/*! 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 98 In a diesel engine, the fuel is ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 98

In a diesel engine, the fuel is ignited without a spark plug. Instead, air in a cylinder is compressed adiabatically to a temperature above the ignition temperature of the fuel; at the point of maximum compression, the fuel is injected into the cylinder. Suppose that air at \(20^{\circ} \mathrm{C}\) is taken into the cylinder at a volume \(V_{1}\) and then compressed adiabatically and quasi-statically to a temperature of \(600^{\circ} \mathrm{C}\) and a volume \(V_{2} .\) If \(\gamma=1.4, \quad\) what is the ratio \(V_{1} / V_{2} ?\) (Note: static.) In an operating diesel engine, the compression is not quasi-

Short Answer

Expert verified
The ratio of the initial volume \(V_1\) to the final volume \(V_2\) in the adiabatic compression process of a diesel engine is approximately \(4.47\).

Step by step solution

01

Express P1 and P2 in terms of T1 and T2

Using the ideal gas law, we can express pressure \(P_1\) in terms of initial temperature \(T_1\) and \(P_2\) in terms of the final temperature \(T_2\): \(P_1V_1 = nRT_1\) \(P_2V_2 = nRT_2\) Dividing both equations we obtain: \(\frac{P_1}{P_2} = \frac{T_1}{T_2}\)
02

Confirm the adiabatic process relationship

For an adiabatic process, the relationship between pressure, volume, and temperature is: \(PV^\gamma = K\) where \(K\) is a constant and \(\gamma\) is the adiabatic index (given as 1.4).
03

Write the equation for the initial and final states of the process

Now let's consider the given initial and final states of the process in terms of the adiabatic relationship and the ideal gas law: \(P_1V_1^\gamma = P_2V_2^\gamma\) Since we found the relationship between the pressures and temperatures: \(\frac{P_1}{P_2} = \frac{T_1}{T_2}\) Then, \(P_1 = \frac{T_1}{T_2}P_2\) Substitute the expression for \(P_1\) in the adiabatic relationship: \[\frac{T_1}{T_2}P_2V_1^\gamma = P_2V_2^\gamma\]
04

Solve for the volume ratio V1/V2

Now, let's solve for the ratio \(\frac{V_1}{V_2}\). Firstly, divide both sides of the equation by \(P_2\): \[\frac{T_1}{T_2}V_1^\gamma = V_2^\gamma\] Rearrange the equation for the desired ratio: \[\frac{V_1}{V_2} = \left(\frac{T_2}{T_1}\right)^\frac{1}{\gamma}\] Now, plug in the given values of \(T_1 = 20^\circ \mathrm{C} + 273.15 \mathrm{K} = 293.15 \mathrm{K}\), \(T_2 = 600^\circ \mathrm{C} + 273.15 \mathrm{K} = 873.15 \mathrm{K}\), and \(\gamma = 1.4\): \[\frac{V_1}{V_2} = \left(\frac{873.15 \mathrm{K}}{293.15 \mathrm{K}}\right)^\frac{1}{1.4}\]
05

Calculate the ratio V1/V2

Using a calculator, we have: \[\frac{V_1}{V_2} \approx 4.47 \] Therefore, the ratio of the initial volume \(V_1\) to the final volume \(V_2\) is approximately \(4.47\).

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.

Adiabatic Process
The term 'adiabatic process' plays a crucial role in understanding the operations of diesel engines. This is a process in which no heat is transferred to or from the working substance, in this case, air, during compression or expansion. It鈥檚 a fundamental concept within thermodynamics and characterizes a situation where all the work done on the gas is used to change the internal energy of the gas, leading to a change in temperature.

An adiabatic process is mathematically expressed by the formula: \(PV^\gamma = K\), where \(P\) is pressure, \(V\) is volume, \(\gamma\) is the specific heat ratio, which is a property of the gas, and \(K\) is a constant. For diesel engines, this process is essential during the compression stroke, when the air is compressed and its temperature increases, leading to the auto-ignition of the injected fuel.

This understanding can improve engagement with the original exercise by highlighting the realistic application within the engines and the significance of why no heat exchange takes place during this crucial step of the diesel engine's cycle.
Ideal Gas Law
Grasping the Ideal Gas Law is key when studying the adiabatic process in diesel engines. It鈥檚 a fundamental equation of state for a hypothetical ideal gas and is often formulated as \(PV = nRT\). Here, \(P\) stands for pressure, \(V\) is volume, \(n\) represents the number of moles of gas, \(R\) is the universal gas constant, and \(T\) signifies temperature in Kelvins.

