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Feasibility Hypothesis

The flash cascade provides a two parameter (0 and D) model ((6.14) and (6.15)). The iterates depend primarily on the value of D. The pinch points towards which these iterates evolve depend only on a single parameter, (Da/0). Therefore, the solution structure is not dependent on the value of 0, and we may choose any value of 0 (we pick 0 = 0.5). This simply rescales the value of Da at which bifurcations occur.

Hypothesis: The trajectories of the flash cascades lie in the feasible product regions for continuous RD.

We test this hypothesis with an example. Consider the esterification of acetic acid with isopropanol at a pressure of 1 atm.

6 Feasibility and Process Alternatives for Reactive Distillation | 157 IPA + HOAc ^ IPOAc + H20 (6.16)

We present the kinetics by a homogeneous model. r = kf ^«hoac«ipa - -—-J (6.17)

where fef is approximately independent of temperature, (kf/kf ref) = 1, and the reaction equilibrium constant has a value of 8.7 in the temperature range of interest [30]. The vapor-liquid equilibrium (VLE) was modeled using the Antoine vapor pressure equation, the NRTL equation for activity coefficients, and including vapor phase dimerization of acetic acid. The physical property models were taken from Table 3 in Venimadhavan et al. [48].

Solutions of (6.14) and (6.15), the rectifying and stripping cascade flash trajectories, can be represented in mole fraction space (three dimensional for the IPOAc system). However, we represent the solutions in transformed composition space, which is two dimensional for IPOAc system (for a derivation and properties of these transformed variables [46]). This transformed composition space is a projection of a three dimension mole fraction space onto a two dimensional transformed composition subspace for the IPOAc system. Even though the correspondence between real compositions and transformed compositions is not one-to-one in the kinetic regime, we will make use of these transforms because of ease of visualization of the trajectories, and because overall mass balance for reactive systems (kinetically or equilibrium limited) can be represented with a lever rule in transformed compositions. We use this property to assess feasible splits for continuous RD.

If we select IPOAc as the reference component, then the definition given in Ung and Doherty [46] gives the transformed variables

Xi — *HOAc + ^IPOAc 1 (6.18) X2 — Xipa + ^IPOAc

Similar expressions are obtained for the transformed variables in the vapor phase, Yi.

The trajectories of the flash cascades for an equimolar feed (xFHOAc — 0.5, xFIPA — 0.5), are shown in Fig. 6.8 for D — 0.25 and D — 0.75. The stripping cascade is solved recursively for this feed composition via (6.14) for a large number of stages until a point is reached where the composition is essentially constant from stage to stage. This pinch point (stable node) is acetic acid. The mole fraction iterates are then converted to the transformed compositions using (6.18). In similar fashion, the rectifying cascade model (6.15) is solved recursively for a large number of stages until a pinch point (stable node) is reached; in this case it is a quaternary mixture. These vapor mole fractions so obtained are then converted to transformed compositions and are shown in Fig. 6.8.

Chemical Equilibrium

Fig. 6.8 Rectifying and stripping cascade tra- formed mole fraction space. For the stripping jectories for a saturated liquid equimolar reac- cascade, X, = xHOAc + *ipoao X2 = xwk + xIPOAc tant feed for the IPOAc system at D = 0.25 and and for the rectifying cascade, Y, = yHOAc + D = 0.75. The trajectories are plotted in trans- YIPOAc, Y2 = yIPA + yIPOAc

Nrtl Equation Graphics

Fig. 6.9 Feasible products from column simu- bottoms and the open squares represent lations with different NT, feed stage location, r, feasible distillates. The rectifying and stripping s, and D recorded in transformed variable cascade trajectories from Fig. 6.8 are also space. The filled squares represent feasible shown for comparison

Fig. 6.9 Feasible products from column simu- bottoms and the open squares represent lations with different NT, feed stage location, r, feasible distillates. The rectifying and stripping s, and D recorded in transformed variable cascade trajectories from Fig. 6.8 are also space. The filled squares represent feasible shown for comparison

Parametric column simulations for the IPOAc system were performed with different Damkohler numbers, reflux ratios, reboil ratios as well as total number of stages, (Nt) and feed tray location, (/). The distillate and bottoms compositions obtained were recorded in transformed composition space. Fig. 6.9 compares the products obtained from column simulations with 30 stages and using different values of r and s at D = 0.25 and D = 0.75. The column feed specification is the same as that to the co-current flash cascade. The flash trajectories provide a good estimate of the product compositions from a continuous column. We also compared the product compositions from column simulations with the flash trajectories in mole fraction space. We found that product compositions from column simulations surrounded the flash trajectories, in agreement with the hypothesis that the flash trajectories lie in the feasible product regions for continuous RD.

