Example Problem with Complete Solution

5E-1 : Charging an Evacuated Vessel From a Steam Line 6 pts
Steam at a pressure of 1.4 MPa and 300oC is flowing through a pipe.  Connected to this pipe through a valve is an evacuated tank.  The valve is opened and the tank fills with steam until the pressure in the tank is also 1.4 MPa.  Then, the valve is closed.  The process is adiabatic and changes in kinetic and potential energies are negligible.  Determine the final temperature of the steam in the tank.
               
Read : Figure 1
Use the  contents of the tank as the system.  Most of the key assumptions for this problem are given in the problem statement.  One additional assumption is that no shaft work crosses the boundary of the system during the process.  Another crucial assumption is that this is a uniform flow, uniform state process.  These assumptions allow us to simplify the 1st Law dramatically.  We can use a transient mass balance to show that the mass in the tank in the final state is equal to the mass that was added to the tank.  This probably seems obvious since the tank was initially empty.  We can determine the specific enthalpy of the steam entering the tank from the steam tables because we know both Tin and Pin.  We will be able to determine the internal energy of the steam in the tank in the final state from the 1st Law.  Then, we can use the Steam Tables to determine the value of T2 using U2 and the given value of P2.
Given : Pin 1400 kPa Q 0 kJ
Tin 300 oC WS 0 kJ
P2 1400 kPa
Find : T2 ??? oC
Figure 2
Diagram : Initial :
Final : Equation 1
Assumptions :
1 - Although this is a transient process, it can be analyzed as a uniform flow, uniform state problem because the properties of the steam entering the tank are constant.
2 - Changes in kinetic and potential energies are negligible.
3 - No shaft work crosses the boundary of the system, which consists of the content of the tank.
4 - The process is adiabatic.
Equations / Data / Solve :
The integral form of the transient energy balance equation for a single-input, single-output system in which kinetic and potential energies are negligible is :
Equation 2
Eqn 1
In our process, no shaft work occurs, there is no mass leaving the system and there is no mass inside the system initially, so Eqn 1 can be simplified and solved for Q :
Equation 3
Eqn 2
The integral form of the transient mass balance on the tank is :
Equation 4
Eqn 3
Eqn 3 can be simplified because there is no mass leaving the sytstem and there is no mass initially inside the system.
Equation 5 Eqn 4
We can use Eqn 4 to further simplify Eqn 2 :
Equation 6 Eqn 5
Next, we need to determine Hin.  First, we need to determine the state of the system.
In the NIST Webbook, we can find the Psat(Tin) for steam:
Psat(Tin) 8.5879 MPa
Because Pin < Psat(Tin), we must consult the Superheated Steam Tables to evaluate Hin.
From the NIST Webbook, we can obtain : Hin 3040.9 kJ/kg
Eqn 5 gives us : U2 3040.9 kJ/kg
Now, we know the values of two intensive properties at state 2: pressure and specific internal energy.  The state is completely determined and we can use the Steam Tables to evaluate any other intensive property, such as T2.  To do so, we must first detemine which phase or phases are present in state 2.
At P = 1.4 MPa, the NIST Webbook tells us that :
Usat liq 828.36 kJ/kg Usat vap 2591.8 kJ/kg
Because U2 > Usat vap(P2), we must consult the Superheated Steam Tables to evaluate T2.
At P = 1.4 MPa, the NIST Webbook tells us that : T (oC) U (kJ/kg)
450 3037
T2 3040.9
460 3053.9
T2 452.31 oC
Verify : None of the assumptions made in this problem solution can be verified.
Answers : T2 452.3 oC
The temperature of the steam in the tank in the final state is greater than the temperature of the steam in the feed line because the surroundings did flow work on the system as it filled with steam.  This flow work caused the internal energy of the steam in the tank to exceed the internal energy of the steam in the feed line.  Consequently, the temperature of the steam in the tank in the final state must be greater than the temperature of feed.