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# Reversed Heat Engines

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Reversible heat engines use the same working principle as heat engines that include a heat transfer between a cold region and a hot one. However, reverse heat engines also transfer energy in the reverse direction. Instead of transferring energy from a higher temperature region to a lower one, reverse heat engines transfer energy from the colder reservoir to the higher temperature system by adding work.

Reversible engines work by the principle of the second law of thermodynamics, which states:

Heat transfer naturally occurs only from higher temperature bodies to lower temperature ones but never in the reverse direction. Heat transfer in the reverse direction is only possible with the addition of external work.

A heat engine converts thermal energy from a heat source into mechanical work, while some part of the heat is lost to the surroundings. The work done by a heat engine is defined as the difference between the heat transferred from the hot reservoir and the heat absorbed by the environment, which is at a lower temperature.

The differences between heat and reverse heat engines are listed in the table below.

 Reverse heat engines Heat engines Heat is transferred from a cold region to a hot one. Heat is transferred from a hot region to a colder one. Work is added into the system to transfer energy from cold to hot regions (i.e., work is done on the system). The heat transfer produces work (which is an output from the system).

Reverse engines convert thermal energy into mechanical work by transferring energy between a hot reservoir and a cold reservoir using a cyclical process. The flow of energy is shown below in figure 1, where the energy is transferred from a lower temperature region to the surroundings (a higher temperature region) by adding work. As this process cannot happen naturally, an electric compressor is used to pump heat out of the system.

Figure 1. Reversible heat engine energy flow diagram. Source: Georgia Panagi: StudySmarter.

The amount of energy discharged into the surroundings (QH) in Joules by a reverse heat engine is expressed as the sum of the work (W) and the heat transfer from a lower temperature (QC) measured in Joules.

What is the difference between heat and work if they are both measured in Joules?

• Heat is the kinetic energy due to the disordered movement of atoms at a microscopic level, where the individual vector of the velocity of the atoms cancels out, resulting in zero momentum.
• Work is the kinetic energy due to the ordered motion of atoms with velocity vectors at a microscopic level that has the same direction, causing the momentum to be non-zero at a macroscopic level.
• They are both forms of energy, but they have different characteristics in their motion.

## Applications of reverse heat engines

There are two main reverse heat engine applications: heat pumps and refrigerators, which are designed to remove heat from a cold region and transfer it to a hotter one.

### Refrigerators

Refrigerators and air conditioners are used to cool down a space by removing heat. Work is done on the system, using a motor to pump warm air inside the fridge to the environment, which is a higher temperature region. The process involves a fluid that is circulated through a closed system:

1. The fluid passes through a nozzle and expands, which turns it into a gas that cools down. This is known as an adiabatic expansion, where energy or mass is not transferred to the environment.
2. The cooler gas is transferred into the inner space of the fridge, which has a higher temperature. Heat is transferred from the fridge to the gas, increasing its temperature. This is known as an isobaric expansion, which means that the pressure is constant.
3. The gas is then transferred to a compressor, which adds work to the system. The compressed gas is heated and becomes a liquid again.
4. The hot liquid goes through coils located on the outside of the fridge, and heat is transferred to the room. This is known as isobaric compression.

These steps can be used to construct a p-v diagram, as shown below in figure 2. The amount of heat removed from the fridge per work is given by the coefficient of performance (COPref). It is a measure of the amount of heat transfer from the cold region compared to the work input to the system.

Figure 2. P-V diagram of a refrigerator. Source: Georgia Panagi, StudySmarter.

Using the relation between work and heat transfer in a reverse heat engine, we get the following equation for the coefficient of performance:

For an ideal refrigerator, we assume that the amount of heat transfer in each region is equal to the temperature of the region, which gives us the following expression for COP:

Power is the work done per unit time measured in Watts or Joules/second.

### Heat pumps

Heat pumps are used to warm up a room. The system is usually comprised of compressed gas, and the sequential working process of a heat pump is as follows:

1. The electrically driven compressor inputs mechanical work into the system. This raises the temperature and pressure of the gas, which is thus forced to enter some condenser coils located in the higher temperature region.
2. As the temperature of the gas is higher than that of the surroundings, thermal energy is transferred to the room, and the gas condenses into a liquid.
3. The liquid goes through an expansion valve, which reduces its temperature before it returns to the evaporator coils to continue the cycle.
4. The last two steps are usually done in reverse order to operate a cooling cycle, and the refrigerant fluid passes through the expansion valve reducing its temperature to the evaporator transferring heat from indoors to outdoors which cools down the refrigerant.

The amount of heat transferred (QH) into a space per unit work input (W) is the coefficient of performance of a heat pump COPhp.

