FL: Thermodynamic Cycles

Thermodynamic Cycles

We would like to be able to convert thermal energy to work in order to operate machines of all sorts. Further, we have shown that we can obtain the maximum work from a reversible expansion (as opposed to an irreversible one).

  • Wanted: $Q \rightarrow W$
  • Known: $W_{rev}>W_{irrev}$
  • Therefore: Use reversible expansion to build an "engine"
rev_irr_noW
Note:
  • $Q_{exp} = - W_{exp}$
  • $Q_{comp} = - W_{comp}$
  • $W_{exp} = - W_{comp}$!! (no net work)

However, we recall that our irreversible process did not cancel out. Why not use that as an "engine"?

rev_irr1
Note:
  • $- Q_{exp} = W_{exp}$
  • $- Q_{comp} = W_{comp}$
  • $W_{exp} < - W_{comp}$
Definition:

A Carnot cycle is the most efficient engine possible.

carnot

$W_{net} = W_{12} + W_{23} + W_{34} + W_{41}$

Since the work from $2 \to 3$ exactly cancels the work from $4 \to 1$, we get

$W_{net} = W_{12} + W_{34} = nRT_1\ln \left (\frac{P_2}{P_1} \right ) + nRT_3\ln \left (\frac{P_4}{P_3} \right )$
$Q_{net} = Q_{12} + Q_{34} = Q_H - |Q_C|$
$\Delta U_{net} = 0$

Note:

Here we take advantage of the fact that $W_{12}$ and $W_{34}$ have differing signs (since one is work "out" and the other is "in"). Similarly, the signs of $Q_{12}$ and $Q_{34}$ differ, so that the net value is obtained by adding them. When we switch to the $Q_{H}$ and $Q_{C}$ notation, we explicitly use the absolute value so that we can emphasize that the net heat flow is the difference between the amount absorbed from the hot reservoir and that expelled to the cold reservoir.

Definition:

Running a Carnot cycle "forward" is called an engine and has an efficiency of :

$\eta = \frac{W_{net}}{Q_{H}}$

Running a Carnot cycle "backward" is called a refrigeration cycle and has an efficiency of:

$COP = \frac{Q_{C}}{W_{net}}$ [$COP \equiv$ "coefficient of performance"]

Outcome:

Identify, formulate, and solve simple engineering problems (such as expansion/compression and power cycles)