A perfect gas expands in a frictionless nozzle from stagnation conditions p 0 = 4 MPa, T0 = 2500 K to ambient pressure 0.1 MPa. Given that the expansion is isentropic, determine the following conditions at the final pressure: (1) velocity, (2) Mach number, (3) temperature, and (4) area per unit mass flow. How does the final flow area compare with the throat area for a PROBLEMS 89 given mass flow? The specific heat ratio y is 1.4, and the molecular weight M is 30.

The answers to the question are as follows

(1) velocity = 164.108 m/s

(2) Mach number, Ma = 3.05696

(3) temperature  = 7172.51 K

(4) Area per unit mass flow = 0.1254 m²

Explanation:

(a) To solve the question, we list out the variables as follows

P₀ = 4 MPa,

T₀ = 2500 K

P₂ = 0.1 MPa

γ = 1.4

Molecular weight = 30

R = 0.287 kJ/kg

= 1.005 kJ/kg

Also

or 40 =  or c = 164.108 m/s

(b) The mach number can be calculated from

 That is or

Ma = 3.05696

(c) Since the process is isentropic, we have

or T = 2500×

= 7172.51 K

(d)  From A = mv₂/C₂

The area for unit mass flow rate =

v = RT/P = 0.287×7172.5×1000/100000 = 20.59 m³/kg

(Note: The 1000 product equation above is because the units were changed from kJ/kg to J/kg and so also for the 100000 in the denominator, that represents the pressure unit change from MPa to Pa)

and A = mv/C which for unit mass flow rate =

v/C Therefore A = 1kg/s ×20.59 m³/kg /164.108 m/s = 0.1254 m²

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