Steady states multiple

A certain class of initial value problems exhibits multiple steady states. Depending on the initial condition, the problems converge to different steady state values. 

Let s examine this result more carefully. In order to fix the position of both G(T) and R(T), we specified aU of the following variables the CSTR volume, V each of the inlet molar flow rates, F,o, the rate equation, all of the constants in the rate equation as functions of temperature, the inlet temperature. To, the coolant temperature, Tc, and the product of the exchanger area and the overall heat-transfer coefficient, UA. Despite fixing all of these parameters, the graphical analysis shows that, at steady state, the CSTR can operate at any one of three combinations of xa and T. Quite remarkable  

A phenomenon that arises particularly with continuous stirred reactors is the occurrence of more than one steady state. This becomes apparent from the heat and material balances. Heat generation is made up of the heat of reaction plus any heat transfer, and the heat removal is the sensible and latent heat change of the reaction products. In problem P4.10.13, for instance, both the heat generation and the heat removal are plotted against the temperature. The two lines intersect at three points which represent the steady states. A point at which the slope of the heat generation line is 

02 has a case where the energy balance line is not quite straight, as it need not be. 

Endothermic reactions are always stable. On the figure of P4.10.13, for instance, when the slope of the heat generation line is negative, evidently only one intersection with the other curve is possible. 

Which particular steady state will prevail can be established by choice of inlet conditions. As problem P4.10.ll shows adjustment of either 

Control systems may produce small fluctuations of the process variables, as in the sinusoidal cases of problem P4.09.34. When they occur while the system is at an unstable point, the temperature will migrate to that at a neighboring steady condition. In problem P4.10.01, as the unstable condition is approached (T = 280, C = 2.4), the profiles of temperature and concentration become erratic and eventually degenerate to the conditions at the stable point to the right. 

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. 

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. 

If we start at a small value of refiux and gradually increase the fiowrate, the resulting trajectories of the benzene and water compositions of the bottoms are shown in the solid lines. For small reflux flowrates, there is not enough benzene being fed, so there is a high concentration of water xb(W) in the bottoms as shown in the lower graph. Of course, there is also very little benzene in the bottoms. 

As the reflux is gradually increased, a point is reached at about 315 kmol/h where the water concentration in the bottoms drops abruptly to very small values, while the concentration of benzene begins to gradually increase. Now there is enough benzene in the column to drive the water overhead however, to have a high-purity ethanol product, there should be very small amounts of both benzene and water. 

The dashed lines correspond to the case when the reflux has been slowly decreased to 300 kmol/h from a higher value. Now there is little water in the bottoms but increasing values of benzene. Temperatures are low throughout the column. 

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Reactions in porous catalyst pellets are Invariably accompanied by thermal effects associated with the heat of reaction. Particularly In the case of exothermic reactions these may have a marked influence on the solutions, and hence on the effectiveness factor, leading to effectiveness factors greater than unity and, In certain circumstances, multiple steady state solutions with given boundary conditions [78]. These phenomena have attracted a great deal of interest and attention in recent years, and an excellent account of our present state of knowledge has been given by Arls [45]. 

Phenomena of multiple steady states and instabilities occur particularly with nonisothermal CSTRs. Some isothermal processes with hyperbohc rate equations and processes with porous catalysts also can have such behavior. 

Plug flow reactors with recycle exhibit some of the characteristics of CSTRs, including the possibility of multiple steady states. This topic is explored by Penmutter Stah dity of (%emical Reactors, Prentice-Hall, 1972). 

FIG. 23-17 Multiple steady states of CSTRs, stable and unstable, adiabatic except the last item, (a) First-order reaction, A and C stable, B unstable, A is no good for a reactor, the dashed line is of a reversible reaction, (h) One, two, or three steady states depending on the combination Cj, Ty). (c) The reactions A B C, with five steady states, points 1, 3, and 5 stable, (d) Isothermal operation with the rate equation = 0 /(1 -I- C y = (C o Cy/t. 

Figure 6-21. Multiple steady states for exothermic reaction in a CFSTR.

In a tubular reactor system, the temperature rises along the reactor length for an exothermic reaction unless effective cooling is maintained. For multiple steady states to appear, it is necessary that a... 

