Speed of sound in the fluid

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. 

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. 

Velocity considerations are important in rotating or reciprocating machinery systems, because, if the compressible fluid velocity exceeds the speed of sound in the fluid, shock waves can be set up and the results of such conditions are much different than the velocities below... 

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. 

Elsewhere in this book attention is focused on particles whose Mach and Knudsen numbers are small. The Mach number is defined as the ratio of the relative velocity between the particle and the fluid to the speed of sound in the fluid ... 

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. 

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. 

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. 

A convenient parameter often used in compressible flow equations is the Mach number,, which is defined as the ratio of i>, the speed of the fluid in the conduit, to man > Ihe speed of sound in the fluid at the actual flow conditions. 

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