ADVANCES in DYNAMICAL SYSTEMS and CONTROL The . Influence Factors and Evaluation

TEODOR LUCIAN GRIGORIE Avionics Department University of Craiova 107 Decebal Street, 200440 Craiova ROMANIA [email protected] http://www.elth.ucv.ro

LIVIU DINCA Avionics Department University of Craiova 107 Decebal Street, 200440 Craiova ROMANIA [email protected] http://www.elth.ucv.ro

JENICA-ILEANA CORCAU Avionics Department University of Craiova 107 Decebal Street, 200440 Craiova ROMANIA [email protected] http://www.elth.ucv.ro

OTILIA GRIGORIE Carol I, High School 2 Ioan Maiorescu Street, 200418 Craiova ROMANIA [email protected]

Abstract: - The paper presents a method to determinate the with an electronic flight instrument system. A brief review of the flight is performed, and the calculus relations of the density altitude are developed. The first two atmospheric layers (0÷11 Km and 11÷20 Km) are considered. For different indicated barometric altitudes an evaluation of the density altitude, as a function of non-standards variations and of dew point value, is realized.

Key-Words: - standard ; atmospheric layers; ; density-altitude; evaluation

1 Introduction altitude information on board can be made directly by Regarded as one of the most important parameters that the measuring system or through an Electronic Flight must known by the pilot during the flight, the altitude is Instrument System (EFIS). If it is determined using a defined as the distance between the centre of mass of the GPS than it can be defined as the aircraft elevation from aircraft and the corresponding point on the surface of the the reference geoid WGS 84 surface (World Geodetic Earth, considered by the vertical ground [1]. In general, System) [2]. at the board, the altitude is measured directly using the Relative to the position of the ground point taken as a altimeter, but can be also calculated by means of reference or to the corrections introduced in the complementary systems, such as, for example, the Air measuring system, the flight altitude behaves different Data Computer (ADC), the Inertial Navigation System names [3]:

(INS) or Global Positioning System (GPS). Showing • true altitude (Hnm): is real altitude of aircraft above

ISSN: 1790-5117 44 ISBN: 978-960-474-185-4 ADVANCES in DYNAMICAL SYSTEMS and CONTROL

mean (MSL); • decreasing of the aircraft maximum flight altitude, • relative altitude (Hr): is the altitude reported at an affecting their capacity to fly above the higher airfield level on which the aircraft performs take off obstacles. or landing manoeuvres; its value depends on the altimeter adjustment.

• absolute altitude (Ha): in this case the reference point 2 Vapor Determination is considered at the intersection of the local vertical The density of the atmospheric air, considered dry, can and the Earth surface, so its calculation takes into be calculated with the relation account the overflowed forms of relief. This altitude is considered to have the greatest importance for ρ = µ RTp ),/( (1) flight safety. the static pressure p being measured directly. In relation

• barometric altitude () (Hb): this is the (1) T is the air temperature, µ - molecular mass of the air altitude indicated by the altimeter when this is tuned (µ=0.0289644 kg/mol), and R is the universal on a basic pressure, so the reference point is constant (R=8.31432 N·m/(mol·K)). In real conditions, positioned on a baric surface. If the reference pressure this density is affected by the humidity in the air, which is 760 mmHg it corresponds to the mean sea level in is considered to be a mixture of dry air molecules and the International Standard Atmosphere (ISA) [4]. water vapor. So, the measured barometric pressure is in

• density altitude (Hρ): is the barometric altitude fact the mixture pressure not the dry air pressure. Thus, corrected for non-standard temperature variations. to determinate the density of the dry air the Dalton This corresponds to the altitude at which the density partial law must be used. In this way, the of air is equal to ISA air density evaluated for the mixture pressure is expressed as the sum of the dry air current flight conditions. and water vapor pv pressures • indicated altitude (H): is the altitude displayed on the = + ppp . (2) dashboard. am v One of the altitudes of great importance, calculated from It results the barometric altitude, is the density altitude (Hρ).

