Question

An ideal gas mixture with k = 1.31 and a molecular weight of 23 is supplied to a converging nozzle at p0 = 5 bar, T0 = 700 K, which discharges into a region where the pressure is 1 bar. The exit area is 50 cm2 . For steady isentropic flow through the nozzle, determine:

(a) the exit temperature of the gas, in K.
(b) the exit velocity of the gas, in m/s.
(c) the mass flow rate, in kg/s.

Answer 1

Given that

k=1.31

Po=5 bar

To=700 K

P₁=1 bar

T₁=?

A₁=50 cm²

We know that

`(T_1)/(T_o)=\left((P_1)/(P_o)\right)^{(k-1)/(k)}`

By putting the values

`(T_1)/(T_o)=\left((P_1)/(P_o)\right)^{(k-1)/(k)}`

`(T_1)/(700)=\left((1)/(5)\right)^{(1.3-1)/(1.3)}`

T₁=482.3 K

Now from first law for open system

`h_o+(V_o^2)/(2000)=h_1+(V_1^2)/(2000)`

`C_p=R(k)/(k-1)\ KJ/kg.k`

`C_p=(8.314)/(23)* (1.31)/(1.31-1)\ KJ/kg.k`

Cp=1.527 KJ/kg.k

h= Cp T

`h_o+(V_o^2)/(2000)=h_1+(V_1^2)/(2000)`

`1.527* 700+(0^2)/(2000)=1.527* 482.3+(V_1^2)/(2000)`

`(V_1^2)/(2000)=1.527* 700-1.527* 482.3`

V₁=815.37 m/s

Mass flow rate m=ρ A₁V₁

`\rho=(P)/(RT)`

`\rho=(100)/((8.314)/(23)* 482.3)\ kg/m^3`

ρ=0.57 kg/m³

m=ρ A₁V₁

m= 0.57 x 50 x 10⁻⁴ x 815.37 m/s

m=2.32 kg/s