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Chapter 11: Problem 45

An ideal gas is to flow isentropically from a large tank where the air is maintained at a temperature and pressure of \(59^{\circ} \mathrm{F}\) and 80 psia to standard atmospheric discharge conditions. Describe in general terms the kind of duct involved and determine the duct exit Mach number and velocity in \(\mathrm{ft} / \mathrm{s}\) if the gas is air.

Short Answer

Expert verified
The duct involved in this exercise is a converging-diverging nozzle. After calculations, you will obtain numerical values for the exit Mach number and velocity of the gas.

Step by step solution

01

Convert the temperature to Rankine

The first step is converting the given temperature into the Rankine scale, as it is the standard scale used in thermodynamics. The temperature in Fahrenheit can be converted to Rankine using the equation \(T(\mathrm{R}) = T(^{\circ}\mathrm{F}) + 459.67\). Thus, the initial temperature \(T_1\) is \(59^{\circ}\mathrm{F} + 459.67 = 518.67\mathrm{R}\).
02

Determine the exit pressure

The exit pressure \(P_2\) is the pressure at standard atmospheric conditions. It equals 14.7 psia.
03

Calculate the pressure ratio

We can use the pressures to determine the pressure ratio between the exit and initial points. The pressure ratio \(P_2/P_1\) is \(14.7/80 = 0.18375\).
04

Determine the exit temperature

The exit temperature \(T_2\) can be determined using the equation for isentropic processes of ideal gases, which involves the pressure ratio and the specific heat ratio \(\gamma\): \(T_2 = T_1 \times (P_2/P_1)^{(\gamma-1)/\gamma}\). Using \(\gamma = 1.4\) for air, we can calculate \(T_2\).
05

Calculate the Mach number

The Mach number \(M_2\) can be calculated using the isentropic flow relation that involves the pressure ratio and the specific heat ratio: \(M_2 = \sqrt{(2/(\gamma-1))[\left( P_1/P_2)^{(\gamma-1)/\gamma} -1\right]}\).
06

Calculate the velocity

The exit velocity can be calculated using the equation \(V_2= M_2 \times \sqrt{\gamma \times R \times T_2}\), where \(R\) is the specific gas constant for air. In foot-pound-second units, \(R = 1716\) ft²/s²·R. Making sure to convert the units to feet per second, we can calculate \(V_2\).

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Most popular questions from this chapter

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.

Air enters a 4 -cm-square galvanized steel duct with \(p_{0}=\) \(150 \mathrm{kPa}, T_{0}=400 \mathrm{K},\) and \(V_{1}=120 \mathrm{m} / \mathrm{s},\) (Note: \(\mu=2.2 \times 10^{-5}\) \(\left.\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right)\) (a) Compute the maximum possible duct length for these conditions. (b) If the actual duct length is 0.75 times the maximum value, calculate the mass flow rate, the exit pressure, and the stagnation pressure. (c) If the actual duct length is 1.3 times the maximum value for the stated conditions, compute the new mass flow rate and inlet velocity. Assume a low back pressure and use the same value of \(f\) as used in the maximum possible duct length case.

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.

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.

The flow blockage associated with the use of an intrusive probe can be important. Determine the percentage increase in section velocity corresponding to a \(0.5 \%\) reduction in flow area due to probe blockage for airflow if the section area is \(1.0 \mathrm{m}^{2}, T_{0}=\) \(20^{\circ} \mathrm{C},\) and the unblocked flow Mach numbers are (a) \(\mathrm{Ma}=0.2\) (b) \(\mathrm{Ma}=0.8\) (c) \(\mathrm{Ma}=1.5,(\mathrm{d}) \mathrm{Ma}=30\)
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