A jet-propelled aircraft is flying at a speed of 1100 km/hour at

SOLUTION

Concept:

Dimensionless Numbers:

These are the numbers which are obtained by dividing the inertia force by viscous force or gravity force or pressure force or surface tension force or elastic force.

As this is the ratio of one force to another force, it will be a dimensionless number. These are also called Non-dimensional Parameters.

The following are the important dimensionless numbers –

1. Reynold’s number

2. Froude’s Number

3. Euler’s Number

4. Weber’s Number

5. Mach’s Number

Mach Number:

It is the ratio of inertia force to the elastic force of flowing fluid.

\(M = \sqrt {\frac{{{F_i}}}{{{F_e}}}} = \sqrt {\frac{{\left( {mass × acceleration} \right)}}{{Elastic\;Stress × Area}}} = \sqrt {\frac{{ρ QV}}{{K.A}}} = \sqrt {\frac{{{V^2}}}{{\frac{K}{ρ }}}} = \frac{V}{{\sqrt {\frac{K}{ρ }} }} = \frac{V}{C}\)

Here, \(C = \sqrt {\frac{K}{ρ }}\)

where

Elastic stress (K) = P × r

P = Adiabatic pressure = ρRT (T is absolute temeperature in Kelvin)

r = Adiabatic index

\(C = \sqrt {\frac{P\times r}{ρ }} =\sqrt {\frac{\rho\times R\times T\times r}{ρ }} = \sqrt {rRT}\)

Calculation:

Given,

Speed of aircraft, V = 1100km/h = 305.55 m/s

Temperature, T = 273+20 = 293°K

Adiabatic Index, r = 1.4 (For diatomic air)

Gas Constant, R = 287 J/kg.K

Since, jet propulsion is an adiabatic process.

Hence, For adiabatic process          ---(1) (P = ρRT)

\(C = \sqrt {\frac{{rρ RT}}{ρ }} = \sqrt {rRT} = \sqrt {1.4 × 287 × \left( {293} \right)} = 343.113\;m/s\)

 \(\therefore M = \frac{V}{C} = \frac{{305.55}}{{343.113}} = 0.89\)