The Sonic Velocity

It has been suggested by Morgan [30] and others [28] that the sonic modulus (i.e. the extensional modulus measured at high frequencies by a wave-propagation technique) can be used to obtain a direct measure of molecular orientation in a manner analogous to the derivation of the so-called optical orientation function /o = (1 - sin2 0) from the birefringence. 

This suggests that, except for high degrees of orientation, both the terms cos 9833 and 

Remembering that the birefringence is given by A = A max(l - f sin 9), it can be seen that the extensional compliance, the reciprocal of the extensional modulus, should be directly related to the birefringence through sin 9 independent of the mechanism of molecular orientation [90], To this degree of approximation, it then follows that 

We would therefore predict a linear relationship between the extensional compliance 533 and the birefringence An, which extrapolates to zero extensional compliance at the maximum birefringence value. 

Samuels [91] has carried the sonic velocity analysis one stage further by recognising the two-phase nature of a crystalline polymer. The natural extension of Equation (8.21) would then be 

This prediction of a linear relation between S3 3 and birefringence, extrapolating to zero compliance at maximum birefringence was tested 

The considerations of Read and Dean are, however, moving away from the spirit of the aggregate model, which is essentially a single phase model, and it is appropriate to consider them as a link with composite structure models, which will now be discussed in some detail. 

The Takayanagi model was developed to account for the viscoelastic relaxation behaviour of two phase polymers, as recorded by dynamic mechanical testing. It was then extended to treat both isotropic and oriented semi-crystalline polymers. The model does not deal with the development of mechanical anisotropy on drawing, but attempts to account for the viscoelastic behaviour of either an isotropic or a highly oriented polymer in terms of the response of components representing the crystalline and amorphous phases. Hopefully, comparisons between the predictions of the model and experimental results may throw light on the molecular processes occurring. 

In its original form the model sought to derive the temperature dependence of the relaxation behaviour of a composite amorphous polymer having two distinct phases in terms of the properties of the individual components. The resultant response would depend on whether the components were in parallel or series (Fig. 5). For the parallel model the complex modulus is given by  

Trials with composite films of polyvinyl chloride intimately bonded to nitrile butadiene rubber, but with distinctly separate phases, confirmed the predictions over a temperature range covering a major rela.xation for each component. If, however, the components were dispersed so that they were separated by a layer containing a mixture of each polymer the experimentally determined relaxations were broader than predicted. 

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The basics of the method are simple. Reflections occur at all layers in the subsurface where an appreciable change in acoustic impedance is seen by the propagating wave. This acoustic impedance is the product of the sonic velocity and density of the formation. There are actually different wave types that propagate in solid rock, but the first arrival (i.e. fastest ray path) is normally the compressional or P wave. The two attributes that are measured are... 

Deflagration A propagating chemical reaction of a substance in which the reaction front advances into the unreacted substance at less than the sonic velocity in the unreacted material. Where a blast wave is produced that has the potential to cause damage, the term explosive deflagration may be used. 

Calculate the sonic velocity using Equation 2.32 where... 

This is a low value, therefore, the possibility exists of an up-rate relative to any nozzle flow limits. At this point, a comment or two is in order. There is a rule of thumb that sets inlet nozzle velocity limit at approximately 100 fps. But because the gases used in the examples have relatively high acoustic velocities, they will help illustrate how this limit may be extended. Regardless of the method being used to extend the velocity, a value of 150 fps should be considered maximum. When the sonic velocity of a gas is relatively low, the method used in this example may dictate a velocity for the inlet nozzle of less than 100 fps. The pressure drop due to velocity head loss of the original design is calculated as follows ... 

Since the pressure drop is quite high, there is a possibility of approaching sonic velocity in the line. This will result in a potential noise problem. Hence, it is a good practice to limit the velocity to 60 percent of the sonic velocity or a 0.6 Mach number. 

As a first trial, an inside pipe diameter is assumed based on 60 percent of the sonic velocity corresponding to the pressure and temperature at the base of the stack, i.e., at 2 psig and temperature =T (upstream temperature since isothermality is assumed). 

