in pounds or newtons. You also saw that pitch, yaw and roll are moments and are expressed in inch-pounds, foot-pounds or newton-meters. So when we talk about our test data we may say, "The wing generated 12 pounds of lift," or "The engine generated 3000 pounds of thrust." That will work fine as long as we are willing to talk only about one specific wing, or one specific engine. What do we do when we want to compare a wing that was tested at one scale and set of conditions, and apply it to another scale and a different set of conditions? Suppose we tested a scale model of a jet transport Model X in the wind tunnel, and now we want to know if the full-scale airplane based on this model will be able to fly from San Francisco to London. The following scenario will show how dimensionless coefficients can help address this problem.

The wind tunnel tests run on the model gave us the following data:

wind tunnel tests run on the model gave us the following data." border=1 cellspacing=0 cellpadding=0>

Model X Data	Test 1	Test 2	Test 3	Test 4	Test 5	Test 6	Test 7	Test 8
Angle of Attack (deg)	0.0	0.5	1.0	1.5	2.0	3.0	5.0	10.0
Lift (pounds)	11.9	238.0	476.4	714.1	952.5	1785.9	2381.4	4048.3
Drag (pounds)	2.5	23.8	42.9	59.6	78.4	118.8	238.1	476.4

The model tested is 5% of full-scale, and the wind tunnel was operated at sea level pressure and a velocity of 300 knots. It is anticipated that the real airplane will fly at 35,000 feet at a velocity of 550 knots. The actual airplane is anticipated to weigh about 550,000 pounds (without fuel) and have a wing area of about 4,800 square feet. The aircraft designers believe that the airplane will have about 200,000 pounds of fuel available for the cruise portion of the flight. The weight of the aircraft (550,000 pounds) plus the weight of the fuel (200,000 pounds) equals a total weight of 750,000 pounds. Engineers anticipate using 2 engines that together, burn about 25,000 lbs of fuel each hour while creating 50,000 pounds of thrust. Does the wind tunnel data show that the airplane would make the 5,300-nautical mile trip?

This is exactly the kind of problem that aerospace engineers evaluate every day. How can we take data from a wind tunnel test and apply it to the real vehicle, especially if that vehicle has yet to be built? Fortunately, we have a hundred years of aeronautical experimentation to back us up. We will use the non-dimensional coefficients that were implemented over those one hundred years to solve the problem.

Over the years the engineers have determined that much of the data can be non-dimensionalized into coefficient form. This means that coefficients will not have units of pounds or foot-pounds like the original data; they will have no units attached. So what this means is that a quantity that was measured in pounds will be divided by something else that has units of pounds, and since pounds divided by pounds results in '1' the units effectively disappear. For example:
3 pound / 1 pound = 3

Treat 'pound' as you would any other number. Now the '3' is a non-dimensional or 'dimensionless' coefficient. Now we just need to find something that has units of 'pounds' to divide by. However, we can't just divide by any old 'pounds' since we are really looking for a way to account for scale, pressure, and velocity effects so we can use our wind tunnel data to predict how the actual airplane will perform.

Instead of proving to ourselves the statement below, this time we will take advantage of the hundred years of aeronautical data that came before us and accept the following statement as fact:

The dynamic pressure "q" and wing area "S" put together will give us the necessary quantity that will "non-dimensionalize" the force measurements, and permit them to be scaled (Think: "Cancel out the units."). The unit part is easy to verify. See the calculations below.

q = pound / foot2
S = foot2
so q * S = pound

There are literally hundreds of reports and research papers that put the scaling effect under scrutiny, so let us just say, that for the most part, and especially for what we are trying to show, non-dimensionalizing forces by dynamic pressure and wing area give the appropriate coefficients. (There are instances and conditions where this may not be true, but those are beyond the scope of this web site.) Similarly, moments are non-dimensionalized by dynamic pressure, wing area, and mean aerodynamic chord. "C" which is usually expressed in feet.

Let’s take another look at the table:

The wind tunnel tests run on the model gave us the following data with the appropriate coefficients added. 

wind tunnel tests run on the model gave us the following data with the appropriate coefficients added." border=1 cellspacing=0 cellpadding=0 >

The wind tunnel tests run on the model gave us the following data with the appropriate coefficients added.

Model X Data	Test 1	Test 2	Test 3	Test 4	Test 5	Test 6	Test 7	Test 8
Angle of Attack (deg)	0.0	0.5	1.0	1.5	2.0	3.0	5.0	10.0
Lift (pounds)	11.9	238.0	476.4	714.1	952.5	1785.9	2381.4	4048.3
Drag (pounds)	2.5	23.8	42.9	59.6	78.4	118.8	238.1	476.4
Dynamic Pressure 'q' in pounds/ft^2 at 300 knots and0 ft alt.	301	301	301	301	301	301	301	301
Wing Area ft^2	12	12	12	12	12	12	12	12
Coefficient of Lift CL	.0033	.0659	.1319	.1977	.2637	.4944	.6593	1.1208
Coefficient of Drag CD	.0007	.0066	.0119	.0165	.0217	.0329	.06593	.1319

We have added some more data, and have expressed the lift and drag in coefficient form. We will now use those coefficients to compute and create some predicted values for lift and drag on the actual airplane.

