Isentropic equation of state

We have divided both numerator and denominator by the mol number to introduce now the molar quantities. Here we observe a mysterious and miraculous effect of the energy law. It is interesting to note that the isentropic application of pressure itself is not directly associated with a change of volume, however, with a change of temperature. The change of temperature effects a change in volume. So the change of volume under pressure and consequently the compression work is done in an indirect way. 

In the case of a positive expansion coeflhcient with increasing pressure a rise in temperature is predicted, in the case of a negative expansion coefficient, the temperature decreases, and in the case of a zero thermal expansion coefficient the temperature does not change at all. All these three cases are realized in the case of water. The thermal expansion coefficient is 0 at 4 °C. At 2 °C, the thermal expansion coefficient is —1.6 and at 6 °C the thermal expansion coefficient is 1.6The 

the numerical value is 1.06 x 10 Km J , which means that for an increase of the pressure of 100 atm, a temperature change of 0.01 K will be observed. The unit m J is the same as Pa  

We discuss finally the case of a thermal expansion coefficient of zero. In this case, a pressure change does not effect a temperature change of the material and also no change in volume. So no input of compression energy is achieved. In contrary to the adiabatic application of pressure in the isothermal case in fact no change in volume will occur. Therefore, the incompressible body should be more accurately addressed as the isothermal incompressible body. 

---

Moreover, upon comparing (4.32) with (4.14), it can be seen that (Jeanloz and Grover, 1988) the Birch-Murnaghan equation (4.32) with a2 = 0 describes the isentropic equation of state provided the linear shock-particle velocity relation (4.5) describes the Hugoniot. In combination, these require that... 

The term BT S, V)/dV is the temperature change at adiabatic (or more exactly isentropic) expansion. Moreover, Eq. (4.33) is the starting point for the derivation of the isentropic equation of state. 

Extensive studies of nonlinear propagation in fluids have been made (36). Although derived for fluids, the results have been applied to a number of sohd polymers (37,38). In these studies, the isentropic equation of state for pressure p, in terms of density p, is expanded to the form... 

The available isentropic head is usually calculated by computer, using any of the various equations of state. In the absence of such facihty, a quick and reasonably reliable calculation follows. In fact, this calculation is valuable as a cross-check on other methods because it is likely to be accurate within a few percent. 

Because the Griineisen ratio relates the isentropic pressure, P, and bulk modulus, K, to the Hugoniot pressure, P , and Hugoniot bulk modulus, K , it is a key equation of state parameter. 

Ruoff (1967) first showed how the coefficients of the shock-wave equation of state are related to the zero pressure isentropic bulk modulus, and its first and second pressure derivatives, K q and Kq, via... 

Understanding such interaction is important both in predicting the amplitudes of shock waves transmitted across interfaces (in the case where the equations of state of all materials are known), and in determining release isentropes or reflected Hugoniots (when measurement of the equation of state is needed). Consider first a shock wave in material A being transmitted to a... 

For many condensed media, the Mie-Gruneisen equation of state, based on a finite-difference formulation of the Gruneisen parameter (4.18), can be used to describe shock and postshock temperatures. The temperature along the isentrope (Walsh and Christian, 1955) is given by... 

With incompressibile fluids, the value of the acoustic speed tends toward infinity. For isentropic flow, the equation of state for a perfect gas can be written ... 

It has been seen in deriving equations 4.33 to 4.38 that for a small disturbance the velocity of propagation of the pressure wave is equal to the velocity of sound. If the changes are much larger and the process is not isentropic, the wave developed is known as a shock wave, and the velocity may be much greater than the velocity of sound. Material and momentum balances must be maintained and the appropriate equation of state for the fluid must be followed. Furthermore, any change which takes place must be associated with an increase, never a decrease, in entropy. For an ideal gas in a uniform pipe under adiabatic conditions a material balance gives ... 

