5C4 :  Expansion of Steam Through a Throttling Valve  5 pts 

A throttling
valve is used to reduce the pressure in a steam line from 10 MPa to 300 kPa. If the steam enters the throttling valve at 500^{o}C, determine… 

a.) The steam temperature at the outlet of the throttling valve b.) The area ratio, A_{2}/A_{1}, required to make the kinetic energy the same at the inlet and the outlet. 

Read :  We know the values of two intensive variables for the inlet steam, so we can determine the values of all of its other properties, including the specific enthalpy, from the Steam Tables. If changes in kinetic and potential energy are negligible and the throttling device is adiabatic, then the throttling device is isenthalpic. In this case, we then know the specific enthalpy of the outlet stream. The pressure of the outlet stream is given, so we now know the values of two intensive properties of the outlet stream and we can determine the values of any other property using the Steam Tables. Part (b) is an application of the 1st Law. The area must be greater at the outlet in order to keep the velocity the same because the steam expands as the pressure drops across the throttling device.  
Diagram: 


Given:  P_{1}  10000  kPa  Find:  T_{1}  ???  ^{o}C  
T_{1}  500  ^{o}C  A_{2} / A_{1}  ???  
P_{2}  300  kPa  
Assumptions:  1   The throttling device is adiabatic.  
2   Changes in potential energy are negligible.  
3   Changes in kinetic energy are negligible because the crosssectional area for flow in the feed and effluent lines have been chosen to make the fluid velocity the same at the inlet and the outlet.  
Equations / Data / Solve:  
Begin by looking up the specific enthalpy of the feed in the steam tables.  
At a pressure of 10,000 kPa, the saturation temperature is :  T_{sat}  311.00  ^{o}C  
Because T_{1} > T_{sat}, we conclude that the feed is superheated steam and we must consult the Superheated Steam Tables. Because 10,000 kPa is listed in the table, interpolation is not required.  
V_{1}  0.032811  m^{3}/kg  H_{1}  3375.1  kJ/kg  
The 1st Law for a throttling device that is adiabatic and causes negligible changes in kinetic and potential energies is :  

Eqn 1  
Because the pressure drops in the throttling device and the feed is a superheated vapor, the effluent must also be a superheated vapor. So, to answer part (a), we must use the Superheated Steam Tables to determine the temperature of 300 kPa steam that has a specific enthalpy equal to H_{2}.  
At 300 kPa :  H (kJ/kg)  T (^{o}C)  V (m^{3}/kg)  
3275.5  400  1.0315  
3375.1  447.2  1.1048  T_{2}  447.2  ^{o}C  
3486.6  500  1.1867  V_{2}  1.1048  m^{3}/kg  
Part b.)  We need to use the definition of kinetic energy to determine how much the area of the outlet pipe must be greater than the area of the inlet pipe in order to keep the kinetic energy (and therefore the velocity) constant.  

Eqn 2 

Eqn 3  
Because the mass flow rate at the inlet and outlet is the same, Eqn 3 simplifies to : 

Eqn 4  
Next, we need to consider the relationship between velocity, specific volume and crosssectional area.  


Eqn 5  
Now, substitute Eqn 5 into Eqn 4 to get : 

Eqn 6  
Solve for the area ratio, A_{2} / A_{1} : 

Eqn 7  
Plugging values into Eqn 7 yields :  A_{2} / A_{1}  33.672  
Verify:  None of the assumptions made in this problem solution can be verified.  
Answers :  T_{2}  447  ^{o}C  A_{2} / A_{1}  33.7 