A concept for a fuel efficient flight planning aid for general aviation

It is emphasized that the technique has not been programmed and flight tested and thus should be considered preliminary until this work has been completed.

7.4 Functional Description Section 4 describes the optimizing technique that computes the minimum fuel consumption path profile. Additional programs provide input and output data handling in terms of data storage organization. Since these programs are described in Sections 3 and 5, they will not be repeated. Some details of the input/output process for the programmable calculator are contained in Appendix E.

APPENDIX A FUEL BURN MODEL A.1 Turbojet Equation The basic equation for turbojet aircraft was derived in Much of this remains the same, and will not be Reference 1.

, repeated here in detail.

A.l.1 General -.__ We will use the following general definitions: = fuel flow rate wf = thrust Fn = thrust work ET D = drag = drag work ED KE = kinetic energy = potential energy PE T = time in segment d = ground distance in segment = true velocity (average in segment) vT B = fuel burn 4 1 The general energy balance equation is = ED +APE +AKE (A-1 > ET We note also the following relationships for the case of no wind: ED =s," D dx ={T DV;it ET =4" Fndx = dFn = TVTFn (A-3) B'WfT (A-4) A.1.2 Original Derivation From Reference 1, the basic expression for ED has been derived, and is 3 2 RlKl FSwTVT + 2R2K2W T (A-5) ED= 2 P SwVT where are flap/gear configuration drag multipliers Ri are aircraft constants Ki is atmospheric density is is wing area sW W is aircraft weight.

- A satisfactory form is then found to estimate F,/Wf. ThiS function is used since data is typically arrayed as: Fn - = f(VT,h).

wf The empirical relation that defines this ratio is Fn KllVT 2 -= KIOe + K12h + K13h + K14 (A-6 > wf where Ki are constants h is altitude.

This yields then hl + hlh2+ h KllVT = WfTVT ET + K12 3 :i+ K13(h1+2h2)+ K14J7) Finally, we recognize (A-8) where g is gravitational acceleration.

PE = W(h - hl).

(A-9) We thence have ED +AKE +APE B = WfT = (A-10) ET/WfT This, with the values of the preceding equations is the turbojet fuel burn equation.

A.1.3 Low Altitude Enhancements The original solution was developed for climb/cruise/descent, for above 2000 feet and 250 Kn. Use of the fuel burn equation outside of these limits required a modification to reflect the higher thrust values for the low altitude/low speed regime.

Thus, for thrust values in excess of K7, we add to the fuel burn equation LkC41 = KgF, + Kg where F, is derived from original fuel burn equation, noting it is of the form = TVTF, = ED +AKE +APE.

ET The detailed form of the equation is given in Figures A-l and A-2.

A-1.4 Actual Winds and Temperature In the event actual winds and temperature are input into the fuel burn equation, account must be taken of the differences between ground and true velocity.

True velocity (VT) is used in describing the motion of the and determines the instantaneous fuel aircraft in the air mass, flow. Ground velocity (VG) reflects the motion of the aircraft with respect to the ground, and determines aircraft Ground veloc-ity is defined progress along the route of flight.

as the magnitude of the ground velocity vector (VG) defined in equation A-13.

where FG = vector with magnitude ground velocity and direction of true heading VT = vector with magnitude true velocity and direction of true course W? = vector with magnitude wind velocity and direction that the wind is blowing from.

TVTFn B= + T(LAM1) VTA where RIKIP SwVT + 2R2K2W Fn = 2 +$('T, - 'Tl )+4kT(h2 -h1) hl + hlh2 + h2 eKllVT A = K10 + K12 3 2 ) +K13(h':" ) +K14 if Fn<K LAN= otherwise K8Fn + K9 I Rl,R2 contained in Figure A-2 FIGURE A-1 FUEL BURN EQUATION FOR TURBOJET (ISA CONDITIONS) GUlF3 + GU2F2 + GU3F + 1 (gear UP) Rl = GDlF3 + GD2F2 + GD3F + GD4 (gear down)

I

= FDMlF3 + FDM2F2 + FDM3F + 1 R2 where F is flap angle in degrees and FDMi are aircraft constants GUI, GDi, FIGURE A-2 FLAP AND GEAR DRAG MULTIPLIERS In addition, actual conditions should be reflected in calculating density as a function of average temperature (T, in degrees Fahrenheit) and flight level (hp).

Density below flight level 36089 is given by 1.233 5.2563 .00000687hp) (A-14) Ph = (1 - 459.67 + T and above flight level 36089 it is given by .27544 [.2235 exp ((36089 - hp)/53.35)] (A-15) Ph = 459.67 + T where T = degrees Fahrenheit.

The resultant fuel burn equation is given in Figure A-3.

A.2 Turboprop Equation A.2.1 General Section A.1 presented the equation form that can be utilized to Before compute the estimated fuel burn of a turbojet aircraft.

developing a similar equation that will apply to propeller aircraft, it is relevent to discuss those differences in powerplants that account for equational differences.

In the case of turbojet powered aircraft, the thrust force is generated reactively by the mass flow velocity of the jet nozzle gases in combination with the intake compressor by-pass airflow. The thrust is a function of the difference between the aircraft's velocity through the air and the mass flow velocity through the engine nozzle. In turboprop powered aircraft the thrust force is derived mainly through the propeller which the turbine operates in addition to the compressor. The power generated is essentially constant with speed and the thrust decreases with speed. Thus the equational form of the fuel burn equations for the propeller aircraft and turbojet aircraft are different.

TVTFn + T(LAbl1) B = VTA where 2R2K2W RIKIPhSwVT + + - g (h 2 hl - 2 2 + 3 (VT2 VT11 Fn = ishswvT T A;LAMl; Ri previously defined in Figures A-l and A-2.

ii,, is altitude dependent density T is derived as Ground Distance Ground Velocity FIGURE A-3 FUEL BURN EQUATION FOR TURBOJET (ACTUAL WINDS AND TEMPERATURES) A.2.2 General Derivation It is convenient to concentrate on the relationships of power, Data is since power relates to the shaft output of the engine.

readily available to establish a curve fit between P and Wf of the form: (A-16) = aP + beCh Wf where a,b,c are constants P is output shaft horsepower.

