Expectations You will understand when and how to calculate mission performance

1

• Lesson objective - to discuss
• Air vehicle performance
• including
• Mission segment performance
• Breguet range and endurance
• and typical design applications

Expectations - You will understand when and how
to calculate mission performance
21-1
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Discussion subjects

• Mission segments
• Start
• Taxi
• Takeoff
• Climb
• Cruise
• Loiter
• Acceleration
• Turn performance
• Descend and land

21-2
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Overall approach

• We will develop air vehicle performance
  equations for a typical mission from engine start
  through landing
• - Some of the equations will be exact, most will
  be approximate to simplify analysis
• Performance will be calculated by mission segment
  
• - Using simplified performance methods
• For each segment we will first discuss
  methodology and then address application
• - Using our example UAVs from the previous
  chapter
• The methods will be applied to the turboprop
  (TBProp) UAV example
• - We will take it through full mission analysis

21-3
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Mission definition
21-4
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Start-taxi-takeoff fuel

• At a detail level, fuel requirements for engine
  start taxi and takeoff (Wfstto) are defined by
  power settings and times - typical examples
• - Engine start and taxi 20 minutes at idle
  power
• - Idle fuel flow X WdotF0
• - Takeoff 1 minute at maximum power
• Therefore
• Wfstto/W0 (20X1)/60(T0/W0)TSFC0
  (21.1)
• - Or for a typical transport where T0/W0 .3,
• TSFC0 .4, X .1, we calculate Wfstto
  0.006W0
• - While for a typical non A/B fighter T0/W0 .5,
  TSFC0 1.0, X .1, we calculate Wfstto
  0.025W0
• These values compare to RayAD Equation 6.8
  suggested values where Wfstto (0.01-0.03)W0

where
and
21-5
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Typical application

• The resized/converged example TBProp UAV has a
  takeoff gross weight (W0) of 2152 lb (chart
  20-30), fuel fraction of 0.175, wing reference
  area (Sref) of 71.7 sqft, a takeoff
  power-to-weight (Bhp0/W0) of 0.092, and SLS
  specific fuel consumption (SFC0) of 0.73, where
• - Engine start and taxi 30 minutes at idle
  power
• Idle fuel flow (X/100)WdotF0 where x 10
• - Takeoff 1 minute at maximum power
• From Eq 21.1
• Wfstto/W0 (30X1)/60(BHp0/W0)SFC0
  .0045
• and
• Wfstto 0.0045W0 10 lbm
• Weight at liftoff (W3) 2152 - 10 2143 lbm or
  W3/Sref 2143/71.7 29.89 psf

21-6
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Example UAV - takeoff performance

• Takeoff distance is estimated from RayAD Fig. 5.4
  
• - In chapter 18, we calculated takeoff at 110
  stall speed where Clto-max 1.8 (plain flap)
  and
• (W/Sref)/Clto qto .00339Vto2
  (21.2)
• where Vto takeoff speed in knots
• - For our TBProp example, Clto 1.8/(1.12)
  1.49, qto 29.89/1.49 20.1 psf and Vto
  77 kts
• - Takeoff parameter (TOP) was previously
  estimated at 220 which is consistent with a 1500
  ft ground roll and an approximate balanced field
  length (BFL) of 3000 ft
• See RayAD page 98 for additional information

21-7
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Climb performance

• Generalized performance for any non-equilibrium
  flight condition (including climb) is defined by
• Hdot/V Vdot/g (Ta-D)/W
  (21.3)
• where (See RayAD Eq. 5.4 and Chapter 17.6)
• Hdot Rate of climb (fps)
• V Speed (fps)
• Vdot/g Axial acceleration (gs)
• Ta Thrust available (lbf)
• D Drag
• W Weight (lbm)
• For a typical climb Vdot/g ltlt Hdot/V and L ? W
  so that
• Hdot V(Ta-D)/W V(Ta/W - 1/(L/D))
  (21.4)

