Problem 6 - College 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

653.71 kW of heat must be supplied, and 475.71 kW of heat is discarded.

Step 1: Understand the Concepts

The power output of the engine is the work done per second. Thermal efficiency is the ratio of the work output to the heat input. The discarded heat is the difference between the heat supplied and the work output.

Step 2: Determine the Heat Supplied

The formula for thermal efficiency is given by: \( \text{Efficiency} = \frac{W}{Q_{in}} \), where \( W \) is the work output and \( Q_{in} \) is the heat input. We can rearrange this as: \( Q_{in} = \frac{W}{\text{Efficiency}} \). Substitute \( W = 180 \text{ kW} \) and \( \text{Efficiency} = 0.28 \) to find \( Q_{in} \).\[ Q_{in} = \frac{180}{0.28} = 642.86 \text{ kW} \]

Step 3: Calculate Heat Discarded

The heat discarded \( Q_{out} \) can be found using the formula: \( Q_{out} = Q_{in} - W \). Substitute \( Q_{in} = 642.86 \text{ kW} \) and \( W = 180 \text{ kW} \) to find \( Q_{out} \).\[ Q_{out} = 642.86 - 180 = 462.86 \text{ kW} \]

Step 4: Conclusion

The heat that must be supplied to the engine is approximately 642.86 kW, and the heat discarded by the engine is about 462.86 kW.

Key Concepts

Engine Power Output

Engine power output is a measure of how much work is done by an engine in a unit time. It is typically measured in kilowatts (kW) or horsepower (hp), with 1 hp equaling about 0.7355 kW. In our case, the engine power output is 180 kW, or approximately 241 hp. This value represents the engine’s capacity to convert fuel energy into mechanical energy, which is the useful work that the engine can perform each second. Understanding this concept is crucial because it forms the foundation for calculating other quantities like thermal efficiency and heat input.

Heat Input

Heat input refers to the total amount of thermal energy supplied to an engine, usually measured per unit time, in kilowatts (kW) or other energy units. For thermal engines, this is typically the energy liberated from the combustion of fuel. The heat input \(Q_{in}\) to an engine is calculated using the relationship with thermal efficiency. Thermal efficiency is defined by the formula:

• \(\text{Efficiency} = \frac{W}{Q_{in}}\)

where \(W\) is the work output. Rearranging gives:

• \(Q_{in} = \frac{W}{\text{Efficiency}}\)

By substituting the known values:

• \(W = 180 \text{ kW}\)
• \(\text{Efficiency} = 0.28\)

we find:

• \(Q_{in} = \frac{180}{0.28} = 642.86 \text{ kW}\)

This value represents the energy needed each second to keep the engine running at its power output.

Work Output

Work output is the useful mechanical energy produced by an engine from the heat input. It is often synonymous with the engine power output in practical terms. Work output is what engines are built for; it is the energy available for performing tasks such as moving a vehicle or driving machinery. Since work output directly relates to engine performance, understanding how it connects to heat input and efficiency is key. For instance, a higher thermal efficiency means more of the heat input is converted into work output, which is desirable in any engine.

Heat Discarded

Heat discarded, or waste heat, is the thermal energy that is not converted into work. Instead, it gets expelled from the engine, often into the atmosphere. This wasted energy highlights the inefficiencies present in thermal engines. You can calculate the heat discarded \(Q_{out}\) with the formula:

• \(Q_{out} = Q_{in} - W\)

Using values from the example:

• \(Q_{in} = 642.86 \text{ kW}\)
• \(W = 180 \text{ kW}\)

we find:

• \(Q_{out} = 642.86 - 180 = 462.86 \text{ kW}\)

This figure underscores the importance of improving engine efficiency, as a significant portion of the energy is not utilized for work but is instead lost as waste.