EE6900 Flight Management Systems

“Relevant Avionics Summary” (remember from ENAV …)

Dr. Maarten Uijt de Haag (Thanks to Dr. Wouter Pelgrum)

Ohio University Information Requirements

• Information is provided by various avionics “boxes” – Air Data Computer (ADC or ADRS) – Inertial Reference System (IRS or ARU) – Global Navigation Satellite System (GNSSS) – Distance Measuring Equipment (DME) – VHF Omnidirectional Range (VOR) – Radio (Radar) Altimeter (RADALT) – Flight Management System (FMS) • Does some “sensor integration” – Data Links – Multifunction Control Display Unit (MCDU) • Human input – Flight Control Computer (FCC) – Thrust Control Computer (TCC)

More on those in the “Avionics Summary” 2 Air Data – Aircraft

3 Air data: Atmospheric Fundamentals

Atmosphere: ~78% nitrogen ~21% oxygen ~1% other gas including water vapor

Ideal Gas Law (Boyle-Gay Lussac): 푝 = 𝜌푅푇

2 2 푝: atmospheric pressure (lbs/ft or N/m ) Dry air: 푅 = 287.05 푚2/(푠2퐾) 𝜌: air density (slug/ft3 or kg/m3) 푝: atmospheric pressure (lbs/ft2 or N/m2) 푅: individual/specific gas constant (ft lb/slug oR or J/(kg OK) or m2/(s2K) 푇: absolute temperature (OR or OK); (Rankine)

OR = OF + 459.7O Dalton’s law of partial pressures: 푝 = 푝푎 + 푝푣 OK = OC + 273.15O water vapor pressure OC = (5/9)(OF -32) dry air pressure 1 slug has a mass of 14.593903 kg

4 Air data: Atmospheric Fundamentals

Dalton’s law of partial pressures: 푝 = 푝푎 + 푝푣

water vapor pressure dry air pressure 푑푦 Then: 푝 − 푝푣 푝푣 𝜌 = 𝜌푎 + 𝜌푣 = + 푑ℎ 푅푇 푅푣푇 1 푅 𝜌 = 푝 − 푝푣 1 − 푅푇 푅푣 Air mass in cube is equal to mass of dry Dry air: 푅 = 286.9 퐽/(푘푔푂퐾) air and the water Water vapor or steam: 푅 = 461.5 퐽/(푘푔푂퐾) 푑푥 vapor mass. 푣

푂 5 2 Assume 푇 = 305.372 퐾, 푝 = 1.013 ∙ 10 푁 푚 and 푝푣 = 4.816 ∙ 103 푁 푚2 𝜌 = 𝜌푎 + 𝜌푣 Then: 𝜌 = 1.1354 푘푔/푚3 compared to 𝜌 = 1.1662푘푔/푚3 for dry air (1.8% reduction in air density)

5 ICAO Standard Atmosphere

International Standard Atmosphere

Thermosphere Mesopause (84.852km or 278,386ft)

Mesosphere 71km or 232,940ft

Mesosphere 51km or 167,323ft

Mesosphere Stratopause (47km or 154,199ft)

Stratosphere 32km or 104,987ft

Stratosphere 20km or 65,617ft

Stratosphere Tropopause (11km or 36,089 ft)

Troposphere

Mean Sea-Level (MSL)

International Civil Aviation Organization (ICAO) atmosphere model = US standard model below 65,617 ft altitude. 6 ICAO Standard Atmosphere

For subsonic aircraft only the troposphere and the stratosphere are important.

