THE EFFICIENCY OF THE MORSE POTENTIAL ON THE STIRLING CYCLE

A heat engine also known as classical engine is a system that converts heat to usable energy, particularly mechanical energy, which can then be used to do mechanical work. While originally conceived in the context of mechanical energy, the concept of the heat engine has been applied to various other kinds of energy, particularly electrical, since at least the late 19th century. The heat engine does this by bringing a working substance from a higher state temperature to a lower state temperature. A heat source generates thermal energy that brings the working substance to the higher temperature state. The working substance generates work in the working body of the engine while transferring heat to the colder sink until it reaches a lower temperature state. During this process some of the thermal energy is converted into work by exploiting the properties of the working substance. The working substance can be any system with a non-zero heat capacity, but it usually is a gas or liquid. During this process, some heat is normally lost to the surroundings and is not converted to work. Also, some energy is unusable because of friction and drag[1].

In general, an engine is any machine that converts energy to mechanical work. Heat engines distinguish themselves from other types of engines by the fact that their efficiency is fundamentally limited by Carnot's theorem of thermodynamics. Although this efficiency limitation can be a drawback, an advantage of heat engines is that most forms of energy can be easily converted to heat by processes like exothermic reactions (such as combustion), nuclear fission, absorption of light or energetic particles, friction, dissipation and resistance. Since the heat source that supplies thermal energy to the engine can thus be powered by virtually any kind of energy, heat engines cover a wide range of applications [1].

Heat engines are often confused with the cycles they attempt to implement. Typically, the term "engine" is used for a physical device and "cycle" for the models.

Figure 1.1 Heat Engine Illustration

In general terms, the larger the difference in temperature between the hot source and the cold sink, the larger is the potential thermal efficiency of the cycle. On Earth, the cold side of any heat engine is limited to being close to the ambient temperature of the environment, or not much lower than 300 kelvin, so most efforts to improve the thermodynamic efficiency of various heat engines focus on increasing the temperature of the source, within material limits. The maximum theoretical efficiency of a heat engine (which no engine ever attains) is equal to the temperature difference between the hot and cold ends divided by the temperature at the hot end, each expressed in absolute temperature[2].

The efficiency of various heat engines proposed or used today has a large range: 3% (97 percent waste heat using low quality heat) for the ocean thermal energy conversion (OTEC) ocean power proposal, 25% for most automotive gasoline engines, 49% for a supercritical coal-fired power station such as the Avedore Power Station, 60% for a combined cycle gas turbine

The efficiency of these processes is roughly proportional to the temperature drop across them. Significant energy may be consumed by auxiliary equipment, such as pumps, which effectively reduces efficiency [3].

Consider two bodies that the engine’s working substance can interact with as we investigate heat engines. One of these is the hot reservoir, which serves as the heat source, it gives the working substance a large amount of heat at constant temperature equation T_H. The other body, the cold reservoir serves to absorb discarded heat from the engine at a constant lower temperature T_C. We represent the quantities of heat transferred from the hot and cold reservoirs as Q_H and Q_C, respectively.

The network W done by the working substance is given as,

All of the heat Q_H would ideally be converted into work; consequently, we would have Q_H  = W and Q_C  = 0. This is impossible there is always some heat wasted Q_C is never zero. The thermal efficiency of an engine, indicated by ղ, is defined as 

One of the practical examples is a gasoline engine, which uses the model of the Otto cycle., Theoretically the thermal efficiency is ղ = 56%. The Otto cycle is a highly idealized model. It assumes that the mixture behaves as an ideal gas; it ignores friction, turbulence, loss of heat to cylinder walls, and many other effects that reduce the efficiency of an engine. Efficiencies of real gasoline engines are typically around 35%.

1.2 QUANTUM HEAT ENGINE

A quantum heat engine is a device that generates power from the heat flow between hot and cold reservoirs. The operation mechanism of the engine can be described by the laws of quantum mechanics. The first realization of a quantum heat engine was pointed out by Scovil and Schulz-DuBois in 1959, showing the connection of efficiency of the Carnot engine and the 3-level maser[4]. Quantum refrigerators share the structure of quantum heat engines with the purpose of pumping heat from a cold to a hot bath consuming power first suggested by Geusic, Schulz-DuBois, De Grasse and Scovil. When the power is supplied by a laser the process is termed optical pumping or laser cooling, suggested by Wineland and Hänsch [5].

 Surprisingly heat engines and refrigerators can operate up to the scale of a single particle thus justifying the need for a quantum theory termed quantum thermodynamics.

