20-3 Refrigerators and Real Engines

20-3 Refrigerators and Real Engines#

Prompts

  • How does a refrigerator differ from a heat engine? What is the direction of energy flow?

  • Define the coefficient of performance \(K\) for a refrigerator. What do we “want” and what do we “pay for”?

  • Write \(K_C\) for a Carnot refrigerator in terms of \(T_L\) and \(T_H\). When is \(K\) larger?

  • Why is a “perfect refrigerator” (transferring heat from cold to hot with no work) impossible?

Lecture Notes#

Overview#

  • A refrigerator uses work to transfer heat from a cold reservoir to a hot reservoir—the reverse of a heat engine.

  • Coefficient of performance \(K = |Q_L|/|W|\)—heat removed per unit work input.

  • A Carnot refrigerator (reversible) has maximum \(K_C = T_L/(T_H - T_L)\).

  • A perfect refrigerator (no work) is impossible—violates the second law.

  • Real engines cannot exceed Carnot efficiency.

Refrigerator operation#

  • Work \(W\) is done on the refrigerator.

  • \(|Q_L|\) is extracted as heat from the cold reservoir.

  • \(|Q_H|\) is discharged as heat to the hot reservoir.

First law (one cycle): \(|W| = |Q_H| - |Q_L|\).

Coefficient of performance#

(200)#\[ K = \frac{|Q_L|}{|W|} = \frac{\text{heat removed}}{\text{work input}} \]

  • Carnot refrigerator (reversible; maximum \(K\)):

(201)#\[ K_C = \frac{|Q_L|}{|Q_H| - |Q_L|} = \frac{T_L}{T_H - T_L} \]

  • Larger \(K\) when \(T_L\) and \(T_H\) are closer (smaller \(T_H - T_L\)).

  • Typical: household refrigerator \(K \approx 5\); room air conditioner \(K \approx 2.5\).

Perfect refrigerator (impossible)#

A perfect refrigerator would transfer heat from cold to hot with no work input. That would give \(\Delta S < 0\) (entropy of cold reservoir decreases more than hot increases), violating the second law.

Clausius statement of the second law: No process is possible whose sole result is the transfer of heat from a colder to a hotter body without work.

Real engines vs Carnot#

No real engine operating between \(T_H\) and \(T_C\) can have efficiency greater than a Carnot engine. Irreversibilities (friction, heat loss) always reduce efficiency below \(\varepsilon_C\).

Poll: Refrigerator

To increase the coefficient of performance of a Carnot refrigerator, you could:

  • (A) Lower \(T_L\) (make cold reservoir colder)

  • (B) Raise \(T_L\) (make cold reservoir warmer)

  • (C) Lower \(T_H\) (make room cooler)

Example: Refrigerator—find \(Q_C\) and \(K\) from diagram

What are (a) the heat extracted from the cold reservoir and (b) the coefficient of performance for the refrigerator shown in the figure?

[FIGURE: Refrigerator energy-transfer diagram with \(W\), \(Q_H\), \(Q_C\) labeled]

Solution: (a) Read \(Q_C\) from diagram. (b) \(K = Q_C / (Q_H - Q_C) = Q_C / W\).

Summary#

  • Refrigerator: \(|W| = |Q_H| - |Q_L|\); work moves heat from cold to hot.

  • \(K = |Q_L|/|W|\); \(K_C = T_L/(T_H - T_L)\) for Carnot.

  • Perfect refrigerator (no work) violates the second law.

  • Real engines \(\varepsilon < \varepsilon_C\); real refrigerators \(K < K_C\).