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Chapter 11: Problem 41

Air flows isentropically through a duct to a section where \(p_{1}=25 \mathrm{kPa}, T_{1}=300 \mathrm{K},\) and \(V_{1}=900 \mathrm{m} / \mathrm{s} .\) For these conditions: (a) Determine the stagnation conditions for the flow. (b) What is the Mach number at station \(1 ?\) Show a \(T-s\) diagram displaying stagnation and static conditions. (c) Is the flow choked? Is the throat behind or ahead of section \(1 ?\) Label this state on the \(T-s\) diagram.

Short Answer

Expert verified
The solution includes the calculation of stagnation temperature and pressure, the Mach number, and determination of the flow being choked or not. All these results are then represented on a T-s diagram.

Step by step solution

01

Calculating the Stagnation Temperature

For an isentropic process, the total (or stagnation) temperature \(T_0\) can be found using the equation: \[T_0=T_1+\frac{V_1^2}{2c_p}\] where \(-V_1\) is the velocity, and \(c_p\) is the specific heat at constant pressure. Using the given values, the stagnation temperature \(T_0\) can be calculated.
02

Calculating the Stagnation Pressure

The total (or stagnation) pressure \(p_0\) can be found using the isentropic flow relation: \[\frac{p_0}{p_1}=\left(\frac{T_0}{T_1}\right)^{k/c_p}\] where \(k\) is the ratio of specific heats. Substituting the values for the temperatures and the specific heats, the stagnation pressure \(p_0\) can be determined.
03

Finding the Mach Number

The Mach number \(M_1\) at station 1 can be found using the formula: \[M_1=\frac{V_1}{a_1}\] where \(a_1=\sqrt{kRT_1}\) is the speed of sound in the medium. Substituting the given values, the Mach number can be calculated.
04

Checking for Choked Flow and Locating the Throat

The flow is considered to be choked (therefore at critical condition) if \(M_1 \geq 1\). Additionally, if \(M_1 > 1\), the section is a downstream of the throat, otherwise it's upstream. Next step is to use this knowledge to label the T-s diagram.
05

Drawing the T-s Diagram

Draw a T-s diagram displaying the static conditions (T, P) and stagnation conditions (T0, P0). Indicate the position of the throat in relation to section 1 based on the result from step 4.

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Most popular questions from this chapter

At some point for air flow in a duct, \(p=20\) psia, \(T=500^{\circ} \mathrm{R}\) and \(V=500 \mathrm{ft} / \mathrm{s}\). Can a normal shock occur at this point?

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)?

The stagnation pressure in a Mach 2 wind tunnel operating with air is 900 kPa. A 1.0 -cm-diameter sphere positioned in the wind tunnel has a drag coefficient of \(0.95 .\) Calculate the drag force on the sphare.

The Pitot tube on a supersonic aircraft (see Video \(\mathbf{V} 3.8\) ) cruising at an altitude of 30,000 ft senses a stagnatior pressure of 12 psia. If the atmosphere is considered standard, determine the airspeed and Mach number of the aircraft. A shock wave is present just upstream of the probe impact hole.

An aircraft cruises at a Mach number of 2.0 at an altitude of \(15 \mathrm{km} .\) Inlet air is decelerated to a Mach number of 0.4 at the engine compressor inlet. A normal shock occurs in the inlet diffuser upstream of the compressor inlet at a section where the Mach number is \(1.2 .\) For isentropic diffusion, except across the shock, and for standard atmosphere, determine the stagnation temperature and pressure of the air entering the engine compressor.
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