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Under a matrix form this gives: 2426 2427 ⎝ ⎜ ⎛ 𝑆𝑆𝑆𝑆𝑆𝑆1 𝑆𝑆𝑆𝑆𝑆𝑆2 𝑆𝑆𝑆𝑆𝑆𝑆3 ⋮ 𝑆𝑆𝑆𝑆𝑆𝑆12⎠ ⎟ ⎞ = ⎝ ⎜ ⎛ 1 𝑒𝑒1 𝑒𝑒1² 1 𝑒𝑒2 𝑒𝑒2² 1 𝑒𝑒3 𝑒𝑒3² ⋮ ⋮ ⋮ 1 𝑒𝑒12 ⋯ 𝑒𝑒12²⎠ ⎟ ⎞ �𝑏𝑏0𝑏𝑏1 𝑏𝑏2 � + ⎝ ⎜ ⎛ 𝐹𝐹1 𝐹𝐹2 𝐹𝐹3 ⋮ 𝐹𝐹12⎠ ⎟ ⎞ 2428 2429 SAR = M b + e 2430 2431 where: 2432 2433 SAR = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛
0.932
0.925
0.913
0.889
0.868
0.848
0.838
0.830
0.806
0.815
0.779
0.788⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ M = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 1 17800 17800² 1 18200 18200² 1 18620 18620² 1 18890 18890² 1 19350 19350² 1 19610 19610² 1 19920 19920² 1 20510 20510² 1 20790 20790² 1 21220 21220² 1 21480 21480² 1 22100 22100²⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ b = � 𝑏𝑏0 𝑏𝑏1 𝑏𝑏2 � e = ⎝ ⎜ ⎛ 𝐹𝐹1 𝐹𝐹2 𝐹𝐹3 ⋮ 𝐹𝐹n⎠ ⎟ ⎞. 2434 2435 The least square principle consists in looking for the parameter values of vector B, minimizing the sum of 2436 the squares of residuals, i.e.: 2437 2438 Min ∑ 𝐹𝐹i²𝑖𝑖=12𝑖𝑖=1 = min ∑ (𝑆𝑆𝑆𝑆𝑆𝑆(i) − 𝑏𝑏0 − 𝑏𝑏1 𝑒𝑒i − 𝑏𝑏2 𝑒𝑒i2)𝑖𝑖=12𝑖𝑖=1 ². 2439 2440 It is equivalent to look for the solutions of 𝜕𝜕(∑ 𝑟𝑟i²) 𝜕𝜕𝑏𝑏j = 0 for j = (0, 1, 2). 2441 2442 The solution B = � 𝐵𝐵0 𝐵𝐵1 𝐵𝐵2 � is given by: 2443 A-1 M’ SAR (see A4-2) 2444 2445 where: 2446 2447 M’ = transpose of M; and 2448 2449 A-1 = (M’ M)-1 = inverse of (M’ M). 2450 2451 Finally, B = �
3.26727172 −0.000205692
4.21798 𝑥𝑥10-09 �; and 2452 2453 SARav = 3.26727172 – 0.000205692 m + 4.21798.10-9 m². 2454 2455 Figure A4-14 provides a representation of the mean SAR model as a function of the airplane gross mass. 2456 2457 2458
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2459 Figure A4-14. SAR model (trend line) versus gross mass — Example 4 2460 2461 2462 The mean SAR values at each of the three reference gross masses of the FEM are as follows: 2463 2464 Table A4-15. 2465 2466 Mean SAR value — Example 4 2467 2468 Reference mass Mass value (kg) Mean SAR value (km/kg) Low mass 17,825 0.94100 Mid mass 19,953 0.84238 High mass 22,080 0.78198 2469 Estimate of the standard deviation(s): 2470 2471 𝑒𝑒 = �∑ (SARi−SARav(i))2𝑖𝑖=𝑛𝑛𝑖𝑖=1 𝑛𝑛−𝐾𝐾−1 = 0.01050 km/kg 2472 2473
0.95
0.90
0.85
0.80
0.75 17 000 18 000 19 000 20 000 21 000 22 000 23 000 Gross mass - m (kg) Example 4 — SAR model (trend line) versus gross mass SAR (km/kg) Measured data points High mass Low mass Mid mass Trend line
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where: 2474 2475 the number of data points (n) = 12; 2476 2477 K = 2 for a second order polynomial regression (see A4-2); and 2478 2479 the degrees of freedom (n-K-1) = 9. 2480 2481 2482 Confidence interval determination 2483 2484 The 90 percent confidence interval (CI90) at an airplane gross mass m0 is calculated as follows (see A4-2485
2): 2486 2487 𝐶𝐶𝐶𝐶90 = SARav(m0) ± 𝑒𝑒(.95 ,𝑛𝑛−𝐾𝐾−1) 𝑒𝑒 �m0 A-1 m0' 2488 2489 where: 2490 2491 the Student t-distribution for 90 percent confidence and 9 degrees of freedom t(.95, 9) = 1.833 (See Table 2492 A4-1); 2493 2494 m0 = (1 m0 m0²) and m0’ = � 1 𝑒𝑒0 𝑒𝑒0² �; and 2495 2496 A-1 = (M’ M)-1 = inverse of (M’ M). 2497 2498 Figure A4-16 provides a representation of the 90 percent confidence interval as a function of airplane 2499 gross mass. 2500 2501
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2502 Figure A4-16. 90 percent confidence interval versus gross mass — Example 4 2503 2504 The 90 percent confidence intervals at each of the three reference gross masses of the CO2 emissions 2505 evaluation metric are as follows: 2506 2507 Table A4-17. 2508 2509 Confidence intervals — Example 4 2510 2511 Reference mass Mass value (kg) 90% confidence interval (kg/km) Low mass 17,825 CI90 = 0.94100 ± 0.01399 Mid mass 19,953 CI90 = 0.84238 ± 0.00823 High mass 22,080 CI90 = 0.78198 ± 0.01505 2512 Check of confidence interval limits 2513 2514 For each of the three reference gross masses of the FEM, the confidence interval extends around the 2515 mean SAR value by an amount provided in Table A4-18. 2516 2517
