Problem 2 - University Physics with Modern Physics

Question

An aircraft engine takes in 9000 J of heat and discards 6400 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% efficiency.

Step 1: Identify Given Information

The problem states that the engine takes in 9000 J of heat energy (let's denote it as \( Q_{ ext{in}} = 9000 \, \text{J} \)) and discards 6400 J of heat energy (denoted as \( Q_{ ext{out}} = 6400 \, \text{J} \)) each cycle.

Step 2: Calculate Mechanical Work Output

The work output \( W \) of the engine is the difference between the heat energy taken in and the heat energy discarded. We use the formula:\[W = Q_{ ext{in}} - Q_{ ext{out}}\]Plug in the given values:\[W = 9000 \, \text{J} - 6400 \, \text{J} = 2600 \, \text{J}\]Thus, the mechanical work output is 2600 J per cycle.

Step 3: Calculate Thermal Efficiency

The thermal efficiency \( \eta \) of the engine is given by the ratio of the work output to the heat input, expressed as a percentage:\[\eta = \left( \frac{W}{Q_{ ext{in}}} \right) \times 100\%\]Substitute the values found into the efficiency formula:\[\eta = \left( \frac{2600 \, \text{J}}{9000 \, \text{J}} \right) \times 100\% \approx 28.89\%\]This means that the thermal efficiency of the engine is approximately 28.89%.

Key Concepts

Mechanical Work

In thermodynamics, mechanical work is an important concept that describes the energy transfer that results in motion. It plays a crucial role in understanding engines and how they convert heat into useful work. For an aircraft engine, like the one in our exercise, mechanical work is determined by the difference between the heat energy input and the heat energy output during each cycle.

The formula for calculating mechanical work output is:

• The work done by the engine, \( W \), equals the heat taken in, \( Q_{\text{in}} \), minus the heat discarded, \( Q_{\text{out}} \).

Thus, in simpler terms, it is the leftover energy after the engine has used up part of the heat absorbed. In our problem, the engine takes in 9000 J and discards 6400 J, resulting in a work output of 2600 J. This indicates the usable energy for propulsion or other mechanical tasks performed by the engine.

Thermal Efficiency

Thermal efficiency is a measure of how effectively a heat engine converts the heat energy it absorbs into mechanical work. It is expressed as a percentage, representing the portion of input energy that is transformed into useful work.

In our context, thermal efficiency \( \eta \) is computed with the formula:

• \( \eta = \left( \frac{W}{Q_{\text{in}}} \right) \times 100\% \)

Here, \( W \) is the work output and \( Q_{\text{in}} \) is the heat input. From our problem, with a work output of 2600 J and heat input of 9000 J, we calculate the thermal efficiency to be approximately 28.89%.

This percentage indicates that only about 29% of the absorbed heat is converted to work, with the remainder lost as waste heat. It's good to remember that no heat engine is 100% efficient due to the inevitable loss of some energy, often as heat to the surroundings.

Heat Engine Cycle

A heat engine cycle refers to the series of processes that a heat engine uses to convert heat energy into mechanical work, often involving repeated sequences. In each cycle of our example engine, heat energy is absorbed and a portion is transformed into work while the rest is expelled.

Key elements of the heat engine cycle are:

• **Heat Intake:** Absorption of heat \( Q_{\text{in}} \), initiating the cycle with energy input.
• **Work Output:** Transformation of part of the absorbed heat into work \( W \).
• **Heat Rejection:** Expulsion of the remaining heat \( Q_{\text{out}} \) to complete the cycle.

Each cycle's outcome governs how effectively an engine operates, dictating things like fuel consumption and energy efficiency. In engineering and real-world applications, understanding these cycles helps in making engines that are more efficient and better for their purpose, such as in aircraft powerplants.