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A Theoretical Conversion of a Boeing 737800 Jet to Battery Power with respect to Power Requirements and Aerofoil Design Chris Ratcliffe

Contents Contents .................................................... 1 Overview .................................................. 1

Introduction .............................................. 2 Power Requirements ................................. 2 Application ............................................... 3 The Power Supply .................................... 4 Limitations ................................................ 6 Numerical Model ...................................... 7 Aerofoil Design ........................................ 9 References .............................................. 12 Appendix ................................................ 13 A: Power Calculations ............................ 13 B: Numerical model ............................... 15 C: Aerofoil Design ................................. 19

Overview In this project I calculated the minimum power and energy requirements for flight of a Boeing 737800 airplane with the aim to investigate if an electric conversion is possible and what changes to the wing design would be necessary. This involved:

- Two methods of calculation - Review of limitations - Research into power supply - Redesign of aerofoil

Firstly, by calculation, I found flight velocity for minimum power usage which resulted in a power requirement for level flight at cruising altitude to be 7.91 MW and required power for the climb phase to be 14.7 MW. Therefore, for a flight from Cork Ireland to London Stanstead with a 15 minute climb and 15 minute cruise, the jet would need batteries with energy density 270 Wh/kg. However, after research of current battery technology this energy density is not yet achievable. Although, these calculations had many assumptions including 100% efficiency, standard atmosphere conditions, constant mass and most importantly a hugely approximated climb phase. For greater accuracy, I used a numerical method to calculate power requirements for the climb phase of a flight, which resulted in an overall reduction in energy requirement over the flight from Cork to London Stanstead. With this improved calculation method, I found a battery energy density of 221 Wh/kg was necessary which is well within current Lithium Ion battery capability. Finally, with a reduction in cruise velocity required to minimise power usage a new wing had to be designed with a greater lift coefficient to produce more lift with a lower velocity. I modelled this on a software called Xfoil and designed a wing with lift coefficient greater than 1.02 which was required by the lower velocity. Within my assumptions I can conclude that the conversion of the Boeing 737800 to battery power is theoretically possible for short flights with current battery technology. However, practically I believe battery technology is still far off the requirements that a real-world situation would demand.

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Introduction Around the world the aviation industry produces around 2% of all man-made carbon dioxide emissions. A round trip from New York to London produces the same emissions as powering a residential home for a year per passenger. Air Transport Action Group, ATAG, a non-profit organisation with members including Airbus, Boeing, Rolls Royce and CFM have the goal to cut net aviation carbon emissions by 50% by 2050. In order to meet this target all major players in the aviation industry will have to make significant developments into renewable flight. [1] I will be investigating the possibilities and limitations of converting a Boeing 737-800 jet (B7378) into a battery powered plane to be used for the flight from Cork Ireland to London Stanstead. The 737 is the best-selling commercial aircraft in the history of aviation. At the start of this year Boeing had over 7000 orders for the 737NG’s and the 800 model made up 70% of those orders [2]. With all major airports in the world equipped to hold the 737’s it seemed logical to investigate a widely used aircraft. I chose the flight from Cork to London Stanstead because it is one of the only current regular flights of the Boeing 737800 to the UK due to the Corona virus outbreak. Furthermore, it is a relatively short flight at only 59 minutes [3]. Being a shorter flight, it gives the battery powered concept the best chance of success as the technology is still primitive compared to what would be required for flights halfway around the globe.

Power Requirements {See App.A for larger graphs and data} T –Thrust D –DragV –VelocityP –Power𝜌 –AirDensityS –WingplanAreaCDo –DragCoefficientCL –LiftCoefficiente –OswaldEfficiencynumberAR –AspectRatioW –WeightE –EnergyGPE –Gravitationalpotentialenergy In order to calculate the energy requirements and specifications of the necessary batteries capable of powering a B7378 I first calculated the power needed for each part of the flight [4]. For level flight at a constant velocity;

𝑇 = 𝐷 𝑃 = 𝑇 × 𝑉 𝑃 = 𝐷 × 𝑉

The drag component can be split into two: parasitic drag and induced drag. Parasitic drag is drag due to friction between the air and surface of the plane and is defined as;

𝐷!"#" =12𝜌𝑉$𝑆𝐶%&

Induced drag is the drag due to lift on the wings and is defined as;

𝐷'() = 𝐶*$ P1

𝜋𝑒𝐴𝑅U

In level flight;

𝐶* =𝑊

12𝜌𝑉

$𝑆

Therefore, induced Drag in level flight;

𝐷'() =𝑊$

12𝜌𝑉

$𝑆P

1𝜋𝑒𝐴𝑅U

And so, to calculate total drag;

𝐷+&, =12𝜌𝑉$𝑆𝐶%& +

𝑊$

12𝜌𝑉

$𝑆P

1𝜋𝑒𝐴𝑅U

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Figure 1, Graph showing Drag against Aircraft speed [4]

Figure 1 shows how total drag and it’s two component factors vary with flight speed and how there is an optimum speed of V for minimal drag. Since;

𝑃 = 𝐷 × 𝑉 The Total power required for level flight is defined as;

𝑃+&, =12𝜌𝑉-𝑆𝐶%& +

𝑊$

12𝜌𝑉𝑆

P1

𝜋𝑒𝐴𝑅U

Figure 2, Graph showing Power required against Aircraft speed [4]

Figure 2 shows us how the power required for level flight varies with flight speed. The minimum stationary point is therefore the minimum velocity required for level flight and can be found by the following calculation; ∆𝑃!"#∆𝑉 =

