Thrust and drag curves

AVIATION ENGINEERING & TECHNOLOGY AVIMA 2010

1

Table of Contents 1 Introduction .................................................................................................................................... 3

2 Problem description ........................................................................................................................ 4

3 Procedure ........................................................................................................................................ 5

3.1 Air speed ................................................................................................................................. 5

3.2 Lift coefficient ......................................................................................................................... 5

3.3 Drag coefficient ....................................................................................................................... 5

3.4 Drag forces .............................................................................................................................. 5

3.5 Thrust ...................................................................................................................................... 5

4 Calculating drag and thrust for each height ................................................................................... 6

4.1 For H=0 .................................................................................................................................... 6

4.2 For H= 4 km ............................................................................................................................. 7

4.3 For H=6 Km .............................................................................................................................. 8

4.4 For H=8 Km .............................................................................................................................. 9

4.5 For H=10 km .......................................................................................................................... 10

4.6 For H=11 km .......................................................................................................................... 11

5 Drag and thrust curves .................................................................................................................. 12

6 Calculating minimum drag ............................................................................................................ 14

7 Calculating speed limits ................................................................................................................ 15

8 Conclusion ..................................................................................................................................... 16

List of tables

Table 1-1, Airbus A300B2 Specifications .......................................................................................3

Table 2-1, Technical data of Airbus A300B2 ......................................................................................... 4

Table 2-2, values dependent to height for H= 0 to 11 km .................................................................... 4

Table 4-1, values for H=0 ...................................................................................................................... 6




Table 4-5, values for H=10 ................................................................................................................... 10

Table 4-6, values for H=11 ................................................................................................................... 11


2

Table 6-1, Minimum drag and the correspondence speed for H=0 to 11 km ..................................... 15

Table 7-1, Maximum and minimum speeds for positive and negative amounts near to T-D=0 ......... 15

List of figures

Figure 5-1, Induced and parasite drag for H=0 .................................................................................... 12

Figure 5-2, drag curves for H=0 to 11 km ............................................................................................ 12

Figure 5-3, Thrust curves for H=0 to 11 km ......................................................................................... 13

Figure 5-4, Pénaud’s diagram for H=0 to 11 km .................................................................................. 14

Figure 7-2, speed limits for H=0 to 11 km ............................................................................................ 16

Symbols and mathematical signs

• CD Drag Coefficient

• CD0 Zero Drag Coefficient

• CL Lift Coefficient

• CTP Static thrust load coefficient

• D Drag Force

• g Acceleration due to gravity

• H Height

• K Induced Drag Coefficient

• K1 Engine parameter

• m Mass

• Ma Mach number

• nF Engine parameter

• S Wing Area

• ST Engine area

• T Thrust

• V Air Speed

• ρ Air Density


3

1 Introduction

In this case study a uncelebrated and level flight of Airbus A300B2 will be investigated. The

most important assumption for a horizontal flight is that all horizontal and vertical forces

are in balance:

� = �� = �

In addition the thrust vector is in the same direction as airflow and the flight is in no-wind

condition.

The first step is to calculating drag and thrust for different height from sea level to 11 km

and shows them in separate diagrams. Also minimum drag surplus will be calculated and

maximum thrust will be demonstrated in the tables. Both drag and thrust curves (thrust

required and thrust available) will be plotted in Pénaud’s diagram, for each height. Finally,

the speed limits will be demonstrated in tables and will be plotted as two curves for the

most nearest negative and positive values to the T=D, where the intersections of the both

curves and the so-called speed limits are.

Airbus A300B4

Engines

Two 227kN General Electric CF6-

50Cs / 236kN Pratt & Whitney

JT9D-9 turbofans

High Cruise

speed 917 km/h

Max range 3430 km

Operating

weight 85,910 kg

Max takeoff

weight 142,000 kg

Wing span 44.48 m

Length 53.62 m

Height 16.53 m

Table 1-1, Airbus A300B2 Specifications


4

2 Problem description

� Calculating the following values for height 0, 4, 6, 8, 10, 11 Km and for Mach number

between 0.1 to 1.0:

1. Air speed (V)

2. Lift coefficient (CL)

3. Drag coefficient (CD)

4. Drag (D)

5. Static thrust load coefficient (CTp)

6. Thrust (T)

7. Thrust surplus (T-D)

� Show the results for drag and thrust in a common diagram (Pénaud’s diagram).

� Determine the maximum thrust surplus for each height.

� Determine the minimum drag for each height.