Hdot/V is defined as climb gradient, (Ta-D)/W
as flight path acceleration (FPA) and FPAV as
specific excess power (Ps)
21-8
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Climb methodology

• Raymer suggests use of a weight fraction (RayAD
  Table 3.2) or a Mach number parametric (RayAD
  Eqs 6.9 and 6.10) to estimate fuel required for
  climb
• These equations, however, do not take into
  account key design features such as T/W, AR or
  speed
• Therefore, we need a better climb methodology
• Our options are to develop a parametric that
  includes key performance features or to simply
  calculate it
• Performance calculation is actually straight
  forward
• We can make a first order climb speed
  approximation by assuming climb at (L/D)max
  (LoDmax) where
• q_at_LoDmax (W/S)/sqrt(?AReCd0)
• and Chapter 17 aerodynamic and Chapter 18
  engine models are used to estimate drag, thrust
  and fuel flow

21-9
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Climb speed and distance

• Even though climb performance can be integrated
  numerically, it is a little awkward for
  spreadsheet computations and we will use a
  simplified approach
• We will assume climb at constant EAS at LoDmax
  and calculate performance at 2 conditions
• Right after takeoff (assumed to be at sea level)
  and again at cruise altitude (hcr)
• - Performance will be averaged between the two
• - We will assume that the average climb air speed
  and ground speed are about the same
• Climb distance can be calculated from time to
  climb (TTC) and climb speed (V-clmb)
• We impose a climb stall margin 1.25 and assume
  Clmax 1.2 or Clmax useable 1.2/(1.252)
  0.768

21-10
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Climb parametric

• There are few parametrics available for
  estimating climb performance
• There are many variables that affect performance
• Data scatter is relatively large but they can
  still be used to check calculations
• Some examples

21-11
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Example UAV application - climb

• Our TBProp at W3/Sref 29.89, Bhp0/W0 0.092
  climbs to 27 Kft, Swet/Sref 4.68 and b2/Swet
  4.27, ?
• For Cfe 0.0035, e 0.8, from equations
  16.6-16.9
• - LoDmax 0.5sqrt(?0.8/0.0035)(4.27)
  27.7
• - Cd0 Cfe(Swet/Sref) .00354.68 .016and
• - Cl _at_ LoDmax .91 which exceeds Clmax 0.768
  so
• q3 29.89/.768 38.9 psf , V3 107 kts
  and
• Cdi Cl2/?Ae 0.012 and Cd Cd0Cdi
  0.028
• From our chapter 18 TBProp model (spreadsheet) at
  sea level and V3 107 kts (M 0.16, q 38.9
  psf)
• Bhp3 229, Ta3 556 lbf and WdotF3 144 pph
• By definition
• D3 Cdq3Sref 0.02838.971.7 78 lbf and
  Hdot3 V3(Ta3-D3)/W3 40.3 fps or 2418 fpm

21-12
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UAV application - climb
If the TBProp climbed at a constant Hdot 2418
fpm and WdotF 146 pph, time to climb (TTC) to
27Kft would be 11.2 min and fuel required would
be 28 lbm - We add this to Wfssto and make a
first approximation estimate of initial cruise
weight (W4est) ? 2115 lbm At 27 Kft we
recalculate performance at KEAS 107 kts (q
38.9 psf) or KTAS 166 kts and M 0.28 - From
our TBP spreadsheet model, Bhp 100, Ta 156,
WdotF 49 pph - At W4est 2115 lbm, Cl 0.78,
Cd 0.028, D 78 lbf - Hdot V(Ta-D)/W
1661.689(156-78)/2115 10 fps or 600
fpm The averages of the sea level and 27 Kft
climb performance parameters are Hdotavg 1522
fpm, WdotFavg 98 pph, KTASavg 137kts
21-13
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Overall climb performance

• Using the averages
• - Time to climb (TTC) 27 Kft/1522 fpm 17.7
  min
• - Fuel to climb (17.7/60)98 29 lbm
• - Distance to climb (Dcl) (17.7/60)137 40 nm
• - Initial cruise weight (W4) 2114 vs. est.
  2115 lbm
• Fuel to takeoff and climb 9.8
• Or Kttoc 0.098

