Performance Concepts Speaker: Randall L. Brookhiser Performance means how fast will it go? How fast will it climb? How quickly it will take-off and land? How far it will go? Let s start with the phase of flight that we spend the most time in level flight cruise. In level flight performance is measured by the maximum cruising speed for the most part. Economy and endurance are also performance factors to consider but let s face it how fast we can get somewhere is a big reason for flying airplanes rather than driving a car. Maximum cruising speed as far as the airframe is considered is determined by the total drag characteristics which determines power required in cruise at various airspeeds. The second factor that determines Maximum Cruising Speed is the powerplant and propulsion system or power available. The Power Required Curve is nothing more than a representation of Total Drag Curve in terms of power units. Power is simply the rate of doing work. Work is accomplished in producing lift in level flight. Let s review some of the simple equations used in aerodynamics. Power equals work per unit of time. Since work is force times distance power can be expressed as force times distance divided by time. The force is thrust; therefore thrust horsepower is thrust times distance divided by time. Velocity or rate is distance divided by time. Therefore: Thrust Horsepower = Thrust times Velocity. Thrust has to be a value equal to drag to maintain level flight therefore thrust horsepower is also equal to drag times velocity. Here we see the Drag-Thrust curves verses true airspeed on the top and the thrust horsepower required verses true airspeed on the bottom. Total drag or thrust curve highlighted here has a similar shape to the thrust horsepower required curve shown here. To derive thrust horsepower Thrust in pounds is multiplied by velocity in feet / second to get ft-lbs./sec power units. The units are converted to horsepower units by dividing by 550. The velocity has been converted and plotted as knots true airspeed. Notice how the thrust horsepower required curve slopes up steeply at higher speed. Power required is so high that all available power is needed to maintain that speed. The exponentially increasing parasite drag causes the power required curve to approach infinity values. The total airframe drag sets a practical limit on maximum cruise speed. The power-required curve has nothing to do with the actual power being produced. The curve reflects the requirements for power to maintain level flight. Thrust equals drag and drag equals the coefficient of drag times one-half rho or air density times the velocity squared. Power equals thrust times velocity. Therefore power equals the coefficient of drag times one-half rho times velocity cubed or the 3rd power. To go twice as fast, two to the third power, or eight times the power is required. Looking at the formula, lowering Coefficient of Drag CD is the only way to lower power required for any given air density rho. When flying in cruise flight it is possible to fly at two different airspeeds with the same power when the power used is above minimum power. At minimum power, one unique airspeed must be maintained. At the bottom of the power curve is the minimum power airspeed and divides the chart into two regions. The reverse command or back side of the power curve shown in red here is found at speeds lower than Page 1 of 5
minimum power airspeed. This region is called reverse command because lower airspeed results in more power required. The normal command or front side of the power curve is found at airspeeds above minimum power airspeed and is shown in blue here. On the front side of the power required curve lower airspeed requires less power and higher airspeed requires more power. Shown here is the thrust horsepower available. Break horsepower is the power coming off the propeller shaft and is represented by a straight line. The power at the propeller for propulsion is a curved line that is lower on the chart. THRUST horsepower is less because some power is lost when converting from the power at the engine shaft to power from the prop. Just like a wing the rotating propeller blade has an angle of attack and velocity where the Lift to Drag Ratio is the highest and the blade is most efficient. In this example peak efficiency is 80% at one unique ratio of forward Velocity to rotational Velocity or at one unique angle of attack. Forward velocity is your airspeed and rotational velocity is determined by your RPM setting. The speed for maximum propeller efficiency is close to but not exactly the same as the speed for best airframe efficiency. Reviewing, the thrust horsepower required for level flight curve is lower and shaped similar to the total drag curve. The shaded area represents the difference between thrust horsepower available and thrust horsepower required at various airspeeds. This difference is what we call excess thrust horsepower. Rate of Climb, or vertical speed, is equal to excess thrust horsepower divided by weight times 33,000 to convert the units into feet per minute. Here is the formula using math variables. The triangle symbol means change or difference in power and W means weight. If values of excess thrust horsepower are plugged into the rate of climb formula a rate of climb curve is created as shown by the lower graph green curve. Notice that as the excess changes rate of climb values change proportionally. At the point where the greatest amount of excess thrust horsepower or difference between power required for level flight and what is available the