Project no. 6 Airplane Performance in Powered Steady Flight

(1)

Project no. 6

Airplane Performance in Powered Steady Flight

6.1 Introduction

Following physical parameters are called as basic steady flight performance parameters:

• maximum level flight speed Vmax,

• minimum flight speed Vmin,

• maximum climb speed (vertical speed) wmax,

• flight speed for wmax , Vw,

• maximum path angle γmax,

• flight speed for γmax, Vγ ,

• absolute (theoretical ) ceiling ht,

• service ceiling hs,

• minimum climb time form ground (h=0) to an altitude h< ht.

All parameter listed above we can calculate using simple algorithm based on steady state equations of motion.

6.2 The engine-propeller propulsion systems Let assume:

• an aircraft is flying in vertical plane with constant speed V and with no rotation along any axis (see fig. 1),

• propulsion thrust force T is parallel to the flight speed V,

• path angle γ is small, less than 15 … 20 degree,

Due to above assumptions the equations of airplane's motion can be expressed as

(1) Forces equations (first set) have the scalar form as follows (fig 1):

(2) (3)

Multiplying the equation ( 2 )by flight speed we are go to the following power balance:

(4)

∑

^^Fj=0 ,

∑

^^Mc j≡0 .

D+ m⋅g⋅sin γ−T =0 , L−m⋅g⋅cos γ=0 . P_level + P_Q − P_avail = 0 ,

Figure 1:

L

T D

P_level + P_Q − P_avail = 0 ,

(2)

where:

P_level=D⋅V - the power required for balancing of the aerodynamic drag, P_Q=V⋅m⋅g⋅sin  - the power for vertical lifting of the airplane,

P_avail=n_engines⋅T⋅V - the power of propulsion system (see project no. 5), n_engines - number of the engines in the propulsion system.

Using the power balance equation ( 5 ) we have:

w=P_avail−P_level

m⋅g ,  =P_avail−P_level

V⋅m⋅g (5)

Flight speed can be calculated using the equation ( 3 ) and using assumption of small path angle γ (cos γ ≈ 1):

V =



^⋅S^2⋅m⋅g^W^⋅^C^L ^, ⁽⁶⁾

Using ( 6 ) for estimating of the power required for steady level flight Plevel , we obtain:

P_level=D⋅V =1

2⋅⋅S_w⋅C_D⋅V³=m⋅g⋅



²^^⋅^m⋅g^S^w ^⋅^C^C^D²^L³⁼^m⋅g⋅

^

^²^⋅^m⋅g^S^w^⋅¹^E ⁽⁷⁾

The power Plevel depends on flight altitude, aircraft's parameters (mass, wing area) as well as the aerodynamic energy function E (see project No. 3).

Taking into account formulas (5) thru (7), the algorithm of computations performance parameters: climb speed w=w(V,h) and path angle γ=γ(V,h) can be specified as follows (see table 6.1):

• assume a set of flight altitudes, same as altitudes used for calculations of propulsion system characteristics Pavail(V,h),

• for each of flight altitude assume a series of CL values starting from CL max until CL close to zero when climb speed and path angle goes to negative values,

• from aerodynamic characteristics of the aircraft (project No. 3) for each value of CL take the value of power function E,

• compute required power Plevel ,

• compute flight speed V,

• from propulsion characteristics for calculated value of flight speed V take proper value of Pavail ,

• compute ΔP= Pavail -Plevel, w and γ,

• reapeat all calculations until for last fligth altitude h all values of w and γ will be negative,

• present result of calculation on graphs (see fig. 2 and 3 , light airplane with no supercharged engine).

Table 6.1 Basic performances of the aircraft in powered flight Flight altitude h0=0 km

CL[-] E [-] V [m/s] Pn [W] Pr [W] ∆P[W] w [m/s] γ [degree]

CL-max

...

CL-Vmax

Flight altitude h1=2 km

CL[-] E [-] V [m/s] Pn [W] Pr [W] ∆P[W] w [m/s] γ [degree]

CL-max

...

CL-Vmax

....

(3)

Flight altitude hend > habsolute

Using above results of calculations (graphs and table) estimate following additional performance parameters:

• maximum level flight speed Vmax (w(Vmax ,h) = γ( Vmax ,h) =0 !),

• minimum flight speed Vmin (CL max or w = 0) Figure 2

30 40 50 60 70 80 90 100 110 120

-20 -15 -10 -5 0 5 10

Performane of the aircraft

m=2000 kg, S = 15 m^2, P_0 = 300 kW

H=0 km H=2 km H=4 km H=6 km H=8 km

V [m/s]

w [m/s]

Figure 3

30 40 50 60 70 80 90 100 110 120

-10 -8 -6 -4 -2 0 2 4 6 8 10

Performane of the aircraft

m=2000 kg, S = 15 m^2, P_0 = 300 kW

H=0 km H=2 km H=4 km H=6 km H=8 km

V [m/s]

gamma [deg]

(4)

• maximum climb speed wmax,

• maximum climb speed flight speed Vw,

• maximum path angle γmax,

• maximum path angle flight speed V_γ ,

• absolute (theoretical) ceiling ht (wmax=0),

• service ceiling hs (wmax=0.5 m/s),

• time of faster climbing from ground to a flight altitude th :

(8)

• time of climbing to the service ceiling th s .