21-14
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Parametric comparison

• We can compare our performance estimates to the
  previous climb parametric
• - Our calculated sea level rate of climb is 2418
  fpm
• - Distance to climb to 27 Kft is 40 nm or 1.5
  nm/Kft
• - The T0/W0 parametric is based on uninstalled
  sea level static where Ta 1129 lbf or T0/W0
  0.48
• The parametric is for jets but the relative climb
  performance still looks reasonable

21-15
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Cruise

• Cruise speed and altitude are selected to
  maximize range factor (VL/D/SFC) or nm/lb of
  fuel
• Assuming sufficient thrust (Ta) is available,
  cruise speed (Vcr) and altitude (hcr) are driven
  by lift coefficient (Cl) and wing loading (W/S) -
  see RayAD Eqs 5.12-14
• Cruise lift coefficient is determined by airfoil
  design (see RayAD Chapter 4.2, airfoil selection)
  and requirements to operate at or near LoDmax
• Wing loading drives cruise altitude (see chart
  20-9)
• Cruise speed is typically a requirement or a
  limit
• Propeller aircraft engines are typically sized by
  speed
• Jet aircraft typically cruise just below some
  limit such as transonic drag rise (0.6ltMddlt0.95)
  or flutter

21-16
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Breguet range - review

• Basic form of the equation (see RayAD Eq. 3.5)
• R V/TSFC?L/D ?LnWi-1/WI
  (21.5)
• where
• R Cruise range (nm)
• V Cruise speed (KTAS)
• TSFC Thrust specific fuel consumption
  (lbm/hr-lbf)
• L/D (LoD) Cruise lift-to-drag ratio
• Wi-1 Weight at beginning of cruise segment
• Wi Weight at end of cruise segment
• Cruise any unaccelerated flight segment
• The basic form can also be expressed in terms of
  horsepower (Bhp) using the definition
• HP T(lbf)?V(KTAS)/325.64?p
  (21.6)
• where
• ?p propeller efficiency

21-17
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Breguet range (horsepower)

• In Bhp form, specific fuel consumption is
  expressed in terms of HP where
• SFC lbm/hr-hp
  (21.7)
• where by definition
• TSFC/SFC lbm/hr-lbf/lbm/hr-hp hp/lbf
  (21.8)
• or from Eq (21.5)
• TSFC SFC?V(KTAS)/325.64?p (21.9)
• By substituting this into the Breguet equation
  for jet aircraft (21.5) we develop the Breguet
  range equation for propeller driven aircraft
• R 325.64??p/SFC?L/D?LnWi-1/WI
  (21.10)

21-18
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Loiter

• The objective of a loiter segment is to maximize
  endurance (L/D/SFC) - see next chart
• - Fuel flow (WdotF), therefore, is minimized
• Loiter speed typically occurs at or near L/Dmax
• - Even though Raymer and Roskam focus on
  different L/D strategies for prop and jet
  aircraft, we will assume both loiter (or cruise)
  at or near L/Dmax
• - See RayAD Eq. 5.13-16 for issues
• - We wont get hung up on the issues and simply
  try to maximize overall performance
• Loiter (and cruise) are easy mission segments to
  analyze since, by definition
• Ta D and L W
• For either segment, a Breguet type equation is
  used to estimate performance

Why?
21-19
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Breguet endurance - review

• A similar form expression is used to calculate
  endurance (See RayAD Eq. 3.7)
• E 1/TSFC?L/D?LnWi-1/WI
  (21.11)
• where
• E Endurance (hrs)
• TSFC Thrust specific fuel consumption
  (lbm/hr-lbf)
• L/D Loiter lift-to-drag ratio
• Wi-1 Weight at beginning of loiter segment
• Wi Weight at end of loiter segment
• Loiter unaccelerated minimum fuel flow flight
  condition (from LW, TSFCD fuel flow)
• or expressed in terms of horsepower
• E (325.64?p)/(V?SFC)?L/D?LnWi-1/WI
  (21.12)
• where
• V Loiter speed (kts)

