# What causes Mach No. to increase with altitude at a constant cost index?

If a jet aircraft flies at a constant cost index but changes flight level the Mach no required by the FMC will increase with increasing altitude and decrease with decreasing altitude. Why is this? (Disregard aircraft weight changes due to fuel burn etc)

• This sounds an awful lot like a homework assignment. – egid Jul 26 '17 at 3:02
• In other words: does the aircraft need to burn less fuel at higher altitude? – Koyovis Jul 26 '17 at 4:22

Cost index is a function of Time Cost/Fuel Cost and is more related to True Air Speed than Mach Speed since Mach Speed varies with altitude.

At FL300 M.85 equals 501 knots but at FL400 M.85 is only 487 knots. In order to maintain 501 knots at FL400 you need to increase your Mach speed to M.87

Both of these terms are not related. There is no cause effect relation between them. That is the reason why there is a constant cost index even though the mach number changes as the altitude increases. This is sufficient to answer your question directly. If you are wondering how these data are unrelated, continue reading.

This may get quite technical. Please bear with me.

# COST INDEX

As said above, cost index is the ratio of unit cost of time to the unit cost of fuel. This is used to optimize the aircraft's airspeed.

Aircraft's airspeed is the speed of the aircraft relative to air in it's vicinity and this is different from ground speed (which is defined as the relative speed of the aircraft with respect the ground). Please note that there are 3 types of airspeed (true, indicated, calibrated). For now, just consider the airspeed.

## Jet Engine Theory

What jet engines do are quite simple:

1. Absorb the air
2. Compressor compresses the air and pushes it backwards
3. Due to compression, air heats up and pressurizes and wants to expand
4. The nozzle is convergent and the ambient air pressure is low. So this gives an extra boost to expanding air.
5. Newton's third law. The air expelled propels the engine forward

## Temperature v/s Altitude

Temperature of the air decreases with increasing altitude

## Efficiency of the jet engine

The net change in pressure created by the compressor of the jet engine

• at ground level, is quite high (Sorry, no numbers. But it is high enough to easily provide enough velocity for lift-off).
• at higher altitudes, this is even higher. Why? The lower temperature of air gives room for higher change in air temperature. And thus higher air pressure. And thus, more thrust.

## How does this related to cost index?

Well, in order to optimize fuel consumption, it is necessary to maintain near constant airspeed (which is optimized based on individual flight parameters). So, now you may understand, why sometimes you are flying slower on a map and why sometimes faster. This is because, the wind-speeds vary a lot. And airlines try to maintain a constant airspeed (not constant ground speed). Saves a lot of fuel.

# Mach Number

$$Mach\space Number = Aircraft\space Airspeed / Speed_{\space Sound\space in\space Air}$$

As explained above, the aircraft's airspeed is maintained as constant as possible coping to changes in external wind speeds. However, what actually makes mach number higher, is the denominator part.

$$Speed_{\space Sound\space in\space Air} = \sqrt{\gamma\space R\space T}$$

Speed of sound in air decreases as the altitude increases. This is due to changes in properties of air with increasing altitude.

And thus, mach number increases even though the airspeed is almost constant.

Hope this helps in discerning how cost index and mach number are independent of each other. Cheers.