Air flows from a very large chamber with constant Pc 500 kP

Air flows from a very large chamber with constant Pc = 500 kPa and constant Tc = 500 K through a converging nozzle with an exit area of 50 cm². Assume isentropic, steady flow in the nozzle. Determine the mass flow flow rate through the nozzle and exit plane static pressure for back pressures of 0, 125, 250, and 375 kPa.

Solution

Convergent Nozzle probem:

Given Data,

P_c = 500 kPa

T_c= 500 K

A_e = 50 cm²

P_b = 0,125,250,375 kPa

Steady, Isentropic flow

Find m (mass flow rate) and P_e (Static Pressure at exit plane)

Solution:

As the chamber is very large we an assume the pressure and temperature will not change in the chamber

and velocity at the inlet of the converging nozzle is very small.

Critical pressure ratio for air = 0.5283

i.e. P^*/P_o = 0.5283. Where P_o is the stagbation pressure at the inlet and P^* is the critical pressure at the exit of nozzle.

P_o = P_c (As velocity is small)

So. P^* = 0.5283 x 500 = 264.15 kPa

P_e = Exit Pressure = P_b (Back Pressure) if P_b >= P^*

= P^* if P_b<P^*

So, for the given back pressures (0,125,250,375 kPa)

P_e = 264.15 kPa for P_b = 0 kPa,125 kPa, 250 kPa

= 375 kPa for P_b = 375 kPa

So, T_e = Exit temperature = T_c x (P_e/P_c)^(k-1)/k = 500 x (264.15/500)^(1.4-1)/1.4 = 416.64 K

[for P_b = 0 kPa,125 kPa, 250 kPa]

And T_e = 500 x (375/500)^(1.4-1)/1.4 = 460.5 K [for Pb = 375 kPa.]

Exit velocity, V_e = Ma_t x (kRT_e)^0.5

Where Ma_t is the Mach Number at the throat = 2.2495 when Pb = 0 kPa

= 1.402 wheb pb = 125 kPa

= 1.0386 when Pb = 250 kPa

= 0.6871 when Pb = 375 kPa

V_e= 2.2495 x (1.4 x 287 x 416.64)^0.5 = 920.43 m/s  [for P_b = 0 kPa]

V_e = 1.402 x (1.4 x 287 x 416.64)^0.5 = 573.65 m/s [for P_b = 125 kPa]

V_e = 1.0386 x (1.4 x 287 x 416.64)^0.5 = 424.96 m/s [for P_b = 250 kPa]

V_e = 0.6871 x (1.4 x 287 x 460.5)^0.5 = 281.14 m/s  [for P_b = 375 kPa.]

Exit air density, r_e = P_e/RT_e = 264.15/(0.287 x 416.64) = 2.21 kg/m³.  [for Pb = 0 kPa,125 kPa, 250 kPa]

r_e = 375/(0.287 x 460.5) = 2.84 kg/m³. [for Pb = 375 kPa.]

Mass flow rate through the nozzle, m_e = r_eA_eV_e = 2.21 x 50 x 10^(-4) x 920.43 = 10.17 kg/s

   = 2.21 x 50 x 10^(-4) x 573.65 = 6.33 kg/s

= 2.21 x 50 x 10^(-4) x 424.96 = 4.69 kg/s

= 2.84 x 50 x 10^(-4) x 281.14 = 3.99 kg/s