I'm an highschooler who's very intrested in physics. I've stumbled on a problem on my textbook ("Dalla mela di Newton al bosone di Higgs volume III", or translated "From Newton's apple to Higgs' boson volume III" by Ugo Amaldi) about thermal conductivity; the book is in italian since I'm an italian student, thus I'm going to try to translate the text in the most rigorous way I can:

> 
> Two identical bars, one made of copper and the other made of aluminium,
> are aligned in parallel such that the starting point of the first one
> matches the one of the second.
> 
> 
> 

Thus they are tangent and they have in common just a segment of length $d$, which is the common length of the two bars. The requests are:

> 
> -> How is it possible to calculate the system's thermal conductivity constant $\lambda$?   
> 
> -> What is the value of $\lambda$?   
> 
> -> Suppose that the $\lambda$ coefficient has to be less than $500 W m^{-1} k ^{-1}$. Which material has the aluminium to be switched with?$   
> 
> [Suggestion: the temperature difference is equal for each one of the
> two bars]
> 
> 
> 

**Edit**:
Original version of the problem (not translated)

> 
> Due barre identiche una di rame e una di alluminio, vengono poste come nella figura.  
> 
> Come si ricava la formula per il coefficiente $\lambda$ di conducibilità termica?  
> 
> Quanto vale $\lambda$?  
> 
> Si vuole avere una conducibilitàtermica minore di $500 W/(m K)$. Quale materiale bisogna inserire al posto dell'alluminio?
> 
> 
> 

My reasoning is the following: during the process of heat propagation, the two bars will interact thermodinamically exchanging heat, thus it should be possible to write the following heat balance equation:

$Q\_{Cu} + Q\_{Al} = Q\_{tot}$

From which applying the formula for heat conduction in a stationary system:

${Q \over \Delta t} = \lambda S {\Delta T \over d}$

where $\Delta t$ is the time interval, $\Delta T$ is the temperature difference, $d$ is the thickness of the bar, in this case it corresponds to its length, $S$ is the section which the heat flows into, and $\lambda$ is the thermal conductivity coefficient. Sobstituting into the heat balance equation:

$\lambda\_{Cu} S {\Delta T \over d} + \lambda\_{Al} S {\Delta T \over d} = \lambda\_{tot} S' {\Delta T \over d}$

where $S'$ is the section of the system of the two bars, thus equal to $2S$. The expression above leads to:

$\lambda\_{tot} = {\lambda\_{Cu} + \lambda\_{Al} \over 2}$

which is not correct since plugging the values for $\lambda\_{Cu}$ and $\lambda\_{Al}$ results in a total conductivity constant of $320 W m^{-1} k ^{-1}$ whereas the result of the book gives me $640 W m^{-1} k ^{-1}$ as result. The problem is the factor of $1 \over 2$ in the final formula.
This might derive from the fact that I was wrong to consider either the system section $S'$ as $2S$ (but to me this is not the case) or to consider the section through which the heat flows - in the expression for $Q\_{Cu}$ and $Q\_{Al}$ - as the section of the copper or aluminium bar alone (this could be the error). In either case I can't understand why it is necessary not to consider $S' = 2S$ or to consider $S'$ to be always the section that has to be used.   

I've also pondered the fact that the book could be wrong in its result, but this option seems to be very unlikely since the third question asks me to modify the coefficient based on the result I had obtained in the second one.

Thanks a lot for your time and effort in answering me and excuse my English which still has to be practised.