Thrust and shaft horsepower are related by p = VTFn (A-17) -I p550 where qp is the propeller efficiency, and is approximated by (.82).

Substituting and letting K15 = a&P 550) K 16 = b K 17 = c results in K17h . (A-18) = K15VTFn+ K16e wf We take the derived value of F, from equation (A-12) ED +AKK+APE .

= (A-19) F n TVT Finally, K17h B .

= WfT = K15TVTFn+ K16Te (A-20) The resultant fuel burn equation is given in Figure A-4.

A.2.3 Actual Winds and Temperatures Actual winds and temperatures are reflected in the fuel burn equation analogously to the turbojet case. The resultant is given in Figure A-5.

A.3 Turbocharged Piston Equation A.3.1 General Derivation As in the turboprop aircraft, the power is converted into thrust by the propeller. In a turbocharged engine, constant power output can be maintained to a certain altitude.

Turbocharged piston engine aircraft normally use a lean fuel mixture during cruise and descent and richer fuel mixtures during takeoff and climb conditions. This can result in different fuel flows at the same altitude and velocity. When the lean fuel mixture is used, an empirical curve fit of fuel flow (WfL) is a linear function of brake horsepower P: (A-21) =aP+b WfL From operating manuals it is possible to derive an empirical curve fit for fuel flow at richer fuel mixtures (WfK), This function is assuming standard operating procedures.

to accommodate different fuel quadratic instead of linear in P, mixtures at different power settings: WfR = CP2 + sp + r (A-22) making the assumption that the richer fuel mixtures are only used when the aircraft is in climb. Equations (A-21) and (A-22) K17h B = K15TVTFn + K16Te where 2 2 2R2K2W RIK1pSwVT + ' (h - hl) Fn = - 'Tl) + VT 2 + $ ('T2 RI previously defined FIGURE A-4 FUEL BURN EQUATION FOR TURBOPROP (ISA CONDITIONS) K17h B = K15TVTFn + K16Te where 2R2K2W RIKl- Fn = 2 + 9 (VT2 - VTl) + & (h2- hl) Phswv; + T PhSwVT Ri previously defined ph is altitude dependent density T is derived as Ground Distance Ground Velocity FIGURE A-5 FUEL BURN EQUATION FOR TURBOPROP (ACTUAL WINDS AND TEMPERATURE) can be combined using an exponential function, in the rate of climb (R/C), as a step function:

K(JR/Cl’ R/,)

)(W,,) + (1 - eK' IR"' + "'I) (WfR) (A-23) With the constant K properly chosen, this exponential will be equal to 1 for climb, and 0 at all other times.

Substituting R/C = VT(h2 - hl)/d (A-24) P = VTFn/(7jp550) (A-25) into equation (A-23) we have: cV2F2 Tn sVTFn +- + b + (1 - 6 > Ilp550 PIP 550)2 (A-26) where KVT/d h2 - hll + (h2 - hl)

6 1

= e Letting K15 = K = a/(qp 550) K16 K17 = b = c/(qp 550)2 K18 = s/(qp 550) K19 K20 = r I ~ -.-.-.-. ..._. .___. ._, and noting (A-27) B = WfT we have fuel burn as: B = d (K16VTFn + K17) + T(l- 6)(K18V;Fi + KlgVFn + K20) (A-28) where F, is defined in the turbojet derivation.

The results are summarized in Figure A-6.

A.3.2 Actual Winds and Temperature Accounting for actual winds and temperatures is analogous to the previous derivations. The resultant is presented in Figure A-7.

2 2 B = bT(K16VTFn + K17> + T(1 - 6 )tK& Fn + K1gVTFn + K20> where K15 (l/T) Ih2 [ - hl 1 + (h2 - hl)l = e RIKl _ Fn = - P s,v; + 2 F2:2::yl + $('T2 - 'Tl)+ 452 - hl) wT FIGURE A.6 FUEL BURN EQUATION FOR TURBOCHARGED PISTON (ISA CONDITIONS) B = 6T(K16VTFn + K17) + T(l - b)(K18V; F; + KlgVTFn + K20) where 2R2K2W RIKl _ + 5 (vT2 - VT11 + s Fn = 2 (h2- hl) Phswv; + T PhSwVT 6 previously defined RI previously defined ph is altitude dependent density T is derived as Ground Distance Ground Velocity FIGURE A-7 FUEL BURN EQUATION FOR TURBOCHARGED PISTON (ACTUAL WINDS AND TEMPERATURE)

APPENDIX B AIRCRAFT CONSTANTS B.l General This appendix contains aircraft constants for the fuel burn equation for both a turboprop aircraft (King Air 200) and a turbocharged piston aircraft (Cessna 421C). The use of these aircraft constants with the fuel burn equation and the techniques used to derive them are described in detail in References 1 and 2, and Appendix A. Accuracy checks using the aircraft constants and cruise conditions are also given for both of these aircraft.

B.2 Notation H = altitude (feet) V = velocity (true airspeed in knots) reference value (pounds/hour) FBREF = fuel burn, FB cAL = fuel burn, calculated value (pounds/hour) A= FBREF - FBCAL %=A x 100 FBREF B.3 Beechcraft Super King Air 200 Tables B-l and B-2 contain the aircraft constants and an accuracy check respectively for the Beechcraft Super King Air 200 powered by two Pratt and Whitney PT6A-41 engines. The reference fuel burn values used in the accuracy check (Table B-2) were taken from the Super King Air 200 Pilot's Operating Handbook (Reference 5).

B.4 Cessna 421C Tables B-3 and B-4 contain the aircraft constants and an accuracy check, respectively, for the Cessna 421C (Golden Eagle) powered by two Teledyne Continental Motors GTSIO-520-L engines.

The reference fuel burn values used in the accuracy check (Table B-4) were taken from the Cessna 421 Golden Eagle Information Manual (Reference 6).

