Question

low-velocity steam (with negligible kinetic energy) enters an adiabatic nozzle at 300 C and 3 MPa. the steam leaves the nozzle at 2 MPa with a velocity of 400 m/s. the mass flow rate is 0.4 kg/s. determine the quality or temperature (C) of the steam leaving the nozzle and the exit area of the nozzle in mm^2.

Answer 1

Answer:

the quality of the steam = 1.05

the temperature at  $P_2 = 2 \ MPa$ =   $380.733^0 \ C$

the exit area of the nozzle = $146.33 \ mm^2$

Explanation:

Given that:

the inlet temperature of steam $T_1$ = $300^0 \ C$

Inlet pressure of steam $P_1= 3 \ MPa$

Initial Velocity $v_1$ at the inlet = 0 m/s

Exit pressure of the steam $P_2 = 2 \ MPa$

Exit velocity of steam $v_2 \ = 400 \ m/s$

Mass flow rate m = 0.4 kg/s

Now, from steam tables at $T_1$ = $300^0 \ C$ and $P_1= 3 \ MPa$

$h_1 = 2992. 35 \ kJ/kg\\s_1 = 6.53535 \ kJ/kg$

At   $P_2 = 2 \ MPa$

$sf_2 = 2.4474 kJ/kgK\\s_g = 6.3409 \ kJ/kg$

To determine the condition of the steam at exit ;

$s_1 = s_2$

therefore,

$6.53535 = s_{f2} +x_2sg-sf_2$

$6.53535 = 2.4474 +x_2(6.3409-2.4474)$

$x_2 = 1.05$

Thus , the quality of the steam = 1.05

However, By using the energy balance equation to determine the temperature of the steam; we have:

$h_1 + \frac{v^2_1}{2}= h_2 + \frac{v^2_2}{2}$

$h_1-h_2 = \frac{v^2_2}{2}$

$h_2 = h_1 - \frac{v^2_2}{2}$

$h_2 = 2992.35 - \frac{250^2}{2000}$

$h_2 = 2912.35 \ kJ/kg$

From steam tables ; at enthalpy $h_2$ and $P_2 = 2 \ MPa$; the corresponding temperature $T_2 = 380.733^0 \ C$

Thus, the temperature at  $P_2 = 2 \ MPa$ =   $380.733^0 \ C$

Finally, to calculate the exit area of the nozzle in mm²

we use the mass flow rate relation:

$m = \frac{A_2v_2}{V_2}$

Making A the subject of the formula; we have:

$A_2 = \frac{mv_2}{V_2}$

From the superheated steam tables at pressure $P_2 = 2 \ MPa$; specific volume $v_2 = 0.14633 \ m^3/kg$

We have:

$A_2 = \frac{mv_2}{V_2}$

$A_2 = \frac{0.4*0.14633}{400}$

$A_2 = 1.4633*10^{-4} m^2$

$A_2 = 146.33 \ mm^2$

Thus, the exit area of the nozzle = $146.33 \ mm^2$