Strongly influenced by changes in temperature and to a = − ppp vam (3) lesser extent by changes in air humidity, the real air density can provides to the pilot vital information for the and the relation (1) for the density calculation becomes flight safety. It is known that the is the vam µ−=ρ RTpp )./()( (4) most important factor influencing the performances of aircraft both at the lift forces level, and at the thrust As a consequence, the dry air density calculation forces level generated by the propulsion systems. supposes to know the actual air pressure (the mixture Density altitude is such a simple way to give the pilot pressure pam), the water vapor pressure pv and the local information on air density, this parameter being temperature T of the atmospheric air. The Mixture calculated such as the pilot can make a coarse pressure and the temperature of local atmospheric air is assessment of aircraft performance in the current flight determined relatively easily by placing the sensors conditions [5]. For example, the increase of the outside the aircraft. Instead, the vapor pressure temperature or of the humidity of the outdoor determination involves the performing of numerical environment in which is operated aircraft lead to lower calculations using information on relative humidity and air density [5], [6], which are as a consequence (as dew point [8]. defined by the density altitude) the indication of a According to [8], two relations often used in the density altitude higher than the current indicated calculation of vapor pressure from the dew point: barometric altitude. a) The first relation shows a high accuracy and is based Therefore, the pilot should expect the following effects on a polynomial development on the aircraft [5], [6], [7]: C (5) pv = 8 , • lift decreasing and the need to increase the flight 0 1 2 3 4 5 6 7 ((((((((( 8 ⋅+++++++++ ctctctctctctctctctc 9 ))))))))) speed to maintain the desired lift; o where t is the dew point expressed in C, C=610.78, c0= • decreasing of the propulsion systems power, and, -2 -4 =0.99999683, c1=-0.90826951·10 , c2=0.78736169·10 , thus, decreasing of the thrust forces; -6 -8 c3=-0.61117958·10 ,c4=0.4388418·10 ,c5=-0.29883885 • runway acceleration decreasing because of the low -10 -12 -14 ·10 ,c6=0.2187442·10 , c7=-0.1789232·10 , c8=0.111 thrust; 1201·10-16, c = -0.30994571·10-19. 9 • increasing of the take off distance and decreasing of b) The second relation, much easier, is generally used in the climb speed; applications that require a lower accuracy

ISSN: 1790-5117 45 ISBN: 978-960-474-185-4 ADVANCES in DYNAMICAL SYSTEMS and CONTROL

tctc )/( 21 +⋅ layer 0). In this situation, the barometric formula, that v Cp ⋅= ,10 (6) gives the dependence of the static pressure p by the with C=610.78, c1=7.5 and c2=237.3. For both relations altitude Hb, is expressed under the form [1] the pressure pv measure unity is . Rg τµ− 0 )/( In Fig. 1 are represented graphically the dependences ( 00 −τ+= bb THHpp 00 ) ,/)(1 (8) p =f(t) between vapor pressure and dew point, given by v p =101325N/m2=760mmHg is the static pressure for the the relation (5), respectively the difference ε between the 0 altitude Hb0. So, for the atmospheric layer 0, the pressure pv values calculated with relations (5) and (6). To be more suggestive the characteristics were designed following equations system is obtained for pressures represented in mmHg. ρ = µ RTp ,/ Dependence pv=f(t) analysis combined with the equation −τ+= HHTT ),( (9) (4) confirming earlier observations that the most 00 bb 0 Rg τµ− 0 )/( disadvantageous circumstances for flying are those with ( 00 −τ+= bb THHpp 00 ) ./)(1 high humidity (positive dew point). For these cases the Successively eliminating the temperature and the vapor pressure can reach up to 42 mmHg, the growth pressure between the three equations is obtained the being exponential. On the other hand, on observes that formula for calculating the density altitude in the form the differences between the two methods of calculation (H becomes H ) are very small (up to 0.012 mmHg). Analyzing both b ρ RgR )/( graphs shows that for positive dew points can be used THH ( pRT µρ−τ−= )/(1)[/( ) 0 τ+µτ− 0 ], (10) without problems relation (6) in high accuracy ρ b 000 00 applications. Instead, in such applications, for negative which, numerically, is values of dew point, especially for those below -20oC, it H 4 953434.01[10433076.4 ρ⋅−⋅⋅= 234969.0 .] (11) is necessary to use formula (5) because the relative error ρ caused by the simplified formula is high, reaching up to In previous relation the density values are entered in 4%. kg/m3, and the resulting density altitude values are in m.