As seen from the equation above, heat pumps seem to have a greater performance when the temperature difference is small. The coefficient of performance is the ratio of heating to required work. Hence, a higher COP means that the heat pump provides the same work with less energy. Therefore, the higher the COP, the higher the efficiency.

### Efficiency

The efficiency of a reverse heat engine is the amount of heat transfer that is actually converted into work. This is determined by dividing work by the heat transfer QH. Then, a relation can be written for the COPhp, and the efficiency determined, as seen below.

Since the efficiency of a heat engine is always less than 1 (there will always be some heat lost), COPhp is always greater than 1 (see the equations below). Therefore, a heat pump has more heat transfer Qh than work put into it.

There is also a relation between the refrigerator coefficient of performance and the heat pump coefficient of performance. This can be derived using the equation of work and the heat pump coefficient formula, as seen below.

We begin using the equation that describes the heat transfer in a heat pump:

Then, we use the heat pump coefficient of performance and refrigerator coefficient of performance equations and re-arrange them in terms of QH and QC, respectively:

We now substitute them into the heat transfer equation mentioned earlier and divide by the work on both sides of the equation, which gives us:

A refrigerator has a COP of 4.8 and uses 400 J of work. Determine the heat transferred and the efficiency of the refrigerator.

We use the COP formula and substitute the value of the coefficient of performance and work to find the heat transferred.

To determine the efficiency, we need to find QH. Hence, we need to use the heat transferred and the work to calculate it. Then, we can use the efficiency formula to calculate efficiency, using heat lost QH and work.

## Reversed Heat Engines - Key takeaways

• Reversible heat engines use the same working principle as heat engines, which includes heat transfer between a cold reservoir and a hot one, but reversible heat engines transfer energy in the reverse direction.
• The reverse engine works according to the second law of thermodynamics.
• The second law of thermodynamics states that heat transfer naturally only occurs from higher temperature bodies to lower temperature ones. A reverse direction transfer is only possible with the addition of external work.
• Refrigerators and air conditioners are used to cool down spaces by removing heat.
• The amount of heat removed from the fridge per work is given by the coefficient of performance (COPref).
• The amount of heat is transferred (QH) into a space per unit work input (W) is the coefficient of performance of a heat pump COPhp.

It is an engine that does work by transferring energy from a lower temperature body to a higher temperature body.

In a heat engine, heat is transferred from hot reservoir to a colder one, whereas in a reverse heat engine, heat is transferred from a colder to a hotter region.

Yes, reversible heat engines are possible by providing additional external work.

A reversible heat engine is an engine that transfer heat from a lower temperature object to a higher temperature object by adding external work, usually using electric motors.

The efficiency of a reverse heat engine is found by the dividing the work by the heat transfer  as shown below.

η= W/ QH

## Final Reversed Heat Engines Quiz

Question

What is a reversible heat engine?

It is a heat engine that transfers energy in the reverse direction, i.e., from a lower to a higher temperature region, with the use of additional work.

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Question

What is the working principle of a reverse engine?

The second law of thermodynamics states that a heat transfer occurs naturally only from higher temperature bodies to lower temperature ones but never in the reverse direction. A reverse direction transfer is only possible with the addition of external work.

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Question

What are the main differences between heat and reverse heat engines in terms of the heat transfer?

In a heat engine, heat is transferred from a hotter to a colder reservoir, whereas in a reverse heat engine, it is transferred from a colder to a hotter region.

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Question

What are the differences between heat engines and reverse heat engines in terms of work?

In a heat engine, the heat transfer produces work, while in a reverse heat engine, work is input into the system for the heat transfer.

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Question

What is the direction of heat transfer and work in a reverse heat engine?

The energy is transferred from a lower temperature region to the surroundings, which is at a higher temperature region, by adding work.

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Question

Why are both heat and work measured in Joules?

They are both forms of energy but have a different characteristic with regard to their motion: heat is a disordered motion of atoms, while work is an ordered motion of atoms in one direction.

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Question

What is a refrigerator and how does it work?

Refrigerators are used to cool down a region and use a motor to pump warm air from inside the fridge to the environment, which is a higher temperature region.

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Question

What does COPref express?

It is a measure of the amount of heat transfer from the cold region compared to the work input to the system.

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Question

Determine the coefficient of performance of a refrigerator if the coefficient of performance of the same engine is 3.9 when working as a heat pump.

2.9.

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Question

Determine the coefficient of performance of an ideal refrigerator if the  temperature of the hot region is 330K and the temperature of the cold region is 305K.

12.2.

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Question

A heat pump has a coefficient of performance of 3.4 and a power capability of 4000W. Determine the amount of energy transferred into a room per second.

13600J.

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