Effective speed of sound

Here ceff represents the effective speed of sound, c0 is the actual speed of sound in free space, y is the specific heat ratio, and kt is the isothermal compressibility of the fluid. A fraction of shell volume occupied by tubes, solidity o can be easily calculated for a given tube pattern. For example, o = 0.9069(d lp,)2 for an equilateral triangular tube layout, and o = 0.7853(d /p,)2 for a square layout. Coefficients a, are the dimensionless sound frequency parameters associated with the fundamental diametrical acoustic mode of a cylindrical volume. For the fundamental mode al = 1.841, and, for the second mode, a2 = 3.054 [122],... 

The effective speed of sound in a bubbly liquid may differ greatly from that of the liquid or the gas, respectively. In water, for example, the speed of sound equals 1500m/s, in air the speed of sound is 332m/s but in water containing only a very small fraction of bubbles values for the effective speeds of sound may fall to 40m/s. This is due to the nonlinear motion of the bubble in the liquid that alters the compressibility drastically. Damping of the sound field originates from this nonlinearity, too. 

If we combine the above-mentioned information then we are able to calculate the effective speed of sound in a bubbly liquid and the damping coefficient as a function of the local sound pressure. These pieces of information can directly be coupled to the sound-field equation and may be calculated by any three-dimensional code capable of solving the wave equation for the sound pressure with variable coefficients. 

The shape of the curve for the effective speed of sound shows a decrease for higher sound pressures. The damping in the liquid is most pronounced for the... 

Figure 8.1.14 Effective speed of sound c = cu/Re in a bubbly liquid as a function of local sound pressure for water with air bubbles at ambient temperature and ambient static pressure.

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. 

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. 

As will be outlined below, the computation of compressible flow is significantly more challenging than the corresponding problem for incompressible flow. In order to reduce the computational effort, within a CED model a fluid medium should be treated as incompressible whenever possible. A rule of thumb often found in the literature and used as a criterion for the incompressibility assumption to be valid is based on the Mach number of the flow. The Mach number is defined as the ratio of the local flow velocity and the speed of sound. The rule states that if the Mach number is below 0.3 in the whole flow domain, the flow may be treated as incompressible [84], In practice, this rule has to be supplemented by a few additional criteria [3], Especially for micro flows it is important to consider also the total pressure drop as a criterion for incompressibility. In a long micro channel the Mach number may be well below 0.3, but owing to the small hydraulic diameter of the channel a large pressure drop may be obtained. A pressure drop of a few atmospheres for a gas flow clearly indicates that compressibility effects should be taken into account. 

That is, as P2 decreases, the mass velocity will increase up to a maximum value of G, at which point the velocity at the end of the pipe reaches the speed of sound. Any further reduction in the downstream pressure can have no effect on the flow in the pipe, because the speed at which pressure information can be transmitted is the speed of sound. That is, since pressure changes are transmitted at the speed of sound, they cannot propagate upstream in a gas that is already traveling at the speed of sound. Therefore, the pressure inside the downstream end of the pipe will remain at P 2, regardless of how low the pressure outside the end of the pipe (P2) may fall. This condition is called choked flow and is a very important concept, because it establishes the conditions under which maximum gas flow can occur in a conduit. When the flow becomes choked, the mass flow rate in the pipe will be insensitive to the exit pressure but will still be dependent upon the upstream conditions. 

When the gas velocity reaches the speed of sound, choked flow occurs and the mass flow rate reaches a maximum. It can be shown from Eq. (10-45) that this is equivalent to a maximum in YX 2, which occurs at Y = 0.667, and corresponds to the terminus of the lines in Fig 10-21. That is, XT is the pressure ratio across the valve at which choking occurs, and any further increase in X (e.g., AP) due to lowering P2 can have no effect on the flow rate. 

The damage effects from an explosion depend highly on whether the explosion results from a detonation or a deflagration. The difference depends on whether the reaction front propagates above or below the speed of sound in the unreacted gases. For ideal gases the speed of sound or sonic velocity is a function of temperature only and has a value of 344 m/s (1129 ft/s) at 20°C. Fundamentally, the sonic velocity is the speed at which information is transmitted through a gas. 

