Equation for the Continuous Phase Vector

Recall that the continuous phase variables were described by the vector field Y(r, t). In general, the components of this vector field should encompass all continuous phase quantities that affect the behavior of single particles. These could include all dynamic quantities connected with the motion of the continuous phase, the local thermodynamic state variables such as pressure and temperature, concentrations of various chemical constituents, and so on. Clearly, this general setting is too enormously complex for fruitful applications so that it is necessary to suitably constrain our domain of interest. In this connection, the reader may recall our exclusion of the fluid mechanics of dispersions, so that we shall not be interested in the equation 

The continuous phase vector Y(r, i) will include temperature (under nonisothermal situations), and concentrations of various chemical components that may be involved in transport between the continuous phase and the particles, and in chemical reactions in either phase. Liquid-liquid dispersed phase reactors are a common feature of the chemical process industry where the preceding processes are encountered. For the present we shall assume isothermal conditions and consider only concentration components in Y(r, t). Alternatively, this strategy would be appropriate even for nonisothermal situations if temperature were to be isolated as another variable to be dealt with through an energy transport equation. 

The particles are viewed as point sources because of the assumption made earlier that the variation of Y(r, t) occurs over length scales considerably larger than that of particle size. The transport equation for Y(r, t) may now be written as 

Except for the third term due to the presence of particles on the left-hand side, Eq. (2.9.1) is a familiar transport equation (e.g., Bird et ai, 1960). The population balance equation (2.7.9) must in general be considered together with the above continuous phase equation. 

10 Note that jy is a vector whose elements are scalars while Jy is a vector whose elements are spatial vectors. 

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