Potential Singular Point Surface

The right-hand side of Eqs. (3) and (4) consist each of two terms. The first term represents the effect of distillation (separation vector), and the second the effect of the chemical reaction (reaction vector). For nonreactive systems only the separation vector plays a role. For kinetically controlled systems, both vectors can dominate the system behavior, depending on the value of Da. For Da — °°, the liquid mixture approaches chemical equilibrium. 

As condition for a singular point, the composition remains unchanged. For the reactive condenser, the vapor phase composition y does not change in time, i.e., 

Rearranging Eq. (9) produces the following rate-independent conditions for the singular points of the reactive condenser  

To fix the singular points, additional NR conditions are needed. For NR = 1 - that is, a single-reaction system - there is exactly one additional governing equation which is simply Eq. (7) (condenser) or Eq. (8) (reboiler) with i = k, where k is the reference 

---

Fig. 4.2. Potential singular point surfaces (dashed-dotted curve) for an ideal ternary system with single reaction A + B C. (a) Ellipse-type system (b) hyperbola-type system. RA = reactive azeotrope solid curve = chemical equilibrium surface.

Fig. 4.6. Potential singular point surface and chemical equilibrium surface for MTBE synthesis at 8.11 X 105 Pa.

Fig. 4.7. Reactive reboiler. Intersections of potential singular point surface with reaction kinetic surfaces at four different Damkohler numbers Da, MTBE synthesis at 8.11 x 105 Pa.

Fig. 4.9. Potential singular point surface, liquid-liquid envelope and chemical equilibrium surface for methanol dehydration at two different pressures.

Fig. 4.10. Potential singular point surface for isopropyl acetate (IPOAc) reaction system at 1.01 X 105 Pa. (a) Liquid phase composition space in mole fractions x, (b) representation in transformed composition space.

Inserting Eq. (82) into Eq. (81) gives the qualifying Eq. (83) to predict the locus curve X],uz = l ),ky(xi,az) at which possible reactive arheotropes can be found (i.e., the potential singular point surface, PSPS). The second curve is given by the equation describing the chemical equilibrium x>az = Fchem(xi az)-... 

Fig. 4.31. Potential singular point surfaces and stable node bifurcation behavior of reactive membrane separation at different mass transfer conditions B + C< > A Keq = 5 ccba = 5.0, acA = 3.0.

Table 4.3. Parameters used tor the potential singular point surfaces presented in Fig. 4.32.

Fig. 4.33. Potential singular point surface and bifurcation behavior for reactive distillation 1,4-BD —> THF + Water p = 5 atm.

Figure 4.33 illustrates the PSPS and bifurcation behavior of a simple batch reactive distillation process. Qualitatively, the surface of potential singular points is shaped in the form of a hyperbola due to the boiling sequence of the involved components. Along the left-hand part of the PSPS, the stable node branch and the saddle point branch 1 coming from the water vertex, meet each other at the kinetic tangent pinch point x = (0.0246, 0.7462) at the critical Damkohler number Da = 0.414. The right-hand part of the PSPS is the saddle point branch 2, which runs from pure THF to the binary azeotrope between THF and water. 

Indeed, it satisfies Laplace s equation everywhere except at the point p, since it describes up to a constant the potential of a point mass located at the point p. Also, it has a singularity at this point and provides a zero value of the surface integral over the hemisphere when its radius r tends to infinity. Correspondingly, we can write... 

Fits are to HF/3-21G(" ) electrostatic potential calculated at 1000 points in a shell between 1.0 and 3.0 times the van der Waals surface of the molecule. The rank estimate is taken as the number of singular values for which the ratio s to Sj does not exceed 10. ... 

---