Algebraic equations multiple steady states

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

In Section 3.4 we study several systems that have no multiple steady states and we introduce several transcendental and algebraic equations of chemical and biological engineering import. As always, the students and readers should find their own MATLAB codes for the various problems first before relying on those that are supplied and before solving the included exercises. 

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). 

Since the CSTR equation is simply a system of nonlinear algebraic equations, it is possible to obtain multiple CSTR steady states for a fixed feed concentration and residence time. For example, if there are polynomial terms in r(C), then more than one concentration will exist as roots to the equation. Even if these roots do not bear any physical significance to the system, they must still be known in order for the appropriate construction of the AR to be carried out. More complex expressions are also valid (and common) in modern-day rate expressions. Be wary of this when attempting to solve for CSTR solutions. The presence of multiple steady states presents... 

The nonlinear algebraic equations that describe a steady-state distiUalion column consist of component balances, energy balances, and vapor-liquid phase equilibrium relationships. These equations are nonlinear, particularly those describing the phase equilibrium of azeotropic systems. Unlike a linear set of algebraic equations that have one unique solution, a nonlinear set can give multiple solutions therefore, the possibility of multiple steady states exists in azeotropic distillation. 

In order to apply the concepts of modern control theory to this problem it is necessary to linearize Equations 1-9 about some steady state. This steady state is found by setting the time derivatives to zero and solving the resulting system of non-linear algebraic equations, given a set of inputs Q, I., and Min In the vicinity of the chosen steady state, the solution thus obtained is unique. No attempts have been made to determine possible state multiplicities at other operating conditions. Table II lists inputs, state variables, and outputs at steady state. This particular steady state was actually observed by fialsetia (8). 

In addition to openness and feedback, a third condition is often found in conjunction with oscillation, though it has not yet been proven to be necessary the existence of multiple stationary states. Most chemists are aware of the steady state condition, wherein the rate of change of the concentration of an intermediate can be equated to zero. Thus, the concentration of this intermediate can be obtained by solving an algebraic, rather than a differential equation. The familiar solution to this equation is a single concentration for a given set of initial conditions. [The reader may be familiar with a mechanism k k... 

The multi-mode model for a tubular reactor, even in its simplest form (steady state, Pet 1), is an index-infinity differential algebraic system. The local equation of the multi-mode model, which captures the reaction-diffusion phenomena at the local scale, is algebraic in nature, and produces multiple solutions in the presence of autocatalysis, which, in turn, generates multiplicity in the solution of the global evolution equation. We illustrate this feature of the multi-mode models by considering the example of an adiabatic (a = 0) tubular reactor under steady-state operation. We consider the simple case of a non-isothermal first order reaction... 

Balakotaiah, V., Luss, D., and Keyfitz, B., Steady State Multiplicity Analysis of Lumped-Parameter Systems Described by a Set of Algebraic Equations, Chem. Eng. Comm. 36 (1982) 121-147. 

---