2.1 Distance definitions

The shortest path between two points on the Earth’s surface is the great circle distance, $s_{\textrm{gc}} $ . This is the ideal circumferential trajectory of an aircraft between two airports and can be calculated with the Haversine formula⁽⁶⁾. In practice, all flights are subjected to various route inefficiencies such as holding patterns, alignment with runways, diversions and other air traffic control (ATC) enforced procedures. Therefore, the projection of the actual aircraft’s path down onto the Earth’s surface, known as the ground track distance, s _g, is always greater than the great circle distance. There is also the distance the aircraft moves relative to the surrounding air, which is known as the air track distance, s. This represents the distance “perceived” by the aircraft, including any headwind or tailwind. As shown in Ref. (Reference Randle, Hall and Vera-Morales1) the air track distance is the most appropriate distance to use in the analysis of aircraft performance. Note that the air distance travelled during the descent phase is denoted s _d.

2.2 A breakdown of mechanical energy use during descent

The mechanical energy of an aircraft in flight is ultimately dissipated as work done against drag. During descent, the work done against drag comes from work done on the aircraft by the engines, any gravitational potential energy released by reducing altitude and any reduction in aircraft kinetic energy. This energy balance can be written for a small increment in air distance ds as follows:

(1) \begin{equation} Dds = {X_N}ds - mgdh - mVdV \end{equation}

The above equation can also be obtained from a simple force balance in the direction of flight. It can be integrated from the top of descent (TOD) to any air distance, s, during the descent. This can be written as the following breakdown of mechanical energy:

(2) $$\matrix{ {g\int\limits_0^s {{{mds} \over {L/D}}} } \hfill & { = LCV\int\limits_0^{{t_s}} {{{\dot m}_f}{\eta _{ov}}dt} + {m_{EOC}}g\left( {{h_{EOC}} - {h_s}} \right) + {1 \over 2}{m_{EOC}}\left( {V_{EOC}^2 - V_s^2} \right){\rm{ }}} \hfill \cr {{E_d}\;\;\;\;\;\;} \hfill & { = \;\;\quad \;{E_{fuel}}\quad \quad \;\;\; + \quad \quad \;\;\Delta GPE\quad \;\;\;\;\; + \quad \quad \Delta KE} \hfill \cr } $$

The term on the left-hand side of Equation (2) is the total mechanical energy dissipated through drag, ${E_d}$ . This cannot be directly determined because the aircraft drag variation is not known accurately during descent. The first term on the right-hand side, ${E_{fuel}}$ , represents the mechanical energy provided by burning fuel. This is the term that, in general, we would like to minimise. It has been expressed in terms of the fuel flow rate using the definition for engine overall efficiency, ${\eta _{ov}} = {{{X_N}V} / {{{\dot m}_f}LCV}}$ . For current descent operations ${E_{fuel}}$ is highly variable in magnitude, representing 10−60% of the total mechanical energy dissipated. The second term is the gravitational potential energy released, $\Delta GPE$ , which represents 30−80% of the total. This is the term that we would like to make best use of during descent as it is “free” mechanical energy that should be used effectively. The third term is the reduction in the aircraft kinetic energy, $\Delta KE$ , and typically represents only 5−10% of the total.

Each of the terms in Equation (2) can be evaluated throughout descent using FDR data, if the engine overall efficiency is known. In Section 3, an engine model is derived that enables ${\eta _{ov}}$ to be computed directly from FDR parameters. This enables ${E_{fuel}}$ , $\Delta GPE$ and $\Delta KE$ in Equation (2) to be determined so that a descent mechanical energy breakdown can be completed, as detailed in Section 6.

