Mach Number Formula

Mach number is calculated by finding the ratio of the speed of an object to the speed of sound in the surrounding medium. Mach number is a unitless (dimensionless quantity).

The formula for calculating the Mach number is M = u/c, where 'M' is Mach number, 'u' is the velocity of an object (in meters per second, feet per second, etc.) and 'c' is the the speed of sound in the medium (in meters per second, feet per second, etc.

What is the Mach number?

Mach number is a fundamental concept in aerodynamics that compares the speed of an object to the speed of sound in the surrounding medium.

• It is a dimensionless quantity denoted by 'M' and calculated as the ratio of the object's velocity to the speed of sound.
• The Mach number classification includes subsonic (M < 1), sonic (M = 1), supersonic (M > 1), and hypersonic (M >> 1) speeds.

Formula,

The Mach number (M) is calculated using the formula:

M = u / c

where,

• M is the Mach number,
• u is the velocity of the object,
• c is the speed of sound in the medium.

⇒The speed of sound is referred to as Mach 1.

⇒Mach 0.75 represents 75% of the speed of sound, classified as subsonic.

⇒Mach 1.65 is 65% faster than the speed of sound, classified as supersonic.

⇒The Mach number is influenced by the local speed of sound, which depends on surrounding conditions like temperature and pressure.

⇒The flow can be considered incompressible using the Mach number.

⇒The medium could be a liquid or gas, and flow can occur with either a stable or traveling boundary.

⇒Both the medium and boundary can travel at specific speeds, but the velocity relative to each other is important.

⇒The medium may flow through devices like wind tunnels or be immersed in another medium.

⇒The Mach number is a dimensionless ratio of two speeds, making it not associated with a specific unit.

⇒The term "Mach number" is named after Austrian philosopher and physicist Ernst Mach.

⇒As a dimensionless quantity, the Mach number is written after the term "Mach," such as Mach 4, rather than 4 Mach.

Key Points

• Subsonic (M < 1): Object moves slower than the speed of sound.
• Sonic (M = 1): Object's speed equals the speed of sound.
• Supersonic (M > 1): Object exceeds the speed of sound.
• Hypersonic (M >> 1): Object travels at extremely high speeds.

Mach Numbers Classification

⇒Mach numbers categorize speeds of objects relative to the speed of sound in a particular medium.

⇒Speed below the speed of sound is termed as subsonic whereas speed above the speed of sound is termed as supersonic, although scientists practising aerodynamics most of the time use these terms to describe a particular range of Mach values.

The various regimes of Mach values are:

Subsonic (Mach number < 1):

• Speeds slower than the speed of sound.
• Common for most everyday objects and vehicles.
• Example: Passenger airplanes during cruising.

Transonic (Mach number ≈ 1):

• Speeds near the speed of sound.
• Objects experience both subsonic and supersonic airflow.
• Example: Some fighter jets during maneuvers.

Supersonic (Mach number > 1):

• Speeds faster than the speed of sound.
• Shockwaves form due to compressed air in front of the object.
• Example: Concorde passenger aircraft.

Sonic ( Mach number =1)

• Speed equal to Mach 1, the speed of sound.
• No shockwaves ahead of the object, but shockwaves form at the surface.
• Example: A jet reaching the speed of sound in level flight.

Hypersonic (Mach number > 5):

• Extremely high speeds, several times faster than the speed of sound.
• Significant aerodynamic heating due to air compression.
• Example: Spacecraft during re-entry into the Earth's atmosphere.

Hypervelocity (Mach number > 10):

• Speeds over Mach 10, more than 10 times the speed of sound.
• Intense aerodynamic heating due to air compression and friction
• Example: Spacecraft re-entering Earth's atmosphere at hypersonic speeds.

Mach regimes are classified based on the Mach number, ranging from subsonic (Mach < 1) to hypersonic (Mach > 5), indicating different speed characteristics relative to the speed of sound :

Solved Examples

Question 1: If the speed of an object is given as 480 m/s at a temperature of 30 degrees, then calculate the Mach number for the object.