50 0.015 Representing graphically the density altitude for the

40 0.01 change ∆t of environmental temperature outside the aircraft (in the range -30oC÷30oC from the standard 30 0.005 values in ISA) and for different values of dew point t [mmHg] [mmHg] v 20 0 o o ε p (between -50 C÷35 C), the three-dimensional character- 10 -0.005 ristics in Fig. 2 are obtained. These were presented for

0 -0.01 four values of air pressure considered to be in the -60 -40 -20 0 20 40 -60 -40 -20 0 20 40 t [oC] t [oC] mixture with water vapor (pam). These values correspond Fig. 1 Evaluation of formulas for calculating the vapor to altitude readings on barometric altimeter equals to 0m, pressure 3000m, 6000m and 9000m.

H = 0 m H = 3000 m b b Once pressure pv determined, with formula (4) can be calculated the dry air density. Equalling the calculated 2 5 density with the density of air in ISA is equivalent to 1 4 0 3 [km] [km] ρ ρ combine the relation (1) with the formulas that give the H H temperature dependence of altitude and with the -1 2 -2 1 40 40 barometric formulas for the atmospheric layers. 20 40 20 40 20 20 0 0 0 0 ∆t [oC] ∆t [oC] -20 -20 o -20 -20 o -40 t [ C] -40 t [ C] -40 -60 -40 -60

H = 6000 m H = 9000 m 3 Density Altitude Evaluation for the b b 9 12

Atmospheric Layer 0 8 11 For atmospheric layer 0 (0÷11 km), the relation which 7 10 [km] [km] ρ 6 ρ 9 gives the dependence of the temperature by the altitude H H 5 8 is [1] 4 7 40 40 20 40 20 40 20 20 0 0 0 0 ∆t [oC] ∆t [oC] = + τ − HHTT ),( (7) -20 -20 o -20 -20 o 00 bb 0 -40 t [ C] -40 t [ C] -40 -60 -40 -60 o where T0=15 C=288.15K (temperature at the lower limit o Fig. 2 Density altitude for layer 0 at the temperature and of the layer 0), τ0=-6.5 /km (temperature gradient of the humidity variations layer 0), and Hb0=0km (altitude at the lower limit of the ISSN: 1790-5117 46 ISBN: 978-960-474-185-4 ADVANCES in DYNAMICAL SYSTEMS and CONTROL

−µ− RTHHg )/()( The maximum and the minimum values of the density bb 11 = 1epp , (13) altitude, calculated in the simulated conditions, are 2 presented in Table 1. p1=22632.1N/m =169.75mmHg is the static pressure for the altitude Hb1. Starting from the relations (1), (12) and Table 1 Maximum and minimum values of the density (13), the following equations system is obtained altitude for the conditions simulated in Fig. 2 ρ = µ RTp ,/

Indicated barometric Maximum value of the Minimum value of the o No. TT 1 =−== K,66.216C5.56 (14) altitude [m] density altitude [m] density altitude [m] −µ− bb RTHHg 11 )/()( 1 0 1596.8135 -1159.4360 = 1epp . 2 3000 4799.5158 1834.7880 3 6000 8106.6341 4828.0847 Successively eliminating the temperature and the 4 9000 11595.6104 7820.2263 pressure between the three equations is obtained the