0.95
0.90
0.85
0.80
0.75 17 000 18 000 19 000 20 000 21 000 22 000 23 000 Gross mass - m (kg) Example 4 — 90% confidence interval versus gross mass SAR (km/kg) Measured data points High mass 90% CI line - upper 90% CI line - lower Low mass Mid mass Trend line
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2518 Table A4-18. 2519 2520 Check of confidence intervals — Example 4 2521 2522 Reference mass Mass value (kg) 90% confidence interval (percentage of mean SAR) Low mass 17,825 (0.01399/0.94100) x 100 = 1.52% Mid mass 19,953 (0.00823/0.84238) x 100 = 0.98% High mass 22,080 (0.01505/0.78198) x 100 = 1.93% 2523 The 90 percent confidence intervals at the low and high reference gross masses of the FEM are above 2524 the confidence interval limit of 1.5 percent, as defined in § A38.5.3.4. 2525 2526 In such case, a penalty equal to the amount that the 90 percent confidence interval exceeds ±1.5 percent 2527 is applied to the mean SAR values as follows: 2528 2529 2530 Table A4-19. 2531 2532 Corrected SAR values — Example 4 2533 2534 Reference mass Mass value (kg) Corrected SAR value (km/kg) Low mass 17,825 0.94100 x [1 - (1.52-1.5)/100] =
0.94081 High mass 22,080 0.78198 x [1 - (1.93-1.5)/100] =
0.77862 2535 2536 As a result, the following mean SAR values associated to each of the three reference masses of the FEM 2537 can be used for the metric determination. 2538 2539 Table A4-20. 2540 2541 Mean SAR values — Example 4 2542 2543 Reference mass Mass value (kg) Mean SAR value (km/kg) Low mass 17,825 0.94081 Mid mass 19,953 0.84238 High mass 22,080 0.77862 2544 2545
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Appendix 5: References 2546 2547 2548 2549
1. International Standards and Recommended Practices, Annex 16 to the Convention 2550 on International Civil Aviation, Environmental Protection, Volume III – Aeroplane CO2 2551 Emissions, First Edition, July 2017, Applicable 1 January 2018, Including Amendment 2552 1 of January 1, 2021 2553 2554
2. International Standards and Recommended Practices, Annex 16 to the Convention 2555 on International Civil Aviation, Environmental Protection, Doc 9501 Environmental 2556 Technical Volume III, Procedures for the CO2 Emissions Certification of Aeroplanes, 2557 First Edition, 2018 2558 2559
3. ASTM International D1655-15 entitled “Standard Specification for Aviation Turbine 2560 Fuels”. This ASTM International publication may be obtained from the ASTM 2561 International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA, 2562 19428-2959 USA, www.astm.org. 2563 2564
4. UK MOD DEF-STAN 91-91, Issue 7, Amendment 3, entitled “Turbine Fuel, Kerosene 2565 Type, Jet A-1”. This United Kingdom Ministry of Defence Standard may be obtained 2566 from Defence Equipment and Support, UK Defence Standardization, Kentigern 2567 House, 65 Brown Street, Glasgow G2 8EX, UK. 2568 2569
5. ASTM International D4809-13 entitled “Standard Test Method for Heat of Combustion 2570 of Liquid Hydrocarbon Fuels by Bomb Calorimeter (Precision Method)”. This ASTM 2571 International publication may be obtained from the ASTM International, 100 Barr 2572 Harbor Drive, PO Box C700, West Conshohocken, PA, 19428-2959 USA, 2573 www.astm.org. 2574 2575
6. ASTM International D4052-11 entitled “Standard Test Method for Density and 2576 Relative Density and API Gravity of Liquids by Digital Density Meter”. This ASTM 2577 International publication may be obtained from the ASTM International, 100 Barr 2578 Harbor Drive, PO Box C700, West Conshohocken, PA, 19428-2959 USA, 2579 www.astm.org. 2580 2581
7. ASTM International D445-15 entitled “Standard Test Method for Kinematic Viscosity 2582 of Transparent and Opaque Liquids (and Calculation of Dynamic Viscosity)”. This 2583 ASTM International publication may be obtained from the ASTM International, 100 2584 Barr Harbor Drive, PO Box C700, West Conshohocken, PA, 19428-2959 USA, 2585 www.astm.org. 2586 2587 2588 2589 2590 2591 2592