Systems with Interfacial Mass-Transfer Resistances

Recently, the effect of interfacial mass-transfer resistances on the location of the attainable products of a reactive separation process was analyzed considering the simple model A + B o C [23]. The analysis was based on the calculation of the top and bottom products of a separative reactor of infinite height at infinite reflux ratio («>,^-analysis). The products of this reactor are obtained from the following equations in matrix-vector notation

Bottom 0 = - [fc'l](ye(xB)-xB) + (v-xbvt) ■ Da ■ r\xB) (5.46)

Distillate 0 = [fc*](ye(xD)-xD) + (v-yevT) ■ Da ■ r*(xD)

with the reactor coordinate r, the vapor phase equilibrium composition ye, the vector of reaction rates Da-r*, the matrix of multicomponent mass-transfer coefficients [fc*], the matrix of stoichiometric coefficients v, and vector of total mole changes vT. The [fc*] elements can be obtained from the Maxwell-Stefan equations with suitable bootstrap conditions [24].

If [fc*] is a diagonal matrix, the attainable products will not be affected by interfacial mass-transfer phenomena. In contrast, if the off-diagonal elements of the [fc*] matrix are not vanishing, mass transfer can have a strong influence on the products. As a consequence, the attainable product region of a membrane reactor, for instance a pervaporative reactor, can be shifted by tailoring the membrane material and/or its pore structure.

Fig. 5.24 shows some illustrative results of a fixed-point continuation for the reversible reaction A + B o C carried out in a pervaporative reactor with internal reflux. The Damkohler number Da is used as continuation parameter. The boiling sequence psA > ps„ > psC was assumed so that the desired product C is obtainable as a bottom product. The reaction kinetics are described by the power law expression r* = xAxB - xC/Kwith the chemical equilibrium constant K as in Section 5.3.2. The role of mass transfer can be studied in terms of the dimensionless quantities fcAC and fcBC, that is the relative membrane permeabilities between A/C and B/C, respectively. For the ternary system considered, the elements of the mass-transfer coefficient matrix [fc*] are calculated as follows [24]

fcAA = fcAc[PAfcBC + (1-yA)]/S


fcAB = [PAfcBc(fcAC-1)]/S


fc'BA = [PBfcAc(fcBC-1)]/S


fc'BB = fcBc[PBfcAC + (1-yB)]/S


When fcAC = fcBC = 1, the off-diagonal elements of [fc*] are equal to zero and the RD process discussed in Section 5.3.2 (case b) is recovered. Then, the location of the attainable product composition only depends on the vapor-liquid equilibria. In Fig. 5.24, singular point curves for the attainable bottom products of the depicted counter-current pervaporative reactor are given as a function of fcAC (Fig. 5.24a) and fcBC (Fig. 5.24b). For each set of membrane permeabilities there is one curve of possible singular points. The full circles on the singular point curves indicate stable nodes, that is they mark attainable bottom products. For the system considered, stable nodes are only obtained below the chemical equilibrium line.

As can be seen from Fig. 5.24a, by increasing fcAC the fixed-point curves move towards the C/B edge until a border line for fcAC ^ is reached. When fcAC is decreased, the fixed points move towards the C/A edge until another border line for fcAC ^ 0 is achieved that cannot be overcome. The gray area between the two border

Potential singular point surface

Fig. 5.24. Attainable regions of the bottom products (gray) of a pervaporative reactor (left) at different relative membrane permeabilities (dashed line: chemical equilibrium line at K = 5, Damkôhler number: Da = 0 —>

Fig. 5.24. Attainable regions of the bottom products (gray) of a pervaporative reactor (left) at different relative membrane permeabilities (dashed line: chemical equilibrium line at K = 5, Damkôhler number: Da = 0 —>

lines and the chemical equilibrium line indicates the resulting attainable region. As illustrated in Fig. 5.24b, a different attainable region is obtained by changing the relative permeability kBC. When kBC decreases, the stable node branch approaches the B/C edge and at kBC ^ 0 even includes the pure B vertex.

These results are valid for the bottom products and are based on (5.46). A similar analysis can be carried out for the attainable top products by use of (5.47) [23].

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