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In-flight performance is calculated from first principles using step-by-step procedures, based on the mass and aerodynamics of the current plane and the characteristics of the engine linked to that plane.
Piano can derive the design range (corresponding to the MTOW and design payload) or analyse 'off-design' missions at specific weights or over required distances. It can examine stand-alone climb, cruise, or descent segments, all-engines or engine-out operational ceilings, speed envelopes, or instantaneous performance characteristics. It can generate payload-range diagrams and various other plots or tables of mission performance.
All these features are described in this Chapter and are based on normal operational procedures. You can also use Piano to create your own arbitrary in-flight manoeuvres or sequences of manoeuvres. The 'Flight Manoeuvres' feature is very flexible and is treated as a separate subject in Chapter#16section01 et seq.
The standard atmospheric model is used throughout performance calculations (see Chapter#05section02 ).
The basic design range is calculated via the 'Range' item (keyboard equivalent Command-R) under the 'Report' menu. All range calculations are conducted according to the current settings of the 'Range Modes...' dialog, as described below:
Use the 'Range Modes...' dialog (keyboard equivalent Command-5, see 'Report' menu) to specify the cruise Mach number, the cruise Altitude(s), and optionally the scheduled climb and descent speeds or some other secondary settings. Your choices will affect all subsequent range calculations for the current aircraft, as follows:
The cruise altitude or altitudes can be any of the following:
- A nominal 'Operating Ceiling'. This is a simple mode, at an altitude matching the residual rate of climb ('Reference RoC') in the dialog, typically 100 feet/min at cruise rating and initial mass.
- An idealised 'Drift-up Profile', for which altitude increases continuously. This follows an optimum line corresponding to maximum SAR (Specific Air Range, the air distance travelled per unit of fuel burn) as a function of the reducing cruise weight.
- One or more 'Flight Level(s)'. This is the recommended altitude mode and corresponds to realistic operational procedures, as follows:
Flight Levels (FL, measured in feet/100) represent standard altitudes (specifically, pressure altitudes) available in controlled airspace. You can choose the traditional Westbound / Eastbound settings (intended to avoid traffic conflicts), or supply your own values - including mixtures with 'reduced separation' or other values that may usually not be considered 'legitimate' Flight Levels.
Piano automatically picks which FLs to use from the list of specified values. It will try to track the optimum drift-up altitude line as closely as possible, subject to there being enough climb performance. The performance margin is controlled by the stepup-cruise-method , which should normally be left at its default.
To cruise at a constant altitude, you input just one Flight Level. This is the recommended mode for short to medium-range aircraft. For such aircraft, step-cruising has negligible benefits and is operationally impractical. A single Flight Level will speed up range calculations considerably, especially during parametric studies and optimisation.
Use the left/right <arrow> keys on your keyboard if you can't see all Flight Levels in the editable text.
Cruise Mach number can be set to any of the following:
- A nominal 'High-speed' Mach. This uses the max. cruise rating, typically calculated at a reference condition according to 'Reference RoC'. If you specify a single Flight Level (constant-altitude cruise), the high-speed Mach will be calculated at that FL instead.
- A Mach number corresponding to 'max S.A.R.'. This determines the optimum Mach number to maximise Specific Air Range, the distance travelled per unit of fuel burn.
- A Mach number corresponding to '99% max S.A.R.' This increases the above Mach number slightly, until only 99% of the maximum SAR is achieved. It corresponds to typical operational 'Long Range Cruise' or LRC conditions.
- A Mach that you 'Specify' yourself. Cruise Mach is often treated as a predetermined design choice. If you 'Specify' the Mach number, range calculations require fewer iterations. This is also the recommended mode during parametric studies and multivariable optimisation exercises.
The cruise Mach is automatically limited to what can be achieved at the max. cruise rating (MCR). Note that operational procedures are almost invariably based on a single Mach number throughout the cruise (some Mach selections are made at nominal weights). However, there is one special mode that allows Mach number to vary:
- Mach number corresponding to 'Max-speed {+ dM}'. This is similar to the high-speed option except that Mach is allowed to increase as the weight reduces, keeping the thrust at MCR throughout. You can only use this mode if you specify a single Flight Level. Operational applicability is limited; just bizjets are occasionally flown 'flat-out' in this fashion when the CEO is on board.