21-20
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Installation effects

• Installation affects the TSFC and SFC terms when
  calculating Breguet range and endurance
• - The effects are different for the jet and prop
  equations
• First the jet range form
• R V/TSFC?L/D ?LnWi-1/WI
• - The basis of the derivation is that LW and TD
  so that
• V/TSFC?L/D ? V?W/WdotF
• - For T to equal D, TSFC must be based on
  installed thrust so TSFC ? WdotF/T-installed
• In the prop form
• R 325.6??p/SFC?L/D?LnWi-1/WI
• - Here also LW and TD but having ?p in the
  numerator requires that SFC be based on
  uninstalled power or SFC ? WdotF/HP-uninstalled
• The same logic follows for the endurance equations

21-21
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Example

• Notional TBF V 300 kts, W 40Klbm, L/D 10,
  WdotF 2Kpph, installation loss 20
• - By definition RF 400kts?40Klbm/2Kpph 6000
  nm
• - For T D, T(inst) 4Klbf and T(uninst)
  5Klbf or TSFC (installed) 0.5 and TSFC
  (uninstalled) 0.4
• - By inspection V/TSFC?L/D must be based on
  TSFC (installed) 0.5
• Notional TBP V 300 kts, W 40Klbm, L/D 10,
  WdotF 2Kpph, ?p 0.8 (includes all losses)
• - By definition RF 400kts?40Klbm/2Klbmph 6000
  nm
• - For TD, T(reqd) 4Klbf and HP(reqd)
  4Klbf300kts/325.60.8 4607 hp (uninstalled)
• and SFC (uninstalled) 0.434
• - By inspection 325.6??p/SFC?L/D must be
  based on SFC (uninstalled)

21-22
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Simple solution

• We can eliminate the thrust vs. horsepower
  differences by expressing engine performance in
  terms of one or the other conventions
• Example TBProp performance can be expressed in
  terms of thrust (Raymer Table E.3) or horsepower
• The same applies to internal combustion engines
• In the early days of the jet era, some tried to
  describe jet engine performance in terms of
  horsepower
• But it was a problem under static conditions,
  since as V? 0, thrust ? ?
• Therefore, thrust became the standard measure of
  jet engine performance
• Our spreadsheets use this approach, all engine
  performance is calculated in terms of Ta and TSFC
• Including internal combustion (IC) engines
• TBProp and IC input values, however, are in HP

21-23
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Acceleration

• Acceleration is estimated using Eq. 21.3 with
  Hdot 0 or
• Vdot/g (Ta-D)/W (?V/?t)g (21.13)
• Acceleration fuel required (Wfacc) is calculated
  using numerical integration or approximate
  methods similar to climb

• UAVs typically do not have acceleration
  requirements
• - UCAVs could
• Time and distance to accelerate from climb speed
  to cruise speed should be included in mission
  performance analysis

21-24
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Another example

• How long would it take the TBProp UAV to
  accelerate from final climb speed to a cruise
  speed of 180 kts?
• - We use our climb performance spreadsheet to
  estimate thrust (Ta),drag (D), weight (W4 2114
  lbm) and WdotF at 27Kft and 166 kts at maximum
  power or
• Ta 156 lbf, D 78 lbf and WdotF 49 ppm
• - From equation 21.13, Vdot g?(Ta-D)/W ?V/?T
  or
• Vdot 32.174?(156- 78)/ 2114 1.19 ft/sec2
• - At 180 kts and maximum throttle setting, Ta
  149 lbf, D 81 lbf and W 2100 lbm and
• Vdot 32.174?(149 - 81)/ 2100 1.04 ft/sec2
• - We calculate acceleration time using the
  average value of Vdot 1.11 ft/sec2 or
• ?Tacc (180-166)1.689/ 1.11 21.3 sec or 0.36
  min.