32𝜌𝑉2𝑆𝐶𝐷𝑜 − 2

𝑊2

𝜌𝑉2𝑆%

1𝜋𝑒𝐴𝑅

&

Let; ∆𝑃!"#∆𝑉 = 0

0 =32𝜌𝑉2𝑆𝐶𝐷𝑜 − 2

𝑊2

𝜌𝑉2𝑆%

1𝜋𝑒𝐴𝑅

&

𝑉$%&(")*+ = (43𝑊,

𝑆,1𝜌,

1𝐶-"

%1

𝜋𝑒𝐴𝑅&4

./

Application I split my calculation into two sections, the cruise phase and climb phase. I am assuming an idle descent where no engine power is required in order to minimize the required energy storage to optimise the possibility of battery use. Furthermore, I am assuming no wind, constant mass, i.e. no fuel consumption, and 100% engine efficiency. I used the following data [2][3][6][7][8], which are published specifications for the B7378.

Mass aircraft (Max T/O weight) 70535 kg

g 9.81 N kg-1 W 691948 N Wing plan area, S 124.58 m2 Air density at sea level 1.225 kg m-3 Relative density at 12km 0.2546 ρ 0.312 kg m-3 Oswald Efficiency, e 0.755 Aspect Ratio, AR 9.54 Drag Coefficient, CDo 0.016 Cruise Altitude 12000 m

Table 1, Base data for power calculations

Using data in table 1, I calculated the velocity for minimum power needed for the cruise phase, level flight at 12000m, as;

𝑉$%&0")*+ = 185𝑚𝑠1.(3𝑠𝑓) Hence;

𝑃2*3 = 7.91𝑀𝑊

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Figure 3 matches closely the predicted curve for the power equation validating the data I have used for my calculations. For the climb phase I approximated the power requirements as power required to lift the mass of the plane to 12000m in 15 minutes using;

𝐺𝑃𝐸 = 𝑚𝑔∆ℎ

𝑃 =𝐸𝑡

𝑃 = 9.23𝑀𝑊 And power required for level flight at average altitude: Average altitude – 6000m Relative density at 6000m – 0.5389 Air density at 6000m, ρ – 0.660 kg m-3

𝑉$%&0")*+ = 127𝑚𝑠1. 𝑃2*3 = 5.44𝑀𝑊

𝑃45%67!"# = 14.7𝑀𝑊

Using the required power for climb and cruise I can calculated energy needed for a flight with cruise time 1 hour and climb 15 minutes

𝐸 = 𝑃2*3𝑡 𝐸45%67 = 13200𝑀𝐽 𝐸4+8%9* = 28500𝑀𝐽

𝐸!"# = 41700𝑀𝐽 = 11600 kWh

The Power Supply Currently all commercial passenger planes around the world are powered by combustion engines, predominantly fuelled by jet fuel, which is a kerosene grade of fuel suited for flight due to its low freezing point of -47º C. Hydrocarbon fuel is so widely used due to its incredibly high energy density; jet fuel has an energy density of 42.8 MJ/kg [9]. Therefore, despite the inefficiency of the combustion engine huge amounts of power can be generated from the fuel. The B7378 carries 26000L of fuel at max take-off weight [2] which at a density of 0.804 kgl-1 means the mass of fuel is 21000kg. Therefore, in my theoretical conversion of the B7378 to electric power the total mass of batteries must be less than 21000kg. With these parameters the greatest limitation on the feasibility of this conversion will be the energy density of the batteries used. Battery Energy Density

Wh/kg Lead Acid 40 Nickel Cadmium 65 Lithium Ion 114-247

Table 2 [10][11]

From table 2, it is clear the lithium ion battery configuration has the highest energy density of all current practical battery technology. There are many different forms, however, the most common are LiCoO2 or the NMC lithium ion battery (LiNixCoyMnzO2, where x, y and z vary dependant on the total sum of the others). The Tesla model 3 2170 cell is the current best rated lithium ion cell with an energy density of 247 Wh/kg.

Figure 3, Graph showing calculated power required against aircraft speed

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When looking at my desired example flight from Cork to London Stanstead, I calculated for a climb and short cruise time of 15 minutes each [3];

𝐸 = 𝑃2*3𝑡 𝐸45%67 = 13200𝑀𝐽 𝐸4+8%9* = 7100𝑀𝐽

𝐸!"# = 20300𝑀𝐽 = 5650 kWh This equates to a required energy density of 270 Wh/kg. From calculations above I plotted cruise time against required energy density of battery, assuming:

- 15-minute climb - Cruise altitude 12000m - Idle decent

From figure 4 it is clear that current battery technology limits significantly the range of an electric B7378. A flight of 4 times the cruise time (60 minutes) of the flight from Cork to Stanstead requires a battery of energy density of 550 Wh/kg which is over double that for the 15-minute cruise. Even taking into account 100% engine efficiency, no further drag or extra safety requirements for a fly around, an approximated climb phase and idle descent, from my calculation current battery technology would not be capable of powering a B7378 for even a short flight from Cork to London Stanstead.

There is increasing battery research into future battery technology as fossil fuels and the combustion engine become less acceptable. The amount of research into solid state and lithium sulfur has increased rapidly. As shown in figure 5 they are the second and third most researched battery technologies behind only Li ion, which is reaching its theoretical limit.

