The Speed of Sound

In high-velocity gas flow, velocities are often reached that are comparable to the speed of sound, so the speed of sound plays an important part in what follows. The speed of sound is the speed at which a small pressure disturbance moves through a continuous medium. Sound, as our ears perceive it, is a series of small air-pressure disturbances oscillating in a sinusoidal fashion in the frequency range jfrom 20 to 20,000 cycles per second, or hertz (Hz). The magnitude of the pressure disturbances is generally less than 10 Ibf/in absolute (7 Pa) [l ]. 

Suppose that we have a bar of steel 1 mi long. We tap the steel sharply on one end our tap causes the near end of the bar to move 0.001 in. If the steel was absolutely incompressible, the far end of the bar would also move 0.001 in instantly. It does not it moves about one-third of a second after we tap the near end. Nothing in this world is absolutely incompressible. 

Consider a pipe full of some fluid, with pistons at each end. We tap one of the pistons. This causes the pressure adjacent to the piston to rise. This moves the next layer of fluid, whose pressure rises, and so on, causing a small pressure pulse to pass down the pipe. This is shown schematically in Fig. 8.1. It is easier to analyze this problem if we ride along with the pressure pulse. We appear to be standing still, and the walls of the pipe seem to be rushing past us. The fluid in the pipe also is rushing toward us and rushing away behind us. We measure the velocity, pressure, and density of the fluid ahead of us and behind us the values of those ahead are slightly different from the values of those behind. This situation is represented in Fig. 8.2. 

The pressure pulse in assumed to have the small volume shown in the figure. The mass flow into it is the same as the mass flow out, so we can apply the steady-flow mass balance equation 

Dividing by /I, we expand the right-hand side and cancel the pV term to get 

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Detonation. In a detonation, the flame front travels as a shock wave, followed closely by a combustion wave, which releases the energy to sustain the shock wave. The detonation front travels with a velocity greater than the speed of sound in the unreacted medium. 

The flow velocity is thus proportional to the difference in transit time between the upstream and downstream directions and to the square of the speed of sound in the fluid. Because sonic velocity varies with fluid properties, some designs derive compensation signals from the sum of the transit times which can also be shown to be proportional to C. 

Compressible Vlow. The flow of easily compressible fluids, ie, gases, exhibits features not evident in the flow of substantially incompressible fluid, ie, Hquids. These differences arise because of the ease with which gas velocities can be brought to or beyond the speed of sound and the substantial reversible exchange possible between kinetic energy and internal energy. The Mach number, the ratio of the gas velocity to the local speed of sound, plays a central role in describing such flows. 

Increase Sound- Transmission Loss. The only significant iacreases ia sound-transmission loss that can be achieved by the appHcation of dampiag treatments to a panel occur at and above the critical frequency, which is the frequency at which the speed of bending wave propagation ia the panel matches the speed of sound ia air. AppHcation of dampiag treatment to 16 ga metal panel can improve the TL at frequencies of about 2000 H2 and above. This may or may not be helpful, depending on the appHcation of the panel. 

Transport Properties. Viscosity, themial conductivity, the speed of sound, and various combinations of these with other properties are called steam transport properties, which are important in engineering calculations. The speed of sound (Fig. 6) is important to choking phenomena, where the flow of steam is no longer simply related to the difference in pressure. Thermal conductivity (Fig. 7) is important to the design of heat-transfer apparatus (see HeaT-EXCHANGETECHNOLOGy). The viscosity, ie, the resistance to flow under pressure, is shown in Figure 8. The sharp declines evident in each of these properties occur at the transition from Hquid to gas phase, ie, from water to steam. The surface tension between water and steam is shown in Figure 9. 

Many special-purpose electrical thermometers have been developed, either for use in practical temperature measurement, or as research devices for the study of temperature and temperature scales. Among the latter are thermometers which respond to thermal noise (Johnson noise) and thermometers based on the temperature dependence of the speed of sound. 

V/c is the ratio of fluid velocity to the speed of sound or aeoustie veloeity, c. The speed of sound is the propagation velocity of infinitesimal pressure disturbances and is derived from a momentum balance. The compression caused by the pressure wave is adiabatic and frictionless, and therefore isentropic. 

Most often, the Mach number is calculated using the speed of sound evaluated at the local pressure and temperature. When M = 1, the flow is critical or sonic and the velocity equals the local speed of sound. For subsonic flowM < 1 while supersonic flows have M > 1. Compressibility effects are important when the Mach number exceeds 0.1 to 0.2. A common error is to assume that compressibihty effects are always negligible when the Mach number is small. The proper assessment of whether compressibihty is important should be based on relative density changes, not on Mach number. 

There are certain limitations on the range of usefulness of pitot tubes. With gases, the differential is very small at low velocities e.g., at 4.6 m/s (15.1 ft/s) the differential is only about 1.30 mm (0.051 in) of water (20°C) for air at 1 atm (20°C), which represents a lower hmit for 1 percent error even when one uses a micromanometer with a precision of 0.0254 mm (0.001 in) of water. Equation does not apply for Mach numbers greater than 0.7 because of the interference of shock waves. For supersonic flow, local Mac-h numbers can be calculated from a knowledge of the dynamic and true static pressures. The free stream Mach number (MJ) is defined as the ratio of the speed of the stream (V ) to the speed of sound in the free stream ... 

