Data D : Atmosphere and Airspeeds

The performance and other operational characteristics of the aircraft are directly influenced by the properties of the atmosphere in which it is flying. These properties vary depending on weather conditions, seasons of the year, geographical location and particularly with the height above the ground. In order to standardise aircraft analysis, the International Civil Aviation Organisation (ICAO), in agreement with other bodies, has specified an international standard for the atmosphere (ISA). This is now established as the recognised basis for all aircraft analysis. In this standard, height above sea-level is measured as the true (tape-line) distance (the geometric altitude) assuming constant gravitational acceleration (g) with height (i.e. a mean sea-level value go = 9.80665 m/sec2). This height definition is known as the geopotential altitude.

The ISA recognises several altitude regions in which air temperature is defined to be either constant or varying linearly with height. Only the lowest two regions are applicable to current civil aircraft operations. The standard defines the sea-level (quoted as SL and meaning zero altitude) air temperature as 288.15°K (15°C) and then assumes a linear variation with height up to 11 km (about 36089 ft) to a temperature of 216.65°K. This region is called the 'troposphere'. Most civil aviation is flown in this region, or just above it, but as aircraft and engines become more efficient higher altitudes (up to 40,000ft) will be used for optimum cruise conditions.

Above 11 km and up to 20 km the ISA defines a constant temperature with height (at 216.65°K). This constant temperature region is called the 'lower stratosphere'. Above this region the standard atmosphere assumes the temperature to rise and then fall. Since even supersonic transport aircraft operate below 20 km altitude, the upper stratosphere regions are not significant to civil operations.

In the lower regions The atmosphere in the lower regions is assumed to follow normal gas laws. This means that temperature, pressure and density are related by thermodynamic equations and functions. The relationships described on the next page can be used for aircraft analysis

In the operational altitudes appropriate to civil aircraft the variation of temperature with height is defined in the ISA. From a specified sea-level value it reduces linearly up to 11km and then remains constant. The equation defining temperature in the troposphere (below 11km) has the following form:-

T_ALT = T_o - [L. h]

where: T_ALT = Air temperature (°K) at altitude (h)

T_o = Air temperature at sea-level = 288.15 °K

L = 0.0065 °K/m

h = altitude (m), with maximum value of 11 Km.

In the lower stratosphere (above 11 km) the temperature remains constant:-

T_ALT = 216.66°K.

The variation of pressure with altitude is governed by the gas equation shown below:-

P_ALT = P_o [ T_ALT / T_o ] g/R.L

where P_ALT = pressure at altitude (h)

(P_o = pressure at sea-level = 1011325 Pa)

g = gravitational acceleration = 9.81 m/sec² (assumed constant)

R = universal gas constant = 287.3 J/kgK (assumed constant)

(Note: as the temperature is assumed to be constant in the ISA the exponential

[g/RL] in the equation above has a constant value of 5.2561)

The variation of density is directly related to temperature and pressure through gas laws:-.

r_ALT = P_ALT /  R.T_ALT

where: r_ALT = density at altitude h (kg/m³)

(r_o = density at sea-level = 1.225 kg/m³)

Viscosity is needed to determine kinematic viscosity as shown in the next item.

m_ALT = (1.458x10^-6 x T_ALT^3/2) / T_ALT + 110.4  

where: m_ALT = viscosity (kg/ms) at altitude (h)

The coefficient of kinematic viscosity is used in determination of Reynolds Number It is evaluated by the ratio:-

n_ALT = m_ALT / r_ALT 

where: n_ALT = coefficient of kinematic viscosity at altitude (m²/s)

The speed of sound varies with altitude by the following relationship:-

where: a = speed of sound at altitude (h) (m/s)

(a_o = speed of sound at SL = 340.29 m/s)

g = coefficient = 1.4 (assumed constant in the ISA)

R as above

= 1.4 for air

Several non-dimensional coefficients are used in some of the estimating procedures (e.g. engine performance). These are mainly ratios of the conditions at altitude to the sea level (often ISA) values:-

Mach number (M) = V / a

where: V = aircraft true airspeed (m/s)

a = local speed of sound (m/s)

The standard atmosphere approximates to the climatic conditions at about 40°latitude North. To account for other conditions the international standard committee recognised other standard climates (e.g. tropical maximum, arctic minimum, temperate max and min.). It is normal practice not to predict these variations from ISA but assume that the user directly inputs a temperature change to account for local conditions (e.g. + 15°C for hot airport allowance when assessing take-off performance). Such variations are shown in the graph below:-

Note: that although the sea level temperature may vary from the ISA condition,

the lapse rate (change in temperature with height) is assumed to be constant at

0.0065 °K/m from all sea level (SL) temperature values.

All the atmosphere equations can be incorporated into a spreadsheet to determine the ISA properties of the atmosphere at defined heights (h), see below:-

Col A: Altitude (m) with initial (SL) value h = 0 and steps at chosen intervals.

Col B: Ambient temp. (°K) with initial (SL) value T_SL = 288.15

then T_h = 288.15 - 0.0065h

Col C: Temperature ratio q = T_h / T_SL

Col D: Ambient air pressure (N/m²) with initial (SL) value P_SL = 1011325

then P_h = P_SL x q ^5.2561

Col E: Relative pressure ratio d = P_h / P_SL

Col F: Ambient air density (kg/m³) with initial (SL) value r_SL = 1.225

then r_h = r_SL. d/q

Col G: Relative density ratio s = r_h / r_SL

Col H: Speed of sound a = (1.4*287.3*T_h)^0.5

The estimation of the atmospheric properties at a particular height and with temperature variation for ISA is done by inputting suitable values to the spreadsheet:-

Performance calculations are usually quoted at true airspeed (TAS) or at an appropriate Mach number. To eliminate altitude effects aircraft performance calculations are often done at equivalent air speed EAS (see below). Published data may be quoted at other airspeed definitions. It may be necessary to convert between the various speeds so it is important to understand airspeed definitions.

The main ones are listed below:-

IAS = 'indicated airspeed'- this is the speed displayed on the aircraft instruments. As such it is a function of height (static pressure) and forward momentum (dynamic pressure). The value displayed on the instrument will be affected by local atmospheric conditions and by errors from the installation of the sensors on the aircraft.

CAS = 'calibrated airspeed' - this is the speed as determined by the airspeed indicator corrected for positional and instrument errors. Since this correction is often done within the instrument in which case the value is often equivalent to the IAS. However if there are significant interference effect generated by the airframe a further correction will be necessary to show IAS.

EAS = 'equivalent airspeed'- this is the true airspeed corrected for altitude effects:-

V_EAS = (s)^0.5. V_TAS

Equivalent airspeed is a direct measure of the kinetic energy per unit volume of air and therefore is directly related to the forces on the aircraft (lift and drag) and the corresponding structural loading. The correction of EAS from CAS depends on the nature of the correction built into the airspeed indicator.

TAS = 'True airspeed'- this is the speed measured over the ground. The prefix K (KTAS) is sometimes used to indicated that the speed is quoted in knots.

DYNAMIC PRESSURE (q)- this is used in aerodynamic equations to link speed to forces on the aircraft.

For example, Lift is defined as (q S C_L) (where, q = 0.5 r (V_TAS)² = 0.5 r_o (V_EAS)² )