� Determine speed limits for each height and show them in a common diagram H = f(Ma).

Technical data needed for calculating the above mentioned values are as follow:

Parameter Sign Value

Mass m 127,6 t

Wing area S 260 m²

Drag coefficient at zero lift CD0 0,0175

Parameter for induced drag K 0,0510

Engine reference area ST 4,52 m²

Number of engines i 2

Engine parameter (from 0 to 11 km) K1 = 0,192 + 0,015 H[km]

Engine parameter (from 0 to 11 km) nF = -0,292 + 0,020 H[km]

Acceleration of gravity g 9,80665 m/s2

Table 2-1, Technical data of Airbus A300B2

As the values should be calculated for different height, values, which are dependent to

height, are selected from normal atmosphere table:

H (km) P(N/m2) ΡΡΡΡ (kg/m3) A (m/s) K1 nF

0 101325 1.225 340 0.192 -0.292

4 61640.2 0.819 324.58 0.252 -0.212

6 47181 0.659697 316.43 0.282 -0.172

8 35599.8 0.5251 308.06 0.312 -0.132

10 26436.2 0.412706 299.46 0.342 -0.092

11 22632 0.3629 295.07 0.357 -0.072

Table 2-2, values dependent to height for H= 0 to 11 km


5

3 Procedure

All the calculations are based on a function of Mach number. Therefore, in diagrams the Mach

number are represented is a speed measure instead of air speed. The Mach number varies from 0.1

to 1.0 and other data will be calculated based on these mach numbers in each demanded height.

The change in elevation results in change of density, pressure and speed of sound, which are

demonstrated in table 1.

3.1 Air speed

The first step is to calculate the air speed as a function of Mach number:

� = � × � (Eq. 1 − 1)

3.2 Lift coefficient

As described before, due to stationary horizontal flight with mentioned conditions:

�� = �� = �� → �� = � �� × � × �� (Eq. 1 − 2)

3.3 Drag coefficient �� = �� + �� (Eq. 1 − 3)

3.4 Drag forces

Having calculated drag coefficient and air speed, the drag force is equal to: � (!) = ��

� = "2 �� = "2 ��(�� + ��) = "2 �� + "2 �� # �"2 ��$�

(%&. 1 − 4)

3.5 Thrust

For calculating the thrust the following parameters should be calculated:

Static thrust load coefficient: �() = �*�+, (Eq. 1 − 5) �* = �� + �*. (Eq. 1 − 6) 01 = 2� + 2*. (Eq. 1 − 7)

Therefore, the thrust is equal: �4 = �()5�( , 7 = 0892:; <= 9<><;? (Eq. 1 − 8) �A = 7 × �4 (Eq. 1 − 9)


6

4 Calculating drag and thrust for each height

Based on the procedure from 3-1 to 3-5, the corresponding values of drag, thrust, maximum

thrust surplus and the speed limits (Ma) for negative thrust surplus for Mach number

between 0.1 and 1.0 is as follow:

4.1 For H=0

Ma V CL CD D CTP T TG T-D

0.1 34 6.797 2.3738 437008 0.376 172248.8 344497.7 -92510

0.125 42.5 4.350 0.9827 282657 0.352 161383.3 322766.6 40110

0.15 51 3.021 0.4830 200042 0.334 153016.3 306032.6 105990

0.175 59.5 2.220 0.2687 151511 0.319 146281.4 292562.8 141052

0.2 68 1.699 0.1648 121333 0.307 140687.5 281375 160042

0.225 76.5 1.343 0.1094 101996 0.297 135931.1 271862.3 169867

0.25 85 1.088 0.0778 89541 0.288 131812.9 263625.7 174085

0.275 93.5 0.899 0.0587 81724 0.280 128195 256390 174666

0.3 102 0.755 0.0466 77193 0.273 124978.9 249957.9 172765

0.325 110.5 0.644 0.0386 75097 0.267 122091.8 244183.5 169087

0.35 119 0.555 0.0332 74876 0.261 119478.1 238956.3 164080

0.375 127.5 0.483 0.0294 76151 0.256 117095.2 234190.4 158039

0.4 136 0.425 0.0267 78658 0.251 114909.2 229818.4 151161

0.425 144.5 0.376 0.0247 82206 0.246 112892.9 225785.8 143579

0.45 153 0.336 0.0232 86659 0.242 111024.3 222048.7 135389

0.475 161.5 0.301 0.0221 91914 0.239 109285.3 218570.6 126657

0.5 170 0.272 0.0213 97892 0.235 107660.7 215321.3 117429

0.525 178.5 0.247 0.0206 104534 0.232 106137.7 212275.4 107741

0.55 187 0.225 0.0201 111794 0.229 104705.7 209411.4 97617

0.575 195.5 0.206 0.0197 119635 0.226 103355.4 206710.8 87076

0.6 204 0.189 0.0193 128028 0.223 102078.9 204157.9 76130

0.625 212.5 0.174 0.0190 136950 0.220 100869.4 201738.7 64789

0.65 221 0.161 0.0188 146381 0.218 99720.76 199441.5 53061

0.675 229.5 0.149 0.0186 156306 0.215 98627.85 197255.7 40950

0.7 238 0.139 0.0185 166713 0.213 97586.03 195172.1 28460

0.725 246.5 0.129 0.0184 177590 0.211 96591.2 193182.4 15593

0.75 255 0.121 0.0182 188928 0.209 95639.74 191279.5 2351

0.775 263.5 0.113 0.0182 200721 0.207 94728.39 189456.8 -11264

0.8 272 0.106 0.0181 212962 0.205 93854.26 187708.5 -25254

0.825 280.5 0.100 0.0180 225645 0.203 93014.73 186029.5 -39616

0.85 289 0.094 0.0180 238767 0.201 92207.44 184414.9 -54352

0.875 297.5 0.089 0.0179 252322 0.200 91430.25 182860.5 -69461

0.9 306 0.084 0.0179 266307 0.198 90681.24 181362.5 -84945

Table 4-1, values for H=0


7

4.2 For H= 4 km


0.1 32.46 10.054 5.173 580241 0.411 114394 228788 -351453

0.125 40.57 6.435 2.129 373165 0.392 109108 218217 -154948

0.15 48.69 4.469 1.036 261429 0.377 104972 209943 -51486

0.175 56.8 3.283 0.567 194837 0.365 101597 203193 8356

0.2 64.92 2.514 0.340 152421 0.354 98761 197522 45100

0.225 73.03 1.986 0.219 124165 0.346 96325 192651 68485

0.25 81.15 1.609 0.149 104793 0.338 94198 188395 83602

0.275 89.26 1.329 0.108 91311 0.331 92313 184627 93315

0.3 97.37 1.117 0.081 81920 0.325 90626 181252 99333

0.325 105.5 0.952 0.064 75482 0.320 89101 178202 102721

0.35 113.6 0.821 0.052 71252 0.315 87712 175425 104172

0.375 121.7 0.715 0.044 68726 0.310 86439 172877 104152

0.4 129.8 0.628 0.038 67550 0.306 85264 170528 102979

0.425 137.9 0.557 0.033 67471 0.302 84175 168351 100879

0.45 146.1 0.497 0.030 68307 0.298 83161 166323 98016

0.475 154.2 0.446 0.028 69919 0.295 82214 164427 94508

0.5 162.3 0.402 0.026 72205 0.292 81324 162649 90444

0.525 170.4 0.365 0.024 75084 0.289 80488 160975 85891

0.55 178.5 0.332 0.023 78496 0.286 79698 159396 80900

0.575 186.6 0.304 0.022 82390 0.283 78950 157900 75510

0.6 194.7 0.279 0.021 86729 0.281 78241 156482 69753

0.625 202.9 0.257 0.021 91482 0.278 77567 155134 63652

0.65 211 0.238 0.020 96622 0.276 76925 153849 57228

0.675 219.1 0.221 0.020 102129 0.274 76312 152623 50494

