Sonic velocity calculation

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 ... 

When the relieving scenarios are defined, assume line sizes, and calculate pressure drop from the vent tip back to each relief valve to assure that the back-pressure is less than or equal to allowable for each scenario. The velocities in the relief piping should be limited to 500 ft/sec, on the high pressure system and 200 ft/sec on the low pressure system. Avoid sonic flow in the relief header because small calculation errors can lead to large pressure drop errors. Velocity at the vent or flare outlet should be between 500 ft/sec and MACH 1 to ensure good dispersion. Sonic velocity is acceptable at the vent tip and may be chosen to impose back-pressure on (he vent scrubber. 

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. 

Calculate sonic velocity for fluid using Equations 2-84 or 2-85. 

If sonic velocity of step 2 is greater than calculated velocity of step 1, calculate line pressure drop using usual flow equations. If these velocities are equal, then the pressure drop calculated will be the maximum for the line, using usual flow equations. If sonic velocity is less than the velocity of step 1, reassume line size and repeat calculations. 

Determine sonic velocity at oudet conditions and check against a calculated velocity using flow rate. If sonic is the lower, it must be used as limiting, and capacity is limited to that corresponding to this velocity. 

Pandit and King (1982) and Bathe et al. (1984) presented measurements using transducer techniques, which are somewhat different from the accepted values of Kiefte et al. (1985). The reason for the discrepancy of the sonic velocity values from those in Table 2.8 and above is not fully understood. It should be noted that compressional velocity values can vary significantly depending on the hydrate composition and occupancy. This has been demonstrated by lattice-dynamics calculations, which showed that the adiabatic elastic moduli of methane hydrate is larger than that of a hypothetical empty hydrate lattice (Shpakov et al., 1998). 

Volumetric flow rates of different gases are often compared to equivalent volumes of air at standard atmospheric temperature and pressure. The ideal gas law works well when used to size fans or compressors. Unfortunately, the gas law relationship, PV/T = constant, is frequently applied to choked gas streams flowing at sonic velocity. A typical misapplication could then be the conversion to standard cubic feet per minute in sizing SRVs. Whether the flow is sonic or subsonic depends mainly on the backpressure on the SRV outlet. In the API calculations, this is taken into account by the backpressure correction factor. 

The sound velocity in a fiber, and the sonic modulus calculated therefrom, are related to molecular orientation (De Vries ). As shown by Moseley ), the sonic modulus is independent of the crystallinity at temperatures well below the T (which means that the inter- and intramolecular force constants controlling fiber stiffness are not measurably different for crystalline and amorphous regions at these temperatures). An orientation parameter a, calculated from the sonic modulus, is therefore taken as a measure for the average orientation of all molecules in the sample, regardless of the degree of crystallinity. The parameter is called the total orientation , as contrasted to crystalline and amorphous orientation, from X-ray data. 

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. 

Some calculations require knowing the critical pressure at which sonic velocity occurs. This is calculated with Equation 5. 

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... 

Calculating Chapman-Jouguet detonations, proceeding at a velocity at which the reacting gases reach sonic velocity. 

To estimate the critical pressure ratio, we equate the sonic velocity to the outlet velocity calculated via equation (5.34). Substituting p = p, = p2 gives ... 

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 ... 

We will now develop an expression for the sonic speed experienced in the throat/outlet of a convergent-only nozzle and at the throat of a convergent-divergent nozzle when the expansion is frictionally resisted. Sonic conditions will exist in the throat when the velocity calculated by applying equation (14.45) to the convergent section of the nozzle has reached the local speed of sound, i.e. ... 

Basing our calculations on the above assumptions and eliminating the mathematics, we define the relationship describing the sonic velocity in a two-phase mixture as follows ... 

In gas systems, the flow becomes choked when the exit velocity through the orifice plate reaches sonic velocity. The mass flow rate is essentially independent of downstream conditions but can be increased by increasing the upstream pressure or decreasing the temperature. For an ideal gas and isentropic flow, the pressure ratio to calculate the onset of sonic conditions depends on the ratio of the specific heats, g, and is often known as the isentropic expansion factor ... 

The calculation of two-phase isothermal and isentropic compressibilities, two-phase sonic velocity, single-phase sonic velocity, and cooling and heating due to expansions are presented in the second part of this chapter. Cubic equations of state facilitate all of these calculations. One basic assumption in the formulation for the two-phase compressibilities and two-phase sonic velocities is the equilibrium state. In the transition from single-phase to two-phase state, compressibilities and sonic velocity may have a sharp discontinuity, which implies lack of validity of averaging procedures. 

In the following, we will first derive the expressions that can be used to calculate the isentropic two-phase compressibility. The thermodynamic sonic velocity then can be readily calculated from the isentropic compressibility. We could have combined the derivations for the isothermal and isentropic compressibilities, but have decided on separate derivations for the sake of simplicity. 

Using the equations presented in this section, one can calculate the two-phase isentropic compressibility and the two-phase sonic velocity. In the following, some numerical results are presented from Firooza-badi and Pan (1997), who employed the PR-EOS for the calculation of coefficient derivatives. 

Figure 3.28 shows the calculated compressibilities and sonic velocity for a mixture of C1/C3 (30 mole% 70 mole% C3) at 130°F. In Fig. 3.28a, Cj and Cg are plotted us. pressure. This figure indicates that there is a discontinuity in both isothermal and isentropic compressibilities, when the phase boundaries are crossed. From a pressure of 1200 psia to a bubblepoint pressure of about 977 psia, there is a small increase in Cg of the undersaturated liquid the increase is, however, more noticeable. At. the bubblepoint, there is a sudden increase in both Cj and Cg. Similar behavior is also observed at the dewpoint of about 453 psia. It is interesting to note that the compressibilities in the two-phase region approaching the dewpoint are higher than the corresponding gas-phase compressibilities. Figure 3.28a also reveals that the variation of Cg in the two-phase region is less than the variation of C. This figure also provides the experimental isothermal compressibility data of Sage et al. (1933). The results in Fig. 3.28a are for a flat interface...

Figure 3.29 Calculated compressibilities and sonic velocity for the C /nCio mixture (95 mole% Cl and 5 mole% nCio) at leo F (adapted from Firoozabadi and Pan, 1997>...

---