The equation of change

Consider a fluid of molecules Interacting with pair additive, centrally symmetric forces In the presence of an external field and assume that the colllslonal contribution to the equation of motion for the singlet distribution function Is given by Enskog s theory. In a multicomponent fluid, the distribution function fi(r,Vj,t) of a particle of type 1 at position r, with velocity Vj at time t obeys the equation of change (Z)... 

First a derivative is given of the equations of change for a pure fluid. Then the equations of change for a multicomponent fluid mixture are given (without proof), and a discussion is given of the range of applicability of these equations. Next the basic equations for a multicomponent mixture are specialized for binary mixtures, which are then discussed in considerably more detail. Finally diffusion processes in multicomponent systems, turbulent systems, multiphase systems, and systems with convection are discussed briefly. 

Further the pressure and temperature dependences of all the transport coefficients involved have to be specified. The solution of the equations of change consistent with this additional information then gives the pressure, velocity, and temperature distributions in the system. A number of solutions of idealized problems of interest to chemical engineers may be found in the work of Schlichting (SI) there viscous-flow problems, nonisothermal-flow problems, and boundary-layer problems are discussed. 

In Sec. II,A the equations of change are derived by assuming that the fluid is a continuum. A physically more satisfying derivation may be performed in which one starts directly from considerations of the fundamental molecular-collision processes occurring in the fluid. For dilute monatomic gases and gas mixtures one can start... 

In turbulent flow the eddies, which are superimposed on the over-all flow pattern, have dimensions large compared with a mean free path. Hence turbulent motion is macroscopic rather than molecular and the equations of change do apply to turbulent flow. This subject is discussed further in Sec. II,D,2. 

It has already been pointed out that the equations of change are valid for describing turbulent flow. The diffusion of A in a nonreacting binary mixture is described by the equation of continuity ... 

Diffusion problems in systems involving forced and free convection are good illustrations of the importance of presenting all three of the equations of change as a prelude to a general discussion of diffusion. Only a handful of idealized problems of this type have been solved analytically. Since they are, however, of considerable importance in chemical engineering it is worth while to make some general remarks about them. 

In connection with the interphase mass transfer in liquid-liquid and liquid-gas systems, the diffusion equations (and indeed all the equations of change) are valid in both phases. Hence, in principle, diffusion problems in a two-phase system may be solved by solving the diffusion equations in each phase and then choosing the constants of interaction in such a way that the solutions match up at the interface. It is customary to require that the following two conditions be fulfilled at the interface, in a system in which the solute is being transferred from phase I to phase II (1) the flux of mass leaving phase I must equal the flux of mass entering phase II if the diffusion... 

This problem is a good example of the importance of formulating a complex diffusion problem in terms of the equations of change. Hence the simplified treatment given here is discussed in terms of the simplified solutions to the three basic equations. 

The fully filled channel and the isothermal assumptions are not realistic in that, in practice, channels are partially filled and the flow is nonisothermal. The constitutive equation and the equations of change used are ... 

The simulation is for a shear thinning fluid and nonisothermal flow. The equations of change are... 

Expression for production of entropy (8.18) can be now compared with the general results of non-equilibrium thermodynamics, which are known for both non-stationary and stationary cases. It is obvious, that last term in the right-hand side of relation (8.18) corresponds to a non-stationary case and includes the equation of change of internal variables that is relaxation equation. The first two terms in formula (8.18) correspond to a stationary case and should be considered as the products of thermodynamic fluxes and thermodynamic forces (it is possible with any multipliers). When the internal variables are absent, we should write a relation between the fluxes and forces in the form... 

If the collisions of molecules produce a chemical reaction, the Boltzmann equation is modified in obtaining the equations of change these problems are addressed and analyzed in the context of quantum theory, reaction paths, saddle points, and chemical kinetics. Mass, momentum, and energy are conserved even in collisions, which produce a chemical reaction. 

Here Cp is the heat capacity at constant pressure. Thermal diffusivity has the same units as kinematic viscosity, and they play similar roles in the equations of change for momentum and energy. The dimensionless ratio... 

No exact general criterion is available when it is necessary to include the relaxation terms in the equations of change however, relaxation terms are necessary for viscoelastic fluids, dispersed systems, rarefied gases, capillary porous mediums, and helium, in which the frequency of the fast variable transients may be comparable to the reciprocal of the longest relaxation time. 

The equations for conservation of mass, momentum, and energy for a one-component continuum are well known and are derived in standard treatises on fluid mechanics [l]-[3]. On the other hand, the conservation equations for reacting, multicomponent gas mixtures are generally obtained as the equations of change for the summational invariants arising in the solution of the Boltzmann equation (see Appendix D and [4] and [5]), One of several exceptions to the last statement is the analysis of von Karman [6], whose results are quoted in [7] and are extended in a more recent publication [8] to a point where the equivalence of the continuum-theory and kinetic-theory results becomes apparent [9]. This appendix is based on material in [8]. 

The rigorous approach to a kinetic-theory derivation of the fluid-dynamical conservation equations, which begins with the Liouville equation and involves a number of subtle assumptions, will be omitted here because of its complexity. The same result will be obtained in a simpler manner from a physical derivation of the Boltzmann equation, followed by the identification of the hydrodynamic variables and the development of the equations of change. For additional details the reader may consult [1] and [2]. 

The equations of change for each fluid phase and the jump balance conditions that must be met at the interface are summarized in Table 1.5. There is an important restriction on the equations in Table 1.5 the effect of chemical reactions in the bulk fluid phase has been neglected. For all of the applications considered in this book this neglect is justified. 

A this point a few mathematical prerequisites are required before the derivation of the equations of change can be discussed. First, the three kinds of time derivatives involved in this task are defined. Secondly, the transport theorem is introduced. 

Moreover, by use of the continuity equation we obtain the form of the equation of change for kinetic energy which is usually given in the literature (e.g., [32]) ... 

Note also that compared to the momentum equation, the equation of change for kinetic energy and the equation for angular momentum do not introduce additional unknowns, nor do they include new information. 

Bird RB (1957) The equations of change and the macroscopic mass, momentum, and energy balances. Chem Eng Sci 6 123-181... 

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