Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 11: Problem 71

Show that for Rayleigh flow, the maximum amount of heat that may be added to the gas is given by: \\[\frac{q_{\max }}{c_{p} T_{1}}=\frac{\left(\mathrm{Ma}_{1}^{2}-1\right)^{2}}{2(k+1) \mathrm{Ma}_{1}^{2}}\\]

Short Answer

Expert verified
The given formula represents the maximum amount of heat addition that can be made to achieve sonic conditions (Ma=1) under Rayleigh Flow and can be derived by understanding the relationships underlying the perfectly gas and the heat addition process.

Step by step solution

01

Recognize the fundamental relationships

The fundamental governing equations for one-dimensional isentropic flow of a perfect gas can be expressed as: \[T T_{0}=\frac{T_{0}}{T}=\frac{1+k M a^{2}}{2}\] and \[M a^{2}=\frac{2}{k-1}\left(\frac{T_{0}}{T}-1\right)\] where T is the absolute static temperature, T0 is the total (or stagnation) temperature, Ma is the Mach number, and k is the specific heat ratio.
02

Understand the Rayleigh Line and Heat addition

In Rayleigh flow, the addition of heat along the flow moves the state towards the Rayleigh line and eventually towards sonic conditions. Also realize that the maximum heat addition would correspond the flow state reaching Mach number equal to 1 (Ma=1).
03

Calculate the Maximum Heat Addition

To set up an equation for maximum heat addition (q_max), we need to understand that the process starts from the initial Mach number (Ma1), passes through multiple states upon heat addition and ends up at Ma=1, representing maximum heat. For the flow to reach the sonic state upon heat addition, we need to go via the Rayleigh line and can thus obtain the heat addition as: \[ \frac{q_{\max }}{c_{p} T_{1}}=T_{01}-T_{1}\] where T01 is the total temperature at the sonic state upon maximum heat addition.
04

Substitute Values in Equation

Substitute T01 using the equation from Step 1 and solving it appropriately to eventually result into the required form: \[\frac{q_{\max }}{c_{p} T_{1}}=\frac{(Ma_{1}^{2}-1)^{2}}{2(k+1) Ma_{1}^{2}}\]

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Helium at \(68^{\circ} \mathrm{F}\) and 14.7 psia in a large tank flows steadily and isentropically through a converging nozzle to a receiver pipe. The cross- sectional area of the throat of the converging passage is \(0.05 \mathrm{ft}^{2}\). Determine the mass flowrate through the duct if the receiver pressure is (a) 10 psia, (b) 5 psia. Sketch temperatureentropy diagrams for situations (a) and (b).

A nozzle for a supersonic wind tunnel is designed to achieve a Mach number of \(3.0,\) with a velocity of \(2000 \mathrm{m} / \mathrm{s},\) and a density of \(1.0 \mathrm{kg} / \mathrm{m}^{3}\) in the test section. Find the temperature and pressure in the test section and the upstream stagnation conditions. The fluid is helium.

Standard atmospheric air \(\left(T_{0}=59^{\circ} \mathrm{F}, p_{0}=14.7 \mathrm{psia}\right)\) is drawn steadily through a frictionless and adiabatic converging nozzle into an adiabatic, constant cross-sectional area duct. The duct is \(10 \mathrm{ft}\) long and has an inside diameter of \(0.5 \mathrm{ft}\). The average friction factor for the duct may be estimated as being equal to \(0.03 .\) What is the maximum mass flowrate in slugs/s through the duct? For this maximum flowrate, determine the valies of static temperature, static pressure, stagnation temperature, stagnation pressure, and velocity at the inlet [section (1)] and exit [section (2)] of the constant area duct. Sketch a temperature-entropy diagram for this flow.

Prove that, in Rayleigh flow, the Mach number at the point of maximum temperature is \(1 / \sqrt{k}\).

Air enters a frictionless, constant area duct with \(\mathrm{Ma}_{1}=2.0\) \(T_{0,1}=59^{\circ} \mathrm{F},\) and \(p_{0,1}=14.7\) psia. The air is decelerated by heating until a normal shock wave occurs where the local Mach number is 1.5. Downstream of the normal shock, the subscnic flow is accelerated with heating until it chokes at the duct exit. Determine the static temperature and pressure, the stagnation temperature and pressure, and the fluid velocity at the duct entrance, just upstream and downstream of the normal shock, and at the duct exit. Sketch the temperature-entropy diagram for this flow.
See all solutions

Recommended explanations on Physics Textbooks

Rotational Dynamics

Read Explanation

Kinematics Physics

Read Explanation

Translational Dynamics

Read Explanation

Work Energy and Power

Read Explanation

Electricity

Read Explanation

Electric Charge Field and Potential

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.

Sign-up for free