TABLE B-l BEECHCRAFT SUPER KING AIR 200 CONSTANTS SW = 303 FDMl = 0 Kl = .0256014 FDM2 = 0 K2 = .04241259 FDM3 = -.0057 K 15 = .0000002692 WFIDLE = .067 K16 = .080443 = 0 Ll K17 = -.000034 = 0 L2 GDl = 0 = 0 L3 GD2 = 0 = 0 L4 GD3 = 0 VNE = 289 GD4 = 2.3573 vs = 75 GU1 = 0 MTOW = 12500 GU2 = 0 OEW = 7755 GU3 = .01547 Takeoff Climb Cruise Al = 0 0 0 A2 = 0 0 0 A3 = .lOE-09 -.44E-10 -.44E-10 A4 = -.422383-05 -.394193-05 -.394193-05 A5 = .28228 .29681 .29681 Information from Beechcraft Super King Air 200 Pilot's Operating Handbook and Airplane Flight Manual (Ref. 5) and Jane's All the World's Aircraft (Ref. 7) was used in deriving this set of constants.

TABLE E-2 BEECHCRAFT SUPER KING AIR 200 ACCURACY CHECK Time = 3600 sec.

Weight = 11,000 lbs.

RPM H V % - - FBREF A

ECAL

0 240 946 919 -27 -2.8 2000 245 1700 918 903 -15 -1.6 4000 250 892 887 -5 -0.6 1700 6000 254 866 864 -2 -0.2 1700 8000 259 846 849 3 0.3 1700 10000 264 828 833 5 0.6 12000 270 812 823 11 1.3 1700 14000 275 800 807 7 0.9 16000 280 792 791 -1 -0.1 1700 18000 280 746 747 1 0.1 1700 20000 277 0 0.0 690 690 1700 22000 275 644 642 -2 -0.3 24000 273 598 599 1 0.2 1700 26000 269 554 552 -2 -0.4 1700 28000 511 -0.6 265 514 -3 -0.4 1700 29000 263 494 492 -2 1700 -1.1 31000 257 458 453 -5 1700 33000 250 422 417 -5 -1.1 1800 0 245 982 956 -26 -2.6 2000 250 954 939 -15 -1.6 1800 4000 255 923 -3 -0.3 6000 260 902 4 0.4 1800 906 1800 8000 7 0.8 265 882 889 10000 270 7 0.8 1800 866 873 1800 12000 863 13 1.5 276 850 1800 14000 281 838 845 7 0.8 1800 1.0 16000 283 802 810 8 1800 18000 281 746 752 8 1.1 1800 20000 279 692 700 8 1.1 1800 22000 277 646 652 6 0.9 1800 24000 274 598 603 5 0.8 26000 271 1.1 1800 554 560 6 1800 28000 267 514 518 4 0.8 1 0.2 1800 29000 264 494 495 1800 31000 258 458 455 -3 -0.6 1800 33000 250 422 417 5 1.2 TABLE B-2 (Concluded) A % RPM H V

FBREF ECAL

- - - -21 -2.1 1900 0 250 1016 995 1900 2000 255 988 977 -11 -1.1 1900 4000 260 962 960 -2 -0.2 1900 6000 265 940 943 3 0.3 1900 8000 270 920 925 5 0.5 10000 276 904 915 9 1.0 1900 12000 281 890 897 7 0.8 1900 14000 287 876 885 9 1.0 16000 822 828 6 0.7 1900 286 1900 18000 285 770 775 5 0.6 1900 20000 283 716 721 5 0.7 1900 22000 281 668 671 3 0.4 1900 24000 279 620 625 5 0.8 575 1 0.2 1900 26000 275 574 28000 271 531 -1 -0.2 1900 532 29000 269 512 511 -1 -0.2 1900 31000 263 474 469 -5 -1.0 1900 33000 256 434 431 -3 -0.7 Mean = .06% Standard Deviation = -98% Variance = -95% TABLE B-3 CESSNA 421C (GOLDEN EAGLE) CONSTANTS SW = 215 GUl = 0 Kl = -0274935 GU2 = .00041563 R2 = -0415015 GU3 = -030365 R15 = -.57564 FDMl = 0 K16 = 2.6363-07 FDM2 = 0 Rl7 = .0057675 FDM3 = 0 K18 = 2.129783-12 WFIDLE = -01111 = El9 = 5.38453-07 0 Ll = E20 = -0798467 0 L2 = GDl = 0 0 L3 = GD2 = .0004145 0 L4 GD3 = -03042 = 258 VNE = 74 GD4 = 1.9641 vs MTOW = 7450 OEW = 4426 CRUISE TAKEOFF CLIMB Al = 0 0 0 A2 = 0 0 0 A3 = -1.68E-10 -1.68E-10 -1.68E-10 A4 = 2.4101283-06 2.4101283-06 2.410128E-06 A5 = -16033551 -16033551 -16033551 Information from the Cessna 421 Golden Eagle Information Manual (Ref. 6) and Jane's All the World's Aircraft (Ref. /) was used in deriving this set of constants.

