Question

An outboard motor for a boat is cooled by lake water at \(15.0^{\circ} \mathrm{C}\) and has a compression ratio of \(10.0 .\) Assume that the air is a diatomic gas. a) Calculate the efficiency of the engine's Otto cycle. b) Using your answer to part (a) and the fact that the efficiency of the Carnot cycle is greater than that of the Otto cycle, estimate the maximum temperature of the engine.

Short answer

Based on the given information, the efficiency of the engine's Otto cycle is approximately 36.9%. The estimated maximum temperature of the engine is approximately 456.6 K or 183.4°C.

Step-by-step solution

Calculate the efficiency of the engine's Otto cycle

To calculate the efficiency of the Otto cycle, we can use the formula \(\eta = 1 - \frac{1}{r^{(\gamma - 1)}}\), where the compression ratio \(r=10\) and the adiabatic index for a diatomic gas \(\gamma=1.4\). Now we plug in the values and get, \(\eta = 1 - \frac{1}{(10)^{(1.4 - 1)}}\) \(\eta = 1 - \frac{1}{(10)^{0.4}} \approx 1 - \frac{1}{(1.58489)}\approx 0.369\). The efficiency of the engine's Otto cycle is approximately \(0.369\) or \(36.9\%\).

Estimate the maximum temperature of the engine

We know that the efficiency of a Carnot cycle is greater than the efficiency of the Otto cycle. The efficiency of a Carnot cycle can be calculated using the formula \(\eta_C = 1 - \frac{T_C}{T_H}\), where \(\eta_C\) is the efficiency of the Carnot cycle, \(T_C\) is the temperature of the cold reservoir, and \(T_H\) is the temperature of the hot reservoir. As the efficiency of the Otto cycle is less than the Carnot cycle, we can write: \(\eta \leq \eta_C \Rightarrow 1 - \frac{T_C}{T_H} \geq 0.369\) The lake water cooling the outboard motor has a temperature of \(15.0^{\circ} \mathrm{C}\). We should convert this temperature to Kelvin since the efficiencies are calculated using absolute temperatures. The lake water temperature in Kelvin is, \(T_C = 15.0 + 273.15 = 288.15\) K. Now, rearrange the inequality to solve for \(T_H\): \(1-\frac{T_C}{T_H} \geq 0.369 \Rightarrow \frac{T_H - T_C}{T_H} \geq 0.369\) \(T_H \geq \frac{T_C}{1 - 0.369} = \frac{288.15}{0.631} \approx 456.6\) K. Thus, the estimated maximum temperature of the engine is \(\approx 456.6\) K or \(\approx 183.4^{\circ}\mathrm{C}\).

Key concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In particular, it describes how thermal energy is converted to and from other types of energy and how it affects matter. The fundamental laws of thermodynamics govern the principles of energy conservation, entropy, and the spontaneity of processes.

For an internal combustion engine like the one in the exercise, thermodynamics helps us understand how energy is transformed during the operation of the engine. For instance, during the combustion process, the chemical energy in the fuel is converted into thermal energy, which then gets converted into mechanical work that propels the boat forward. The efficiency of these conversions is crucial for the performance of the engine, and optimizing this efficiency is a primary goal in engine design.

Carnot Cycle

The Carnot cycle is a theoretical model that represents the most efficient possible engine cycle. It is composed of two adiabatic processes and two isothermal processes. During the isothermal expansion and compression, the system exchanges heat with the environment in such a way that the temperature remains constant, while during the adiabatic processes, the system is perfectly insulated, so no heat is exchanged.

In the context of the exercise, understanding the Carnot cycle is key to realizing that no real engine can be 100% efficient. Moreover, it provides a benchmark against which other cycles like the Otto cycle can be compared. Knowing that a Carnot engine operates between temperatures of hot and cold reservoirs allows us to do calculations like estimating the maximum temperature that an engine can achieve, based on its efficiency.

Adiabatic Process

An adiabatic process is one where no heat is transferred to or from the working fluid in the system as the fluid undergoes change in volume or pressure. In other words, all of the work done on or by the fluid affects its internal energy, which in turn changes the temperature of the fluid. For example, in the compression stroke of an engine, as the cylinder's volume decreases, it's considered an adiabatic compression if no heat is lost to the surroundings.

Importance in Engines

Understanding adiabatic processes helps in comprehending the behavior of the working fluid in real engines during compression and expansion strokes. While real engines are not perfectly adiabatic, as some heat loss is inevitable, the assumption allows us to simplify calculations and approach the characteristics of an ideal engine, such as the Otto cycle in the exercise we are exploring.

Compression Ratio

The compression ratio of an engine is a crucial parameter that describes the ratio of the volume of its combustion chamber from its largest capacity to its smallest capacity. It significantly influences the efficiency and power output of the engine. Mathematically, it is expressed as the maximum cylinder volume divided by the minimum cylinder volume after a piston stroke.

High compression ratios can lead to greater thermal efficiency because the heat during combustion is concentrated in a smaller space, hence increasing the amount of work done by each unit of fuel. In our exercise, the compression ratio of 10 implies that the volume of the chamber is reduced by a tenth when the piston is at the top of its stroke compared to when it is at the bottom. Understanding this concept is critical when calculating the efficiency of the Otto cycle engine and subsequently estimating maximum temperatures that the engine can safely tolerate.