With future batteries there should be a significant improvement to performance and in particular to energy density. Future Battery Predicted/

Theoretical Energy Density Wh/kg

Solid State Lithium Metal

>300

Lithium Sulfur 2500 Metal Air 11000

Table 3 [10][12]

Using the data in table 3, I could calculate the possible cruise time each type of battery technology could provide by calculating the total energy stored in 21000kg of the battery. Using previous calculations for power requirements;

𝐸!"# = 𝐸-*&9%#: × 21000𝑘𝑔 𝐸4+8%9* = 𝐸!"# − 𝐸45%67

𝑡4+8%9* =𝐸_𝐶𝑟𝑢𝑖𝑠𝑒𝑃2*34+8%9*

Figure 5, Number of publications with keywords in title of abstracts against year from 1976-2018 [10]

Figure 4, Graph showing required battery energy density against cruise time

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Solid state Li Metal - >19.98 min Lithium Sulfur – 6.17 hours Metal Air – 28.73 hours My calculations have shown that with predicted future battery technologies, a plane of size similar to the Boeing 737800 could theoretically be powered by batteries, potentially creating clean commercial flight all the way around the globe.

Limitations To simplify the calculations I have done there are many approximations that I will address now as follows: 100% engine efficiency I have assumed that all the energy supplied by the batteries will be directly transferred into thrust. While electric motors are much more efficient than combustion engines, they are not quite 100%. There is still energy loss through friction between parts, sound and thermal energy produced by the batteries themselves. Perfect flying conditions I have assumed that no head wind or tail wind is affecting my plane, of course a tailwind would help the batteries increase the range of the aircraft, however, a headwind would be detrimental to the performance of the batteries as drag forces increase. However, a headwind could increase lift as the airspeed over the aerofoil is faster possibly reducing the power requirement for lift. Engine design I have assumed that the engine when powered by batteries will still produce the same amount of thrust as when powered by a combustion engine. However, a jet engine relies on the compression of about 20% of the airflow by the combustion components, which would not be possible

without the combustion of jet fuel. In reality an electric plane would most likely have numerous propellers producing thrust to reduce drag and unnecessary weight. But I have not taken this into account. Wave Drag Wave drag occurs when airflow over the wing is supersonic (faster than speed of sound) which creates sonic shocks breaking the airflow boundary layer on the camber of the wing, however this will not occur under the critical Mach number, Mc, for the plane. Even then the wave drag remains relatively small until Drag-divergence Mach number, MDD, is reached when it rapidly increases the drag coefficient. [5] NB. Mach number is a ratio between true airspeed and speed of sound at that altitude. Mc is the lowest Mach number at which airflow over the plane reaches the speed of sound but not exceeding it. MDD is the Mach number at which wave drag rapidly increases form this point on.

Figure 6, Graph showing effect of Mach number on wave drag [5]

Figure 6 shows how below the critical Mach number MC for a subsonic plane like the B7378 wave drag remains negligible until MDD. For the B737800 [13];

𝑀4 = 0.6 𝑀-- = 0.8

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From my previous calculations the optimum cruise velocity for B7378 is 185 ms-1. Speed of sound at 12000m, a is 295 ms-1

[6]

𝑀 =𝑉𝑎

Mach number at cruise; 𝑀4+8%9* = 0.627

As 𝑀4 < 𝑀4+%89* < 𝑀--

Wave drag can be ignored as negligible. Climb Phase My calculations for the climb phase of the flight rely heavily on major approximations of the power requirements. It also ignores the acceleration on the runway completely, which will be one of the most power intensive phases, as the plane must accelerate the quickest and is in the densest air at sea level so parasitic drag is at its highest.

Numerical Model {See App.B for larger graphs and data} To better approximate power required for the whole flight I split my climb phase into 4 further parts: runway, initial climb, secondary climb, tertiary climb. [3][14] This will improve my model as the conditions of the flight change constantly throughout the climb, so the more parts it is split into the better. I will calculate the power required at each phase of the flight to know the max power required by combining the power to accelerate the plane and power to climb in each phase. I can then use this to sum the energy required in each phase of the flight and hence find required energy density of a possible battery to be used over the whole flight. For the 4 climb phases I will use a numerical method to find the power. For

the cruise in level flight I can use my previous calculation;

𝑃1#2345 =12𝜌𝑉-𝑆𝐶%& +

𝑊$

12𝜌𝑉𝑆

P1

𝜋𝑒𝐴𝑅U

Phases of Climb Runway

- Altitude 0 m - Rate of climb 0 ms-1 - Velocity 0 – 80 ms-1 - Time 0.5 minutes - Acceleration 2.67 ms-2 - CL 1.09

Initial Climb - Altitude 0 - 305m - Rate of climb 1.69 ms-1 - Velocity 80 - 85 ms-1 - Time 3 minutes - Acceleration 0.03 ms-2 - CL 1.09

Secondary Climb - Altitude 305 – 4600 m - Rate of climb 14.32 ms-1 - Velocity 85-150 ms-1 - Time 5 minutes - Acceleration 0.217 ms-2 - CL 0.40

Tertiary Climb

- Altitude 4600 – 12000 m - Rate of climb 18.97 ms-1 - Velocity 150 – 180 ms-1 - Time 6.5 minutes - Acceleration 0.09 ms-2 - CL 0.4

In addition, my equation for power had to be modified as previously I had assumed no acceleration and the only force opposing the thrust as drag, in level flight. However, now as shown in figure 7, I must consider the component of weight in the direction of flight and the power required to accelerate the plane.

