A commercial airplane fl ies at 540 mi/h at a standard altitude of 30,000 ft. What is its Mach number?
Estimate ρ and cp of steam at 100 lbf/in 2 and 400 8 F, in English units, ( a ) by the perfect-gas approximation and ( b ) by the ASME Steam Tables [23]
The Correct Answer and Explanation is :
Part 1: Mach Number of a Commercial Airplane
To calculate the Mach number of an airplane, we use the formula:
[
M = \frac{v}{a}
]
Where:
- ( M ) is the Mach number
- ( v ) is the speed of the airplane (540 mi/h)
- ( a ) is the speed of sound at the given altitude (30,000 ft)
Step 1: Determine the speed of sound at 30,000 ft
The speed of sound depends on the temperature of the air. At 30,000 feet, the air temperature is approximately -40°C or -40°F. In the International Standard Atmosphere (ISA) model, the speed of sound at this altitude is roughly 573.5 mph.
Step 2: Calculate Mach number
Now, we can plug the given values into the Mach number equation:
[
M = \frac{540 \, \text{mi/h}}{573.5 \, \text{mi/h}} = 0.941
]
So, the Mach number of the airplane is approximately 0.94.
Part 2: Estimating the Properties of Steam
(a) Perfect-Gas Approximation
To estimate the properties of steam (water vapor) using the perfect-gas approximation, we use the ideal gas law:
[
PV = nRT \quad \text{or} \quad \rho = \frac{P}{RT}
]
Where:
- ( P ) is the pressure in lbf/in²
- ( T ) is the temperature in Rankine (R)
- ( R ) is the specific gas constant for steam (( R = 0.594 \, \text{ft} \cdot \text{lbf} / (\text{lbm} \cdot \text{R}) ))
Given:
- ( P = 100 \, \text{lbf/in}^2 )
- ( T = 400 \, \text{°F} = 400 + 459.67 = 859.67 \, \text{°R} )
Now, we can calculate the density ( \rho ) of the steam:
[
\rho = \frac{P}{R T} = \frac{100}{0.594 \times 859.67} \approx 0.191 \, \text{lbm/ft}^3
]
Next, for the specific heat capacity at constant pressure (( c_p )), we approximate ( c_p ) of steam at this temperature using the general value:
[
c_p \approx 1.996 \, \text{Btu/lbm} \cdot \text{°F}
]
(b) Using the ASME Steam Tables
For more accurate values, we refer to the ASME Steam Tables. From the tables at a pressure of 100 lbf/in² and a temperature of 400°F:
- The density ( \rho ) is approximately 0.174 lbm/ft³
- The specific heat capacity ( c_p ) is approximately 1.97 Btu/lbm·°F
Explanation
The perfect-gas approximation gives an estimate of the steam’s properties under ideal conditions. However, real steam deviates slightly from perfect gas behavior, especially at higher pressures or temperatures, which is why values from the ASME Steam Tables are typically used for more precise engineering calculations. The ideal gas approach assumes that the gas molecules do not interact with each other, whereas the steam tables account for the real behavior of steam, particularly at superheated conditions.
In conclusion, while the perfect gas approximation provides a reasonable estimate, using the ASME Steam Tables yields more accurate results that are essential for high-precision engineering applications.