Basic Model Equations

A batch process is one in which there is no flow into or out of the system. As we saw in Chapter 6,batch processes can be useful for determining reaction rate information. They are also useful for determining rates of mass transfer. As with the batch reactor, we charge the two phases to the stirred vessel at time t= 0. We assume that the two phases are in intimate contact, with one phase uniformly distributed throughout the other; uniformity is easy to achieve in practice for many systems. We further assume that each phase is individually well mixed, so that a sample of either phase drawn at any time will be the same as any other sample of that phase drawn

Phase lI Volume V^II Density ρ^II Concentration c_A^II Phase I

Volume V^I Density ρ^l Concentration c_A^I Figure 10.1. Schematic of a

two-phase system. Each two-phase must be taken as a separate control volume.

from any position in the tank at the same time. We have already seen the importance of the assumption of perfect mixing for single-phase systems, and the experimental achievement of near-perfect mixing for each phase is of comparable importance in our analysis here. We assume that the mass transfer occurs isothermally; this is frequently a valid assumption, and the isothermal analysis is always part of the more complete treatment when thermal effects do have to be taken into account.

As before, we will use volume V, densityρ, and concentration c as characterizing variables for mass. It is conventional when considering interfacial mass transport to measure concentration in mass units (e.g., kg/m³ or lbm/ft³), in contrast to the use of molar units when we were considering reacting systems. This practice causes no inherent difficulty, but it does require some care when analyzing systems in which there is both interfacial mass transfer and chemical reaction, since the reac-tion rates will be expressed most naturally in molar units. Selecreac-tion of the control volume requires some care. It is clear that the entire vessel is not a useful control volume for the two-phase system in most cases. We are interested in transfer between the two phases, hence we must work with two control volumes, one for each phase.

We will arbitrarily designate one phase as the continuous phase (Phase I) and one as the dispersed phase (Phase II). The continuous phase consists of the Swiss cheese-like volume shown inFigure 10.1; the volume of the continuous phase is denoted V^I, the densityρ^I, and the concentration of any species i is denoted c_i^I. The volume of the dispersed phase, V^{I I}, is made up of all the elements of the other phase and, although we treat it as a single volume for modeling purposes, it may consist physically of a number of distinct volumes. The density of the dispersed phase will be denoted by ρ^{I I}, and the concentration of any species i is denoted c_i^{I I}.

We will develop the mathematical description for a general two-phase system, recognizing that with appropriate identification of the continuous and dispersed phases, the basic model equations will apply for batch solid-liquid, solid-gas, liquid-liquid, or liquid-gas systems as long as the assumption of density and concentration uniformity within each phase can be maintained. (Meeting this requirement may be difficult in systems with a solid phase, or systems with a very viscous liquid, where transport within the phase may be very slow.)

To simplify the algebraic manipulations and emphasize the physical processes, we will develop the model equations for the case in which a single component species A is transferred between the phases. A has a concentration c^I_Ain the continuous phase and c^{I I}_A in the dispersed phase and does not react with any component in either phase. It is not difficult conceptually to extend the treatment to any number of species and to include reaction in one or both phases; in fact, we will introduce chemical reactions later in this chapter.

The equations of conservation of mass will require an expression for the rate at which each component species is transferred between the phases. We will use the boldface symbol r to distinguish this rate from the rate of chemical reaction.

Thus, the rate at which A enters Phase I through mass transfer across the inter-face is denoted r^I_A+, whereas the rate at which A is depleted in Phase I by mass transfer across the interface is denoted r^I_A−. Similarly, the rate of accumulation of A in Phase II through interfacial mass transfer is r^{I I}_A+, and the rate at which A is lost from Phase II through interfacial mass transfer is r^{I I}_A−. The dimensions of the rate of mass transfer are mass per area per time. The rate is written on a per-area basis because, all other things being held equal, an increase in the area between the phases will lead to a proportionate increase in the mass transferred. (We have already uti-lized this concept in Chapter 5 in writing the rate of solute transfer through the membrane on an area basis.)

Let a denote the total interfacial area between the phases. The equations for conservation of mass in each of the control volumes are then, respectively,

dρ^IV^I

dt = a[r^I_A+− r^I_A−], (10.1I) dρ^{I I}V^{I I}

dt = a[r^{I I}_A+− r^{I I}_A−]. (10.1II) It is evident for the batch system that the mass that leaves Phase I must go to Phase II, and vice versa. Thus, r^I_A+= r^{I I}_A− and r^I_A−= r^{I I}_A+. For convenience, we define the net rate r_Aas

r_A= r^{I I}A+− r^{I I}A−. (10.2) Then

dρ^IV^I

dt = −arA, (10.3I)

dρ^{I I}V^{I I}

dt = arA. (10.3II)

Application of conservation of mass to the species that is transferred, component A, leads in an identical manner to the component equations:

dc^I_AV^I

dt = −arA, (10.4I)

dc^{I I}_AV^{I I}

dt = arA. (10.4II)

0 50 100 150 200 250 300 450

400

350

300

250

200

150

100

50

c_A^II (acetone in water layer), kg/m³ cA (acetone in C2H3C3ayer) kg/m3

Slope = 2 Figure 10.2. Equilibrium concentration

of acetone in 1,1,2-trichloroethane at 25^◦C as a function of equilibrium con-centration of acetone in water. Data of R. E. Treybal, L. E. Weber, and J.

F. Daley, Ind. Eng. Chem., 38, 817–821 (1946).