Problem 4 - University Physics with Modern Physics

Question

A gasoline engine has a power output of 180 \(kW\)(about 241 hp). Its thermal efficiency is 28.0%. (a) How much heat must be supplied to the engine per second? (b) How much heat is discarded by the engine per second?

Step-by-Step Solution

Answer

(a) 642.86 kW; (b) 462.86 kW.

Step 1: Understanding Power and Efficiency

The power output of the engine is the useful work done per second. The thermal efficiency formula is given by \(\eta = \frac{W}{Q_{in}}\), where \(\eta\) is the efficiency, \(W\) is the work done or power output, and \(Q_{in}\) is the heat energy supplied per second. The engine's power output is 180 kW, and its thermal efficiency is 28%.

Step 2: Calculate Heat Supplied Per Second (Part a)

Rearrange the formula for efficiency to find the heat input: \(Q_{in} = \frac{W}{\eta}\). Substituting the values, \(Q_{in} = \frac{180 \, \text{kW}}{0.28} = 642.86 \, \text{kW}\). Therefore, approximately 642.86 kW of heat must be supplied per second.

Step 3: Calculate Heat Discarded Per Second (Part b)

The heat discarded \(Q_{out}\) can be calculated using the relation \(Q_{out} = Q_{in} - W\). From part a, \(Q_{in} = 642.86 \, \text{kW}\) and \(W = 180 \, \text{kW}\). Therefore, \(Q_{out} = 642.86 \, \text{kW} - 180 \, \text{kW} = 462.86 \, \text{kW}\). So, approximately 462.86 kW of heat is discarded per second.

Key Concepts

Heat Engine

A heat engine is a fascinating piece of technology used to convert thermal energy into mechanical work. Essentially, it captures heat from a high-temperature source, performs work, and then releases some of the heat at a lower temperature. The power process is much like what happens in a gasoline engine.
In this scenario, the gasoline engine receives thermal energy from burning fuel. It uses part of this energy to produce useful mechanical work, such as moving your car. However, not all energy is transformed into work; some energy is inevitably lost as waste heat.
Key features of a heat engine include:

• Source of thermal energy: Typically fuel combustion in gasoline engines.
• Conversion to work: Mechanical power that moves a vehicle.
• Heat rejection: The unwanted heat leaving the engine, often resulting in energy loss.

Understanding how a heat engine works helps in appreciating the importance of efficiency, as engineers constantly seek to minimize waste and maximize work output.

Thermal Efficiency

Thermal efficiency is a crucial measure of a heat engine's performance. It describes how effectively the engine converts heat into useful work. In general, efficiency is about maximizing output for a given input.To calculate thermal efficiency (\(\eta \)), we use the formula:\[\eta = \frac{W}{Q_{in}}\]where:

• \(W\) is the work output (power output in kW).
• \(Q_{in}\) is the heat supplied to the engine (in kW).

For example, if an engine has a thermal efficiency of 28%, it means 28% of the heat energy provided is converted into work, while the rest is lost.
Enhancing thermal efficiency helps engines to perform better, reduce fuel consumption, and lower emissions. Understanding and optimizing efficiency parameters allows engineers to contribute to more environmentally friendly and cost-effective technologies.

Heat Transfer

Heat transfer is a key aspect and inevitable component of heat engines. It involves the movement of heat from one place to another, which can be from a hot source to engine components or from the engine to a cooler sink.
In a gasoline engine, besides converting heat into work, a significant portion of the heat is lost as waste. This process is referred to as heat rejection. Being able to manage this heat transfer efficiently ensures that the engine remains functional without overheating.
Heat transfer occurs in several ways:

• Conduction: Heat moves through solid parts of the engine.
• Convection: Heat transfers to fluids like air or coolant surrounding the engine.
• Radiation: Emission of heat in the form of radiation, especially critical for cooling.

All these heat transfer methods are crucial for the engine's cooling system, which keeps the engine at a safe operating temperature.