From Fig. 2 one observe that the maximum values are formula for calculating the density altitude in the form obtained for ∆t=30oC and t=35oC and the minimum (Hb becomes Hρ) values for ∆t=-30oC and t=-50oC. Consistent with the ρ = b1 − 1 µ ρ µpRTgRTHH 11 )],/(ln[)/( (15) values in Table I, from Fig. 2 can be remarked also a growing of the area curvature with the increasing of the which, numerically, is indicated barometric altitude. According to the tabled H =11000 − ⋅ 2.747995(ln912741.6341 ⋅ρ). (16) maximum and minimum values, the difference between ρ the density altitude and barometric altitude for this layer The density is entered in kg/m3, and the values of the may even exceed 2500 m superposing certain conditions density altitude results in m. of temperature and humidity. Successively neglecting Considering the same values for the non-standard the effects of humidity and of non-standard change of variation ∆t of the environmental temperature outside environment temperature on density altitude, for the four the aircraft (-30oC÷30oC), the variation interval previously considered cases, for the indicated barometric -50oC÷35oC for the dew point t at altitudes under 16000 altitude, result the characteristics in Fig. 3 a., m, and the variation interval -50oC÷25oC for the dew respectively in Fig. 3 b. point t at altitudes over 16000 m, it result the three-

12 12 dimensional characteristics in Fig. 4. For the high

10 H = 9000 m 10 altitudes in this layer the dew point values higher than b H = 9000 m b o 8 25 C temperature are irrelevant physically because in Hb = 6000 m 8 6 Hb = 6000 m 6 these situations the calculated vapor pressure becomes [km] Hb = 3000 m [km] ρ 4 ρ H H 4 greater than the pressure of the mixture. The 2 Hb = 3000 m H = 0 m b characteristics were presented for four values of air 0 2

Hb = 0 m -2 0 pressure considered to be in the mixture with water -40 -20 0 20 40 -60 -40 -20 0 20 40 non-standard temperature change ∆t [oC] dew point t [oC] vapor (pam). These values correspond to altitude readings a. b. on barometric altimeter equals to 11000 m, 14000 m, 17000 m and 20000 m. Fig. 3 The evaluation of the influences of the non- The maximum and the minimum values of the density standard temperature variations and humidity on the altitude, calculated in the simulated conditions, are density altitude in the layer 0 presented in Table II.

Table 2 Maximum and minimum values of the density 4 Density Altitude Evaluation for the altitude for the conditions simulated in Fig. 4 Atmospheric Layer 1 Indicated barometric Maximum value of the Minimum value of the No. For atmospheric layer 1 (11÷20 km), the relation which altitude [m] density altitude [m] density altitude [m] gives the dependence of the temperature by the altitude 1 11000 13633.6955 10056.6798 is [1] 2 14000 18048.2587 13057.7078 3 17000 20657.6839 16059.3595 4 20000 26301.9825 19062.0127 = + τ11 − bb = THHTT 11 ,)( (12)

o where T1=-56.5 C=216.66K (temperature at the lower From Fig. 4 one observe that the maximum values are o o o limit of the layer 1), τ1=0 /km (temperature gradient of obtained at ∆t=30 C and t=35 C for Hb=11000 m and o o the layer 1), and Hb1=11km (altitude at the lower limit of Hb=14000 m, respectively at ∆t=30 C and t=25 C for the layer 1). In this situation the barometric formula Hb=17000 m and Hb=20000 m. The minimum values are becomes [1] obtained at ∆t=-30oC and t=-50oC for all four situations.