Stratopause (47km or 154,199ft)

Stratosphere 32km or 104,987ft

Stratosphere 20km or 65,617ft

Stratosphere Tropopause (11km or 36,089 ft)

Troposphere

Mean Sea-Level (MSL)

Standard atmosphere defines sea-level properties:

2 푔0 = 9.80665 푚 푠푒푐 2 5 2 5 푝0 = ퟐퟗ. ퟗퟐ 풊풏 푯품 = 2,116.2 푙푏푠 푓푡 = 1.01325 ∙ 10 푁 푚 = 1.01325 ∙ 10 푃푎 = ퟏퟎퟏퟑ. ퟐퟓ풉푷풂 푂 푂 푂 푂 푇0 = 59 퐹 = 518.7 퐹 = 15 퐶 = 288.15 퐾 3 3 𝜌0 = 0.002377 푠푙푢푔푠 푓푡 = 1.225 푘푔 푚

7 ISA – Temperature variation

푇 = 푇𝑖 + 푎𝑖 ℎ − ℎ𝑖

Thermosphere Temperature Gradients: Mesopause (h7 = 84.852km or 278,386ft)

푂 푂 Mesosphere 푇6 = −58.5 퐶, 푎6 = −2.0 퐶/푘푚 h6 = 71km or 232,940ft

Mesosphere 푂 푂 푇5 = −2.5 퐶, 푎5 = −2.8 퐶/푘푚 h5 = 51km or 167,323ft

푂 푂 Mesosphere 푇4 = −2.5 퐶, 푎4 = +0.0 퐶/푘푚 Stratopause (h4 = 47km or 154,199ft)

푂 푂 Stratosphere 푇3 = −44.5 퐶, 푎3 = +2.8 퐶/푘푚 h3 = 32km or 104,987ft

푂 푂 Stratosphere 푇2 = −56.5 퐶, 푎2 = +1.0 퐶/푘푚 h2 = 20km or 65,617ft

푂 푂 Stratosphere 푇1 = −56.5 퐶, 푎1 = 0 퐶/푘푚 Tropopause (h1 = 11km or 36,089 ft)

Troposphere 푂 푂 푇0 = 15 퐶, 푎0 = −6.5 퐶/푘푚 h0 = 0km or 0ft Mean Sea-Level (MSL)

푂 8 푎0 = −0.0065 퐶/푚 Atmospheres (Wikipedia)

U.S. Standard Atmosphere

Kármán line, at an altitude of 100 km (62 mi) above sea 9 level, is used as the start of outer space (informally) ISA – Pressure variation

“Pressure is force per unit area applied in a direction perpendicular to the surface of an object” Force equilibrium yields: 푝 + 푑푝 푑푥푑푦 푝푑푥푑푦 − 푝 + 푑푝 푑푥푑푦 − 𝜌푔푑푥푑푦푑ℎ = 0 −푑푝푑푥푑푦 = 𝜌푔푑푥푑푦푑ℎ 푑푦 푑푝 = −𝜌푔푑ℎ 푑ℎ Pressure decreases with increasing altitude 퐹 Substitute the ideal gas law:

퐹 = 푚푔 = 𝜌푑푥푑푦푑ℎ 푔 푑푝 푔푑ℎ = − 푝푑푥푑푦 푝 푅푇 푑푥

Below 36,089 ft: 푇 = 푇0 + 푎 ℎ − ℎ0 푇 = 푇0 + 푎ℎ

1 1 푝 1 ℎ 푔 푝 ℎ 푔 푑푝 = − 푑ℎ 푑푝 = − 푑ℎ 푙푛 = − 푑ℎ 푝 푅푇 푝 푅푇 푝 푅 푇 + 푎ℎ 푝0 ℎ0 0 ℎ0 0

10 ISA – Pressure variation

Below 36,089 ft: 푇 = 푇0 + 푎0 ℎ − ℎ0 푇 = 푇0 + 푎0ℎ 푑푦 푝 ℎ 푔 푙푛 = − 푑ℎ 푝0 ℎ 푅 푇0 + 푎0ℎ 푑ℎ 0 푝 푔 ℎ 1 푙푛 = − 푑(푇 + 푎 ℎ) 퐹 푝 푎 푅 푇 + 푎 ℎ 0 0 0 0 ℎ0 0 0 푝 푔 푙푛 = − 푙푛 푇 + 푎 ℎ ℎ 0 0 ℎ0 푝푑푥푑푦 푝0 푎0푅 푑푥 푝 푔 푇 + 푎 ℎ 푙푛 = − 푙푛 0 0 푝0 푎0푅 푇0