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- 𝛼 – Angle of attack - γ – Angle of climb - V – Velocity direction vector - F – Force direction Vector - Mg – Weight of aircraft - D – Drag force vector

S𝐹 = 𝑚𝑎 = 𝐹𝑐𝑜𝑠(𝛼) − 𝐷 −𝑚𝑔𝑠𝑖𝑛(𝛾) However, 𝛼is often less than 10° [5] so;

cos(𝛼) ≈ 1 And;

𝐹 = 𝐷 +𝑚𝑔𝑠𝑖𝑛(𝛾) + 𝑚𝑎 𝐷 = 𝐷;&< + 𝐷0=+=

To find the power required I plotted a graph of drag against velocity and GPE against time and found the area under and gradient of the line respectively. Using a numerical method with values as shown under the phase headings and constants used previously, I calculated force required at various time intervals using; 𝐹 =

12𝜌𝑉

%𝑆𝐶&' + 𝐶(% *1

𝜋𝑒𝐴𝑅/ +𝑚(𝑔𝑠𝑖𝑛(𝛾) + 𝑎)

Where 𝜌was found by interpolation along a graph of relative air density against altitude.[6] GPE was calculated using;

𝐺𝑃𝐸 = 𝑚𝑔ℎ

Analysis

Figure 8, Graph showing Force against velocity

To calculate the area under figure 8, I used the trapezium rule by interpolating force values using a velocity interval of 1 ms-1. Results of this are in table 4. The graph is separated due to the force required to accelerate the plane (𝑚𝑎 term in the equation), as I assumed different constant accelerations for each phase.

Power to accelerate Runway 14.58 MW Initial 0.14 MW Secondary 1.82 MW Tertiary 0.72 MW Cruise 7.91 MW

Table 4, Power to accelerate the plane

Figure 9, Graph showing change in GPE vs Time

I calculated the gradient of each phase of climb from figure 9 to find the power required to lift the plane to 12000m. Results for this is shown in table 5.

Figure 7, Diagram showing component forces acting on climbing aircraft [5]

0.0

50000.0

100000.0

150000.0

200000.0

250000.0

0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 200.0

Forc

e/ N

Velocity/ ms-1

Force vs Velocity

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 100 200 300 400 500 600 700 800 900 1000

GPE

Gain

ed/ M

J

Time/ s

GPE vs Time

9

Power to climb

Runway 0.00 MW Initial 1.17 MW Secondary 9.91 MW Tertiary 13.13 MW Cruise 0.00 MW

Table 5, Power for plane to climb

Therefore, the total power required for flight;

Total power

Runway 14.58 MW Initial 1.32 MW Secondary 11.73 MW Tertiary 13.85 MW Cruise 7.91 MW

Table 6, Total power of whole flight

From table 6, the max power output of a battery used would have to be 14.58 MW as the power to accelerate the plane on the runway is the most power intense phase of the flight. For the flight from Cork to London Stanstead, of climb 15 minutes and cruise 15 minutes the total energy requirements are;

Table 7, Energy requirements for exemplar flight

From table 7, converting from MJ to kWh, the total energy for the flight from Cork to Stanstead is 4643 kWh, which with 21000kg of batteries would need a battery of energy density 221 Wh/kg. This is within current battery technology limits. My numerical model has produced a smaller value for the power required in the climb phase of the flight than my previous calculation which I believe to be more accurate within my parameters.

Therefore, within my assumptions I can conclude that the conversion of the Boeing 737800 to battery power is possible for short flights alike Cork to London Stanstead with current battery technology. However, practically I believe battery technology is still far off the requirements that a real-world situation would demand.

Aerofoil Design [See App.C for graphs and data} The B7378 is designed to cruise at about 230 m/s however to reduce power consumption based on figure 3, I determined that the electric converted plane would cruise at 185 m/s for most efficient power use. The required lift coefficient of the wing at 𝛼 = 0° (level flight) can be calculated using;

𝐶*657 =𝑊𝑐𝑜𝑠𝛾12𝜌𝑉

$𝑆

Using data from table 1; 𝐶>2*3 = 1.02

However, I must ensure the drag coefficient of the wing remains at 0.016 or less to ensure no more power is required to overcome drag. To design this wing, I used a software called Xfoil [15]. This allowed me to create an aerofoil and produce pressure polars to give the lift, drag and moment coefficients of the wing and the terminal point.

Energy for Cork to Stanstead

Runway 437 MJ Initial 237 MJ Secondary 3519 MJ Tertiary 5402 MJ Cruise 7120 MJ Total 16716 MJ

leading edge

trailing edge

lower surface

upper surface

Figure 10, Aerofoil diagram

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I started with the NACA wings [16] developed by NASA in the 1920’s and changed, max camber, max thickness and position along the aerofoil of max camber and thickness. For example, the NACA 2210 wing, has max camber 0.02 at x = 0.20 and max thickness 0.10 at x=0.30, I adjusted these variables to understand their effect on lift drag ratio. From this I found moving the max camber towards the trailing edge will increase the lift drag ratio. Moving the max thickness towards the trailing edge can reduce the drag however has little effect on the lift drag ratio. Increasing camber will increase the lift drag ratio and changing thickness has little noticeable effect. Although I did find that a thinner leading edge allows for a better lift drag ratio. Xfoil can calculate for viscous liquids so I calculated the Reynolds number and mach number for the plane at cruising altitude [17]. Reynolds – 1.19x106

Mach – 0.44 I used the NACA 5330 wing as a base for my final design. (See appendix C for earlier iterations) Final design Base wing NACA 5330 Max Camber

0.01 at x = 0.32

Max Thickness

0.15 at x = 0.29

Table 8, Final Wing geometrics designed on Xfoil

Figure 11, Designed aerofoil

Table 9, Output of Xfoil calculations

This shape aerofoil, in figure 11, provides a CL greater than required at 1.027 and has the drag coefficient, CD, of 0.0157. Which is equal to the value of CD for the original wing 0.016. The top and bottom xtr values in table 9 indicate where boundary layer separation occurs along the aerofoil, this is called the terminal point. Boundary layer separation is the separation of laminar airflow from the surface of the aerofoil into turbulent flow. The closer to the trailing edge this occurs the less drag experienced by the wing. [5]

Figure 12, Overlay of real and my design of aerofoil {18}

Figure 12 shows my designed aerofoil in blue and the actual B7378 aerofoil in white. It has a much higher camber than the original B7378 wing. This allows for higher lift as a higher air pressure can build on the bottom edge as shown by the pressure polar, figure 13. This polar is reversed to have the pressure of the upper side at the top. However, the upper side has lower negative pressure. The greater the difference between the upper and lower curves the more lift is created.

