Question

A diesel engine performs 2200 J of mechanical work and discards 4300 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?

Short answer

(a) 6500 J, (b) 33.85% efficiency.

Step-by-step solution

Understand the Problem

We have a diesel engine performing mechanical work and losing some heat. We need to find out how much heat energy was supplied to it and its thermal efficiency.

Identify Known Values

The engine performs 2200 J of work, and 4300 J of heat is discarded in each cycle. The heat supplied to the engine is unknown.

Apply the First Law of Thermodynamics

According to the first law of thermodynamics, the heat supplied (Q_{in}) must equal the sum of the work done (W) and the heat discarded (Q_{out}):\[ Q_{in} = W + Q_{out} \]Substitute the given values:\[ Q_{in} = 2200 \, \text{J} + 4300 \, \text{J} \]

Calculate Heat Supplied

Perform the calculation from the previous step:\[ Q_{in} = 2200 \, \text{J} + 4300 \, \text{J} = 6500 \, \text{J} \]So, 6500 J of heat must be supplied to the engine in each cycle.

Formula for Thermal Efficiency

Thermal efficiency (\eta) is the ratio of the work done by the engine to the heat supplied to it, given by:\[ \eta = \frac{W}{Q_{in}} \]

Calculate Thermal Efficiency

Substitute the known values into the efficiency equation:\[ \eta = \frac{2200 \, \text{J}}{6500 \, \text{J}} \approx 0.3385 \]Convert it to percentage:\[ \eta \times 100\% \approx 33.85\% \]

Finalize the Solution

The heat supplied to the engine is 6500 J, and its thermal efficiency is approximately 33.85%.

Key concepts

First Law of Thermodynamics

The First Law of Thermodynamics is a fundamental principle in physics that addresses how energy is conserved within a system. It is essentially a restatement of the law of conservation of energy. This law posits that energy can neither be created nor destroyed, only transformed from one form to another. In the context of engines, such as a diesel engine, this transformation usually occurs between heat energy and mechanical energy.

For a working engine, the energy input is heat energy (often from burning fuel), and the output includes mechanical work and heat lost to the surroundings. Mathematically, the First Law of Thermodynamics for a closed cycle is expressed as:

• The heat added to the system, denoted as \( Q_{in} \), equals the work done by the system, \( W \), plus the heat lost, \( Q_{out} \): \[ Q_{in} = W + Q_{out} \]

This equation ensures that all energy introduced into the system is accounted for as either work or heat loss. It is a vital concept in determining the efficiency of thermal systems.

Diesel Engine

A diesel engine is a type of internal combustion engine noted for its higher efficiency compared to gasoline engines. It converts the chemical energy in diesel fuel into mechanical energy that can do work, like moving a vehicle or powering machinery. Diesel engines work on a cycle that involves:

• Intake of air and fuel mixture
• Compression of the mixture
• Ignition and expansion that does mechanical work
• Exhaust of combustion gases

Key to their operation is the compression phase, where the air-fuel mixture is compressed to high pressures, raising its temperature until it combusts spontaneously (without a spark plug).
This process is efficient because of the high compression ratio, meaning the engine can extract more energy per unit of fuel than a gasoline engine, thus running with better thermal efficiency. The diesel engine continually converts heat into mechanical work, which is analyzed using the principles discussed in the First Law of Thermodynamics.

Mechanical Work

Mechanical work in the context of engines refers to the useful energy output generated by the engine that can be used to perform tasks, such as moving a car or generating electricity. In engines, work is produced during the power or expansion stroke when the burning fuel-air mixture pushes the piston down the cylinder.

The mechanical work done by an engine can be calculated by using the formula:

• Mechanical Work (\( W \)) is the result of converting heat energy from fuel (or from another form) into kinetic energy:
  For the diesel engine in our example:\( W = 2200 \, \text{J} \)

The mechanical work is one of the components in determining the engine's efficiency and also helps apply the First Law of Thermodynamics to examine energy usage.

Heat Transfer

Heat transfer in engines is the movement of thermal energy resulting from a difference in temperature between the engine and its surroundings. In diesel engines, heat transfer plays a crucial role, as it affects the efficiency and performance. There are generally three modes of heat transfer:

• Conduction: Heat flows through materials.
• Convection: Heat circulates through fluids like air or water.
• Radiation: Heat radiates from hot surfaces like an engine block.

In the working cycle of a diesel engine, not all heat energy can be converted into mechanical work. A significant portion is discarded as waste heat, which is accounted for using the First Law of Thermodynamics (\( Q_{out} \)). Efficient heat management helps in maximizing work done and improving thermal efficiency. This involves maintaining optimal heat transfer to achieve balance between engine power and waste heat.