The term 'speed schedule' refers to standard operational variations of airspeed with altitude. During the initial climb, calibrated speed (CAS) is constant, but above a certain altitude the climb continues at constant Mach number. Under the default setting (marked 'Calculate'), Piano picks a typical CAS that approximately maximises the rate of climb, followed by a Mach slightly below divergence conditions. The 'Specify' option lets you input any alternative speed schedule.
Below 10,000 feet, a speed limit of 250 kts is always applied, to comply with standard Air Traffic Control (ATC) rules. You can change this via the parameter air-traffic-speed-limit .
The descent schedule is assumed to use the same values as the climb (in reverse), unless you also fill-in the corresponding editable text boxes (leave them blank to follow the climb values). Descent is conducted at idle thrust, unless you specify a cabin re-pressurisation limitation in the box marked 'Restrict cabin descent'. Some partial thrust may then be used at the higher altitudes. Details of climb and descent procedures are given in later sections.
The design range is calculated via the 'Range' item (key equivalent Command-R, see 'Report' menu). It corresponds to the MTOW and design payload. On some rare occasions, the nominal design payload quoted by a manufacturer is 'fuel capacity limited'. In this case, it is possible to either ignore any restrictions on fuel capacity, or to reduce the fuel load and therefore calculate the design range at less than MTOW. The choice depends on the setting of ignore-fuel-vol-violations (see also Chapter#04section14 ).
Sample Range Report
Calculating Design Range...Done. RANGE REPORT _____________ {Takeoff mass 162040.lb./ Fuel 40874.lb./ Payload 30093.lb.} Range mode: fixed mach, step-up cruise Climb schedule: 250.kcas, 274.kcas, 0.729 mach Cruise at Mach = 0.780 {FL 350 390} ICA 35000.feet, 450.ktas, 264.kcas, CL=0.62, 4562.lbf./eng=MCR-18% FCA 39000.feet, 447.ktas, 241.kcas, CL=0.61, 3796.lbf./eng=MCR-19% Distance Time Fuel (n.miles) (min.) (lb.) _________ ______ _______ Climb 165. 27. 4117. {S.L to ICA} Cruise 2594. 348. 28589. {ICA to ICA} Descent 110. 19. 409. {ICA to S.L} _________ ______ _______ Trip total 2870. 394. 33116. Block total ========= 410. 34129. Emissions: taxi,t/o climb cruise descent app,taxi total (lb.NOx) 6.7 44.8 172.2 1.4 2.9 228.1 (lb.HC) 0.27 0.19 3.31 0.63 0.29 4.67 (lb.CO) 4.4 8.9 111.0 9.6 4.7 138.5 Manoeuvre allowances: taxi-out 203. lb. {extra to t/o mass} 8.min. takeoff 295. lb. 1.min. approach 380. lb. 3.min. taxi-in 135. lb. {taken from reserves} 5.min. Diversion distance 200. n.miles Diversion mach 0.702 Diversion altitude 28570. feet Diversion fuel 2822. lb. Holding time 30. minutes Holding mach 0.331 Holding altitude 5000. feet Holding fuel 2571. lb. Contingency fuel 1690. lb. {5.% of mission fuel} Total Reserve fuel 7083. lb.
You can analyse off-design missions at arbitrary combinations of takeoff mass and payload through the 'Mission @Mass...' item (see 'Study' menu).
Alternatively, you can specify a required off-design range with a given payload through the 'Mission @Range...' item. Piano then calculates the necessary fuel load and takeoff mass to match the target range using iterative procedures.
The 'Missions Table...' ('Study' menu) lets you create extensive and very detailed tabulations of off-design mission performance. You can specify a list of payloads in various formats and another list for either range targets or takeoff masses. Piano will then run the entire matrix of combinations. The output is shown in a separate window and can be saved as a standard text file that may also be read by a spreadsheet. It includes full details of times, distances and fuel burns for individual mission segments, TOFLs, operating costs, pollutant emissions etc. The block fuel can also be plotted.