Given:
Object speed (u) = 480 m/s
Temperature (T) = 30°C = 303 K (conversion for temperature unit)
Step 1: Calculate the speed of sound (c) at 30°C
The speed of sound in air varies with temperature. We can use the following formula:
c = sqrt(k * R * T)
Where :
k = specific heat ratio (1.4 for air)
R = gas constant (287 J/kg*K)
T = temperature in Kelvin
Calculation:
c = sqrt(1.4 ×287 × 303) ≈ 347 m/s
Step 2: Calculate the Mach number (M)
M = u / c
Calculation:
M = 480 m/s / 347 m/s ≈ 1.38
Therefore, the Mach number of the object at 30°C is approximately 1.38.

Question 2: An Airplane travels in air at 20°C at a speed of 2400 km/hr. Find the Mach number. Take k=1.4 and R=287 J/Kg K.

Given:
Airplane speed (u) = 2400 km/hr
Temperature (T) = 20°C (needs conversion to Kelvin)
Specific heat ratio (k) = 1.4
Gas constant (R) = 287 J/kg*K
Step 1: Convert speed from km/hr to m/s
u = 2400 km/hr × (1000 m/km) × (1 hr/3600 s) = 667 m/s
Step 2: Convert temperature from Celsius to Kelvin
T = 20°C + 273.15 = 293.15 K
Step 3: Calculate the speed of sound (c) using the formula:
c = sqrt(k * R * T)
c = sqrt(1.4 × 287 J/kg*K × 293.15 K) ≈ 343.11 m/s
Step 4: Calculate the Mach number (M):
M = u/ c
M = 667 m/s / 343.11 m/s ≈ 1.94

Question 3: If the speed of an object is 500 m/s at a temperature of 10 degrees and the Mach number for the object is 2. Then calculate the velocity of sound in air?

Given:
Object speed (u) = 500 m/s
Mach number (M) = 2
Formula:
M = u / c
Rearranging for c:
c = u / M
Calculation:
c = 500 m/s / 2 = 250 m/s
Therefore, the speed of sound in air in this scenario is approximately 250 m/s.
To properly calculate the speed of sound in air at 10°C, we should use the formula for the speed of sound, which depends on the temperature:
c=√k⋅R⋅T
where,
k=1.4k (specific heat ratio for air)
R=287J/kg\cdotpK (specific gas constant for air)
T=10^∘C=283.15K (temperature in Kelvin)
c=√1.4 ×287×283.15​≈337.18m/s
So, the speed of sound at 10°C is approximately 337.18 m/s,

Question 4: If the aircraft is moving with a speed of 5600 m/s and its velocity of sound in air is 800 m/s. Calculate the Mach number for that aircraft?

Given:
Aircraft speed (u) = 5600 m/s
Speed of sound (c) = 800 m/s
Formula:
Mach number (M) = u / c
Calculation:
M = 5600 m/s / 800 m/s = 7
Therefore, the Mach number of the aircraft is 7. This indicates that the aircraft is traveling seven times faster than the speed of sound, placing it firmly in the hypersonic regime.

Question 5: A pipeline transports methane gas (CH₄) at 40°C with a velocity of 560 m/s. What is the Mach number of methane gas and what type of flow does it have?

Given that, 
Velocity u = 560 m/s 
Temperature T = 40°C → T=313 K
⇒T = (40 + 273)K = 313 K.
where k = specific heat of methane gas = 1.264
R= gas constant of methane 
= 518.25 J/ Kg K 
Therefore c= √(1.264)(518.25)(313)
c = 452.81 m/s.
Mach number of methane gas 
M = u/c 
= 560/452.81
= 1.23
Therefore Mach number of methane gas is 1.23
Mach number of methane > 1
Therefore the flow is supersonic.

Conclusion

Mach numbers provide a key framework for understanding the velocity of objects relative to the speed of sound. This classification system, spanning from subsonic to trans hypersonic, guides the design and exploration of vehicles across various industries, from aviation to space travel. As technology evolves, Mach numbers will remain a cornerstone for pushing the boundaries of speed and advancing aerospace engineering.

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