푎 푅 푎 푅 − 0 − 0 푝 𝑔 푇 + 푎 ℎ 푇 푝 𝑔 푙푛 = 푙푛 0 0 ℎ = 0 − 1 푝0 푇0 푎0 푝0

𝑔 𝑔 𝑔 − − − 푎 푅 푎 푅 푝 푇 + 푎 ℎ 푎0푅 푝 푇0 + 푎0ℎ 0 푇 0 0 0 = = 푙푛 = 푙푛 푝 푇 푇 푝0 푇0 0 0 0

11 Altitudes – Pressure Altitude

푎 푅 − 0 푇 푝 𝑔 ℎ = 0 − 1 푎0 푝0

12 ISA – Pressure variation

푑푦 Below 36,089 ft:

𝑔 푑ℎ − 푝 푇 푎0푅 = 퐹 푝0 푇0

𝜌 푅푇0 푝 푇0 푝 Air density ratio: = = 푝푑푥푑푦 𝜌0 푅푇 푝0 푇 푝0 푑푥

𝑔 𝑔 − −1 − −1 𝜌 푇 푎0푅 𝜌 푇 + 푎 ℎ 푎0푅 = = 0 0 𝜌0 푇0 𝜌0 푇0

𝑔 − −1 푎0푅 푎 푅 − 𝜌 푇0 + 푎0ℎ 0 푇 𝜌 𝑔+푎0푅 = 0 𝜌 푇 ℎ = − 1 0 0 푎 𝜌0

13 ISA – Pressure variation

Between 11-20km (lower stratosphere): 푑푦 푇 = 푇1 푝 ℎ 푔 푑ℎ 푙푛 = − 푑ℎ 푝1 ℎ 푅푇1 퐹 1 푝 푔 ℎ 푙푛 = − 푑ℎ 푝 푎 푅푇 1 1 1 ℎ1 푝푑푥푑푦 푝 푔 푙푛 = − ℎ − ℎ1 푑푥 푝1 푎1푅푇1

푎1푅푇1 푝 ℎ = ℎ1 − 푙푛 푔 푝1

𝑔 푝 − ℎ−ℎ1 = 푒 푎1푅푇1 푝1

14 Measuring Altitude

• Barometric altimeter (baro-altimeters) use a pressure gauge to measure the pressure and then convert that measured pressure into an altitude reading

푎 푅 − 0 푇 푝 𝑔 ℎ = 0 − 1 푎 푝0

15 Altitudes - According to FAA

• Indicated Altitude – is the altitude shown on the altimeter. • True Altitude – is height above mean sea level (MSL). • Absolute Altitude – is height above ground level (AGL). • Pressure Altitude – is the indicated altitude when an altimeter is set to 29.92 in Hg (1013 hPa in other parts of the world). It is primarily used in aircraft performance calculations and in high-altitude flight (> 18,000 ft). • Density Altitude – is formally defined as “pressure altitude corrected for nonstandard temperature variations.” – Needed for aircraft performance calculations (margins etc.)

