Problem 14 - An Introduction to Thermal Physics

Question

A heat pump is an electrical device that heats a building by pumping heat in from the cold outside. In other words, it's the same as a refrigerator, but its purpose is to warm the hot reservoir rather than to cool the cold reservoir (even though it does both). Let us define the following standard symbols, all taken to be positive by convention: \(T_{h}=\) temperature inside building \(T_{c}=\) temperature outside \(Q_{h}=\) heat pumped into building in 1 day \(Q_{c}=\) heat taken from outdoors in 1 day \(W=\) electrical energy used by heat pump in 1 day (a) Explain why the "coefficient of performance" (COP) for a heat pump should be defined as \(Q_{h} / W\) (b) What relation among \(Q_{h}, Q_{c},\) and \(W\) is implied by energy conservation alone? Will energy conservation permit the COP to be greater than \(1 ?\) (c) Use the second law of thermodynamics to derive an upper limit on the \(\mathrm{COP},\) in terms of the temperatures \(T_{h}\) and \(T_{\mathrm{c}}\) alone. (d) Explain why a heat pump is better than an electric furnace, which simply converts electrical work directly into heat. (Include some numerical estimates.)

Step-by-Step Solution

Answer

(a) COP is defined as \( Q_{h} / W \) because it shows efficiency. (b) By energy conservation, \( Q_{h} = Q_{c} + W \); COP can be >1. (c) \( \text{COP}_{\text{max}} = \frac{T_{h}}{T_{h} - T_{c}} \). (d) Heat pump's COP can exceed 1, unlike an electric furnace.

Step 1: Understanding COP for Heat Pumps

The Coefficient of Performance (COP) for a heat pump is defined as the ratio of useful heat output to the work input. Therefore, the COP is given by \( \text{COP} = \frac{Q_{h}}{W} \). This definition is chosen because the purpose of a heat pump is to provide as much heat as possible to the building (\( Q_{h} \)) for a given amount of work (\( W \)) from the electrical energy used.

Step 2: Applying Energy Conservation

By the First Law of Thermodynamics (energy conservation), the total energy input into the system must equal the total energy output. Thus, we have \( Q_{h} = Q_{c} + W \). This implies that the heat pumped into the building (\( Q_{h} \)) is the sum of the heat extracted from the outside (\( Q_{c} \)) and the work done by the heat pump (\( W \)). Because \( Q_{h} > W \), the COP can be greater than 1.

Step 3: Finding Upper Limit on COP (Using Second Law)

The Second Law of Thermodynamics provides an upper limit to efficiency which is described by the Carnot cycle for a reversible process. The maximum COP is given by the reversible condition: \( \text{COP}_{ ext{max}} = \frac{T_{h}}{T_{h} - T_{c}} \). Here, \( T_{h} \) and \( T_{c} \) must be in absolute temperatures (Kelvin). This shows that the closer \( T_{h} \) and \( T_{c} \) are, the higher the possible COP.

Step 4: Comparing Heat Pump with Electric Furnace

An electric furnace directly converts electrical energy into thermal energy with an efficiency close to 1 (every unit of work results in one unit of heat: \( \text{COP} \approx 1 \)). However, a heat pump can have a COP greater than 1 because it transfers additional energy from the external environment. For example, if a heat pump has a COP of 3, it can deliver 3 units of heat energy per unit of electrical energy, making it more efficient than an electric furnace.

Key Concepts

Coefficient of Performance

In the world of heat pumps, the Coefficient of Performance (COP) is a crucial measurement. It tells us how efficiently the system is working by comparing the heat output to the energy input. For a heat pump, we calculate this as

• \( \text{COP} = \frac{Q_{h}}{W} \),

where \(Q_{h}\) is the heat delivered to the building and \(W\) is the electrical energy used. This ratio is significant because it shows the system's ability to transfer heat relative to the energy it consumes. A higher COP means the heat pump is more efficient at producing more heat with less energy, directly influencing its cost-effectiveness and energy-saving capabilities.

There’s another important aspect to keep in mind: the practical range of COP values. For a heat pump, the COP can exceed 1, meaning it delivers more heat energy than the electrical energy it uses. This is contrary to many other systems, where energy efficiency typically maxes out at a ratio of 1.

First Law of Thermodynamics

The First Law of Thermodynamics, also known as the principle of energy conservation, plays a pivotal role in understanding heat pumps. It tells us that the total energy entering a system should equal the total energy leaving it. For heat pumps, this principle translates to:

• \( Q_{h} = Q_{c} + W \).

This equation explains that the heat delivered inside the building \(Q_{h}\) comes from two sources:

• the heat extracted from the outside \(Q_{c}\),
• and the work done by the heat pump \(W\).

By this law, as long as \(Q_{h}\) is greater than \(W\), the COP can be larger than 1. This shows the power of heat pumps: they can warm spaces efficiently, using external heat rather than relying solely on electrical energy, which sets them apart from traditional heating methods.

Second Law of Thermodynamics

The Second Law of Thermodynamics helps us understand the efficiency limits of heat pumps. It provides insights into why no machine can be perfectly effective and sets a theoretical maximum for the COP based on temperature differences. This maximum is described using the Carnot cycle. The equation:
\[\text{COP}_{\text{max}} = \frac{T_{h}}{T_{h} - T_{c}}\]
shows us the highest efficiency a heat pump can achieve when running in a reversible process. This formula takes into account:

• \(T_{h}\) - the temperature inside the building,
• \(T_{c}\) - the temperature outside,

expressed in Kelvin. As the outside and inside temperatures become closer, the COP increases. Although achieving the Carnot efficiency practically is impossible due to irreversible losses, this formula gives valuable insights into optimizing heat pump performance.

Carnot Cycle

The Carnot cycle represents an idealized thermodynamic cycle that sets an upper limit for the efficiency of heat engines. Understanding the Carnot cycle is instrumental for designing efficient heat pumps. It utilizes four key processes: two isothermal (constant temperature) and two adiabatic (no heat exchange) processes. In simplest terms, it outlines the most efficient way possible to convert heat into work and vice versa.

When we consider heat pumps, the Carnot efficiency informs us about the theoretical best performance we can aim for. It tells us that if we approach this efficiency, we minimize energy waste. For practical purposes, while real-world devices cannot reach Carnot efficiency due to factors like friction and heat loss, this cycle is the benchmark.

Thus, using the principles derived from the Carnot cycle, engineers strive to make devices as efficient as possible by reducing energy losses and maximizing the heat transfer between reservoirs effectively.

Energy Efficiency

Energy efficiency in the context of heat pumps refers to their ability to use less energy to achieve the desired heating effect. The COP provides an indication of efficiency, positioning heat pumps as more efficient alternatives to conventional heating systems, such as electric furnaces. This high efficiency stems from their capability to move heat, instead of generating it directly from electrical work.

To illustrate this: if a heat pump has a COP of 3, it means it delivers three times more heat than the electrical energy it consumes. Contrast this with an electric furnace, whose efficiency usually peaks at a COP of 1, delivering heat equivalent to the input of electrical energy.

So, in terms of raw numbers, the higher COP indicates more heat output per unit of energy, representing not only a cost-effective but also a more environmentally sustainable heating solution. This efficiency can help lower energy bills and decrease the carbon footprint of heating systems in buildings, aligning well with modern environmental goals.