Problem — Drawing a Carnot cycle (engine and refrigerator)

Problem

On the $P$-$V$ diagram, draw a Carnot cycle between two isotherms ($T_H$ and $T_C$) and two adiabatics, indicating the direction of travel if the machine is a heat engine. Show qualitatively where the heat $Q_H$ enters, where $Q_C$ leaves, and which area corresponds to the net work. Finally, draw on the same diagram how the cycle changes if it is reversed to make a refrigerator.

Solution

Structure of the cycle. The Carnot cycle consists of four alternating reversible stages: two isotherms (hyperbolas $P = nRT/V$, a hot one at $T_H$, a cold one at $T_C$) and two adiabatics ($PV^\gamma = \text{const}$), steeper than the isotherms because $\gamma > 1$.

Engine (clockwise direction). Travelling A→B→C→D→A clockwise, the net work is the enclosed area and is positive ($L_\text{net} > 0$):

• A→B isothermal expansion at $T_H$: the gas absorbs $Q_H$ (in contact with the hot source);
• B→C adiabatic expansion: $Q = 0$, the gas cools from $T_H$ to $T_C$;
• C→D isothermal compression at $T_C$: the gas releases $Q_C$ to the cold source;
• D→A adiabatic compression: $Q = 0$, the gas heats from $T_C$ to $T_H$.

$\ev{L_\text{net} = Q_H - |Q_C| = \text{enclosed area}, \qquad \text{clockwise} \Rightarrow \text{engine}}$

Carnot cycle clockwise (engine): the enclosed area is the positive net work.

Refrigerator (reversed direction, anticlockwise). Travelling the same cycle in reverse, A→D→C→B→A anticlockwise, the net work changes sign ($L_\text{net} < 0$): now work is spent from outside. The gas absorbs $Q_C$ from the cold source (along the cold isotherm, now travelled from D to C) and releases $Q_H$ to the hot source: heat is “pumped” from cold to hot.

$\ev{\text{anticlockwise} \Rightarrow \text{refrigerator / heat pump}, \qquad L_\text{net} < 0}$

The curve in the $P$-$V$ plane is the same: only the direction of travel changes. Engine and refrigerator are the same Carnot cycle travelled in the two opposite directions.

Links

Topics: Heat engines Concepts: Carnot cycle · Refrigerator and heat pump · Thermodynamic transformations Skills: Reading graphs Methods: P-V and ln P – ln V diagram Objects: Heat engine · Refrigerator