0.7 227.2 0.205 0.020 107986 0.272 75726 151451 43465

0.725 235.3 0.191 0.019 114179 0.270 75164 150329 36149

0.75 243.4 0.179 0.019 120696 0.268 74626 149252 28556

0.775 251.5 0.167 0.019 127528 0.266 74109 148218 20691

0.8 259.7 0.157 0.019 134664 0.264 73612 147224 12560

0.825 267.8 0.148 0.019 142099 0.262 73133 146266 4167

0.85 275.9 0.139 0.018 149827 0.261 72672 145344 -4483

0.875 284 0.131 0.018 157841 0.259 72227 144453 -13388

0.9 292.1 0.124 0.018 166138 0.258 71797 143593 -22545



8

4.3 For H=6 Km


0.1 31.64 14.574 10.851 931611 0.419 89362 178725 -752886

0.125 39.55 9.328 4.455 597617 0.403 85998 171995 -425622

0.15 47.46 6.478 2.157 416762 0.391 83343 166685 -250077

0.175 55.38 4.759 1.173 308310 0.381 81162 162324 -145987

0.2 63.29 3.644 0.695 238537 0.372 79319 158638 -79899

0.225 71.2 2.879 0.440 191332 0.364 77728 155457 -35875

0.25 79.11 2.332 0.295 158208 0.358 76332 152665 -5543

0.275 87.02 1.927 0.207 134352 0.352 75091 150183 15830

0.3 94.93 1.619 0.151 116868 0.347 73976 147952 31084

0.325 102.8 1.380 0.115 103928 0.342 72964 145929 42001

0.35 110.8 1.190 0.090 94333 0.338 72040 144080 49747

0.375 118.7 1.036 0.072 87270 0.334 71190 142381 55111

0.4 126.6 0.911 0.060 82172 0.330 70404 140809 58637

0.425 134.5 0.807 0.051 78633 0.327 69674 139348 60715

0.45 142.4 0.720 0.044 76357 0.324 68993 137985 61628

0.475 150.3 0.646 0.039 75124 0.321 68354 136708 61584

0.5 158.2 0.583 0.035 74767 0.318 67754 135507 60740

0.525 166.1 0.529 0.032 75158 0.315 67187 134375 59216

0.55 174 0.482 0.029 76198 0.313 66652 133304 57105

0.575 181.9 0.441 0.027 77809 0.310 66144 132288 54480

0.6 189.9 0.405 0.026 79927 0.308 65662 131324 51397

0.625 197.8 0.373 0.025 82503 0.306 65202 130405 47902

0.65 205.7 0.345 0.024 85495 0.304 64764 129528 44033

0.675 213.6 0.320 0.023 88872 0.302 64345 128690 39818

0.7 221.5 0.297 0.022 92605 0.300 63944 127887 35283

0.725 229.4 0.277 0.021 96671 0.298 63559 127118 30447

0.75 237.3 0.259 0.021 101051 0.296 63189 126379 25327

0.775 245.2 0.243 0.021 105730 0.295 62834 125668 19938

0.8 253.1 0.228 0.020 110694 0.293 62492 124984 14290

0.825 261.1 0.214 0.020 115930 0.291 62162 124324 8394

0.85 269 0.202 0.020 121430 0.290 61844 123687 2257

0.875 276.9 0.190 0.019 127184 0.289 61536 123072 -4112

0.9 284.8 0.180 0.019 133186 0.287 61239 122477 -10709



9

4.4 For H=8 Km


0.1 30.81 19.316 19.046 1233832 0.4228 68036 136073 -1097760

0.125 38.51 12.362 7.812 790698 0.4105 66062 132123 -658575

0.15 46.21 8.585 3.776 550417 0.4008 64491 128981 -421436

0.175 53.91 6.307 2.046 405986 0.3927 63192 126383 -279602

0.2 61.61 4.829 1.207 312709 0.3859 62088 124175 -188534

0.225 69.31 3.815 0.760 249235 0.3799 61130 122259 -126976

0.25 77.02 3.091 0.505 204317 0.3747 60285 120571 -83746

0.275 84.72 2.554 0.350 171575 0.37 59532 119064 -52511

0.3 92.42 2.146 0.252 147170 0.3657 58852 117704 -29466

0.325 100.1 1.829 0.188 128680 0.3619 58233 116467 -12213

0.35 107.8 1.577 0.144 114516 0.3584 57667 115333 817

0.375 115.5 1.374 0.114 103601 0.3551 57144 114287 10686

0.4 123.2 1.207 0.092 95183 0.3521 56659 113318 18135

0.425 130.9 1.069 0.076 88724 0.3493 56207 112415 23691

0.45 138.6 0.954 0.064 83831 0.3467 55785 111570 27739

0.475 146.3 0.856 0.055 80214 0.3442 55388 110776 30563

0.5 154 0.773 0.048 77650 0.3419 55014 110029 32379

0.525 161.7 0.701 0.043 75971 0.3397 54661 109322 33351

0.55 169.4 0.639 0.038 75044 0.3376 54327 108653 33609