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Figure 13, Pressure polar for 𝛼 = 0

When comparing the lift and drag coefficients at varying angles of attack, as shown in figure 14, the highest lift drag ratio of 70.4 is achieved at 𝛼 = 1.5°. Showing that my aerofoil is most efficient at 𝛼 = 1.5°, however will still function at a required standard when in level flight. By plotting a CL against 𝛼graph I can find the angle of attack for max lift and at what angle stall occurs. Stall is a loss of control caused by flow separation from the boundary layer on the upper surface of the aerofoil. As angle of attack increases the air pressure on upper surface decreases until separation occurs. [5] From figure 15 , CLmax = 2.016 at 𝛼 =14°, beyond this point CL decreases, this is where stall occurs. As most commonly 𝛼 < 10° stall should typically not be a problem for my wing.

Figure 15, Graph showing CL against 𝛼

My aerofoil design is for the midspan of the wing only and so root and tip aerofoil modelling would also be required along with redesign of all aerodynamic surfaces. This would include the tail and fin but also flaps and ailerons among other moveable aerodynamic parts.

Figure 14, Pressure polar for 𝛼 =−5𝑡𝑜5in 0.5° steps

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References [1] Air Transport Action Group, https://www.atag.org/facts-figures.html [2] The Boeing 737 technical site http://www.b737.org.uk/techspecsdetailed.htm [3] UK Flight Aware flight tracker https://uk.flightaware.com/live/flight/RYR906/history/20200501/1752Z/EICK/EGSS [4] MIT education Thermodynamics and Propulsion https://web.mit.edu/16.unified/www/SPRING/propulsion/notes/node97.html [5] Future Learn ABC flight mechanics, Eric Poquillon [6] Engineering Tool box, International standard atmosphere, https://www.engineeringtoolbox.com/international-standard-atmosphere-d_985.html

[7]Estimating the Oswald factor from basic aircraft geometrical parameters, Mihaela Nita, 12/09/12, https://www.fzt.haw-hamburg.de/pers/Scholz/OPerA/OPerA_PRE_DLRK_12-09-10_MethodOnly.pdf

[8] Response to the 2009-2010 AIAA Foundation Undergraduate Team Aircraft Design Competition, Jason Henn, http://www.dept.aoe.vt.edu/~mason/Mason_f/VTTEnvironauticsEN-1UGACD.pdf

[9] Alternative Jet Fuels, William Greenbaum, 11/12/12 http://large.stanford.edu/courses/2012/ph240/greenbaum1/

[10] Challenges and perspectives for new material solutions in batteries, Vittorio Pellegrini, https://arxiv.org/pdf/2004.02114.pdf

[11]Energy Density of Cylindrical Li-Ion Cells: A Comparison of Commercial 1850 to the 21700 Cells, Jason B. Quinn, 19/10/18

https://iopscience.iop.org/article/10.1149/2.0281814jes

[12] Inward-growth plating of lithium driven by solid-solution based alloy phase for highly reversible lithium metal anode, Song Jin, https://arxiv.org/pdf/1910.13159.pdf

[13] Drag Estimation, Deiter Scholz, https://www.fzt.haw-hamburg.de/pers/Scholz/materialFM1/DragEstimation.pdf

[14]Boeing 737800 performance data https://www.skybrary.aero/index.php/B738

[15]Xfoil download https://web.mit.edu/drela/Public/web/xfoil/

[16]NACA Wings https://occamsracers.com/2019/05/23/naca-wing-shapes-and-airfoil-tools/

[17]Mach and Reynolds number calculator http://www.aerospaceweb.org/design/scripts/atmosphere/

[18] Boeing 737800 midspan aerofoil shape http://airfoiltools.com/polar/details?polar=xf-b737b-il-1000000

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Appendix

A: Power Calculations

Figure 3

Velocity /kts 145 150 155 160 165 170 175 180 185 190Velocity /ms-1 74.59 77.17 79.74 82.31 84.88 87.46 90.03 92.60 95.17 97.74 Power Parasitic drag/ MW 0.13 0.14 0.16 0.17 0.19 0.21 0.23 0.25 0.27 0.29 Power Induced drag/ MW 14.74 14.25 13.79 13.36 12.95 12.57 12.21 11.87 11.55 11.25 Total power requried/ W 14.87 14.39 13.95 13.53 13.14 12.78 12.44 12.12 11.82 11.54

195 200 205 210 215 220 225 230 235 240100.32 102.89 105.46 108.03 110.61 113.18 115.75 118.32 120.89 123.47

0.31 0.34 0.36 0.39 0.42 0.45 0.48 0.51 0.55 0.5910.96 10.69 10.43 10.18 9.94 9.72 9.50 9.29 9.09 8.9111.27 11.03 10.79 10.57 10.36 10.17 9.98 9.81 9.64 9.49