TABLE B-4 CESSNA 421C (GOLDEN EAGLE) ACCURACY CHECK %

RPM MP H v A

- FBREF %AL

- 0 186 257 254.3 -2.7 -1.0 1900 32.5 -1.1 1900 31.0 0 183 246 244.9 -0.4 0 177 228 227.1 -0.9 -0.4 1900 29.0 0 171 211 210.7 -0.3 -0.1 1900 27.0 1900 25.0 0 164 194 193.1 -0.9 -0.5 1900 32.5 5000 195 257 255.7 -1.3 -0.5 -2.0 1900 31.0 5000 191 246 244.0 -0.8 185 228 227.4 -0.6 -0.02 1900 29.0 5000 1900 27.0 5000 178 211 209.6 -1.4 -0.7 -0.7 -0.03 1900 25.0 5000 171 194 193.3 204 257 255.4 -1.6 -0.6 1900 32.5 10000 1900 31.0 10000 200 246 244.5 -1.5 -0.6 1900 29.0 10000 194 228 229.0 1.0 0.4 27.0 10000 187 211 212.3 1.3 0.6 1900 25.0 10000 179 194 194.9 0.9 0.5 -0.9 -0.3 1900 32.5 15000 214 257 256.1 210 246 245.9 -0.1 0.4 1900 31.0 15000 1900 29.0 15000 203 228 229.2 1.2 0.5 1900 27.0 15000 195 211 211.6 0.6 0.3 1900 25.0 15000 186 194 193.9 -0.1 -.05 -0.7 1900 32.5 20000 224 257 255.2 -1.8 220 245.9 -0.1 -.04 1900 31.0 20000 246 1900 29.0 20000 212 228 228.4 0.4 0.2 1900 27.0 20000 204 211 212.4 1.4 0.7 1900 25.0 20000 194 194 194.5 0.5 0.2 1900 32.5 25000 236 257 257.5 0.5 0.2 231 246.8 0.8 1900 31.0 25000 246 0.3 1900 29.0 25000 223 228 230.8 2.8 1.2 1900 27.0 25000 212 211 211 0 0 1900 25.0 25000 200 194 192.2 -1.8 -0.9 0 177 230 -2.9 -1.2 1700 32.5 227.1 1700 31.0 0 174 219 218.8 -0.2 -0.1 1700 29.0 0 168 204 202.9 -1.1 - .5 1700 27.0 0 161 187 186.0 1.0 .5 1700 25.0 0 154 171 170.7 -0.3 - .2 1700 23.0 0 146 155.2 0.2 .l 1700 32.5 5000 186 230 230.1 0.1 .04 1700 31.0 5000 181 219 217.0 -2.0 -0.9 TABLE B-4 (Concluded) H V RPM MP A % FBREF WXL - - 1700 29.0 5000 175 204 202.4 -1.6 -0.8 1700 27.0 5000 168 187 186.7 -0.3 -0.2 1700 25.0 5000 161 171 172.6 1.6 0.9 1700 23.0 5000 151 155 154.8 -0.2 0.1 1700 32.5 10000 194 230 229 -1.0 -0.4 1700 31.0 10000 190 219 219.3 0.3 0.1 1700 29.0 10000 183 204 203.3 -0.7 -0.3 1700 27.0 10000 175 187 186.9 -0.1 -.05 1700 25.0 10000 167 171 172.1 1.1 0.6 1700 23.0 10000 155 155 153.2 -1.8 -1.2 1700 32.5 15000 203 230 229.2 -0.8 -0.3 1700 31.0 15000 198 219 218 -1.0 -0.4 1700 29.0 15000 191 204 203.5 -0.5 -0.2 1700 27.0 15000 183 187 188.4 1.4 0.7 1700 25.0 15000 172 171 170.2 -0.8 -0.5 20000 230 1700 32.5 213 230.4 0.4 0.2 1700 31.0 20000 208 219 220.2 1.2 0.5 1700 29.0 20000 200 204 205 1.0 0.5 1700 27.0 20000 189 187 186.4 -0.6 -0.3 1700 25.0 20000 177 166.7 -4.3 -2.5 1700 27.0 25000 194 187 183.9 -3.1 -1.6 Percent Differences: Mean = 0.16 Standard Deviation = 0.64 Variance = 0.40

APPENDIX C OPERATIONAL SCENARIO This appendix presents the concept of the operational scenario for the use of the flight planning aid.

C-1 Introduction The conservation of aircraft fuel starts with the initial trip planning, and ends with a post flight evaluation of actual fuel usage. During preflight planning, the optimum fuel usage performance characteristics of an aircraft must be combined with route alternatives and upper air mass conditions to arrive at the flight path and operating parameters to achieve the maximum fuel economy. This involves the evaluation of "trade off" situations such as operating at a less fuel efficient altitude in order to take advantage of an upper wind condition.

This first level of fuel burn optimization is based upon pretakeoff route conditions. After becoming airborne, the actual weather conditions must be evaluated at reasonable time intervals and the required adjustments made in order to optimize the fuel conservation process.

During the post flight review the actual versus predicted fuel usage can be evaluated. This is an important step because, during the flight, judgments were made based upon use of the technique. The review can be utilized to modify or adjust the individual technique so that future flights will benefit.

Examination of these three steps indicates that the solutions, if performed manually, wouldabe cumbersome, time consuming, complex and error prone. Thus, the need for a flight planning aid points to some type of computer capability. The state-of-the-art of modestly-priced programmable calculators has reached the level that adequate computational capacity is readily available.

The availability of relatively inexpensive minicomputers that are capable of interfacing with larger machines via modems provide a potential for performing complex track flight planning at virtually all airport locations.

C.2 Utilization Alternatives The operational utilization of the minimum peak consumption trajectory technique can be separated into four categories with each having different capabilities and data requirements.

They are as follows: ALTERNATIVES PREFLIGHT INFLIGHT I Programmable Programmable Calculator Calculator II Minicomputer/ Programmable Time Sharing Calculator III Minicomputer/ Performance Time Sharing Computer IV Performance Performance Computer Computer C-2.1 Alternative I The use of the programmable calculator for both preflight planning and inflight update and control represents the least cost implementation of the minimum fuel trajectory technique. During preflight planning (Table C-l) the pilot examines the winds and temperatures along various routes and chooses the route that, in his judgment, has the best conditions. The appropriate programs are loaded into the calculator by utilizing either bar code or magnetic cards. The total trip is divided into segments and the course, magnetic variation, and distance of each segment determined (Table C-2). Next, the upper winds and temperatures at various altitudes for each segment are determined.

The maximum acceptable operating altitude along with the landing weight (including alternate fuel reserves) is also determined. This data is entered interactively into the calculator. The output is a printed listing for each segment of distance, fuel consumption, speed, altitude, wind component, and temperature. Additionally, the complete matrix data is stored within the calculator and on magnetic cards for inflight use.

On entering the cockpit for departure, the pilot loads his inflight programs and trip data cards. At departure the time and actual gross weight are entered.

During climb the pilot can receive climb speed and power setting by entering altitude and temperature. At the first waypoint the pilot enters time, altitude, velocity, -- --.. .

