Two Carnot engine $C_1$ and $C_2$ are arranged in series as shown. In the arrangement heat given by $C_1$ to common reservoir is used by $C_2$ to produce work.

 

 

 

 

If both the engines produce equal amounts of work, temperature of the common reservoir is

Assuming temperature of common reservoir equal to T. 

$Q=$heat supplied by hot reservoir to$C_1$

$Q\text{'}=$ heat supplied by$C_{1}to{C}_2$via common reservoir

$Q\text{'}\text{'}=$ heat supplied to cold reservoir by$C_2$

$W_1=$work output by $C_1=Q-Q\text{'}$

$W_2=$work output by$C_2=Q\text{'}-Q\text{'}\text{'}$

Given$W_1=W_2$

$\begin{array}{l}\Rightarrow Q-Q\text{'}=Q\text{'}-Q\text{'}\text{'}\\ \Rightarrow Q+Q\text{'}\text{'}=2Q\text{'}\\ \Rightarrow \frac{Q}{Q\text{'}}+\frac{Q\text{'}\text{'}}{Q\text{'}}=2\\ \Rightarrow \frac{3T_0}{T}+\frac{T_0}{T}=2\\ \Rightarrow T=2T_0\end{array}$