Batch Conservation Equations

The population balance, which was put forward by Randolph and Larson (1962) and Hulbert and Katz (1964), serves as the basis for characterizing the CSD in suspension crystallization systems. For a batch crystallizer or a semibatch crystallizer with no net inflow or outflow of crystals, the population balance can be written as (Randolph and Larson 1988) 

Equation (10.3) requires a boundary condition and an initial condition. The boundary condition n(0, l) is the nuclei population density ( o), which can be related to the nucleation rate (So) by 

The initial population density n L, 0) for a batch crystallizer is not well defined. For a crystallizer seeded externally, n(L, 0) may be denoted by an initial seed distribution function hs L). However, in an unseeded system, initial nucleation can occur by several mechanisms, and one cannot realistically use a zero initial condition for the size distribution. To overcome this difficulty, Baliga (1970) suggested the use of the size distribution of crystals in suspension at the time of the first appearance of crystals as the initial population density. 

In addition to the population balance equation, appropriate kinetic equations are needed for evaluation of the nucleation rate (.So) and the growth rate (G). Empirical power-law expressions are frequently used to correlate G and 5o (Randolph and Larson 1988). For the growth rate. 

The bulk solute concentration (c) can be determined from the mass balance for batch system (Baliga 1970) 

---

The structure and interrelationship of the batch conservation equations (population, mass, and energy balances) and the nucleation and growth kinetic equations are illustrated in an information flow diagram shown in Figure 10.8. To determine the CSD in a batch crystallizer, all of the above equations must be solved simultaneously. The batch conservation equations are difficult to solve even numerically. The population balance, Eq. (10.3), is a nonlinear first-order partial differential equation, and the nucleation and growth kinetic expressions are included in Eq. (10.3) as well as in the boundary conditions. One solution method involves the introduction of moments of the CSD as defined by... 

Figure 10.8 Information flow diagram showing interrelationships of batch conservation equations, nucleation and growth kinetic equations, and the resulting CSD in batch suspension crystallizers. (Reproduced by permission of Gordon and Breach Science Publishers S.A. from Wey 1985.)...

The analysis of batch crystallizers normally requires the consideration of the time-dependent, batch conservation equations (e.g., population, mass, and energy balances), together with appropriate nucleation and growth kinetic equations. The solution of these nonlinear partial differential equations is relatively difficult. Under certain conditions, these batch conservation equations can be solved numerically by a moment technique. Several simple and useful techniques to study crystallization kinetics and CSDs are discussed. These include the thermal response technique, the desupersaturation curve technique, the cumulative CSD method, and the characterization of CSD maximum. 

In general, solutions are obtained by couphng the basic conservation equation for the batch system, Eq. (16-49) with the appropriate rate equation. Rate equations are summarized in Table 16-11 and 16-12 for different controlhng mechanisms. 

The conservation equation for B in the liquid phase is a batch-reactor model ... 

Mathematical models derived from mass-conservation equations under unsteady-state conditions allow the calculation of the extracted mass at different bed locations, as a function of time. Semi-batch operation for the high-pressure gas is usually employed, so a fixed bed of solids is bathed with a flow of fluid. Mass-transfer models allow one to predict the effects of the following variables fluid velocity, pressure, temperature, gravity, particle size, degree of crushing, and bed-length. Therefore, they are extremely useful in simulation and design. 

Batch Reactor Since there is no addition or removal of reactants, the mass and energy conservation equations for a batch reactor with a constant reactor volume are... 

The mass conservation equations for a batch reactor are as follows ... 

Rawlings JB, Ray WH (1988) The Modeling of Batch and Continuous Emulsion Polymerization Reactors. Part 1 Model Formulation and Sensitivity to Parameters. Polymer Engineering and Science 28(5) 237-256 Reyes Jr JN (1989) Statistically derived conservation equations for fluid particle flows. Proc ANS Winter Meeting. Nuclear Thermal Hydraulics, 5th Winter Meeting... 

In this section we must be careful to respect our prior concern about the definition of rate with regard to the volume of reaction mixture involved. Further, since we wish to concentrate attention on the kinetics, we shall study systems in which the conservation equation contains the reaction term alone, which is the batch reactor of equation (1-12). It is convenient to view this type of reactor in a more general sense as one in which all elements of the reaction mixture have been in the reactor for the same length of time. That is, all elements have the same age. Since the reactions we are considering here occur in a single phase, the relationships presented below pertain particularly to homogeneous batch reactions, and the systems are isothermal. 

Unsteady material and energy balances are formulated with the conservation law, Eq. (7-68). The sink term of a material balance is and the accumulation term is the time derivative of the content of reactant in the vessel, or 3(V C )/3t, where both and depend on the time. An unsteady condition in the sense used in this section always has an accumulation term. This sense of unsteadiness excludes the batch reactor where conditions do change with time but are taken account of in the sink term. Startup and shutdown periods of batch reactors, however, are classified as unsteady their equations are developed in the Batch Reactors subsection. For a semibatch operation in which some of the reactants are preloaded and the others are fed in gradually, equations are developed in Example 11, following. 

Storage Tanks The equations for batch operations with agitation may be applied to storage tanks even though the tanks are not agitated. This approach gives conservative results. The important cases (nonsteady state) are ... 

There is of course an equation for each species, subject to conservation of total mass given by the continuily equation, and we implicitly used conservation of total mass as stoichiometric constraints, Njo — Nj)lvj are equal for all 7 in a batch (closed) system or (Fjo — Fj)/vj are equal for aU j in a steady-state continuous reactor. 

Thus far, we have considered the reaction engineering of molecules in containers that conserve mass. The same equations describe the populations of living systems in their environment, a container operating in batch or flow mode with inputs, outputs, disturbances, catalysts, internal recirculation patterns, and even diffusion (svvirtirtiirig, walking, flying) (Figure 8-19). 

If the processes just described are assumed to characterize the transfer of mass and energy in a fixed-bed adsorber, the conservation principles may be applied to them to describe the temperature and concentration as a function of time and position. Presenting the equations for a fixed-bed geometry has the advantage of including also equations, as special cases, for transient adsorption in single particles or groups of particles in batch systems. 

The mathematical model of the batch reactor consists of the equations of conservation for mass and energy. An independent mass balance can be written for each chemical component of the reacting mixture, whereas, when the potential energy stored in chemical bonds is transformed into sensible heat, very large thermal effects may be produced. 

Theories are not used directly, as in the discussion presented in Sect. 3.1, but allow building a mathematical model that describes an experiment in the unambiguous language of mathematics, in terms of variables, constants, and parameters. As an example, when considering the identification of kinetic parameters of chemical reactions from isothermal experiments performed in batch reactors, the relevant equations of mass conservation (presented in Sect. 2.3.1) give a set of ordinary differential equations in the general form... 

This equation can be applied to the total mass involved in a process or to a particular species, on either a mole or mass basis. The conservation law for mass can be applied to steady-state or unsteady-state processes and to batch or continuous systems. A steady-state system is one in which there is no change in conditions (e.g., temperature, pressure) or rates of flow with time at any given point in the system the accumulation term then becomes zero. If there is no chemical reaction, the generation term is zero. All other processes are classified as unsteady state. 

---