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Economics for Electric Take-Off to 10,000m AltitudeYou can use this spreadsheet to experiment with the parameters for an electric-takeoff system.

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Electricity Cost EstimatesValueUnitNotes

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Electricity Cost80$/MWh

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Tow Aircraft Power Required13.3876190674319MWSee power calculation below

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Hourly Cost1071.00952539455$/h

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Weight of electric tow aircraft10000KGAssume no seats, no aviation fuel, retrofitted electric engines

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Drag from Power Cable

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Width of cable (w)0.05mHMPE rope and aluminium conductor. Assume aluminium and HMPE are packaged in aerodynamic sheath.

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Length of electric cable (xmax)10770.329614269mAssume 10km altitude, 4km radius for tow plane

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Area of cable1538.51648071345m2

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Coefficent of drag for cable with aerodynamic sheath/fairing0.02http://en.wikipedia.org/wiki/Drag_coefficient#Aircraft

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Air density (10000m altitude)0.4135kg/m3http://www.aerospaceweb.org/design/scripts/atmosphere/

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Velocity450kmhFastest point of cable

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Max Cable Velocity (m/s) (Vmax)125m/s

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Cable Velocity at point x on cable: Vx = Vmax * x / xmaxVelocity on cable is proportional to radius of circle (i.e. distance x from center of circle)

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Substitute Cable Velocity equation into drag equationDrag(x) = 1/2 * rho * (Vmax * x / xmax)^2 * cD * w

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Integrate from x=0 to x=xmax to get total drag11597.7377486985Total drag = 1/2 * rho * 1/3 x^3 * Vmax^2 / xmax ^2 * cD * w

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Cable drag11597.7377486985Nhttp://en.wikipedia.org/wiki/Drag_equation

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Cable drag for tow cable + two pulley cables34793.2132460956N

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Aluminum weight (1km)150KG/km

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HMPE cable weight (1km) (184 MT breaking load)1340kg/kmhttp://www.samsonrope.com/offshore-pdf/ofs-neutron-8-specs.pdf

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Aluminum total weight3,231KGTwo conductors to tow height for high voltage direct current power (HVDC)

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HMPE + aluminum cable total weight 43431.0988842807KGAssume three tethers to tow plane height

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Freight Cost per Tonne-Mile

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Payload (max. weight) - includes aircraft and cargo35,092KGNote payload weight is halved to account for dual downward forces on the pulley (winch and payload)

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Cargo weight17545.9441883686KGAssume 50% cargo, 50% aircraft weight.

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Lift duration3min

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Lifts per hour20Assumes continuous operation

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Cargo per hour (weight)350,919KG

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Energy Cost per 1000KG payload (fixed)$3.05$/1000KGAssumes continuous operation

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Winch Lifting Velocity250kmh

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Lifting Velocity (m/s)69.44m/s

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Lifting Power at max payload23.8819795897239MWVertical distance per second multiplied by weight * 9.8 (gravity). Ignores friction of pulley and winch motor.

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Lifting Power Cost per 1000KG (gravitational potential energy)$5.44$/1000KG

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Lift distance10,000m

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Lift distance (feet)33,333feet

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Glide distance300000m

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Total cost per 1000KG (Tonne)$8.50$/1000KG

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Fuel cost (Electric Air Freight)$0.05tonne-mileMultiply electric ascent height by glide ratio to calculate glide distance. Assumes insignificant landing or taxi fuel cost. Assume freight aircraft stops at intermittent airports after each glide for electric ascent.

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Freight revenue (Conventional Trucking)$0.27tonne-milehttp://www.bts.gov/publications/national_transportation_statistics/html/table_03_17.html

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Freight revenue (Conventional Air Freight)$0.82tonne-milesee above

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Freight revenue (Rail)$0.03tonne-milesee above

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Freight cost per ton - kg$0.03ton - km

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Lift Calculation

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Angle of Attack15Optimized for maximum lift (see diagram on right)

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Lift Coefficient (CL)1.5http://www.aerospaceweb.org/question/aerodynamics/q0015b.shtml

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Air density (10000m altitude) (p)0.4135KG/m3http://www.aerospaceweb.org/design/scripts/atmosphere/

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Wing area (A)250m2Multiple tow planes may be used or a single tow plane with large wing area. Boeing 737 has 125 m2 wing area for comparison. See http://www.airliners.net/aircraft-data/stats.main?id=96

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Speed450kmh

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Speed (m/s)125m/s

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Lift = 1/2 p V^2 A CL1211425.78125N

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Lift (KG)123614.875637755KGhttp://www.aerospaceweb.org/question/aerodynamics/q0015b.shtml

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Glide Distance Calculation

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Glide ratio for freight aircraft30 Note aircraft designed for glide may achieve a ratio of 30 or more.http://en.wikipedia.org/wiki/Gliding_%28flight%29#Glide_ratio

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Lift to Drag ratio of tow aircraft20

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Induced drag of tow aircraft60571.2890625N

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Drag from cable34793.2132460956N

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Parasitic drag of tow aircraft (guess)2000N

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Total drag = thrust required97364.5023085956N

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Tow aircraft power12.1705627885744MWhttp://en.wikipedia.org/wiki/Thrust#Thrust_to_power

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Electrical transmission losses (10%)1.21705627885744MW

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Total tow aircraft power13.3876190674319MW