2.1.1. Maximum Range Mach Number (MMR) 

Figure F1 illustrates the specific range as a function of Mach number for a given weight at a constant altitude. 

As a result, for a given weight, a maximum specific range value exists and the corresponding Mach number is called Maximum Range Mach number (MMR). 

The advantage of the Maximum Range Mach number is that the fuel consumption for a given distance is at its minimum. It also corresponds to the maximum distance an aircraft can fly with a given fuel quantity. 

During cruise, the aircraft’s weight decreases due to fuel burn. At the same time, the specific range increases, but MMR decreases (Figure F2). The Mach number must therefore be adjusted to correspond to weight changes during the entire flight at constant altitude. 

. Pressure Altitude Influence 

Variations of the maximum range Mach number are summarized as follows: 

2.1.2. Long-Range Cruise Mach Number (MLRC) 

An alternative to MMR is to increase cruise speed with only a slight increase in fuel consumption. Typically, the long-range cruise Mach number (MLRC) provides this possibility. 

At the long-range cruise Mach number, the specific range corresponds to 99% of the maximum specific range (Figure F4). Economically speaking, the 1% loss compared to the maximum specific range is largely compensated by the cruise speed increase due to the flatness of the curve. 

MLRC > MMR 

SRlong range = 0.99 x SRmax range 

In relation to the Maximum Range Mach number, the long-range Cruise Mach number also decreases when weight decreases, as shown in Figure F5. 

2.1.3. Economic Mach Number (MECON) 

Long-range Cruise Mach number was considered as a minimum fuel regime. If we consider the Direct Operating Cost instead, the Economic Mach number (MECON), can be introduced. 

As indicated in §1.1, DOCs are made up of fixed, flight-time related and fuel-consumption related costs. As a result, for a given trip, DOC can be expressed as: 

As DOCs are calculated per nautical mile, it is possible to plot fuel-related costs, flight-time related costs, and direct operating costs based on Mach number (Figure F6). 

Minimum fuel costs correspond to the Maximum Range Mach number. The minimum DOC corresponds to a specific Mach number, referred to as Econ Mach (MECON). 

The MECON value depends on the time and fuel cost ratio. This ratio is called cost index (CI), and is usually expressed in kg/min or 100lb/h: 

When CT is fixed and CF increases, it becomes interesting to decrease fuel consumption. Therefore, when CI decreases, Econ Mach decreases. 

The extreme CI values are: 

CI = 0: Flight time costs are null (fixed wages), so MECON = MMR (lowest boundary). 

CI = CImax: Flight time costs are high and fuel costs are low, so MECON = MAX SPEED in order to have a trip with a minimum flight time. The maximum speed is generally (MMO - 0.02) or (VMO - 10kt). 

For instance, a cost index of 30 kg/min means that the cost of one flightminute is the same as the cost of 30 kg of fuel. This does not mean the fuel flow is 30 kg/min. 

2.1.4. Constant Mach Number 

The aircraft is often operated at a constant Mach number. 

Nevertheless, as the aircraft weight decreases, the gap between the selected Mach and the MMR increases. As a result, fuel consumption increases beyond the optimum. 

3. ALTITUDE OPTIMIZATION 

3.1. Optimum Cruise Altitude 

3.1.1. At a Constant Mach Number 

In examining SR changes with the altitude at a constant Mach number, it is apparent that, for each weight, there is an altitude where SR is maximum. This altitude is referred to as “optimum altitude” (see Figure F8). 

When the aircraft flies at the optimum altitude, it is operated at the maximum lift to drag ratio corresponding to the selected Mach number (as in Figure F9). 

When the aircraft flies at high speed, the polar curve depends on the indicated Mach number, and decreases when Mach increases. So, for each Mach number, there is a different value of (CL/CD)max, that is lower as the Mach number increases. 

When the aircraft is cruising at the optimum altitude for a given Mach, CL is fixed and corresponds to (CL/CD)max of the selected Mach number. As a result, variable elements are weight and outside static pressure (Ps) of the optimum altitude. The formula expressing a cruise at optimum altitude is: 

The optimum altitude curve, illustrated in Figure F10, is directly deduced from Figure F8. 

Getting to Grips with Aircraft Performance  CRUISE 

Summary: ISO Mach number optimum altitude curves are all quasi-parallel (Figure F11).