Output multiplicity

Urine. The kidneys filter the entire cardiac output multiple times each day, and thus provide a large opportunity for the removal of chemicals from the bloodstream. How much of a xenobiotic is actually excreted is dependent on three factors or processes. 

This is the most general heat-balance equation for a multiple input, multiple output, multiple reactions (and of course multi-components) system. 

Equations (4.16) and (4.17) are required to explore issues around output multiplicity and steady-state sensitivity. 

The need for nonlinearity is easy to see. A linear equation has no more than a single solution. A quadratic equation may have two solutions, etc. The describing equations for a reactor must therefore be nonlinear to show output multiplicity. 

We next turn to process feedback. W e mentioned earlier that a plug-flow reactor can be viewed as a string of small batch reactors. We also pointed out that the result of each batch is uniquely determined by the fresh feeds since the solution to the batch equations is a forward integration in time. A plug-flow7 reactor cannot by itself show output multiplicity or open-loop instability. This picture changes when we... 

The graphic illustration of output multiplicity focuses on the steady-state solutions of Eq. (4.17). This equation can be viewed as a tradeoff between a nonlinear heat generation term, Q iT), and a linear heat removal expression ... 

Inverse response creates control difficulties. Assume, for example, that we wish to control the exit temperature of an adiabatic plug-flow reactor by manipulating the inlet temperature as shown in Fig. 4.13. From a steady-state viewpoint this is a perfectly reasonable thing to consider, since there are no issues of output multiplicity or open-loop instability, assuming the fluid is in perfect plug flow and there is no... 

In Chap. 4 we mentioned that the simplest reactor type from a control viewpoint is the adiabatic plug-flow reactor. It does not suffer from output multiplicity, open-loop instability, or hot-spot sensitivity. Furthermore, it is dominated by the inlet temperature that is easy to control for an isolated unit. The only major issue with this reactor type is the risk of achieving high exit temperatures due to a large adiabatic temperature rise. As we recall from Chap. 4, the adiabatic temperature rise is proportional to the inlet concentration of the reactants and inversely proportional to the heat capacity of the feed stream. WTe can therefore limit the temperature rise by diluting the reactants with a heat carrier. 

So far we have been discussing output multiplicity as the primary source of open-loop instability. It is also possible to have input multiplicity in distillation systems. Input multiplicity means that we can get the same output for different levels of the input variables. 

J DeCicco and A Cinar. Empirical modeling of systems with output multiplicities by multivariate additive NARX models. Ind. Engg. Chem. Research, 39(6) 1747-1755, 2000. 

M Pottmann and R Pearson. Block-oriented NARMAX models with output multiplicities. AIChE J., 44(1) 131-140, 1998. 

Most emphasis has been on output multiplicity as well as on sustained oscillations in chemical process systems. The role of input multiplicity compared to output multiplicity has been treated for RD processes in [26, 98], and the notion of pseudomultiplicity was introduced in [98]. This corresponds to a situation where ...molar inputs (rather than mass or volume inputs that would result from control valves) produce an output multiplicity . Since this behavior can only be observed via simulation and is not associated with actually operating columns we will focus in this chapter on input and output multiplicity and do not treat pseudomultiplicity. 

The three solution branch dominating the nonlinear behavior of the process can also be observed in a conventional process with a reactor-separator recycle as recently reported by Blagov et al. [12]. This flowsheet is analogous to a column with one reactive and one non-reactive column section with a single product stream, the multiplicity behavior of which has been studied Cerafimov and coworkers [49, 83]. In these papers it is shown that this type of output multiplicity is a generic phenomenon for reaction systems with competing irreversible reactions and a similar distribution of volatUities between reactants and products. 

Temperature profiles after a stepwise increase of the heating rate are shown on the right side of Eig. 10.29. These profiles show distinct nonlinear wave characteristics as discussed in the previous section. Therefore the process is sensitive to disturbances and composition control is required. Again focus is on an inferential control scheme. Measured variables are the temperatures on trays 4 and 60, which are located within the upper and the lower ware front. Hence, they show good sensitivity to disturbances. Manipulated variables are the heating rate and the reflux ratio. Eor this process neither input nor output multiplicities occur. 

Output multiplicities a column of a given design exhibits different column profiles and product compositions at steady state for the same set of inputs and the same values of operating parameters. This type of multiplicity will be addressed during the course of this section. 

Prom a mathematical point of view, the sufficient and necessary conditions for output multiplicity are given by the following inequalities. 

Steady-state behavior. In this first section, a reactive flash with a rather ideal system (mildly exothermic isomerization reaction and light-boiling reactant) was considered. This simplified system was selected to account exclusively for the effect of phenomena interaction on the occurrence of input and output multiplicity. Therefore, any effect related to unit configuration is considered outside the scope of the analysis. 

More than one steady state for the same set of specified variables (output multiplicity) is one of the interesting features of azeotropic distillation. Simple distillation columns with ideal vapor-liquid equilibrium, however, may also show MSS (Jacobsen and Skogestad, 1991). The existence of output multiplicities in distillation were first reported on the ternary ethanol-water-benzene (EWB) system. Earlier simulation-based studies had reported two distinct steady states depending on the starting guesses (Bekiaris et al.. 

SG output (multiple choice saturated steam, superheated steam, N/A) 1 ... 

When dealing with nonlinear systems, the boimdaries of a given input space do not necessarily map to the boundaries of the output region. The existence of either input or output multiplicities at steady state are causes for parts of the boundary of an input region not to map... 

The simulations indicate that the modifications have the ability to reject larger disturbances in the inlet temperature Tin than the base case design does. It is worth noticing that both the base case and the modifications can eventually be stabilised by the feedback control since there only exists output multiplicity in the MWav-Fc loop. But, a sudden change in the output could happen for a large disturbance during operation. 

With exactly the same input variables fixed (feed flow and composition, reflux flowrate, and distillate flowrate), there may be completely different values for the compositions and temperatures throughout the column. This is called output multiplicity. If this occurs it adds significant complexity to the design and control of these systems. Problems in converging the steady-state program in Aspen Plus frequently are encountered and can be challenging to overcome. 

NOTE The Edit Text box can be resized in terms of length and width during the design phase so that it can accommodate the welcome statement. Keep in mind that this type of control accepts only one line statement. To output multiple lines, the designer needs to use the ListBox control. 

Nonlinearity and Output Multiplicity. Once the manipulated variables are determined, we can evaluate the process nonlinearity for these three different types of flowsheets. The tray temperatures are treated as the state variables. The manipulated variables are the heat input and feed ratio FR. The upper and lower bounds of the steady-state... 

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