Example Problem with Complete Solution

5C-7 : Heat Losses From a Steam Compressor 6 pts
A 25 kW compressor  is used to increase the pressure of saturated steam at 140oC to 1.2 MPa. The compressor effluent is at 280oC. 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 steady-state 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 P2 1200 kPa
0.0617 kg/s T2 280 oC
x1 1.00 kg vap/kg total W -25 kW
T1 140 oC
Find: Q ??? kW
Diagram:
Assumptions: 1 - Changes in kinetic and potential energy are negliqible.
Equations / Data / Solve:
Part a.) Begin by writing the steady-state 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 H1 and H2.  We can lookup H1 in the NIST Webbook.
H1 2733.4 kJ/kg
For state 2, we must first determine the phase.
Psat(T2) 6416.6 kPa P2 < Psat, therefore we must consult the Superheated Steam Tables.
From the NIST Webbook, we can obtain : H2 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 50oC 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
MWH2O 18.016 g/mole DH 272.4 kJ/kg
Now, plug this value into Eqn 2 to evaluate QIG : QIG -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.
QIG -8.20 kW
%error 2.4%