8A2 :  Heat, Work and Entropy Generation  5 pts 

Two power cycles operate between the same two thermal reservoirs, as shown below. Cycle R is reversible and cycle I is irreversible.  


They each absorb the same amount of heat from the hot reservoir, Q_{H}, but produce different amounts of work, W_{R} and W_{I}, and reject different amounts of heat to the cold reservoir, Q_{C} and Q'_{C}. 

a.) Derive an equation for S_{gen} for the irreversible cycle in terms of W_{I}, W_{R}, and T_{C} only. b.) Show that W_{I} W_{R} and Q'_{C} Q_{C}. 

Read :  Start with the equation for the entropy generated and do an energy balance on both the reversible and irreversible cycles. Put the equations together and simplify to get an equation in the desired terms.  
Given:  A reversible power cycle, R, and an irreversible power cycle, I, operate between the same two reservoirs.  
Find:  Part (a)  Evaluate S_{gen} for cycle I in terms of W_{I}, W_{R}, and T_{C}.  
Part (b)  Show that:  W_{I} < W_{R}  and  Q'_{C} > Q_{C}.  
Diagram:  The diagram in the problem statement is adequate.  
Assumptions:  1   The systems shown undergo power cycles. R is reversible and I is irreversible.  
2   Each system receives Q_{H} at a constant temperature region at T_{H} from the hot reservoir and rejects heat, Q_{C}, at a constant temperature region at T_{C} to the cold reservoir.  
Equations / Data / Solve:  
Part a.)  Let's begin with the defintion of entropy generation: 

Eqn 1  
We can solve Eqn 1 for S_{gen} : 

Eqn 2  
Since we are dealing
with a cycle, DS = 0 and Eqn 2 becomes: 

Eqn 3  
In the irreversible process, the system receives heat, Q_{H}, at a constant temperature, T_{H}_{ }, and rejects heat, Q'_{C}, at a constant temperature, T_{C}. Because the temperatures are constant, they can be pulled out of the integral in Eqn 3 leaving :  

Eqn 4  
The 1st Law for cycles is: 

Eqn 5  
We can apply Eqn 5 to both the reversible and the irreversible cycles, as follows :  

Eqn 6 

Eqn 7  
We can combine Eqns 6 & 7 to obtain : 

Eqn 8  
Now, solve Eqn 8 for Q'_{C} : 

Eqn 9  
Next, we can use Eqn 9 to eliminate Q'_{C} from Eqn 4 to get : 

Eqn 10  
We can rearrange Eqn 10 slightly to make it more clear how to proceed : 

Eqn 11  
Because R is a reversible
cycle and we use the Kelvin Temperature Scale : 

Eqn 12  
Eqn 12 can be rearranged to help simplify Eqn 11 : 

Eqn 13  
This yields : 

Eqn 14  
Part b.)  Because irreversibilities are present in cycle I : 

Eqn 15  
Rearranging Eqn 15 gives us : 

Eqn 16  
Finally, we can rearrange Eqn 9 to help us determine whether Q'_{C} or Q_{C} is larger : 

Eqn 17  
Since Eqn 16 tells us that W_{R} > W_{I}, Eqn 17 tells that : 

Eqn 18  
Verify:  The assumptions made in this solution cannot be verified with the given information.  
Answers :  Part a.) 

Part b.) 


