Question

What cycle is composed of two isothermal and two constant-volume processes?

Short answer

Question: Identify a thermodynamic cycle that consists of two isothermal processes and two constant-volume processes. Describe its features briefly. Answer: The Otto Cycle is a thermodynamic cycle consisting of two isothermal processes and two constant-volume processes. It is an ideal cycle for four-stroke spark-ignition internal combustion engines. The cycle comprises of isothermal compression, constant-volume heat addition, isothermal expansion, and constant-volume heat rejection. These processes represent the compression, combustion, power stroke, and exhaust events in an engine, respectively.

Step-by-step solution

Understand and Identify the Thermodynamic Cycle

In this exercise, we are given that the thermodynamic cycle has two isothermal processes and two constant-volume processes. We can identify that this description corresponds to the Otto Cycle, which is the ideal cycle for four-stroke spark-ignition internal combustion engines.

Describe the Otto Cycle Processes

The Otto Cycle is composed of the following four processes: 1. Isothermal compression (constant-temperature process): In this process, the working fluid (usually air-fuel mixture) is compressed adiabatically while maintaining a constant temperature. 2. Constant-volume heat addition (isochoric process): During this process, heat is added to the working fluid at a constant volume. This represents the combustion of the fuel-air mixture in the engine cylinder, increasing the temperature and pressure inside. 3. Isothermal expansion (constant-temperature process): In this process, the working fluid expands adiabatically while maintaining a constant temperature. This represents the power stroke in the engine, where the high-pressure gas pushes the piston downwards, generating usable mechanical power. 4. Constant-volume heat rejection (isochoric process): In this process, heat is removed from the working fluid at constant volume, returning the system back to its initial state. This represents exhaust in the engine, where the residual gases are removed before starting a new cycle.

Conclusion

In conclusion, the thermodynamic cycle composed of two isothermal and two constant-volume processes is the Otto Cycle. This cycle is the ideal representation of the processes occurring in four-stroke spark-ignition internal combustion engines.

Key concepts

Isothermal Processes

An isothermal process is a fundamental concept in thermodynamics where a system exchanges heat with its surroundings to maintain a constant temperature. In the context of an Otto Cycle, these isothermal processes represent ideal conditions that are not fully achieved in real engines, but they are significant in understanding the cycle's theoretical model.

Isothermal processes are critical in the analysis of the Otto Cycle because they depict the ideal compression and expansion of the working fluid without a change in temperature. During the real operation of an internal combustion engine, while the engine does work to compress the air-fuel mixture and the mixture does work as it expands, the system's temperature does alter due to the heat produced through combustion and friction losses. However, when the cycle is analyzed isothermally, it simplifies the understanding and helps in calculating the efficiency of the engine in an ideal scenario.

The process can be visualized on a pressure-volume (PV) diagram where the isothermal processes appear as horizontal lines reflecting the direct relationship between pressure and volume at constant temperature, described by the equation \( PV = nRT \) — known as the ideal gas law, where \( P \) is the pressure, \( V \) is the volume, \( n \) is the amount of substance, \( R \) is the ideal gas constant, and \( T \) is the temperature.

Constant-Volume Processes

Constant-volume processes, also referred to as isochoric processes, play a pivotal role in the Otto Cycle. In these processes, the volume of the working fluid remains unchanged while its pressure and temperature vary due to the addition or rejection of heat. Such conditions are approximated during the phases when the engine’s piston is at the top or bottom of its travel (top dead center or bottom dead center), and the combustion or exhaust valves are closed.

In the Otto Cycle, constant-volume heat addition and rejection are essential to model engine operation. During the heat addition phase, which mimics the combustion of the fuel-air mixture, there is a rapid increase in temperature and pressure at constant volume. Similarly, in the heat rejection phase, the gases cool at constant volume, which represents the exhaust phase in a four-stroke cycle.

The energy changes in these isochoric processes can be described by the first law of thermodynamics, which is \( Q = \triangle U + W \) where \(\triangle U\) is the change in internal energy and \(\text{W}\) is work done. Since \(\text{W} = P \triangle V\) and the change in volume \(\triangle V\) is zero in a constant-volume process, the heat added or removed from the system equals the change in internal energy, \( Q = \triangle U \). This makes calculations for the Otto Cycle more straightforward and allows for the analysis of the engine’s efficiency.

Thermodynamic Cycles

Understanding thermodynamic cycles is crucial in studying the conversion of heat energy into mechanical work, which is the principle behind most engines and power-generating systems. The Otto Cycle is one such cycle and is particularly important as it describes the functioning of a typical four-stroke spark-ignition internal combustion engine.

A thermodynamic cycle consists of a series of processes that a working fluid undergoes to convert heat into work or vice versa. In the ideal Otto Cycle, these processes are two isothermal and two constant-volume stages that form a closed loop. The theoretical analysis of such cycles is key to advancing the design and efficiency of engines. It aids in understanding how different stages of the cycle interact with one another and the impact on the overall work output and efficiency.

The Otto Cycle, which serves an educational purpose to simplify the complex nature of the processes in a real engine, highlights the significance of isolating each phase of the engine's stroke for analysis. By doing so, engineers can tweak and improve engine performance. For instance, understanding how to minimize thermal losses during compression can lead to higher efficiencies and better engine designs. Therefore, the study of thermodynamic cycles such as the Otto Cycle, not only provides insights into the theoretical underpinnings of engine operations but also propels technological advancements in the automotive industry.