Problem 1 - University Physics with Modern Physics

Question

A diesel engine performs 2200 \(\mathrm{J}\) of mechanical work and discards 4300 \(\mathrm{J}\) of heat each cycle. (a) How much heat must be supplied to the engine in each cycle? (b) What is the thermal efficiency of the engine?

Step-by-Step Solution

Answer

Heat supplied: 6500 J; Efficiency: 33.8%.

Step 1: Understand the Problem

We know the work done by the engine and the heat discarded during one cycle. We need to calculate the heat input required for one cycle and the thermal efficiency of the engine.

Step 2: Apply Energy Conservation Principle

By the first law of thermodynamics, the heat supplied to the engine (\(Q_{in}\)) is equal to the sum of the work done (\(W\)) and the heat discarded (\(Q_{out}\)). So, \(Q_{in} = W + Q_{out}\).

Step 3: Calculate Heat Supplied

Substitute the given values into the equation from Step 2: \(Q_{in} = 2200 \, \mathrm{J} + 4300 \, \mathrm{J}\). Compute this to find \(Q_{in}\).

Step 4: Perform the Calculation

Calculate \(Q_{in}\):\[Q_{in} = 2200 \, \mathrm{J} + 4300 \, \mathrm{J} = 6500 \, \mathrm{J}\].

Step 5: Define Thermal Efficiency

The thermal efficiency \(\eta\) of an engine is defined as the ratio of work output to heat input: \(\eta = \frac{W}{Q_{in}}\).

Step 6: Calculate Thermal Efficiency

Substitute the values from previous steps: \[\eta = \frac{2200 \, \mathrm{J}}{6500 \, \mathrm{J}}\].Calculate \(\eta\) to find the engine's efficiency.

Step 7: Perform the Calculation for Efficiency

Calculate \(\eta\):\[\eta = \frac{2200}{6500} \approx 0.338\].Convert this to percentage form by multiplying by 100, which gives \(\eta \approx 33.8\%\).

Key Concepts

First Law of Thermodynamics

The First Law of Thermodynamics is a fundamental concept in physics that deals with energy conservation. It states that energy cannot be created or destroyed; it can only be transformed from one form to another. In simple terms, for any given system, the change in internal energy equals the heat added minus the work done by the system. This can be represented as:
\[ \Delta U = Q - W \]However, in many problems like the one with our diesel engine, the focus shifts to find how much heat needs to be supplied to perform a certain amount of work. In this case, we focus on the relationship:
\[ Q_{in} = W + Q_{out} \]

• **\(Q_{in}\)** is the heat supplied to the system.
• **\(W\)** is the work done by the system.
• **\(Q_{out}\)** is the heat discarded by the system.

By understanding this process, we see that the diesel engine converts heat into work while discarding some heat inevitably due to inefficiencies. The amount of heat supplied is crucial as it determines how much work the engine can perform.

Thermal Efficiency

Thermal efficiency is a key measure of how well a system uses energy. In the context of engines, it represents the portion of heat energy that is converted into mechanical work. The formula for thermal efficiency, represented as \(\eta\), is:
\[ \eta = \frac{W}{Q_{in}} \]This equation gives a ratio that shows the effectiveness of the engine in utilizing the input heat. It is generally expressed as a percentage:
- **If \(\eta\) = 100%**, no heat is lost, and all supplied energy is converted into work (an ideal but impossible scenario).- Engines will usually have a lower efficiency due to unavoidable losses like friction and heat dissipation.
In our example, the diesel engine achieves a thermal efficiency of approximately \(33.8\%\). This indicates that only a third of the heat energy supplied goes into performing mechanical work, reflecting the realistic performance of practical engines.

Heat Transfer

Heat transfer is the process of thermal energy moving from a hotter area to a cooler one. In engines, this process is inevitability part of energy conversion cycles. There are three modes of heat transfer:

• **Conduction**: Direct molecular transfer through a medium.
• **Convection**: Motion of heat in fluids like air and water.
• **Radiation**: Energy transfer through electromagnetic waves, without needing a medium.

During a diesel engine cycle, some heat is transferred from the combustion chamber to the surroundings due to conduction and convection. This 'heat discarded' (\(Q_{out}\)) is energy not used for work. Effective management and minimization of this heat loss can improve an engine's efficiency. Understanding the principles of heat transfer can help in designing better engines with improved performance and less wasted energy.