Numerous typical thermodynamic cycles, including the Carnot, Joule-Brayton, and Otto cycles, have been examined in their quantum form. Where different working substances like free particles, harmonic oscillators, Poschl-Teller, wood-Saxon, Etc. This potential is applied in the quantum approach of thermodynamic cycles and proven to have an efficiency of 100%, this is because the cycle processes are reversible [4]. The Carnot engine consists of a cylinder of ideal gas that is alternately placed in thermal contact with the high-temperature and low-temperature heat reservoirs whose temperatures are T_H and T_C, respectively. Carnot showed that the efficiency η of such a reversible heat engine is

The Stirling cycle consists of four processes, each of which is reversible. First, the gas in the cylinder undergoes an isothermal expansion at temperature T_H while it is in contact with the high-temperature reservoir. Second, the gas continues to expand in an isochoric process in thermal isolation until its temperature drops to T_C. Third, the gas is compressed isothermally in contact with the low-temperature reservoir [6]. we construct an idealized reversible heat engine that consists of a single quantum-mechanical particle contained in a potential well. Rather than having an ideal gas in a cylinder, we allow the walls of the confining potential to play the role of the piston by moving in and out. We show that there exist quantum equivalents of isothermal and adiabatic reversible thermodynamic processes. However, in place of the temperature variable in classical thermodynamics, we use the energy as given by the pure-state expectation value of the Hamiltonian. Not surprisingly, since this engine is reversible, its efficiency is identical to the classical with T replaced by the expectation value of the Hamiltonian. In our formulation, we do not use the concept of temperature [7]. In a classical thermodynamic system, such as an ideal gas in a cylinder, the temperature is determined by the average velocity of a large number of gas molecules [8].

1.3 Aims and Objectives of the Study

Aims:

To analyze the impact of the Morse potential on the thermodynamic efficiency of the Stirling cycle.

Objectives:

1. To derive the thermodynamic equations governing the Stirling cycle using the Morse potential.
2. To conduct simulations that model the Stirling cycle's performance when incorporating the Morse potential.
3. To evaluate the effect of molecular bond strength and distance parameters in the Morse potential on engine efficiency.
4. To compare the energy outputs and efficiencies between Morse potential-based Stirling cycles and those using classical potentials, identifying key advantages or limitations.

1.4 Scope of the Study

This study focuses on evaluating the efficiency of the Stirling cycle when incorporating the Morse potential, which models molecular interactions. The research will investigate how this potential impacts thermodynamic processes within the Stirling cycle, specifically analyzing its influence on performance metrics like energy output, heat transfer, and overall efficiency. The study is confined to the theoretical analysis and simulation of the Stirling engine cycle under varying conditions, such as different temperature ranges and engine configurations, to assess how the Morse potential can optimize energy conversion.

The scope is limited to examining the Stirling engine in the context of energy systems, with a particular focus on ideal gas behavior and molecular interactions described by the Morse potential. While the study will provide insight into the potential application of this model to real-world Stirling engines, it will not include experimental testing or the engineering design of actual Stirling engine prototypes. Instead, the research is computational and simulation-based, using thermodynamic principles to explore the potential benefits and limitations of using the Morse potential in this context.

1.5 Definition of Terms

1. Morse Potential: A mathematical model used to describe the interaction between atoms in a molecule, particularly in the context of diatomic molecules. It accounts for the energy changes due to the distance between atoms, providing a more accurate representation of molecular bonds than the harmonic oscillator model.
2. Stirling Cycle: A thermodynamic cycle that describes the operation of a heat engine. The Stirling cycle consists of four processes: two isothermal (constant temperature) processes and two isochoric (constant volume) processes, used to convert heat energy into mechanical work.
3. Thermodynamic Efficiency: The ratio of the useful work output of a heat engine to the heat energy input. It measures how effectively an engine converts thermal energy into mechanical work.
4. Ideal Gas Behavior: A simplified model of gas behavior where the gas particles are assumed to have no volume and no intermolecular forces. The behavior of the gas can be described by the ideal gas law, which relates pressure, volume, and temperature.
5. Heat Transfer: The process of energy transfer due to a temperature difference between systems. In the context of the Stirling cycle, heat transfer occurs during the isothermal expansion and compression phases, playing a critical role in the engine's performance.
6. Thermodynamic Processes: Describes the changes in state variables such as temperature, pressure, and volume in a system. The four processes in a Stirling cycle include isothermal expansion, isochoric heat addition, isothermal compression, and isochoric heat rejection.

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