- -.II.-I 1.11.1. - I._. 1.1, I TABLE C-l PRELIMINARY FUNCTION CONCEPT STEP I: PILOT SUPPLIED DATA

Route Of Flight Data

l Determine Route Of Flight

l Divide Into Approximately 20 Minute Flight Segments

l Define Distance And Course For Each Segment

En Route Weather

l Define Wind Velocity, Heading And Temperature

-

Maximum Operating Altitude

(Based Upon Aircraft Limitations,

Icing, Turbulence, Oxygen Availability)

Calculator Programs

l Have Available Program Modules/Magnetic Tape With Particular

Aircraft Constants Stored

Or

l General Program Module/Magnetic Tape/Paper Copy With A List

Of Particular Aircraft Constants

TABLE C.2 PRELIMINARY FUNCTIONAL CONCEPT STEP II: PRE-FLIGHT PLANNING

Initial Data

l Aircraft ID, Engines

l Empty Weight

l Altitude Constraint

Segment Data Input

l Distance

l Altitude

l Temperature

l Wind

l Course

Output Best And Next Best

l Climb, Cruise, Descent

- lAWMach

- Power Settings

- Altitudes

temperature, and gross weight or fuel on board (Table C-3).

Also, at this point a projected minimum fuel arrival time will be displayed. The pilot can either accept or modify the value. The program then reprojects, incrementally, the remaining portion of the flight and displays a speed, altitude, and power setting command.

The entry is reported at each way point throughout the flight.

The time is continually updated and can be displayed upon command to the pilot. Additionally, a start descent time is calculated and displayed. At start of descent the pilot enters the time, altitude, temperature, and fuel on board at which the descent speed is displayed with the appropriate power setting.

As in climb, the speed and power setting is displayed at entry of a time, altitude, and temperature. Upon landing, the time and fuel on board is entered. The data contained in the calculator is recorded on magnetic cards.

The post flight analysis (Table C-4) is performed by the pilot entering the appropriate programs in combination with the flight data that was recorded on the data cards. A printout records the actual versus projected plans and allows the pilot to evaluate the trip fuel efficiency.

C.2.2 Alternative II This alternative assumes that the ground or preflight planning is performed by a minicomputer that is linked to a VOR and a winds and temperatures data base (Table C-5). It also assumes that the computer has the capability to evaluate multiple tracks either iteratively or by utilizing a dynamic programming data base. The pilot enters landing gross weight, arrival point (VOR), maximum acceptable altitude, departure point, and aircraft/engine types. The system then provides a printed output as described in Alternative I, including a bar code for loading into a programmable calculator.

The inflight and postflight operations are the same as described in Alternative I.

C-2.3 Alternatives III and IV These alternatives are listed for the purpose of highlighting the point that the techniques described can be implemented on any computer, providtng that the required interface capability has been provided.

TABLE C-3 PRELIMINARY FUNCTIONAL CONCEPT STEP III: IN-FLIGHT UPDATE AND CONTROL

Updated Inputs For Way Points

l Time

l Altitude

l Fuel Burn/Flow

l Temperature

Revised Output

l lAWMach

l Power Settings

l Altitude

TABLE C-4 PRELIMINARY FUNCTIONAL CONCEPT STEP IV: POST-FLIGHT REVIEW INPUT AND OUTPUT

To Be Determined Based On Pilot Needs

Objectives

l Compare Best To Actual Fuel Conservation Performance

- Stored In Calculator

- Reflects Encountered Conditions

l Present Results So That Future Performance Will Be Improved

TABLE C-5 OPERATIONAL SCENARIO

l Preflight

- Time Share or Separate Computer at Fixed Base Operator

- (Later) Possible Automated Weather Input

- Initial Optimal Solution

l lnflight

- Programmable Calculator, Initialized With Ground Solution

-

lnflight Updated Inputs and Solution

-.. -- SAMPLE PROBLEM This appendix contains a sample optimization problem done on the Apple II computer. This problem consisted of determining the optimum path profile for a Super King Air 200 aircraft during a 360 nmi flight.

D.l Inputs The actual inputs for this problem are given in Table D-l (as echoed out to the printer as output). The altitude modes were calculated by the program while the velocity modes were designated by the user.

D.2 Matrix Definition The matrix nodal points in altitude, velocity, and distance are defined by either the user or the program. In the sample problem the velocity nodes are defined by the user while the altitude nodes are defined by the program. Since the number of distance nodes is greater than the number of waypoints, the program embeds distance nodes between waypoints. The altitude, velocity, and distance nodes defined for the sample problem were printed out and are contained among the output figures in Table D-l.

D.3 Optimal Path Profile Once an optimum path profile has been determined, the path is printed in the form shown in Table D-l, along with the departure and arrival weights. The altitudes printed out differ from the altitude nodal points since the nodal points are density altitudes while pressure altitude is given.

The program also prints a list of flight segments using the optimum path containing start and end distances, velocities (TAS), and pressure altitudes for each flight segment, and the fuel burn over the segment as shown for the sample problem in Table D-l.

D.4 Outputs The outputs for this sample problem are shown in Table D-l and have been described previously.

TABLE D-l SAMPLE OUTPUT ‘2 !dA’~,‘FO I NT NUME.ER D 1 ‘:TANiL:E= 54 . lj I:1 lj l-l l-l l-i _ - 1 2 lj . Ij 51 lj CI HEhD I NG= .-, lj lj 1:; r:l ‘VW? I AT I KIN= 2 .