Explosions are either deflagrations or detonations. The difference depends on the speed of the shock wave emanating from the explosion. If the pressure wave moves at a speed less than or equal to the speed of sound in the unreacted medium, it is a deflagration if it moves faster than the speed of sound, the explosion is a detonation. 

Deflagration to Detonation Transition A reaction front that starts out with velocities below the speed of sound and subsequently accelerates to velocities higher than the speed of sound in the unreacted material is said to have undergone a Deflagration to Detonation Transition. The possibility of transition is enhanced by confinement/turbulence generators in the path of the reaction front. 

When an isotropic material is subjected to planar shock compression, it experiences a relatively large compressive strain in the direction of the shock propagation, but zero strain in the two lateral directions. Any real planar shock has a limited lateral extent, of course. Nevertheless, the finite lateral dimensions can affect the uniaxial strain nature of a planar shock only after the edge effects have had time to propagate from a lateral boundary to the point in question. Edge effects travel at the speed of sound in the compressed material. Measurements taken before the arrival of edge effects are the same as if the lateral dimensions were infinite, and such early measurements are crucial to shock-compression science. It is the independence of lateral dimensions which so greatly simplifies the translation of planar shock-wave experimental data into fundamental material property information. 

In Chapter 13 we showed that, if a material contains a crack, and is sufficiently stressed, the crack becomes unstable and grows - at up to the speed of sound in the material -to cause catastrophically rapid fracture, or fast fracture at a stress less than the yield stress. We were able to quantify this phenomenon and obtained a relationship for the onset of fast fracture... 

When vapor bubbles eollapse inside the pump the liquid strikes the metal parts at the speed of sound. This is the elicking and popping noise we hear from outside the pump when we say that eavitation sounds like pumping marbles and roeks. Sound travels at 4,800 ft per second in water. The velocity head formula gives a elose approximation of the energy contained in an imploding cavitation bubble. Remember that implosion is an explosion in the opposite direction. 

For compressible fluids one must be careful that when sonic or choking velocity is reached, further decreases in downstream pressure do not produce additional flow. This occurs at an upstream to downstream absolute pressure ratio of about 2 1. Critical flow due to sonic velocity has practically no application to liquids. The speed of sound in liquids is very liigh. See Sonic Velocity later in this chapter. 

A relationship that is useful in compressor and compressor systems i> the speed of sound of the gas at the flowing conditions. The actntMK velocity, a, can be calculated using the following equation ... 

It has played a dual role, one in Equation 2.18 on specific heat ratio and the other as an isentropic exponent in Equation 2.53. In the previous calculation of the speed of sound. Equation 2.32, the k assumes the singular specific heat ratio value, such as at compressor suction conditions. When a non-perfect gas is being compressed from point 1 to point 2, as in the head Equation 2.66, k at 2 will not necessarily be the same as k at 1. Fortunately, in many practical conditions, the k doesn t change very... 

Critical and Subcritical Flow - The maximum vapor flow through a restriction, such as the nozzle or orifice of a pressure relief valve, will occur when conditions are such that the velocity through the smallest cross-sectional flow area equals the speed of sound in that vapor. This condition is referred to as "critical flow" or "choked flow . 

A detonation shock wave is an abrupt gas dynamic discontinuity across which properties such as gas pressure, density, temperature, and local flow velocities change discontinnonsly. Shockwaves are always characterized by the observation that the wave travels with a velocity that is faster than the local speed of sound in the undisturbed mixtnre ahead of the wave front. The ratio of the wave velocity to the speed of sound is called the Mach number. 

Overdriven Detonation The unstahle condition that exists during a defla-gration-to-detonation transition (DDT) before a state of stable detonation is reached. Transition occurs over the length of a few pipe diameters and propagation velocities of up to 2000 m/s have been measured for hydrocarbons in air. This is greater than the speed of sound as measured at the flame front. Overdriven detonations are typically accompanied by side-on pressure ratios (at the pipe wall) in the range 50-100. A severe test for detonation flame arresters is to adjust the run-up distance so the DDT occurs at the flame arrester, subjecting the device to the overdriven detonation impulse. 

Initial shock-wave overpressure can be determined from a one-dimensional technique. It consists of using conservation equations for discontinuities through the shock and isentropic flow equations through the rarefaction waves, then matching pressure and flow velocity at the contact surface. This procedure is outlined in Liepmatm and Roshko (1967) for the case of a bursting membrane contained in a shock tube. From this analysis, the initial overpressure at the shock front can be calculated with Eq. (6.3.22). This pressure is not only coupled to the pressure in the sphere, but is also related to the speed of sound and the ratio of specific heats. 

The ratio of the speed of sound in the compressed nitrogen to the speed of sound in the ambient air, ada, is approximately 1. 

The speed of sound Oq of the contained gas at failure temperature must be calculated ... 

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