Isothermal sonic velocity

It is shown later in Section 4.3 that this corresponds to the velocity at which a small pressure wave will be propagated under isothermal conditions, sometimes referred to as the isothermal sonic velocity , though the heat transfer rate will not generally be sufficient to maintain truly isothermal conditions. 

Since the derivation of governing equations may be found elsewhere (S6, W6), we need only state them briefly in order to establish our notation and to point out some further simplifications. Let c be the isothermal sonic velocity, then the equation of state in terms of compressibility factor Z is... 

The explicit methods avoid the need of solving large sets of equations and can therefore be used on smaller computers. However, these methods tend to be unstable unless the step sizes are kept small and an artificial constraint is introduced on the variables. In most formulations the time step must be less than the reach length divided by the isothermal sonic velocity (see Section V,B,1). Explicit methods are used in PIPETRAN (D7) and SATAN (G4). 

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). 

The rate of flow of gas under adiabatic conditions is never more than 20 per cent greater than that obtained for the same pressure difference with isothermal conditions. For pipes of length at least 1000 diameters, the difference does not exceed about 5 per cent. In practice the rate of flow may be limited, not by the conditions in the pipe itself, but by the development of sonic velocity at some valve or other constriction in the pipe. Care should, therefore, be taken in the selection of fittings for pipes conveying gases at high velocities. 

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... 

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. 

Equation (5.51) is indeterminate at = 1.0, but nevertheless we can gain an insight into the behaviour by setting n = 1.01, when pdP = 0.4978. (A more complete treatment of the theoretical case of sonic velocity in an isothermal expansion is given in Section 5.4.3.)... 

The observed effects of heat transfer on the flow in micro-nozzles are readily explained as follows. From compressible Rayleigh flow, it is known that removing heat from a supersonic flow acts to accelerate the flow. At steady-state, the bulk of the flow in the micro-nozzle expander is supersonic, and thus, heat transfer acts to further accelerate the supersonic flow. Concurrently, as the flow is cooled, the exit density p increases. The overall effect is an increase in thrust. Heat extraction from the flow into the substrate increases performance from the subsonic layer point of view as well. For low nozzle wall temperatures, the local sonic velocity is diminished and the near-wall Mach number increases. This phenomenon is the force driving the reduction in subsonic layer size for micro-nozzle flows with heat removal. In fact, with sufficient heat extraction from the flow, the subsonic layer can be reduced to the point where the competing effects of viscous forces and nozzle geometry cause the optimum expander half angle to be shifted from 30° to a more traditional expander half angle of 15°. This is demonstrated in Fig. 7 for isothermal wall temperatures less than 700 K. 

Using the same procedures for finding a maximum flow that were used in the isothermal case, the maximum flow occurs when the velocity at the downstream end of the pipe is the sonic velocity for adiabatic flow. This is... 

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 previous section, the two-phase isothermal compressibility of multicomponent systems was formulated using the equilibrium assumption that there is no gradient of chemical potentials in the systems. In this section, the two-phase isentropic compressibility and the two-phase sonic velocity for multicomponent systems will be formulated. Again, we will make the assumption of equilibrium, which implies the gradients of chemical potentials are zero. The equilibrium assumption regarding the two-phase compressibilities depends on the problem and may or may not be justified. As long as there is no supersaturation, and adequate time is allowed to reach the state of equilibrium, then the equilibrium criterion can be invoked. 

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. 

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...

In isentropic flow (just as in isothermal flow), the mass velocity reaches a maximum when the downstream pressure drops to the point where the velocity becomes sonic at the end of the pipe (e.g., the flow is choked). This can be shown by differentiating Eq. (9-25) with respect to P2 (as before) or, alternatively, as follows... 

Now, we assume an isothermal siphon flow outside the photosphere which passes trough a sonic point designated by subscript s. Then the velocity vr of the mass flow at the surface of the secondary is determined from... 

It has been reported that for a given wave Mach number at small scales, the resulting particle velocities are lower but the pressures are higher [9]. Also, it has been shown that at small scales, the weak wave solution is no longer sonic and isentropic but is isothermal and can be subsonic. 

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