/*! 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 6 At a section in a passage, the p... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 6

At a section in a passage, the pressure is \(150 \mathrm{kPa}(\mathrm{abs})\) the temperature is \(10^{\circ} \mathrm{C},\) and the speed is \(120 \mathrm{m} / \mathrm{s}\). For isentropic flow of air, determine the Mach number at the point where the pressure is \(50 \mathrm{kPa}\) (abs). Sketch the passage shape.

Short Answer

Expert verified
The Mach number at the given point of pressure \(50 Kpa\) is calculated through a step by step process by first finding the initial Mach number, then calculating the pressure ratio and finally finding the Mach number at the given point. The passage shape should be diverging for the scenario of an isentropic flow of air where pressure decreases and velocity increases.

Step by step solution

01

Calculation of initial Mach number

We first calculate the initial Mach number, using the equation \(M = V/a\), where \( V \) is the speed and \(a\) is the speed of sound. The speed of sound in air is given by \(a = \sqrt{纬RT}\), where \( 纬 \) (gamma) is the ratio of specific heat capacities, \( R \) is the gas constant for air and \( T \) is the absolute temperature in Kelvin, we find that \( T = 10^{\circ}C + 273 = 283K \), \( R = 287 J/kg.K \) and \( 纬 = 1.4 \), hence use these values to calculate \( a \) and then \( M \)
02

Calculation of pressure ratio

We then need to find the pressure ratio using the equation \(\frac{P2}{P1} = (\frac{1 + 纬M1虏}{1 + 纬M2虏})^{纬/纬-1}\), where \( P2 \) is the pressure at the point we want to find the Mach number, \( P1 \) is the initial pressure and \( M1 \) is the initial Mach number. \( P2 \) is given as \(50 kPa\), \( P1 \) as \(150 kPa\), \( M1 \) we calculated in the previous step and \( 纬 \) as before is \(1.4\), so we can calculate the pressure ratio.
03

Calculation of Mach number at given point

Finally, we can calculate the Mach number at the given point \( M2 \) using the equation derived from step 2 to isolate \( M2 \), that is: \( M2 = \sqrt{[(1+ 纬M1虏)/(\frac{P2}{P1})^{纬-1/纬}]-1}/纬 \), we then insert all the values and get the desired Mach number \( M2 \).
04

Sketch the Passage Shape

The passage shape cannot be sketched in this textual format. However, in an isentropic flow of air, where the flow is subsonic initially (Mach number less than 1), and the pressure decreases along the direction of flow, the speed of the flow (velocity) increases. For the flow to accelerate, the passage needs to increase its area, hence the passage shape would be diverging.

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!

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

Stagnation pressure and temperature probes are located on the nose of a supersonic aircraft. At \(35,000 \mathrm{ft}\) altitude a normal shock stands in front of the probes. The temperature probe indicates \(\bar{T}_{0}=420^{\circ} \mathrm{F}\) behind the shock. Calculate the Mach number and air speed of the plane. Find the static and stagnation pressures behind the shock. Show the process and the static and stagnation state points on a \(T s\) diagram.

A converging-diverging nozzle is attached to a large \(\operatorname{tank}\) of air, in which \(T_{0_{1}}=300 \mathrm{K}\) and \(p_{0_{1}}=250 \mathrm{kPa}(\mathrm{abs}) . \mathrm{At}\) the nozzle throat the pressure is \(132 \mathrm{kPa}(\text { abs }) .\) In the diverging section, the pressure falls to \(68.1 \mathrm{kPa}\) before rising suddenly across a normal shock. At the nozzle exit the pressure is 180 kPa. Find the Mach number immediately behind the shock. Determine the pressure immediately downstream from the shock. Calculate the entropy change across the shock. Sketch the \(T s\) diagram for this flow, indicating static and stagnation state points for conditions at the nozzle throat, both sides of the shock, and the exit plane.

Air is flowing steadily through a series of three tanks. The first very large tank contains air at \(650 \mathrm{kPa}\) and \(35^{\circ} \mathrm{C}\). Air flows from it to a second tank through a converging nozzle with exit area \(1 \mathrm{cm}^{2}\). Finally the air flows from the second tank to a third very large tank through an identical nozzle. The flow rate through the two nozzles is the same, and the flow in them is isentropic. The pressure in the third tank is 65 kPa. Find the mass flow rate, and the pressure in the second tank.

Air flows isentropically through a converging-diverging nozzle attached to a large tank, in which the pressure is 251 psia and the temperature is \(500^{\circ} \mathrm{R}\). The nozzle is operating at design conditions for which the nozzle exit pressure, \(p_{e},\) is equal to the surrounding atmospheric pressure, \(p_{a}\). The exit area of the nozzle is \(A_{e}=1.575\) in. \(^{2}\). Calculate the flow rate through the nozzle. Plot the mass flow rate as the temperature of the tank is progressively increased to \(2000^{\circ} \mathrm{R}\) (all pressures remaining the same). Explain this result (e.g., compare the mass flow rates at \(500^{\circ} \mathrm{R}\) and \(2000^{\circ} \mathrm{R}\) ).

A converging-diverging nozzle, with a throat area of 2 in. \(^{2}\) is connected to a large tank in which air is kept at a pressure of 80 psia and a temperature of \(60^{\circ} \mathrm{F}\). If the nozzle is to operate at design conditions (flow is isentropic) and the ambient pressure outside the nozzle is 12.9 psia, calculate the exit area of the nozzle and the mass flow rate.
See all solutions

Recommended explanations on Physics Textbooks

Kinematics Physics

Read Explanation

Space Physics

Read Explanation

Physical Quantities And Units

Read Explanation

Thermodynamics

Read Explanation

Force

Read Explanation

Translational Dynamics

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.