0.575 177.1 0.584 0.035 74766 0.3356 54009 108018 33251

0.6 184.8 0.537 0.032 75054 0.3338 53706 107412 32358

0.625 192.5 0.494 0.030 75842 0.332 53418 106835 30993

0.65 200.2 0.457 0.028 77075 0.3303 53142 106284 29209

0.675 207.9 0.424 0.027 78709 0.3286 52878 105755 27047

0.7 215.6 0.394 0.025 80708 0.327 52624 105249 24541

0.725 223.3 0.367 0.024 83042 0.3255 52381 104763 21721

0.75 231 0.343 0.024 85685 0.3241 52147 104295 18610

0.775 238.7 0.322 0.023 88616 0.3227 51922 103844 15229

0.8 246.4 0.302 0.022 91817 0.3213 51705 103410 11593

0.825 254.1 0.284 0.022 95273 0.32 51495 102991 7718

0.85 261.9 0.267 0.021 98971 0.3188 51293 102586 3615

0.875 269.6 0.252 0.021 102899 0.3175 51097 102194 -705

0.9 277.3 0.238 0.020 107047 0.3164 50907 101815 -5233



10

4.5 For H=10 km


0.1 29.95 26.009 34.516 1660648 0.423 50508 101017 -1559631

0.125 37.43 16.645 14.148 1063592 0.414 49482 98964 -964627

0.15 44.92 11.559 6.832 739586 0.407 48659 97318 -642268

0.175 52.41 8.493 3.696 544556 0.401 47974 95948 -448609

0.2 59.89 6.502 2.174 418319 0.397 47388 94776 -323543

0.225 67.38 5.137 1.364 332125 0.392 46877 93755 -238371

0.25 74.87 4.161 0.901 270831 0.389 46425 92850 -177981

0.275 82.35 3.439 0.621 225846 0.385 46020 92040 -133806

0.3 89.84 2.890 0.443 192001 0.382 45653 91306 -100695

0.325 97.32 2.462 0.327 166035 0.379 45318 90636 -75399

0.35 104.8 2.123 0.247 145808 0.377 45010 90020 -55788

0.375 112.3 1.849 0.192 129871 0.374 44725 89450 -40420

0.4 119.8 1.626 0.152 117209 0.372 44460 88921 -28288

0.425 127.3 1.440 0.123 107100 0.370 44213 88426 -18674

0.45 134.8 1.284 0.102 99016 0.368 43981 87963 -11053

0.475 142.2 1.153 0.085 92562 0.366 43763 87526 -5035

0.5 149.7 1.040 0.073 87441 0.365 43557 87114 -327

0.525 157.2 0.944 0.063 83426 0.363 43362 86724 3298

0.55 164.7 0.860 0.055 80339 0.361 43177 86354 6015

0.575 172.2 0.787 0.049 78040 0.360 43001 86001 7962

0.6 179.7 0.722 0.044 76416 0.358 42833 85665 9249

0.625 187.2 0.666 0.040 75380 0.357 42672 85344 9964

0.65 194.6 0.616 0.037 74858 0.356 42518 85037 10178

0.675 202.1 0.571 0.034 74791 0.355 42371 84742 9951

0.7 209.6 0.531 0.032 75130 0.353 42229 84459 9329

0.725 217.1 0.495 0.030 75833 0.352 42093 84187 8353

0.75 224.6 0.462 0.028 76868 0.351 41962 83924 7056

0.775 232.1 0.433 0.027 78205 0.350 41836 83672 5466

0.8 239.6 0.406 0.026 79820 0.349 41714 83428 3607

0.825 247.1 0.382 0.025 81693 0.348 41596 83192 1499

0.85 254.5 0.360 0.024 83805 0.347 41482 82963 -842

0.875 262 0.340 0.023 86142 0.346 41371 82743 -3399

0.9 269.5 0.321 0.023 88691 0.345 41264 82528 -6162



11

4.6 For H=11 km


0.1 29.51 30.381 47.089 1939540 0.421 43105 86210 -1853330

0.125 36.88 19.444 19.298 1241971 0.415 42418 84836 -1157134

0.15 44.26 13.502 9.316 863319 0.409 41865 83730 -779589

0.175 51.64 9.920 5.036 635291 0.405 41403 82806 -552486

0.2 59.01 7.595 2.959 487588 0.401 41007 82014 -405575

0.225 66.39 6.001 1.854 386626 0.397 40660 81321 -305305

0.25 73.77 4.861 1.223 314716 0.394 40353 80706 -234010

0.275 81.14 4.017 0.841 261824 0.392 40077 80154 -181669

0.3 88.52 3.376 0.599 221912 0.389 39827 79654 -142258

0.325 95.9 2.876 0.439 191170 0.387 39598 79196 -111974

0.35 103.3 2.480 0.331 167101 0.385 39387 78775 -88326

0.375 110.7 2.160 0.256 148008 0.383 39192 78384 -69623

0.4 118 1.899 0.201 132709 0.381 39010 78021 -54688

0.425 125.4 1.682 0.162 120359 0.380 38841 77681 -42678

