Question

An Otto engine has a maximum efficiency of \(20.0 \%\) find the compression ratio. Assume that the gas is diatomic.

Short answer

Answer: The compression ratio of the Otto engine is approximately 2.378.

Step-by-step solution

Write down the efficiency formula for an ideal Otto cycle

The efficiency of an ideal Otto cycle is given by: \(\eta = 1 - \frac{1}{r^{(\gamma - 1)}}\) where \(\eta\) represents the efficiency of the engine, \(r\) is the compression ratio, and \(\gamma\) is the specific heat ratio of the gas.

Determine the specific heat ratio for diatomic gas

For a diatomic gas, the specific heat ratio \(\gamma\) is given by: \(\gamma = \frac{C_p}{C_v} = \frac{7}{5}\)

Rewrite the efficiency formula for an Otto engine using the given efficiency and specific heat ratio

Substitute the given efficiency and the specific heat ratio for a diatomic gas into the formula for the efficiency of an ideal Otto cycle: \(0.20 = 1 - \frac{1}{r^{(\frac{7}{5}- 1)}}\)

Simplify the formula and solve for the compression ratio

Simplify the formula as follows: \(0.20 = 1 - \frac{1}{r^{(\frac{2}{5})}}\) \(0.80 = \frac{1}{r^{(\frac{2}{5})}}\) Now, we need to solve for the compression ratio \(r\): \(r^{(\frac{2}{5})} = \frac{1}{0.80}\) \(r^{(\frac{2}{5})} = 1.25\) To get rid of the exponent, raise both sides of the equation to the power of \(\frac{5}{2}\): \((r^{(\frac{2}{5})})^{\frac{5}{2}} = (1.25)^{\frac{5}{2}}\) \(r = 1.25^{\frac{5}{2}}\)

Calculate the compression ratio

Perform the calculation to find the value of \(r\): \(r = 1.25^{\frac{5}{2}} \approx 2.378\) Thus, the compression ratio of the Otto engine is approximately \(2.378\).

Key concepts

Compression Ratio

Imagine squeezing a balloon with your hands. The more you squeeze, the more its volume decreases and this is similar to how compression ratio works in an engine. The compression ratio is a measure of how much the engine compresses the air-fuel mixture. It is defined as the ratio of the volume before compression to the volume after compression. This is often expressed in mathematical terms as:

• The volume ratio of the cylinder when the piston is at the bottom (bottom dead center) to when it is at the top (top dead center).

A higher compression ratio will usually mean more engine efficiency. That's because the air-fuel mixture burns more completely when compressed more tightly. More complete burning means more power from the same amount of fuel.
In the exercise above, the Otto engine's efficiency relates directly to this compression ratio. Finding the value of the compression ratio gives insight into how efficiently the engine operates.

Specific Heat Ratio

The term specific heat ratio can sound complicated, but it's quite simple once you break it down. It's a measure of the heat capacity of a gas and how much it can absorb or lose under varying conditions. In essence, it tells us how heat behaves with gases as they are heated or compressed. For gases, this ratio is symbolized by the Greek letter gamma (\( \gamma \)).

• It is the ratio of specific heat at constant pressure \( C_p \) to specific heat at constant volume \( C_v \).

For diatomic gases, like the air in our Otto cycle problem, \( \gamma \) is approximately 1.4 or \(\frac{7}{5}\) as outlined in the solution steps. This value plays a crucial role in calculating the efficiency of cycles like the Otto because it determines how readily the gas exchanges heat, thus influencing efficiency outcomes.

Diatomic Gas

A diatomic gas is a gas that is made up of molecules consisting of two atoms. These can be either of the same element or different ones. The most common diatomic gases in nature include oxygen (\(O_2\)), nitrogen (\(N_2\)), and hydrogen (\(H_2\)).
Diatomic gases carry certain distinct properties, such as specific heat capacities and enthalpies, which differ from those of monatomic gases. In the context of thermodynamics and the Otto cycle, these properties are crucial as they determine how the gas will behave under compression and expansion.

• For example, the specific heat ratio \( \gamma \) is crucial for efficiency determination.

Understanding the behavior of diatomic gases, such as their heat capacity, helps predict how they react under the changing pressures and temperatures inside an engine's cylinder.

Thermodynamics

Thermodynamics is the science behind heating and energy transfer, especially as it pertains to engines like the Otto Cycle. It focuses on how heat energy is converted into mechanical work, like moving pistons in an engine.
The Otto cycle, a core part of thermodynamics, describes how an engine transforms energy through compression and ignition of the air-fuel mixture.

• By applying the laws of thermodynamics, we calculate the efficiency of an engine cycle.

The study of thermodynamics involves various laws that define energy transfer, including the First and Second Laws of Thermodynamics. Both of these pertain to the conservation of energy and the efficiency of energy conversions, respectively, making them vital for understanding and calculating how efficient an engine can be.