Solute movement theory

All the solute velocities in a countercurrent system can be determined using the solute movement theory developed in Section 14.1. The solute velocities developed In Eqs. (14.1-I). (14.1-3). and (14.1-5) are all with respect to the solid. Thus, in the countercurrent system ihe appropriate fluid velocity v is... 

The solute movement theory developed in Section 14,1 is applied easily to the simulated moving-bed process,1 In between shifts in the port location each section is a fixed bed. The solute velocities are then given by Eq. (14.1-3) or (14.1-5). The solute movement for a linear system can be plotted as shown in Fig. 14.3-2. Feed is introduced at all locations labeled A + B, but most of the solute waves are not shown... 

Solute movement theory for simulared countercurrent process.1 Reprinted by permission Renner s Association, ]nc. 

A 1-meter column is packed with activated alumina. The column initially contains pure liquid cyclohexane solvent. At time t = 0 a feed pulse that contains 0.0001 mol/L anthracene and 0.0002 mol/L naphthalene in cyclohexane is input for 10.0 minutes. The superficial velocity is 20 cm/min for both feed and solvent steps. Use the solute movement theory to predict the outlet concentrations. 

F. Generalize. Since no spreading phenomena are included in the simple solute movement theory, the oudet peaks are predicted to be square waves tFigure 18-6Bi. When mass transfer and axial dispersion are included, the curves are spread out more as was illustrated in Figure 18-5C. The solute movement theory correcdy predicts the behavior of an average molecule. Thus, the time for the center of the peaks is correctly predicted. Note that the dominant term in the denominator for both solutes is the adsorption term This will be the case when there is relatively strong adsorption. 

Since purge cycles use large amounts of solvent, other regeneration methods have been developed. These methods and their analysis with the solute movement theory is the topic of this section. 

As a first approximation, and solute movement theory is a first approximation, the thermal wave velocity is independent of concentration and tenperature. Tenperature is constant along the lines with a slope equal to the numerical value of 11. ... 

In TSA processes the purpose of increasing the tenperature is to remove the adsorbate from the adsorbent. This happens when the thermal wave intersects the solute wave. Assume that the column is initially at a uniform tenperature Tf, concentration Cf, and adsorbent loading q. Fluid at tenperature T2 is fed into the column. This tenperature change causes the concentration and adsorbent loading to change to C2 and q2 (currenfly both unknown). Since the solute movement theory assumes local equilibrium, C2 and q2 are in equilibrium at T2. The control volume shown in Figure 18-7B fWankat. 1986) will be used... 

Assume Wall heat capacities can be ignored, heat of adsorption is negligible, no adsorption of n-heptane, and system is at cyclic steady state. Using the solute movement theory... 

The simple Skarstrom cycle for PSA shown in Figure 18-llA has constant pressure (isobaric) periods and periods when pressure is changing. We will assume that a very dilute gas stream containing trace amounts of adsorbate A in an weakly adsorbed carrier gas is being processed and that over the concentration range of interest the linear isotherm, Eq. fl8-5bl. is accurate. If mass transfer is very rapid, then the solute movement theory can be applied. Since the system is very dilute, the gas velocity is constant and the system is assumed to be isothermal. In more concentrated PSA systems neither of these assunptions are true, and a more conplicated theory must be used fRuthvenetal.. 19941. 

Application of the solute movement theory will be illustrated in Exanple 18-4. Before doing this, we note that axial dispersion is normally significant in gas systems. Thus, we expect that the solute movement theory will over-predict the separation that occurs. Alternatively, the value of y required in Eq. (18-26) for a given product purity will be larger in a real system than predicted by the solute movement theory (y = 1 for linear systems). For separations based on differences in equilibrium isotherms, if the solute movement theory predicts that a separation is not feasible or will not be economical, more detailed calculations will rarely inprove the results. 

Usually the desorbent must be removed from the A and B product streams. Increasing the amount of desorbent will increase the cost for this removal and will also increase the diameters of the columns requiring more adsorbent. Thus, the ratio of desorbent to feed, D/F, often controls the cost of SMB systems. For an ideal system with no zone spreading (no axial dispersion and very fast mass transfer rates) the solute movement theory can be used to calculate D/F by solving Eqs. tl8-29al to tl8-29dl simultaneously with Eq. (18-15) and the mixing mass balances with constant density. 

The TMB shown in Figure 18-14A is also of interest and can be analyzed using solute movement theory. 

A 100.0 cm long column is packed with activated alumina. The column is initially totally saturated at c = 0.011 mol/L anthracene in cyclohexane solvent. It is then eluted with pure cyclohexane solvent (c = 0) at a superficial velocity of 30.0 cm/min. Predict and plot the oudet concentration profile using solute movement theory. 

A. Define. Find the values of the outlet concentration at different times using the solute movement theory. 

For an isotherm with a Langmuir shape, if the column is initially loaded at some low concentration, c q, (ciow = 0 if the column is clean) and is fed with a fluid of a higher concentration, C] (see Figure 18-IZA), the result will be a shock wave. The feed step in adsorption processes usually results in shock waves. Experiments show that when a shock wave is predicted the zone spreading is constant regardless of the column length (a constant pattern wave). With the assunptions of the solute movement theory (infinitely fast rates of mass transfer and no axial dispersion), the wave becomes infinitely sharp (a shock) and the derivative dq/dc does not exist. Thus, the Aq/Ac term in the denominator of Eq. f 18-141... 

The solute movement theory developed in Sections 18.3. and 18.4. is easily extended to ion movement. For gel-type ion-exchange resins, which are most popular, there are no permanent pores and p = 0. The development of the solute movement theory from Eqs. fl8-l01 to tl8-l41 is modified by setting Cp = 0,... 

Note that when z = u t, which is the solute movement solution, the argument of the error function is zero, erf is zero, and c/cp = Vi. Thus, the solute movement theory predicts the center of the spreading wave. 

Experimental results (Figure 18-IS) and the shockwave analysis showed that the wave shape for constant pattern waves is independent of the distance traveled. This allows us to decouple the analysis into two parts. First, the center of the wave can be determined by analyzing the shock wave with solute movement theory. Second, the partial differential equations for the column mass balance can be sirtplified to an ordinary differential equation by using a variable = t - z/u jj that defines the deviation from the center of the wave. This approach is detailed in more advanced sources (e.g., Ruthven. 1984 Sherwood et al.. 1975 Wankat. 19901. 

The LUB approach is used for the adsorption step. During desorption a proportional pattern (diffuse) wave usually results as shown in Figure 18-15B (monovalent-divalent ion exchange can be an exception to this—see Exanple 18-8). Since the shape of the pattern changes with length, the LUB approach cannot be used for desorption. However, the results of the solute movement theory for diffuse waves are often quite accurate. Thus, the diffuse wave predictions can be used for preliminary design. The desorption step should be checked with experimental data. 

Use solute movement theory to predict the oudet concentration profile of A (Cqu vs. time). (You can report this as a graph or as a table or as both.)... 

Note The solute movement theory works just as well for mole balances as for mass balances. 

FIGURE 14.3-2 Solute movement theory for simulated countercurrent process. Reprinted by peimisriion of Com Refiner s Association. Inc. 

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