Unfortunately, the extensive work on gas detonations has had little impact on the development of the theory for condensed phas es. The reason is the lack of a reliable equation of state for these. While fundamental significance must be achieved eventually, an empirical fit to actual performance would be helpful at present. A sophisticated general equation for the isentrope, which is a C-J isentrope, is called for. Above 150-200 kbar, the polytropic (gamma-law) equation of state is valid (Skidmore Hart, in Ref 19, p 47). 

In "concluding remarks", Lutzky stated that the calculation of C-J T with the help of LSZK equations, assuming cv=0.3cal/g (approx average value for deton products), gave results which were too low at high densities (See Table 2). The reason for this is not known - probably it is due to incompleteness of LSZK theory. In any case, it is believed that in all applications where the ealen of T is not needed, and only an (e, p, v) equation of state is required (such as the calculation of the non-reactive, isentropic expansion of detonation products by means of hydrodynamic computer codes), the LSZK equation of state, in particular ... 

Kamlet parameter that depends on expl compn (see article on Velocity of Detonation in this Vol). Their correlation is shown in Fig 17 Kamlet Finger (Ref 23) propose a somewhat simpler empirically-fitted correlation, namely, y/ZE1 = 0.887 >asp0° 4. It should be emphasized that this correlation, as well as the one in Fig. 17, are based on isentrope expansion calcns with all their inherent uncertainties as to the equation of state to be used for the expand-ign detonation products... 

Although Equation (4.1) is useful, particularly for isentropic changes of state for which dS = 0, the entropy appears as an independent variable. This circumstance is inconvenient from an experimental point of view, because no meters or devices exist that measure entropy. A new function in which the temperature is an independent variable rather than the entropy is obtained by subtracting... 

This chapter addresses the various phenomena indicated. In addition, the thermodynamic laws governing physical properties of the gas-solid mixture such as density, pressure, internal energy, and specific heat are introduced. The thermodynamic analysis of gas-solid systems requires revisions or modifications of the thermodynamic laws for a pure gas system. In this chapter, the equation of state of the gas-solid mixture is derived and an isentropic change of state is discussed. 

The equation of state in an isentropic process of a gas-solid mixture can be obtained in terms of an energy conservation relationship. When applying the first law of thermodynamics to a gas-solid mixture, we have... 

Another approach for estimating am is based on the pseudothermodynamic properties of the mixture, as suggested by Rudinger (1980). The equation for the isentropic changes of state of a gas-solid mixture is given by Eq. (6.53). Note that for a closed system the material density of particles and the mass fraction of particles can be treated as constant. Hence, in terms of the case for a single-phase fluid, the speed of sound in a gas-solid mixture can be expressed as... 

ISENTROPIC PENG-ROBINSON EQUATION OF STATE CALCULATION... 

Shock compression is an irreversible adiabatic compression that heats the material behind the front [1]. The temperature rise can be divided into two parts. The minimum temperature rise would result if shock compression were slow enough that it approximated a reversible adiabatic compression from Vq to Vj. This process, where zlS = 0, is also called an isentropic compression [1]. Due to the irreversible nature of shock compression, an additional rise is produced that results from the entropy increase zl5,vr across the shock front. This additional rise depends on the detailed nature of the shock front. Shock compression is hotter than isentropic compression. The new temperature Tj cannot be determined from the Hugoniot-Rankine equations alone. Some kind of equation of state (EOS) is also needed (for state-of-the art examples see Refs. [12-14]), and the usual choice is a Griineisen equation of state. The temperature Ti is given by [1],... 

Kuznetsov, N.M. 1981. Two-phase water-steam mixture Equation of state, sound velocity, isentropes. Doklady USSR Acad. Sci. 257 858. 

Adapt the program PRl, or one of the other Peng-Robinson programs, or develop a program of your own using the Peng-Robinson equation of state to do the calculations for an isentropic expansion of a liquid under pressure to produce a vapor-liquid mixture... 

---