21-25
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Turn performance

• Typically two types of turn performance are of
  most interest - turn rate and time to turn (both
  instantaneous and sustained)
• - Sustained turns are at constant speed (and
  altitude)
• - Maximum sustained turn rate is at corner
  speed, the speed for LoDmax (See RayAD Fig 17.6)
• - By definition Ta D
• - In instantaneous turns, speed (or altitude) can
  be lost
• - Maximum turn rate is at Clmax or max gs (Nz)
• In a level turn (RayAD Fig 17.5)
• Lcos(?) W or ? arccos(1/Nz)
  (21.14)
• and d?/dt (g/V)sqrt(Nz2-1)
  (21.15)
• where ? bank angle and d?/dt turn rate
• Time to turn time to bank turn angle (?)
  (d?/dt)

21-26
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UAV application

• Although UCAVs may have maneuver and dash
  requirements to allow them to operate with manned
  aircraft, UAVs currently do not
• - In the future,however, we should anticipate UAV
  requirements on turn performance
• - The requirements will probably be driven by
  platform reaction time, for example, to turn X
  degrees in Y seconds in order to position a
  sensor on a target
• Consider a loitering UAV with a forward looking
  sensor, flying a race track pattern.

21-27
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Example

• We assume that an incident occurs when the UAV is
  looking directly away from it
• - How long does it take for a 3 g, 180 kt UAV to
  put its sensor on target assuming a ? 30 degree
  field of regard, that is the UAV has to turn 150
  degrees, assuming a roll rate of 30
  degrees/second
• From 21.14, ? arccos(1/Nz) ? arccos(1/3)
  70.5?
• - Time to bank 70.5/30 2.4 seconds
• - d?/dt (g/V)sqrt(Nz2-1) (g/(1801.689))(9-
  1).5
• 0.3 rad/sec 17.15 deg/sec
• Therefore, time to turn 2.4 150/17.15 11.1
  sec

30 deg
21-28
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Descent and landing

• For pre-concept design, mission rules typically
  give no range credit (and no fuel penalty) for
  descent
• We will use this ground rule for pre-concept
  studies
• For conceptual and preliminary design, glide
  (idle thrust) performance can be calculated at
  L/Dmax
• Mission rules typically specify landing fuel
  reserves in terms of endurance plus a percentage
  of total fuel
• For small UAVs we will use a 10 reserve (0.1Wf)
• For UAVs operating from manned airfields we will
  use one hour endurance (Ello 1 hr) 5 fuel
  (0.05Wf)
• Landing distance typically is about equal to
  takeoff
• Specifying balanced field length for takeoff
  assures that landing requirements will not be
  critical
• See RayAD 17.9 for landing analysis methodology

21-29
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Example cruise range

• In order to calculate cruise range for the
  example TBProp UAV we need to know two (2)
  weights
• 1. The weight at the beginning of cruise
• - Which we assume is the weight at the end of
  climb
• i.e. we ignore fuel required to accelerate from
  climb speed (166 kts) to cruise speed (180 kts)
• 2. The weight at the end of cruise
• Which we assume is equal to the weight at the
  beginning of the landing loiter
• To calculate landing loiter weight, we work
  backwards from the landing weight (W19) which, by
  definition is empty weight Wmisc landing fuel
  reserves or
• W19 W0 0.95?Wf 1794 lbm
• Which we assume is also the weight at the end of
  loiter

21-30
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Landing loiter and final cruise

• Loiter speed at LoDmax 27.7 and 1794 lbm
  (W18/Sref 25) is 98 kts
• - From Cllo(max) .768 and q18 25/.768 32.5
  psf or Mlo 0.148 and Vlo 98 kts
• From the TBProp engine model, installed sea level
  TSFC at 98kts is 0.246 or LoD/TSFC 112.6
• The endurance requirement is 1 hour
• - Fuel required (equation 21.11) is calculated at
  16 lbm
• - Weight at the start of loiter (W18) is 1811 lbm
• By our mission rules W17 W18
• Working backward and forward we now have final
  and initial cruise weights
• W/Sref at start of cruise 2114/71.7 29.5 psf,
  L/D 26.1
• At end of cruise W/Sref 1811/71.7 25.2 psf,
  L/D 24.6