Source codes: Relevant functions are find-range , off-design-performance and off-design-iteration .
Use the 'Range Iteration...' item (see 'Study' menu) to find a value of mto-mass (the MTOW) that corresponds to a particular design range.
Unlike the range-related features described above, this one will change the design characteristics of the current plane: Each point examined involves a complete redesign (see Chapter#08section07 ) at a different MTOW. Piano needs lower and upper estimates for the mto-mass in order to initiate the search. These estimates should 'bracket' the solution (one below, one above) to ensure convergence. The default values shown in the dialog are good enough in most cases, and Piano tries a number of intermediate restarts if the iteration fails. It is still possible that a given range simply can't be achieved with a given configuration, no matter what the starting points for the search.
Source codes: The basic procedure is range-iteration .
Complete payload-range diagrams can be produced via the 'Payload-Range...' item (see 'Study' menu). The boundaries of these diagrams are determined by the MTOW, the max. payload or the MZFW, and the fuel capacity (see max-payload/design-payload , fuel-vol-adjustment , Chapter#04section14 , Chapter#03section16 ).
Normally Piano calculates just three ranges, corresponding to (max.payload @ MTOW), the capacity limit or 'kink point', and the zero payload case. These can be connected by straight lines. Some more intermediate points will be calculated if you tick the option marked 'Use extra points between kinks'. This is normally unnecessary, except perhaps for long-range aircraft with large fuel capacities.
For more precise calculations, use the off-design range features described earlier. If you have already analysed other planes during your current session, you can tick the option labelled 'Compare with previous aircraft' to superimpose all payload-range diagrams.
Source codes: See payload-range .
The following parameters control all fuel reserves and allowances:
Diversion to an alternate airport at a specified diversion-distance is followed by a holding pattern for a period hold-time-mins at a typical hold-altitude and a Mach number given by hold-mach (a 'calculable' parameter, set near to minimum drag conditions by default). An amount of contingency fuel is retained according to contingency-fuel-fraction , which, depending on the setting of contingency-definition , will be a fraction of the mission fuel, the total fuel, the flight time, or the mto-mass . (The last option is useful if you want to freeze the reserves). Allowances for the initial taxi, takeoff, approach, and final taxi phases can be added through the parameters taxi-out-time , takeoff-time , approach-time , and taxi-in-time . For a missed approach and overshoot following the end of the basic mission, use missed-approach-time .
Suitable combinations of the above parameters will match different reserve rules. For example, 'Typical International' reserves for medium/long-haul aircraft correspond to a 200 n.m. diversion, 30 minutes hold, plus 5% contingency on mission fuel. These are the default settings. You can create your own combinations, or load some frequently used sets of rules (U.S. short-haul, European short-haul, NBAA-IFR, or Douglas rules) through the palette's 'Load Values...' feature (see Chapter#08section04 ).
You can limit the diversion altitude via diversion-altitude-limit . The diversion-mach is usually slightly below cruise mach. You also have a choice of diversion-method : This parameter lets you use a simplified calculation (based on correlations with the basic mission profile) or a full analysis of the complete diversion mission to determine the diversion fuel requirement. The latter implementation is more accurate (and a bit slower); the former is retained for backward compatibility.
The calculated fuel reserves and allowances can also be adjusted through the following parameters: user-factor-on-diversion-fuel , user-factor-on-hold-fuel , user-factor-on-taxi-out-fuel , user-factor-on-taxi-in-fuel , user-factor-on-takeoff-fuel , and user-factor-on-approach-fuel .
A note on ramp fuel allowance: Most design calculations are based on the takeoff mass (see mto-mass ). The maximum ramp mass is usually not relevant. However, you can add an adjustment via ramp-fuel-allowance , to match quoted ramp weights for existing aircraft. If ramp-fuel-allowance is input (i.e. non-zero), taxi-out fuel will be set to this value instead of being calculated from taxi-out-time and the idle fuel flow. Also, fuel volume requirements will correspond to the ramp (not takeoff) condition.