rate of climb curve peaks out. Maximum Rate of Climb occurs at Maximum Excess Thrust Horsepower airspeed and angle of attack. At airspeeds below and above the speed for maximum rate of climb the rate of climb is less. At maximum cruise airspeed rate of climb possible reduces to zero as the power available equals the power required. For example looking at the highest value on the rate of climb curve we find the best rate of climb designated V y at 1300 feet per minute. This speed is found where we have the greatest excess thrust horsepower. The best climb angle is found by a line starting at the origin and becoming tangent to the rate of climb curve. Notice that best angle of climb rate is only 1100 feet per minute. Here we can compare best angle with the lower rate with best rate of climb that has the lower angle. Angle of climb is determined by the ratio of the forward speed to the vertical speed. Let s look at the difference between the two types of climb in a more practical sense. Climbing at Best Rate of Climb is not the same as climbing at Best Angle of Climb. The next series of slides will compare the slow rate high angle climb of a light airplane to the high rate shallow angle climb of a jet. The jet will reach an altitude of 1,000 feet above ground faster than the little cub but has to fly through the terrain. In this example the small cub is climbing at 500 ft./min and takes 2 minutes to clear the 1,000 feet terrain. The jet takes only 30 seconds climbing at 2,000 ft./min. However, the jet has a shallow climb angle or angle of the flight path to the horizon. The ratio of the vertical speed to the forward speed Page 2 of 5
determines the climb angle and in this case the ratio is much lower than the ratio on the cub. As shown here the jet has too low of a climb angle to clear the trees and to clear the terrain. The cub has a much lower climb rate but also a much slower forward speed. The ratio of vertical speed to forward speed is much higher resulting in a much steeper climb angle. However, the pitch attitude of the airplane is shallow. The flight path angle to the ground is what counts. The cub is so much slower in its forward speed that it has much more time to climb over the obstacle before reaching the obstacle. Comparing the two we can distinguish between rate of climb and angle. An airplane flown at best rate will get to altitude the fastest but will have a shallower angle and slower rate of climb when operating at best angle. Rate of Climb varies inversely with weight. Higher Gross Weight shifts the Required THP Curve up and there is less Excess THP and less rate of climb. Now back to normal weight. Lower Gross Weight shifts the Required THP Curve down and there is more Excess THP and more rate of climb. Since an airplane is able to climb on the excess a 50% loss in power may result in a 70% loss in climb performance! Here you can see that cutting power in half results in the excess reduced much more than half. An increase in altitude will shift the power required curve to the right and lower the power available and rate of climb. Best Rate of Climb V y (True Airspeed) is slightly higher at altitude. Since the chart is in True Airspeed V y is higher but Indicated V y is lower. Absolute Ceiling occurs when THP Available is just enough to equal THP Required. A more practical limit is Service Ceiling where there is still 100 ft/min rate of climb. Here we see how rate of climb decreases with altitude. Once again service ceiling is the point where rate of climb is still 100 ft per minute. Let s look at twin-engine performance. The power needed is divided between two engines and climb performance drops off dramatically flying on one engine. Here we see the power available with both engines at the top and power required when two engines are operating at the bottom. The airframe drag is lowest when both engines are running as opposed to single-engine operation. On two engines the excess thrust horsepower is significant. In this example two engine operation results in 80 excess thrust horsepower and 1500 feet per minute rate of climb. The single-engine available and required curves are shown as dashed lines. Notice that power required for single engine operation is greater due to the extra drag associated with one engine operation. Two engine excess power results in a healthy 1500 ft per minute climb rate. Operating on one engine the excess power is greatly reduced and the airplane can barely climb. On one engine the excess thrust horsepower is reduced to 16 horsepower resulting in only a 300 feet per minute rate of climb. The excess changes from 80 to 16 horsepower. The power loss is 50 percent on one engine but the performance loss is 80% in this example. This change is typical for many airplanes. If proper procedures such as feathering and stopping the dead engine propeller and retracting gear and flaps are not followed most likely the airplane will not even maintain altitude. Next we need to look at sink rates but first let s define the lower and upper end of the normal range of airspeeds. At the speed V a shown power available and required are equal at a speed just above stall. On the high end we have maximum cruise speed designated as V b for the purpose of our discussion where power available and required are also equal. If thrust horsepower available was less than that required between V a and V b or power available was zero a loss of altitude at a Page 3 of 5