0.45 132.8 1.500 0.132 110340 0.378 38681 77362 -32978

0.475 140.2 1.347 0.110 102194 0.377 38531 77062 -25133

0.5 147.5 1.215 0.093 95573 0.375 38389 76777 -18795

0.525 154.9 1.102 0.079 90210 0.374 38254 76508 -13701

0.55 162.3 1.004 0.069 85897 0.373 38126 76252 -9645

0.575 169.7 0.919 0.061 82472 0.372 38004 76009 -6464

0.6 177 0.844 0.054 79805 0.370 37888 75776 -4029

0.625 184.4 0.778 0.048 77790 0.369 37777 75554 -2236

0.65 191.8 0.719 0.044 76343 0.368 37670 75341 -1002

0.675 199.2 0.667 0.040 75394 0.367 37568 75136 -258

0.7 206.5 0.620 0.037 74887 0.366 37470 74940 53

0.725 213.9 0.578 0.035 74773 0.365 37375 74751 -22

0.75 221.3 0.540 0.032 75013 0.364 37284 74568 -444

0.775 228.7 0.506 0.031 75573 0.364 37196 74393 -1180

0.8 236.1 0.475 0.029 76425 0.363 37111 74223 -2202

0.825 243.4 0.446 0.028 77545 0.362 37029 74058 -3487

0.85 250.8 0.420 0.027 78913 0.361 36950 73899 -5013

0.875 258.2 0.397 0.026 80509 0.360 36873 73745 -6764

0.9 265.6 0.375 0.025 82321 0.360 36798 73596 -8725


AVIATION ENGINEERING

5 Drag and thrust curves

Based on the equation in 1

first part is

is formed from sum of these parts

50000

100000

150000

200000

250000

300000

350000

400000

450000

Dra

g (

D)

{N}

50000

100000

150000

200000

250000

300000

350000

Dra

g (

D),

{N

}


Drag and thrust curves

n the equation in 1

part is called parasite drag and the second


0

50000

100000

150000

200000

250000

300000

350000

400000

450000

0

Induced drag, parasite drag and drag curves in H=0 km

50000

100000

150000

200000

250000

300000

350000

0 0.1

AVIATION ENGINEERING & TECHNOLOGY


n the equation in 1-4, drag is consisted of two parts:

called parasite drag and the second


Figure 5-

Figure

0.2


0.1 0.2 0.3

& TECHNOLOGY


, drag is consisted of two parts:

called parasite drag and the second

is formed from sum of these parts:

-1, Induced and parasite drag for H=0

Figure 5-2, drag curves for H=0 to 11 km

0.4

Mach number (Ma) {


0.3 0.4

Mach Number, (Ma) {

Drag diagram

, drag is consisted of two parts:

called parasite drag and the second part is induced drag, which the drag curved

1, Induced and parasite drag for H=0

2, drag curves for H=0 to 11 km

0.6

Mach number (Ma) {-}


0.5 0.6

Mach Number, (Ma) {-}

Drag diagram

AVIMA 2010

, drag is consisted of two parts: �1) �� induced drag, which the drag curved


2, drag curves for H=0 to 11 km

0.8


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AVIMA 2010

) �� 0Cinduced drag, which the drag curved


1


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�0C �2) �D�E�FGE. The

induced drag, which the drag curved

Parasite Drag

Induced darg

Drag

D, H=0

D, H=4

D, H=6

D, H=8

D, H=10

D, H=11

The

induced drag, which the drag curved


So, by increasing the Mach number in each altitude, the first part, parasite drag, increases

proportional to

the drag decreases to its minimum point (minimum drag speed).

the density reduces and therefore, the speed should increases in the

the dynamic pressure (

each height will be the same

Therefore, for each higher altitude, the curve is

right side. Only at higher altitude, the increasing rate of parasite drag is decreased, since the

density has a direct influence on parasite drag.

figure 5-2.

The curves for thrust in each altitude also show that with increasing the

available is decreased and also remain almost constant with increasing the Mach number.

