Caloric Equations of State

In general the flow of a pure fluid is described by the equation of continuity, the three equations of motion, and the equation of energy balance. In addition, one has to specify boundary and initial conditions and also the dependence of p on p and T (the thermal equation of state) and the dependence of Cv or U on p and T (the caloric equation of state). 

The first term depends on what is sometimes called the caloric equation of state, describing how intramolecular properties, the properties within the molecules, are a function of the state variables. The expression in brackets requires the mechanical equation of state, which expresses the dependency of a property, for example, V on the infermolecular interactions, the interactions... 

Equations (1.18.13) represent the caloric equations of state. Note how the Maxwell relations were used to eliminate the entropy terms, and that the equations of state P - P(T,V) and V - V(T,P) permit the determination of the state functions E - E(T,V) and H - H(T,P) examples of this procedure are to be furnished in the exercises for this section. 

Closure of such differential equations requires the definitions of both constitutive relations for hydrodynamical functions and also kinetic relations for the chemistry. These functions are specified by recourse both to theoretical considerations and to rheological measurements of fluidization. We introduce the ideal gas approximation to specify the gas phase pressure and a caloric equation-of-state to relate the gas phase internal energy to both the temperature and the gas phase composition. It is assumed that the gas and solid phases are in local thermodynamic equilibrium so that they have the same local temperature. 

A solid phase internal energy is related, again through a caloric equation-of-state, to the temperature. The solid phase pressure is defined as a function of the solid volume fraction where the functional relationship (cf., M) is based upon the fluidized bed stability measurements of Rietma and his coworkers ( ). 

The differential forms of the conservation equations derived in the appendixes for reacting mixtures of ideal gases are summarized in Section 1.1. From the macroscopic viewpoint (Appendix C), the governing equations (excluding the equation of state and the caloric equation of state) are not restricted to ideal gases. Most of the topics considered in this book involve the solutions of these equations for special flows. The forms that the equations assume for (steady-state and unsteady) one-dimensional flows in orthogonal, curvilinear coordinate systems are derived in Section 1.2, where specializations accurate for a number of combustion problems are developed. Simplified forms of the conservation equations applicable to steady-state problems in three dimensions are discussed in Section 1.3. The specialized equations given in this chapter describe the flow for most of the combustion processes that have been analyzed satisfactorily. 

This provides a caloric equation of state for the system. It follows directly from the fact that U is the average energy of the system, that is. 

The quantities Cy and Cp are defined as shown in (1.13.15), and are known as heat capacities at constant volume or at constant pressure. These can be experimentally determined as a function of T over wide temperature ranges, normally by standard calorimetric methods (see also Section 1.16). Integration of the experimental heat capacities with respeet to temperature then yields 2 , //, or S as a function of T (see Section 1.17), with either K or 7 as parameters. Alternatively, by inserting the equation of state into (1.13.16) or (1.13.17), followed by integration, one can find the dependence of on T or /f on P, with T as a parameter. Eqs. (1.13.16) and (1.13.17) are known as caloric equations of state. 

We begin by inserting the constitutional equation of state in the caloric equation of state, Eq. (1.13.16) this leads to the important finding that (dE/dV)T = 0, regarded as a second criterion to be imposed on ideal gases. Thus, the energy of an ideal gas depends solely on temperature. As a result we now write out the differential energy in the abbreviated form dE = dE/dT)y dT, whence... 

One should note that Eq. (5.3.8c) represents an extension of the ordinary caloric equation of state, Eq. (1.13.16) the above relation applies when mole numbers are allowed to vary. 

The caloric equations of state can be made to mimic those of Briant and Burton and of Etters and Kaelberer, if the curves for the two forms are connected by a horizontal line at the temjjerature for which the two have the same free energy jjer particle. However, the question of the loops, like those of the van der Waals equation, which Briant and Burton report, is posponed until the third section of this chapter, which deals with simulations. 

In order to be able to solve the energy equation (3.71) or (3.75), some constitutive equations are required. We will now consider equation (3.71) for pure substances. This necessitates the introduction of the caloric equation of state u = u( t), v). By differentiation we obtain... 

At this stage we have to realize that there are still eight unknown variables left. Therefore we introduce the caloric equation of state together with the temperature T as a new variable ... 

The summation convention over repeated indices is assumed. Here an asterisk denotes some constant reference value taken, say, at the undisturbed interface. Besides, q denotes density and p, pressure. Vi and T denote the i-th component of the velocity field and temperature, respectively, T] is the shear or dynamic viscosity, rj = gu with u the kinematic viscosity, Cp the specific heat at constant pressure, g is the acceleration due to gravity and e the unit normal vector along z. a is the coefficient of thermal expansion. In eq. (3) we have also used the simplest caloric equation, of state between energy and temperature. To a first approximation the heat (J ) flux is... 

Equation (1.12.7b) is the so-caUed caloric equation of state, into which we had introduced the appropriate Maxwell relation [8] of Table 1.12.11. Thus, if the equation of state P = P(T,V,ni) is known for a given system, then its energy at a given temperature and composition is found by integration with respect to V. This provides a second method for finding the energy function of a system. 

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