Problem 60 - University Physics with Modern Physics

Question

Engine Thrbochargers and Intercoolers. The power output of an automobile engine is directly proportional to the mass of air that can be forced into the volume of the engine's cylinders to react chemically with gasoline. Many cars have a turbochargers which compresses the air before it enters the engine, giving a greater mass of air per volume. This rapid, essentially adiabatic compression also heats the air. To compress it further, the air then passes through an intercooler in which the air exchanges heat with its surroundings at essentially constant pressure. The air is then drawn into the cylinders. In a typical installation, air is taken into the turbocharger at atmospheric pressure \(\left(1.01 \times 10^{5} \mathrm{Pa}\right)\) , density \(p=1.23 \mathrm{kg} / \mathrm{m}^{3}\) , and temperature \(15.0^{\circ} \mathrm{C}\) . It is compressed adiabatically to \(1.45 \times 10^{5} \mathrm{Pa}\) . In the intercooler, the air is cooled to the original temperature of \(15.0^{\circ} \mathrm{C}\) at a constant pressure of \(1.45 \times 10^{5} \mathrm{Pa}\) (a) Draw a \(p V\) -diagram for this sequence of processes. b) If the volume of one of the engine's cylinders is 575 \(\mathrm{cm}^{3}\) , what mass of air exiting from the intercooler will fill the cylinder at \(1.45 \times 10^{5} \mathrm{Pa} 2\) Compared to the power output of an engine that takes in air at \(1.01 \times 10^{5} \mathrm{Pa}\) at \(15.0^{\circ} \mathrm{C}\) , what percentage increase in power is obtained by using the turbocharger and intercooler? (c) If the intercooler is not used, what mass of air exiting from the turbocharger will fill the cylinder at \(1.45 \times 10^{5} \mathrm{Pa} ?\) Compared to the power output of an engine that takes in air at obtained by using the turbocharger alone?

Step-by-Step Solution

Answer

Using the intercooler increases engine power output significantly more than the turbocharger alone by allowing a higher air mass into the cylinders.

Step 1: Drawing the pV Diagram

The process involves two main stages: adiabatic compression by the turbocharger and cooling via the intercooler. Start at point A, representing the initial state with pressure \(1.01 \times 10^5 \mathrm{Pa}\), density \(1.23 \mathrm{kg/m^3}\), and temperature \(15.0^{\circ} \mathrm{C}\). Upon adiabatic compression, move to point B at \(1.45 \times 10^5 \mathrm{Pa}\). The compression process on a \(pV\) diagram is a steep curve moving rightward. From point B, the air is cooled at constant pressure back to \(15.0^{\circ} \mathrm{C}\), moving left to point C. This process is horizontal on the diagram since pressure remains constant.

Step 2: Calculating Mass of Air after Intercooling

To find the mass of air from the intercooler that will fill a cylinder of volume \(575 \mathrm{cm}^3\) at \(1.45 \times 10^5 \mathrm{Pa}\), use the ideal gas law: \(pV = nRT\). First, convert the volume to \(m^3\) by dividing by 1,000,000. Calculate mols \(n\) using \(n = \frac{pV}{RT}\), where \(R\) is the specific gas constant \(287 \mathrm{J/kg \, K}\) and \(T\) is in Kelvin \(288.15 \mathrm{K}\). Substitute values: \(n = \frac{1.45 \times 10^5 \times 0.000575}{287 \times 288.15}\), then convert moles to mass using the molar mass of air \(0.029 \mathrm{kg/mol}\).

Step 3: Calculating Percentage Increase in Power

Compare the mass after cooling with the mass initially at atmospheric conditions. Similarly, use \(pV = nRT\) for air intake at \(1.01 \times 10^5 \mathrm{Pa}\). Calculate the initial mass of air \(n_0 = \frac{1.01 \times 10^5 \times 0.000575}{287 \times 288.15}\). Determine the mass ratio \(\frac{mass_{intercooler}}{mass_0}\) and compute the power increase percentage: \((\frac{mass_{intercooler}}{mass_0} - 1) \times 100\%\).

Step 4: Calculating Mass of Air without Intercooling

For the condition without using an intercooler, the compressed air remains at a higher temperature due to adiabatic compression. Use the adiabatic equation \(T_2 = T_1 \left( \frac{p_2}{p_1} \right)^{\frac{\gamma - 1}{\gamma}}\), where \(\gamma = 1.4\) for air. Find the new temperature \(T_2\) and use this in the ideal gas law to find the new mass \(n_2 = \frac{1.45 \times 10^5 \times 0.000575}{287 \times T_2}\). Convert to mass using the molar mass.

Step 5: Calculating Power Increase without Intercooler

Compare the mass of air without an intercooler to the initial mass. Similarly, calculate the percentage increase \((\frac{mass_{adiabatic}}{mass_0} - 1) \times 100\%\) to determine the effect on power output using the turbocharger alone.

Key Concepts

Adiabatic Compression

Adiabatic compression is a thermodynamic process where a gas is compressed without any heat exchange with its surroundings. This means that all the energy put into the system goes into compressing the gas, causing its temperature to rise. This type of compression is very common in applications like turbochargers in engines.
During adiabatic compression, the temperature increase happens because the kinetic energy of the gas molecules increases as they are forced closer together.
To understand this process, you need to know a bit about how gases behave. The general behavior of gases during adiabatic processes is described by the equation: \( pV^\gamma = \text{constant} \) where \(p\) is pressure, \(V\) is volume, and \(\gamma\) is the heat capacity ratio. For air, \(\gamma\) is approximately 1.4.
The equation shows that when the pressure increases, the volume decreases, but in such a way that their product raised to the power of the heat capacity ratio remains constant. This tells us how much the temperature will change when pressure changes due to compression.

Ideal Gas Law

The Ideal Gas Law is a formula used to relate the pressure, volume, and temperature of an ideal gas. The equation is: \( pV = nRT \)
Here, \(p\) is the pressure, \(V\) is the volume, \(n\) represents the number of moles of gas, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.
This equation helps describe how a gas will behave under different conditions. For example, if you know any three of the variables, you can calculate the fourth. This makes it very useful in engineering applications like calculating the mass of air compressed in an engine cylinder.
Using the Ideal Gas Law, it's possible to calculate the mass of air filling a space after processes like adiabatic compression, since you can find the number of moles \(n\) and then convert this to mass using the molar mass of the gas, which for air is about 29 g/mol. Understanding this basic principle is key to solving real-world problems related to gas behavior in engines and other systems.

Engine Efficiency

Engine efficiency is fundamentally about how well an engine converts the energy from fuel into useful work. It's expressed as a percentage, representing how much of the fuel's energy ends up as mechanical energy rather than being lost as heat.
Turbochargers and intercoolers are used to improve engine efficiency. A turbocharger compresses incoming air, increasing its pressure and allowing more air—and hence more fuel—to enter the engine, leading to greater power output.

• Adiabatic compression by turbochargers increases air temperatures, which isn't always beneficial as hotter air contains less oxygen per volume.
• An intercooler helps here, cooling the air after compression without reducing its pressure, so it retains more oxygen per volume upon reaching the engine.

Improving efficiency in this way can lead to a greater percentage increase in power output. For instance, comparing the mass of air entering the engine with and without these devices shows how much additional air can be utilized, which directly correlates to increased power. Understanding these concepts will help you appreciate the role of thermodynamics in enhancing engine performance.