Isentropic curve

Figure 4.7. Internal reflection of a shock wave from a free surface, (a) Reflection of a shock wave from a free surface causes a reflected rarefaction wave. As indicated in (b), this increases the velocity of the shocked material from u, to Uf. The path upon shocking is Rayleigh line 0-1, whereas unloading occurs along release isentrope curve I -O, (c) Release isentrope path in P- V plane is indicated.

The basic equations for describing the detonahon characteristics of condensed materials are fundamentally the same as those for gaseous materials described in Sections 3.2 and 3.3. The Rankine-Hugoniot equations used to determine the detonation velocities and pressures of gaseous materials are also used to determine these parameters for explosives. Referring to Sechon 3.2.3, the derivative of the Hugoniot curve is equal to the derivative of the isentropic curve at point J. Then, Eq. (3.13) be-... 

As discussed in Section 3.2.3 of Chapter 3, the derivative of the Hugoniot curve is equal to the derivative of the isentropic curve at point J. Then, Eq. (3.13) becomes... 

If the state point moves down from point P or P, the state point is an isentropic curve, which conflicts with the second law of thermodynamics (principle of entropy increase for any spontaneous process). If the state point moves up, it compresses the... 

Figure 1 shows the chemical equilibrium time as a function of pressure for different states on the isentropic curve of the products of explosion at a constant volume of an acetylene-oxygen mixture (T = 298 K and p = 0.1 MPa). Line 1 represents calculations by formula (18) while points 2... 

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

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.)...

By plotting Hugoniot curves in the pressure-particle velocity plane (P-u diagrams), a number of interactions between surfaces, shocks, and rarefactions were solved graphically. Also, the equation for entropy on the Hugoniot was expanded in terms of specific volume to show that the Hugoniot and isentrope for a material is the same in the limit of small strains. Finally, the Riemann function was derived and used to define the Riemann Invarient. 

Figure 4.2. Pressure-volume compression curves. For isentrope and isotherm, the thermodynamic path coincides with the locus of states, whereas for shock, the thermodynamic path is a straight line to point Pj, V, on the Hugoniot curve, which is the locus of shock states.

An important application of the impedance match method is demonstrated by the pressure-particle velocity curves of Fig. 4.9 for various explosives. Using the above method, the pressure in shock waves in various explosives is inferred from the intersection of the explosive Hugoniot with the explosive product release isentropes and reflected shock-compression Hugoniots (Zel dovich and Kompaneets, 1960). The amplitudes of explosively induced shock waves which can be propagated into nonreacting materials are calculable using results such as those of Fig. 4.9. 

Figure 4.18. Sketch of Hugoniots of material with different porosities. Isentrope for material is indicated by curve S, along OA. Principal Hugoniot, H, indicated by curve OB. Porous Hugoniots are indicated with initial volumes. shown at O, O", and O " and curves OC, OC", and OC". Normal Hugoniots are indicated for K > k where = (2/ o + 1) 2nd k = Foo/K whereas anomalous Hugoniots are shown for K < k.

Figure 9.42 shows the typical characteristic curve of a centrifugal fan, where the blades are backward curved. The figure also shows the theoretical characteristic curve when the slip factor is 1 and when it is smaller than 1. Characteristic curves for a real fan are closer to the isentropic one at the design point. At this point the efficiency is maximum. 

At the instant a pressure vessel ruptures, pressure at the contact surface is given by Eq. (6.3.22). The further development of pressure at the contact surface can only be evaluated numerically. However, the actual p-V process can be adequately approximated by the dashed curve in Figure 6.12. In this process, the constant-pressure segment represents irreversible expansion against an equilibrium counterpressure P3 until the gas reaches a volume V3. This is followed by an isentropic expansion to the end-state pressure Pq. For this process, the point (p, V3) is not on the isentrope which emanates from point (p, V,), since the first phase of the expansion process is irreversible. Adamczyk calculates point (p, V3) from the conservation of energy law and finds... 

Figure 2.12 A set of parallel, isentropic surfaces ordered so that S, > S2 > S3. The solid curve marked 6 rev = 0 represents a reversible adiabatic path that connects two states that lie on the entropy surface. Si. The dashed curves marked 6qm = 0 are irreversible paths that connect states on different entropy surfaces. Only one of these two paths will be allowed the other will be forbidden.

The curve of maximum entropy is the locus of end points, in the v,P plane, of all possible Rayleigh transformations starting with the "spike state and representing release of a given quantity of chemical energy Q. Oppenheim (Ref 1) calls this the Q-curve. Its equation is derived on the assumption that the combustion products move at the local velocity of sound with respect to the detonation front. Thus the Rayleigh-Mikhel son line at the C-J point, where it reaches the Q-curve, has the same slope as the isentrope at that point... 

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... 

These thermodynamic processes, as they occur in useful machines, arc not often of the exact polytropic form desired. For example, an isentropic process, which is exemplified, at least theoretically, by expansion of the burned gases after the explosive combustion in the gasolme engine, is modified slightly by die interchanging of heat between gases and cylinder wall, whereas a true isentropic has no heat either added or rejected in this way. ITic particular polytropic curve that would suit these conditions of expansion would depart somewhat from the adiabatic form. 

Referring to curve (B) of Fig 1, note that the isentrope thru the CJ point always lies between the Hugoniot and the Rayleigh line tangent to the Hygoniot at the CJ point... 

The operation of TET. First, the gas flow is cooled in heat exchanger under approximately constant pressure, then gas flow works in the turbine and is cooled further. (Note that this process is not isentropic, i.e. even in ideal case the losses are inevitable). Thus it founds itself under the condensation curve and the condensation process takes place. Further, gas again passes through heat exchanger and finally is additionally compressed by compressor. 

The exit Mach number M2 may not exceed unity. M2 = 1 corresponds to choked flow sonic conditions may exist only at the pipe exit. The mass velocity G in the charts is the choked mass flux for an isentropic nozzle given by Eq. (6-118). For a pipe of finite length, the mass flux is less than G under choking conditions. The curves in Fig. 6-21 become vertical at the choking point, where flow becomes independent of downstream pressure. 

It would appear that by a finite number of successive applications of the magnetization-demagnetization process shown in Fig. la, 0 K could be achieved. The only thing that could prevent this goal, as required by the third law, is if, as the temperature is reduced, the two curves approach each other, as shown in Fig. lb. In this case, the temperature change for the isentropic demagnetization approaches zero as T approaches 0 K ... 

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