5C7 :  Heat Losses From a Steam Compressor  6 pts 

A 25 kW compressor is used to increase the pressure of saturated steam at 140^{o}C to 1.2 MPa.
The compressor effluent
is at 280^{o}C. If the steam flow rate
is 3.7 kg/min… 

a.) Calculate the rate of heat loss from the compressor b.) Assume the steam behaves as an ideal gas and calculate the % error in the heat loss that results from the ideal gas assumption. 

Read :  This is a straightforward application of the steadystate form of the 1st Law.  
In part (a), we can lookup properties in the Steam Tables.  
In part (b), we must use the Ideal Gas Heat Capacity from the NIST Webbook to evaluate DH.  
Given:  m  3.7  kg/min  P_{2}  1200  kPa  
0.0617  kg/s  T_{2}  280  ^{o}C  
x_{1}  1.00  kg vap/kg total  W  25  kW  
T_{1}  140  ^{o}C  
Find:  Q  ???  kW  
Diagram: 


Assumptions:  1   Changes in kinetic and potential energy are negliqible.  
Equations / Data / Solve:  
Part a.)  Begin by writing the steadystate form of the 1st Law for open systems in which changes in kinetic and potential energy are negligible.  

Eqn 1  
We can solve Eqn 1 for Q : 

Eqn 2  
Now, we need to determine H_{1} and H_{2}. We can lookup H_{1} in the NIST Webbook.  
H_{1}  2733.4  kJ/kg  
For state 2, we must first determine the phase.  
P_{sat}(T_{2})  6416.6  kPa  P_{2} < P_{sat}, therefore we must consult the Superheated Steam Tables.  
From the NIST Webbook, we can obtain :  H_{2}  3002.6  kJ/kg  
Now, we can plug values into Eqn 2 :  Q  8.404  kW  
Part b.)  Eqn 2 still applies if the steam is treated as an ideal gas.  
The difference from part (a) lies in how we evaluate the change in the specific enthalpy of the steam.  
In part (b) we evaluate the change in the enthalpy using :  

Eqn 3  
The Shomate Equation for the ideal gas heat capacity is : 

Eqn 4  
NIST Webbook :  where : 

Eqn 5  
Temp (K)  500.  1700.  and : 

Eqn 6  
A  30.092  
B  6.832514  Let's use these Shomate constants even though 50^{o}C is outside of the recommended temperature range. These are the best values available to us.  
C  6.793435  
D  2.53448  
E  0.082139  
Combining Eqns 1, 2 and 3 and integrating yields :  

Eqn 7  
Plug in values for the temperatures and the constants to get :  DH  4908  J/mol  
MW_{H2O}  18.016  g/mole  DH  272.4  kJ/kg  
Now, plug this value into Eqn 2 to evaluate Q_{IG} :  Q_{IG}  8.200  kW  
We can calculate the %error due to assuming that the steam is an ideal gas using :  

Eqn 8  
%error  2.42%  
Verify:  The assumption made in this problem solution cannot be verified.  
Answers :  Q  8.40  kW  Note that the negative sign indicates that heat transfer is from the compressor to the surroundings.  
Q_{IG}  8.20  kW  
%error  2.4% 