All these data should be collected in a table (see table 6.2) and then should be presented on the offer graph (fig. 4 and 5 ).

Table 6.2... ...Aircraft performances - the offer graph h

[km]

Vmin

[km/h]

Vmax

[km/h]

wmax

[m/s]

Vw

[km/h] γmax

[degree]

V_γ [km/h]

th

[s]

0 2 ...

hs th s

ht Vh t Vh t 0 Vh t 0 Vh t -

hmax -

Figure 4

-2 0 2 4 6 8 10

0 1 2 3 4 5 6 7 8 9

Performance of the aircraft

the offer chart - climb performance

w_max gamma_max h_t

h_p

w_max [m/s], gamma_max [deg]

h [km]

∫

= ^h

h w

t d

0 max(ζ) ζ

(5)

Remarks:

• maximum level speed Vmax can be estimated using a simple linear interpolation formula

V_max=w_n−1⋅V_n−w_n⋅V_n−1 wn−1−wn

, (9)

where index "n" and "n-1" denotes two close calculation points where wn is the first negative value of climb speed in the set [Vj , wj ], see table 6.1 and figure 3, pairs [90.0, +0.042] and [100.0, -3.9] for altitude h=0 km; similar expression may be used for calculation of the absolute and service ceilings:

h_t=w_{max n-1}⋅h_n−w_{max n}⋅h_n−1

w_{max n-1}−w_{max n} , h_s=



^wmax n-1−0.5



^⋅^hn−



^wmax n−0.5



^⋅^hn−1

w_{max n-1}−w_{max n} , (10)

• for light airplanes with piston engine without supercharging the climb speed wmax at any flight altitude can be approximated with linear function:

(11)

and for this case the integral for th give us the expression:

(12)

Figure 5

0 10 20 30 40 50 60 70 80 90 100

0 1 2 3 4 5 6 7 8 9

The aircraft performance

the offer chart -- characteristic flight speeds

V_min V_gamma V_w V_max h_t h_p t_h

speed [m/s], climb time [min]

h [km]

) 1 (

* ) 0 ( )

( _max

max

ht

w h h

w = −













−

=

t t

h

h w h

t h

1 ln 1 )* 0

max(

(6)

• if the aircraft is equipped with single-stage supercharged piston engine (see Project #5), the climb time should be calculated using formulas:

for flight altitude h ≤ hnom

t_{0 h}= h_nom

w_max(h_nom)−w_max(0)⋅ln

(

¹⁺^w^max⁽^h^nom^hnom^)−w^max⁽⁰⁾

⋅ h

w_max(0)

)

^, ⁽¹³⁾

for flight altitude h > hnom

t_h=t_{0 h}+ h_t−h_nom

w_max(h_nom)⋅ln

(

^h^h^t⁻t−h^h^nom

)

^. ⁽¹⁴⁾

Note: hnom is the piston engine nominal altitude where the engine's power achieves maximum value (for example, hnom =3.0 km, fig. 2 in the description of project no. 5); results calculation of the climb time using formulas (13) and (14) are shown bellow (fig. 6 ).

6.3 The turbojet propulsion systems

If the airplane is powered by turbojet propulsion system (one or more engines), very similar method as described in the previous paragraph 6.2 can be used, but instead of power, the thrust of engine (engines) must be used. Therefore:

w=



^Tavail−D



^⋅^V

m⋅g , =T_avail−D

m⋅g (15)

Of course, Tavail = nengines T, where T is the thrust of single turbojet engine.

Usually, jet airplanes rather are operating in high subsonic range of the flight speed (Vmax > 720 km/h, 200 m/s) and because this the influence of air compressibility (Mach number effect) must be taken into account. It can be done using following algorithm:

Figure 6

0 500 1000 1500 2000 2500 3000 3500

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Climb time t_h

t_h [sec]

h_t

w_max*100 w_max*100 h_nom

time [sec]

flight altitude [m]

(7)

• assume a set of flight altitudes, same as for the aircraft engine;

• for each of flight altitudes and for each of values of airplane's lift coefficient CL and drag coefficient CD compute Mach number using formula:

(16)

where as is the sound speed at the flight altitude h; calculations should be stopped if for given CL the Mach number is greater than 1;

• compute corrected drag coefficient from Ludvig Prandtl formula:

(17)

• continue calculations of performances using corrected drag coefficient C_D instead of regular low speed coefficient CD .

An example drag characteristics calculated using above method is showed on fig. 7.

Remark: the method described above may be applicable for jet airplanes with relatively large mass-to-thrust ratio (more than 300 kg of take-off mass per 1 kN of all engines thrust); this limitation is derived directly from the limitation of path angle γ ; if the requirement is not satisfied, for example the airplane is a high-performance military fighter, more accurate methods (ie. Joukowsky thrusts method) must be used.

Figure 7

0,00 0,02 0,04 0,06 0,08 0,10

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Lilienthal altitude polars (influence of air compressiblity)

incompr.

H=0 H=3 km H=6 km H=9 km

C_D

C_L

Ma=



^ρ⋅S⋅C^2⋅m⋅g^L

a_s , a_s=a_sh

C_D= C_D



^1−Ma² ^,