# Example Problem with Complete Solution

5C-1 : Cross-Sectional Area Requirement for an Adiabatic Nozzle 6 pts
Steam at 3.5 MPa and 350oC flows steadily into an adiabatic nozzle at a mass flow rate of 3.15 kg/s. The steam leaves the nozzle at 500 kPa with a velocity of 571 m/s.
If the inlet velocity of the steam is 5 m/s, determine the exit diameter of the nozzle in cm.

Read : 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 state 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 3500 kPa P2 500 kPa
T1 350 oC v2 571 m/s
v1 5 m/s mdot 3.15 kg/s
Find: D2 ??? cm
Diagram:
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.
Eqn 1
Based on the assumptions listed above, we can simplify Eqn 1 as follows :
Eqn 2
The only unknown in Eqn 2 is H2 because we can lookup H1 and the velocities are both given.
So, let's look up H1 and solve Eqn 2 for H2 :
Eqn 3 H1 3104.8 kJ/kg
H2 2941.8 kJ/kg
We could use H2 and P2 to determine T2 using the Steam Tables, but we are more interested in V2 because :
Eqn 4 or : 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 500 kPa :
T (oC) H (kJ/kg) V (m3/kg)
200 2855.8 0.42503
T2 2941.8 V2 T2 240.9 oC
250 2961.0 0.47443 V2 0.46541 m3/kg
Now, plug V2 into Eqn 5 : A2 2.568E-03 m2
Determine D2 from A2 : Eqn 6
Solving for D2 yields: Eqn 7
Plug values into Eqn 7: D2 0.05718 m
5.718 cm
Verify: None of the assumptions made in this problem solution can be verified.
Answers : D2 5.72 cm2