## Parallels Between the Energy Balance and the Unit Singularities Causes

The explanations for multiplicity caused by unit singularities and by energy balance effects are fundamentally different but there are similarities between the two mechanisms which suggest that a unified method of analysis might be available. In fact, equations (8.4) and (8.5) (the conditions for multiplicity due to unit singularities) can be combined with inequalities (8.6) and (8.7) (the conditions for multiplicity due to the influence of the energy balance) according to equations (8.10) and (8.11). These equations yield new, more general conditions for multiplicity (inequalities 8.12 and 8.13) and do not include any molar flow terms.

rn

_UJ

JdD

U&,,

Uf.

(SB Ï

<0

m

ULJ

m dB

Any of the partial derivatives on the right hand sides of equations (8.10) and (8.11) can be negative and, thereby, cause multiplicity. If two of the derivatives are negative at the same time, the effects will cancel each other and the multiplicity will only be observed for molar units. An example of this is provided by the MTBE column described in Table 8.2. Figure

\dVJ

< 0 . The two curves have the same shape so that dD.

> 0 for all values of the boilup rate. Boilup (mol/min)

Figure 8.10 - Singularities Between the Reboiler Duty and the Molar Boilup

Boilup (mol/min)

Figure 8.10 - Singularities Between the Reboiler Duty and the Molar Boilup

Unfortunately, conditions (8.12) and (8.13) can only realistically be evaluated using a bifurcation analysis with a rigorous simulation model of the system under study. This restricts the application of these equations but other tools (especially those that rely on molar inputs) are not guaranteed to be accurate in all cases.

### 8.2.6 Mechanistic Interpretation

The multiplicity shown in the ETBE column due to unit singularities (Figure 8.3) was investigated further in order to interpret the differences between the parallel steady states from a mechanistic perspective. It was considered that an understanding of the changes and differences would have practical importance. Other multiplicities could be investigated similarly although the mechanism might be quite specific.

A comparison between the two steady states described in Table 8.4 is provided by Figures 8.11-8.13 which compare the temperature profiles (Figure 8.11), the composition profiles (Figure 8.12) and the reaction rate profiles (Figure 8.13) for points A and B. The most striking contrasts are the stripping section temperatures; (b) the stage-to-stage compositions in the stripping section; and (c) the reaction rate on the lowermost reactive stage. Figure 8.11 - Temperature Profiles in the ETBE Column

100%

—X— Ugh Conversion SS —□—Low Conversion SS

100%

—X— Ugh Conversion SS —□—Low Conversion SS Figure 8.12 - Selected Composition Profiles in the ETBE Column

Figure 8.12 - Selected Composition Profiles in the ETBE Column

3000 2500 2000 1500

a a 1000

High Conwrsion Steady State o o-

-500

-1000

-1500 12 Stage

Figure 8.13 - Reaction Rate Profiles in the ETBE Column

Figures 8.11 and 8.12 indicate that the rectifying section of the column is similar in both the high and the low conversion steady states. However, slight changes in the stage-to-stage compositions are magnified by the different reaction rate profiles (Figure 8.13) so that there are substantial differences in the temperature and compositions on the lower reactive stages. The effect of fractionation in the reaction zone is to concentrate ethanol (if present) near the lower reactive stages. This promotes the synthesis reaction due to the increased reactant availability but eventually initiates decomposition if the elevated phase temperature (caused by the increasing ethanol concentration and the formation of ETBE) reduces the reaction equilibrium constant too much. If there is no ethanol present in the reaction zone, the synthesis reaction is suppressed even though the reaction equilibrium constant is higher.

Thus, the crucial consideration is the supply of ethanol to the reaction zone. This is dependent on the fractionation in the stripping section and the stage-to-stage compositions in the stripping section. It is proposed that a high concentration of n-butylenes induces a pseudo-binary separation between C4s and ethanol/ETBE, while a low concentration of n-butylenes allows a pseudo-binary separation of ethanol (and all lighter components) and ETBE. It is important to realise that ETBE is the heavy boiler in this system even though the boiling point of ethanol is higher than the boiling point of ETBE as the vapour pressure curves of ethanol and ETBE intersect at around 300 kPa. Although the azeotrope between ethanol and ETBE creates a distillation boundary where pure ethanol is the stable node in one distillation region, in this instance, both steady states (and the third, unstable, steady state) are in the other distillation region (i.e. where pure ETBT is the stable node). The main effect of the ethanol-ETBE azeotrope is to increase the curvature of the distillation lines in the vicinity of the azeotrope. There is no evidence to suggest that the azeotrope causes the multiplicity.

The two steady states arise since the additional production rate of ETBE which results from maintaining an adequate supply of ethanol to the reaction zone can exactly balance the flow of ethanol that would otherwise leave the column with the bottoms product. That is, there is a singularity in the relationship between the bottoms mass flow and the bottoms molar flow. This is fully consistent with the fundamental explanation given in Section 8.2.1.