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Solutions to problems for
Aircraft Performance: An Engineering Approach, Mohammad Sadraey, 2
nd ed., CRC, 2023
Chapter 1-
Ch. 1
The software package Mathcad is used to solve problems.
1.1. Determine the temperature, pressure and air density at 5,000 m and ISA condition.
There are two methods:
a. Using appendix:
From Appendix A:
- Temperature: 255.69 K
- Pressure: 54,048 Pa
- Air density: 0.7364 kg/m
3
b. Calculations:
Same results.
Sea level:
5000 m: (Equ 1.6)
(Equ 1.16)
(Equ 1.23)
h 5000 m ISA L1^ 6.^
K
1000 m
R1 287
J
kg K
P
o
101325 Pa
T
o
( 15 273 ) K 288K
T
T
o
L1 h 255.5K
P
P
o
T
T
o
54000.3Pa
P
R1 T
kg
m
3
1.2. Determine the pressure at 5,000 m and ISA-10 condition.
1.3. Calculate air density at 20,000 ft altitude and ISA+15 condition.
Sea level:
5000 m: (Equ 1.6)
(Equ 1.16)
Sea level:
20000 ft: (Equ 1.6)
(Equ 1.16)
(Equ 1.23)
h 5000 m ISA 10 L1 6.
K
1000 m
R1 287
J
kg K
P
o
101325 Pa
T
o
( 15 273 10 ) K 278K
T
T
o
L1 h 245.5K
P
P
o
T
T
o
52714.2Pa
h 20000 ft ISA 15 L1 2
K
1000 ft
R1 287
J
kg K
P
o
101325 Pa
T
o
[( 15 273 ) 15 ] K 303K T
o
545.4 R
T
T
o
L1 h 263K T 20
473.4 R
P
P
o
T
T
o
48143.9Pa P 20
lbf
ft
2
P
R1 T
kg
m
3
slug
ft
3
1.5. Determine relative density () at ISA-20 condition and 80,000 ft altitude.
1.6. Determine the temperature at 70,000 ft and ISA condition.
Sea level:
Second layer:
(Equ 1.16)
(Equ 1.23)
(Equ 1.25)
Sea level:
Second layer:
h 80000 ft ISA 20 R1 287
J
kg K
P
o
101325 Pa o
slug
ft
3
T
o
[( 15 273 ) 20 ] K 268K T
o
482.4 R T
o
5.15 °C
T
( 56 273 20 ) K T
354.6 R T
197K
P
P
o
T
T
o
20098.1Pa P 80
lbf
ft
2
P
R1 T
kg
m
3
slug
ft
3
o
h 70000 ft ISA
T
o
( 15 273 ) K 288K T
o
518.4 R
T
( 56 273 ) K T
390.6 R T
217K
1.7. An aircraft is flying at an altitude at which its temperature and pressure are 255 K and 4.72 × 10
4
Pa.
Calculate:
a. pressure altitude
From Appendix A, for T = 255 K; h = 5,000 m
b. temperature Altitude
From Appendix A, for P = 4.72 × 10
4
Pa; h = 6,000 m
c. density altitude
Interpolation:
From Appendix A, for = 0.645 kg/m
3
; h = 6,215 m
(Equ 1.23)
T1 255 K P1^ 4.72 10
4 Pa R1 287
J
kg K
P
R1 T1
kg
m
3
given h 6500
7000 h
find (h ) 6215.
1.10. An aircraft is flying at 20,000 ft altitude with speed of 400 km/hr and ISA condition. What is
aircraft Mach number?
1.11. Calculate air density at sea level and ISA condition, when humidity is 100%.
Sea level:
(Equ 1.6)
(Equ 1.35)
Sea level:
Table 1.
