Three Carnot
heat engines operate between three thermal reservoirs, as shown in the
diagram, below. 








Derive an equation for the thermal efficiency of HE_{C} (h_{C}), in terms of the thermal
efficiency of HE_{A} (h_{A}) and the thermal efficiency of HE_{B} (h_{B}). 





Read : 
The key to this
problem is the fact that all three heat
engines are Carnot
Engines and their efficiencies are completely determined by the temperatures of the three reservoirs. Our goal is to algebraically manipulate the three equations for the three efficiencies in order to eliminate all three
temperatures. 


Given: 
HE_{1} absorbs heat from a reservoir at T_{1} and rejects heat to
a reservoir at T_{2}. 


HE_{2} absorbs heat from a reservoir at T_{2} and rejects heat to
a reservoir at T_{3}. 


HE_{3} absorbs heat from a reservoir at T_{1} and rejects heat to
a reservoir at T_{3}. 


Find: 



Diagram: 
The diagram in the
problem statement is adequate. 


Assumptions: 
1  
All three heat engines are Carnot Heat Engines. 


Equations
/ Data / Solve: 



Begin by applying the
equation relating the thermal efficiency of a Carnot Engine to the temperatures of the reservoirs to all three heat
engines. 




Eqn 1 




Eqn 2 
















Eqn 3 



Rearrange Eqns
1 & 2 as follows : 




Eqn 4 



Eqn 5 



Multiply Eqn 4 by Eqn
5 to get : 

Eqn 6 



Now, substitute Eqn 6 back into Eqn 3 : 

Eqn 7 


Simplify Eqn 7 : 


Eqn 8 


Finally : 



Eqn 9 


Verify: 
We cannot verify that
the heat engines are Carnot Engines, but the problem
statement instructed
us to make this assumption. 



Answers : 














