Refrigerators and heat pumps Knowledge-Based Study

Study notes

1. Reversed Heat Engines

   According to the Clausius statement of the second law of thermodynamics, heat cannot spontaneously flow from a colder reservoir to a warmer reservoir without external work being performed on the system. Devices designed to move thermal energy in this non-spontaneous direction are known as reversed heat engines or vapor-compression cycles.

   Both refrigerators and heat pumps operate on cyclic thermodynamic processes where a working fluid (refrigerant) absorbs heat from a low-temperature region, receives mechanical work from a compressor, and rejects thermal energy to a high-temperature region.

   First Law Energy Conservation

   For a complete cycle, the net change in internal energy of the working fluid is zero ($\Delta U = 0$). Applying the first law of thermodynamics gives:

   $$Q_H = Q_C + W$$

   where:

   • $Q_C$: Heat extracted from the cold reservoir (low-temperature space at temperature $T_C$).
   • $W$: Net external work supplied to the compressor.
   • $Q_H$: Heat rejected to the hot reservoir (high-temperature space at temperature $T_H$).

   The Objective of a Refrigerator

   The primary purpose of a refrigerator (or air conditioner) is to maintain a refrigerated space at a temperature lower than that of the surrounding environment by continuously removing heat from that cold space.

   Consequently, the desired thermal effect is the heat removed from the cold reservoir, $Q_C$, and the required input is the net work supplied, $W$.

   Coefficient of Performance ($\text{COP}_R$)

   Thermal efficiency is not defined for these cycles because the energy of interest is heat transfer rather than work output. Instead, performance is evaluated using the Coefficient of Performance ($\text{COP}$):

   $$\text{COP}_R = \frac{\text{Desired Output}}{\text{Required Input}} = \frac{Q_C}{W} = \frac{Q_C}{Q_H - Q_C}$$

   Carnot or Reversible Limit

   For an ideal, reversible cycle operating between absolute temperatures $T_C$ and $T_H$ (measured in kelvins), the heat ratio equals the absolute temperature ratio ($\frac{Q_C}{Q_H} = \frac{T_C}{T_H}$). The maximum possible coefficient of performance is:

   $$\text{COP}_{R,\text{rev}} = \frac{T_C}{T_H - T_C}$$

   Real refrigerators always operate below this theoretical ceiling due to irreversible processes such as friction, fluid turbulence, and finite-temperature-difference heat transfer.

   The Objective of a Heat Pump

   A heat pump operates on the exact same thermodynamic cycle as a refrigerator, but its operational objective is different: it is designed to deliver heat to a high-temperature space (such as the interior of a building during winter) by extracting thermal energy from a colder outdoor environment.

   Therefore, the desired thermal effect for a heat pump is $Q_H$, while the cost remains the work input $W$.

   Coefficient of Performance ($\text{COP}_{HP}$)

   The performance of a heat pump is defined as:

   $$\text{COP}_{HP} = \frac{\text{Desired Output}}{\text{Required Input}} = \frac{Q_H}{W} = \frac{Q_H}{Q_H - Q_C}$$

   Relationship Between $\text{COP}_{HP}$ and $\text{COP}_R$

   Using the energy conservation equation $Q_H = Q_C + W$, we can express $\text{COP}_{HP}$ directly in terms of $\text{COP}_R$ for identical operating reservoirs:

   $$\text{COP}_{HP} = \frac{Q_C + W}{W} = \frac{Q_C}{W} + 1 = \text{COP}_R + 1$$

   Because $\text{COP}_R > 0$, it follows that $\text{COP}_{HP} > 1$. Even under degraded conditions, a heat pump delivers more thermal energy to the heated space than the electrical energy consumed by the compressor.

   Reversible Limit for Heat Pumps

   For an ideal reversible cycle:

   $$\text{COP}_{HP,\text{rev}} = \frac{T_H}{T_H - T_C}$$

   Summary of Key Differences

   • Refrigerator: Objective is removing $Q_C$; $\text{COP}_R = \frac{Q_C}{W}$.
   • Heat Pump: Objective is supplying $Q_H$; $\text{COP}_{HP} = \frac{Q_H}{W}$.
   • General Relation: $\text{COP}_{HP} = \text{COP}_R + 1$ when operating across identical source and sink temperatures.

   Impact of Temperature Lift

   Both $\text{COP}_R$ and $\text{COP}_{HP}$ depend inversely on the temperature difference $\Delta T = T_H - T_C$, known as the temperature lift:

   • As $T_H - T_C$ increases, the work required to transfer a unit of heat increases, causing both $\text{COP}_R$ and $\text{COP}_{HP}$ to decrease significantly.
   • For example, an air-source heat pump becomes less effective in extremely cold winter weather because $T_C$ drops, expanding the temperature gap between outdoors and indoors.