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Chapter 13: Problem 83

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.

Short Answer

Expert verified
The Mach number of the plane is calculated following the step-by-step procedure detailed above. The air speed of the plane is then obtained multiplying the Mach number with the speed of sound. The static and stagnation pressures behind the shock are also determined with the given formulas. Finally, the state points are shown on a \(T s\) diagram.

Step by step solution

01

Conversion of units and determination of stagnation temperature

Given that the stagnation temperature behind the shock, \(T_0\), is \(420^{\circ} F\), it needs to be converted into Kelvin. The conversion can be done using the relation \(K = (F - 32) * 5/9 + 273.15\). After converting, determine the stagnation temperature before the shock, \(T_{01}\), using the equation \(T_{01} = T_0/(1 + 0.2M_2^2)\), where \(M_2\) is the Mach number after the shock.
02

Estimation of Mach number

Use initial isentropic flow relations to determine the initial Mach number, \(M_1\), assuming the flight is at the total condition. For example, use the relation \(T = T_{01}/(1 + 0.2M_1^2)\) where, \(T\) is the temperature of air which can be estimated from the provided altitude.
03

Calculation of air speed

Once the Mach number is calculated, the air speed \(V_1\) can be obtained by multiplying the Mach number with the speed of sound at the given altitude. This can be calculated using the formula \(V_1 = M_1 * a\), where \(a\) is the speed of sound which is about \(\sqrt{\gamma * R * T}\), where \(R\) is the gas constant and \(\gamma\) is the ratio of specific heats.
04

Find the static and stagnation pressures

The static and stagnation pressures behind the shock, \(P\) and \(P_0\) respectively, can now be calculated using normal shock relations. Use the equations \(P = P_1*\frac{2\gamma M_1^2 - (\gamma - 1)}{(\gamma + 1)}\) and \(P_0 = P_1*\frac{(\gamma + 1)M_1^2}{2 + (\gamma = 1)M_1^2}*\frac{1 - (\gamma - 1)/(2\gamma)M_1^2}{1 - (\gamma - 1)/(2\gamma)}\), where \(P_1\) is the pressure before the shock and can be calculated from the given altitude.
05

Prepare the \(T s\) diagram

On a \(T s\) diagram, the process can be illustrated as an initial isentropic process that increases with Mach number, followed by a vertical process illustrating the shock that results in sudden change in pressure and temperature.

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Most popular questions from this chapter

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.

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 \(A_{1} / A_{2}\).

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.

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

Air flows adiabatically through a duct. At the entrance, the static temperature and pressure are \(310 \mathrm{K}\) and \(200 \mathrm{kPa}\) respectively. At the exit, the static and stagnation temperatures are \(294 \mathrm{K}\) and \(316 \mathrm{K},\) respectively, and the static pressure is \(125 \mathrm{kPa}\). Find (a) the Mach numbers of the flow at the entrance and exit and (b) the area ratio \(A_{2} / A_{1}\).
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