Note that calculating the flash trajectories at 0 = 0.5 does not provide the entire feasible product regions for continuous RD, but instead generates a subset of the feasible products. Selecting an iterate on the stripping cascade trajectory as a potential bottoms and an iterate on the rectifying cascade trajectory as a potential distillate does not guarantee that these products can also be obtained simultaneously from a RD column. This is simply because these product compositions may not simultaneously satisfy the overall mass balance for a reactive column. However, when the flash trajectories are used in conjunction with the lever rule for a continuous reactive column, the feasible splits for continuous RD can be quickly predicted.

Next, we derive the fixed-point criteria for the flash cascades and use bifurcation theory to propose rules to estimate feasible products.

Bifurcation Analysis of the Flash Cascade Model

The fixed points of the flash cascade model are the solutions of equations (6.14) and (6.15) for j ^ œ. in other words, successive liquid and vapor mole fractions reach constant values. The fixed points, x, for the stripping cascade (6.14) are solutions of where x and y are in vapor liquid equilibrium with each other.

Similarly, the fixed points, y, for the rectifying cascade (6.15) are solutions of

The fixed points obtained by solving (6.15) for a large number of iterates are stable nodes in the rectifying cascade. The same fixed points will form a subset of solu tions to equation (6.20). However, to have an analogy between this work and the earlier work done in non-reactive and equilibrium limited RD, we rewrite equation (6.20) as

(1 - D)(x - & - (fcfcfi-) (j^)r(x) = 0 (i = 1... c - 1) (6.21)

where x and y are in vapor liquid equilibrium with each other.

Equation (6.21) has the same fixed points as (6.20); except that their stability is reversed. Thus, a fixed point which is a stable node in equation (6.20) becomes an unstable node fior equation (6.21).

The solutions for equations (6.19) and (6.21) behave as follows. At D = 0 (the non-reactive limit), the fixed point criteria for both the rectifying and stripping cascades reduce to the same equation x - y = 0 (i = 1 ...c - 1) (6.22)

Equation (6.22) is the fixed point criteria for simple distillation and also for a continuous column at total reflux and total reboil. Since there is a symmetry in the rectifying and stripping maps, we can find the fixed points for both the rectifying and stripping cascades from equation (6.22). Thus, in this limit, our model recovers the criterion fior fiixed points in the well-known limit ofi no-reaction. At D = 1 (the chemical equilibrium limit), the fixed point criteria reduce to a single equation r(x) = 0 (6.23)

Equation (6.23) implies that fixed points lie on the reaction equilibrium surface. Equation (6.23), however, is just a necessary condition; the sufficient condition fior fiixed points ofi the rectifiying and stripping cascades can be written in terms ofitransfiormed variables by writing either (6.19) or (6.21) fior component i and a reference component, fc and adding the two, giving

The solutions of equation (6.24) are fixed points for simple RD at chemical equilibrium and also for a continuous RD at total reflux and total reboil. As in the non-reactive case, the fiixed point criteria fior the rectifiying and stripping cascades are the same and is given by equations (6.23) and (6.24). Once again our model reduces to the well-known criteria for chemical equilibrium fixed points.

For 0 < D < 1 (kinetically controlled regime), (6.19) gives the fixed points of the stripping cascade and (6.21) the fixed points of the rectifying cascade. Therefore, in the kinetic regime, there are different fixed point criteria for the rectifying and stripping cascades, a fact that has not been previously recognized. The solutions of equation (6.19) at 0 = 1 are the fixed points for simple RD [48], but the structure of the solutions to equation (6.21) has not been reported. As we shall see, it can be

Chemical Equilibrium
Fig. 6.10 Bifurcation diagrams for the (a) rectifying and (b) stripping cascades. The filled circles denote stable node branches, open circles denote unstable node branches, and the open squares denote saddle branches

quite different than for equation (6.19). Since fixed points for simple RD are equivalent to the fixed points of the stripping cascade at 0 = 1, they can only provide information about the potential bottoms product from a continuous column. Therefore, the distillate product composition from a continuous column for 0 < D < 1 cannot be inferred from a knowledge of fixed points of simple RD. However, it is possible to estimate potential distillates from the fixed points of the rectifying flash cascade (equation (6.21)).

We are interested in investigating the fixed-point branches of the flash cascade model for 0 < D < 1 at 0 = 0.5. A systematic approach is by a bifurcation analysis of the solutions, x(D), for (6.19) and (6.21). The starting points for the analysis are the solutions at D = 0; to calculate these points, a homotopy continuation method is available [14]. We use the mixture boiling point, T(x(D)), to represent the solu-

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