2.3 The aircraft range factor

The basic Breguet Range Equation^{(Reference Bréguet7)} for an aircraft can be expressed as follows:

(3) \begin{equation}s = - \int\limits_{{m_1}}^{{m_2}} {H\frac{{dm}}{m}} = H\ln \left( {\frac{{{m_1}}}{{{m_2}}}} \right)\end{equation}

In this equation, H is the range factor, which is a measure of how effectively an aircraft converts fuel into distance travelled. In performing the integration in Equation (3) this is assumed to be constant between points 1 and 2 in a flight and is defined as:

(4) \begin{equation} H = \frac{{V(L/D)}}{{g\,sfc}} = \frac{{{\eta _{ov}}\,(L/D)\,LCV}}{g} \end{equation}

The range factor encapsulates both the airframe aerodynamics and the engine performance in one parameter that determines fuel burn. For the cruise phase of flight the aircraft altitude, Mach number and engine operating point remain relatively constant and thus the integrated form of Equation (3) is accurate. It is therefore possible to derive the cruise range factor from FDR data using the following:

(5) \begin{equation} {H_{cr}} = {{{s_{cr}}} \over {{\rm{In}}\,({m_{TOC}}/{m_{TOD}})}} \end{equation}

Note that the masses ${m_{TOC}}$ and ${m_{TOD}}$ are determined using the recorded aircraft mass at takeoff minus the cumulative mass of fuel burned at top of climb and at top of descent, respectively. A range factor could be determined in a similar way during descent, but this has less meaning since both the engine efficiency and airframe performance would be continuously varying.

2.4 Recovered fuel

As stated in the introduction, Torenbeek^{(Reference Torenbeek3)} assumed that during descent, the fuel burn was equivalent to that during cruise. The fuel burn that would be obtained if an aircraft continued operating at the cruise condition over the same air distance can be calculated from simply rearranging Equation (3):

(6) \begin{equation} {\left( {{{{m_{f,d}}} \over {{m_{TOD}}}}} \right)_{(3)}} = 1 - \exp ( - {\rm{ }}{s_d}/{H_{cr}}){\rm{ }} \end{equation}

The subscript (3) in Equation (6) is included to indicate the fuel burn during descent determined according to Equation (3). The recovered fuel, as defined in Ref. (Reference Randle, Hall and Vera-Morales1), is the descent fuel burn given by Equation (6) minus the actual descent fuel burn, as given by the FDR data:

(7) \begin{equation} {{{m_{f,rec}}} \over {{m_{TOD}}}} = \left[1 - \exp ( - {s_d}/{H_{cr}})\right] - {{{m_{f,d}}} \over {{m_{TOD}}}} \end{equation}

Recovered fuel for descent is comparable to Torenbeek’s lost fuel for takeoff^{(Reference Torenbeek3)}. It quantifies the change in total mission fuel relative to that obtained if the aircraft was operated at the cruise condition for the entire flight.

3.1 Flight data analysed

The FDR data was provided by Swiss International Air Lines and included several thousand flights for Airbus and Boeing aircraft during 2008. The focus for this paper is the Airbus A320-214 flights because complete, detailed datasets were available for this aircraft type. All these aircraft were fitted with CFM56-5B unmixed turbofan engines. In total, 162 flights were available, but it was not possible to identify a steady cruise phase for 70 of these flights, and therefore the statistical results in this paper are based on 92 flights that could be processed with a consistent approach. The flights that were discarded were usually short distance and included frequent changes in altitude during the flight. Figure 1 shows the flight trajectories of the 92 flights processed in this paper. As expected, the flights all land or take off at Zurich Airport. There is a range of stage lengths included in the sample, although these are all significantly shorter than the design range of the A320-214 (5,700km or 3,100NM).

Figure 1. Trajectories of the A320 flights included in the current study.

For each of the flights processed 86 FDR parameters were recorded at regular intervals. These parameters included engine measurements as well as a range of aircraft data including speed, mass and location parameters. A period of 5s between measurements was used during the more variable phases of flight, i.e. taxi, climb, descent and landing; and the frequency reduced to every 60s during well-established cruise. The accuracy of the FDR measurements is largely unquantified. The readings of fuel flow are deemed correct to within 1% of cruise fuel flow, see Vera-Morales and Hall^{(Reference Vera-Morales and Hall8)}, while the altitude, latitude and longitude are assumed to be highly accurate since they are calculated from barometric altimeter data and the Global Positioning System (GPS).

3.2 Identification of the descent phase

The descent phase was isolated by searching in the FDR data for where the absolute rate of descent increases above 10ft/s for a period of at least 60s. This value was chosen by comparing plots of descent rate against ground-track distance for several flights. Using 10ft/s as a threshold ensured a significant rate of descent, whilst allowing for small perturbations in cruise altitude, due to factors such as turbulence.

