6 Cycle Devices

For cycle devices, the normal sign conventions (heat in and work out are positive) are ignored. Instead, the signs of heat and work are defined such that $\dot Q_H$, $\dot Q_L$ and net work $\dot W$ are normally positive. The pictures below show the directions of positive heat fluxes and work rates for a heat engine as compared to a heat pump (or refrigeration cycle).


\begin{picture}(1392,460)(0,0)
\par
\put(196,0){
\begin{picture}(400,460)(0,0)
...
...put(200,0){\makebox(0,0){Low temperature $T_L$}}
\end{picture}
}
\end{picture}

The first law for a cycle:

\begin{displaymath}
\mbox{Net work } {\mbox{out} \atop \mbox{in}} =
\mbox{Ne...
...t}}\mbox{:}
\qquad W_{\mbox{\scriptsize (net)}} = Q_H - Q_L
\end{displaymath}

The Kelvin-Planck Statement: Any working heat engine must dump some waste heat to a lower temperature than the input heat is provided at.

The Clausius Statement: Any working heat pump must have positive work going in.

Thermal efficiency:

\begin{displaymath}
\eta_{\mbox{\scriptsize thermal}} \equiv
\frac{\dot W_{\...
..._{\mbox{\scriptsize thermal, Carnot}} =
\frac{T_H-T_L}{T_H}
\end{displaymath}

Refrigeration coefficient of performance:

\begin{displaymath}
\beta \equiv
\frac{\dot Q_L}{\dot W_{\mbox{\scriptsize n...
...d
\beta_{\mbox{\scriptsize Carnot}} =
\frac{T_L}{T_H-T_L}
\end{displaymath}

Heating heat pump coefficient of performance $\beta' = 1 + \beta$:

\begin{displaymath}
\beta' \equiv
\frac{\dot Q_H}{\dot W_{\mbox{\scriptsize ...
...
\beta'_{\mbox{\scriptsize Carnot}} =
\frac{T_H}{T_H-T_L}
\end{displaymath}

Ideal gas Carnot cycle:

\begin{displaymath}
q_H = \vphantom{q}_1q_2 = R T_H \ln\left(\frac{v_2}{v_1}\r...
...v_3}{v_4}\right)
\qquad
\frac{v_2}{v_1} = \frac{v_3}{v_4}
\end{displaymath}