Flash Cascade Model

In a distillation column, each section (rectifying or stripping) can be effectively represented by a counter-current cascade of flashes. This cascade can be simplified for the purpose of estimating feasible product compositions, by removing the counter-current recycling of liquid and vapor flows among successive flash units, leaving a co-current arrangement shown in Fig. 6.7. The compositions of the liquid and vapor streams in the flash cascade arrangement are close to those of the counter-current cascade and by extension, to those in a column section. The two arrangements (counter-current against co-current) differ mainly in that the recovery of the key components in the counter-current cascade is much higher than in the flash cascade arrangement [25]. In a feasibility analysis, it is the product compositions that we are trying to estimate, so we use the simpler co-current flash cascades to study the feasibility of reactive mixtures in continuous columns.

We formulate the reactive flash model for an equimolar chemistry. Next, we hypothesize a condition under which the trajectories of the flash cascade model lie in the feasible product regions for continuous RD. This hypothesis is tested for an example mixture at different rates of reaction. The fixed point criteria for the flash cascade are derived and a bifurcation analysis shows the sharp split products from a continuous RD.

A schematic of the co-current flash cascade arrangement is shown in Fig. 6.7. There are two sections. In the rectifying cascade, vapor from each flash reactor

is partially condensed and fed to the next unit. This vapor cascade is similar to the rectifying section of a continuous distillation column but without any liquid recycling. The opposite is done for the stripping cascade shown in the bottom half of the figure, where the liquid stream from each flash device is partially vaporized and sent as feed to the next unit in the series.

Each flash device in Fig. 6.7 is a two phase CSTR reactor-separator with chemical reaction occurring only in the liquid phase. For simplicity we begin with a single reaction, an equimolar chemistry, a saturated liquid feed, and steady state conditions [5]. The flash devices operate under isobaric conditions, so that the temperature changes from stage to stage according to the boiling point of the stage composition.

The overall mass balance for the /h unit in the stripping cascade in Fig. 6.1 is

The material balance for the jth component is Lj-iXij-i = VjYij + LjXij - vikfHf(Xj) (i = 1 ■■■ c - 1, j = 1 ■■■ N) (6.6)

where I and Vare liquid and vapor molar flows; X; and are the liquid and vapor phase mole fractions for component i; is the stoichiometric coefficient for component i (vi < 0 for reactants, v; > 0 for products); kf is the forward rate constant with the dimensions of reciprocal time; Hj is the molar liquid holdup in the jth flash unit and r(x) is the reaction driving force

\ reactants products )

where Keq is the chemical equilibrium constant and ai is the activity for component i. For liquid phase reactions, activities are represented by the product of the activity coefficient, yi and the liquid phase composition, ai = yi xi. Eliminating Lj from (6.6) leads to

V , Hj xi,j-i - Xj = (Yij - xij) - Vikf—^- r(xf) (i = 1... c - 1,j = 1... N) (6.8)

We define two dimensionless parameters:

• < = Vj/Lj_1, the fraction of feed vaporized in the jth unit.

• Daj = (Hj/Lj_1)/(1/ki lei), the Damkohler number for unit j.

This is the ratio of the characteristic liquid residence time to a characteristic reaction time. kf, ref is the forward rate constant at a reference temperature, Tref. No reaction occurs in the limit Daj ^ 0 and reaction equilibrium is achieved as Daj ^ At intermediate Daj, the stage operates in the kinetically controlled regime. Incorporating < and Daj in (6.8), we find

Xij-1 - Xij = (Yij - Xij) - Vi\T^—) Dajr(Xj) (i = 1... c - 1, j = 1... N) (6.9)

<j and Daj are two independent parameters for each flash unit. We study the cases in which the same fraction of feed is vaporized in each stage

• <1 = <2 = ... = = ... <n = < (6.10)

each flash stage has the same residence time

These two assumptions imply that the vapor rate and the liquid hold-up both decrease along the cascade for a fixed feed flow rate. This produces a policy of decreasing vapor rate along the cascade similar to a decreasing vapor rate policy in simple (batch) distillation, which keeps the instantaneous value of Da approximately constant [47]. The model for the stripping cascade becomes

Xj-1 = < Yij + (1 - <)Xij - n (t^MDar(Xj) (i = 1... c - 1, j = 1,2 ... N) (6.12)

It is convenient to introduce a dimensionless parameter [48]

156 | S. B. Gadewar, N. Chadda, M. F. Malone, and M. F. Doherty Da

where D varies from 0 for the case of no reaction to unity in the limit of reaction equilibrium.

Including (6.13) in (6.12), gives the final formulation of the stripping cascade

„,_ 0 N + (i - „(£)(£)«(, _ i..„ - 1,, _ 1.2...,

Equation (6.14) can be solved recursively for given values of the parameters, 0 and D starting with the initial condition, ao = af. The solution is a trajectory of liquid compositions for the stripping cascade. We can also derive a model for the rectifying cascade

Yij-1 = 0 Yij + (1 - 0Kj - (¿) (J-d)^) (l = 1 ■■■ c - 1' J = 2' 3 ■■■M)

where y1 = y{, and y[ is the vapor stream composition from the first flash device of the stripping cascade shown in Fig. 6.7. The solution to (6.15) is a trajectory of vapor phase compositions along the rectifying cascade.

Equations (6.14) and (6.15) model the co-current stripping and rectifying cascades, respectively. For a given feed composition, these cascades are solved recursively for N, M ^ until there is no change in successive iterates, that is, until a stable fixed point is reached.

We demonstrate the use of this model on an example and then characterize its properties in terms of fixed points and bifurcations.

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