245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00126.04 128.61 131.18 133.76 136.33 138.90 141.47 144.04 146.62 149.19

0.62 0.66 0.70 0.74 0.79 0.83 0.88 0.93 0.98 1.038.72 8.55 8.38 8.22 8.07 7.92 7.77 7.63 7.50 7.379.35 9.21 9.08 8.96 8.85 8.75 8.65 8.56 8.48 8.40

295.00 300.00 305.00 310.00 315.00 320.00 325.00 330.00 335.00 340.00151.76 154.33 156.91 159.48 162.05 164.62 167.19 169.77 172.34 174.91

1.09 1.14 1.20 1.26 1.32 1.39 1.45 1.52 1.59 1.667.25 7.12 7.01 6.89 6.79 6.68 6.58 6.48 6.38 6.298.33 8.27 8.21 8.16 8.11 8.07 8.03 8.00 7.97 7.95

345.00 350.00 355.00 360.00 365.00 370.00 375.00 380.00 385.00 390.00177.48 180.06 182.63 185.20 187.77 190.34 192.92 195.49 198.06 200.63

1.74 1.81 1.89 1.97 2.06 2.14 2.23 2.32 2.42 2.516.20 6.11 6.02 5.94 5.86 5.78 5.70 5.62 5.55 5.487.93 7.92 7.91 7.91 7.91 7.92 7.93 7.95 7.97 7.99

395.00 400.00 405.00 410.00 415.00 420.00 425.00 430.00 435.00 440.00203.21 205.78 208.35 210.92 213.49 216.07 218.64 221.21 223.78 226.36

2.61 2.71 2.81 2.92 3.02 3.14 3.25 3.36 3.48 3.615.41 5.34 5.28 5.21 5.15 5.09 5.03 4.97 4.91 4.868.02 8.05 8.09 8.13 8.17 8.22 8.28 8.34 8.40 8.46

445.00 450.00 455.00 460.00 465.00 470.00 475.00 480.00228.93 231.50 234.07 236.64 239.22 241.79 244.36 246.93

3.73 3.86 3.99 4.12 4.26 4.39 4.54 4.684.80 4.75 4.70 4.65 4.60 4.55 4.50 4.458.53 8.61 8.68 8.77 8.85 8.94 9.04 9.13

14

Figure 4

15

Figure 5

B: Numerical model

Climb for 15 mins 13197.49662 MJPower required 7911460.771 WMass fuel 20932.14 kgcruise time / minutes 3 5 10 15 30 45 60 75 90 105Required Energy for cruise + 15 mins climb /MJ 14,621.56 15,570.93 17,944.37 20,317.81 27,438.13 34,558.44 41,678.76 48,799.07 55,919.38 63,039.70 Wh 4,061,544.32 4,325,259.68 4,984,548.08 5,643,836.48 7,621,701.67 9,599,566.86 11,577,432.06 13,555,297.25 15,533,162.44 17,511,027.63

Requried energy desity of battery / Whkg-1 194.03 206.63 238.13 269.63 364.11 458.60 553.09 647.58 742.07 836.56

120 135 150 165 180 195 210 225 240 25570160.01 77280.33 84400.64 91520.96 98641.27 105761.59 112881.90 120002.22 127122.53 134242.85

19488892.83 21466758.02 23444623.21 25422488.40 27400353.60 29378218.79 31356083.98 33333949.18 35311814.37 37289679.56931.05 1025.54 1120.03 1214.52 1309.01 1403.50 1497.99 1592.48 1686.97 1781.46

270 285 300 315 330 345 360 375 390 405141363.16 148483.48 155603.79 162724.11 169844.42 176964.73 184085.05 191205.36 198325.68 205445.99

39267544.75 41245409.95 43223275.14 45201140.33 47179005.52 49156870.72 51134735.91 53112601.10 55090466.30 57068331.491875.95 1970.43 2064.92 2159.41 2253.90 2348.39 2442.88 2537.37 2631.86 2726.35

420 435 450 465 480 495212566.31 219686.62 226806.94 233927.25 241047.57 248167.88

59046196.68 61024061.87 63001927.07 64979792.26 66957657.45 68935522.642820.84 2915.33 3009.82 3104.31 3198.80 3293.29

16

Runway Total elapsed time/ s Time/ s Altitude/ m Rho/ kgm-3 Velocity/ ms-1Force/ N GPE/ MJ0 0 0.0 1.2250 0.0 0.1 0.002 2 0.0 1.2250 5.3 188128.1 0.004 4 0.0 1.2250 10.7 188232.3 0.006 6 0.0 1.2250 16.0 188405.9 0.008 8 0.0 1.2250 21.3 188649.0 0.00

10 10 0.0 1.2250 26.7 188961.6 0.0012 12 0.0 1.2250 32.0 189343.6 0.0014 14 0.0 1.2250 37.3 189795.0 0.0016 16 0.0 1.2250 42.7 190315.9 0.0018 18 0.0 1.2250 48.0 190906.3 0.0020 20 0.0 1.2250 53.3 191566.1 0.0022 22 0.0 1.2250 58.7 192295.4 0.0024 24 0.0 1.2250 64.0 193094.1 0.0026 26 0.0 1.2250 69.3 193962.3 0.0028 28 0.0 1.2250 74.7 194900.0 0.0030 30 0.0 1.2250 80.0 195907.0 0.00

Initial Climb Total elapsed time/ s Time/ s Altitude/ m Rho/ kgm-3 Velocity/ ms-1Force/ N GPE/ MJ0 0.0 1.2250 80.0 195907.0 0.00