ISSN: 1790-5117 47 ISBN: 978-960-474-185-4 ADVANCES in DYNAMICAL SYSTEMS and CONTROL

According to the maximum and minimum values above this way, mathematical relations were developed for the table, the difference between the density altitude and both 0 and 1 atmospheric layers (0÷11km, respectively barometric altitude for this layer may even exceed 6000 11÷20 km). m superposing certain conditions of temperature and For the layer 0, values between -30oC÷30oC for non- humidity. standard temperature variations, and between - o o H = 11000 m H = 14000 m 50 C÷35 C for the dew point, was considered. Also, the b b evaluation was performed for four barometric altitude 14 19 18 values: 0m, 3000m, 6000m and 9000m. The maximum 13 17 deviation of the density altitude from the indicated 12 16 [km] [km] ρ ρ H H 15 barometric altitude value is 2595.6104 m, and was 11 14 o o 10 13 obtained for ∆t=30 C and t=35 C at Hb=9000m. 40 40 o o 20 40 20 40 20 20 For the layer 1, values between -30 C÷30 C for non- 0 0 0 0 ∆t [oC] ∆t [oC] -20 -20 o -20 -20 o -40 t [ C] -40 t [ C] -40 -60 -40 -60 standard temperature variations was considered. For the dew point, values between -50oC÷35oC for barometric H = 17000 m H = 20000 m b b altitudes fewer than 16000 m, and between -50oC÷25oC 21 28 for barometric altitudes over 16000 m, was considered. 20 26 The evaluation was realised for the next values of the 19 24 [km] [km] ρ 18 ρ 22

H H indicated barometric altitude: 11000 m, 14000 m, 17000 17 20 m, and 20000 m. In this case, the maximum deviation of 16 18 40 40 30 30 the density altitude from the indicated barometric 20 20 20 20 0 0 0 0 ∆t [oC] -20 ∆t [oC] -20 -20 t o -20 t o altitude value is 6301.9825 m, and was obtained for -40 [ C] -40 [ C] -40 -50 -40 -50 o o ∆t=30 C and t=25 C at Hb=20000m. Fig. 4 Density altitude for layer 1 at the temperature and For both layers was depicted the individual effects of the humidity variations non-standard temperature variation, respectively of the air humidity, on the density altitude (Fig. 3, respectively Individual effects of non-standard change of Fig. 5). environment temperature and humidity on the density altitude, for the four previously considered cases for the indicated barometric altitude, can be seen in Fig. 5 a., References: respectively in Fig. 5 b. [1] I. Aron, Aparate de bord pentru aeronave. Editura Tehnică, Bucureşti, 1984. 22 26 [2] National Imagery and Mapping Agency Technical 24 20 H = 20 km 22 Report NIMA TR8350.2, Third Edition - Department of b H = 20 km 18 b 20 H = 17 km Defence World Geodetic System 1984 - Its Definition b Hb = 17 km 16 18 [km] [km] ρ ρ

H H and Relationships with Local Geodetic Systems, 3 16 14 H = 14 km H b b = 14 km 14 January 2000. 12 12 [3] M. J. Mahoney, A Discussion of Various Measures of Hb = 11 km H = 11 km 10 10 b -40 -20 0 20 40 -60 -40 -20 0 20 40 Altitude. [Online]. Available: http://mtp.jpl.nasa.gov/ non-standard temperature change ∆t [oC] dew point t [oC] a. b. notes/altitude/altitude.html. [4] International Civil Aviation Organization. Manual of Fig. 5 The evaluation of the influences of the non- the ICAO Standard Atmosphere (extended to 80 standard temperature variations and humidity on the kilometers (262500 feet)). Third edition 1993. density altitude in the layer 1 [5] J. Brandon (2000-2008), Flight Theory Guide. [Online]. Available: http://www.auf.asn.au. For the other layers, formulas for calculating the density [6] J. Williams, Density Altitude - Why airplanes like altitude are derived similarly, but they are less important cool days better, Flight Training Magazine, July 2003. for aviation. [7] J. Williams, Understanding air density and its effects. [Online]. Available: http://www.usatoday.com/ weather/wdenalt.htm. 5 Conclusions [8] R. Shelquist, Air density and density altitude A method to calculate the density altitude starting from calculations. [Online]. Available: http://wahiduddin.net/ the static pressure, non-standard air temperature calc/density_altitude .htm, Longmont, Colorado 2 variations and dew point information was presented. In December 2008.

ISSN: 1790-5117 48 ISBN: 978-960-474-185-4