You can analyse the climb phase in great detail, independently of range calculations, through the 'Climb To...' item (keyboard equivalent Command-H, 'Report' menu). This lets you input any combination of speed schedule, initial and final altitude, initial mass and delta-ISA temperature. Engine rating is set to max. climb (MCL). Detailed results will be produced in a tabular format if you tick the 'history' option, and a plot of time versus altitude is also shown.
Unless you supply a speed schedule, Piano selects a constant CAS that approximately maximises the rate of climb. Below 10,000 feet, climb speed is restricted by the value of air-traffic-speed-limit . A brief phase of level flight is introduced to allow for the necessary acceleration at this altitude. Beyond that point, Mach number increases with altitude until it reaches a preset limit, close to divergence. The climb then continues at constant Mach. In the troposphere, this implies a reducing TAS, and a slight 'boost' in performance will therefore result as more energy is channelled into the rate of climb. (You can force the switch-over to happen above some specific altitude via the parameter climb-schedule-switch-alt. , but this is normally not required).
Source codes: The rate of climb at arbitrary point conditions is calculated by the function instantaneous-rate-of-climb-at . The basic performance equation for moderate climb angles is used, R.o.C. = V * (Thrust - Drag) / Weight, with additional acceleration corrections given by climb-accel-term , which depend on the climb technique (constant CAS or constant Mach). The default schedule is determined through the golden-search procedure by find-climb-schedule , and used by scheduled-rate-of-climb-at . Step-by-step integration of the entire climb is done in detail by execute-climb-from-to . Level accel at 10,000 ft. is done by do-level-accel-fl10 . Stand-alone climbs independent of range are calculated by do-special-climb , using more steps ( *number-of-climb-steps* or *dense-climb-steps* ).Sample Climb Report
You can analyse any constant-altitude cruise segment in detail, independently of range calculations, through the 'Cruise At...' item (keyboard equivalent Command-7). Given the initial mass, altitude, and Mach number (or airspeed), Piano generates the fully integrated performance to match a final cruise mass, or distance, or time, as required.
Source codes: The relevant function is do-special-cruise . Cruise performance in range calculations is calculated separately.
You can analyse a descent between any two altitudes in detail, independently of range calculations, through the 'Descent From...' item (keyboard equivalent Command-8). Just as in a climb, a plot of time versus altitude is shown and there is a 'history' option.
The default speed schedule is the same as the climb schedule, in reverse. Thrust will be at idle throughout, unless you tick the option marked 'Restrict cabin descent'. In this case, partial thrust may be needed at the start of the descent to constrain the rate of cabin repressurisation (dp/dt) and avoid passenger discomfort. Typically, the dp/dt is limited to the equivalent of 300 feet/min at sea level (dh/dt). Thrust is used for long enough to ensure that the cabin has time to repressurise fully at the final altitude (to zero differential). At the start, the cabin is assumed to be pressurised to its maximum pressure differential. This structural limit is derived from the cabin-altitude (default 8,000 feet) and the max-operating-altitude , and is taken into account throughout the descent.
Typically, a descent starting with a cabin altitude of 8,000 ft, using -300 feet/min repressurisation, and arriving at sea level with zero differential must imply a time requirement of 23.7 minutes. The initial cabin pressure may be higher if descent starts below the max. operating altitude, and operators may aim for pressure equalisation at 1500 ft rather than sea level, reducing the time. Operational descents have to balance fuselage fatigue considerations, air conditioning usage, passenger comfort, and ATC requirements.
Source codes: Rate of descent is found by rate-of-descent-at . The main functions are execute-descent-from-to and execute-descent-aux . If there is a dp/dt restriction, two separate descents are conducted, one at idle and one with part power, and the results suitably combined by execute-descent-crossover to ensure enough time for repressurisation. The speed limit is handled by do-level-decel-fl10 . Stand-alone descents independent of range are calculated by do-special-descent , using more steps ( *number-of-descent-steps* or *dense-descent-steps* ).
