/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 The Otto-cycle engine in a Merce... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 20: Problem 7

The Otto-cycle engine in a Mercedes-Benz SLK230 has a compression ratio of 8.8 (a) What is the ideal efficiency of the engine? Use \(\gamma=1.40 .\) (b) The engine in a Dodge Viper GT2 has a slightly higher compression ratio of \(9.6 .\) How much increase in the ideal efficiency results from this increase in the compression ratio?

Short Answer

Expert verified
(a) Ideal efficiency for SLK230: Approximately 52.4%. (b) Increase in efficiency: Approximately 1.2%.

Step by step solution

01

Formula for Ideal Efficiency

The ideal efficiency \((\eta)\) of an Otto cycle engine can be determined using the formula: \[ \eta = 1 - \frac{1}{r^{\gamma - 1}} \] where \(r\) is the compression ratio and \(\gamma\) is the specific heat ratio, which is given as \(1.40\) in this problem.
02

Calculate Efficiency for SLK230

For the Mercedes-Benz SLK230, with a compression ratio \(r = 8.8\), substitute the values into the efficiency formula: \[ \eta_{SLK230} = 1 - \frac{1}{8.8^{1.40 - 1}} \]. Compute this to find the efficiency.
03

Calculate Efficiency for Viper GT2

For the Dodge Viper GT2, with a compression ratio \(r = 9.6\), substitute the values into the efficiency formula: \[ \eta_{Viper} = 1 - \frac{1}{9.6^{1.40 - 1}} \]. Compute this to find the efficiency.
04

Find Increase in Efficiency

Subtract the efficiency of the SLK230 from that of the Viper GT2 to find the increase in efficiency: \[ \text{Increase} = \eta_{Viper} - \eta_{SLK230} \]. Substitute the calculated efficiencies to find the result.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Compression Ratio
The compression ratio is fundamental in understanding the function of an Otto cycle engine. It is the ratio of the total volume of the cylinder when the piston is at the bottom (known as the bottom dead center) to the volume remaining when the piston is at the top (known as the top dead center). This ratio influences the pressure and temperature inside the cylinder. A higher compression ratio usually means a more efficient engine, as it generally results in better fuel combustion, leading to more energy being extracted. The compressive ability allows the Otto cycle to convert more of the fuel energy into useful work.
Thermodynamics
The principles of thermodynamics are crucial in analyzing engine cycles like the Otto cycle. Thermodynamics involves the study of heat and energy transfer and their effects on physical systems. In an Otto cycle engine, energy from fuel is converted into mechanical work. This is achieved through a series of thermodynamic processes including adiabatic compression and expansion, and isochoric heating and cooling. Understanding these processes helps in analyzing how efficiently an engine converts fuel into work, and is fundamental to improving engine designs.
Heat Engine Efficiency
Heat engine efficiency assesses how well an engine converts heat energy from fuel into mechanical energy. The efficiency of an Otto cycle engine can be determined using the formula: \[ \ \eta = 1 - \frac{1}{r^{\gamma - 1}} \]where \(r\) is the compression ratio and \(\gamma\) is the specific heat ratio. This formula highlights that efficiency improves with a higher compression ratio and a specific heat ratio. An ideal efficiency assumes no energy loss during conversion processes. However, real engines have inherent losses making their efficiency lower.
Specific Heat Ratio
The specific heat ratio \((\gamma)\) is significant in thermodynamics and heat engines. It is the ratio of the specific heat capacity at constant pressure \(C_p\) to the specific heat capacity at constant volume \(C_v\). This ratio affects engine efficiency calculations, as seen in the formula for the Otto cycle's efficiency. For air and most diatomic gases, this value is approximately 1.4, which is a common assumption for many engine efficiency problems. A higher \(\gamma\) typically suggests increased efficiency, as it indicates a greater capacity to do work with the same heat input.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A box is separated by a partition into two parts of equal volume. The left side of the box contains 500 molecules of nitrogen gas; the right side contains 100 molecules of oxygen gas. The two gases are at the same temperature. The partition is punctured, and equilibrium is eventually attained. Assume that the volume of the box is large enough for each gas to undergo a free expansion and not change temperature. (a) On average, how many molecules of each type will there be in either half of the box? (b) What is the change in entropy of the system when the partition is punctured? (c) What is the probability that the molecules will be found in the same distribution as they were before the partition was punctured-that is, 500 nitrogen molecules in the left half and 100 oxygen molecules in the right half?

Entropy Change from Digesting Fat. Digesting fat produces 9.3 food calories per gram of fat, and typically 80\(\%\) of this energy goes to heat when metabolized. (One food calorie is 1000 calories and therefore equals 4186 \(\mathrm{J} . )\) The body then moves all this heat to the surface by a combination of thermal conductivity and motion of the blood. The internal temperature of the body (where digestion occurs) is normally \(37^{\circ} \mathrm{C},\) and the surface is usually about \(30^{\circ} \mathrm{C} .\) By how much do the digestion and metabolism of a \(2.50-\mathrm{g}\) pat of butter change your body's entropy? Does it increase or decrease?

An air conditioner operates on 800 \(\mathrm{W}\) of power and has a performance coefficient of 2.80 with a room temperature of \(21.0^{\circ} \mathrm{C}\) and an outside temperature of \(35.0^{\circ} \mathrm{C}\) . (a) Calculate the rate of heat removal for this unit. (b) Calculate the rate at which heat is discharged to the outside air. (c) Calculate the total entropy change in the room if the air conditioner runs for 1 hour. Calculate the total entropy change in the outside air for the same time period. (d) What is the net change in entropy for the system (room + outside air)?

As a budding mechanical engineer, you are called upon to design a Carnot engine that has 2.00 mol of a monatomic ideal gas as its working substance and operates from a hightemperature reservoir at \(500^{\circ} \mathrm{C}\) . The engine is to lift a 15.0 -kg weight 2.00 \(\mathrm{m}\) per cycle, using 500 \(\mathrm{J}\) of heat input. The gas in the engine chamber can have a minimum volume of 5.00 \(\mathrm{L}\) during the cycle. (a) Draw a \(p V\) -diagram for this cycle. Show in your diagram where heat enters and leaves the gas. (b) What must be the temperature of the cold reservoir? (c) What is the thermal efficiency of the engine? (d) How much heat energy does this engine waste per cycle? (e) What is the maximum pressure that the gas chamber will have to withstand?

An experimental power plant at the Natural Energy Laboratory of Hawaii generates electricity from the temperature gradient of the ocean. The surface and deep-water temperatures are \(27^{\circ} \mathrm{C}\) and \(6^{\circ} \mathrm{C},\) respectively. (a) What is the maximum theoretical efficiency of this power plant? (b) If the power plant is to produce 210 \(\mathrm{kW}\) of power, at what rate must he extracted from the warm water? At what rate must heat be absorbed by the cold water? Assume the maximum theoretical efficiency. (c) The cold water that enters the plant leaves it at a temperature of \(10^{\circ} \mathrm{C}\) . What must be the flow rate of cold water through the system? Give your answer in \(\mathrm{kg} / \mathrm{h}\) and in \(\mathrm{L} / \mathrm{h}\) .
See all solutions

Recommended explanations on Physics Textbooks

Famous Physicists

Read Explanation

Dynamics

Read Explanation

Radiation

Read Explanation

Engineering Physics

Read Explanation

Medical Physics

Read Explanation

Fields in Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.