The Ideal Gas Law is pivotal because it provides a relationship between the pressure, volume, and temperature of a gas that can be used to describe its state at any moment. This relationship guided Step 1 in our original exercise, helping us express the initial and final states of pressure in terms of the temperatures \(T_1\) and \(T_2\).

By understanding this principle, students can appreciate why changing one variable (like temperature during compression in a diesel engine) will affect the others. The step-by-step solution becomes more relatable when students see how real-world engines abide by this law.
Volume Ratio
The volume ratio is a measure that compares the change in volume of a substance from one state to another. In the context of a diesel engine, this is the ratio of the volume of air before compression \(V_1\) to the volume after compression \(V_2\), expressed as \(V_1/V_2\). It鈥檚 a key factor in the efficiency of an engine because a higher compression ratio can lead to a higher thermal efficiency.

Incorporating the volume ratio in our calculations allows us to determine the extent of compression the air undergoes inside the engine cylinder. The final step in the original exercise used this concept to solve for the volume ratio using the temperatures and the specific heat ratio. \[\frac{V_1}{V_2} = \left(\frac{T_2}{T_1}\right)^\frac{1}{\gamma}\]

Understanding the volume ratio helps students visualize the effects of the compression stroke in a diesel engine, connecting the abstract concepts of pressure, temperature, and volume with the tangible changes taking place within the confines of an engine cylinder. This connection enhances the educational value of the exercise.

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 monatomic ideal gas undergoes a quasi-static process that is described by the function \(p(V)=p_{1}+3\left(V-V_{1}\right),\) where the starting state is \(\left(p_{1}, V_{1}\right)\) and the final state \(\left(p_{2}, V_{2}\right) .\) Assume the system consists of \(\mathrm{n}\) moles of the gas in a container that can exchange heat with the environment and whose volume can change freely. (a) Evaluate the work done by the gas during the change in the state. (b) Find the change in internal energy of the gas. (c) Find the heat input to the gas during the change. (d) What are initial and final temperatures?

The state of 30 moles of steam in a cylinder is changed in a cyclic manner from a-b-c-a, where the pressure and volume of the states are: a \((30 \mathrm{atm}, 20 \mathrm{L}), \mathrm{b}(50 \mathrm{atm}, 20\) L), and \(c(50 \text { atm, } 45\) L). Assume each change takes place along the line connecting the initial and final states in the pV plane. (a) Display the cycle in the pV plane. (b) Find the net work done by the steam in one cycle. (c) Find the net amount of heat flow in the steam over the course of one cycle.

In a quasi-static isobaric expansion, \(500 \mathrm{J}\) of work are done by the gas. The gas pressure is 0.80 atm and it was originally at 20.0 L. If the internal energy of the gas increased by 80 J in the expansion, how much heat does the gas absorb?

Pressure and volume measurements of a dilute gas undergoing a quasi-static adiabatic expansion are shown below. Plot ln p vs. \(\mathrm{V}\) and determine \(\gamma\) for this gas from your graph. $$\begin{array}{cc} \mathbf{P}(\mathrm{atm}) & \mathbf{V}(\mathbf{L}) \\\\\hline 20.0 & 1.0 \\\17.0 & 1.1 \\\14.0 & 1.3 \\\11.0 & 1.5 \\\8.0 & 2.0 \\\5.0 & 2.6 \\ 2.0 & 5.2 \\\1.0 & 8.4\end{array}$$

Steam to drive an old-fashioned steam locomotive is supplied at a constant gauge pressure of \(1.75 \times 10^{6} \mathrm{N} / \mathrm{m}^{2}\) (about 250 psi) to a piston with a 0.200 -m radius. (a) \(\mathrm{By}\) calculating \(p \Delta V\), find the work done by the steam when the piston moves \(0.800 \mathrm{m}\). Note that this is the net work output, since gauge pressure is used. (b) Now find the amount of work by calculating the force exerted times the distance traveled. Is the answer the same as in part (a)?
See all solutions

Recommended explanations on Physics Textbooks

Rotational Dynamics

Read Explanation

Magnetism

Read Explanation

Solid State Physics

Read Explanation

Fluids

Read Explanation

Mechanics and Materials

Read Explanation

Particle Model of Matter

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.