# Energy and Power of Flying

## Introduction

 Fig. 1: Boeing 737-300. (Source: Wikimedia Commons)

Heavier-than-air aircraft go through a continual process of design change in order to meet the ever changing mission objectives. To accommodate a future without fossil fuels, aircraft will need to be re-engineered to accommodate synthetic fuels and be more energy efficient in order to continue to transport people and commodities in an efficient manner. Recent reports from different companies estimate future air-traffic growth to vary between 4%-6% per year. [1] In order for this prediction to turn into reality, both the power requirements and design factors for jet aircrafts have to be understood.

## Power Requirement to Keep a Jet Aircraft in the Air

The total energy of an aircraft flying in the atmosphere can be calculated using equation 1. [2]

E = ½ m v2 + mgh

A Boeing 737-300 has a maximum takeoff weight of 5.65 × 104 kg, a cruise altitude of h = 10,195 m, and cruise speed of 221 m/sec. Inserting these numbers into the above equation, we obtain 7.03 GJ for the energy at cruise conditions. [3] However, the engines mounted onto the wings of the plane are required to provide additional energy per time, power, in order to keep the aircraft flying at a constant altitude and speed. This extra power needed to overcome the drag acting on the airplane and is calculated using

P = (dh/dt) m g

where (dh/dt) is the decent speed required to keep the plane's speed constant if the engines fail. Assuming that a Boeing 737-300 has a similar glide ratio to a Boeing 747 of 17:1, a Boeing 737-300 has a descent speed of approximately (dh/dt)= (221m/sec)/17 = 13 m/sec. The power required to keep a Boeing 737-300 flying at a constant altitude and speed is 7.2 × 106 watts. The rate of fuel burn necessary for the engines to produce enough power to keep the airplane flying can be calculated from the total power requirement. Assuming the efficiency of the engines adding thrust to the aircraft is roughly 30% and the specific gravity of jet fuel is about 0.82 kg per liter, a Boeing 737-300 burns approximately 0.665 liters per second. A Boeing 737-300 has a standard fuel capacity of 2.0104 × 104 liters. [3] Assuming that takeoff, climb, and landing burns approximately one third of the total fuel, and at cruise conditions with a fuel burn rate of 0.665 liters per second, a Boeing 737-300 can fly for approximately (2/3) × 8.4 hours= 5.6 hours at cruise conditions before running out of jet fuel.

If all the engines were to completely fail, jet aircrafts would not have any of the required power to keep flying at a constant altitude, and thus descend to the ground. As the aircraft is gliding towards the ground the glide angle, glide slope, and glide ratio can be calculated based on the aircraft specifications. The glide angle (γ) is the angle of the aircraft's nose relative to the horizon. [4] In cruise conditions, the smallest glide angle occurs when the drag per unit weight is minimized and/or the lift-to-drag ratio is maximized. [4] The glide ratio of an airplane, with no power, is the ratio of the forward travel to the loss of altitude. [5] The glide ratio of a Boeing 747 is 17:1 meaning that for every 1,000ft of altitude lost, the Boeing 747 will travel 17,000ft. [6] The glide angle (γ) is related to the glide slope by tan(γ) and the glide ratio is calculated as the inverse of the glide slope, cot(γ). [6] Using glide ratio of 17, the glide angle (γ) is calculated using the relationship, glide ratio= cot(γ). The glide angle for the Boeing 747 is approximately 3.4 degrees.

## Design Considerations

Since about 1959, the commercial jet aircraft geometrical design has not changed much. [1] However, the energy efficiency of the aircraft has and still does continually improve in small increments. [1] Due to a 40% engine efficiency improvement, the overall jet aircraft efficiency improved by about 1.5% per year between the years 1959-1995. [1] Although the engine efficiency improved by 40% the new engine design did negatively impact the aerodynamics of the aircraft. [1] For example, the engine improvement was due to the implementation of the high bypass engine which caused the engine diameter to increase in size and thus increase the engine weight and aerodynamic drag. [1] To lessen the aerodynamic drag, the aircraft designers were able to create improved wing designs and engine to airframe integration. [1] The overall aerodynamic efficiency continually increases and between the same period of years, 1959-1995, the aerodynamic efficiency improved an average of 0.4% annually. [1] The use of computational and experimental design tools have significantly helped to improve the designing of more energy efficient aircrafts. [1]

The other design aspect to consider is the material selection for reducing the weight of the aircraft while maintaining the component's structural integrity. Up until the creation of Boeing's 787 Dreamliner, approximately 90% of the commercial jet aircrafts are made from aluminum parts. [1] Composite materials are much lighter than metals and can be designed to have equal or better structural strength compared to metals. A composite airframe can potentially reduce the weight of the airframe by 30%. [7]

## Future of Flying

I believe there will be a continued need for transport aircrafts in the future in order to move both people and cargo in a timely and efficient manner. In order for the aircraft industry to survive in a nonexistent fossil fuel world, alongside the need for commercially produced synthetic fuels, aircrafts will need to be designed with energy efficiency as a major priority.

© Andrea Eller. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

## References

[1] J. J. Lee et al., "Historical and Future Trends in Aircraft Performance, Cost, and Emissions," Annu. Rev. Energy Environ. 26, 167 (2001).

[2] C. Borst et al., "Ecological Approach to Support Pilot Terrain Awareness After Total Engine Failure," J. Aircraft 45, 159 (2008).

[3] "Boeing 737-300," IHS Jane's 00433799, 15 Oct 13.

[4] B.N. Pamadi, "Gliding Flight," in Performance, Stability, Dynamics, and Control of Airplanes, ed. by J. A. Schetz (AIAA, 2004), P. 75.

[5] "Airplane Flying Handbook," U.S. Federal Aviation Administration, FAA-H-8083-3B, May 2004.

[6] B. Lautrup, "Unpowered Steady Descent," in Physics of Continuous Matter, Second Edition: Exotic and Everyday Phenomena in the Macroscopic World ed. by T. Spicer, (IOP, 2005), p. 519.

[7] D.L. Greene, "Energy-Efficiency Improvement Potential of Commercial Aircraft," Annu. Rev. Energy Environ. 17, 551 (1992).