Figure 2 shows the variation of altitude and mass of fuel burned for three Flights, A, B, and C, which cover the descents that are analysed in detail later in this paper. Note that the descent air distance is the same for each of these flights, and the mass fraction of fuel burned during takeoff and cruise is similar. However, the fuel fraction burned during the descent phase is significantly different in each case.

Figure 2. Flight profiles with corresponding fuel burn showing descent variations.

3.3 A correlation for descent fuel burn

Figure 3 shows the 92 descents considered in this paper on a non-dimensional plot of descent fuel burn against non-dimensional descent air distance. This plot shows that there is significant variation in descent distance (H _cr is almost constant) and a large variation in the scaled descent fuel burn. It also shows that the scaled descent fuel burn correlates well with non-dimensional air distance. Any correlation with great circle distance or ground track distance is much weaker since the flight trajectory and wind conditions are only accounted for in the air distance. The black line is the result of using Equation (6), assuming the aircraft descent performance is the same as at cruise, following Torenbeek^{(Reference Torenbeek3)}. Points below this line represent a positive recovered fuel.

The FDR data shows that the potential for recovered fuel reduces with increasing descent air distance. This is as expected since the engine and airframe performance are worse during descent. This leads to the line of best-fit through the data having a higher gradient than Equation (6) because more fuel is needed to cover additional distance in descent than in cruise. Based on the line of best fit, an empirical improvement to Equation (6) for estimating descent fuel burn is given by:

(8) \begin{equation} {m_{f,d}}/{m_{TOD}} \cong 1.4({s_d}/{H_{cr}}) - 0.0078. \end{equation}

This result and Fig. 3 confirm that the fuel burn during descent scales with the aircraft top of descent mass and that it depends mainly on the ratio of descent air distance to range parameter, ${s_d}/{H_{cr}}$ . However, there is still significant scatter in the results, and Flights A, B and C, with very similar values of ${s_d}/{H_{cr}}$ , have almost 50% variation in their non-dimensional descent fuel burn. These descents were selected for further investigation, and the precise causes of the fuel burn differences are explored in the Section 5.

4.1 Modelling the engine off-design

The challenge is creating an engine model that is reliable in predicting performance when operating at the extremely low power settings typical of descent and final approach. Off-design performance component characteristics are required and the component that shows the greatest variation is the fan. GasTurb was chosen for this purpose because it was found to be highly flexible and could incorporate off-design component characteristics from literature. The fan map used for the engine model was based on Ref. (Reference Freeman and Cumpsty11), re-scaled to give a design FPR of 1.7 at 100% speed with a polytropic efficiency of 0.89 at this condition (as given in Table 1). During approach, the fan operating point can move from near 100% rotational speed down to below 40% speed. At lower rotational speeds, the efficiency and pressure ratio are greatly reduced and combined with low flight Mach number the fan working line moves towards instability.

The actual in-service performance of the fan on the CFM56 may be different from that in Ref. (Reference Freeman and Cumpsty11) in terms of the detailed variations in flow, speed and FPR. The engine model was tested with a range of component characteristics that gave different variations of fan speed and efficiency with flow and FPR. However, the trends in performance were the same in all cases, and the fan map derived from Ref. (Reference Freeman and Cumpsty11) was found to give the widest range of stable operation.

Table 1 Parameters for the model of the CFM56-5B at the design point

Figure 4. Contours of overall efficiency for the engine model.

Contours of engine overall efficiency as a function of flight Mach number and OPR are shown in Fig. 4, determined by running the engine model off-design at a range of flight Mach numbers and engine shaft speeds. It should be noted that GasTurb was sometimes unable to converge for OPR < 5 and in this region, far from the design point, extrapolated values have been used. This enabled a complete look-up table to be produced according to the functional dependence:

(9) \begin{equation} {\eta _{ov}} = {\eta _{th}}{\eta _{prop}} = f\left( {OPR,{M_\infty }} \right)\end{equation}

As expected, for a given Mach number, the overall efficiency generally increases with OPR, due to increasing core thermal efficiency. The OPR giving peak efficiency is below the maximum value, which is a result of the efficiency variations described by the component characteristics. At a given OPR, the overall efficiency increases with flight Mach number because the ratio of jet velocity to flight speed will reduce thus increasing propulsive efficiency: ${\eta _{prop}} = 2/(1 + {V_j}/{V_\infty })$ .

