In isothermal CSTRs

Examples include multiple steady states in isothermal CSTRs, predator-prey fluctuations, the Belousov-Zhabotinsky reaction, and a test for stability of quasi-stationary states in reactions with a self-accelerating intermediate steps. 

Figure 13.19 One-reactant, n -order reaction in isothermal CSTR-Separator-Recycle system...

Rumschitzky, D. S. (1984). On the theory of multiple steady states in isothermal CSTRs (Ph.D. thesis). University of Rochester, Rochester, New York. 

To understand this, consider the following case which is limited to a homogeneous reaction executed in an isothermal CSTR as shown in Figure 9.1.1. Isothermal here means feed and discharge are at the reaction temperature, and all heat is removed by heat transfer through the walls. 

The most comprehensive simulation of a free radical polymerization process in a CSTR is that of Konopnicki and Kuester (15). For a mechanism which includes transfer to both monomer and solvent as well as termination by combination and disproportionation they examined the influence of non-isothermal operation, viscosity effects as well as induced sinuoidal and square-wave forcing functions on initiator feed and jacket temperature on the MWD of the polymer produced. 

Example 14.1 Consider a first-order reaction occurring in a CSTR where the inlet concentration of reactant has been held constant at uq for f < 0. At time f = 0, the inlet concentration is changed to Up Find the outlet response for t > 0 assuming isothermal, constant-volume, constant-density operation. 

Example 14.1 shows how an isothermal CSTR with first-order reaction responds to an abrupt change in inlet concentration. The outlet concentration moves from an initial steady state to a final steady state in a gradual fashion. If the inlet concentration is returned to its original value, the outlet concentration returns to its original value. If the time period for an input disturbance is small, the outlet response is small. The magnitude of the outlet disturbance will never be larger than the magnitude of the inlet disturbance. The system is stable. Indeed, it is open-loop stable, which means that steady-state operation can be achieved without resort to a feedback control system. This is the usual but not inevitable case for isothermal reactors. 

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. 

Suppose the following reactions are occurring in an isothermal CSTR ... 

Example 15.9 Use residence time theory to predict the fraction unreacted for an isothermal, homogeneous, first-order reaction occurring in a CSTR and aPFR. 

We do this for isothermal constant-density conditions first in a BR or PFR, and then in a CSTR. The reaction conditions are normalized by means of a dimensionless reaction number MAn defined by... 

A gas-phase hydrogenation reactioa C2H (A) + H2 - C2H6, is conducted under isothermal and isobaric conditions in a CSTR The feed, consisting of equimolar amounts of each reac-... 

A performance comparison between a BR and a CSTR may be made in terms of the size of vessel required in each case to achieve the same rate of production for the same fractional conversion, with the BR operating isothermally at the same temperature as that in the CSTR. Since both batch reactors and CSTRs are most commonly used for constant-density systems, we restrict attention to this case, and to a reaction represented by... 

Calculate the ratio of the volumes of a CSTR and a PFR ( Vst pf) required to achieve, for a given feed rate in each reactor, a fractional conversion (/A) of (i) 0.5 and (ii) 0.99 for the reactant A, if the liquid-phase reaction A - products is (a) first-order, and (b) second-order with respect to A. What conclusions can be drawn Assume the PFR operates isothermally at the same T as that in the CSTR. 

Data obtained in continuous stirred tank reactors have the merits of isothermicity and of an algebraic relation between the variables rather than a differential one. At steady state In a CSTR the material balance on a reactant A is... 

Find the time required for 70% conversion (a) at isothermal operation of 20 C, (b) under adiabatic conditions, (c) In a CSTR find temperature and conversion at residence times of 5, 10 and 20 min. 

A reaction with rate equation, r = k C2/ (1 + k2C), is to be conducted in an isothermal CSTR, Examine the possibility of the occurrence of more than one steady state conversion. 

Isothermal Reactor with Complex Reaction 265 Continuous Stirred-Tanks, Tracer Experiment 273 Deactivating Catalyst in a CSTR 268 Distribution of an Insecticide in an Aquatic Ecosystem 581... 

Benzene is nitrated in an isothermal CSTR in three sequential irreversible reactions ... 

The equations describing the series of three isothermal CSTRs were developed in Sec. 3.2. 

Example 9Ji, Consider the isothermal CSTR of Example 6.6. The equation describing the system in terms of perturbation variables is... 

If a proportional controller is used in the three-isothermal CSTR process, a controller gain of 22.6 gives a phase margin of 45°. A gain of 20 gives a maximum closedloop log modulus of +2 dB with a closedloop resonant frequency of 1.1 radian per minute. 

The final test of the proposed nonlinear robust controllers to be presented in this chapter is its implementation in a highly nonlinear reactor an non-isothermal homopolymerization carried out in a CSTR. In this case, we develop a multivariable version of the nonlinear regulator, in contrast to the SISO versions used in the previous examples. 

We took the 4- sign on the square root term for second-order kinetics because the other root would give a negative concentration, which is physically unreasonable. This is true for any reaction with nth-order kinetics in an isothermal reactor There is only one real root of the isothermal CSTR mass-balance polynomial in the physically reasonable range of compositions. We will later find solutions of similar equations where multiple roots are found in physically possible compositions. These are true multiple steady states that have important consequences, especially for stirred reactors. However, for the nth-order reaction in an isothermal CSTR there is only one physically significant root (0 < Ca < Cao) to the CSTR equation for a given T. ... 

AUTOCATALYSIS IN WELL-STIRRED OPEN SYSTEMS THE ISOTHERMAL CSTR... 

Gray, P. and Scott, S. K. (1985). Sustained oscillations and other exotic patterns of behavior in isothermal reactions. J. Phys. Chem., 89, 22-32. Lin, K. F. (1981). Multiplicity, stability and dynamics for isothermal autocatalytic reactions in CSTR. Chem. Eng. Sci., 36, 1447-52. 

REACTION IN A NON-ISOTHERMAL CSTR STATIONARY STATES AND SINGULARITY THEORY... 

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