Thrust available is dependent on pressure and two engine parameters, which vary

different altitude. Thrust coefficient,

altitude than se�+, →amounts of trust available in higher altitude is lower than those near the sea level.

results for thrust curves are also demonstrated in figure 5

As a result, the combination of all drag (thrust required) and thrust available for each

altitude are plotted and demonstrated in one diagram, Pénaud’s diagram.

50000

100000

150000

200000

250000

300000

350000

Th

rust

(T

) {N

}



proportional to �� and the induced drag is reduced, since it is proportional to



the dynamic pressure (

each height will be the same




2.

ves for thrust in each altitude also show that with increasing the




altitude than sea level, since the n1. In addition the pressure decreases as the altitude

amounts of trust available in higher altitude is lower than those near the sea level.




50000

100000

150000

200000

250000

300000

350000

0 0.1



and the induced drag is reduced, since it is proportional to



the dynamic pressure (�� ) constant.

each height will be the same, but the minimum drag speed increases with altitude increase.








level, since the n

. In addition the pressure decreases as the altitude



Figure



0.1 0.2 0.3

& TECHNOLOGY





) constant. As it will be discussed in

, but the minimum drag speed increases with altitude increase.







different altitude. Thrust coefficient, �() =level, since the nF has a negative amount and




Figure 5-3, Thrust curves for H=0 to 11 km



0.3 0.4

Mach Number, (Ma) {

Thrust diagram





As it will be discussed in


Therefore, for each higher altitude, the curve is almost the same, but only just shifted to the


density has a direct influence on parasite drag. The result for all drag curves are shown in



Thrust available is dependent on pressure and two engine parameters, which vary = �*�+, , is almost remains constant in higher

has a negative amount and




, Thrust curves for H=0 to 11 km



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Thrust diagram

AVIMA 2010





As it will be discussed in part 6,


almost the same, but only just shifted to the


The result for all drag curves are shown in




, is almost remains constant in higher

has a negative amount and is decreased

. In addition the pressure decreases as the altitude increases and therefore, the


results for thrust curves are also demonstrated in figure 5-3.

, Thrust curves for H=0 to 11 km



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Thrust diagram

AVIMA 2010


and the induced drag is reduced, since it is proportional to Gthe drag decreases to its minimum point (minimum drag speed). As the altitude increase

the density reduces and therefore, the speed should increases in the same time to maintain

part 6, minimum drag for





ves for thrust in each altitude also show that with increasing the altitude, the thrust




is decreased to zero, then:

increases and therefore, the




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AVIMA 2010

So, by increasing the Mach number in each altitude, the first part, parasite drag, increases*GE. Therefore

As the altitude increase

same time to maintain

minimum drag for





altitude, the thrust


Thrust available is dependent on pressure and two engine parameters, which vary in


to zero, then:


amounts of trust available in higher altitude is lower than those near the sea level. The


T, H=0

T, H=4

T, H=6

T, H=8

T, H=10

T, H=11


. Therefore,

As the altitude increase,

same time to maintain

minimum drag for





altitude, the thrust


in


to zero, then:


The


6 Calculating minimum drag

Minimum drag in each

flight. The minimum drag remains constant in each altitude, but the

minimum drag, occurs, increases as the altitude increase, since as the Eq. 1

on density:

� = "2 ��For obtain the minimum drag, its derivation should be equal to the zero:

→ C� =→ �H = The results for each

from the results, the amount of minimum drag rem

amounts, which are highlighted in the above table as minimum drag (without calculations)

are almost equal in each height.

50000

100000

150000

200000

250000

300000

350000

Dra

g (

D),

Th

rust

(T

) {N

}


Calculating minimum drag

Minimum drag in each

The minimum drag remains constant in each altitude, but the


on density:

�� = "2 ��For obtain the minimum drag, its derivation should be equal to the zero:

"�� − 4"�4��"�� (EqThe results for each



re almost equal in each height.

50000

100000

150000

200000

250000

300000

350000

0 0.1


Figure 5


Minimum drag in each height is, the minimum thrust requir



��(�� + �For obtain the minimum drag, its derivation should be equal to the zero:4��"��I = 0

Eq. 1 − 10)

The results for each altitude and the equivalent drag are gathered in the table ….. As it clear



re almost equal in each height.

