Expansion isentropic

Note that widely different mixts all follow a single curve. The horizontal lines I and II are theoretical computations. Une I is based on a frozen sound velocity and line U is based on an equilibrium sound velocity. Clearly the former provides a better fit (at large d/a) to the exptl data than the latter. Frozen sound velocity is computed under the assumption that compn and entropy remain constant, while for equilibrium sound velocity one assumes the chemical reaction manages to follow the changes in the expansion isentrope. Vasil ev et al suggest that the larger-than-theoretical values of (c—u)/D at small d/a are due to an increase in c because... 

For a converging/diverging nozzle with negligible entrance velocity in which expansion isentropic, sketch graphs of mass flow rate m, velocity u, and area ratio A/A, vs. the pressure i P/ P,. Here, A is the cross-sectional area of the nozzle at the point in the nozzle where the press" is P, and subscript 1 denotes the nozzle entrance. 

Pinegree, M., Aveille, Leroy, J., Leroy, M., Protat, J. C., Cheret, R., and Camarcat, N., Expansion Isentropes ofTATB Compositions Released into Argon, in Proceedings of the Eighth Symposium (International) on Detonation, Albuquerque, New Mexico, July 1985. 

The expansion isentrope for hexogen/trinitrotoluene composition (50/50) obtained by the above method is shown in Figure 4.58. 

Specific impulse value may be approximately calculated if the pressure-time dependence is known, assuming that there is no lateral scattering of detonation products. Assuming that the expansion isentrope equation for the detonation products has the form (Eq. (4.50))... 

Fig. 1 Equilibration time as a function of pressure on the expansion isentropic curve for explosion products), of a stoichiometric acetylene-oxygen mixture.

Experimental data are required in evaluating any model of the detonation process. Data that relate to the detonation properties are detonation velocity as a function of density, C-J pressures and temperatures, and state points on the shock Hugoniot and expansion isentrope of the detonation products. 

The conservation conditions require that the pressure and particle velocity be identical across the interface between an explosive and an inert. Since the Hugoniots for many materials have been measured as shown in Figure 2.2, and an experimentally determined state point for an explosive-inert interface determines a point on the detonation product shock Hugoniot if the match is above the C-J point, and a point on the detonation product expansion isentrope if it is below the C-J point. The C-J point usually is determined by the intersection of the Rayleigh line (line of constant detonation velocity) with the experimentally measured isentrope or Hugoniot. 

Figure 2.2 Hugoniots for several materials and BKW-calculated Composition B detonation product Hugoniot, Rayleigh line, and expansion isentrope through C-J point.

Considering that real detonations are not steady-state and that chemical equilibrium is not necessarily achieved, it is difficult to evaluate any equation of state. One can probably consider an equation of state adequate for engineering purposes if, over a wide range of density and composition, the computed and experimental pressures and temperatures agree to within 20% and the detonation velocities to within 10%. Such explosive systems usually exhibit small changes in detonation parameters with diameter, have small failure diameters, and behave like most high-energy explosive systems. It is also important that the expansion isentrope be accurately reproduced to within 5%. 

One may define a nonideal explosive as having a C-J pressure, velocity, or expansion isentrope significantly different from those expected from equilibrium, steady-state calculations such as BKW. Pressure differences of 50 kbar and velocity differences of 500 m/sec are probably nonideal. A significant isentrope difference depends upon the application, but if the experimental and calculated air isentrope values differ by more than 0.1 cm/psec, the explosive can be nonideal. Nonideal explosives often exhibit other differences, such as larger sensitivity to diameter or confinement. 

The expansion isentrope of the inert metal-loaded explosive must be less steep than for completely reacted ideal explosives of the same detonation pressure since additional decomposition occurs behind the detonation shock front. The high pressure expansion isentrope would result in larger plate dents and greater aquarium bubble expansions than those characteristic of ideal explosives with the same detonation pressure. 

The addition of ammonium salts to explosives results in increased non-ideality with increased ammonium salt concentration. The amount of ammonium salt that reacts defines both the C-J state and the expansion isentrope for the explosive mixture. 

Since the effect of the form of the carbon is so important to the performance of an explosive, it is important that we examine the evidence for diamond and other forms of carbon in the detonation products. Experiments where the detonation products are recovered at ambient conditions do not represent the composition of the detonation products at the C-J state and do not furnish us the history of the products along the expansion isentrope. The recovered products will have had a complicated history of expansion, being reshocked after interaction with the container walls and associated non-equilibrium composition changes. If diamond is seen, the diamond phase can probably be associated with the higher pressures near the C-J state. Graphite can be formed near the C-J state and during the release process. 

Turbine outlet conditions for isentropic expansion to 150°C from steam tables are... 

For a single-stage expansion with isentropic efficiency of 85 percent, from Eq. (6.2),... 

Figure A2.1.4. Adiabatic reversible (isentropic) paths that do not intersect. (The curves have been calculated for the isentropic expansion of a monatomic ideal gas.)...

Fig. 28. Rankine cycle for superheat, where ( ) represents adiabatic (isentropic) compression ( ), isobaric heating ( ), vaporization (x x x x), superheating turbine expansion and (-----), heat rejection. To convert kPa to psi, multiply by 0.145. To convert kj to kcal, divide by 4.184.

These derivatives are also of interest for free expansions or isentropic changes. 

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. 

It is not uncommon to utilize both the isentropic and isenthalpic expansions in a cycle. This is done to avoid the technical difficumes associated with the formation of liquid in the expander. The Claude or expansion engine cycle is an example of a combination of these meth-... 

Efficiency for a turboexpander is calculated on the basis of isentropic rather than polytropic expansion even though its efficiency is not 100 percent. This is done because the losses are largely introduced at the discharge of the machine in the form of seal leakages and disk friction which heats the gas leaking past the seals and in exducer losses. (The exducer acts to convert the axial-velocity energy from the rotor to pressure energy.)... 

Size, rotating speed, and efficiency correlate well with the available isentropic head, the volumetric flow at discharge, and the expansion ratio across the turboexpander. The head and the volumetric flow and rotating speed are correlated by the specific speed. Figure 29-49 shows the efficiency at various specific speeds for various sizes of rotor. This figure presumes the expansion ratio to be less than 4 1. Above 4 1, certain supersonic losses come into the picture and there is an additional correction on efficiency, as shown in Fig. 29-50. 

The fact that shock waves continue to steepen until dissipative mechanisms take over means that entropy is generated by the conversion of mechanical energy to heat, so the process is irreversible. By contrast, in a fluid, rarefactions do not usually involve significant energy dissipation, so they can be regarded as reversible, or isentropic, processes. There are circumstances, however, such as in materials with elastic-plastic response, in which plastic deformation during the release process dissipates energy in an irreversible fashion, and the expansion wave is therefore not isentropic. 

In Seetion 2.8, we noted that most expansion waves are isentropie. It was shown in Seetion 2.4 that the differenee between the Hugoniot and isentrope is small for hydrodynamie materials at small strains. Thus we ean also represent relief waves in the P-u plane with the same eurve used to represent shoek waves, if the strains are not too large. 

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