(Equ 1.29)
(Equ 1.30)(Equ 1.23)
h 20000 ft ISA L1^2
K
1000 ft
V1 400
km
hr
V1 111.
m
s
1.4 R1 287
J
kg K
T
o
( 15 273.15) K 288.15K T
o
518.67 R
T
T
o
L1 h 248.15K T 20
446.67 R
a R1T 20
m
s
M
V
a
R1 287
J
kg K
100% P
o
101325 Pa
T
o
( 15 273.15) K 288.15K P
g
1.7051 kPa
P
v
P
g
P
v
P
g
3 Pa
P
a
P
o
P
v
4 Pa
P
a
R1 T
o
kg
m
3
1.12. Determine the speed of sound at sea level and
a. ISA condition
b. ISA+
c. ISA-
Sea level:
(Equ 1.34)
(Equ 1.34)
(Equ 1.34)
ISA 1.4 R1 287
J
kg K
T
o
( 15 273.15) K 288.15K T
o
518.67 R
a R1T o
m
s
T 12 K T
b
T
o
T 300.15K
a R1T b
m
s
T 18 K T
c
T
o
T 270.15K
a R1 T c
m
s
1.15. Fighter aircraft Mig-31 is able to fly with Mach 2.2 at an altitude of 60,000 ft altitude. What is the
dynamic pressure if flying at ISA+20 condition?
Atmospheric pressure is not a function of temperature, so ISA+20 has no effect and is not used.
1.16. The temperature at the summit of a mountain at ISA condition is -
o
C. What is the height of this
summit from sea level?
From Appendix A, the altitude is almost 3000 m.
1.17. The highest peak of Mount Everest has the elevation of 29,035 ft. Calculate temperature, pressure
and air density at this peak.
(Equ 1.21)
(Equ 1.36)
Sea level:
(Equ 1.6)
App B: Interpolation:
(Equ 1.23)
P
o
101325 Pa h 60000 ft M1 2.
P
60
P
o 0.2234 e
36089 ft h
20807 ft
7173.267 Pa
q 0.7 P 60
M
2
24303Pa q 24.303kPa
T1 ( 5 273.15) K 268.15K
h 29035 ft R1 287
J
kg K
L1 2
K
1000 ft
T
o
( 15 273.15) K 288.15K T
o
15°C
T1 T
o
L1 h 230.08K T1 43.07 °C
h 29035 ft
P1 2000
629.7 P
30000 h Find P1( ) 657.
P1 657.
lbf
ft
2
P1 3.149 10
4 Pa
P
R1 T1
kg
m
3
slug
ft
3
1.18. The fighter aircraft F-15C is capable of flying at the altitude of 12,000 m with the speed of 2443
km/hr. What is its speed in terms of Mach number at ISA condition?
Second layer:
(Equ 1.35)
h 12000 m ISA V1 2443
km
hr
V1 678.
m
s
1.4 R1 287
J
kg K
T
( 56 273 ) K T
390.6 R T
217K
a R1T 12
m
s
M
V
a
b. ISA+20 condition
c. ISA-20 condition.
1.21. Determine the air viscosity for an altitude of 10,000 m and ISA condition.
(Equ 1.6)
(Equ 1.34)
(Equ 1.35)
(Equ 1.6)
(Equ 1.34)
(Equ 1.35)
Sea level:
(Equ 1.6)
(Equ 1.26)
T 20 K T
b
T
o
T 308.15K
T
T
b
L1 h 268.15K T 20
482.67 R
a R1T 20
m
s
M
V
a
V1 M1 a 590.
m
s
V1 1321.
mile
hr
T 20 K T
c
T
o
T 268.15K
T
T
c
L1 h 228.15K T 20
410.67 R
a R1T 20
m
s
M
V
a
V1 M1 a 545
m
s
V1 1219.
mile
hr
h 10000 m ISA L1 6.
K
1000 m
T
o
( 15 273.15) K 288.15K T
o
518.67 R
T
T
o
L1 h 223.15K T 10
401.67 R
a 1.485 10
6
kg
m s K
1
2
b 110.4 K
a T 10
b
T
5
kg
m s
1.22. The Earth is rotating around itself once a day. Calculate its velocity at a city at Equator in terms
of Mach number, if speed of sound is assumed to be 340 m/sec. The average radius of Earth is about
6,400 km.
1.23. A Boeing 767 is flying at the altitude of 40,000 ft. How much air pressure must be increased by
its air pressure system in order to provide pressurized air of 0.8 atm for the passengers inside cabin
and cockpit?
