/*! 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 68 Air flows into a converging duct... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 68

Air flows into a converging duct, and a normal shock stands at the exit of the duct. Downstream of the shock, the Mach number is \(0.54 .\) If \(p_{2} / p_{1}=2,\) compute the Mach number at the entrance of the duct and the area ratio \(A_{1} / A_{2}\).

Short Answer

Expert verified
The Mach number at the entrance of the duct or upstream of the shock is obtained from Step 2. The ratio of the areas A1 and A2 of the duct is obtained from Step 3. Both values can be obtained by inserting the values from the problem into the given equations and performing the calculations.

Step by step solution

01

Identify the given variables

Here, downstream Mach number M2 = 0.54 and the pressure ratio p2/p1 = 2 are given.
02

Use Normal Shock Relations to find upstream Mach number

Normal shock relations can be used to find the Mach number upstream of the shock. The relation connecting M1, M2 and p2/p1 is \(M_{1}=\sqrt{[(M_{2}^2(纬-1)+2) / (2纬M_{2}^2-纬+1)]}\), where 纬 is the heat capacity ratio value of a specific gas (For air, 纬=1.4). By substituting the values, we get, \(M1 = \sqrt{(0.54^2(1.4-1)+2) / (2*1.4*0.54^2-1.4+1)}.\)
03

Calculation of the area ratio A1/A2

The area ratio A1/A2 can be obtained through the isentropic flow equations. The relation is: \(A_{1}/A_{2}=(M_{2}/M_{1}) * sqrt[(2+(纬-1)*M_{1}^2) / (2+(纬-1)*M_{2}^2)]*\[(1+0.5*(纬-1)*M_{2}^2)/(1+0.5*(纬-1)*M_{1}^2)\]^(纬/(纬-1))\). Upon substitution of the values for M1, M2, and 纬, we can compute the value of A1/A2.

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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 supersonic aircraft cruises at \(M=2.7\) at \(60,000 \mathrm{ft}\) altitude. A normal shock stands in front of a pitot tube on the aircraft; the tube senses a stagnation pressure of 10.4 psia. Calculate the static pressure and temperature behind the shock. Evaluate the loss in stagnation pressure through the shock. Determine the change in specific entropy across the shock. Show static and stagnation states and the process path on a \(T s\) diagram.

A normal shock stands in a constant-area duct. Air approaches the shock with \(T_{0_{1}}=550 \mathrm{K}, p_{0_{1}}=650 \mathrm{kPa}(\mathrm{abs})\) and \(M_{1}=2.5 .\) Determine the static pressure downstream from the shock. Compare the downstream pressure with that reached by decelerating isentropically to the same subsonic Mach number.

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 \(2000^{\circ} \mathrm{R}\) ).

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.
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