45 15 25.4 1.2221 80.4 9835.7 17.5960 30 50.8 1.2191 80.8 9898.4 35.1775 45 76.3 1.2162 81.3 9961.2 52.7690 60 101.7 1.2133 81.7 10024.0 70.35

105 75 127.1 1.2103 82.1 10086.8 87.94120 90 152.5 1.2074 82.5 10149.6 105.52135 105 177.9 1.2045 82.9 10212.5 123.11150 120 203.3 1.2015 83.3 10275.3 140.70165 135 228.8 1.1986 83.8 10338.2 158.28180 150 254.2 1.1957 84.2 10401.1 175.87195 165 279.6 1.1927 84.6 10463.9 193.46210 180 305.0 1.1898 85.0 10526.8 211.04

Secondary Total elapsed time/ s Time/ s Altitude/ m Rho/ kgm-3 Velocity/ ms-1Force/ N GPE/ MJ0 305.0 1.1898 85.0 10526.8 211.04

225 15 519.8 1.1651 88.3 24326.1 359.64240 30 734.5 1.1412 91.5 24805.1 508.24255 45 949.3 1.1173 94.8 25279.9 656.83270 60 1164.0 1.0941 98.0 25755.3 805.43285 75 1378.8 1.0711 101.3 26226.6 954.02300 90 1593.5 1.0485 104.5 26694.2 1102.62315 105 1808.3 1.0264 107.8 27158.7 1251.22330 120 2023.0 1.0043 111.0 27615.0 1399.81345 135 2237.8 0.9830 114.3 28070.6 1548.41360 150 2452.5 0.9617 117.5 28515.3 1697.00375 165 2667.3 0.9410 120.8 28957.3 1845.60390 180 2882.0 0.9206 124.0 29389.7 1994.20405 195 3096.8 0.9004 127.3 29813.9 2142.79420 210 3311.5 0.8807 130.5 30230.8 2291.39435 225 3526.3 0.8611 133.8 30634.6 2439.98450 240 3741.0 0.8422 137.0 31036.5 2588.58465 255 3955.8 0.8233 140.3 31422.5 2737.17480 270 4170.5 0.8049 143.5 31802.7 2885.77495 285 4385.3 0.7867 146.8 32168.7 3034.37510 300 4600.0 0.7689 150.0 32524.9 3182.96

17

Figure 8

Tertiary Total elapsed time/ s Time/ s Altitude/ m Rho/ kgm-3 Velocity/ ms-1Force/ N GPE/ MJ0 4600.0 0.7689 150.0 32524.9 3182.96

525 15 4884.6 0.7458 151.3 23356.1 3379.90540 30 5169.2 0.7233 152.7 23136.5 3576.84555 45 5453.8 0.7011 154.0 22909.8 3773.78570 60 5738.5 0.6797 155.4 22685.5 3970.72585 75 6023.1 0.6585 156.7 22451.1 4167.66600 90 6307.7 0.6381 158.1 22220.4 4364.60615 105 6592.3 0.6180 159.4 21982.7 4561.54630 120 6876.9 0.5985 160.8 21747.0 4758.48645 135 7161.5 0.5795 162.1 21508.0 4955.41660 150 7446.2 0.5608 163.5 21263.5 5152.35675 165 7730.8 0.5427 164.8 21021.4 5349.29690 180 8015.4 0.5248 166.2 20770.6 5546.23705 195 8300.0 0.5078 167.5 20527.8 5743.17720 210 8584.6 0.4909 168.8 20278.0 5940.11735 225 8869.2 0.4746 170.2 20030.2 6137.05750 240 9153.8 0.4586 171.5 19780.3 6333.99765 255 9438.5 0.4430 172.9 19527.0 6530.93780 270 9723.1 0.4280 174.2 19278.9 6727.87795 285 10007.7 0.4132 175.6 19024.0 6924.81810 300 10292.3 0.3990 176.9 18775.6 7121.75825 315 10576.9 0.3849 178.3 18521.1 7318.68840 330 10861.5 0.3714 179.6 18271.0 7515.62855 345 11146.2 0.3568 181.0 17975.3 7712.56870 360 11430.8 0.3413 182.3 17634.1 7909.50885 375 11715.4 0.3265 183.7 17303.7 8106.44900 390 12000.0 0.3119 185.0 16968.1 8303.38

18

Figure 9

Air density interpolation

19

C: Aerofoil Design Iterations: No.1 – Base shape NACA 2210, adjusted

No.2 - Base shape NACA 7410, adjusted

20

No.3 - Base shape NACA 5330, adjusted

21

Final Design – e-B7378, base aerofoil NACA 5330 Figure 10

22

Figure 11

Figure 12

23

Figure 13

24

Original Polars

Figure 14

25

Calculated polar for: e-B7378

1 1 Reynolds number fixed Mach number fixed

xtrf = 1 (top) 1 (bottom)Mach = 0.44 Re = 1.196 e 6 Ncrit = 9

alpha CL CD CDp CM Top_Xtr Bot_Xtr------ -------- --------- --------- -------- -------- --------

-20 0.0812 0.16793 0.16634 -0.1015 0.9516 0.0086-19.5 0.0954 0.16417 0.16253 -0.1039 0.9409 0.0086-19 0.1105 0.16034 0.15866 -0.1062 0.9293 0.0087

-18.5 0.1274 0.15698 0.15525 -0.1084 0.9164 0.0089-18 0.1441 0.15376 0.15198 -0.1107 0.902 0.0091