certain sink rate would occur depending on the airspeed. The lowest point on the THP Required Curve results in the Minimum Sink Rate or Maximum Time in the air. The lowest point on the THP Required Curve results in the lowest change or deficiency in power with power off. Similar to the rate of climb formula the Rate of Sink is equal to the deficiency in power divided by the weight and times 33,000 to convert to feet per minute units. Using math variables the equation becomes RS equals delta P divided by weight times 33,000. However, the speed for Minimum Sink Rate or Maximum Time does NOT result in the most efficient glide or glide distance! Although vertical speed is lowest the forward speed is so slow that the ratio of forward speed to vertical speed results in steep angle and short glide distance. Gliding at minimum sink rate is not the same as gliding at best glide speed to achieve maximum glide distance. The flight path on the left represents gliding at an airspeed and angle of attack that result in a minimum sink rate of 1,000 feet per minute. From 3,000 feet above ground the glide to the ground would take 3 minutes. The flight path on the right represents gliding at an airspeed for maximum glide distance or maximum lift to drag ratio. The sink rate at best glide distance is higher at 1500 feet per minute and the glide from 3,000 feet only takes 2 minutes. The best glide angle is the lowest angle below the horizon and it will produce the greatest glide distance. Like the best angle climb the ratio of forward speed to vertical speed is the key. Best glide occurs at the highest ratio of forward speed to vertical descent speed or sink rate. Stated another way it could be described as the minimum ratio of sink rate to forward speed. At the speed for minimum sink rate the ratio of forward speed to sink rate is lower than when flying at best glide speed. Time in the air is maximized at the expense of much less gliding distance. If your engine quits and you want to make the field glide at best glide. If your engine quits and you want more time to get it started again then fly at minimum sink rate airspeed. The glide angle for best glide is shaded yellow. The glide for flight at minimum sink rate is shown as red. Once again best glide maximizes distance and minimum sink rate airspeed maximizes time. Best efficiency as far as the airframe is concerned is found by looking at the total drag curve and the resulting power required curve. The airspeed for minimum total drag aligns with the tangent line shown here. The tangent line begins at the graph origin and becomes tangent near the bottom of the power required curve. At any point on the power required curve the values of power on the vertical axis can be compared with the forward velocity on the X axis. The tangent point of the line starting at the origin would be the lowest amount of power required for the forward airspeed. At this tangent point the ratio of power required to airspeed is the lowest. Forward airspeed is maintained at the lowest cost of power. This aligns with minimum total drag where lift is produced at the least cost of drag. Maximum Lift to Drag Ratio produces the minimum fuel consumption per mile traveled or best range, however, the best Lift to Drag speed is too low for cruise. Minimum Power Required airspeed maximizes TIME. Minimum Power Required is the lowest power required but has a greater ratio of power to forward speed than minimum drag point. Thus it is less efficient. However ENDURANCE or time is maximized. Endurance is simply the time in the air or Range divided by Velocity. Maximum Endurance found at the speed for minimum fuel consumption rate should be used when holding waiting for a clearance or for the weather to improve. Page 4 of 5
Reviewing now, Best Range or Distance consumes more fuel than the speed for Minimum Power Required and Minimum Fuel Consumption which maximizes TIME. Minimum Power Required is the lowest power required but has a greater ratio of power to forward speed than minimum drag point. Thus it is less efficient. However ENDURANCE or TIME is maximized. Best Range speed is higher than Endurance airspeed. Also, we can state that the speed for maximum glide distance is greater than the speed for minimum sink rate. Comparing forward speed to total drag instead of power required we can find that minimum fuel consumption per unit of SPEED is found at the maximum Velocity to Drag ratio. This speed at the maximum Velocity to Drag Ratio is known as Cruise Optimum Airspeed or VC. VC gives the best return in increase in airspeed for excess fuel consumption above the optimum. Remember that the minimum fuel consumption per unit of DISTANCE is found at the minimum Thrust Horse Power required to Velocity Ratio not the drag to velocity ratio. Maximizing cruise efficiency involves going to the highest possible altitude where the desired percentage of power can be maintained where the air is less dense and the drag in producing true airspeed is less. Here we see the variation of maximum and cruising speeds for various power settings. Looking at the 55 percent power line a maximum speed of 117 knots true airspeed can be achieved at 12,000 feet and full throttle. Once again, a higher TRUE airspeed is achieved at a higher altitude due to less drag at the higher altitude. At the lower altitude a lower TRUE airspeed of 107 is found at 2,000 ft. We can save fuel by flying at 55% at a higher altitude rather than flying at 65% at a lower altitude. In this example 118 knots true airspeed is obtained at 11,700 feet and 55% power. The same true airspeed is achieved at 2,500 feet with 65% power an increase in power of 10%. Randall L. Brookhiser and Indian Hills Community College Page 5 of 5