Solute movement analysis concentrated systems

To study TSA systems with the solute movement analysis we must determine the effect of tenperature changes on the solute waves, the rate at which a tenperature wave moves in the column, and the effect of temperature changes on concentration. The first of these is easy. As tenperature increases the equilibrium constants, and K, both decrease, often following an Arrhenius type relationship as shown in Eq. fl8-7). If the effect of temperature on the equilibrium constants is known, new values of the equilibrium constants can be calculated and new solute velocities can be determined. 

In general, we cannot obtain analytical solutions of the complete mass and energy balances for nonlinear systems. One exception to this is for isothermal systems when a constant pattern wave occurs. Constant pattern waves are concentration waves that do not change shape as they move down the column. They occur when the solute movement analysis predicts a shock wave. 

The analysis procedure developed in the previous section for gas permeation forms the basis for analyzing RO. However, the RO analysis is more complicated because of 1) osmotic pressure, which is included in Eq. fl7-12T and 2) mass transfer rates are much lower in liquid systems. Since the mass transfer rates are relatively low, the wt frac of solute at the membrane wall x will be greater than the wt frac of solute in the bulk of the retentate x, . This buildup of solute at the membrane surface occurs because the movement of solvent through the membrane carries solute with it to the membrane wall. Since the solute does not pass through the semipermeable membrane, its concentration will build up at the wall and it must back diffuse from the wall to the bulk solution. This phenomenon, concentration polarization, is illustrated in Figure 17-10. Concentration polarization has a major effect on the separations obtained in RO and UF (see next section). Since concentration polarization causes x > Xp the osmotic pressure becomes higher on the retentate side and, following Eq. fl7-12). the flux declines. Concentration polarization will also increase Ax in Eq. tl7-13 and flux of solute may increase, which is also undesirable. In addition, since concentration polarization increases solute concentration, precipitation becomes more likely. 

Diffusion is the mass transfer caused by molecular movement, while convection is the mass transfer caused by bulk movement of mass. Large diffusion rates often cause convection. Because mass transfer can become intricate, at least five different analysis techniques have been developed to analyze it. Since they all look at the same phenomena, their ultimate predictions of the mass-transfer rates and the concentration profiles should be similar. However, each of the five has its place they are useful in different situations and for different purposes. We start in Section 15.1 with a nonmathematical molecular picture of mass transfer (the first model) that is useful to understand the basic concepts, and a more detailed model based on the kinetic theory of gases is presented in Section 15.7.1. For robust correlation of mass-transfer rates with different materials, we need a parameter, the diffusivity that is a fundamental measure of the ability of solutes to transfer in different fluids or solids. To define and measure this parameter, we need a model for mass transfer. In Section 15.2. we discuss the second model, the Fickian model, which is the most common diffusion model. This is the diffusivity model usually discussed in chemical engineering courses. Typical values and correlations for the Fickian diffusivity are discussed in Section 15.3. Fickian diffusivity is convenient for binary mass transfer but has limitations for nonideal systems and for multicomponent mass transfer. 

The tracer diffusion coefficients for cations, anions, and water are shown as a function of salt concentration in Table 4.4. From the table, it can be seen that the diffusion coefficients for the ions and water molecules decrease with increasing salt concentration, and close analysis of the data in Table 4.4 indicates that, for the Csl solution, the decrease in diffusion coefQcients with concentration is the smallest and largest for the LiCl solution. In both cases, as the concentration increases, the diffusion coefficients of the cation and anion become more nearly alike. A similar observation has been made in previous studies of aqueous solutions of NaCl and KCl at high concentrations (Allen and Tildesley 1987 Chowdhuri and Chandra 2001 Koneshan and Rasaiah 2000). As for the systems containing large ions (e.g., Cs+, Rb+, and I" ions) with weaker electric fields, the observed decrease of ion diffusivity also can be explained in terms of inCTeased ion pairing, which slows down the movement of ions and water molecules bound to them. 

The solution of the initial value problem described by Eq. (2.9) can be visualised so that the calculated concentrations are plotted as a function of time as shown in Fig. 2.1a. Another possibility is to explore the solution in the space of concentra-tirais as in Fig. 2.1b. In this case, the axes are the concentrations and the time dependence is not indicated. The actual concentration set is a point in the space of concentrations. The movement of this point during the simulation outlines a curve in the space of cOTicentrations, which is called the trajectory of the solution. This type of visuaUsation is often referred to as visualisation in phase space. In a closed system, the trajectory starts from the point that corresponds to the initial value and after a long time ends up at the equilibrium point. In an open system where the reactants are continuously fed into the system and the products are continuously removed, the trajectory may end up at a stationary point, approach a closed curve (a limit cycle in an oscillating system) or follow a strange attractor in a chaotic system. It is not the purpose of this book to discuss dynamical systems analysis of chemical models in detail, and the reader is referred to the book of Scott for an excellent treatment of this topic (Scott 1990). 

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