-17.5 0.1604 0.15056 0.14869 -0.113 0.8849 0.0096-17 0.1763 0.14737 0.14542 -0.1152 0.8683 0.0102

-16.5 0.1905 0.1442 0.14217 -0.1174 0.8552 0.0108-16 0.2026 0.14155 0.13944 -0.1197 0.8419 0.0109-15 0.2363 0.13344 0.13119 -0.1241 0.8096 0.0111

-14.5 0.255 0.1296 0.12726 -0.1262 0.7913 0.0114-14 0.2731 0.12563 0.12319 -0.1283 0.7723 0.0118

-13.5 0.29 0.12147 0.11894 -0.1302 0.7526 0.0125-13 0.3057 0.11677 0.11415 -0.1321 0.7317 0.0133

-12.5 0.3187 0.11186 0.10914 -0.1343 0.7103 0.0135-12 0.3336 0.10685 0.10402 -0.136 0.6857 0.0135

-11.5 0.3529 0.1024 0.09946 -0.1371 0.6587 0.0138-10.5 0.3887 0.09358 0.09037 -0.1399 0.5992 0.0147-10 0.405 0.0888 0.08547 -0.1413 0.5682 0.0159-9.5 0.4166 0.08363 0.08016 -0.143 0.5387 0.0162-9 0.377 0.09371 0.09012 -0.1431 0.5473 0.0162

-8.5 0.3946 0.08881 0.08507 -0.1451 0.5136 0.0163-8 0.4173 0.08497 0.08108 -0.146 0.478 0.0166

-7.5 0.4387 0.08132 0.0773 -0.1472 0.4493 0.0172-7 0.4588 0.07742 0.07332 -0.1486 0.4287 0.0181

-6.5 0.4716 0.0729 0.06877 -0.1507 0.4148 0.0191-6 0.4766 0.06898 0.06486 -0.1505 0.4043 0.0192

-5.5 0.4961 0.06558 0.06144 -0.15 0.394 0.0195-5 0.5158 0.06236 0.05821 -0.1508 0.3861 0.0201

-4.5 0.5354 0.05878 0.05464 -0.152 0.3801 0.0213-4 0.5557 0.05337 0.04922 -0.1572 0.3765 0.0223

-3.5 0.5598 0.05008 0.04596 -0.1523 0.3722 0.0226-3 0.5812 0.04729 0.04315 -0.1505 0.3672 0.023

-2.5 0.6385 0.04171 0.03744 -0.1586 0.3635 0.0257-1.5 0.7543 0.03294 0.02842 -0.1674 0.3571 0.025-1 0.895 0.01857 0.01287 -0.1899 0.3533 0.0277

-0.5 0.9704 0.01577 0.00923 -0.1935 0.3485 0.0310 1.0294 0.0158 0.00909 -0.1925 0.3465 0.0333

0.5 1.0868 0.01604 0.00929 -0.1913 0.3443 0.03641 1.1425 0.01641 0.00962 -0.19 0.3419 0.0394

1.5 1.1961 0.01698 0.01016 -0.1885 0.3395 0.04332 1.2467 0.01775 0.01093 -0.1866 0.337 0.0476

2.5 1.2946 0.0187 0.01189 -0.1844 0.3343 0.05313 1.3406 0.01979 0.013 -0.1822 0.3313 0.0596

3.5 1.3911 0.02072 0.01399 -0.1806 0.33 0.06794 1.4399 0.02177 0.01511 -0.1789 0.3284 0.0778

4.5 1.4872 0.02293 0.01634 -0.1771 0.3266 0.09085 1.5325 0.02424 0.01774 -0.1751 0.3247 0.1069

5.5 1.5766 0.02568 0.01925 -0.1731 0.3228 0.12616 1.619 0.02726 0.02091 -0.171 0.3208 0.15

6.5 1.6577 0.0291 0.02285 -0.1686 0.3184 0.18467 1.7003 0.03013 0.02527 -0.1673 0.3159 1

7.5 1.7411 0.03202 0.02721 -0.1653 0.3148 18 1.779 0.03412 0.02937 -0.1631 0.3134 1

8.5 1.8146 0.03643 0.03176 -0.1608 0.3117 19 1.8484 0.03893 0.03433 -0.1585 0.3097 1

9.5 1.8802 0.04163 0.0371 -0.1562 0.3076 110 1.9099 0.04452 0.04006 -0.1538 0.3057 1

10.5 1.9359 0.04773 0.04335 -0.1513 0.3037 111 1.9568 0.05123 0.04692 -0.1484 0.3013 1

11.5 1.9743 0.0553 0.05113 -0.1459 0.2997 112 1.9876 0.0599 0.0559 -0.1436 0.2975 1

12.5 1.9966 0.06489 0.06104 -0.1412 0.295 113 2.0034 0.07014 0.06643 -0.1391 0.2925 1

13.5 2.0117 0.07535 0.07174 -0.1372 0.29 114 2.0155 0.08085 0.07733 -0.1352 0.287 1

14.5 1.9736 0.09114 0.08788 -0.1334 0.2835 115 1.9347 0.1015 0.09844 -0.1325 0.2787 1

15.5 1.9362 0.10772 0.10474 -0.132 0.2752 116 1.8032 0.12918 0.12646 -0.1343 0.2643 1

16.5 1.8259 0.13323 0.13055 -0.1344 0.2613 117.5 1.6868 0.16502 0.16258 -0.1435 0.2378 118 1.7233 0.1673 0.16491 -0.1439 0.2358 1

18.5 1.7678 0.16858 0.16623 -0.1439 0.232 119 1.7729 0.17528 0.173 -0.1464 0.2237 1

26