We will now use those coefficients to compute and create some predicted values for lift and drag on the actual airplane.

We will now use those coefficients to compute and create some predicted values for lift and drag on the actual airplane.

Aircraft X Data	Test 1	Test 2	Test 3	Test 4	Test 5	Test 6	Test 7	Test 8
Angle of Attack (deg)	0.0	0.5	1.0	1.5	2.0	3.0	5.0	10.0
Coefficient of Lift CL	.0033	.0659	.1319	.1977	.2637	.4944	.6593	1.1208
Coefficient of Drag CD	.0007	.0066	.0119	.0165	.0217	.0329	.06593	.1319
Dynamic Pressure 'q' in pound/ft^2 at 550 knots and 35K ft alt.	316	316	316	316	316	316	316	316
Wing Area ft^2	4800	4800	4800	4800	4800	4800	4800	4800
Lift (pounds)	5,000	100,000	200,000	300,000	400,000	750,000	1,000,000	1,700,000
Drag (pounds)	1,000	10,000	18,000	25,000	33,000	50,000	100,000	200,000

We can see within the column marked "Test 6" that the airplane needs to maintain an angle of attack of at least 3 degrees to hold its own weight in the air. We can also see that it generates 50,000 pounds of drag at that condition which matches those engines our designers pick out. We are now very close to determining whether our airplane can fly from San Francisco to London.

We need one more equation: the Breguet Range. This equation is used to calculate the range of an aircraft given a specific flight plan or profile. A typical flight profile includes taxi and takeoff, steep initial climb, climb, cruise, descent, approach and landing. The changing weight of the aircraft as it burns fuel is also taken into consideration. Furthermore, reserves are also calculated as part of a contingency plan for when an approach is missed, which would require additional maneuvers and flight time.

To find the range is much more complex than simply dividing the total fuel by the average hourly fuel consumption and then multiplying by the velocity. That's why we need the Breguet Range equation to predict how far an airplane will fly accounting for flight performance and the change in weight of the airplane as it burns or uses its fuel.

The Breguet Range equation is given below.

Range = (V/C) * (L/D) * ln(Wi/Wf)

V= Velocity in knots

C= Specific fuel consumption in pounds fuel per pounds thrust per hour

L= Lift in pounds

D= Drag in pounds

ln= Natural logarithm, or loge

Wi= Weight of aircraft at start of cruise in pounds

Wf= Weight of aircraft at end of cruise in pounds

The Breguet Range equation will give the flying range of our actual aircraft in nautical miles. For our aircraft we hope the range will be greater than 5,300 nautical miles for our hypothetical trip.

Specific fuel consumption was alluded to earlier, even though it was not specifically named. It was mentioned in the line: "using 2 engines that together, burn about 25,000 pounds of fuel each hour while creating 50,000 pounds of thrust." That comes out to a specific fuel consumption of 0.5.

C = Fuelflow/Thrust
C = (25000 pounds/hour)/(50000 pounds) = 0.5/hour 

Notice that "C" isn't quite a non-dimensional coefficient, since it has units of "per hour." We can also see that L/D could be replaced with CL/CD since we could divide both the numerator and denominator by q * S and not change the equation since (q*S)/(q*S) is like multiplying by 1.0. We can now solve the Breguet range equation. See the solution below.

The aircraft will not make the 5,300 nm trip. Using the Breguet equation range with the data collected from the wind tunnel tests on the model, it appears we need to either land in Scotland or make some changes to our airplane. From the Breguet equation we know that we can increase range by increasing the values of V, L or Wf, and by reducing the values of C, D, and Wi. Changing these values will affect other design aspects of the aircraft that have problems and complications associated with them in real life. Since Wi and Wf are manipulated by the ln function, the impact of changing these parameters is less dramatic. This is referred to as modifying the "fuel fraction." This will involve reducing the structural weight so a larger percentage of the total weight is fuel. This can be very tricky and expensive since it has the potential to affect just about everything on the aircraft. For example, we really cannot increase the lift on our actual aircraft since we want to maintain a steady altitude. To increase the lift in this case it would mean a change in the structural weightof the airplane. We could increase the speed, but that might also increase the drag if we go too fast.

Our best choices for improvement really come with specific fuel consumption and drag. If we can refine our design a bit more, we may get a reduction in drag. If we can get some help from the engine manufacturer, they might be able to reduce the amount of fuel needed to generate the same amount of thrust. We only need 3.4% improvement in either one or in a combination of both to meet our goal. Of course, that may not sound like a lot, but it is enough to financially make or break an aircraft company.