True moving beds (TMB):

Chromatographic separation processes such as chromatographic columns are intrinsically dynamic process and usually operating in batch. A true and simulated moving bed is an alternative to batch operation and combines the distinct process steps in a single continuously operating unit. This has

Figure 1: True for the separation of i-pentane and 22DMB using a simulated moving bed . The coloring represent the composition distribution and was obtained by simulation.

high economical potential and made it possible to apply chromatographic separations in fine chemistry applications such as separations of natural products, pharmaceuticals and aromas.

The process principles of a true moving bed are shown in Figure 1. The solid phase is not stationary but moving counter current to the vapor phase. At a feed F1, the mixture is injected to the TMB unit. In our example the mixture contains 50% of 22DMB and 50% of a weaker adsorbing i-pentane. While the feed mixture is carried in the gas phase towards the outlet P1, the more strongly adsorbing component, 22DMB, is adsorbed and carried in the solid phase to the outlet P2; see section II and III. If the operation parameters are correctly chosen all 22DMB should be adsorbed before the gas phase reaches the outlet P1. Therefore, the outlet P1 will only contain i-pentane and no 22DMB. The true moving bed has another feed F2 that supplies inert solvent (desorbent) to the column. The inert solvent has the tasks to remove 22DMB from the solid phase (and so to refresh/desorb the bed in section I) and to carry the desorbed 22DMB in the gas phase to the outlet P2 where it is withdrawn. Moving the regenerated adsorbing phase towards the outlet P1 closes the cycle. In section IV the remaining i-pentane is adsorbed and "recycled" by carrying it within the solid phase to section III. There it desorbs since the competing component 22DMB adsorbs stronger and i-pentane is carried back in the gas phase to the outlet P1.

Notes on the implementation:

In case of a true moving bed (TMB) the solid phase is moving with a constant interstitial velocity, u_s, whereas the feeds and withdrawals are placed at fixed locations. Our strategy was to adapt the discretized fixed bed model. The fixed bed model accounts for intra-particle diffusion resistance and incorporates several adsorption isotherms such as multicomponent Langmuir and IAS theory. When simulating a TMB unit we connect the outlet of the original fixed bed model with its inlet and add additional nodes describing the feeds and withdrawals. Furthermore we have to adapt the component mass balance in the gas phase. In order to do so we redefine the dimensionless adsorber length by taking into account that the solid phase is continuously moving at a constant interstitial velocity, u_s. At any location z* along the TMB unit we obtain for the solid phase the component mass balance:

where z* is the location of the solid in time. In case of a SMB or a fixed bed unit, the solid phase does not move and so the location does not change in time, z*=z. However, if we model a counter-currently moving solid phase, we have to account for this by modifying the flow path coordinate, z*=z-u_st . Therefore, the mass balances in the solid phase yields:

and at steady state we obtain

Since the solid and gas phases are operating counter-current, we applied a downwind discretization scheme for the gas phase and an upwind discretization scheme for the solid phase. The resulting set of algebraic equations can be solved by a Newton method.

Nodes describe the feeds or withdrawals. For each node the user has to specify the feed or withdrawal mixture flow rates and additionally for feed the inlet compositions.

Figure 2: Schematic drawing of a general node. The yellow marked parameters are specified by the user. The inert gas is implicitly added or removed depending on the specification of uI.

Another input parameter is the downstream gas velocity, which allows fixing the flow rate ratios in each section; see Figure 2. Note, the specified gas velocity serves as an initial condition for each equation subset describing a section (the fixed bed between two nodes). Since we allow specifying both feed/withdrawal flow rates and the initial gas velocity of the section, the user has to carefully check consistency. If no inert gas is added or removed at a feed the mass balance relates the two parameters with each other. On the other hand this way of specifying the flow rates, gives the user the freedom to modify the critical flow rate ratios independently by adding implicitly inert gas (recall: the inert component is not defined as a mixture component). For instance, one can withdraw the entire stream at the end of a section and by specifying a gas velocity implicitly feeding inert gas to the next section. Instead of specifying explicitly the flow rates, we input the initial gas velocity and the withdrawal or feed ratios; see Figure 2. The feed ratio r^F is given by the ratio between the feed flow rate and the inlet flow rate of the subsequent section. The withdrawal ratio r^W is defined with regard to the outlet flow rate of the previous section. Due to these definitions the values of both input parameters will range between 0 and 1.

For initialization we assume that the solid phase is stagnant. Hence, the concentration profiles and gas velocities are constant too (considering a non-reactive mixture). In many cases the Newton solver converges quickly to the solution with this very simple initialization scheme. If not, the user can apply a homotopy method or initialize the solver with a known solution.

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