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Chapter 4: Problem 31

In steady-state flow, downstream the density is \(1 \mathrm{kg} / \mathrm{m}^{3}\) the velocity is \(1000 \mathrm{m} / \mathrm{sec},\) and the area is \(0.1 \mathrm{m}^{2}\). Upstream, the velocity is \(1500 \mathrm{m} / \mathrm{sec},\) and the area is \(0.25 \mathrm{m}^{2}\). What is the density upstream?

Short Answer

Expert verified
The upstream density is around \(0.267 kg/ m^{3}\).

Step by step solution

01

Identify the variables

First, identify and write down the given elements of the equation: downstream density \(\rho_1 = 1 kg / m^{3}\), downstream velocity \(V_1 = 1000 m / sec\), downstream area \(A_1 = 0.1 m^{2}\), upstream velocity \(V_2 = 1500 m / sec\), and upstream area \(A_2 = 0.25 m^{2}\). The unknown is the upstream density, \(\rho_2\).
02

Write down the continuity equation

The continuity equation in this context is \(\rho_1 V_1 A_1 = \rho_2 V_2 A_2\). This equation represents the conservation of mass in steady-state flow.
03

Substitute the known values into the equation

Substitute the known values into the equation: \(1 kg / m^{3} * 1000 m / sec * 0.1 m^{2} = \rho_2 * 1500 m / sec * 0.25 m^{2}\).
04

Solve for the unknown

Solve the equation for \(\rho_2\): \(\rho_2 = (1 kg / m^{3} * 1000 m / sec * 0.1 m^{2}) / (1500 m / sec * 0.25 m^{2})\). Upon calculation, you will find that the upstream density is \(\rho_2 = 0.267 kg / m^{3}\).

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Most popular questions from this chapter

\( \mathrm{A}\) small solid-fuel rocket motor is fired on a test stand. The combustion chamber is circular, with \(100 \mathrm{mm}\) diameter. Fuel, of density \(1660 \mathrm{kg} / \mathrm{m}^{3},\) burns uniformly at the rate of \(12.7 \mathrm{mm} / \mathrm{s}\). Measurements show that the exhaust gases leave the rocket at ambient pressure, at a speed of \(2750 \mathrm{m} / \mathrm{s}\). The absolute pressure and temperature in the combustion chamber are \(7.0 \mathrm{MPa}\) and \(3610 \mathrm{K},\) respectively. Treat the combustion products as an ideal gas with molecular mass of 25.8 . Evaluate the rate of change of mass and of linear momentum within the rocket motor. Express the rate of change of linear momentum within the motor as a percentage of the motor thrust.

Obtain expressions for the rate of change in mass of the control volume shown, as well as the horizontal and vertical forces required to hold it in place, in terms of \(p_{1}, A_{1}, V_{1}, p_{2}\) \(A_{2}, V_{2}, p_{3}, A_{3}, V_{3}, p_{4}, A_{4}, V_{4},\) and the constant density \(\rho\).

A rocket sled with initial mass of 3 metric tons, including 1 ton of fuel, rests on a level section of track. At \(t=\) \(0,\) the solid fuel of the rocket is ignited and the rocket burns fuel at the rate of \(75 \mathrm{kg} / \mathrm{s}\). The exit speed of the exhaust gas relative to the rocket is \(2500 \mathrm{m} / \mathrm{s}\), and the pressure is atmospheric. Neglecting friction and air resistance, calculate the acceleration and speed of the sled at \(t=10\) s.

\(\mathrm{A}\) rocket sled traveling on a horizontal track is slowed by a retro- rocket fired in the direction of travel. The initial speed of the sled is \(U_{0}=500 \mathrm{m} / \mathrm{s}\). The initial mass of the sled is \(M_{0}=1500 \mathrm{kg}\). The retro-rocket consumes fuel at the rate of 7.75 kgls, and the exhaust gases leave the nozzle at atmospheric pressure and a speed of \(2500 \mathrm{m} / \mathrm{s}\) relative to the rocket. The retro- rocket fires for 20 s. Neglect acrodynamic drag and rolling resistance. Obtain and plot an algebraic expression for sled speed \(U\) as a function of firing time. Calculate the sled speed at the end of retro-rocket firing.

\(\mathrm{A}\) -home-made" solid propellant rocket has an initial mass of \(20 \mathrm{lbm} ; 15 \mathrm{lbm}\) of this is fuel. The rocket is directed vertically upward from rest, burns fuel at a constant rate of \(0.5 \mathrm{lbm} / \mathrm{s},\) and ejects exhaust gas at a speed of \(6500 \mathrm{ft} / \mathrm{s}\) relative to the rocket. Assume that the pressure at the exit is atmospheric and that air resistance may be neglected. Calculate the rocket speed after 20 s and the distance traveled by the rocket in 20 s. Plot the rocket speed and the distance traveled as functions of time.
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