Temperature gradients within the porous catalyst could not be very large, due to the low concentration of combustibles in the exhaust gas. Assuming a concentration of 5% CO, a diffusion coefficient in the porous structure of 0.01 cms/sec, and a thermal conductivity of 4 X 10-4 caI/sec°C cm, one can calculate a Prater temperature of 1.0°C—the maximum possible temperature gradient in the porous structure (107). The simultaneous heat and mass diffusion is not likely to lead to multiple steady states and instability, since the value of the 0 parameter in the Weisz and Hicks theory would be much less than 0.02 (108). 

Linear control theory will be of limited use for operational transitions from one batch regime to the next and for the control of batch plants. Too many of the processes are unstable and exhibit nonlinear behavior, such as multiple steady states or limit cycles. Such problems often arise in the batch production of polymers. The feasibility of precisely controlling many batch processes will depend on the development of an appropriate nonlinear control theory with a high level of robustness. 

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. 

Microbial kinetics can be quite complex. Multiple steady states are always possible, and oscillatory behavior is common, particularly when there are two or more microbial species in competition. The term chemostat can be quite misleading for a system that oscillates in the absence of a control system. 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

Procedures enabling the calculation of bifurcation and limit points for systems of nonlinear equations have been discussed, for example, by Keller (13) Heinemann et al. (14-15) and Chan (16). In particular, in the work of Heineman et al., a version of Keller s pseudo-arclength continuation method was used to calculate the multiple steady-states of a model one-step, nonadiabatic, premixed laminar flame (Heinemann et al., (14)) a premixed, nonadiabatic, hydrogen-air system (Heinemann et al., (15)). 

However, all rate data for this reaction are not explained simply by this rate expression. At pressures above 10 Torr the rate exhibits multiple steady states, long transients, and rate oscillations ]). Clearly other processes are Involved than those Implied by the simple one state, constant parameter LH model. 

Model instability is demonstrated by many of the simulation examples and leads to very interesting phenomena, such as multiple steady states, naturally occurring oscillations, and chaotic behaviour. In the case of a model which is inherently unstable, nothing can be done except to completely reformulate the model into a more stable form... 

Choose a multiple steady-state case and try to upset the reactor by changing AO, F, TO or TJ interactively during a run. Only very small changes are required to cause the reactor to move to the other steady state. Plot as time and phase-plane graphs. 

For the parameters used in the program, the system is characterised by multiple steady-state operating conditions. For example, four of them are as follows... 

Based on a linearisation approach as applied to the model equations, Clough and Ramirez (1971) predict multiple, steady-state solutions for the reactor. Is this confirmed and if not, why not ... 

The problem of ignition and extinction of reactions is basic to that of controlling the process. It is interesting to consider this problem in terms of the variables used in the earlier discussion of stability. When multiple steady-state solutions exist, the transitions between the various stable operating points are essentially discontinuous, and hysteresis effects can be observed in these situations. 

Cutlip and Kenney (44) have observed isothermal limit cycles in the oxidation of CO over 0.5% Pt/Al203 in a gradientless reactor only in the presence of added 1-butene. Without butene there were no oscillations although regions of multiple steady states exist. Dwyer (22) has followed the surface CO infrared adsorption band and found that it was in phase with the gas-phase concentration. Kurtanjek et al. (45) have studied hydrogen oxidation over Ni and have also taken the logical step of following the surface concentration. Contact potential difference was used to follow the oxidation state of the nickel surface. Under some conditions, oscillations were observed on the surface when none were detected in the gas phase. Recently, Sheintuch (46) has made additional studies of CO oxidation over Pt foil. 

A reaction which follows power-law kinetics generally leads to a single, unique steady state, provided that there are no temperature effects upon the system. However, for certain reactions, such as gas-phase reactions involving competition for surface active sites on a catalyst, or for some enzyme reactions, the design equations may indicate several potential steady-state operating conditions. A reaction for which the rate law includes concentrations in both the numerator and denominator may lead to multiple steady states. The following example (Lynch, 1986) illustrates the multiple steady states... 

Figure 14.4 Multiple steady states in a CSTR (Example 14-7)...

Particular forms of the rate equation can give rise to multiple steady states even under isothermal conditions, as in problem P4.10.09 where... 

It is suspected that this nonlinear rate form, which has a maximum value, may cause certain regions of unstable operation with multiple steady states. How should the operation be conducted to ensure unique steady conditions ... 

Enhancement of a rate by temperature can counteract the effect of falling concentration. Exothermic reaction rates in pores, as a consequence, can be much greater than at the surface condition. Another peculiarity that can arise with adiabatic reactions is multiple steady states. 

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