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& TECHNOLOGY

5-4, Pénaud’s diagram


height is, the minimum thrust requir



��) = "2 ��For obtain the minimum drag, its derivation should be equal to the zero:

altitude and the equivalent drag are gathered in the table ….. As it clear



0.3 0.4

Mach Number, (Ma) {

Pénaud’s diagram

Pénaud’s diagram for H=0 to 11 km

height is, the minimum thrust requir



�� + "2 ��For obtain the minimum drag, its derivation should be equal to the zero:


from the results, the amount of minimum drag remains the same for each altitude. Also, the


0.5 0.6


Pénaud’s diagram

AVIMA 2010

for H=0 to 11 km

height is, the minimum thrust required to for a steady and level



�� # �"2 ��For obtain the minimum drag, its derivation should be equal to the zero:


ains the same for each altitude. Also, the


0.7 0.8

Pénaud’s diagram

AVIMA 2010

for H=0 to 11 km

ed to for a steady and level

The minimum drag remains constant in each altitude, but the speed, i

minimum drag, occurs, increases as the altitude increase, since as the Eq. 1-10, it depends

�$�

(%&. 1 −For obtain the minimum drag, its derivation should be equal to the zero:




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AVIMA 2010


speed, in which the

10, it depends

− 9)




D, H=0

T, H=0

D, H=4

T, H=4

D, H=6

T, H=6

D, H=8

T, H=8

D, H=10

T, H=10

D, H=11

T, H=11


n which the

10, it depends





15

H ρρρρ Vmd Dmin

0 1.225 109.9519 67383.35

4 0.819 134.471 67383.35

6 0.6596 149.8409 67383.35

8 0.5251 167.9382 67383.35

10 0.4127 189.432 67383.35

11 0.3639 201.7342 67383.35

Table 6-1, Minimum drag and the correspondence speed for H=0 to 11 km

7 Calculating speed limits

For each drag and thrust curves in each height there are two intersection points, where T=D,

or thrust required is equal thrust available. That is for satisfying the level flight condition

T=D, two speed limitations are defined, which flying beyond these limits leads to an

unsteady flight:

1. Lift limit: The aircraft in each altitude cannot fly below stall speed. Stall speed

happens when lift coefficient increases till it reaches the stall amount. ��KLM�� decreases the speed to �KLM��.

2. Mach limit: The second limit is the highest speed, where the thrust required and

thrust available are the same. Again here flying with a speed more than this amount

leads to unsteady flight.

Based on the data in the above table for each altitude, the maximum and minimum speed

for the 2 nearest positive and negative thrust surpluses to T-D=0 are shown in the following

table:

H Negative amount Positive amount

Vmin Vmax Vmin Vmax

0 0.1 0.775 0.125 0.75

4 0.15 0.85 0.175 0.825

6 0.25 0.875 0.275 0.85

8 0.325 0.875 0.35 0.85

10 0.5 0.85 0.525 0.825

11 0.675 0.7 0.725 0.7

Table 7-1, Maximum and minimum speeds for positive and negative amounts near to T-D=0,

For H=0 to 11 km


16

Based on the values of the table 7-1, the diagram of height vs. Mach number, is as follow:

Figure 7-2, speed limits for H=0 to 11 km

The curves for values T=D should be between the dark and light blue / red curves. In the

figure on area outside of the level-flight envelope is the area in which the aircraft cannot fly

in that region in straight and level flight.

8 Conclusion

Based on the above discussed issues, it can be summarized that for a level horizontal flight

without acceleration, with no wind situation and thrust vector in airspeed direction, the

following situation happens with changes in altitude and Mach speed:

1. The drag curves, consists of two parasites and induced parts. With increasing the

Mach number, the drag first reaches its minimum value at speed for minimum drag.

That is, the pilot needs by flying in this speed minimum required drag. After that the

drag curve has an increasing trend. But the steep in each higher altitude decreases,

since it is proportional to V and speed also varies with changing by altitude, eq.1-1.

2. The thrust values in low altitude decrease with increasing the Mach number but for

higher altitudes the values remain constant for different speed. Also, the thrust

values are decreasing for higher altitude, since the thrust is proportional mainly with

pressure, which decreases as the altitude increases.

3. In the Pénaud’s diagram, drag and thrust curves in each altitude have two

intersection points, which denotes is the speed limits points. The flight envelope

diagram shows these speed limits for different altitudes. If a pilot flies with speed

lower than minimum speed (stall speed) and maximum speed, the flight would be

not stable and horizontal anymore. At high altitude the margin for stable and level

flying is too small and the curves meet each other at Q-point.

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

He

igh

t (k

m)

Speed (m/s)

Speed limits

Stall limit Buffet limit

Flight envelope

Stall limit

Q-Corner

Flight envelope

Thrust and drag curves

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