4.2 Engine model comparison with flight data

Figure 5 shows a plot of non-dimensional fuel flow and core compressor polytropic efficiency as a function of engine OPR for both the FDR data and the engine model. These parameters can be determined from the FDR data directly using the following non-dimensional relations:

(10) \begin{equation} {\eta _{poly}} = {{\gamma - 1} \over \gamma }{{\ln \left( {{{{p_{03}}} \mathord{\left/ {\vphantom {{{p_{03}}} {{p_{02}}}}} \right.} {{p_{02}}}}} \right)} \over {\ln \left( {{{{T_{03}}} \mathord{\left/ {\vphantom {{{T_{03}}} {{T_{02}}}}} \right. } {{T_{02}}}}} \right)}},\,\,{{\tilde{\dot m}}_f} = {{{{\dot m}_f}LCV} \over {{p_{02}}{A_2}\sqrt {{c_p}{T_{02}}} }},\,\,OPR = {{{p_{03}}} \over {{p_{02}}}} \end{equation}

In Equation (10) A ₂ is the fan face area in the engine model and standard properties of atmospheric air are used. Figure 5 includes output data from the engine model for a range of flight Mach numbers of 0.2−0.8 with each red curve representing a different flight Mach number. The FDR data shown includes measurements from five descents, including those for Flights A, B and C.

Figure 5. Comparison of engine model (red curves) and flight data (black points).

The lower plot of fuel-flow rate shows how the non-dimensional treatment collapses the engine performance and the FDR data onto what appears to be a single curve. If the propelling nozzles are choked, the relationship between any two non-dimensional engine parameters would indeed be a single curve, as is true for high OPR and high M. At lower OPR, the bypass and core nozzles unchoke and there is some variation with Mach number. The real variation of fuel flow with OPR is well matched by the model. Note that during descent the engine typically operates at OPR < 15 and drops to values as low as 3. The upper plot in Fig. 5 shows the model and FDR measurements of the polytropic efficiency of the core compression process. Firstly, the measured values appear realistic – around 90% at cruise and dropping off to as low as 70% at OPR < 5. There is more scatter in the measured data relative to the model than for the non-dimensional fuel flow. However, this is to be expected due to the accuracy and response times of the temperature measurements involved in calculating the polytropic efficiency (see below). It is also worth noting that the efficiency is slightly under-predicted at high OPR, but over-predicted at low OPR, probably due to the particular shapes of the component characteristics that are used in the model.

Figure 6 compares FDR data to the GasTurb engine model for a single aircraft descent. Instantaneous values of OPR and M from the FDR data have been used in combination with look-up tables to “simulate” a flight using the engine model. The variation in OPR is thus identical in the model and the real descent. The derived variation in non-dimensional fuel flow closely follows the FDR values in accordance with the results in Fig. 5. There is some discrepancy visible where the engine is operating at very low pressure ratios, which is unsurprising given the difficulties the model has in predicting performance in this region.

Figure 6. Comparison of flight data and engine model results for a single descent (Flight A).

The FDR data include some erratic variations in the polytropic efficiency. These arise because the measurement of T₀₃ responds more slowly than the measurement of p₀₃. Any acceleration or deceleration of the engine causes a rapid change in pressure ratio, but the measured temperature ratio takes more time to respond, leading to errors in the calculated polytropic efficiency of the compressor, Equation (10). The less extreme variations in the model values of polytropic efficiency are more realistic. The bottom plot of Fig. 6 shows the variation of the predicted overall efficiency during the descent. The range of values (from about 5% to 30%) corresponds to values in Fig. 4, and the variations are synchronised with the fluctuations in engine operating point.

4.3 Engine descent performance

Table 2 summarises descent-averaged engine parameters for Flights A, B and C. The average OPR and ${\tilde{\dot m}_f}$ are found to vary in the same way as the average low-pressure (LP) shaft speed. Section 6 examines how the variation of the engine setting during descent relates to the overall descent fuel burn and makes further use of the results in Table 2. This shows that the lower the engine OPR and engine speed during descent, the lower the fuel burn.

Table 2 Average engine parameters during descent for Flights A, B and C