6F2 :  Determining Whether a Power Cycle is Reversible, Irreversible or Impossible  4 pts 

Consider the power cycle described by the diagram, below. Consider each of the following cases and determine whether the power cycle in each case is reversible, irreversible or impossible.  


a.) Q_{H} = 1150 kJ, W_{cycle} = 988 kJ b.) Q_{H} = 1122 kW, Q_{C} = 207 kW c.) W_{cycle} = 1660 kJ, Q_{C} = 499 kJ d.) h = 74% 

Read :  What is the relationship between Q_{C}/Q_{H} and T_{C}/T_{H} for a reversible cycle?  
Read about the Kelvin temperature scale. How does this allow you to determine the maximum theoretical efficiency of a thermodynamic cycle from T_{C} and T_{H}?  
Note that the maximum efficiency by definition is associated with a reversible cycle.  
Any real cycle always possesses some losses or friction and these irreversibilities reduce h below h_{max}.  
Diagram:  The diagram in the problem statement is adequate.  
Given:  T_{H}  1870  K  
T_{C}  345  K  
a.)  Q_{H}  1150  kJ  c.)  W_{cycle}  1660  kJ  
W_{cycle}  988  kJ  Q_{C}  499  kJ  
b.)  Q_{H}  1122  kJ  d.)  h  74%  
Q_{C}  207  kJ  
Find:  Reversible ?  Irreversible ?  Impossible ?  
Assumptions:  1   The system shown undergoes a power cycle.  
Equations / Data / Solve:  
To determine if each case is reversible, irreversible, or impossible we need to compare the actual efficiency of the case to the maximum efficiency. There are 3 possibilities :  
If the efficiency of the process equals the maximum efficiency, then the process is reversible.  
If the efficiency of the process is less than the maximum efficiency, then the process is irreversible.  
If the efficiency of the process is greater than the maximum efficiency, then the process is impossible.  
Since the maximum efficiency, by definition, is associated with a 'reversible' cycle, the maximum thermal efficiency for any power cycle operating between thermal resevoirs T_{H} and T_{C} is:  

Eqn 1  
The efficiency for each case is determined by: 

Eqn 2  
Only in part (a) do we know W_{cycle} and Q_{H} but realizing: 

Eqn 3  
or : 

Eqn 4  
Combining Eqns 2 and 4 gives an equation we can use to resolve parts (b) and (c) : 

Eqn 5  
Plug values into Eqn 1 to determine the maximum thermal efficiency :  h_{max}  0.816  
a.)  Plug values into Eqn 2 to determine h :  h  0.859  
b.)  Plug values into Eqn 5 to determine h :  h  0.816  
c.)  Plug values into Eqn 5 to determine h :  h  0.769  
d.)  Given :  h  0.740  
Verify:  The assumptions made in this problem cannot be verified with the given information.  
Answers :  Part (a)  The process is impossible because the efficiency of the process is greater than the maximum efficiency for any power cycle operating between thermal resevoirs at T_{H} and T_{C}.  
Part (b)  The process is reversible because the efficiency of the process equals the maximum efficiency for any power cycle operating between thermal resevoirs at T_{H} and T_{C}.  
Part (c)  The process is irreversible because the efficiency of the process is less than the maximum efficiency for any power cycle operating between thermal resevoirs at T_{H} and T_{C}.  
Part (d)  The process is irreversible because the efficiency of the process is less than the maximum efficiency for any power cycle operating between thermal resevoirs at T_{H} and T_{C}.  