The value of C in equation 10.39 depends on the way in which the pipe is restrained but for practical purposes a value of unity is adequate. In this equation, E is Young s modulus of elasticity of the pipe, d, the internal diameter of the pipe and tw its wall thickness. The value of E for steel is about 2 x 10s MPa and K for water is about 2 x 103 MPa thus K/E is about 10-2. It will be seen that the elasticity of the pipe has a negligible effect with thick-wall pipes but with thin-wall ones (say djtw > 40) the propagation speed a will typically be reduced to about 70 per cent of the speed of sound c in the liquid. 

For our purpose, it is convenient to classify the measurements according to the format of the data produced. Sensors provide scalar valued quantities of the bulk fluid i. e. density p(t), refractive index n(t), viscosity dielectric constant e(t) and speed of sound Vj(t). Spectrometers provide vector valued quantities of the bulk fluid. Good examples include absorption spectra A t) associated with (1) far-, mid- and near-infrared FIR, MIR, NIR, (2) ultraviolet and visible UV-VIS, (3) nuclear magnetic resonance NMR, (4) electron paramagnetic resonance EPR, (5) vibrational circular dichroism VCD and (6) electronic circular dichroism ECD. Vector valued quantities are also obtained from fluorescence I t) and the Raman effect /(t). Some spectrometers produce matrix valued quantities M(t) of the bulk fluid. Here 2D-NMR spectra, 2D-EPR and 2D-flourescence spectra are noteworthy. A schematic representation of a very general experimental configuration is shown in Figure 4.1 where r is the recycle time for the system. 

The accurate calculation of the resonance frequencies of a tripod scanner is a complicated problem. The flexing modes are effectively coupled with the stretching modes. An evaluation of the lowest resonance frequency of the flexing mode provides an order-of-magnitude estimation of the lowest resonance frequency of the tripod scanner. For a piezo made of PZT-5A, 20 mm long and 2 mm thick, the radius of gyration is 2 mm/y/ 2 = 0.577 mm. The speed of sound is about 2.8 km/sec. Using Eq. (9.44), the resonance frequency is found to be 3.3 kHz, which is close to the values often observed experimentally. 

Shock waves can be produced in a number of ways, such as movement of projectiles or other objects thru air at supersonic speeds, or pushing out of the air by the products of a detonation, which expand at many times the speed of sound. The latter type of shock wave is much stronger and is known as a blast wave (See under BLAST EFFECTS IN AIR, EARTH AND WATER in Vol 2 of Encycl, pp B180 to B184)... 

These results indicate that in the present linear elastic model, the limiting velocity for the screw dislocation will be the speed of sound as propagated by a shear wave. Even though the linear model will break down as the speed of sound is approached, it is customary to consider c as the limiting velocity and to take the relativistic behavior as a useful indication of the behavior of the dislocation as v — c. It is noted that according to Eq. 11.20, relativistic effects become important only when v approaches c rather closely. 

Einstein showed that when a reversible reaction is present sound dispersion occurs at low frequency the equilibrium is shifted within the time of oscillation, the effective specific heat is at a maximum, and the speed of sound c0 is at a minimum. At high frequency the oscillations occur so rapidly that the equilibrium has no time to shift (it is frozen ). The corresponding Hugoniot adiabate (FHA) is shown in the figure. Here the effective heat capacity is minimal, the speed of sound c is maximal cx > c0. From consideration of the final state and the theory of shock waves it follows that C>c0. 

The nondimensionalized pressure term, pf/pU2, can be ignored when the gas velocity is low compared to the speed of sound or when the effects of pressure on the thermodynamic... 

Equation (6.54) accounts for the change of state in the isentropic processes of a gas-solid mixture in which the effect of a finite particle volume is considered. An example using this equation to obtain the speed of sound in a gas-solid mixture is introduced in the next section. 

Since the maximum fluid velocity obtainable in a converging nozzle is speed of sound, a nozzle of this kind can deliver a constant flow rate into a regi of variable pressure. Suppose a compressible fluid enters a converging nozzle pressure Pi and discharges from the nozzle into a chamber of variable press P2. If this discharge pressure is P)t the flow is zero. As P2 decreases below the flow rate and velocity increase. Ultimately, the pressure ratio P2/Pi reach a critical value at which the velocity in the throat is sonic. Further reduction i P2 has no effect on the conditions in the nozzle. The flow remains constant, ah the velocity in the throat is that given by Eq. (7.21), regardless of the value P2/P , provided it is always less than the critical value. For steam, the criti value of this ratio is about 0.55 at moderate temperatures and pressures. 

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