This section summarises the methodology, source codes, and approach taken when combining the climb, cruise and descent phases into integrated range calculations. You can skip it if you are not interested in the mechanics of performance estimation.
Source codes: In analysing a complete mission profile, Piano generally starts with the climb and descent portions and, having made the necessary adjustments for reserves and allowances, evaluates the cruise segment last, using the remaining fuel to find the total distance travelled.
The function set-cruise-alt-and-mach determines an initial cruise altitude and the (constant) Mach number. It sets the nominal operating ceiling through find-ceiling using the reference rate of climb specified in 'range modes'. The 'high speed' cruise Mach is calculated by max-cruise-mach via the regula-falsi solver procedure, matching cruise drag and available thrust. In the case of 'max. SAR' or '99% max. SAR', the search is conducted by mach-for-max-sar or mach-for-99%-max-sar respectively. Point calculations at arbitrary flight-conditions use the function flight-sar-at-mass and the golden-search procedure to maximise specific air range.
When the altitude mode is 'drift-up profile', Piano determines the optimal cruise altitude variation through altitude-for-max-sar and subject to any ceiling restrictions dictated by find-ceiling . The integrated cruise performance is then found by driftup-cruise-distance-at . If the altitude mode is 'Flight Levels', a complex search procedure is used to try and follow the optimum altitude-versus-mass line as closely as possible whilst selecting the next step-up altitude from the available Flight Levels. Calculations are handled by stepup-cruise-distance-revised and include calls to enroute-climb-from-to that allow for the necessary step climbs. If only one Flight Level has been specified, the simpler flat-cruise-distance-at procedure is used.
Climb and descent calculations were described in an earlier section and use the functions execute-climb-from-to and execute-descent-from-to . Complete range calculations are always from sea-level to sea-level. Note that when outputting detailed range reports, Piano only shows a descent phase from the Initial Cruise Altitude (ICA) back down to sea level. The reason for this is that any initial descent from the Final Cruise Altitude (FCA) down to the ICA is treated internally as an extended cruise. This is a sufficiently accurate and marginally conservative approach that avoids the need for additional iterations. You can use the 'descent from..' item if you need detailed data at the higher altitudes.
Reserve fuel is estimated by find-reserves . The function execute-hold calculates the minimum drag conditions and associated fuel flow. Diversion fuel is calculated by execute-diversion . To avoid unnecessary complexity, diversion climb and descent fuel burns can be derived from the existing main-mission profile data, suitably factored for the reduced flight mass. Alternatively, a full diversion mission analysis can be carried out, depending on your choice of diversion-method . Diversion cruise altitude may be restricted to ensure that at least 50% of the total diversion distance is in the form of a cruise. The simplified diversion analysis still yields accurate estimates for the fuel burn, which is all that is required from this part of the mission. Finally, the contingency-fuel is found by simple factoring, to yield a total value for the reserve-fuel .
Operating ceilings for various conditions can be calculated through the 'Ceiling At...' item ('Report' menu). You can input the mass directly or apply a factor to the value shown (the MTOW by default). When all engines are operative, the rating can be either 'max. climb' or 'max. cruise'. Ceilings are defined to match a residual rate of climb. Representative combinations are 300 feet/min at MCR or 100 feet/min at MCL rating.
The airspeeds used to calculate ceilings are normally taken from the climb schedule in the 'Range Modes...' dialog. Alternatively, you can specify your own ('off-schedule') Mach number. This can be used to evaluate the Initial Cruise Altitude Capability (ICAC), which is typically done at the cruise Mach number, MCR rating, zero feet/min, and something like 98% of MTOW.
If one engine is inoperative, the ceiling is calculated to match a residual gradient, instead of a rate of climb. According to FAR-25 rules this gradient should be 1.1% for twin-engined aircraft, 1.4% for three-engined, or 1.6% for four-engined aircraft, but you can also specify any other value. The rating is set to max. continuous (MCO), and the airspeed is calculated to maximise the gradient. Asymmetric and windmilling drag contributions are also calculated (see Chapter#05section19 ) and included in the output together with the instantaneous value of the specific air range.
