Speed of sound in a fluid

The speed uw with which a small pressure wave propagates through a fluid can be shown [Shapiro (1953)] to be related to the compressibility of the fluid dp/dP by equation 6.65  

Assuming that the pressure wave propagates through the fluid polytropi-cally, then the equation of state is 

The propagation speed uw of the pressure wave is therefore given by 

In practice, small pressure waves (such as sound waves) propagate virtually isentropically. The reasons for this are that, being a very small disturbance, the change is almost reversible and, by virtue of the high speed, there is very little heat transfer. Thus the speed of sound c is equal to the speed at which a small pressure wave propagates isentropically, so from equation 6.69 

As will be seen in Section 6.10, there is a fundamental difference between flow in which the fluid s speed u is less than c, ie subsonic flow, and that 

---

Fig. 5. Diagram illustrating an ultrasonic instrument designed to measure the speed of sound in a fluid under known shea conditions. The design is based on a combination of a pulse-echo ultrasonic reflectometer and a controlled-strain concentric cylinder rheometer.

It is shown in the theory of hydrodynamics that the speed of sound in a fluid, Ug, is given by... 

The sonic or critical velocity (speed of sound in the fluid) is the maximum velocity which a compressible fluid can attain in a pipe [3]. 

This maximum velocity of a compressible fluid in a pipe is limited by the velocity of propagation of a pressure wave that travels at the speed of sound in the fluid [3]. This speed of sound is specific for each individual gas or vapor or liquid and is a function of the ratio of specific heats of the fluid. The pressure reduces and the velocity increases as the fluid flows downstream through the pipe, wdth the maximum velocity occurring at the downstream end of the pipe. WTien, or if, the pressure drop is great enough, the discharge or exit or outlet velocity will reach the velocity of sound for that fluid. 

If the valve is closed more quickly, the pressure rise will be correspondingly greater. It might be thought that if the valve were closed instantly the pressure rise would be infinite. This is not the case. When a valve is closed suddenly, a pressure wave propagates upstream at approximately the speed of sound in the fluid and only the fluid through which the pressure wave has passed is decelerated thus the pressure rise is finite because the speed of sound is finite. 

Another approach for estimating am is based on the pseudothermodynamic properties of the mixture, as suggested by Rudinger (1980). The equation for the isentropic changes of state of a gas-solid mixture is given by Eq. (6.53). Note that for a closed system the material density of particles and the mass fraction of particles can be treated as constant. Hence, in terms of the case for a single-phase fluid, the speed of sound in a gas-solid mixture can be expressed as... 

Equations (7.14), (7.15), and (7.20), combined with the relations between the thermodynamic properties at constant entropy, determine how the velocity varies with cross-sectional area of the nozzle. The variety of results for compressible fluids (e.g., gases), depends in part on whether the velocity is below or above the speed of sound in the fluid. For subsonic flow in a converging nozzle, the velocity increases and pressure decreases as the cross-sectional area diminishes. In a diverging nozzle with supersonic flow, the area increases, but still the velocity increases and the pressure decreases. The various cases are summarized elsewhere.t We limit the rest of this treatment of nozzles to application of the equations to a few specific cases. 

Figure 1. Dominant echo generated by cylinder A The echo is normalised with respect to the amplitude of the incident wave packet. The distance X is normalised to three wavelengths of the incident wave packet. I is the incident wave packet (shown for reference only) R2 is the second, or dominant, echo generated by the cylinder T2 the second transmitted wave packet the ratio of the speed of sound in cylinder fluid to the speed of sound in surrounding water.

Mach number M is the ratio of the speed of fluid in the duct to the speed of sound in the fluid. The derivatives in these equations are rates of change with length as the fluid passes through a duct. Equation (4-160) relates the pressure derivative, and Eq. (4-161), the velocity derivative, to the entropy and area derivatives. According to... 

Here, u represents a characteristic velocity of the flow and usotmd is the speed of sound in the fluid at the same temperature and pressure. It may be noted that usound for air at room temperature and atmospheric pressure is approximately 300 m/s, whereas the same quantity for liquids such as water at 20°C is approximately 1500 m/s. Thus the motion of liquids will, in practice, rarely ever be influenced by compressibility effects. For nonisothermal systems, the density will vary with the temperature, and this can be quite important because it is the source of buoyancy-driven motions, which are known as natural convection flows. Even in this case, however, it is frequently possible to neglect the variations of density in the continuity equation. We will return to this issue of how to treat the density in nonisothermal flows later in the book. 

The Mach number, denoted by is defined as the ratio of u, the speed of the fluid, to a, the speed of sound in the fluid under conditions of flow. 

ANL s ultrasonic viscometer is a nonintrusive in-line device that measures both fluid density and viscosity. The design of the viscometer is based on a technique that measures acoustic and shear impedance. The technique was first applied by Moore and McSkimin (1970) to measure dynamic shear properties of solvents and polystyrene solutions. The reflections of incident ultrasonic shear (1-10 MHz) and longitudinal waves (1 MHz), launched toward the surfaces of two transducer wedges that are in contact with the fluid, are measured. The reflection coefficients, along with the speed of sound in the fluid, are used to calculate fluid density and viscosity. Oblique incidence was commonly used because of better sensitivity, but mode-converted waves often occur in wedges that do not exhibit perfect crystal structure and lack well-polished surfaces. For practical applications, we use the normal-incidence arrangement. 

If a fluid wei e absolutely incompressible (which no materials. known to humans are), then the speed of sound in that fluid would be infinite. What happens to the pressure rise in Example 7.11 as we replace the water with fluids which are less and less compressible ... 

Mach number The ratio of the relative speeds of a fluid and a rigid body to the speed of sound in that fluid under fhe same condifions of temperature and pressure, if the Mach number exceeds 1 the fluid or body is moving at a supersonic speed, if the Mach number exceeds 5 it is said to be hypersonic. The number is named after Ernst Mach (1838-1916). 

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. 

If the velocity of the gas/fluid equals or exceeds the speed of sound, shock waves are set up, and vibrations and other mechanically related problems may result, compared to the conditions when velocities are below the speed of sound. For a Mach of 1.0, the gas velocity equals the velocity of sound in the fluid. 

This is identical to the equation derived in physics for the speed of so in the fluid. Therefore, the maximum fluid velocity obtainable in a pipe of const cross-sectional area is the speed of sound. This does not imply that higt velocities are impossible they are, in fact, readily obtained in converg diverging nozzles (Sec. 7.3). However, the speed of sound is the maximum val that can be reached in a conduit of constant cross section, provided the entran velocity is subsonic. The sonic velocity must be reached at the exit of the pi-If the pipe length is increased, the mass rate of flow decreases so that the so velocity is still obtained at the outlet of the lengthened pipe. 

---