Sonic velocity will be established at a restricted point in the pipe, or at the outlet, if the pressure drop is great enough to establish the required velocity. Once the sonic velocity has been reached, the pressure drop in the system will not increase, as the velocity will remain at this value even though the fluid may be discharging into a vessel at a lower pressure than that existing at the point where sonic velocity is established. 

In general, the sonic or critical velocity is attained for an outlet or downstream pressure equal to or less than one half the upstream or inlet absolute pressure condition of a system. The discharge through an orifice or nozzle is usually a limiting condition for the flow through the end of a pipe. The usual pressure drop equations do not hold at the sonic velocity, as in an orifice. Conditions or systems exhausting to atmosphere (or vacuum) from medium to high pressures should be examined for critical flow, otherwise the calculated pressure drop may be in error. 

General design practice avoids using gas velocities near or greater than the sonic velocity. Figure 12-72 indicates the effect of temperature on the sonic velocity. 

With a converging-diverging nozzle, the velocity increases beyond the sonic velocity only if the velocity at the throat is sonic and the pressure at the outlet is lower than the throat pressure. For a converging nozzle the rate of flow is independent of the downstream pressure, provided the critical pressure ratio is reached and the throat velocity is sonic. 

The value of w given by equation 4.43 is the critical pressure ratio wc given by equation 4.26a. Thus the velocity at the throat is equal to the sonic velocity. Alternatively, equation 4.42 may be put in terms of the flowrate (G/A2) as ... 

A nozzle is correctly designed for any outlet pressure between P[ and PE in Figure 4.5. Under these conditions the velocity will not exceed the sonic velocity at any point, and the flowrate will be independent of the exit pressure PE = Pb- It is also correctly designed for supersonic flow in the diverging cone for an exit pressure of PEj. 

In Figure 4.6 values of w2A i, u2/ Ptvi, and (A2/Cj) Jp v which are proportional to v2, u2, and A2 respectively are plotted as abscissae against P2jP. It is seen that the area A2 decreases to a minimum and then increases again. At the minimum cross-section the velocity is equal to the sonic velocity and P2/P1 is the critical ratio. is... 

For constant upstream conditions, the maximum flow through the pipe is found by differentiating with respect to v2 and putting (dG/dn2) equal to zero. The maximum flow is thus shown to occur when the velocity at the downstream end of the pipe is the sonic velocity y/yP2v2 (equation 4.37). 

It will now be shown from purely thermodynamic considerations that for, adiabatic conditions, supersonic flow cannot develop in a pipe of constant cross-sectional area because the fluid is in a condition of maximum entropy when flowing at the sonic velocity. The condition of the gas at any point in the pipe where the pressure is P is given by the equations ... 

In a similar analysis (McWilliam and Duggins, 1969), the sonic velocity was shown to decrease with increasing bubble size and decreasing pressure. Van Wijin-gaarden (1966) derived equations to show that there is a dispersion of the acoustic wave that is,... 

The ratio of the sonic velocity in a homogeneous two-phase mixture to that in a gas alone is cm/c = Pg/ePm = Vpl/ATO — ). This ratio can be much smaller than unity, so choking can occur in a two-phase mixture at a significantly higher downstream pressure than for single phase gas flow (i.e., at a lower pressure drop and a correspondingly lower mass flux). 

Isothermal flow of gas in a pipe with friction is shown in Figure 4-15. For this case the gas velocity is assumed to be well below the sonic velocity of the gas. A pressure gradient across... 

Levenspiel13 showed that the maximum velocity possible during the isothermal flow of gas in a pipe is not the sonic velocity, as in the adiabatic case. In terms of the Mach number the maximum velocity is... 

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. 

For a deflagration the energy from the reaction is transferred to the unreacted mixture by heat conduction and molecular diffusion. These processes are relatively slow, causing the reaction front to propagate at a speed less than the sonic velocity. 

Flares are sometimes used after knockout drums. The objective of a flare is to burn the combustible or toxic gas to produce combustion products that are neither toxic nor combustible. The diameter of the flare must be suitable to maintain a stable flame and to prevent a blowout (when vapor velocities are greater than 20% of the sonic velocity). 

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. 

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 ... 

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