My girls and I just can\'t get this one. Prove that the efficiency of two Carnot

Question

My girls and I just can't get this one.
Prove that the efficiency of two Carnot engines operating intandem, one going from a cold reservoir of the tempTc to a hot reservoir of Ti thenthe second going from a cold reservoir of tempTi to a hot reservoir of tempTh must be less than that of a single Carnot enginegoing from Tc to Th.
Prove that the efficiency of two Carnot engines operating intandem, one going from a cold reservoir of the tempTc to a hot reservoir of Ti thenthe second going from a cold reservoir of tempTi to a hot reservoir of tempTh must be less than that of a single Carnot enginegoing from Tc to Th.

Explanation / Answer

   Efficiency of firstengine   I    =   1- Tc /T1   =   (T1   -   Tc)/ T1    Efficiency of secondengine   II    =   1- T1 /Th   =   (Th   -   T1)/ Th    overall efficiency of two engines intandem   1   =   I*II   =   {(T1   -   Tc)/ T1} *{(Th   -   T1)/ Th}       1   =   (T1* Th   +   Tc *T1   -   Tc *Th   -   T12)/ T1 * Th    Efficiency of secondengine   2   =   1   -   Tc/Th   =   (Th   -   Tc)/ Th    2   -   1   =   (Th   -   Tc)/Th   -      (T1* Th   +   Tc *T1   -   Tc *Th   -   T12)/ T1 * Th                      =   (Tc/ T1)   +   (T1/ Th)    since   2   -   1 is a positivequantity,   2   >   1    since   2   -   1 is a positivequantity,   2   >   1    

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