Sonic Velocity and Mach Number

Using the equation of state, Eq. (1.8), and the expression for adiabatic change, Eq. (1.14), one gets 

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Fig. 2. Sonic velocity and Mach number in two-phase hydrogen.

Lines of constant values of sonic velocity and Mach number are superimposed on the temperature-entropy diagram in Fig. 2, The sonic velocity of the mixture was determined from an equation developed by Holzman [ ]. For the above conditions, the sonic velocity of the mixture is approximately 430 ft/sec, which corresponds to a Mach number of about 0.35. 

Air passes from a large reservoir at 70°F through an isentropic converging-diverging nozzle into the atmosphere. The area of the throat is 1 cm2, and that of the exit is 2 cm2. What is the reservoir pressure at which the flow in the nozzle just reaches sonic velocity, and what are the mass flow rate and exit Mach number under these conditions ... 

Detonation, Mach Number in. Mach (pronounced as Makh) Number, designated as M or Mc, is the ratio of the shock velocity to the sonic velocity and for an ideal gas may be expressed as ... 

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. 

Equation (6-128) does not require fric tionless (isentropic) flow. The sonic mass flux through the throat is given by Eq. (6-122). With A set equal to the nozzle exit area, the exit Mach number, pressure, and temperature may be calculated. Only if the exit pressure equals the ambient discharge pressure is the ultimate expansion velocity reached in the nozzle. Expansion will be incomplete if the exit pressure exceeds the ambient discharge pressure shocks will occur outside the nozzle. If the calculated exit pressure is less than the ambient discharge pressure, the nozzle is overexpanded and compression shocks within the expanding portion will result. 

The temperature or enthalpy of the gas may then be plotted to a base of entropy to give a Fanno line.iA This line shows the condition of the fluid as it flows along the pipe. If the velocity at entrance is subsonic (the normal condition), then the enthalpy will decrease along the pipe and the velocity will increase until sonic velocity is reached. If the flow is supersonic at the entrance, the velocity will decrease along the duct until it becomes sonic. The entropy has a maximum value corresponding to sonic velocity as shown in Figure 4.11. (Mach number Ma < 1 represents sub-sonic conditions Ma > 1 supersonic.)... 

Equation (1.54) indicates that A/A becomes minimal at M = 1. The flow Mach number increases as A/A decreases when M < 1, and also increases as A/A increases when M > 1. When M = 1, the relationship A = A is obtained and is independent of Y- It is evident that A is the minimum cross-sectional area of the nozzle flow, the so-called nozzle throat", in which the flow velocity becomes the sonic velocity, furthermore, it is evident that the velocity increases in the subsonic flow of a convergent part and also increases in the supersonic flow of a divergent part. 

WAVE IN. A shock wave is a violent disturbance moving with a loud bang along a medium (such as air, water, or earth), at a speed greater than that of sound ("supersonic speed ). Velocity of sound, c, in air is 331.9 m/sec (1088ft/sec) and the ratio of shock velocity to sonic velocity is known as Mach Number, M... 

Since Mak s Isothermal flow chart is intended for relief manifold design, it supports calculations starting with P2, the outlet pressure, that is atmospheric at the flare tip, and back-calculates each lateral s inlet pressure. Pi. These inlet pressures are the individual relief valves back pressures. The chart parameter is M2, the Mach number at the pipe outlet. Having M2 is very useful in monitoring proximity to sonic velocity, a common problem in compressible flow. 

The graphs in Fig. 6-21 are based on accurate calculations, but are difficult to interpolate precisely. While they are quite useful for rough estimates, precise calculations are best done using the equations for one-dimensional adiabatic flow with friction, which are suitable for computer programming. Let subscripts 1 and 2 denote two points along a pipe of diameter D, point 2 being downstream of point 1. From a given point in the pipe, where the Mach number is M, the additional len h of pipe required to accelerate the flow to sonic velocity (M = 1) is denoted and may be computed from... 

A typical lifting transient has been found to be in the order of 50 msec. In this time, the disk of a 20 mm nominal bore full-lift valve will move approximately 5 mm, while a particle of air travelling at sonic velocity would move in excess of 15 m. Even at a Mach number of 0.1 these rough figures indicate that a quasisteady assumption is reasonable, as Bicen, Vafidis, and Whitelaw also discovered for internal combustion engine valves. 

The skin friction coefficient varies with Re, the Mach number, M, (when it involves incompressible fluids of local speeds less than the sonic speed, 1), and the character of the boundary layer. The momentum transferred between the air and the body surface appears as a velocity deficit in the viscous wake behind the body. 

The introduction of the valve into the pipe described in Section 6.7 has decreased the flow rate, even at fully open, from the value 2.626 kg/s calculated in Section 6.7 to 2.28 kg/s now. However, the flow at the outlet from the pipe is still sonic, and hence the pressure just inside the pipe outlet, p3, is greater than the atmospheric pressure that exists just outside the pipe, p4. As the valve is closed, however, the pressure drop across it increases, until, 17 seconds into the transient, the throat to inlet pressure ratio falls to the critical value needed for sonic flow in the valve. At this point we have the interesting phenomenon that the flow is sonic in the valve throat, then reduces to subsonic at the valve outlet, only to accelerate to sonic velocity at the pipe outlet. This is shown most clearly in Figure 10.3, which plots the Mach numbers at various points in the pipe. For about 3 seconds in the middle of the transient, the Mach number at the throat of the valve is equal to unity, as is the Mach number at the pipe outlet ... 

But, as shown previously, RkT IM is the square of the speed of sound at state 1, or Cl, so the left side is (1/,/c,). The ratio V/c is called the Mach number M in honor of the Austrian physicist Ernst Mach. This ratio plays a crucial role in the study of high-velocity gas flows (and is widely reported in the press describing the speed of supersonic aircraft). It is the ratio of the local flow velocity to the local speed of sound. For subsonic flows M is less than 1 for sonic flows it equals 1 for supersonic flows it is greater than 1. Making this definition, we can rearrange Eq. 8.15 to... 

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