Multiple Steady States MSS

In this section we consider the steady-state operation of a CSTR in which a first-order reaction is taking place. An excellent experimental investigation that demonstrates the multiplicity of steady states was carried out by Vejtasa and Schmitz. They studied the reaction between sodium thiosulfate and hydrogen peroxide, 

When more than one intersection occurs, there is more than one set of conditions that satisfy both the energy balance and mole balance Xeb - 

We begin by recalling Equation (12-24), which applies when one neglects shaft work and ACp (i.e ACp = 0 and therefore = A/7r, ). 

Using the CSTR mole balance X=—. Equation (12-24) may be rew ritten as 

The right-hand side of Equation (12-28) is referred to as the beaf-removed tenn (by flow and heat exchange) R T)  

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Multiple steady states (MSS) in conventional distillation have been known from simulation and theoretical studies dating back to the 1970s and has been a topic of considerable interest in the distillation community. Taylor and Krishna [1] cite the papers highlighting MSS in RD. We single out just a few of these papers for mention here. 

We noted that Equation 4.77 is very important in the nonisothermal operation of a CSTR. This algebraic equation has more than one solution and leads to the concept of multiple steady states (MSS). On the other hand, the differential equation that characterizes a PER has only one solution, that is, the PER operates at a single steady state. Multiple steady states are of particular concern to us because they can occur in the physically realizable range of variables, between zero and infinity, and not at absurd values such as negative concentration or temperature (which would then be no more than a mathematical artifact). 

Then, if > 0, Eq. (7) has one real positive root and a pair of complex conjugate roots. On the other hand, if < 0, Eq. (7) has three real positive roots that is, DE(4) admits more than one steady state solution in the positive quadrant. Multiple steady state (MSS) solutions will appear for all 5 satisfying... 

Multiple steady states (MSS) are not detected automatically by the state-of-the-art commercial simulators, despite the fact that extensive capabilities are available for complex distillation columns and the solution algorithms used are very robust and efficient. Since the implications of multiplicity could be numerous, it is of great importance to detect MSS in distillation by a systematic procedure. 

Derive equations that lead to multiple steady states (MSS) in continuous reactors. 

Basic PFR equation Design equations Nonisothermal operation Perfectly mixed flow reactor (MFR) Basic CSTR equation Nonisothermal operation Multiple steady states MSS In a CSTR Adiabatic CSTR... 

It is relevant to mention that the disturbance scenarios are defined in such a way that multiplicities are avoided. Since a detailed MSS analysis in RD has been already addressed in section 7.2.1, this section focuses exclusively on the dynamic behavior of the RD unit around an already known steady state. Thus, the control structure is defined so that no steady-state transitions are occurring. The effect of large methanol feed disturbances (AFmsOH > 0.1) on the operational steady state has been covered by Schrans et al. (1996) Sneesby et al. (1998a,fe). 

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.. 

The implications of multiplicity on distillation design, synthesis, simulation and control can be critical (Bekiaris and Morari, 1996). Using an acetone-benzene-heptane example, Bekiaris et al. (1993) have illustrated how column profiles may jump from one steady state (99% acetone) to another (93% acetone) for some feed disturbance. Seider and Kovach (1987) performed simulations and experiments on dehydration of sec-butanol and conjectured that the observed erratic column behavior is a consequence of MSS. Moreover, the sensitivity and dynamic characteristics of each steady state may differ and affect column controllability. The existence of MSS raises other problems related with the start-up strategy that would drive the column to the desired steady state. Thus, it is essential to detect all possible steady states of a given column. Since the popular commercial simulators do not have provision to find MSS directly, the aim of this work is to develop a systematic procedure to achieve this. A very recent work in this direction is due to Vadapalli and Seader (2001). 

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