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How to calculate the Mach Number?

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  • Last Updated : 11 Apr, 2022

Mach number is used to compare the speed of any object with the speed of sound. It is a parameter that is used to denote the speed of a flying object in an air medium. In fluid dynamics, it is a very important quantity. Mach number is mainly used to know the proper idea of the motion of airplanes and rockets. Mach number is a dimensional less quantity expressing the ratio of the velocity of an object in a medium to speed of sound.

Mach Number

When an object is moving in the sky the air particles that are nearer to the body of the object get disturbed. If the object has a higher velocity it provides a compressing effect on the surrounding air. Therefore, by Mach number, we can say whether the fluid is compressible or not. Mach number has no dimensions (units); because it is the ratio of two velocities that are having the same dimensions.

For example, the design of aircraft is done based on the Mach number to get the best results and we know that sound moves in the form of waves. In an air medium, the movement of these sound waves depends on various factors like temperature and pressure of the air. We know that when a fluid flows over a solid surface, a thin membrane of fluid sticks to the surface and acquires the same velocity as the surface.

Classification of Mach Number

According to the variation of Mach number, fluid flow is classified.

  1. Subsonic: The Mach number remains below 0.8 in subsonic flow. Commercial airplanes flowing in this range have a round nose and leading edges.
  2. Sonic: The Mach number is one in sonic flow. In these types of flow, the velocity of an object is equal to the velocity of sound in the local medium.
  3. Supersonic: If an object is flowing with greater velocity than the sound it is said to have supersonic velocity. It has a Mach number greater than one. Airplanes have proper designs to undergo supersonic speed.
  4. Hypersonic: If the speed of an object is higher than the speed of sound, then the flow is called hypersonic. In this type of flow, the Mach number is five to ten.

How to calculate the Mach Number?

Solution:

The Mach number is represented with the symbol ‘M’.

Mach Number= Speed of object / Speed of sound

or

M = v/c

where v is the speed of the object in a medium with respect to some boundary either internal or external, c is the speed of sound in the local medium and the speed of sound varies with the thermodynamic temperature as:

C = √KRT

where K is the ratio of specific heat of a gas at a constant pressure to heat at a constant volume (1.4 for air), R is the specific gas constant for air and T is the static air temperature.

Sample Questions

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.

Answer: 

Given that,

The speed of an object, v is 480 m/s.

The speed of light, c is 350 m/s at a normal temperature of 30 degrees.

Formula of mach number is:

M = v/c

= 480/350

= 1.37

Therefore, Mach number is 1.37.

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.

Answer:

Given that,

The Temperature, t = 20˚ C = 20 + 273 = 293 K

The Speed, v = 3600 km/hr = (3600×1000)/(60×60) =1000m/s 

K = 1.4

R = 287 J/kg K.

The velocity of sound is,

C = √KRT

=√(1.4×287×293)

= 343.11 m/s

Therefore, the Mach Number is 

M = v/c 

= 1000/343.11 

= 2.91

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?

Answer:

Given that,

v= 500 m/s

Mach number, M = 2

Formula of Mach number is:

M = v/c

2 = 500/c

or

c = 500/2

= 250 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?

Answer:

Given that,

The speed of the aircraft, v is 5600 m/s.

The speed of sound, c is 800 m/s.

Formula of mach number is:

M = v/c

= 5600/800

= 7

Therefore, Mach number is 7.

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?

Answer:

Given that, 

Velocity v = 560 m/s 

Temperature T = 40°C

T = (40 + 273)K = 313 K.

Local speed sound in air in an ideal gas c = √kRT

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 = v/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.

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