16 Example

• Problem: a standard altimeter indicates 15,000ft when the ambient temperature is 35OF. Calculate the density altitude. – Note: make sure to use SI units in your calculations

Is this a “standard” atmosphere?:

At a height of 15,000ft = 4,572m: 푇 = 푇0 + 푎0ℎ = 258.4퐾 = 5.5퐹 non-standard atmosphere

The Baro-altimeter will read the “correct” pressure form its pressure gauge:

𝑔 − 0 푝 푇 + 푎 ℎ 푎0푅 = 0 0 푝 = 57181.6 푁 푚2 푝0 푇0 2 푝0 = 101325 푁 푚 Air density at that altitude: 푇0 = 288.15퐾 푝 3 푎0 = −0.0065 퐾 푚 𝜌 = = 0.725 푘푔 푚 푅 = 287.05 푚2/(푠2퐾) 푅푇푎푚푏 2 푔0 = 9.80665 푚 푠 푇푎푚푏 = 35퐹 = 274.8퐾

17 Example - Continued

• Problem: a standard altimeter indicates 15,000ft when the ambient temperature is 35OF. Calculate the density altitude. – Note: make sure to use SI units in your calculations

Now, revisit the previously derived equations:

푝 3 푝0 3 𝜌 = = 0.725 푘푔 푚 and 𝜌0 = = 1.225 푘푔 푚 푅푇푎푚푏 푅푇0 𝜌 = 0.592 𝜌0 𝑔 푎0푅 − −1 − 푎푅 𝑔 +푎 푅 𝜌 푇0 + 푎0ℎ 푇0 𝜌 0 0 = ℎ푑푒푛푠𝑖푡푦 = − 1 = 5,124푚 = 16,871푓푡 𝜌0 푇0 푎0 𝜌0

From temperature alone, you would get an altitude way off: 274.82 − 288.15 푇 = 35퐹 = 274.82퐾 = 푇 + 푎 ℎ ℎ = = 2,051푚 = 6,728푓푡 0 0 −0.0065

18 Altitudes – On Flight Deck

• Altitude (QNH) – is height above mean sea level (MSL) – QNH is the barometric pressure causing altimeter to read height above MSL • Altitude (QFE) – is height above ground level or a specific aerodrome – QFE is the barometric pressure causing altimeter to read height above ground level or a specific aerodrome • Flight Level – Is the nominal altitude (pressure altitude) in hundreds of feet assuming a standard MSL pressure of 1013.25 hPa; – Example: FL150 = Flight Level 150 = 15,000ft

“The abbreviation QNH originates from the days much communication was done by Morse Code. To avoid the need for long Morse transmissions, many of the most commonly used communications were incorporated into a Q code. To ask for atmospheric pressure at sea-level (i.e., at zero altitude) the letters 'QNH' would be transmitted. A common mnemonic for QNH is "Nautical Height", (whereas the mnemonic often used for QFE is "Field Elevation").”

19 Altitudes - Other

20 Altitude on PFD

QNH selected Set barometric reference

QFE selected Set barometric reference

Select on EFIS control panel

21 Pressure Gauge

From: http://en.wikipedia.org/wiki/File:Faa_pitot_static_system.JPG

22 Static Ports – Airbus Example

flush-mounted hole on the fuselage of an aircraft, and is located where it can access the air flow in a relatively undisturbed area

23 What about Airspeed?

Pitot Tubes

24 Pitot/Static/Pitot-static Tubes

http://www.centennialofflight.net/essay/Theories_of_Flight/Ideal_Fluid_Flow/TH7G3.htm 25 Pitot Static Tube

Static pressure 푝

푉

Static lead

Total pressure Pitot lead or stagnation pressure

푝푡

When operating subsonic (<200kts), the incompressible Bernoulli equation applies: Pressure gauge, calibrated 1 as an airspeed indicator 푝 = 푝 + 𝜌푉2 푡 2 1 1 푝 = 푝 + 𝜌푉2 푝 − 푝 = 𝜌푉2 = 푞 Dynamic pressure 푡 2 푡 2

26 Airspeed – Incompressible Air

When operating subsonic (<200kts), the incompressible Bernoulli equation applies: 1 2 푝 − 푝 푝 = 푝 + 𝜌푉2 푉 = 푡 푡 2 𝜌

True Airspeed

Using the standard air density at sea-level instead of the true air density yields:

2 푝푡 − 푝 𝜌0 1 푉푒 = 푉 = 푉푒 = 푉푒 𝜌0 𝜌 𝜎

푝 Temperature Equivalent Airspeed 𝜌 = 푅푇

27 Airspeed – Compressible Air

When operating subsonic (>200 kts and <400kts), the compressible Bernoulli equation applies: 훾−1 훾 푝 훾 푝 1 훾 푝 푝 훾 푡 = + 푉2 푉2 = 2 푡 − 1 훾 − 1 𝜌푡 훾 − 1 𝜌 2 훾 − 1 𝜌 푝

Uses: True Airspeed 푝 푝푡 퐶푝 훾 = 훾 = 푐표푛푠푡푎푛푡 and 훾 = ≈ 1.4 (푎𝑖푟) 𝜌 𝜌푡 퐶푣 (ratio of specific heat at constant volume and constant pressure)

훾푝 Speed of sound 푉 = = 훾푅푇 푎 𝜌 훾−1 훾−1 2 훾푝 푝 2푉2 푝 푉2 = 푡 훾 − 1 = 푎 푡 훾 − 1 훾−1 휌 푝 훾−1 푝

28 Mach Number

훾−1 2 훾 Temperature 푉 2 푝푡 2 free 푀 = 2 = − 1 푉푎 훾 − 1 푝

Local/ambient speed of sound 훾푝 푉 = = 훾푅푇 푎 𝜌

• Subsonic: 푀 < 1.0 • Supersonic: 푀 > 1.0

29 Temperature

TAT probe measures the so-called Total Air Temperature (TAT) or stagnation temperature, rather than the Static Air Temperature, SAT (also referred to Outside Air Temperature, OAT).

Due to frictional heating and compression of the air impinging on the thermometer. 훾 − 1 푇 = 푇 1 + 휂푀2 푚 푠 2

TAT SAT Recovery factor (0.7 and up) (frictional heating, re-radiation, and non isentropic compression of air)

Required for calculating of true airspeed

30 Airspeeds …

• Instrument indicated airspeed, 푉𝑖 Doesn’t provide a spot that is • Indicated Airspeed (IAS) , 푉퐼 always provides ambient static pressure – 푉퐼 = 푉𝑖 + Δ푉𝑖 correction for instrument error • Calibrated Airspeed (CAS), 푉퐶 – 푉퐶 = 푉퐼 + Δ푉푝 correction for position error due to incorrect static port location

• Equivalent Airspeed (EAS), 푉퐸 correction for adiabatic flow at the particular altitude – 푉퐸 = 푉퐶 − Δ푉푐 (compressibility correction) • True Airspeed (TAS) ,푉

휌 – 푉 = 푉 0 퐸 휌

31 Airspeeds …

• True Airspeed (TAS) ,푉 훾−1 훾 푝 푝 훾 푉2 = 2 푡 − 1 훾 − 1 𝜌 푝

• Calibrated Airspeed (CAS), 푉퐶 훾−1 훾 2 훾 푝0 푝푡 푉퐶 = 2 − 1 훾 − 1 𝜌0 푝

32 Pitot Tubes and Ice

• Air France 447 accident (2009) resulted from the following succession of events: – Temporary inconsistency between the measured airspeeds, likely following the obstruction of the Pitot probes by ice crystals that led in particular to autopilot disconnection and a reconfiguration to alternate law, – Inappropriate control inputs that destabilized the flight path, – The crew not making the connection between the loss of indicated airspeeds and the appropriate procedure, – The PNF’s late identification of the deviation in the flight path and insufficient correction by the PF, – The crew not identifying the approach to stall, the lack of an immediate reaction on its part and exit from the flight envelope, – The crew’s failure to diagnose the stall situation and, consequently, the lack of any actions that would have made recovery possible.

33 Angle-of-Attack (AOA)

34 Ports – Airbus 330

35