- #1
Telemachus
- 835
- 30
Homework Statement
A thermodynamic engine is operated between two cooler bodies A and B, extracting work until the two cooler bodies reach a common temperature [tex]T_f[/tex]. This work is then used as the input to a heat pump, extracting heat from the cooler pair and heating the hot body C. Find the final temperature [tex]T_{fC}[/tex] C if work is maximum.
The initial temperatures for the cooler bodies A and B are: [tex]T_{A0}=300ºK,T_{B0}=350ºK[/tex], and for the hot body C: [tex]T_{C0}=400ºK[/tex]
The equation of state for the three bodies is: [tex]U=aT[/tex]
So, this is what I did till now.
[tex]U=aT\Rightarrow S=S_0+ln\left(\displaystyle\frac{U}{U_0}\right)[/tex]
[tex]\Delta U_{AB}=U_f-U_0=a(T_{fA}+T_{fB})-a(T_{A0}+T_{B0})[/tex]
[tex]T_{fA}=T_{fB}=T_f[/tex]
[tex]\Delta U_{AB}=U_f-U_0=a(2T_f}-T_{A0}-T_{B0})[/tex]
[tex]W=-\Delta U_{AB}=a(-\left(2T_f}+T_{A0}-T_{B0})[/tex]
[tex]\Delta S_{AB}=2a \ln\left(\displaystyle\frac{T_f}{\sqrt{T_{A0}T_{B0}}}\right)[/tex]
As the work is maximum: [tex]T_f=\sqrt{T_{A0}T_{B0}[/tex]
So [tex]W=a(T_{A0}+T_{B0}-2\sqrt{T_{A0}T_{B0}[/tex]
How do I get [tex]T_{Cf}[/tex] from here?
Bye there, and thanks for posting.
Attachments
Last edited: