Nonisothermal Stirred Tank Reactors

Nonisothermal stirred tanks are governed by an enthalpy balance that contains the heat of reaction as a significant term. If the heat of reaction is unimportant, so that a desired Tout can be imposed on the system regardless of the extent of reaction, then the reactor d5mamics can be analyzed by the methods of the previous section. 

This section focuses on situations where Equation (14.3) must be considered as part of the design. Even for these situations, it is usually possible to control a CSTR at a desired temperature. If temperature control can be achieved rapidly, the isothermal design techniques again become applicable. Rapid means on a time scale that is fast compared with reaction times and composition changes. 

Example 14.8 The styrene polymerization example of Example 5.7 shows three steady states. The middle steady state with aoui = 0.738 and Tout = 403 K is unstable. Devise a control system that stabilizes operation near it. 

Solution There are several theoretical ways of stabilizing the reactor, but temperature control is the normal choice. The reactor in Example 5.7 was adiabatic. Some form of heat exchange must be added. Possibilities are to control the inlet temperature, to control the pressure in the vapor space thereby allowing reflux of styrene monomer at the desired temperature, or to control the jacket or external heat exchanger temperature. The following example regulates the jacket temperature. Refer to Example 5.7. The component balance on styrene is unchanged from Equation (5.29)  

A heat exchange term is added to the energy balance. Equation (5.30), to give 

The heat transfer group is dimensionless. Assume its value is 0.02. 

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Nonisothermal Stirred-Tank Reactor with Irreversible Exothermic Reaction A B ... 

P. Albertos and M. Perez Polo. Selected Topics in Dynamics and Control of Chemical and Biochemical Processes, chapter Nonisothermal stirred-tank reactor with irreversible exothermic reaction A B. 1.Modelling and local control. LNCIS. Springer-Verlag, 2005 (in this volume). 

In contrast to the previous case study, this one (a) includes design variables within the search space, and (b) utilizes a continuous-time formulation. The problem formulation follows closely that described in Section 5.2.4. The study was presented originally in [29] and is based on a nonisothermal stirred tank reactor system analyzed in [30]. 

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 order to reduce the disparities in volume or space time requirements between an individual CSTR and a plug flow reactor, batteries or cascades of stirred tank reactors ard employed. These reactor networks consist of a number of stirred tank reactors confiected in series with the effluent from one reactor serving as the input to the next. Although the concentration is uniform within any one reactor, there is a progressive decrease in reactant concentration as ohe moves from the initial tank to the final tank in the cascade. In effect one has stepwise variations in composition as he moves from onfe CSTR to another. Figure 8.9 illustrates the stepwise variations typical of reactor cascades for different numbers of CSTR s in series. In the general nonisothermal case one will also en-... 

N. Watanabe, H. Kurimoto, M. Matsubara, and K. Onogi. Periodic control of continuous stirred tank reactor II Case of a nonisothermal single reactor. Chem. Eng. Sci., 37 745-752, 1982. 

The reactor system may consist of a number of reactors which can be continuous stirred tank reactors, plug flow reactors, or any representation between the two above extremes, and they may operate isothermally, adiabatically or nonisothermally. The separation system depending on the reactor system effluent may involve only liquid separation, only vapor separation or both liquid and vapor separation schemes. The liquid separation scheme may include flash units, distillation columns or trains of distillation columns, extraction units, or crystallization units. If distillation is employed, then we may have simple sharp columns, nonsharp columns, or even single complex distillation columns and complex column sequences. Also, depending on the reactor effluent characteristics, extractive distillation, azeotropic distillation, or reactive distillation may be employed. The vapor separation scheme may involve absorption columns, adsorption units,... 

One of the simplest practical examples is the homogeneous nonisothermal and adiabatic continuous stirred tank reactor (CSTR), whose steady state is described by nonlinear transcendental equations and whose unsteady state is described by nonlinear ordinary differential equations. 

The rational design of a reaction system to produce a desired polymer is more feasible today by virtue of mathematical tools which permit one to predict product distribution as affected by reactor type and conditions. New analytical tools such as gel permeation chromatography are beginning to be used to check technical predictions and to aid in defining molecular parameters as they affect product properties. The vast majority of work concerns bulk or solution polymerization in isothermal batch or continuous stirred tank reactors. There is a clear need to develop techniques to permit fuller application of reaction engineering to realistic nonisothermal systems, emulsion systems, and systems at high conversion found industrially. A mathematical framework is also needed which will start with carefully planned experimental data and efficiently indicate a polymerization mechanism and statistical estimates of kinetic constants rather than vice-versa. 

Example 2—Unstable CSTR with bounded output. Consider the reaction R P occurring in a nonisothermal jacket-cooled continuous stirred tank reactor (CSTR) with three steady states. A, B, C, corresponding to the intersection points of the two lines shown in Fig. 3 (Stephanopoulos,... 

In this chapter we are concerned only with the rate equation for the i hemical step (no physical resistances). Also, it will be supposed that /"the temperature is constant, both during the course of the reaction and in all parts of the reactor volume. These ideal conditions are often met in the stirred-tank reactor (see-Se c." l-6). Data are invariably obtained with this objective, because it is extremely hazardous to try to establish a rate equation from nonisothermal data or data obtained in inadequately mixed systems. Under these restrictions the integration and differential methods can be used with Eqs. l-X and (2-5) or, if the density is constant, with Eq. (2-6). Even with these restrictions, evaluating a rate equation from data may be an involved problem. Reactions may be simple or complex, or reversible or irreversible, or the density may change even at constant temperatur (for example, if there is a change in number of moles in a gaseous reaction). These several types of reactions are analyzed in Secs. 2-7 to 2-11 under the categories of simple and complex systems. 

Example 14. /. Multiple steady states and hysteresis in a nonisothermal continuous stirred-tank reactor (CSTR) [1,2]. In a CSTR, the curve for the temperature dependence of heat loss to the cooling coil is linear (loss proportional to temperature difference) while that for heat generation by the reaction is S-shaped (Arrhenius ex-... 

Adiabatic or nonisothermal operation of a stirred tank reactor presents a different physical situation from that for plug flow, since spatial variations of concentration and temperature do not exist. Rather, reaction heat effects manifest themselves by establishing a temperature level within the CSTR that differs from that of the feed. Thus, when we use the terms adiabatic or nonisothermal in reference to CSTR systems, it will be understood to imply analysis where thermal effects are included in the conservation equations but not to imply the existence of thermal gradients. 

Multiple steady-state behavior is a classic chemical engineering phenomenon in the analysis of nonisothermal continuous-stirred tank reactors. Inlet temperatures and flow rates of the reactive and cooling fluids represent key design parameters that determine the number of operating points allowed when coupled heat and mass transfer are addressed, and the chemical reaction is exothermic. One steady-state operating point is most common in CSTRs, and two steady states occur most infrequently. Three stationary states are also possible, and their analysis is most interesting because two of them are stable whereas the other operating point is unstable. 

The most important feature of a CSTR is its mixing characteristics. The idealized model of reactor performance presumes that the reactor contents are perfectly mixed so that the properties of the reacting fluid are uniform throughout. The composition and temperature of the effluent are thus identical with those of the reactor contents. This feature greatly simplifies the analysis of stirred-tank reactors vis-h-vis tubular reactors for both isothermal and nonisothermal... 

Stationary Conditions for a Nonisothermal Continuous Stirred Tank Reactor... 

Many chemical reactions are nonisothermal. In order to carry out a particular reaction, we must maintain the temperature of the reaction volume within some limits to avoid species decomposition. In particular, exothermic reactions are problematic. We need to be able to eliminate the generated energy so that we can maintain the reaction under control. Let s consider a system of a jacketed stirred tank reactor where we need to control the temperature by means of the flow rate through the jacket and that there is a perturbation in the flow that feeds the reactor. In our system, we have a controller, a valve, the actuator, the system, our reactor, the perturbation, and a sensor see Figure 4.33. 

Continuous stirred tank reactor polymerization reactors can also be subject to oscillatory behavior. A nonisothermal CSTR free radical solution polymerization can exhibit damped oscillatory approach to a steady state, unstable (growing) oscillations upon disturbance, and stable (limit cycle) oscillations in which the system never reaches steady state and never goes unstable, but continues to oscillate with a fixed period and amplitude. However, these phenomena are more commonly observed in emulsion polymerization. High-volume products such as styrene-butadiene rubber (SBR) often are produced by continuous emulsion polymerization. As noted earlier, this is... 

Although the system we have considered here is a relatively simple one involving a first-order reaction, it has revealed the existence of some fascinating and exotic phenomena. Such phenomena are not limited to catalytic reactions but arise in other nonisothermal systems, for example, in continuous-flow stirred tank reactors, and even in isothermal systems. Their common feature is that the performance curve describing the system has to exhibit an inflection. Such inflections have also been observed in a biological context, where they play the role of a biological switch, which is activated in response to particular stimuli. 

Figure 2.6 A nonisothermal continuous stirred-tank reactor.

Chapter 2 presents calorimeters for measuring accurately the rate of heat release during discontinuous and continuous reactions versus time under isothermal and nonisothermal conditions. In addition, the chapter contains a description of an apparatus that can be used to record online the rate of heat release within a stirred tank reactor during a reaction. 

We demonstrate the use of arclength.continuation.m for the study ofmultiple steady states in a nonisothermal CSTR. Consider a perfectly-mixed stirred-tank reactor with A reacting to form B... 

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