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Chapter 11: Problem 72

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.

Short Answer

Expert verified
The magnitude of the heat exchange can be calculated using the initial and final energies and the direction of heat exchange is determined by whether the energy of the air increased or decreased.

Step by step solution

01

Write down the known variables

The given values are: initial pressure in the tank \( P1 = 40 \, \text{psia} \), initial temperature \( T1 = 500^{\circ} \mathrm{R} \), exit pressure \( P2 = 15 \, \text{psia} \), and throat-exit area ratio \( A* = 2.5 \).
02

Convert pressures to absolute units

To use the ideal gas law, pressures need to be in absolute units. Therefore, convert psia to lbf/ft^2. So, \( P1 = 40 \, \text{psia} = 5760 \, \text{lbf/ft}^2 \) and \( P2 = 15 \, \text{psia} = 2160 \, \text{lbf/ft}^2 \).
03

Calculate initial energy

The energy of a gas is given by the equation \( E = \frac{1}{2} \cdot m \cdot v^2 + \frac{3}{2} \cdot n \cdot R \cdot T \), where \( m \) is the mass, \( v \) is the velocity, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature. In this case, the velocity of the air inside the tank is negligible, and the number of moles can be calculated using the ideal gas law. Therefore, the initial energy \( E1 \) is: \( E1 = \frac{3}{2} \cdot \frac{P1 \cdot V1}{R \cdot T1} \cdot R \cdot T1 = \frac{3}{2} \cdot P1 \cdot V1 \), where \( V1 \) is the initial volume of the air and can be calculated as \( V1 = A* \cdot P1 \).
04

Calculate final energy

Similarly, the final energy \( E2 \) is: \( E2 = \frac{3}{2} \cdot P2 \cdot V2 \). Here, \( V2 \) is the final volume and can be calculated using the exit-to-throat area ratio and the exit pressure as \( V2 = A* \cdot P2 \).
05

Determine the magnitude and direction of heat exchange

The magnitude of the heat exchange \( Q \) is the difference between the final and initial energy, \( Q = E2 - E1 \). If the value of \( Q \) is positive, then heat is absorbed by the air, if it's negative, heat is released by the air.

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Most popular questions from this chapter

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.

The static pressure to stagnation pressure ratio at a point in a gas flow field is measured with a Pitot-static probe as being equal to \(0.6 .\) The stagnation temperature of the gas is \(20^{\circ} \mathrm{C}\). Determine the flow speed in \(\mathrm{m} / \mathrm{s}\) and the Mach number if the gas is air. What error would be associated with assuming that the flow is incompressible?

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)?

Helium at \(68^{\circ} \mathrm{F}\) and 14.7 psia in a large tank flows steadily and isentropically through a converging nozzle to a receiver pipe. The cross- sectional area of the throat of the converging passage is \(0.05 \mathrm{ft}^{2}\). Determine the mass flowrate through the duct if the receiver pressure is (a) 10 psia, (b) 5 psia. Sketch temperatureentropy diagrams for situations (a) and (b).

Prove that, in Rayleigh flow, the Mach number at the point of maximum temperature is \(1 / \sqrt{k}\).
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