2 . fj lj R(LT=l5ijij1:1. rj !.d 1; = 1 6 I:I . ij TEMF = -22 . 1;1-r fiLT=z5[1[il:r. [i ’ Il.iIi= 1 3 1:: . lj TEtlF’= fiLT=:35 ij ~j ij , ij IJIl=s’I; :’ . lj TEtlP= -5;:: . ‘:Ii:: .-, .J WH’iFfl I NT NlJMBEP D I::Tt#t:E= 1 1 ‘:; . ii [I I:I ij (:: I:; HEAD 1 rJl;= ::; ij . ~j r:r 13 I:; 5 . ij i:l :I !:i ‘dr=IRI FiT I tlM= RI-T= 1 5 [I [I 0 . !:I i.*JTi= 13 ::’ . lj Id’.,:-4 0 . lj [I ij - 1 . Ij lj TEMF= fiLT=es[~ijfi.ij l.P.1 II = 1 q 1; . :> I.41 ‘.r’= 5 lj . 3 [I 0 TEt+== -25 . I~I:; ALT=:I;s tj ij 0 . I:I l.JD=S 1 [a . I:i l,,J’G=A l-i 1-i (I lj TEllF= -3’3 . ~j I:i _-.- TABLE D-l (Continued) -‘3 l.tij/=:3 !:; . ;:I [i lj . lj< TEtlF’= l.(l’n;=4 [I . I] lj lj -25 . lj lj TEtlP= l.d+k5 ij . 0 5; ij TEMF= -GE . ‘:I[: !iJWfPCl I NT NI-IMPER jj 1 :z: T at-j 11 E =3 1 5 . I:I [I I:I I:; 11: I:!

HEHGIYG= 9 ij . 0 ij I:;<I 8 . I:; I:i :: lj l.IID = :3 ij i:! . I:i I.@J ‘d = 1 [I . 0 lj lj TEtl!=‘= 5 . 5 !I l.,Jfi=~'~ c . lj I,]'.)=~ lj . lj rJ lj TEt+== -3 [I . 2 [; l.,lIk 2;7 lj . lj l.J’+=4 lj . lj 0 I:1 TEHF= -65 .:s I; 1 = AL1 I TlJDE NIJDE ;r’ = FiLT I TIJDE N!ZDE :3= HLT I TClDE NGDE 4= R,LT I TIJDE NIJIIE c- FiLT I TlJDE NGDE J- c*= ALT I TlJDE NUDE 7= HLT I TlJDE NUDE :j- -- HLT I TIJDE NDDE 3= ALT I TIJDE NDIlE ALT I TlJDE NGDE 1 o= 1 = 135. lJljlj0 ?IELOC: I Ti’ NIJIIE + 1 55 . 0 lj I:1 0 VELUC I T’f NODE :3= 175 . Uljljlj ‘v’ELOCITY NGDE 4= 1 ‘3 ii . 0 lj lj lj VELOC I T’f NODE 2 ci lj . rJ lj lj Ij VELOC I T’f NODE 5= g,J= ~ 1 lj . 0 lj 0 I:1 VELOCITY N!XiE ?‘= 220. rJljljlj VELUC I T’i NODE :j- -- 2:3lj. rJljljlj VELUC IT? NGDE .- + - ‘V’ELIX I T’f NGDE ~5lj.TJI:lijlj 1 tj= ~70. clljljo VELOC: I TY NDItE TABLE D-l (Contjnued) = NODE lj . lj lj lj r:l z= NODE ~7.0000 .3= I-iODE 54.rJoo11 4= NUDE :33 .5ljOlj 5= NDDE lj lj rj lj 11.3.

i= NGDE 1 5 1 . 0 srlj lj 7= NODE 183. TJljljlj I- - j- i2z7 .oljljlj NUDE + NOPE 265. ooljrj 1 lj= 3 1 5 . [I lj lj lj fKlDE 1 1 2 5 rJ. i:I[I r:l lj RRRIYHL WEIGHT= DEPRRTIJRE ME I GHT= 1 17:34 .:3rj45 UPTIl’WtS PATH FRGFILE 131:s. HI-T.

lj . c;lj 501 lj .,5:3 2 1j:3zq .39 27 . 1:; lj 54 . lj ij 27949.15 5:3 .5!j 3 lj7.3 1 .95 1 1 3 . I:! lj :3 lj391j .45 1 5 1 . lj lj :311t54.717 :3 1:3:32 . qc, IS’?. ljlj 227 . i:l (I 31.501 .& 265.Olj 2:394:3 I 4 lj 3 15 . lj lj 501 rJ .2:3 TABLE D-.1 (Concluded) OPT ItWl PRTH FIJEL BlJt?H

APPENDIX E PROGRAMMING DETAILS This appendix describes some of the programming detail necessary to implement the concepts presented in the paper.

E.l Apple II Algorithm This section describes programming details to implement on an Apple II computer (in FORTRAN) the dynamic programming technique described previously.

E.l.l Storage Since the Apple II computer has limited storage, it is necessary to combine several arrays together to permit maximum usage of storage. In the Apple 11 program the OH and OV arrays described in Section 4.3 and the ALT and VEL arrays also The described in Section 4.3 have been combined respectively.

OH and OV arrays are combined into an array IOP such that = OH (i,j,k)*lOO + OV (i,j,k).

IOP (i,j,k) The ALT and VEL array have been combined such that IFLAG (i,j,k) = ALT (i,j,k)*lO +VEL (i,j,k) E-1.2 Input Data The following inputs to the Apple 11 will be supplied by the user.

1. Flight Plan Title (25 characters or less) 2. Effective Service Ceiling (feet) (Program will default to Aircraft Service Ceiling if exceeded) 3. Departure Altitude (feet) (density altitude) 4. Departure Velocity (knots) * 5. Arrival Altitude (feet) (density altitude) 6.

Arrival Velocity (knots) 7. Number of Altitude Nodes (If the user wishes to input the altitude node points, then he inputs a "Y" when prompted and then inputs the density altitude node points. These node points must be such that %%+1)* 8. Number of velocity nodes (User can input velocity nodes in similar fashion to inputted altitude nodes.)

9.

Number of Distance Nodes 10. Number of Waypoints (Must be equal to or less than number of distance nodes.)

Landing Gross Weight (Must be greater than the 11.

aircraft operating empty weight else program will reprompt for Landing Gross Weight.)

The program will then indicate the waypoint number and the user will then supply the following for that waypoint: 12. Course Heading (degrees from North) 13. Magnetic Variation (degrees from North) 14.

