Training Aerospace Engineering Atmosphere & Flight Performance
4 / 5

Atmosphere & Flight Performance

24 min Aerospace Engineering

Atmosphere & Flight Performance

Aircraft performance depends on atmospheric conditions, which change with altitude. The International Standard Atmosphere (ISA) model provides the baseline.

ISA Sea-Level Conditions

$T_0 = 288.15$ K, $P_0 = 101{,}325$ Pa, $\rho_0 = 1.225$ kg/m³. Temperature lapse rate in the troposphere: $\lambda = 6.5$ K/km.

Temperature & Pressure vs. Altitude

In the troposphere ($h < 11$ km):

$$T = T_0 - \lambda h \qquad \frac{P}{P_0} = \left(\frac{T}{T_0}\right)^{g/(R\lambda)} \approx \left(1 - \frac{\lambda h}{T_0}\right)^{5.256}$$

Mach Number

$$M = \frac{v}{a} \qquad a = \sqrt{\gamma R T}$$

where $a$ = speed of sound, $\gamma = 1.4$ for air, $R = 287$ J/(kg·K).

Example 1

Find the temperature, pressure, and density at $h = 5$ km altitude (ISA).

$T = 288.15 - 6.5(5) = 255.65$ K

$P/P_0 = (255.65/288.15)^{5.256} = 0.8873^{5.256} = 0.5334$

$P = 0.5334 \times 101{,}325 = 54{,}027$ Pa

$\rho = P/(RT) = 54{,}027/(287 \times 255.65) = 0.736$ kg/m³

Example 2

Find the speed of sound at 10 km altitude.

$T = 288.15 - 6.5(10) = 223.15$ K

$$a = \sqrt{1.4 \times 287 \times 223.15} = \sqrt{89{,}571} = 299.3 \text{ m/s}$$

Example 3

An aircraft flies at 250 m/s at 8 km altitude. Find its Mach number.

$T = 288.15 - 52 = 236.15$ K

$a = \sqrt{1.4 \times 287 \times 236.15} = 308$ m/s

$$M = 250/308 = 0.812$$

This is transonic flight (approaching Mach 1).

Practice Problems

1. Find the ISA temperature at $h = 3$ km.
2. Find the speed of sound at sea level.
3. An aircraft flies at Mach 0.85 at 11 km ($T = 216.65$ K). Find its true airspeed.
4. Find the pressure ratio $P/P_0$ at 8 km.
5. Air density at 6 km using ISA.
6. Range equation: $R = (v/c)(L/D)\ln(W_0/W_1)$ where $c$ = specific fuel consumption. If $v = 250$ m/s, $c = 0.06$ kg/(N·s), $L/D = 18$, $W_0/W_1 = 1.3$. Find range.
7. Endurance: $E = (1/c)(L/D)\ln(W_0/W_1)$. Find endurance using values from #6.
8. Rate of climb: $RC = (T - D)v/W$. If excess thrust $= 20$ kN, $v = 80$ m/s, $W = 100$ kN. Find RC.
9. Service ceiling: altitude where $RC = 0.5$ m/s. If RC decreases linearly from 10 m/s at sea level, estimate the ceiling.
10. True airspeed vs. indicated airspeed: $v_{\text{true}} = v_{\text{IAS}} \sqrt{\rho_0/\rho}$. At 5 km, $\rho = 0.736$. If IAS = 200 knots, find TAS.
11. Takeoff distance: $s = v_{\text{TO}}^2/(2a)$ where $a = (T-D-\mu W)/m$. Estimate for $v_{\text{TO}} = 70$ m/s, $a = 2$ m/s².
12. In the stratosphere ($h > 11$ km), temperature is constant at 216.65 K. Find the speed of sound there.
Show Answer Key

1. $T = 288.15 - 19.5 = 268.65$ K $= -4.5°$C

2. $a = \sqrt{1.4 \times 287 \times 288.15} = 340.3$ m/s

3. $a = \sqrt{1.4 \times 287 \times 216.65} = 295.1$ m/s; $v = 0.85 \times 295.1 = 250.8$ m/s

4. $T = 236.15$ K; $P/P_0 = (236.15/288.15)^{5.256} = 0.3519$

5. $T = 249.15$ K; $P = 0.4660 \times 101{,}325 = 47{,}217$ Pa; $\rho = 47{,}217/(287 \times 249.15) = 0.660$ kg/m³

6. $R = (250/0.06)(18)\ln 1.3 = 4{,}167 \times 18 \times 0.2624 = 19{,}677$ m $\approx 19.7$ km (per unit consistent; check units)

7. $E = (1/0.06)(18)(0.2624) = 78.7$ s (units depend on $c$ definition)

8. $RC = 20{,}000 \times 80/100{,}000 = 16$ m/s

9. Linear decrease: ceiling $\approx$ where $RC = 0.5$; fraction $= 1 - 0.5/10 = 0.95$; ceiling depends on the reference altitude

10. $v_{\text{true}} = 200\sqrt{1.225/0.736} = 200 \times 1.290 = 258$ knots

11. $s = 70^2/(2 \times 2) = 1{,}225$ m

12. $a = \sqrt{1.4 \times 287 \times 216.65} = 295.1$ m/s

Rocket Propulsion — Thrust & Δv Placement Test Practice — Aerospace Engineering