Example Problem with Complete Solution

5C-1 : Cross-Sectional Area Requirement for an Adiabatic Nozzle 6 pts
Steam enters a nozzle operating at steady-state with P1 = 40 bar, T1 = 400oC and a velocity of 10 m/s.  The steam flows through the nozzle with negligible heat transfer and no significant change in potential energy.  At the exit, P2 = 15 bar and the velocity is 665 m/s.  The mass flow rate is 2 kg/s.  Determine the exit area of the nozzle in m2.
Read : Figure 1
The key here is that we know both the mass flow rate and velocity of the effluent stream.  If we can determine the specific volume of the effluent, we can determine the cross-sectional area for flow at the effluent, A2.  We are given the value of one intensive variable for the effluent, P2, but we need to know another in order to completely determine the state of the effluent.  Once know the sate of the effluent, we can use the steam tables to determine the specific volume and then the cross-sectional area.  We must apply the steady-state form of 1st Law for open systems to this process.  If we assume that heat transfer and changes in potential energy are negligible and that no shaft work occurs, we can solve for the specific enthalpy of the effluent and thereby fix the state of the system.  This allows us to complete the problem.
Given: P1 4000 kPa Find:
T1 400 oC A2 ??? m2
v1 10 m/s
P2 1500 kPa
v2 665 m/s
mdot 2 kg/s
Diagrams : Figure 2
Equation 1
Assumptions:
1 - The nozzle operates at steady-state.
2 - Heat transfer is negligible.
3 - No shaft work crosses the system boundary.
4 - The change in the potential energy of the fluid from the inlet to the outlet is negligible.
Equations / Data / Solve :
Let's begin by writing the steady-state form of the 1st Law for open systems.
Equation 2
Eqn 1
Based on the assumptions listed above, we can simplify Eqn 1 as follows :
Equation 3
Eqn 2
The only unknown in Eqn 2 is H2 because we can look up H1 and the velocities are both given.
So, let's look up H1 and solve Eqn 2 for H2 :
Equation 4
Eqn 3 H1 3214.5 kJ/kg
H2 2993.4 kJ/kg
We could use H2 and P2 to determine T2 using the Steam Tables, but we are more interested in V2 because :
Equation 5
Eqn 4
or : Equation 6
Eqn 5
Once we know the specific volume at state 2, we can use Eqn 5 to determine the cross-sectional area of the effluent pipe.
Interpolating on the Steam Tables at 1.5 MPa :
T (oC) H (kJ/kg) V (m3/kg)
250 2923.9 0.15201
T2 2993.4 V2 T2 280 oC
300 3038.2 0.16971 V2 0.16278 m3/kg
Now, plug V2 into Eqn 5 : A2 4.896E-04 m2
Verify : None of the assumptions made in this problem solution can be verified.
Answers : A2 4.90E-04 m2