Two power cycles are available for your
use and analysis. One is reversible (R) and
one is irreversible (I). You also have two thermal reservoirs at your disposal.
You connect both cycles
to both reservoirs, as
shown below. 







a.) If each cycle receives the same amount of heat from the hot reservoir, show that the irreversible cycle rejects more heat to the cold reservoir than the reversible cycle does. 

b.) If each cycle produces the same net
amount of work, show
that the irreversible cycle must absorb more heat from the hot reservoir than the reversible cycle does. 





Read : 
Between reversible and irreversible cycles, the Carnot Corollaries indicate that h_{Rev} > h_{IRR}. 


Use the 1st Law and the Carnot Corollaries to demonstrate
these two points. 












Given: 
A reversible
power cycle R and an irreversible power cycle I
operate between the same
two thermal reservoirs. 












Find: 
Show that: 




^{} 





Part (a) 
for Q_{H} = Q'_{H} 

Q'_{C}
> Q_{C} 







Part (b) 
for Q_{R} = W_{I} 

Q'_{H}
> Q_{H} 

















Diagram: 












Assumptions: 
1  
The system R undergoes a reversible power cycle while system
I undergoes an irreversible power cycle. 












Equations
/ Data / Solve: 




















No equations are
needed to answer this problem. 













This problem is a
proof. Therefore, the equations needed will be determined in the answer
questions section. 












Verify: 
The assumptions in
this problem cannot be verified with the given information. 












Answers : 





















Part (a) 
By the first Carnot Corollary, h_{Rev} > h_{IRR}. 


Since both cycles receive the same amount of energy, Q_{H}, it follows that: 




















Eqn 1 













An energy balance on cycle R is: 



Eqn 2 


An energy balance on cycle I
is: 



Eqn 3 


Combining Eqns 1, 2 and 3
yields : 


Eqn 4 













The Q_{H} terms in Eqn 4 cancel out and we obtain: 


Eqn 5 












Thus, not only do actual cycles develop less work
they also discharge more energy by heat transfer to their surroundings, thereby increasing the effect of thermal polution. 












Part (b) 
By the first Carnot Corollary, h_{Rev} > h_{IRR}_{ }and from the problem
statement we know: 









Eqn 6 













Efficiency is defined by: 



Eqn 7 













Therefore, because h_{Rev} > h_{IRR} : 



Eqn 8 












The work terms cancel because both cycles produce the same amount of work. Therefore, Eqn 8 becomes : 









Eqn 9 













Notice that: 




Eqn 10 













If the hot reservoir were maintained by, say, energy from the combustion of a fossil fuel, the irreversible cycle would have the greater fuel
requirement. Also, note the irreversible cycle would also have
the greater energy discharge to the cold reservoir, increasing the magnitude of thermal polution. 


