Source codes: The basic functions are find-ceiling or find-ceiling-at-mach . Engine-out cases are handled by find-eng-out-ceiling , find-eng-out-climb-speed and sar-at-eng-out-ceiling .
The 'Altitude vs Mass' item ('Study' menu) shows various details relating to step-up cruises at multiple Flight Levels. It plots the variation of optimum altitude with mass (the max. SAR line), the absolute ceiling at MCR rating and cruise Mach, and the actual step-altitude profile followed by Piano. Depending on the stepup-cruise-method , there may be another plot of the ceiling at the MCL rating and a matching residual rate of climb (typically 300 feet/min) showing the available performance margin.
The altitude boundary marking the onset of aerodynamic buffet is shown as a dotted line. It corresponds to a margin of 1.3 G. Normally, buffet is only shown for advisory purposes and is not treated as a restriction in any calculations. This is because buffet is effectively impossible to predict without full knowledge of the aerofoil's characteristics. Piano uses its own statistical buffet curve (based on a simple correlation with roof-top-end ). You can adjust this via buffet-cl-adjustment and buffet-mach-adjustment . If you happen to know the entire buffet curve for a given aircraft, you can input it directly via buffet-mach-cl-curve . This parameter holds a list of numbers representing alternately a Mach number and the corresponding CL for buffet onset at 1G conditions.
When you have adjusted Piano's buffet estimates through one of the three parameters above, buffet will be treated as a 'hard restriction' in the altitude capability during cruise calculations. Note that only high-speed buffet is considered (at Mach > 0.6).
If you hold down the Shift key whilst selecting the 'Altitude vs Mass' item, you can also obtain a tabulation of the various lines shown in the plot.
Source codes: See alt-cap , buffet-cl , buffet-altitude .
You can generate a picture of the 'Flight Envelope...' in level flight at any given mass (see 'Study' menu). This is intended primarily to show the variation of maximum cruise speed (and/or Mach number) with altitude. A precise value of the speed can be requested at a given altitude. Other lines show the maximum dynamic pressure (max. Q , corresponding to the design dive speed VD) and the design dive Mach (MD). These are only advisory statistical estimates (see Chapter#04section04 ) and do not restrict the level flight performance at normal operating altitudes. The envelope also includes some high-speed buffet points (see above) and the low-speed stall line in the clean configuration, which forms the left-hand boundary.
Source codes: See flight-envelope .
The Specific Air Range (SAR) is the air distance travelled per unit of fuel burn. It can be plotted via the 'S.A.R. Sketch...' item (see 'Study' menu). Each plot shows curves that correspond to one altitude and several different values of mass. To start with, Piano picks the Initial Cruise Altitude and a selection of five masses, scaling the axes automatically. You can change these settings as required. SAR plots include a line of 'maximum thrust' and the loci of '100% of max SAR' and '99% of max SAR'.
The 'Cruise Table...' item ('Study' menu) lets you create very detailed tabulations of cruise performance at any combination of mass, altitude, and Mach number. Outputs include the drag, throttle setting (as % of MCR), L/D, sfc, SAR, fuel flow, potential rates of climb at the MCL rating, and the onset of buffet in Gs. You can save the results as a standard text file which can then be read by a spreadsheet. Extra points are included to cover the max. SAR condition, 99% of max. SAR, and a 'max.limit' that represents either 100% of the MCR thrust or the limits of engine data availability.
Source codes: See plot-sar .
You can get the instantaneous or 'point' performance at any given Mach number (or CAS/TAS/EAS) and mass (or lift coefficient) via the 'Drag Spot...' item ('Report' menu). Be sure to click the option marked 'Show performance'. The output includes fuel flow, sfc, SAR, available and required thrust, and performance reserves expressed as potential rates of climb.
The 'Flight Manoeuvre...' feature ('Flight' menu) can also be used to derive detailed point performance for arbitrary configurations (See Chapter#16section01 et seq).
Source codes: See report-cd , flight-cond-dialog .Sample Point Performance Report
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