Multiplicity refers to the condition where a one-to-one relationship does not exist between the inputs and outputs of a process. Input multiplicity is present when two or more values of an input variable produce the same output condition with all other members of the set of input variables the same. That is, the process inputs are not unique for a known output condition. Output multiplicity occurs when one complete set (i.e. a set with sufficient members to fully satisfy the degrees of freedom in the system) of input variables maps to two or more distinct and independent sets of output variables. That is, the outputs are not unique for a given set of inputs. Figure 8.1 provides a graphical distinction between input and output multiplicity for arbitrary variables. Input multiplicity is present in the first chart as input variable values of both a and b result in the same value of the output variable. Output multiplicity is present in the second case as an input value of c could result in output values of either d or e.

The differentiation between input and output variables is best made with reference to control structures. Input variables are those which can be manipulated by controllers. The primary input variables for distillation include the reflux rate and the reboiler duty (or boilup), and the distillate and the bottoms rates, and combinations of these (e.g. the reflux ratio). The product draw rates can be considered as inputs because the inventory (level) controllers on the reflux accumulator and the reboiler sump will transmit changes in the product rates directly to the column. Output variables are those which are either controlled or used to describe the conditions of the process (e.g. product or stage temperatures, concentrations and yields). By convention, input variables are plotted on the *-axis while output variables are plotted on the_y-axis.

fitiure 8.1 - Input Multiplicity (left) and Output Multiplicity (right) 8.1.2 Examples of Input Multiplicity

The effects of various operating variables on two hybrid reactive distillation columns were described in Chapter 4, Section 4.1. The reboiler duty (Section 4.1.7) was shown to have a bidirectional effect on the principal performance parameters (i.e. the ether purity and the isobutene conversion): depending on the magnitude of the reboiler duty, an increase in the duty could either increase or decrease the purity or the conversion, as indicated in Figure 4.3 which was constructed using simulation results for the 10 stage ETBE column. Input multiplicity exists in this case since some output conditions (e.g. the bottoms temperature, Tb) map to more than one value of a specific input variable (e.g. the reboiler duty, Qr) while all other input conditions are the same, as shown in Table 8.1.

Steady State A |
Steady State B | |

Output Condition | ||

Bottoms temperature (°C ) |
150 |
150 |

Input Conditions | ||

Hydrocarbon feed composition |
40% isobutene, |
40% isobutene, |

60% n-butene |
60% n-butene | |

Methanol excess (mol%) |
5% |
5% |

Total feed rate (L/min) |
0.76 |
0.76 |

Pressure (kPa-g) |
950 |
950 |

Reflux rate (L/min) |
2.50 |
2.50 |

Reboiler duty (kW) |
8.20 |
8.97 |

A similar, bidirectional dependence of the column performance on the reboiler duty is seen with the MTBE system. Table 8.2 describes the configuration of a 17 stage MTBE column (Nijhuis et al., 1993). Figure 8.2 shows the bidirectional effect of the reflux rate (L) on the bottoms temperature of this column for a constant bottoms yield of 35.0% by volume. Input multiplicity is present since, for example, a bottoms temperature of 145°C results with reflux rates of 130 and 535 m3/hr.

Design Parameter |
Value |

Rectifying stages (including total condenser) |
3 |

Reactive stages |
8 |

Stripping stages (including partial reboiler) |
6 |

Feed stage |
lowest reactive stage |

Hydrocarbon feed composition (mol %) |
36% isobutene, 64% n-butane |

Stoichiometric methanol excess (mol %) |
10% |

Total feed rate |
2752 kmol/hr |

Overhead pressure |
1000 kPa-g |

Reflux ratio |
—7.0 |

The two examples of input multiplicity indicated above can also be described analytically. The ETBE column multiplicity satisfies the condition given by equation (8.1) while the MTBE column multiplicity satisfies equation (8.2). A more general expression of this condition is given by equation (8.3), which describes any stationary point between an input variable (x,) and an output input variable (y) with other inputs (x2) constant.

5Tb |
\ |

àQ, |
L |

sV | |

dLj |
Qr |

dx}J |
Xi |

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