Problem 4 - University Physics with Modern Physics

Question

A gasoline engine has a power output of 180 \(\mathrm{kW}\) (about 241 \(\mathrm{hp}\) ). Its thermal efficiency is 28.0\(\%\) . (a) How much heat must be supplied to the engine per second? (b) How much heat is discarded by the engine per second?

Step-by-Step Solution

Answer

(a) 642.86 kW; (b) 462.86 kW.

Step 1: Understand the Concepts

The problem involves calculating the heat supplied to and discarded by an engine. Thermal efficiency is the ratio of the engine's power output to the input energy it receives, indicating how well the engine converts heat into work.

Step 2: Determine Heat Supplied

The formula for thermal efficiency \( \eta \) is given by: \( \eta = \frac{W_{ ext{out}}}{Q_{ ext{in}}} \). We rearrange this equation to solve for the heat input \( Q_{ ext{in}} \): \[ Q_{ ext{in}} = \frac{W_{ ext{out}}}{\eta} \]Here, \( W_{\text{out}} = 180 \text{ kW} \) and \( \eta = 0.28 \). Thus, \[ Q_{ ext{in}} = \frac{180}{0.28} \approx 642.86 \text{ kW} \].

Step 3: Calculate Heat Discarded

The heat discarded \( Q_{ ext{out}} \) can be determined using the formula: \[ Q_{ ext{out}} = Q_{ ext{in}} - W_{ ext{out}} \].Substituting the known values,\[ Q_{ ext{out}} = 642.86 - 180 = 462.86 \text{ kW} \].

Key Concepts

Heat Engine

A heat engine is a fascinating machine that plays a crucial role in converting heat into useful work. Imagine it like a black box that takes in energy in the form of heat, performs some magic, and then generates power. In the case of a gasoline engine, this magic involves internal combustion. Here, fuel burns in a controlled manner, releasing energy. Some of this energy is used to move pistons and do work, such as powering a car.

Understanding a heat engine helps us appreciate how everyday machines like cars and generators work. The working cycle involves pulling in energy, transforming it, and putting it to work. However, no engine is perfect, as not all the supplied energy gets converted into desirable work; some energy is inevitably lost. This energy not converted into work typically dissipates as waste heat, which is something every engineer tries to minimize. This aspect of energy loss leads us directly into the concept of thermal efficiency.

Power Output

In the context of engines, power output is essentially how much power is being produced for doing useful work. For our gasoline engine example, the power output is measured at 180 kW. This means the engine is capable of producing 180 kilowatts of power at its peak. Once converted, it powers up systems like the drivetrain in a vehicle, allowing the car to move.

Power output is measured in watts, and in some cases, like in cars, it might also be represented in horsepower (hp), where 1 hp is approximately 0.746 kW. Always remember, though, that an engine's power output will be less than the power input due to energy losses. These losses are part of what makes understanding thermal efficiency critical. Power output is the practical result of the energy conversion process happening inside the engine.

Thus, a higher power output means more energy is being efficiently converted into work, allowing for more robust and more powerful applications.

Energy Conversion

Energy conversion is at the heart of a heat engine's purpose. It's all about transforming the heat energy generated by burning fuel into mechanical work. This process isn't completely efficient, and that is why not all input heat energy ends up as useful mechanical energy.

Here's how it works step-by-step:

• Fuel burns and releases energy in the form of heat.
• This heat energy increases the pressure inside the engine's cylinder.
• The high pressure forces the piston to move, converting thermal energy into mechanical work.

But, the conversion process is imperfect. Only a portion of the input energy gets translated into mechanical work—much of it is lost due to friction, exhaust heat, and other factors. That's where the thermal efficiency metric becomes important, as it helps determine how well the engine converts heat into work.

Heat Transfer

Heat transfer is a key concept that helps us understand how energy flows in and out of a heat engine. It represents the movement of thermal energy from one part of the engine to another.

In our engine context:

• Heat is supplied to the engine during the combustion process. This is the input heat energy.
• Some of this energy gets converted to work, powering the engine's output.
• The rest of the energy isn't used for work and is discarded as waste heat.

This waste heat often gets released into the atmosphere or into a cooling system as part of the heat transfer process.

The degree of heat transfer efficiency can impact the engine's thermal efficiency. Less heat wasted means the engine is more effective at converting heat into usable power. Conversely, more heat lost indicates lower efficiency, highlighting the importance of good design in minimizing unnecessary heat transfer.