Problem 3 - College 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 is 6500 J; thermal efficiency is 33.85%.

Step 1: Understanding the Problem

We are given that a diesel engine does 2200 J of mechanical work and discards 4300 J of heat each cycle. Our tasks are to find the amount of heat supplied to the engine in each cycle and calculate its thermal efficiency.

Step 2: Finding Heat Supplied (Q_in)

According to the first law of thermodynamics, the heat supplied to the engine (Q_in) minus the heat discarded (Q_out) is the mechanical work (W) done by the engine. Mathematically, this is represented by the equation:\[ Q_{in} = W + Q_{out} \]Substituting the given values:\[ Q_{in} = 2200 \,\mathrm{J} + 4300 \,\mathrm{J} = 6500 \,\mathrm{J} \]

Step 3: Calculating Thermal Efficiency

The thermal efficiency (\eta) of an engine is calculated using the ratio of work output (W) to heat input (Q_in), expressed as a percentage:\[ \eta = \left( \frac{W}{Q_{in}} \right) \times 100 \]Substituting the known values:\[ \eta = \left( \frac{2200 \,\mathrm{J}}{6500 \,\mathrm{J}} \right) \times 100 \approx 33.85\% \]

Key Concepts

First Law of Thermodynamics

The first law of thermodynamics is a fundamental principle which states that energy cannot be created or destroyed. Instead, energy is transferred or transformed from one form to another. In the context of a heat engine like a diesel engine, this law is expressed as the relationship between heat added to the system, the work done by the system, and the heat discarded by the system.
In a mathematical form, this can be expressed as:

• Input Energy = Output Energy

Or more specifically:\[ Q_{in} = W + Q_{out} \]Where:

• \( Q_{in} \) = Heat supplied to the engine
• \( W \) = Mechanical work done by the engine
• \( Q_{out} \) = Heat exhaustion

This equation signifies that the heat energy supplied to the engine is partly used to perform mechanical work and the rest is lost as exhaust heat. This balanced equation helps in calculating unknown quantities, given the others.

Thermal Efficiency

Thermal efficiency is a measure of how well an engine converts the heat input into useful work. It indicates the engine's performance efficiency.
The formula for thermal efficiency is:\[\eta = \left( \frac{W}{Q_{in}} \right) \times 100\]This equation helps in assessing the performance of engines by comparing work output to the heat supplied:

• \( \eta \) = Thermal efficiency in percentage
• \( W \) = Mechanical work output
• \( Q_{in} \) = Total heat input to the engine

Using this formula, we can estimate what fraction of energy input is being effectively converted into work and how much is dissipated as waste energy. For instance, if an engine has a thermal efficiency of 33.85%, it means only 33.85% of the input energy is converted into mechanical work, while the rest is lost.

Mechanical Work

Mechanical work in thermodynamics refers to the energy transferred by the system to move or exert force. In engines, mechanical work is the useful energy output used to perform tasks like moving a vehicle.
Work is achieved when a force causes displacement, and in engines, this is typically done by pistons moving within cylinders. The work done by a heat engine can be calculated using the first law of thermodynamics, as discussed before, in:

• \( W = Q_{in} - Q_{out} \)

This relationship highlights the interplay between the heat supplied, heat removed, and the work extracted. In the given problem, the mechanical work is 2200 J, indicating the amount of energy being used effectively to produce motion.

Heat Transfer

Heat transfer in thermodynamics deals with the movement of thermal energy from a warm area to a cooler one. In diesel engines, heat transfer is crucial, as the energy from burning fuel is transferred as heat to perform work.
When considering the engine cycle:

• \( Q_{in} \) represents the heat energy supplied, typically from fuel combustion.
• \( Q_{out} \) refers to the heat energy that is expelled, such as exhaust.

The balance of heat input and output determines the amount of work that can be extracted. Proper management of heat transfer maximizes efficiency. Effective heat transfer allows a greater portion of the input energy to be converted into work, which is the goal in optimizing engine performance.