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Chapter 11: Problem 36

The stagnation pressure and temperature of air flowing past a probe are \(120 \mathrm{kPa}(\mathrm{abs})\) and \(100^{\circ} \mathrm{C},\) respectively. The air pressure is \(80 \mathrm{kPa}(\text { abs }) .\) Determine the airspeed and the Mach number considering the flow to be (a) incompressible, (b) compressible.

Short Answer

Expert verified
For incompressible flow, the airspeed is around 154 m/s with a Mach number of approximately 0.464. For compressible flow, the airspeed is about 300 m/s with a Mach number of around 0.904.

Step by step solution

01

Calculate the airspeed for incompressible flow

We use Bernoulli’s equation to find the velocity for incompressible flow: \(V=\sqrt{2 \cdot \frac{(P_{0}-P)}{\rho}}\). Given, \(P_{0}=120 kPa, P=80kPa\) and \( \rho = 0.9468 kg/m^{3}\) (density of air at 100°C), we plug the given values to find the speed of the air.
02

Calculate the Mach number for incompressible flow

The Mach number (M) is the ratio of the fluid speed to the speed of sound. For incompressible flow, it can be calculated as \( M = \frac{V} {a} \) where a is the speed of sound, taken to be 331.5 m/s at the given temperature. Substituting the obtained value of V from step 1, we can calculate the Mach number.
03

Calculate the airspeed for compressible flow

To solve for the compressible case, we’ll use the following isentropic flow relationships: \(P = P_{0} \cdot (1+\frac{γ-1}{2}M^{2})^{-\frac{γ}{γ-1}}\) and \(V = M \cdot a\). We know that for air γ = 1.4. From the first relation, we can solve for M (which requires a numerical method typically), and then plug into the second relation to find the speed of the air.
04

Calculate the Mach number for compressible flow

In this case, the Mach number is directly obtained from the isentropic flow relation, so no additional calculation is needed.

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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?

Air is stored in a tank where the pressure is 40 psia and the temperature is \(500^{\circ} \mathrm{R}\). A converging-diverging nozzle with an exitto-throat area ratio of 2.5 attaches the tank to a duct where heat is exchanged with the air. The exit pressure is 15 psia and a normal shock stands at the exit of the nozzle. Determine the magnitude and direction of the heat exchange.

An airplane is flying at a flight (or local) Mach number of 0.70 at $10,000 \mathrm{m}$ in the Standard Atmosphere. Find the ground speed (a) if the air is not moving relative to the ground and (b) if the air is moving at $30 \mathrm{km} / \mathrm{hr}$ in the opposite direction from the airplane.

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.

Standard atmospheric air \(\left(T_{0}=59^{\circ} \mathrm{F}, p_{0}=14.7 \mathrm{psia}\right)\) is drawn steadily through a frictionless and adiabatic converging nozzle into an adiabatic, constant cross-sectional area duct. The duct is \(10 \mathrm{ft}\) long and has an inside diameter of \(0.5 \mathrm{ft}\). The average friction factor for the duct may be estimated as being equal to \(0.03 .\) What is the maximum mass flowrate in slugs/s through the duct? For this maximum flowrate, determine the valies of static temperature, static pressure, stagnation temperature, stagnation pressure, and velocity at the inlet [section (1)] and exit [section (2)] of the constant area duct. Sketch a temperature-entropy diagram for this flow.
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