21-31
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Cruise range

• By definition cruise speed is 180 kts at 27 Kft
• Installed TSFC at all cruise conditions is 0.332
• Recalling that Range Factor (RF) V(L/D)/TSFC
• Initial outbound cruise RF4 180?26.1/.332
  14147 nm
• If we assume outbound RF remains constant, weight
  at the end of initial cruise (distance 255-40
  215 nm) is given by the Breguet range equation
  or
• W7 W4/exp(215/14150) 2081.9 lbm
• At this weight, LoD7 26.0 and RF7 14075
• Average outbound cruise RF, therefore, is 14111
  nm vs. the previous assumed value of 14147nm
• Using this value we can recalculate W7 2082.0
  nm and conclude that one iteration is adequate to
  converge the end of outbound cruise weight
• Similar logic is used to iterate the initial
  cruise back weight (W14) starting with a known
  value of final cruise weight (W17 1811 lb)
  where we find W14 1845 lb

21-32
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Operational loiter

• Operational loiter is defined to be at 27 Kft and
  we will assume that at this higher altitude, we
  can fly within 10 of our stall speed without
  undue risk so that
• Cllo (max) 1.2/1.21 0.992 and we can now fly
  at our calculated Cl for LoDmax of 0.908 and
• LoDlo LoDmax 27.7
• Weight at the start of loiter has been calculated
  at W7 2082 lb ? qlo 2082/(71.7?0.908) 32
  psf or Vlo 150.4 kts
• At this speed, installed engine TSFC 0.292
• Weight at the end of loiter (W8) weight at the
  start of cruise back (no combat, no ingress, no
  loiter) or W8 W14 1845 lb so qlo 28.3 psf
  and Vlo 141.6 kts
• At this speed, installed engine TSFC 0.280
• Therefore, average Endurance Factor 27.7/(.292
  .280)/2 96.85 hrs and..
• Operational Loiter 96.85?ln(2082/1845) 11.7
  hrs

21-32a
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Performance reconciliation

• For our operating distance of 255 nm, we
  baselined a starting requirement of 12 hrs of
  operational loiter vs. 11.7 hrs calculated
• - Fuel fraction, therefore, needs to increase,
  the air vehicle needs to be resized and all the
  performance needs to be recalculated
• This will be a major task if we continue to work
  through the calculations serially
• Therefore, from this point on we will use an
  integrated performance spreadsheet to
  iterate/calculate weights, geometry and
  performance to meet input requirements
• We will not, however, accept the spreadsheet
  results without careful consideration
• Key performance parameters will be challenged
  (e.g. Cl-cr reasonable?, T gt D?, Waf/Sref and
  TSFC consistent with parametric data?, etc.)

21-32b
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Expectations

• You should now understand
• Parametric performance, range and endurance
• Where they come from
• How they are used
• The limits of their applicability

21-33
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Homework (240 credit)

• Starting with your previous homework estimate of
  maximum takeoff weight (W0) for your air vehicle
  and using Chapter 21 methods, do a hand
  calculation of
• Start taxi and takeoff fuel required
• Time, fuel, and distance to climb
• Time and fuel to cruise out to initial loiter
  location
• Landing loiter fuel for 60 minutes at sea level
  plus 5 landing reserves
• Time and fuel to cruise from loiter location back
  to base
• Available operational loiter performance (time)
• Using the appropriate performance spreadsheet
  program, compare your hand calculations against
  the spreadsheet results
• - Homework credit 6 additional problems

See chart 15-39
21-34
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Recommended reading

• Raymer - Aircraft Design - A Conceptual Approach
• Chapter 17 - Performance and Flight Mechanics
• 17.1 - Introduction
• 17.2 - Steady Level Flight
• 17.3 - Steady Climb/Descent
• 17.4 - Level Turning Flight
• 17.5 - Gliding Flight

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Intermission
21-36