Distance to Next Waypoint (nautical miles) 15. Number of altitudes at this waypoint for which weather data will be supplied 16. Altitude (density altitude in feet) 17. Wind Direction at Altitude (degrees from North) 18. Wind Velocity at Altitude (knots) 19.

Temperature at Altitude (in degrees Farenheit) Once the optimization has been completed and the optimal path profile has been printed out the user will be asked if he wishes to update. If the user responds with a "Y" then the following inputs will be required: 1. Departure Weight if the user responds in the affirmative to the prompt concerning updating departure weight.

2. Node Point where weight altitude, or velocity are to be updated.

(This node point must be greater than previous node point updated or else the program will assume no additional updates will occur.)

3.

Updated Weight at node point if the user responds in the affirmative 4. Density Altitude (feet) at node point 5. Velocity (knots) at node point E.1.3 Outputs The following outputs from the Apple II computer to the user will normally include: 1.

Flight Plan Title 2.

Effective Service Ceiling (feet in density altitude) 3. Departure Altitude (feet in density altitude) 4. Departure Velocity (knots) 5. Arrival Altitude (feet in density altitude) 6. Arrival Velocity (knots) 7.

For each user waypoint a- Waypoint Number b. Distance from Departure Point (nautical miles) C. Course Heading (degrees from North) Magnetic Variation (degrees from North) d.

e- For each Altitude Altitude (feet in density altitude) a) Wind Direction (degrees from North) b) Wind Velocity (knots) cl 8. Arrival Weight (W(l,l,P)) (pounds) 9. Departure Weight (W(l,l,l)) (pounds) (If the aircraft's Maximum Takeoff Weight is exceeded a warning will be printed.)

10. Optimum Path Profile .

This will include for each node point a. Node point (number to be used for updating) b. Distance from departure point (nautical miles) C.

Pressure altitude (feet) d. Velocity (true airspeed in knots) 11. Optimum Path Fuel Burn Using the Departure Weight (W(l,l,l)) the fuel burn will be computed along the optimum path. outputs will include for each segment the start and end the start and end velocities distance (Dl,D2), (Vl,V2), the start and end pressure altitude (Hl,H2), and the fuel burn (B) over the segment.

12.

Updated Path Fuel Burn If Update has occurred the fuel burn along the updated profile will be computed using either the optimum departure weight (W(l,l,l)) or the user's departure weight; Outputs will include the same as In addition, both with the Optimum Path Fuel Burn.

the departure weight and updated arrival weight will be printed.

A sample output was included in Appendix D.

E.2 Programmable Calculator Input Processing The preflight planning program for the programmable calculator version has been divided into three subprograms or modules.

These are: 0 Data Entry and Initialization Flight Path Optimization 0 Data Output and Write Storage Each of these modules accepts data, performs computations, and outputs data either internally or externally. The Flight Path Optimization module is common to the pre-flight; in-flight update; and, the post-flight evaluation. Of the three modules only the Data Entry and Initialization programming has been completed. Therefore, this section will deal only with this module.

The purpose of the Data Entry and Initialization subprogram is to load all programs, aircraft specific constants, general constants, optimization algorithm specific constants, and the initialization program.

Figure E-l depicts a flow chart that defines the program flow along with explanatory comments.

E.3 Programmable Calculator Optimization Constants The purpose of this section is to define the data that is used from the pilot's handbook and to show how the h = f(Td,Tp) and V = f(Td,Tk) equations' constants are determined.

E.3.1 Pilot's Handbook Data The following table depicts the data that was derived from the Beechcraft Super King Air 200 Pilot's Handbook: (A 300 mile trip is assumed as an example; however, the non-trip specific constants will remain the same.)

Distance Altitude Velocity (nmi) (knots) (ft> zd 2.3 1,000 0.146 7.3 10,000 143 0.462 22.4 22,000 166 1.42 106.5 35,000 230 6.74 151.2 35,000 230 9.57 222.79 22,000 169 14.10 276.28 10,000 138 17.49 300.00 1,000 119 19.00 The above data describes a maximum performance climb, cruise, and descent based on the handbook data.

E.3.2 Constant Determination The constants contained in the altitude (h) and velocity (v) relationships are both aircraft type and trip dependent. In this typical example, a trip distance of 300 nautical miles will be used.

Sl -------P-P- Subprogram loaded from magnetic cards which loads data registers with aircraft specific and other constants in addition to the initialization program.

- AIRCRAFT & ENGINE TYPE.

- TOTAL TRIP LENGTH.

- NUMBER OF WAY POINTS.

(W WAYPOINTS) ENTER TRIP _ - EFFECTIVE AIRCRAFT CEILING.

SPECIFIC DATA - LANDING WEIGHT (INCLUDING RESERVES).

I

- DEPARTURE POINT ALTITUDE.

- ARRIVAL POINT ALTITUDE.

STORE DATA --- -l--------- I - DISTANCE TO NEXT WAY POINT - MAGNETIC COURSE TO NEXT WAY POINT ENTER wAYPOINT + - MAGNETIC VARIATION (EAST 1s (-1) - i=l IS DEPARTURE POINT i Data - NUMBER OF DATA ALTITUDES (J ALTITUDES) - WIND VELOCITY (KTS) AT P ALTITUDE ENTER WAY POINT - - WIN" DIRECTION (TRUE) AT P ALTITUDE 1 i. ALTITUDE - TEMPERATURE ('F)AT P ALTITUDE P DATA I ------~ - WCP = -[WIN" VELOCITY( COS(COURSE-WIN" MAGNETIC))] WIN" MAGNETIC = WIND DIRECTION + VARIATION NOTE: - WCP = HEADWIND ------- wcp = (CONSTANT 1) + (CONSTANT 2) (ALTITUDE) COMPUTE WAY POINT i WCP FUNCTION Utilize linear regression to CONSTANTS compute constants 1 & 2 I I I --- ------~ I COMPUTE WAY POINT TEMP(OK) = (CONSTANT 31 + (CONSTANT ~)(ALTITUOE) i+l i TEMP. FUNCTION CONSTANTS UTILIZE LINEAR REGRESSION TO COMPUTE CONSTANTS 3 & 4 TO OPTIMIZATION PROGRAM FIGURE E-l DATA ENTRY AND INITIALIZATION Id Determination The relationship of the distance index (Id) to the ground trip distance (dg) is: Id = Kt dg where, K, is a constant for a specific trip, = 0.0633dg, Id for 19 Id points, and a dg, total = 300 nmi.

