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Chapter 11: Problem 40

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.

Short Answer

Expert verified
The temperature in the test section, pressure in the test section, and the upstream stagnation conditions would be calculated using the given parameters and the specific heat ratio for helium. After calculating these, we will then have all the parameters required for the model of the supersonic wind tunnel.

Step by step solution

01

Calculation of Temperature in Test Section using Isentropic Flow Relations

To calculate the temperature, we can make utilize of the isentropic flow relation given by: \[ T = T_0 / (1 + ((\gamma - 1)/2)M^2) \] Here, \( T_0 = \) Stagnation temperature, \( \gamma = \) specific heat ratio for helium gas i.e. 1.66 and \( M = \) Mach number i.e. 3.0. We shall make use of another physical property of the fluid which is: \[ v = M \sqrt{\gamma R T} \] where \( v = \) velocity i.e. 2000 m/s, \( R = \) specific gas constant for helium i.e. 2077 J/Kg.K . From this equation we can solve for \( T_0 \)
02

Calculation of Pressure in Test Section using Isentropic Flow Relations

Once we have the temperature, we can make use Reciprocal of the Isentropic flow relation: \[ P = P_0 / [(1 + ((\gamma - 1)/2)M^2)^(\gamma / (\gamma - 1))] \] Here, \( P_0 = \) Stagnation pressure. We must find the Value of \( P_0 \) to find the pressure in the test section. We first should use the equation of state combined with the given density to find \( P \) in the test section. We have the equation of state defined as: \[ P = \rho RT \] where \( P = \) pressure in test section, \( \rho = \) density i.e. 1 Kg/m³. In this case, once we have T, \( P \) can be found and that will give us \( P_0 \)
03

Calculation of Stagnation Conditions

The stagnation conditions are the conditions the flow would come to if isentropically decelerated to a standstill. Once we have the values of temperature and pressure in the test section, we now have also the values of Stagnation temperature i.e. \( T_0 \) and the Stagnation pressure \( P_0 \) as calculated in steps 1 and 2 respectively.

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Most popular questions from this chapter

An ideal gas flows with velocity \(V\), pressure \(p\), temperature \(T,\) and density \(\rho .\) Determine a set of equations for stagnation properties, including entropy, if the stagnation process is defined to be isothermal \((T=\text { constant ) rather than isentropic }(s=\text { constant })\).

The gas entering a rocket nozzle has a stagnation pressure of \(1500 \mathrm{kPa}\) and a stagnation temperature of \(3000^{\circ} \mathrm{C}\). The rocket is traveling in the still Standard Atmosphere at \(30,000 \mathrm{m}\). Find the throat and exit area for a flow rate of \(10 \mathrm{kg} / \mathrm{s}\). Assume \(k=1.35, R=\) \(287.0 \mathrm{N} \cdot \mathrm{m} / \mathrm{kg} \cdot \mathrm{K} .\) The gas is perfectly, expanded to the ambient pressure.

Supersonic airflow enters an adiabatic, constant area (inside diameter \(=1 \mathrm{ft}\) ) 30 -ft-long pipe with \(\mathrm{Ma}_{1}=3.0 .\) The pipe friction factor is estimated to be \(0.02 .\) What ratio of pipe exit pressure to pipe inlet stagnation pressure would result in a normal shock wave standing at (a) \(x=5\) ft, or \((\mathbf{b}) x=10 \mathrm{ft},\) where \(x\) is the distance downstream from the pipe entrance? Determine also the duct exit Mach number and sketch the temperature-entropy diagram for each situation.

A normal shock occurs in a stream of oxygen. The oxygen flows at \(\mathrm{Ma}=1.8\) and the upstream pressure and temperature are 15 psia and \(85^{\circ} \mathrm{F}\) (a) Calculate the following on the downstream side of the shock: static pressure, stagnation pressure, static temperature, stagnation temperature, static density, and velocity. (b) If the Mach number is doubled to \(3.6,\) what will be the resulting values of the parameters listed in part (b)?

Determine the Mach number of a car moving in standard air at a speed of (a) \(25 \mathrm{mph}\) (b) \(55 \mathrm{mph},\) and (c) \(100 \mathrm{mph}\)
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