Problem 4 - College Physics

Question

\(\cdot\) An aircraft engine takes in 9000 \(\mathrm{J}\) of heat and discards 6400 \(\mathrm{J}\) each cycle. (a) What is the mechanical work output of the engine during one cycle? (b) What is the thermal efficiency of the engine?

Step-by-Step Solution

Answer

(a) 2600 J; (b) 28.89%

Step 1: Identify Given Values

Determine the values provided in the problem. The heat absorbed, \( Q_{in} = 9000 \) J, and the heat discarded, \( Q_{out} = 6400 \) J, are given.

Step 2: Calculate Work Output

Use the formula for work output, \( W = Q_{in} - Q_{out} \), to find the work done by the engine in each cycle. Substitute the known values: \( W = 9000 \text{ J} - 6400 \text{ J} = 2600 \text{ J} \).

Step 3: Formula for Thermal Efficiency

Thermal efficiency \( \eta \) can be calculated using the formula \( \eta = \frac{W}{Q_{in}} \), where \( W \) is the work output.

Step 4: Substitute Values to Find Efficiency

Substitute \( W = 2600 \text{ J} \) and \( Q_{in} = 9000 \text{ J} \) into the efficiency formula: \( \eta = \frac{2600}{9000} \).

Step 5: Calculate Thermal Efficiency

Perform the calculation: \( \eta = \frac{2600}{9000} \approx 0.2889 \) or \( 28.89\% \).

Key Concepts

Understanding Thermal Efficiency

When we talk about thermal efficiency, we're really looking at how well a heat engine converts heat into work. In simpler terms, it's a measure of how good an engine is at using the energy it receives. The efficiency of an engine is represented as a percentage, which tells us what proportion of the heat energy input is turned into useful work output. This is crucial when evaluating the performance of engines, especially those used in aircraft. To calculate thermal efficiency (\( \eta \)= eta), we use the formula: \[\eta = \frac{W}{Q_{in}}\] where \( W \) is the work output, and \( Q_{in} \) is the heat input. In the exercise example, the formula calculates the efficiency as approximately 28.89%. This means that approximately 28.89% of the heat energy taken in by the engine is being converted into mechanical work. The rest is often lost as waste heat, which impacts the overall performance of the engine.

Calculating Work Output in Engines

Work output is a key concept in understanding heat engines. It refers to the amount of work that the engine performs or delivers in one complete cycle. In our example, this work is simply the difference between the heat input and the heat output. Using the formula for work output: \[ W = Q_{in} - Q_{out} \]We calculate the work output by subtracting the heat discarded \( Q_{out} \) from the heat absorbed \( Q_{in} \). For the given problem, the values are \( Q_{in} = 9000 \) J and \( Q_{out} = 6400 \) J, leading to a work output of 2600 J.This tells us that out of the 9000 J of heat energy taken in, 2600 J is effectively used to perform work. The remaining energy is not converted into work and is typically lost as waste heat.

Exploring Heat Engines

Heat engines are fascinating devices that transform heat into work. They take heat energy from a source, use part of it to do work, and then discard the rest as waste heat. This basic principle helps power many machines, including car engines, power plants, and aircraft engines. A typical heat engine works through a cycle.

• In the cycle, heat is absorbed from a high-temperature source.
• It then performs work.
• Finally, heat is discarded at a lower temperature.

The performance of a heat engine is largely determined by its thermal efficiency, with higher thermal efficiency indicating a better conversion of heat into work. Each component of the cycle is essential for understanding how effectively an engine operates. Knowing how much energy is absorbed and discarded is key to improving engine designs for better performance and reduced energy waste.