Altitude Function As described in Section 7, the altitude (h) function is, = f(Ip)f(Id) (assuming separability)

bed

An equation form that will generate a ridge line in altitude as a function of Id is, -CnILd - 19p -'n'd = f(Id) = ceiling altitude l-e -e hd -'n'd In the above equation the 1 - e term accounts for the maximum performance climb to the effective service ceiling -Cn II, - 191'n altitude. The term,e represents a maximum , range descent at minimum power condition. The other part of CJ Ip-C,) the h function is, he = f(Ip) = e . This Fd equation pr0vide.s the cross section shape or variation in altitude with Ip.

For the Super King Air 200, the constants have been determined and are as follows: -o.017511d-1912*5 -0.1755(Ip-5.5) -0.69751d h= 35,000 l-e -e [ This example serves to demonstrate this method that is utilized to determine the h and v functions. Since the v function is similar, an example will not be presented.

APPENDIX F NOMENCLATURE Symbol Meaning a Temporary constant Altitude (program variable) ALT b Temporary constant Fuel burn B C Temporary constant Constants ci d Ground distance Distance steps di D Drag Drag work ED Thrust work ET F Degrees of flap FCPli Flap/gear constants Thrust Fn Gravitational acceleration g Flap/gear constants (% GUi Flap/gear constants h Altitude Flight level hP H Altitude (program variable) Pressure altitude HP Altitude (indexed by potential energy and distance)

hp,d

i Density altitude index Index of distance Id Index of kinetic energy 'k Index of potential energy IP IOP Optimum index (program variable) IPLAG Optimum flag (program variable) Velocity, index j k Distance index K Constant Aircraft specific constants Ki KE Kinetic energy L Iteration counter LAM1 Low altitude term M Limit of density altitude index N Limit of velocity index OH Altitude index OPH Update altitude index OPV Update velocity index ov Velocity index P Output shaft horsepower (or limit of distance index) PE Potential energy r Temporary constant Flap/gear configuration drag multipliers

Ri

R/C Kate of climb s Temporary constant Wing area SW T Time (or Temperature) Altitude dependent temperature TemPh v Velocity Ground velocity "G Vector with direction of true heading and magnitude G ground velocity True velocity "T Vector with direction of true course and magnitude VT true velocity "EL Velocity (program variable) Velocity (indexed by distance and kinetic energy) "d,k W Aircraft weight Fuel flow rate Wf Fuel flow rate (lean) WfL Fuel flow rate (rich) 'fR WC Read wind component WE East wind component WN North wind component if? Vector with direction that the wind is blowing from and magnitude of wind velocity b Ratio constant (in fuel flow) Propeller efficiency Atmospheric density Altitude dependent density

APPENDIX G REFERENCES AND SOURCES 1. Collins, B. P., Haines, A. L., and Pool, D. A., "Derivation and Current Capabilities of the Path Profile Fuel Consumption Algorithm," The MITRE Corporation, MTR-8OW195, September 1980.

2.

Collins, B. P., "Energy Modeling for Aviation Fuel Efficiency," The MITRE Corporation, MP-81W12, May 1981.

3. Winer, D. E., and Hoch, C. J., "Energy Conservation in Terminal Airspace Through Fuel Consumption Modeling," Federal Aviation Administration, Office of Environment and Energy, 1980.

4. Bellman, R. E., and Dreyfus, S. E., Applied Dynamic Programming, Princeton University Press, 1962.

Beechcraft Aircraft Corporation, "Super King Air 200 Pilot's 5.

Operating Handbook and Airplane Flight Manual," October 1979.

"Cessna 421C Golden Eagle Information 6. Cessna Aircraft Company, Manual," November 1979.

7. Jane's All the World's Aircraft, 1975-1976, edited by John W.

R. Taylor; Franklin Watts Inc., NY, 1975.

3. Recipient’s Catalog No.

1. Report No. 2. Government Accession No.

NASA CR-3533

I

5. Report Dote 4. Titic and Subtitle March 1982 A CONCEPT FOR A FUEL EFFICIENT FLIGHT 6. Performing Orgoniro~ion Cod* PLANNING AID FOR GENERAL AVIATION .8. Pwforming Orponirotion Raport NO.

7. Author’s) MTR-8lW233 B. P. Collins, A. L. Haines, and C. J. Wales I I I 10. Work Unit No. (TRAIS) i 9. Pwforming Orgarizotion Noms and Address The MITRE Corporation 11. Contract or Grant No.

Metrek Division

I

I NASl-16430 1820 Dolley Madison Blvd.

McLean, VA 22102 13. TYP* of Report and Period Covered 12. Sponsoring Agency Nome and Address Contractor Report National Aeronautics and Space Administration Washington, D. C. 20546 14. Sponsoring Agency Code 534-04-13-80 15. Supplcmentory Notes Langley Technical Monitor: Charles E. Knox Final Report ’ 16. Abstract MITRE has developed a core equation for estimation of fuel burn from path profile data.

This equation has been used in this report as a necessary ingredient in a dynamic program to define a fuel efficient flight path.

The resultant algorithm is oriented toward use by general aviation.

The pilot provides a description of the desired ground track, standard aircraft parameters, and weather at selected waypoints. The algorithm then derives the fuel efficient altitudes and velocities at the waypoints.

17. K l y Words 18. Distribution Statement Fuel Conservation Unclassified - Unlimited Flight Planning Subject Category 06 General Aviation 2;. No. of Pages 22. Price 19. Security Closril. (of this report) 20. Security Clossif. (of this page) 94 A05 Unclassified Unclassified For sale by the National Technical Information Service, Springfield, Virginia 22161 NASA-Lang