Isentropic Change of State

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 

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Fig. 4.1-1. Isentropic change of state (CO2) AB, Gases CD, Liquids EF, Supercritical fluids.

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. 

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... 

Herewith, the solution of the integral equation (15.7) is traced back to the calculation of an isentropic change of state of a real gas ... 

As an alternative, Eq. (15.12) can also be integrated if the average values of the parameter Z and II, (Z) and (II), in each case averaged between the stagnation conditions in the entry and the nozzle throat conditions, are introduced. The velocity in the nozzle throat in the case of an isentropic change of state is obtained from this ... 

Equation (6.54) accounts for the change of state in the isentropic processes of a gas-solid mixture in which the effect of a finite particle volume is considered. An example using this equation to obtain the speed of sound in a gas-solid mixture is introduced in the next section. 

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. 

It is usually assumed that the change of state is isentropic (reversibly adiabatic). Hence, one has... 

By means of Eqs. (15.9) and (15.10), the derivation of an analytical equation for the mass flow rate through the nozzle is quite simple if the change of state between the entry and the nozzle throat is assumed to be isentropic (ds = 0) and the frictional losses neglected at this time as well as the heat transfer with the wall is accounted by the velocity coefficient (f otherwise, the equations cannot be analytically integrated. 

Compressible Fluids (Gases). The assumption of no transfer of heat between the fluid and the pipe, which implies no friction, permits assuming that any change of state between sections and 2 is a reversible isentropic change. For such flow conditions, the flow velocity through an orifice is given by... 

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 ... 

During an isentropic process of a closed system between stale 1 and 2, the change in internal energy equals minus the work done between the two states, or... 

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%. 

The change of enthalpy within the isentropic nozzle flow can be calculated by using the first and second laws of thermodynamics, if the equation of state for real gases... 

The thermodynamic state of liquids is a changing polytropic one (close to isentropic) during quick compression (see Fig.4.1-1, for C02) and the corresponding temperature rise is low for liquids and large for gases (see AB in Fig. 4.1-1). The final compression temperature limits the compression ratio per stage with gas compressors. 

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