At an altitude of 40,000 ft, from Appendix B, the pressure is:
To provide pressurized air of 0.8 atm for the passengers inside cabin:
Total Distance traveled; Circumference:
(Equ 1.35)
a 340
m
s
R
E
6400 km
t 24 hr t^ 8.64^10
4 s
X 2 R
E
40212.4 km
V
E
X
t
m
s
M
V
E
a
P
lbf
ft
2
P
18821.7Pa P 40
0.186atm
1.26. The humidity is 80% at a room with the volume of 160 m
3
in a summer day. Determine the water
vapor content (in kg) of this room at ISA+30 condition, if located at 2,000 ft altitude.
T T T Lh T C K
o
ISA o
15 0. 002 2000 30 41 314. 15 2000
P Pa
T
T
T
T
P
P
o o
ISA
o
ISA
101325 0. 9356 94800
- 9356
15 273. 15 30
15
256
256
2000
- 256
The saturation pressure Pg at 41
o
C can be found from table 1.2 as 7.8 kPa.
7. 8 0. 8 6. 24 v g
g
v P P
P
P
P P P kPa a v
94. 8 6. 24 88. 56
kg dry air
kgHO
P
P
a
v (^) 2
0438
56
24
0. 622 0. 622
kg
RT
PV
m m
a
v a
88
287 314. 15
56 160
0438
1.27. A transport aircraft is flying at an altitude of 25,000 ft, ISA condition. The pilot observes that the
ice has been formed on the leading edge of the wing. How much deicing system must increase the
temperature of the leading edge in order to melt the ice.
As long as the temperature of the ice is above 0
o
C, it will be melted.
Sea level:
(Equ 1.6)
h 25000 ft L1^2
K
1000 ft
T
o
( 15 273.15) K 288.15K T
o
15 °C
T
T
o
L1 h 238.15K T 12
35 °C
T
LE
o
C
1.28. The temperatures of a city in a summer day and a winter day are 115
o
F and 15
o
F respectively.
Find the ratio between air densities at a summer day to that of the winter day.
The ratio between air densities is 0.826.
1.29. The humidity of a city in a summer day (ISA+20) is 100%, and in a winter day (ISA-20) is 10%.
Find the ratio between air densities at a summer day to that of the winter day.
Sea level:
(Equ 1.23)
(Equ 1.23)
Sea level:
Summer:
Table 1.
(Equ 1.29) Winter:Table 1.2 (Equ 1.30)(Equ 1.23)(Equ 1.29)(Equ 1.30)(Equ 1.23)
T1 115°F T2 15°F R1^287
J
kg K
P
o
110325 Pa
P
o
R1 T1
kg
m
3
slug
ft
3
P
o
R1 T2
kg
m
3
slug
ft
3
R1 287
J
kg K
P
o
101325 Pa
T 20 100% T
o
( 15 273.15T) K 308.15K T
o
35°C
P
g
5.628 kPa
P
v
P
g
P
v
P
g
3 P Pa a
P
o
P
v
4 Pa
s
P
a
R1 T
o
kg
m
3
T 20 10%
T
o
( 15 273.15T) K 268.15K T
o
P 5 °C
g
0.4 kPa
P
v
P
g
P
v
P
g
P 40Pa a
P
o
P
v
5 Pa
w
P
a
R1 T
o
kg
m
3
s
w
1.31. What is the dynamic pressure when an aircraft is cruising at an altitude of 14,500 m and Mach
number of 0.6, ISA-12 flight condition?
1.32 An aircraft is cruising at an altitude of 40,000 m and a Mach number of 0.93, in ISA +
flight condition. Determine the dynamic pressure.
1.33 Determine the air viscosity for an altitude of 30,000 ft and ISA+20 condition.
(Equ 1.20)
(Equ 1.36)
P
o 101325 Pa h 14500 m M1 0.
P1 P
o 0.2234 e
11000 m h
6342 m
13035.357 Pa
q 0.7 P1 M
2 3284.9 Pa q 3.285kPa
1.34 Determine temperature, pressure, and air relative density in ISA + 30 condition and 50,000 ft
altitude.
Sea level:
(Equ 1.6)
(Equ 1.26)
50000 ft: (Equ 1.7)
(Equ 1.21)
(Equ 1.23)
